# Stability of phase diagram for a gradient ODE with memory

Piotr Kalita, Piotr Zgliczyński  Faculty of Mathematics and Computer Science,
Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider the problem governed by the gradient ODE $x^{\prime}=\nabla F(x)$
in $\mathbb{R}^{d}$ on which we assume that it has a finite number of
hyperbolic equilibria whose stable and unstable manifolds intersect
transversally. This problem is perturbed by the memory term
$x^{\prime}(t)=\nabla F(x(t))+\varepsilon\int_{-\infty}^{t}M(t-s)x(s)\,ds$
where $\varepsilon>0$ is a small constant. The key result is that the
structure of connections between the equlibria of the unperturbed problem is
exactly preserved for a small $\varepsilon>0$.

This work was supported by National Science Center (NCN) of Poland under
project No. UMO2016/22/A/ST1/00077. Work of PK was also partially supported by
Ministerio de Ciencia e Innovación of Kingdom of Spain under project No.
PID2021-122991NB-C21.

###### Contents

  1. 1 Introduction.
  2. 2 The weighted history space and its norm.
  3. 3 Problem setup
  4. 4 Further properties of the weighted history norm.
  5. 5 Continuation of Lyapunov functions and isolating blocks with cone conditions.
    1. 5.1 Lyapunov function for the problem with delay.
    2. 5.2 The continuation of the isolating block
    3. 5.3 Cone condition
  6. 6 Equilibria and their nonexpansion in the limit
  7. 7 Continuation of the intersection of manifolds.
  8. 8 Appendix 1: Asymptotic compactness
  9. 9 Appendix 2: Graph transform for existence of local stable and unstable manifolds.
  10. 10 Appendix 3: $C^{0}$ dependence of local unstable and stable manifolds on parameter.
    1. 10.1 Cone condition with parameter
    2. 10.2 Lipshitz continuous dependence of local unstable manifolds on parameter
    3. 10.3 Lipshitz continuous dependence of local stable manifolds on parameter
  11. 11 Appendix 4: $C^{1}$ smoothness of local stable and unstable manifolds
    1. 11.1 Fibre contraction theorem
    2. 11.2 Cone condition
    3. 11.3 Fixed point procedure for unstable manifold and its derivative
      1. 11.3.1 Graph transform for the unstable manifold
      2. 11.3.2 Graph transform for the derivative of unstable manifold
      3. 11.3.3 A priori bound for $\mathcal{U}(h,M)$
      4. 11.3.4 A priori bound for the difference $\mathcal{U}(h_{1},M_{1})-\mathcal{U}(h_{2},M_{2})$
      5. 11.3.5 A priori bound for the Lipschitz constant for $\mathcal{U}(h,M)$
      6. 11.3.6 Graph transform $(\mathcal{T},\mathcal{U})$ for the unstable manifold and its derivative has an absorbing fixed point.
    4. 11.4 Fixed point argument for construction of the stable manifold and its derivative
      1. 11.4.1 Graph transform for the stable manifold
      2. 11.4.2 Graph transform for the derivative of stable manifold
      3. 11.4.3 A priori bounds for $\mathcal{R}$
      4. 11.4.4 A priori bounds for the difference of two $\mathcal{R}$’s.
      5. 11.4.5 A priori bounds for Lipschitz constant for $\mathcal{R}(v,M)$
      6. 11.4.6 Graph transform $(\mathcal{S},\mathcal{R})$ for the stable manifold and its derivative has an absorbing fixed point.
  12. 12 Appendix 5: Verification of conditions from Appendix 4.
  13. 13 Appendix 6. Continuity of derivatives of stable and unstable manifolds on parameter.
    1. 13.1 Graph transform for the unstable manifold.
    2. 13.2 Graph transform for the stable manifold.

##  1\. Introduction.

This paper deals with the gradient ordinary differential equation in
$\mathbb{R}^{d}$

(1) $x^{\prime}(t)=\nabla F(x(t))\ \ \textrm{for}\ \ F\in
C^{3}(\mathbb{R}^{d}),$

and its perturbation by the linear memory term though which the derivative of
the unknown solution depends not only on the instantaneous value of this
solution but also on its past values

(2) $x^{\prime}(t)=\nabla
F(x(t))+\varepsilon\int_{-\infty}^{t}M(t-s)x(s)\,ds.$

Because of the presence of the distributed delay term, if we study the flow
governed by (2), we need to consider it in an infinite dimensional space,
containing functions defined in the time interval from minus infinity to the
current time instance.

We make the following assumptions on the functions
$F:\mathbb{R}^{d}\to\mathbb{R}$ and $M:[0,\infty)\to\mathbb{R}^{d\times d}$:

  * (1)

There exist constants $\gamma>0$ and $\delta\in\mathbb{R}$ such that
$F(x)\leq-\gamma|x|^{2}+\delta$, cf. (6).

  * (2)

The unperturbed equation (1) has finite number of equilibria, all of them
being hyperbolic, and their stable and unstable manifolds intersect
transversally.

  * (3)

There exists a function $A\in C^{1}([0,\infty);\mathbb{R}^{d\times d})$ with
$A(s)$ being symmetric and positive definite matrix for every $s\geq 0$ such
that

    * (A)

For almost every $s>0$ and for every $x\in\mathbb{R}^{d}$ we have

$\left(\frac{dA(s)}{ds}x,x\right)\leq-C(A(s)x,x),$

with a constant $C>0$ cf. Assumption 2.1.

    * (B)

For every $s\geq 0$ we have

$\frac{\lambda_{max}(A(s))}{\lambda_{min}(A(s))}\leq D,$

with a constant $D>0$, cf. Assumption 2.3,

and

$\|M(s)\|\leq E\lambda_{min}(A(s)),$

with $\int_{0}^{\infty}M(s)\,ds$ symmetric.

The function $A$, which by (3)(A) must decay exponentially to zero as
$t\to\infty$, defines the phase for the memory term, it is the space

$L^{2}_{A}(\mathbb{R}^{+})^{d}=\left\\{\eta:[0,\infty)\to\mathbb{R}^{d}\,:\
\int_{0}^{\infty}(A(s)\eta(s),\eta(s))\,ds<\infty\right\\}.$

Now, the equation (1) defines the gradient dynamical system
$S^{0}(t):\mathbb{R}^{d}\to\mathbb{R}^{d}$ for $t\geq 0$. This dynamical
system has a global attractor which consists of the finite number of
equilibria and their connections. It structure is represented as a graph of
partial order, the vertexes of this graph correspond to the equilibria of the
system. An edge from $e_{i}$ to $e_{j}$ exists in this graph it there exists a
bounded solution of (1) which converges to $e_{i}$ as time tends to minus
infinity and to $e_{j}$ as time tends to plus infinity.

Now, the problem governed by (2) defines a dynamical system for
$\varepsilon>0$ denoted as
$S^{\varepsilon}(t):L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}\to
L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}$, where the space
$\mathbb{R}^{d}$ contains the current state of the system, and
$L^{2}_{A}(\mathbb{R}^{+})^{d}$ its past. This system for $\varepsilon>0$ is
infinite dimensional. The main result of the paper is the following theorem

######  Theorem 1.1.

Assume (1)–(3) above. There exists $\varepsilon_{0}>0$ such that for every
$\varepsilon\in[0,\varepsilon_{0}]$ the dynamical system governed by (2) has a
global attractor consisting of a finite number of equilibria and their
connections. The graph that represents this system coincides with the graph
for the unperturbed finite dimensional system $\\{S^{0}(t)\\}_{t\geq 0}$.

The result is perturbative in its nature, i.e., it assumes that
$\varepsilon>0$ is small. It states that for such small $\varepsilon$ we can
fully determine the structure of the global attractor consisting of the
equilibria and their connections which coincide with that of the unperturbed
problem.

The question is motivated by the results of [2]. There, the authors consider
the infinite dimensional autonomous gradient dynamical system and they prove
that upon small non-autonomous perturbation the structure of its attractor is
preserved, that is the phase diagram of the non-autonomous dynamics coincides
with the autonomous one. Thus, the authors in [2] are able to fully
characterize the non-autonomous dynamics for the problem which is small
perturbation of the autonomous one (see also [1] for a similar result where
the small perturbation is autonomous, but not $C^{1}$ \- only Lipschitz). Our
result is of similar nature as [1, 2], but our main contribution stands in the
fact that the unperturbed system is finite-dimensional and the perturbed one -
infinite dimensional.

The proof that the structure of connections is exactly preserved upon
perturbation consists of three ingredients:

  * (A)

the equilibria of the perturbed problem exist in the vicinity of the
equilibria of the original one, and that these are all equilibria,

  * (B)

no new connections arise when $\varepsilon>0$, i.e. the connections structure
behaves upper-semicontinuously,

  * (C)

the existing connections are preserved upon perturbation, i.e. the connections
structure behaves lower-semicontinuously.

Fundamental ingredient in the proofs of these items is the fact, obtained in
Section 5, that certain dynamical properties of the unperturbed problem can be
continued for $\varepsilon>0$. In particular it is possible to construct the
common Lyapunov function for $\varepsilon\in[0,\varepsilon_{0}]$. Moreover, we
construct sets which isolate the equilibria of (1), which after taking the
Cartesian product with a certain ball in the memory space
$L^{2}_{A}(R^{+})^{d}$ also isolate the equilibria of (2) with entry and exit
behavior on the boundary being uniform with respect to
$\varepsilon\in[0,\varepsilon_{0}]$. We prove that the new, infinite
dimensional variable $\eta\in L^{2}_{A}(R^{+})^{d}$ can be bundled together
with the stable variables in the finite dimensional state
$x\in\mathbb{R}^{d}$. Finally, we prove that the cone condition holds in these
sets with the same system of coordinates and the same quadratic form in the
range of small $\varepsilon\in[0,\varepsilon_{0}]$. This opens the possibility
of using the Hadamard’s graph transform procedure to construct the local
stable and unstable manifolds of the equilibria as the Lipschitz graphs over
the same systems of coordinates in the considered range of $\varepsilon$. The
above assertion (A) follows from the construction of common isolating sets
with the cone condition, and (B) follows from the compactness argument
(similar as in [2]), these results are contained in Section 6. To get the most
involved result (C), we need to prove that the local stable and unstable
manifolds are actually $C^{1}$ close to each other in dependence on
$\varepsilon$. We prove this by differentiation of the graph transform.
Moreover, we transport the smallness of $C^{1}$ distance between the local
unstable manifolds along the flow in order to prove that the transversality of
the intersection implies that this intersection is preserved upon the
perturbation. This result is contained in Section 7.

The fact that the norm of the memory term is weighted by the expression that
decays exponentially to zero is a fundamental fact which allows us to treat
the memory variable as the stable variable in the neighbourhood of the
equilibrium. The key result here is the dissipative estimate (18) on the time
evolution of the norm of memory variable which is obtained in Section 4. This
estimate is derived using the concept from the seminal paper of Dafermos [6],
who proved that in the linear problem of viscoelasticity the memory term is
dissipative and has damping effect on the solution, which decays to zero due
to this term’s presence. Discoveries of Dafermos were later used in the
context of global attractors for the nonlinear problem of viscoelasticity by
Conti and Pata [4], who explored the dissipative nature of the memory term to
obtain the existence of the global attractor. Dissipativity of the memory term
in the context of global attractors has also been explored for the first
order, reaction-diffusion type, problems in [5, 9, 10]. All these results,
however are of global nature. The novel contribution of this paper, is the
exploitation of the dissipative nature of the memory term in the local
argument realized in the neighbourhood of the equilibria and its application
to recover the full intrinsic structure of the global attractor.

##  2\. The weighted history space and its norm.

Let $A:[0,\infty)\to\mathbb{R}^{d\times d}$ be a time dependent matrix
function. This function will define the norm. Assumptions 2.1, 2.3 on the
function $A$ and assumption 2.5 on related matrix function $M$ will be
standing assumptions throughout the whole article.

######  Assumption 2.1.

Assume that $A(s)$ is a symmetric and positive definite matrix for $s\geq 0$,
$[0,\infty)\ni s\mapsto A(s)$ belongs to $C^{1}([0,\infty);\mathbb{R}^{d\times
d})$ and that for almost every $s>0$ and every $u\in\mathbb{R}^{d}$

$\left(\frac{dA(s)}{ds}u,u\right)\leq-C(A(s)u,u).$

######  Lemma 2.2.

Under Assumption 2.1 we have

$\int_{0}^{\infty}\|A(s)\|\,ds<\infty$

###### Proof.

We have

$e^{Cs}\frac{d}{ds}(A(s)u,u)+Ce^{Cs}(A(s)u,u)\leq 0,$

for every $u\in\mathbb{R}^{d}$. Hence

$\frac{d}{ds}(e^{Cs}(A(s)u,u))\leq 0,$

and

$e^{Cs}(A(s)u,u)\leq(A(0)u,u).$

Finally

$(A(s)u,u)\leq e^{-Cs}(A(0)u,u),$

for every $s\geq 0$ and $u\in\mathbb{R}^{d}$. As $A(s)$ is symmetric and
positively definite then for every $s$ we can find a vector $u(s)$ with norm
one such that

$\|A(s)\|=(A(s)u(s),u(s))\leq e^{-Cs}(A(0)u(s),u(s))\leq\|A(0)\|e^{-Cs},$

and the assertion follows. ∎

We define the space $L^{2}_{A}(\mathbb{R}^{+})^{d}$ with the norm
$\|\eta\|^{2}=\int_{0}^{\infty}(A(s)\eta(s),\eta(s))\,ds.$

######  Assumption 2.3.

Assume that for some constant $\overline{D}>0$ and every $x\in\mathbb{R}^{d}$,
$s\in\mathbb{R}^{+}$

(3) $\|A(s)\|\ |x|^{2}\leq\overline{D}^{2}(A(s)x,x).$

In other words

$\frac{\lambda_{max}(A(s))}{\lambda_{min}(A(s))}\leq\overline{D}^{2}\ \
\textrm{for every}\ \ s\geq 0.$

######  Lemma 2.4.

Under Assumptions 2.1 and 2.3 for every $\eta\in
L^{2}_{A}(\mathbb{R}^{+})^{d}$

$\left|\int_{0}^{\infty}A(s)\eta(s)\,ds\right|\leq\left(\overline{D}\sqrt{\int_{0}^{\infty}\|A(s)\|\,ds}\right)\|\eta\|:=D\|\eta\|.$

###### Proof.

$\displaystyle\left|\int_{0}^{\infty}A(s)\eta(s)\,ds\right|^{2}\leq\left(\int_{0}^{\infty}\|A(s)\||\eta(s)|\,ds\right)^{2}.$

By the Hölder inequality

$\displaystyle\left|\int_{0}^{\infty}A(s)\eta(s)\,ds\right|^{2}\leq\left(\int_{0}^{\infty}\sqrt{\|A(s)\|}\sqrt{\|A(s)\|}|\eta(s)|\,ds\right)^{2}$
$\displaystyle\ \
\leq\int_{0}^{\infty}\|A(s)\|\,ds\int_{0}^{\infty}\|A(s)\||\eta(s)|^{2}\,ds\leq\overline{D}^{2}\int_{0}^{\infty}\|A(s)\|\,ds\int_{0}^{\infty}(A(s)\eta(s),\eta(s))\,ds.$

and the proof is complete. ∎

Now consider the function $M:[0,\infty)\to\mathbb{R}^{d\times d}$. Make the
following assumption

######  Assumption 2.5.

Assume that for every $s\geq 0$

${\|M(s)\|}\leq\overline{C_{1}}^{2}\lambda_{min}(A(s)),$

with a constant $\overline{C_{1}}>0$ and

$\int_{0}^{\infty}M(s)\,ds\ \ \textrm{is symmetric}.$

The next result holds analogously to Lemma 2.4

######  Lemma 2.6.

Under Assumptions 2.1 and 2.5 we have

$\int_{0}^{\infty}\|M(s)\|\,ds<\infty$

and

$\left|\int_{0}^{\infty}M(s)\eta(s)\,ds\right|\leq C_{1}\|\eta\|\ \ \
\textrm{for every}\ \ \ \eta\in L^{2}_{A}(\mathbb{R}^{+})^{d},$

for every $\eta\in L^{2}_{A}(\mathbb{R}^{+})^{d}$, where
$C_{1}=\overline{C_{1}}\sqrt{\int_{0}^{\infty}\|M(s)\|\,ds}$.

An example of $A(s)$ which satisfies the above assumptions is $A(s)=e^{-\kappa
s}I$. Then $C=\kappa$, $\overline{D}=1$, and we need, in addition to the
symmetry of the integral of $M$ that

$\|M(s)\|\leq\overline{C_{1}}^{2}e^{-\kappa s}.$

##  3\. Problem setup

We consider the following ODE

(4) $x^{\prime}(t)=f(x(t))\quad\textrm{where}\quad f\in
C^{2}(\mathbb{R}^{d};\mathbb{R}^{d}).$

We assume that the ODE has a gradient form, i.e.

(5) $f(x)=\nabla F(x)\ \ \textrm{where}\ \ F\in C^{3}(\mathbb{R}^{d}).$

Moreover we assume that there exist constants $\gamma>0$ and
$\delta\in\mathbb{R}$ such that

(6) $F(x)\leq-\gamma|x|^{2}+\delta,$

We perturb the above ODE with the additive linear distributed delay term with
a multiplicative parameter $\varepsilon>0$. This yields the equation

(7) $x^{\prime}(t)=f(x(t))+\varepsilon\int_{-\infty}^{t}M(t-s)x(s)\,ds,$

where $M(s)=\\{M_{ij}(s)\\}_{i,j=1}^{d}$ is a time dependent matrix.

Rearranging, we obtain

$\displaystyle
x^{\prime}(t)=f(x(t))+\varepsilon\int_{-\infty}^{t}M(t-s)(x(s)-x(t)+x(t))\,ds$
$\displaystyle\ \
=f(x(t))+\varepsilon\left(\int_{-\infty}^{t}M(t-s)\,ds\right)x(t)+\varepsilon\int_{-\infty}^{t}M(t-s)(x(s)-x(t))\,ds.$

This motivates the system

$\displaystyle
x^{\prime}(t)=f(x(t))+\varepsilon\left(\int_{0}^{\infty}M(s)\,ds\right)x(t)+\varepsilon\int_{-\infty}^{t}M(t-s)(x(s)-x(t))\,ds,$

or

(8) $\displaystyle
x^{\prime}(t)=f^{\varepsilon}(x(t))+\varepsilon\int_{0}^{\infty}M(s)(x(t-s)-x(t))\,ds=f^{\varepsilon}(x(t))+\varepsilon\int_{0}^{\infty}M(s)\eta^{t}(s)\,ds,$

where $\eta^{t}:[0,\infty)\to\mathbb{R}^{d}$ is defined as
$\eta^{t}(s)=x(t-s)-x(t)$, or, more specifically

(9) $\eta^{t}(s)=\begin{cases}x(t-s)-x(t)\ \ \textrm{for}\ \ s\leq t\\\
x(t-s)-x(t)=x_{0}+\eta^{0}(s-t)-x(t)\ \ \textrm{otherwise}.\end{cases}$

and
$f^{\varepsilon}(x)=f(x)+\varepsilon\left(\int_{0}^{\infty}M(s)\,ds\right)x$.

While we skip the standard argument on the existence of global solution for
every initial data $(\eta^{0},x_{0})\in L^{2}_{A}(\mathbb{R}^{+})^{d}\times
R^{d}$, which follows from the fact that $f$ is locally Lipschitz, and from
the Lyapunov function (24), we prove the Lipschitz continuous dependence on
the initial data. Before we pass to this result we prove the boundedness of
the solution.

######  Lemma 3.1.

Assume (5) and (6). Then if only

(10) $\varepsilon\int_{0}^{\infty}\|M(s)\|\,ds<2\gamma,$

then every solution is bounded uniformly on bounded sets of initial data.

###### Proof.

From the Lyapunov function (23) we obtain that

$E\|\eta^{t}\|^{2}-2F(x(t))-\varepsilon\left(\int_{0}^{\infty}M(s)\,dsx(t),x(t)\right)\leq
E\|\eta^{0}\|^{2}-2F(x_{0})-\varepsilon\left(\int_{0}^{\infty}M(s)\,dsx_{0},x_{0}\right)\leq
C(|x_{0}|,\|\eta^{0}\|),$

where $C(\cdot,\cdot)$ is a continuous function independent of $\varepsilon$,
that may change from line to line. This means that

$E\|\eta^{t}\|^{2}+2\gamma|x(t)|^{2}\leq
C(|x_{0}|,\|\eta^{0}\|)+\varepsilon\int_{0}^{\infty}\|M(s)\|\,ds|x(s)|^{2},$

which immediately yields the assertion of the lemma. ∎

######  Lemma 3.2.

Assume (5) and (6). There exists $\varepsilon_{0}$ such that for every
$\varepsilon\in[0,\varepsilon_{0}]$ if $(\eta^{t},x(t))$ and $(\xi^{t},x(t))$
are two solutions with the initial data $(\eta^{0},x_{0})$ and
$(\xi^{0},y_{0})$, respectively, then for every $T>0$ there exists a constant
$L(T)$ such that for every $t\in[0,T]$ we have

$|x(t)-y(t)|+\|\eta^{t}-\xi^{t}\|\leq
L(T)(|x_{0}-y_{0}|+\|\eta^{0}-\xi^{0}\|).$

###### Proof.

In the proof by $D$ we will denote constants (which can vary from line to
line) dependent on the initial data for both problems and by $C_{i}$ constants
independent on these data. Subtracting (8) for the two solutions we obtain

$(x(t)-y(t))^{\prime}=f(x(t))-f(y(t))+\varepsilon\int_{0}^{\infty}M(s)\,ds(x(t)-y(t))+\varepsilon\int_{0}^{\infty}M(s)(\eta^{t}(s)-\xi^{t}(s))\,ds.$

From Lemma 3.1 we deduce that

(11) $\frac{d}{dt}|x(t)-y(t)|\leq D|x(t)-y(t)|+C_{1}\|\eta^{t}-\xi^{t}\|.$

From Lemma 4.2 we obtain

$\frac{d}{dt}\|\eta^{t}-\xi^{t}\|^{2}+C_{2}\|\eta^{t}-\xi^{t}\|^{2}\leq
C_{3}\|\eta^{t}-\xi^{t}\|\ \ \|(x(t)-y(t))^{\prime}\|.$

It follows that

$\frac{d}{dt}\|\eta^{t}-\xi^{t}\|+C_{1}\|\eta^{t}-\xi^{t}\|\leq D\
\|x(t)-y(t)\|+C_{3}\varepsilon\|\eta^{t}-\xi^{t}\|,$

and choosing $\varepsilon>0$ small enough (this choice in independent on the
initial data), we obtain

$\frac{d}{dt}\|\eta^{t}-\xi^{t}\|\leq D\ \|x(t)-y(t)\|.$

This inequality, together with (11) yield the assertion of the Lemma. ∎

The question which we address in the remaining part of the article is the
following. Assume that $x^{\prime}=f(x)$ is a Morse–Smale system. The
Morse–Smale property in our case means that the vector field $f$ has a finite
number of hyperbolic equilibria such that the intersections of stable and
unstable manifolds are transversal. If $\varepsilon>0$ is small, can we say
that the problem with distributed memory has the same structure of the global
attractor as the ODE?

The families of maps $\\{S_{\varepsilon}(t)\\}_{t\geq
0}:L^{2}_{A}(R^{+})^{d}\times\mathbb{R}^{d}\to
L^{2}_{A}(R^{+})^{d}\times\mathbb{R}^{d}$ denote the semiflows the govern the
solutions of the problem (8).

We prove that assumptions (5) and (6) imply that for
$\varepsilon\in[0,\varepsilon_{0}]$ problems have global attractors
$\mathcal{A}_{\varepsilon}\subset
L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}$ such that

(12) $\bigcup_{\varepsilon\in[0,\varepsilon_{0})}\mathcal{A}_{\varepsilon}\ \
\ \textrm{is bounded in}\ L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}.$

We will restrict the analysis of the dynamics to these sets.

######  Lemma 3.3.

Assume (5) and (6). Then there exists $\varepsilon_{0}$ such that for every
$\varepsilon\in[0,\varepsilon_{0}]$ the problems governed by (8)-(9) have
global attractors $\mathcal{A}_{\varepsilon}$, that satisfy (12).

###### Proof.

We are in position to use Lemma 8.1 to deduce that for every bounded set
$\mathcal{B}\subset L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}$ its
$\omega$-limit set $\omega(\mathcal{B})$ is nonempty, compact and attracts
$\mathcal{B}$ in the sense of Hausdorff semidistance in
$L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}$.

The argument now follows the lines of the proof of Theorem A.3 in [4]. Now the
set of equilibria is denoted as $\mathcal{E}_{\varepsilon}$ and for every
equilibrium $\eta=0$, while $x$ belongs to the isolating set around the
equilibrium for $\varepsilon=0$. Lemma 8.1 as well as the existence of the
Lyapunov function imply that for every initial data $(\eta^{0},x_{0})$ we can
find an equilibrium $(0,x^{*})$ such that
$S_{\varepsilon}(t)(\eta^{0},x_{0})\to(0,x^{*})$ as $t\to\infty$. Denote the
Lyapunov function as

(13)
$L_{\varepsilon}(x,\eta)=E\|\eta\|^{2}-F(x)-\varepsilon\left(\int_{0}^{\infty}M(s)\,dsx,x\right),$

and define

$\mathcal{C}_{\varepsilon}=\left\\{(x,\eta)\in\mathbb{R}^{d}\times
L^{2}_{A}(\mathbb{R}_{+})^{d}\,:\
L_{\varepsilon}(x,\eta)<\max_{(y,\xi)\in\mathcal{E}_{\varepsilon}}L_{\varepsilon}(y,\xi)+1\right\\}.$

The set $\bigcup_{\varepsilon\in[0,\varepsilon_{0}]}\mathcal{C}_{\varepsilon}$
is bounded.

If we fix $\mathcal{B}$, then there exists time $t^{*}(\mathcal{B})$ such that
$S_{\varepsilon}(t)\omega(\mathcal{B})\subset\mathcal{C}_{\varepsilon}$ for
$t\geq t^{*}$. Indeed, by continuity of $S_{\varepsilon}(t)$ for every
$p\in\omega(\mathcal{B})$ there exists a neighbourhood $\mathcal{U}_{p}$ and
$t_{p}$ such that for
$S_{\varepsilon}(t_{p})\mathcal{U}_{p}\subset\mathcal{C}_{\varepsilon}$. As
$\mathcal{C}_{\varepsilon}$ is positively invariant the inclusion
$S_{\varepsilon}(t)\mathcal{U}_{p}\subset\mathcal{C}_{\varepsilon}$ holds for
every $t\geq t_{p}$. Sets $\\{\mathcal{U}_{p}\\}_{p\in\omega(\mathcal{B})}$
are open cover of $\omega(\mathcal{B})$. We extract finite subcover,
$\\{\mathcal{U}_{p_{n}}\\}_{n=1}^{N}$ whereas
$t^{*}=\max\\{t_{p_{1}},\ldots,t_{p_{n}}\\}$. Since there exists a function
$\psi(t)>0$ such that $\lim_{t\to\infty}\textrm{dist}_{\mathbb{R}^{d}\times
L^{2}_{A}(\mathbb{R}^{+})^{d}}(S_{\varepsilon}(t)\mathcal{B},\omega(\mathcal{B}))\leq\lim_{t\to\infty}\psi(t)=0$,
for every $t$ and $(x,\eta)\in\mathcal{B}$ there exists
$k(t)\in\omega(\mathcal{B})$ and $q(t)$ such that
$S_{\varepsilon}(t)(x,\eta)=k(t)+q(t)$ and $\|q(t)\|_{\mathbb{R}^{d}\times
L^{2}_{A}(\mathbb{R}^{+})^{d}}\leq 2\psi(t)$. Now
$S_{\varepsilon}(t+t^{*})(x,\eta)=S_{\varepsilon}(t^{*})k(t)+S_{\varepsilon}(t^{*})(k(t)+q(t))-S_{\varepsilon}(t^{*})k(t)$
and $S_{\varepsilon}(t^{*})k(t)\in\mathcal{C}_{\varepsilon}$. Moreover
$S_{\varepsilon}(t^{*})$ is continuous and hence it is uniformly continuous in
a neighbourhood of a compact set, and hence for $t$ large enough

$\|S_{\varepsilon}(t^{*})(k(t)+q(t))-S_{\varepsilon}(t^{*})k(t)\|_{\mathbb{R}^{d}\times
L^{2}_{A}(\mathbb{R}^{+})^{d}}\leq 1.$

This means that, for $t$ large enough $S_{\varepsilon}(t+t^{*})(x,\eta)$
belongs to the ball centered at zero and with radius
$\sup_{(x,\eta)\in\mathcal{C}_{\varepsilon}}\|(x,\eta)\|_{\mathbb{R}^{d}\times
L^{2}_{A}(\mathbb{R}^{+})^{d}}+1$. Together with Lemma 8.1 it is enough to
guarantee the existence of the global attractor $\mathcal{A}_{\varepsilon}$
and the bound (12). ∎

In the next lemma we compare two solutions for original and variational
equations. We always assume that $\varepsilon\in[0,\varepsilon_{0}]$ with
$\varepsilon_{0}$ being sufficiently small.

######  Lemma 3.4.

Consider two solutions: one $(\eta,x)$ of problem with $\varepsilon_{1}$ with
the initial data $(\eta^{0},x_{0})$ and another one $(\xi,y)$ of the problem
with $\varepsilon_{2}$ with the initial data $(\xi^{0},y_{0})$. Then

(14) $|x(t)-y(t)|+\|\eta^{t}-\xi^{t}\|\leq
Ce^{Ct}(\|\eta^{0}-\xi^{0}\|+|x_{0}-y_{0}|+|\varepsilon_{1}-\varepsilon_{2}|),$

for every $t\geq 0$ where the constants $C$ dependin on the initial data
$(\eta^{0},x_{0})$ and $(\xi^{0},y_{0})$ and are bounded on bounded sets of
initial data.

###### Proof.

Then

$\displaystyle(x(t)-y(t))^{\prime}=f(x(t))-f(y(t))+\varepsilon_{1}\left(\int_{0}^{\infty}M(s)\,ds\right)(x(t)-y(t))+(\varepsilon_{1}-\varepsilon_{2})\left(\int_{0}^{\infty}M(s)\,ds\right)y(t)$
$\displaystyle\
+\varepsilon_{1}\int_{0}^{\infty}M(s)(\eta^{t}(s)-\xi^{t}(s))\,ds+(\varepsilon_{1}-\varepsilon_{2})\int_{0}^{\infty}M(s)\xi^{t}(s)\,ds.$

Liapunov function (13) implies that sets $\\{\textrm{conv}\\{x(t),y(t)\\}\ :\
t\geq 0\\}$ and $\\{\textrm{conv}\\{\eta^{t},\xi^{t}\\}\ :\ t\geq 0\\}$ are
bounded by constants depending on the initial data of the problem. We denote
the generic constant depending on the initial data by $C$. Applying the norm
on the both sides of the above equatiom we obtain

$\frac{d}{dt}|x(t)-y(t)|\leq C\
|x(t)-y(t)|+C|\varepsilon_{1}-\varepsilon_{2}|+C\varepsilon_{1}\|\eta^{t}-\xi^{t}\|.$

Using (20) it follows that

$\frac{d}{dt}\|\eta^{t}-\xi^{t}\|^{2}+C\|\eta^{t}-\xi^{t}\|^{2}\leq
C\|\xi^{t}-\eta^{t}\|\,|x^{\prime}(t)-y^{\prime}(t)|\leq
C\|\xi^{t}-\eta^{t}\|\,|x(t)-y(t)|+C|\varepsilon_{1}-\varepsilon_{2}|\,\|\xi^{t}-\eta^{t}\|+C\varepsilon_{1}\|\eta^{t}-\xi^{t}\|^{2}.$

After straightforward calculations, and for sufficiently small
$\varepsilon_{0}$,

$\frac{d}{dt}\|\eta^{t}-\xi^{t}\|\leq
C|x(t)-y(t)|+C|\varepsilon_{1}-\varepsilon_{2}|,$

whence

$\frac{d}{dt}(|x(t)-y(t)|+\|\eta^{t}-\xi^{t}\|)\leq C\
(|x(t)-y(t)|+\|\eta^{t}-\xi^{t}\|)+C|\varepsilon_{1}-\varepsilon_{2}|,$

which yields the assertion of the lemma. ∎

In the next lemma we characterize the derivative of the flow with respect to
the initial data

######  Lemma 3.5.

Consider the mapping
$L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}\ni(\eta^{0},x_{0})\mapsto(\eta(t),x_{t})=S^{\varepsilon}(t)(\eta^{0},x_{0})$
defining the solutions of (8)–(9). The mapping $S^{\varepsilon}(t)$ is Fréchet
differentiable and its derivative at $(\eta^{0},x_{0})$ is defined as the
linear mapping that assigns to $(\xi^{0},w_{0})$ the solution of the
variational problem

(15) $\displaystyle
w^{\prime}(t)=Df(x(t))w(t)+\varepsilon\left(\int_{0}^{\infty}M(s)\,ds\right)w(t)+\varepsilon\int_{0}^{\infty}M(s)\theta^{t}(s)\,ds.$
(16) $\displaystyle\theta^{t}(s)=\begin{cases}w(t-s)-w(t)\ \ \textrm{for}\ \
s\leq t\\\ w_{0}+\xi^{0}(s-t)-w(t)\ \ \textrm{otherwise}.\end{cases}$

$w(0)=w_{0}$, $\theta^{0}=\xi^{0}$.

###### Proof.

We take two initial conditions $x_{0},\overline{x}_{0}$ and
$\eta^{0},\overline{\eta}^{0}$ and call the corresponding solutions
$(\eta^{t},x(t))$ and $(\overline{\eta}^{t},\overline{x}(t))$. Their
difference will be called $z(t)=\overline{x}(t)-x(t)$ and
$\xi^{t}=\overline{\eta}^{t}-{\eta}^{t}$. They satisfy the equations

$\displaystyle
z^{\prime}(t)=f(x(t)+z(t))-f(x(t))+\varepsilon\left(\int_{0}^{\infty}M(s)\,ds\right)z(t)+\varepsilon\int_{0}^{\infty}M(s)\xi^{t}(s)\,ds.$

and

$\xi^{t}(s)=\begin{cases}z(t-s)-z(t)\ \ \textrm{for}\ \ s\leq t\\\
z_{0}+\xi^{0}(s-t)-z(t)\ \ \textrm{otherwise}.\end{cases}$

This motivates the definition (15)–(16) of a variational equation with
unknowns $\theta^{t}$ and $w(t)$. Denote the difference $z-w=p$ and
$\xi^{t}-\theta^{t}=\omega^{t}$. Then

$\displaystyle
p^{\prime}(t)=f(x(t)+z(t))-f(x(t))-Df(x(t))w(t)+\varepsilon\left(\int_{0}^{\infty}M(s)\,ds\right)p(t)+\varepsilon\int_{0}^{\infty}M(s)\omega^{t}(s)\,ds.$

$\omega^{t}(s)=\begin{cases}p(t-s)-p(t)\ \ \textrm{for}\ \ s\leq t\\\ -p(t)\ \
\textrm{otherwise}.\end{cases}$

Rearranging the first equation and using the Taylor formula we obtain

$\displaystyle
p^{\prime}(t)=Df(x(t))p(t)+\frac{1}{2}D^{2}f(x(t)+\lambda(t)z(t))(z(t),z(t))+\varepsilon\int_{0}^{t}M(t-s)p(s)\,ds.$

Integrating and using the fact that $p(0)=0$ we obtain

$|p(t)|\leq C\int_{0}^{t}|p(s)|ds+C\int_{0}^{t}|z(s)|^{2}\,ds+\varepsilon
C\int_{0}^{t}\int_{0}^{s}|p(r)|\,dr\,ds\leq
C\int_{0}^{t}|z(s)|^{2}\,ds+C(1+\varepsilon t)\int_{0}^{t}|p(s)|\,ds.$

We need an estimate for $|z(s)|$. We have

$\displaystyle
z^{\prime}(t)=Df(x(t)+\lambda(t)z(t))z(t)+\varepsilon\int_{0}^{t}M(s)z(t-s)\,ds+\varepsilon\int_{t}^{\infty}M(s)\,dsz_{0}+\varepsilon\int_{t}^{\infty}M(s)\xi^{0}(s-t)\,ds.$

Rewriting, we obtain

$\displaystyle
z^{\prime}(t)=Df(x(t)+\lambda(t)z(t))z(t)+\varepsilon\int_{0}^{t}M(t-s)z(s)\,ds+\varepsilon\int_{t}^{\infty}M(s)\,dsz_{0}+\varepsilon\int_{0}^{\infty}M(s+t)\xi^{0}(s)\,ds.$

Now $x(t)$ is bounded and as is $z(t)$ because $\overline{x}(t)$ is attracted
to the attractor and hence also bounded. We obtain

$|z(t)|\leq(1+\varepsilon Ct)|z_{0}|+\varepsilon Ct\|\xi^{0}\|+C(1+\varepsilon
t)\int_{0}^{t}|z(s)|\,ds.$

By the Gronwall lemma

$|z(t)|\leq((1+\varepsilon Ct)|z_{0}|+\varepsilon
Ct\|\xi^{0}\|)e^{Ct(1+\varepsilon t)}.$

This means that

$|z(t)|^{2}\leq g(t)(|z_{0}|^{2}+\|\xi^{0}\|^{2}),$

where by $g(t)$ we denote a generic increasing and continuous function of $t$.
We deduce that

$|p(t)|\leq g(t)(|z_{0}|^{2}+\|\xi^{0}\|^{2})+C(1+\varepsilon
t)\int_{0}^{t}|p(s)|\ ds.$

By the Gronwall lemma

$|p(t)|\leq g(t)(|z_{0}|^{2}+\|\xi^{0}\|^{2}).$

Moreover,

$\|\omega^{t}\|^{2}=\int_{0}^{t}(A(s)p(t-s),p(t-s))\,ds-\int_{0}^{\infty}(A(s)p(t),p(t))\,ds,$

this means that

$\|\omega^{t}\|^{2}\leq
g(t)\left(|p(t)|^{2}+\int_{0}^{t}|p(s)|^{2}\,ds\right).$

We deduce that

$\|\omega^{t}\|\leq g(t)(|z_{0}|^{2}+\|\xi^{0}\|^{2}).$

We conclude that

$\lim_{|z_{0}|\to 0,\|\xi^{0}\|\to
0}\frac{|p(t)|+\|\omega^{t}\|}{|z_{0}|+\|\xi^{0}\|}\leq\lim_{|z_{0}|\to
0,\|\xi^{0}\|\to
0}\frac{g(t)(|z_{0}|^{2}+\|\xi^{0}\|^{2})}{|z_{0}|+\|\xi^{0}\|}=0.$

This implies the Fréchet differentiability of the flow and the fact that the
derivative with respect to the initial data is a solution of the variational
equation. ∎

The following lemma implies the continuous dependence of the derivative with
respect to the initial data on the parameter $\varepsilon$ on attractors.

######  Lemma 3.6.

Let $(\eta^{0,\varepsilon_{1}},x^{\varepsilon_{1}}_{0})$ and
$(\eta^{0,\varepsilon_{2}},x^{\varepsilon_{2}}_{0})$ be the initial data for
problems with $\varepsilon_{1}$ and $\varepsilon_{2}$, respectively. Moreover,
let $(\theta^{0,\varepsilon_{1}},w^{\varepsilon_{1}}_{0})$ and
$(\theta^{0,\varepsilon_{2}},w^{\varepsilon_{2}}_{0})$ be the initial data for
the variational problem. Then

$\displaystyle\left\|\frac{DS^{\varepsilon_{2}}(t)(\eta^{0,\varepsilon_{2}},x^{\varepsilon_{2}}_{0})}{D(\eta,x)}(\theta^{0,\varepsilon_{2}},w^{\varepsilon_{2}}_{0})-\frac{DS^{\varepsilon_{1}}(t)(\eta^{0,\varepsilon_{1}},x^{\varepsilon_{1}}_{0})}{D(\eta,x)}(\theta^{0,\varepsilon_{1}},w^{\varepsilon_{1}}_{0})\right\|_{L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}}$
$\displaystyle\ \ \leq
Ce^{Ct}(|\varepsilon_{2}-\varepsilon_{1}|+|w^{\varepsilon_{2}}_{0}-w^{\varepsilon_{1}}_{0}|+\|\theta^{0,\varepsilon_{2}}-\theta^{0,\varepsilon_{1}}\|+|x^{\varepsilon_{2}}_{0}-x^{\varepsilon_{1}}_{0}|+\|\eta^{0,\varepsilon_{2}}-\eta^{0,\varepsilon_{1}}\|),$

for every $t\geq 0$, where $C$ is non-decreasing in all arguments and bounded
on bounded sets.

###### Proof.

Denote

$\frac{DS^{\varepsilon_{i}}(t)(\eta^{0,\varepsilon_{i}},x^{\varepsilon_{i}}_{0})}{D(\eta,x)}(\theta^{0,\varepsilon_{i}},w^{\varepsilon_{i}}_{0})=(\theta^{\varepsilon_{i},t},w^{\varepsilon_{i}}(t)).$

We have

$\frac{d}{dt}|w^{\varepsilon_{2}}(t)|\leq
C|w^{\varepsilon_{2}}(t)|+\varepsilon_{2}\|\theta^{\varepsilon_{2},t}\|.$

Moreover, from (18),

$\frac{d}{dt}\|\theta^{\varepsilon_{2},t}\|^{2}+C\|\theta^{\varepsilon_{2},t}\|^{2}\leq
C\|\theta^{\varepsilon_{2},t}\|\,|(w^{\varepsilon_{2}})^{\prime}(t)|.$

It follows that

$\frac{d}{dt}\|\theta^{\varepsilon_{2},t}\|\leq C|w^{\varepsilon_{2}}(t)|,$

and

$\frac{d}{dt}(|w^{\varepsilon_{2}}(t)|+\|\theta^{\varepsilon_{2},t}\|)\leq
C(|w^{\varepsilon_{2}}(t)|+\|\theta^{\varepsilon_{2},t}\|),$

whereas

(17) $|w^{\varepsilon_{2}}(t)|+\|\theta^{\varepsilon_{2},t}\|\leq
e^{Ct}(|w^{\varepsilon_{2}}_{0}|+\|\theta^{\varepsilon_{2},0}\|).$

Now, we have the following equation for the difference between two solutions
of variational equations along the equations on attractors

$\displaystyle(w^{\varepsilon_{2}}(t)-w^{\varepsilon_{1}}(t))^{\prime}=(Df(x^{\varepsilon_{2}}(t))-Df(x^{\varepsilon_{1}}(t)))w^{\varepsilon_{2}}(t)+Df(x^{\varepsilon_{1}}(t))(w^{\varepsilon_{2}}(t)-w^{\varepsilon_{1}}(t))$
$\displaystyle\ \
+(\varepsilon_{2}-\varepsilon_{1})\int_{0}^{\infty}M(s)(s)\,dsw^{\varepsilon_{2}}(t)+\varepsilon_{1}\int_{0}^{\infty}M(s)\,ds(w^{\varepsilon_{2}}(t)-w^{\varepsilon_{1}}(t))$
$\displaystyle\ \
+(\varepsilon_{2}-\varepsilon_{1})\int_{0}^{\infty}M(s)\theta^{\varepsilon_{2},t}(s)\,ds+\varepsilon_{1}\int_{0}^{\infty}M(s)(\theta^{\varepsilon_{2},t}-\theta^{\varepsilon_{1},t})\,ds.$

Denote $w^{\varepsilon_{2}}(t)-w^{\varepsilon_{1}}(t)=z(t)$ and
$\theta^{\varepsilon_{2},t}-\theta^{\varepsilon_{1},t}=\zeta^{t}$. We obtain

$\frac{d}{dt}|z(t)|\leq
C|x^{\varepsilon_{2}}(t)-x^{\varepsilon_{1}}(t)|\,|w^{\varepsilon_{2}}(t)|+C|z(t)|+C|\varepsilon_{2}-\varepsilon_{1}|(|w^{\varepsilon_{2}}(t)|+\|\theta^{\varepsilon_{2},t}\|)+\varepsilon_{1}\|\zeta^{t}\|.$

Using (17) and Lemma 3.4 we obtain

$\frac{d}{dt}|z(t)|\leq
Ce^{Ct}|x^{\varepsilon_{2}}_{0}-x^{\varepsilon_{1}}_{0}|+Ce^{Ct}\|\eta^{0,\varepsilon_{2}}-\eta^{0,\varepsilon_{1}}\|+C|z(t)|+Ce^{Ct}|\varepsilon_{2}-\varepsilon_{1}|+\varepsilon_{1}\|\zeta^{t}\|,$

where the constants $C$ depend on the initial data for original problems and
variational problems. We need to derive the estimate on the difference of the
norms
$\|\theta^{\varepsilon_{1},t}-\theta^{\varepsilon_{2},t}\|=\|\zeta^{t}\|$. To
this end, we use let (20), whence

$\frac{d}{dt}\|\zeta^{t}\|^{2}+C\|\zeta^{t}\|^{2}\leq
C\|\zeta^{t}\|\,|z^{\prime}(t)|.$

It follows that

$\frac{d}{dt}\|\zeta^{t}\|\leq
Ce^{Ct}|x^{\varepsilon_{2}}_{0}-x^{\varepsilon_{1}}_{0}|+Ce^{Ct}\|\eta^{0,\varepsilon_{2}}-\eta^{0,\varepsilon_{1}}\|+C|z(t)|+Ce^{Ct}|\varepsilon_{2}-\varepsilon_{1}|.$

We deduce the estimate

$\frac{d}{dt}(|z(t)|+\|\zeta^{t}\|)\leq
C(|z(t)|+\|\zeta^{t}\|)+Ce^{Ct}(|x^{\varepsilon_{2}}_{0}-x^{\varepsilon_{1}}_{0}|+\|\eta^{0,\varepsilon_{2}}-\eta^{0,\varepsilon_{1}}\|+|\varepsilon_{2}-\varepsilon_{1}|),$

and the Gronwall lemma yields the desired assertion. ∎

##  4\. Further properties of the weighted history norm.

The following lemma plays a crucial role in passing from ODE (4) to (7) as it
shows that the tail $\eta$ can be treated as a "contracting" direction from
the point of view of geometric methods in dynamics.

######  Lemma 4.1.

Let Assumption 2.1 hold and let $(\eta,x)$ solve (7), (9) with
$x_{0}\in\mathbb{R}^{d}$ and $\eta_{0}\in L^{2}_{A}(\mathbb{R}^{+})^{d}$. Then

(18)
$\frac{d}{dt}\|\eta^{t}\|^{2}+C\|\eta^{t}\|^{2}\leq-2\left(\int_{0}^{\infty}A(s)\eta^{t}(s)\,ds,x^{\prime}(t)\right)$

and

(19) $\|\eta^{t_{2}}\|^{2}\leq
e^{-C(t_{2}-t_{1})}\|\eta^{t_{1}}\|^{2}-2e^{-Ct_{2}}\int_{t_{1}}^{t_{2}}e^{Ct}\left(\int_{0}^{\infty}A(s)\eta^{t}(s)\,ds,x^{\prime}(t)\right)\,dt\
\ \textrm{for}\ \ t_{1}<t_{2}.$

###### Proof.

Let $\eta^{0}\in L^{2}_{A}(\mathbb{R}^{+})^{d}$ and $x_{0}\in\mathbb{R}^{d}$.
Define $x(-s)=x_{0}+\eta^{0}(s)$ for $s\leq 0$ and let $x\in
C^{1}([0,\infty))$ be a solution of (7). Moreover, for $t>0$

$\eta^{t}(s)=\begin{cases}x(t-s)-x(t)\ \ \textrm{for}\ \ s\leq t\\\
x(t-s)-x(t)=x_{0}+\eta^{0}(s-t)-x(t)\ \ \textrm{otherwise}.\end{cases}$

The squared norm of $\eta^{t}$ is given by

$\|\eta^{t}\|^{2}=\int_{0}^{\infty}(A(s)(x(t-s)-x(t)),(x(t-s)-x(t)))\,ds=\int_{-\infty}^{t}(A(t-s)(x(s)-x(t)),(x(s)-x(t)))\,ds.$

Let $t\geq 0$ and $h>0$. We calculate the right derivative of the above
squared norm with respect to $t$.

$\displaystyle\frac{\|\eta^{t+h}\|^{2}-\|\eta^{t}\|^{2}}{h}=\frac{1}{h}\int_{t}^{t+h}(A(t+h-s)(x(s)-x(t+h)),(x(s)-x(t+h)))\,ds$
$\displaystyle\ \ \ \
+\int_{-\infty}^{t}\left(\frac{A(t+h-s)-A(t-s)}{h}(x(s)-x(t)),(x(s)-x(t))\right)\,ds$
$\displaystyle\ \ \ \
+h\left(\int_{-\infty}^{t}A(t+h-s)\,ds\frac{x(t)-x(t+h)}{h},\frac{x(t)-x(t+h)}{h}\right)$
$\displaystyle\ \ \ \
+2\left(\int_{-\infty}^{t}A(t+h-s)(x(s)-x(t))\,ds,\frac{x(t)-x(t+h)}{h}\right).$

Passing to the limit with $h\to 0^{+}$, using the mean value theorem for
integrals, the first term in the above sum tends to zero. Moreover, the limit
of the third term is zero. In the second and fourth term we use the Lebesgue
dominated convergence theorem to pass to the limit, whence

$\displaystyle\lim_{h\to 0^{+}}\frac{\|\eta^{t+h}\|^{2}-\|\eta^{t}\|^{2}}{h}$
$\displaystyle\ \ =\lim_{h\to
0^{+}}\int_{0}^{\infty}\left(\frac{A(s+h)-A(s)}{h}(x(t-s)-x(t)),(x(t-s)-x(t))\right)\,ds$
$\displaystyle\ \ \ \ \ \ \ -2\left(\lim_{h\to
0^{+}}\int_{0}^{\infty}A(s+h)(x(t-s)-x(t))\,ds,x^{\prime}(t)\right)$
$\displaystyle\ \
=\int_{0}^{\infty}\left(\frac{dA(s)}{ds}\eta^{t}(s),\eta^{t}(s)\right)\,ds-2\left(\int_{0}^{\infty}A(s)\eta^{t}(s)\,ds,x^{\prime}(t)\right).$

Similar calculation for $t>0$ and $h<0$ leads to the left derivative for
$t>0$. Hence

$\frac{d}{dt}\|\eta^{t}\|^{2}=\int_{0}^{\infty}\left(\frac{dA(s)}{ds}\eta^{t}(s),\eta^{t}(s)\right)\,ds-2\left(\int_{0}^{\infty}A(s)\eta^{t}(s)\,ds,x^{\prime}(t)\right),$

and the assertion (18) follows by Assumption 2.1. After multiplication by the
integrating factor $e^{Ct}$ we deduce

$\frac{d}{dt}e^{Ct}\|\eta^{t}\|^{2}\,dt\leq-2e^{Ct}\left(\int_{0}^{\infty}A(s)\eta^{t}(s)\,ds,x^{\prime}(t)\right)$

Integrating from $t_{1}$ to $t_{2}$ we obtain (19). ∎

Similar argument leads to the following result

######  Lemma 4.2.

Let $(x,\eta)$ and $(y,\xi)$ be two solutions, not necessarily with same
$\varepsilon$. Then

(20)
$\frac{d}{dt}\|\eta^{t}-\xi^{t}\|^{2}+C\|\eta^{t}-\xi^{t}\|^{2}\leq-2\left(\int_{0}^{\infty}A(s)(\eta^{t}(s)-\xi^{t}(s))\,ds,(x(t)-y(t))^{\prime}\right)$

and

$\displaystyle\|\eta^{t_{2}}-\xi^{t_{2}}\|^{2}\leq
e^{-C(t_{2}-t_{1})}\|\eta^{t_{1}}-\xi^{t_{1}}\|^{2}$
$\displaystyle\qquad-2e^{-Ct_{2}}\int_{t_{1}}^{t_{2}}e^{Ct}\left(\int_{0}^{\infty}A(s)(\eta^{t}(s)-\xi^{t}(s))\,ds,x^{\prime}(t)-y^{\prime}(t)\right)\,dt\
\ \textrm{for}\ \ t_{1}<t_{2}.$

##  5\. Continuation of Lyapunov functions and isolating blocks with cone
conditions.

The goal of this section is to show that several dynamical properties of (4)
"survive" as we pass to (7). The dynamical objects discussed here are

  * •

Lyapunov function. For ODE which is a gradient system (i.e. $f=\nabla F$) for
sufficiently small $\varepsilon$ for (7) we construct the Lyapunov function.
This is contained in Lemma 5.1 and, see inequality (23).

  * •

Isolating blocks satisfying cone conditions from (4) "survive" for
sufficiently small $\varepsilon$ in (7). The continuation of isolating block
is established in Section 5.2. The cone conditions are discussed in Section
5.3.

###  5.1. Lyapunov function for the problem with delay.

######  Lemma 5.1.

For every $E>0$ there holds the bound

$\displaystyle\frac{d}{dt}E\|\eta^{t}\|^{2}+2|x^{\prime}(t)|^{2}+EC\|\eta^{t}\|^{2}\leq
2(f(x(t)),x^{\prime}(t))$ (21) $\displaystyle\ \
+2\varepsilon\left(\int_{0}^{\infty}M(s)\,dsx(t),x^{\prime}(t)\right)+2\left(\int_{0}^{\infty}\left(EA(s)-\varepsilon
M(s)\right)\eta^{t}(s)\,ds,x^{\prime}(t)\right).$

In particular if $f=\nabla F$ and $\int_{0}^{\infty}M(s)\,ds$ is symmetric
then

(22)
$\displaystyle\frac{d}{dt}\left(E\|\eta^{t}\|^{2}-2F(x(t))-\varepsilon\left(\int_{0}^{\infty}M(s)\,ds\,x(t),x(t)\right)\right)+2|x^{\prime}(t)|^{2}+EC\|\eta^{t}\|^{2}$
$\displaystyle\ \ \leq 2\left(\int_{0}^{\infty}\left(EA(s)-\varepsilon
M(s)\right)\eta^{t}(s)\,ds,x^{\prime}(t)\right).$

Moreover, there exists $E_{0}>0$ such that for every $E\in(0,E_{0})$ there
exists $\varepsilon_{0}(E)>0$ such that for every
$\varepsilon\in[0,\varepsilon_{0})$ there holds

(23)
$\displaystyle\frac{d}{dt}\left(E\|\eta^{t}\|^{2}-2F(x(t))-\varepsilon\left(\int_{0}^{\infty}M(s)\,ds\,x(t),x(t)\right)\right)+|x^{\prime}(t)|^{2}+E\frac{C}{4}\|\eta^{t}\|^{2}\leq
0$

###### Proof.

Multiply (8) by $2x^{\prime}(t)$. Then

$2|x^{\prime}(t)|^{2}=2(f^{\varepsilon}(x(t)),x^{\prime}(t))+2\varepsilon\left(\int_{0}^{\infty}M(s)\eta^{t}(s)\,ds,x^{\prime}(t)\right).$

Adding this equation to the inequality (18) multiplied by $E>0$ we obtain
(5.1). If $f=\nabla F$, we obtain (22). Choosing $\delta>0$ we estimate the
term on the right-hand side as

$2\left(\int_{0}^{\infty}\left(EA(s)-\varepsilon
M(s)\right)\eta^{t}(s)\,ds,x^{\prime}(t)\right)\leq\delta|x^{\prime}(t)|^{2}+\frac{1}{\delta}\left|\int_{0}^{\infty}\left(EA(s)-\varepsilon
M(s)\right)\eta^{t}(s)\,ds\right|^{2}$

. The last term can be estimated as

$\displaystyle\frac{1}{\delta}\left|\int_{0}^{\infty}\left(EA(s)-\varepsilon
M(s)\right)\eta^{t}(s)\,ds\right|^{2}\leq\frac{1}{\delta}\left(\int_{0}^{\infty}\sqrt{\|EA(s)-\varepsilon
M(s)\|}\sqrt{\|EA(s)-\varepsilon M(s)\|}|\eta^{t}(s)|\,ds\right)^{2}$
$\displaystyle\ \leq\frac{1}{\delta}\int_{0}^{\infty}\|EA(s)-\varepsilon
M(s)\|\,ds\int_{0}^{\infty}\|EA(s)-\varepsilon M(s)\||\eta^{t}(s)|^{2}\,ds.$

Moreover,

$\int_{0}^{\infty}\|EA(s)-\varepsilon M(s)\|\,ds\leq
E\int_{0}^{\infty}\|A(s)\|\,ds+\varepsilon\int_{0}^{\infty}\|M(s)\|\,ds,$

and, using Assumption 2.3 and 2.5

$\displaystyle\int_{0}^{\infty}\|EA(s)-\varepsilon
M(s)\||\eta^{t}(s)|^{2}\,ds\leq
E\int_{0}^{\infty}\|A(s)\||\eta^{t}(s)|^{2}\,ds+\varepsilon\int_{0}^{\infty}\|M(s)\||\eta^{t}(s)|^{2}\,ds$
$\displaystyle\ \
\leq\left(E\overline{D}^{2}+\varepsilon\overline{C_{1}}^{2}\right)\|\eta^{t}\|^{2}$

Choosing $\delta=1$ we get

$\displaystyle\frac{d}{dt}\left(E\|\eta^{t}\|^{2}-2F(x(t))-\varepsilon\left(\int_{0}^{\infty}M(s)\,ds\,x(t),x(t)\right)\right)+|x^{\prime}(t)|^{2}+EC\|\eta^{t}\|^{2}$
$\displaystyle\ \
\leq\left(E\int_{0}^{\infty}\|A(s)\|\,ds+\varepsilon\int_{0}^{\infty}\|M(s)\|\,ds\right)\left(E\overline{D}^{2}+\varepsilon\overline{C_{1}}^{2}\right)\|\eta^{t}\|^{2}.$

Moving all terms to the left, the constant in front of $\|\eta^{t}\|^{2}$ is
equal to

$-E^{2}\overline{D}^{2}\int_{0}^{\infty}\|A(s)\|\,ds+E\left(C-\varepsilon\overline{C_{1}}^{2}\int_{0}^{\infty}\|A(s)\|\,ds-\varepsilon\overline{D}^{2}\int_{0}^{\infty}\|M(s)\|\,ds\right)-\varepsilon^{2}\overline{C_{1}}^{2}\int_{0}^{\infty}\|M(s)\|\,ds.$

This expression can be rewritten as

$-E^{2}G_{1}+E(G_{2}-\varepsilon G_{3})-\varepsilon^{2}G_{4},$

where $G_{1},G_{2},G_{3},G_{4}$ are positive constants. Take
$E_{0}=\frac{G_{2}}{2G_{1}}$. If $E\in(0,E_{0})$, then

$-E^{2}G_{1}+E(G_{2}-\varepsilon G_{3})-\varepsilon^{2}G_{4}\geq
EG_{2}-E\frac{G_{2}}{2}-\varepsilon\frac{G_{3}G_{2}}{2G_{1}}-\varepsilon^{2}G_{4}=E\frac{G_{2}}{2}-\varepsilon\frac{G_{3}G_{2}}{2G_{1}}-\varepsilon^{2}G_{4}.$

Now take $\varepsilon_{0}(E)$ such that

$\varepsilon\frac{G_{3}G_{2}}{2G_{1}}+\varepsilon^{2}G_{4}\leq
E\frac{G_{2}}{4},$

if only $\varepsilon\in(0,\varepsilon_{0})$. This means that

$-E^{2}G_{1}+E(G_{2}-\varepsilon G_{3})-\varepsilon^{2}G_{4}\geq
E\frac{G_{2}}{4}=E\frac{C}{4}.$

The proof is complete. ∎

As a consequence of the above Lemma we obtained to following Lyapunov function
$L:L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}$ valid for every
$E\in(0,E_{0})$ and for every $\varepsilon\in[0,\varepsilon_{0}(E))$

(24)
$L(\eta^{t},x(t))=E\|\eta^{t}\|^{2}-2F(x(t))-\varepsilon\left(\int_{0}^{\infty}M(s)\,ds\,x(t),x(t)\right)$

###  5.2. The continuation of the isolating block

The next result follows from [11, Theorem 26]. We give a short proof for the
completeness of exposition. In this section we use the notation

$B_{u}(\delta)=\prod_{k=1}^{u_{1}}[-\delta,\delta]\times\prod_{k=u_{1}+1}^{u_{1}+u_{2}}\\{(x,y)\,:\,x^{2}+y^{2}\leq\delta^{2}\\},$

and

$B_{s}(\delta)=\prod_{k=u_{1}+u_{2}+1}^{u_{1}+u_{2}+s_{1}}[-\delta,\delta]\times\prod_{k=u_{1}+u_{2}+s_{1}+1}^{u_{1}+u_{2}+s_{1}+s_{2}}\\{(x,y)\,:\;x^{2}+y^{2}\leq\delta^{2}\\}.$

######  Lemma 5.2.

Let $x_{0}$ be such that $f(x_{0})=0$. Assume that this equilibrium is
hyperbolic, that it, that the matrix $Df(x_{0})$ is nonsingular, with $s$
equal to the dimension of its stable space, and $u=d-s$ the dimension of its
unstable space. Let $s=s_{1}+2s_{2}$, where $s_{1}$ is the dimension of the
generalized eigenspace related with real stable eigenvalues, and $2s_{2}$ is
the dimension of the generalized eigenspace related with complex stable
eigenvalues. Analogously, $u=u_{1}+2u_{2}.$ There exists the nonsingular
matrix $T_{\kappa}$ and a number $\delta_{0}>0$ such that for every
$\delta\in(0,\delta_{0})$ the set

$N_{\kappa}(\delta)=T_{\kappa}\left(B_{u}(\delta)\times
B_{s}(\delta)\right)+x_{0}.$

is an isolating block with cones for $\varepsilon=0$, i.e. for equation (4).

###### Proof.

Let $T_{\kappa}$ be an invertible matrix such that
$T_{\kappa}^{-1}Df(x_{0})T_{\kappa}$ is the Jordan form, that on the diagonal
has either real eigenvalues $\lambda$ of $Df(x_{0})$, or blocks
$\begin{pmatrix}\alpha&\beta\\\ -\beta&\alpha\end{pmatrix}$ in case of complex
eigenvalues, and all off-diagonal terms belong to $(0,\kappa)$. Assume that
the eigenvalues in $T_{\kappa}$ are sorted such that: first there are real
positive eigenvalues, then complex eigenvalues with positive real part, then
negative real eigenvalues, and finally complex eigenvalues with negative real
part. Moreover assume that
$T_{\kappa}^{-1}Df(x_{0})T_{\kappa}=\begin{pmatrix}A&0\\\ 0&B\end{pmatrix}$,
where $A+A^{T}\in M^{u\times u}$ is negative definite and $B+B^{T}\in
M^{s\times s}$ is positive definite. For every $\kappa>0$ such change of
coordinates $T_{\kappa}$ exists. We first prove that the set
$N_{\kappa}(\delta)$ is isolating.

If we denote $x=x_{0}+T_{\kappa}y$, we obtain the system

$y^{\prime}=T_{\kappa}^{-1}Df(x_{0})T_{\kappa}y+T_{\kappa}^{-1}f(x_{0}+T_{\kappa}y)-T_{\kappa}^{-1}Df(x_{0})T_{\kappa}y=h(y).$

Now for $y\in B_{u}(2\delta)\times B_{s}(2\delta),$ we deduce, by the Taylor
theorem, as $f\in C^{2}(\mathbb{R}^{d},\mathbb{R}^{d})$ that

$|T_{\kappa}^{-1}f(x_{0}+T_{\kappa}y)-T_{\kappa}^{-1}Df(x_{0})T_{\kappa}y|\leq
C_{\kappa}\delta^{2}$

where $C_{\kappa}$ depends on $\kappa$ but not on $\delta\in(0,\delta_{0})$.
We need to prove that:

  * •

if $y\in B_{u}(2\delta)\times\partial B_{s}(\delta),$ then

(25) $h_{i}(y)y_{i}<0\ \ \textrm{for}\ \
i\in\\{u_{1}+2u_{2}+1,\ldots,u_{1}+2u_{2}+s_{1}\\}$

and

(26) $h_{i}(y)y_{i}+h_{i+1}(y)y_{i+1}<0\ \ \textrm{for}\ \
i\in\\{u_{1}+2u_{2}+s_{1}+1,\ldots,u_{1}+2u_{2}+s_{1}+j,\ldots
u_{1}+2u_{2}+s_{1}+(2s_{2}-1)\\},$

where $j$ are odd numbers,

  * •

if $y\in\left(B_{u}(2\delta)\setminus(\textrm{int}B_{u}(\delta))\right)\times
B_{s}(\delta),$ then

(27) $h_{i}(y)y_{i}>0\ \ \textrm{for}\ \ i\in\\{1,\ldots,u_{1}\\}$

and

(28) $h_{i}(y)y_{i}+h_{i+1}(y)y_{i+1}>0\ \ \textrm{for}\ \
i\in\\{u_{1}+1,\ldots,u_{1}+j,\ldots u_{1}+2u_{2}-1\\},$

where $j$ are odd numbers,

By the Lipschitz continuous dependence on the initial condition, on bounded
sets of initial data and compact time intervals these conditions imply the
isolation in Definition 9.6.

To prove the first assertion observe that for $i\in
u_{1}+u_{2}+1,\ldots,u_{1}+u_{2}+s_{1}$.

$h_{i}(y)y_{i}=\lambda_{i}y_{i}^{2}+G(y),$

where

$|G(y)|\leq(d-1)\kappa 4|\delta|^{2}+2C_{\kappa}|\delta|^{3},$

the first term coming from off diagonal values (at most $d-1$) in
$T_{\kappa}^{-1}Df(x_{0})T_{\kappa}$, and the second one from the remainder
which is a product of number which is dominated by the euclidean norm of a
vector bounded by $C_{\kappa}|\delta|^{2}$ and a number bounded by $2\delta$.
This means that we can choose $\kappa$ small enough (related to the lowest
eigenvalue $\lambda_{i}$) and $\delta_{0}$ (that is chosen according to
$C_{\kappa}$) and to get t(25).

For the complex pairs of eigenvalues the off diagonal term in blocks
$\begin{pmatrix}\alpha&\beta\\\ -\beta&\alpha\end{pmatrix}$ cancels and we
obtain

$h_{i}(y)y_{i}+h_{i+1}(y)y_{i+1}=\textrm{Re}\,\lambda_{i}(y_{i}^{2}+y_{i+1}^{2})+G(y),$

with

$|G(y)|\leq 8(d-2)\kappa|\delta|^{2}+2C_{k}|\delta|^{3},$

and (26) holds analogously as (25). Verification of (27) and (28) follows
analogously.

To see that the cone condition holds it is enough to take the matrix $Q$ such
that $q_{ij}=0$ for $i\neq j$, $q_{ii}=-1$ for $i=1,\ldots,u$ and $q_{ii}=1$
for $i=u+1,\ldots,d$ and see that
$QT_{\kappa}^{-1}Df(x_{0})T_{\kappa}+T_{\kappa}^{-1}Df(x_{0})T_{\kappa}Q$ is
positive definite, which must be preserved on a small neighborhood of $y=0$. ∎

In the subsequent part of this section we will show that it is possible to
choose $\delta$ and as well as $R>0$ such that if $N_{\kappa}(\delta)$ is an
isolating block with cones for (4) then the set $N_{\kappa}(\delta)\times
B_{L_{A}^{2}(\mathbb{R}^{+})^{d}}(0,R)$ is an isolating block with cones for
(7). We start from an estimate. Substitute (8) in (18). Then we obtain

$\displaystyle\frac{d}{dt}\|\eta^{t}\|^{2}+C\|\eta^{t}\|^{2}\leq-2\left(\int_{0}^{\infty}A(s)\eta^{t}(s)\,ds,f(x(t))\right)-2\varepsilon\left(\int_{0}^{\infty}A(s)\eta^{t}(s)\,ds,\left(\int_{0}^{\infty}M(s)\,ds\right)x\right)$
$\displaystyle\ \ \
-2\varepsilon\left(\int_{0}^{\infty}A(s)\eta^{t}(s)\,ds,\int_{0}^{\infty}M(s)\eta^{t}(s)\,ds\right).$

After computations which use Lemmas 2.4 and 2.6 it follows that

$\displaystyle\frac{d}{dt}\|\eta^{t}\|^{2}+C\|\eta^{t}\|^{2}\leq
2D\|\eta^{t}\||f(x(t))|+2\varepsilon
D\|\eta^{t}\|\int_{0}^{\infty}\|M(s)\|\,ds|x(t)|+2\varepsilon
DC_{1}\|\eta^{t}\|^{2}.$

$\displaystyle\frac{d}{dt}\|\eta^{t}\|^{2}\leq\|\eta^{t}\|\left(2D|f(x(t))|+2\varepsilon
D\int_{0}^{\infty}\|M(s)\|\,ds|x(t)|+2\varepsilon
DC_{1}\|\eta^{t}\|-C\|\eta^{t}\|\right).$

The above computation leads is a straightforward way to the following lemma.

######  Lemma 5.3.

Suppose that $f(x_{0})=0$ and that $N\subset\mathbb{R}^{d}$ is a compact set
containing $x_{0}$. Moreover let $\varepsilon<\frac{C}{2DC_{1}}$ and

$R>\frac{2D\left(\sup_{z\in
N}|f(z)|+\varepsilon\int_{0}^{\infty}\|M(s)\|\,ds\cdot\sup_{z\in
N}|z|\right)}{C-2\varepsilon DC_{1}}=\frac{2D\left(\sup_{z\in
N}|f(z)|+\varepsilon D\cdot\sup_{z\in N}|z|\right)}{C-2\varepsilon DC_{1}}.$

Then for $\eta\in\partial B_{L^{2}_{A}(\mathbb{R}^{+})^{d}}(0,R)$, and $y\in
N$ there holds

$\frac{d}{dt}\|\eta^{t}\|^{2}_{L^{2}_{A}(\mathbb{R}^{+})^{d}}<0.$

Let us rewrite the equation (7) in the changed variables $y$.

$y^{\prime}(t)=h(y(t))+\varepsilon
T_{\kappa}^{-1}\left(\int_{0}^{\infty}M(s)\,ds\right)(x_{0}+T_{\kappa}y(t))+\varepsilon
T_{\kappa}^{-1}\int_{0}^{\infty}M(s)\eta^{t}(s)\,ds.$

Now choose $\kappa$ and $\delta_{0}$ such that Lemma 5.2 holds and assume that
$\delta<\frac{\delta_{0}}{2}$. For such $\delta$ let $r(\delta)$ be a smallest
possible number such that $N_{\kappa}(2\delta)\subset B(x_{0},r)$. Note that
$r\to 0$ as $\delta\to 0$. Take $\eta^{t}\in
B_{L^{2}_{A}(\mathbb{R}^{+})^{n}}(0,R)$ and $x=T_{\kappa}y+x_{0}\in
N_{\kappa}(2\delta)$. We rewrite the $i$-th equation of the above system as

$y_{i}^{\prime}(t)=h_{i}(y(t))+g_{i}(y(t),\eta^{t}),$

where

$g_{i}(y(t),\eta^{t})=\varepsilon\left(T_{\kappa}^{-1}\left(\int_{0}^{\infty}M(s)\,ds\right)(x_{0}+T_{\kappa}y(t))\right)_{i}+\varepsilon\left(T_{\kappa}^{-1}\int_{0}^{\infty}M(s)\eta^{t}(s)\,ds\right)_{i}$

hence

(29)
$|g_{i}(y,\eta)|\leq|g(y,\eta)|\leq\varepsilon\|T_{\kappa}^{-1}\|D\|\eta^{t}\|+\varepsilon\|T_{\kappa}^{-1}\|\int_{0}^{\infty}\|M(s)\|\,ds(|x_{0}|+r)\leq\varepsilon\|T_{\kappa}^{-1}\|D(R+|x_{0}|+r).$

######  Theorem 5.4.

There exists $\kappa>0$, $\varepsilon_{0}>0$, $\delta>0$, and $R>0$ such that
for every fixed point $x_{0}$, every $\varepsilon\in(0,\varepsilon_{0})$ the
set $N_{\kappa}(\delta)\times B_{L^{2}_{A}(\mathbb{R}_{+})^{d}}(0,R)$ is an
isolating block for (7), i.e.

  * (I)

if $(y,\eta)\in B_{u}(2\delta)\times\partial B_{s}(\delta)\times
B_{L^{2}_{A}(\mathbb{R}_{+})^{d}}(0,R),$ then

(30) $(h_{i}(y)+g_{i}(y,\eta))y_{i}<0\ \ \textrm{for}\ \ i\in
u_{1}+2u_{2}+1,\ldots,u_{1}+2u_{2}+s_{1}$

and

(31) $(h_{i}(y)+g_{i}(y,\eta))y_{i}+(h_{i+1}(y)+g_{i+1}(y,\eta))y_{i+1}<0\ \
\textrm{for}\ \ i\in u_{1}+2u_{2}+s_{1}+1,\ldots,u_{1}+2u_{2}+s_{1}+j,\ldots
u_{1}+2u_{2}+s_{1}+(2s_{2}-1),$

where $j$ are odd numbers,

  * (II)

if
$(y,\eta)\in\left(B_{u}(2\delta)\setminus(\textrm{int}B_{u}(\delta))\right)\times
B_{s}(\delta)\times B_{L^{2}_{A}(\mathbb{R}_{+})^{d}}(0,R),$ then

(32) $(h_{i}(y)+g_{i}(y,\eta))y_{i}>0\ \ \textrm{for}\ \ i\in 1,\ldots,u_{1}$

and

(33) $(h_{i}(y)+g_{i}(y,\eta))y_{i}+(h_{i+1}(y)+g_{i+1}(y,\eta))y_{i+1}>0\ \
\textrm{for}\ \ i\in u_{1}+1,\ldots,u_{1}+j,\ldots u_{1}+2u_{2}-1,$

where $j$ are odd numbers,

  * (III)

if $(y,\eta)\in B_{u}(2\delta)\times B_{s}(\delta)\times\partial
B_{L^{2}_{A}(\mathbb{R}_{+})^{d}}(0,R)$, then

$\frac{d}{dt}\|\eta_{t}\|^{2}_{L^{2}_{A}(\mathbb{R}_{+})^{d}}\leq 0\ \
\textrm{at}\ \ t=0.$

###### Proof.

We first provide the condition needed for (III) to hold: this is the entry
condition for the variable $\delta$. Following Lemma 5.3 we need that $R>0$
and $\delta_{0}>0$ should satisfy

(34) $R>\frac{2D\left(\sup_{z\in N_{\kappa}(2\delta)}|\nabla
f(z)|r(\delta)+\varepsilon Dr(\delta)\right)}{C-2\varepsilon DC_{1}}.$

We switch to the conditions needed for (I) and (II), that is, for (30)–(33).

We first note that there exists $\delta_{0}$ and a constant $C_{1}>0$ such
that if only $\delta\in(0,\delta_{0})$ and $y\in B_{u}(2\delta)\times\partial
B_{s}(\delta)\cup(B_{u}(2\delta)\setminus\textrm{int}B_{u}(\delta))\times
B_{s}(\delta)$ then

(35) $C_{1}\delta\leq|h_{i}(y)|\ \ \textrm{for indexes corresponding to real
eigenvalues},$

and

(36) $C_{1}\delta^{2}\leq|h_{i}(y)y_{i}+h_{i+1}(y)y_{i+1}|\ \ \textrm{for
indexes corresponding to complex eigenvalues}.$

In order for (30) and (32) we need $h_{i}(y)$ to have the same sign as and
$h_{i}(y)+g_{i}(y,\eta)$ for $y\in B_{u}(2\delta)\times\partial
B_{s}(\delta)\cup(B_{u}(2\delta)\setminus\textrm{int}B_{u}(\delta))\times
B_{s}(\delta)$ and $\eta\in B_{L^{2}_{A}(\mathbb{R}_{+})^{d}}(0,R)$. For the
complex eigenvalues we need, on the other hand, that
$(h_{i}(y)+g_{i}(y,\eta))y_{i}+(h_{i+1}(y)+g_{i+1}(y,\eta))y_{i+1}$ and
$h_{i}(y)y_{i}+h_{i+1}(y)y_{i+1}$ have the same signs. Therefore, in view of
(35) it is sufficient to prove that $|g_{i}(y,\eta)|<C_{1}\delta$ and
$|g_{i}(y,\eta)y_{i}+g_{i+1}(y,\eta)y_{i+1}|<C_{1}\delta^{2}$. Using (29) it
is enough that the following inequality holds

(37) $\varepsilon\|T_{\kappa}^{-1}\|D(R+|x_{0}|+r(\delta))<C_{1}\delta.$

First choose $\varepsilon_{0}$ such that
$C-2\varepsilon_{0}DC_{1}<\frac{C}{2}$. The same inequality holds for
$\varepsilon\in(0,\varepsilon_{0})$ in the denominator of (34). Now pick
$R>0$. We need to choose $\delta$ small enough such that

$\frac{4D\sup_{z\in N_{\kappa}(2\delta)}|Df(z)|r(\delta)}{C}<\frac{R}{2}.$

This is possible, because by decreasing $\delta$ we can make $r(\delta)$
arbitrarily small. Now it is possible to choose sufficiently small
$\varepsilon_{0}$ such that
$\varepsilon_{0}\|T_{\kappa}^{-1}\|D(R+|x_{0}|+r(\delta))<C_{1}\delta$ and
$\frac{\varepsilon_{0}4D^{2}r}{C}<\frac{R}{2}$. Thus both inequalities are
satisfied and the proof is complete. ∎

###  5.3. Cone condition

The goal of this section is to show that cone conditions from (4) "survive"
for sufficiently small $\epsilon$ for (7). To this end assume that for the
equation

$x^{\prime}(t)=f^{\varepsilon}(x(t))=f(x(t))+\varepsilon\int_{0}^{\infty}M(s)\,dsx(t)$

we have a quadratic form $Q$ (a symmetric matrix) and a set $N$ (h-set) such
that on $N$ we have for any $x\in\mathbb{R}^{d}$ and $|\varepsilon|\leq\Delta$
for some $G>0$

(38) $x^{t}(Df^{\varepsilon}(N)^{T}Q+QDf^{\varepsilon}(N))x\geq G|x|^{2}.$

Note that, as

$Df^{\varepsilon}(x)=Df(x)+\varepsilon\int_{0}^{\infty}M(s)\,ds.$

Hence, if $\varepsilon>0$ is small enough, then the same form $Q$ is valid
both for $f$ and for $f^{\varepsilon}$ (with possibly smaller constant $G$).
Let $(\eta,x)$ and $(\xi,y)$ be two solutions of (8)–(9) such $x,y\in N$ and
$\|\eta\|,\|\xi\|\leq R$. Let $E>0$ be any positive constant. We hope that for
the quadratic form

(39) $\widetilde{Q}(\eta,x)=Q(x)-E\|\eta\|^{2}$

we will have cone-conditions on the set (isolated block)

(40) $\widetilde{N}=\\{\|\eta\|\leq R\\}\times N.$

We have

$\displaystyle\frac{d}{dt}\left(Q(x(t)-y(t),x(t)-y(t))\right)=(x^{\prime}-y^{\prime})^{\top}Q(x-y)+(x-y)^{\top}Q(x^{\prime}-y^{\prime}).$

Since

$\displaystyle x^{\prime}-y^{\prime}$ $\displaystyle=$
$\displaystyle\left(f^{\varepsilon}(x)+\varepsilon\int_{0}^{\infty}M(s)\eta(s)ds\right)-\left(f^{\varepsilon}(y)+\varepsilon\int_{0}^{\infty}M(s)\xi(s)ds\right)$
$\displaystyle=$
$\displaystyle(f^{\varepsilon}(x)-f^{\varepsilon}(y))+\varepsilon\int_{0}^{\infty}M(s)(\eta(s)-\xi(s))ds$
$\displaystyle=$
$\displaystyle\overline{Df^{\varepsilon}[x,y]}(x-y)+\varepsilon\int_{0}^{\infty}M(s)(\eta(s)-\xi(s))ds,$

we obtain using Lemma 2.6

$\displaystyle(x^{\prime}-y^{\prime})^{\top}Q(x-y)+(x-y)^{\top}Q(x^{\prime}-y^{\prime})=(x-y)^{\top}\left(\overline{Df^{\varepsilon}[x,y]}^{\top}Q+Q\overline{Df^{\varepsilon}[x,y]}\right)(x-y)$
$\displaystyle\ \ \ \
+\varepsilon\left(\int_{0}^{\infty}M(s)(\eta(s)-\xi(s))ds\right)^{\top}Q(x-y)+\varepsilon(x-y)^{T}Q\int_{0}^{\infty}M(s)(\eta(s)-\xi(s))ds$
$\displaystyle\ \geq G|x-y|^{2}-2\varepsilon
C_{1}\|Q\|\cdot|x-y|\cdot\|\eta-\xi\|.$

From Lemma 4.2 we have

$\displaystyle\frac{d}{dt}\|\eta^{t}-\xi^{t}\|^{2}\leq-C\|\eta-\xi\|^{2}-2\left(\int_{0}^{\infty}A(s)(\eta(s)-\xi(s))ds,(x(t)-y(t))^{\prime}\right)$
$\displaystyle\ \ \
\leq-C\|\eta-\xi\|^{2}-2\left(\int_{0}^{\infty}A(s)(\eta(s)-\xi(s))ds,\overline{Df^{\varepsilon}[x,y]}(x-y)\right)+$
$\displaystyle\ \ \ \ \ \
-2\varepsilon\left(\int_{0}^{\infty}A(s)(\eta(s)-\xi(s))ds,\int_{0}^{\infty}M(s)(\eta(s)-\xi(s))ds\right)$
$\displaystyle\ \ \
\leq-C\|\eta-\xi\|^{2}+2D\|\eta-\xi\|\cdot\|Df^{\varepsilon}(N)\|\cdot|x-y|+\varepsilon
2C_{1}D\|\eta-\xi\|^{2}$

Now we are ready to demonstrate that the cone condition holds. From previous
derivations we obtain

$\displaystyle\frac{d}{dt}\left(Q(x(t)-y(t),x(t)-y(t))-E\|\eta^{t}-\xi^{t}\|^{2}\right)$
$\displaystyle\ \ \ \geq G|x-y|^{2}-2\varepsilon
C_{1}\|Q\|\cdot|x-y|\cdot\|\eta-\xi\|$ $\displaystyle\ \ \ \ \ \
-E\left(-C\|\eta-\xi\|^{2}+2D\|\eta-\xi\|\cdot\|Df^{\varepsilon}(N)\|\cdot|x-y|+\varepsilon
2C_{1}D\|\eta-\xi\|^{2}\right)$ $\displaystyle\ \ \
=G|x-y|^{2}+2\left(-\varepsilon
C_{1}\|Q\|-ED\|Df^{\varepsilon}(N)\|\right)|x-y|\cdot\|\eta-\xi\|+$
$\displaystyle\ \ \ \ \ \ +\left(CE-2\varepsilon
C_{1}D\right)\|\eta-\xi\|^{2}$

The expression on the right-hand side is a quadratic form in terms of
$(|x-y|,\|\eta-\xi\|)$ with the matrix

(41) $B=\left[\begin{array}[]{cc}G&\left(-\varepsilon
C_{1}\|Q\|-ED\|Df^{\varepsilon}(N)\|\right)\\\ \left(-\varepsilon
C_{1}\|Q\|-ED\|Df^{\varepsilon}(N)\|\right)&\left(CE-2\varepsilon
C_{1}D\right)\\\ \end{array}\right]$

Consider first the case with $\varepsilon=0$. Matrix $B$ becomes

(42) $B_{0}=\left[\begin{array}[]{cc}G&-ED\|Df(N)\|\\\ -ED\|Df(N)\|&CE\\\
\end{array}\right]$

It is positive definite provided the following condition holds

(43) $0<\det(B_{0})=CEG-E^{2}D^{2}\|Df(N)\|^{2},$

which is satisfied if

(44) $E<\frac{CG}{D^{2}\|Df(N)\|^{2}}.$

Since $\det B$ depends continuously on $\varepsilon$ we obtain the following
theorem.

######  Lemma 5.5.

For any $E>0$ satisfying

(45) $E<\frac{CG}{D^{2}\|Df(N)\|^{2}}.$

there exists $\varepsilon_{1}=\varepsilon_{1}(E)$, such cone condition holds
for quadratic form (39) for any $\varepsilon\leq\varepsilon_{1}$.

Consider the problem governed by (8)–(9) and denote the solution with
$\varepsilon\in[0,\Delta]$ and initial data $(\eta^{0},x_{0})\in\widetilde{N}$
by
$S^{\varepsilon}(t)(\eta^{0},x_{0})=(\eta^{t}_{\varepsilon,x_{0},\eta_{0}},x_{\varepsilon,x_{0},\eta_{0}})$.
Define

$\displaystyle\widehat{Q}(x,\eta,\varepsilon)=\widetilde{Q}(x,\eta)+L|\varepsilon|^{2}={Q}(x)-E\|\eta\|^{2}+L|\varepsilon|^{2},$

where $L$ can be either positive or negative. In the next lemma we prove that
$\widehat{Q}$ satisfies the assumptions of Lemma 10.1 for the sufficient
choice of $L$.

######  Lemma 5.6.

There exists $L_{0}>0$ and $E_{max}>0$ such that for every $|L|\geq L_{0}$ and
$E\in(0,E_{max})$ there exists $\Delta(E)>0$ such that the cone conditions
with parameter given in (i) and (ii) of Definition 10.1 are satisfied on the
h-set $\widetilde{N}$ with $\varepsilon_{1},\varepsilon_{2}\in[0,\Delta(E)]$

###### Proof.

Assume that $(\eta^{0}_{1},x^{0}_{1},\varepsilon_{1})$ and
$(\eta^{0}_{2},x^{0}_{2},\varepsilon_{2})$ are such that

$\widehat{Q}(x^{0}_{1}-x^{0}_{2},\eta^{0}_{1}-\eta^{0}_{2},\varepsilon_{1}-\varepsilon_{2})=0.$

We must prove that

$\frac{d}{dt}\widehat{Q}(x_{\varepsilon_{1},x^{0}_{1}\eta^{0}_{1}}(t)-x_{\varepsilon_{2},x^{0}_{2}\eta^{0}_{2}}(t),\eta_{\varepsilon_{1},x_{1}^{0},\eta_{1}^{0}}(t)-\eta_{\varepsilon_{2},x^{0}_{2}\eta^{0}_{2}}(t),\varepsilon_{1}-\varepsilon_{2})\geq
0\ \ \textrm{for}\ \ t=0.$

Denoting, for simplicity,
$(\eta_{1}(t),x_{1}(t))=(\eta_{\varepsilon_{1},x_{1}^{0},\eta_{1}^{0}}(t),x_{\varepsilon_{1},x^{0}_{1}\eta^{0}_{1}}(t))$
and
$(\eta_{2}(t),x_{2}(t))=(\eta_{\varepsilon_{2},x_{2}^{0},\eta_{2}^{0}}(t),x_{\varepsilon_{2},x^{0}_{2}\eta^{0}_{2}}(t))$
we should prove that

(46)
$\frac{d}{dt}\left((x_{1}(t)-x_{2}(t))^{\top}Q(x_{1}(t)-x_{2}(t))-E\|\eta_{1}(t)-\eta_{2}(t)\|^{2}\right)\geq
0\ \ \textrm{for}\ \ t=0.$

We estimate both terms from below separately

$\displaystyle\frac{d}{dt}((x_{1}(t)-x_{2}(t))^{\top}Q(x_{1}(t)-x_{2}(t))=(x_{1}^{\prime}(t)-x_{2}^{\prime}(t))^{\top}Q(x_{1}(t)-x_{2}(t))+(x_{1}(t)-x_{2}(t))^{\top}Q(x_{1}^{\prime}(t)-x_{2}^{\prime}(t))$
$\displaystyle\ \ \
=(f(x_{1}(t))-f(x_{2}(t)))^{\top}Q(x_{1}(t)-x_{2}(t))+(\varepsilon_{1}x_{1}(t)-\varepsilon_{2}x_{2}(t))^{\top}\int_{0}^{\infty}M^{\top}(s)\,dsQ(x_{1}(t)-x_{2}(t))$
$\displaystyle\ \ \
\qquad+\left(\varepsilon_{1}\int_{0}^{\infty}M(s)\eta_{1}^{t}(s)\,ds-\varepsilon_{2}\int_{0}^{\infty}M(s)\eta_{2}^{t}(s)\
ds\right)^{\top}Q(x_{1}(t)-x_{2}(t))$ $\displaystyle\ \ \
\qquad+(x_{1}(t)-x_{2}(t))^{\top}Q(f(x_{1}(t))-f(x_{2}(t)))+(x_{1}(t)-x_{2}(t))^{\top}Q\int_{0}^{\infty}M(s)\,ds(\varepsilon_{1}x_{1}(t)-\varepsilon_{2}x_{2}(t))$
$\displaystyle\ \ \
\qquad+(x_{1}(t)-x_{2}(t))^{\top}Q\left(\varepsilon_{1}\int_{0}^{\infty}M(s)\eta_{1}^{t}(s)\,ds-\varepsilon_{2}\int_{0}^{\infty}M(s)\eta_{2}^{t}(s)\
ds\right)$ $\displaystyle\ \ \
=(x_{1}(t)-x_{2}(t))^{\top}(Df(N)^{\top}Q+QDf(N))(x_{1}(t)-x_{2}(t))$
$\displaystyle\ \ \
\qquad+\varepsilon_{2}(x_{1}(t)-x_{2}(t))^{\top}\left(\int_{0}^{\infty}M^{\top}(s)\,dsQ+Q\int_{0}^{\infty}M(s)\,ds\right)(x_{1}(t)-x_{2}(t))$
$\displaystyle\ \ \
\qquad+2\varepsilon_{2}(x_{1}(t)-x_{2}(t))^{\top}Q\left(\int_{0}^{\infty}M(s)(\eta_{1}^{t}(s)-\eta_{2}^{t}(s))\,ds\right)$
$\displaystyle\ \ \
\qquad+2(\varepsilon_{1}-\varepsilon_{2})(x_{1}(t)-x_{2}(t))^{\top}Q\int_{0}^{\infty}M(s)\,dsx_{1}(t)+2(\varepsilon_{1}-\varepsilon_{2})(x_{1}(t)-x_{2}(t))^{\top}Q\left(\int_{0}^{\infty}M(s)\eta_{1}^{t}(s)\,ds\right)$
$\displaystyle\ \ \ =I_{1}+I_{2}+I_{3}+I_{4}.$

Now

$I_{1}+I_{2}\geq G|x_{1}(t)-x_{2}(t)|^{2},$

where $G$ can be chosen uniformly for $\varepsilon\in[0,\Delta].$ Moreover

$I_{3}\geq-2\Delta|x_{1}(t)-x_{2}(t)|\
\|Q\|C_{1}\|\eta_{1}^{t}-\eta_{2}^{t}\|,$

and

$I_{4}\geq-2|\varepsilon_{1}-\varepsilon_{2}|\ |x_{1}(t)-x_{2}(t)|\
\|Q\|\left(\int_{0}^{\infty}\|M(s)\|\,ds\sup_{x\in N}|x|+C_{1}R\right).$

For simplicity we use the following notation for the constant which will
appear several times in the subsequent computations
$\overline{R}=\int_{0}^{\infty}\|M(s)\|\,ds\sup_{x\in N}|x|+C_{1}R$.
Summarizing, we obtain

$\displaystyle\frac{d}{dt}((x_{1}(t)-x_{2}(t))^{\top}Q(x_{1}(t)-x_{2}(t))\geq
G|x_{1}(t)-x_{2}(t)|^{2}-2\Delta|x_{1}(t)-x_{2}(t)|\,\|Q\|C_{1}\|\eta_{1}^{t}-\eta_{2}^{t}\|$
$\displaystyle\ \ \
-2|\varepsilon_{1}-\varepsilon_{2}|\,|x_{1}(t)-x_{2}(t)|\,\|Q\|\overline{R}.$

We estimate the second term in (46) from Lemma 4.2

$\displaystyle\frac{d}{dt}\|\eta_{1}^{t}-\eta_{2}^{t}\|^{2}\leq-C\|\eta_{1}^{t}-\eta_{2}^{t}\|^{2}-2\left(\int_{0}^{\infty}A(s)(\eta_{1}^{t}(s)-\eta_{2}^{t}(s))ds,(x_{1}(t)-x_{2}(t))^{\prime}\right)$
$\displaystyle\ \ \
=-C\|\eta_{1}^{t}-\eta_{2}^{t}\|^{2}-2\left(\int_{0}^{\infty}A(s)(\eta_{1}^{t}(s)-\eta_{2}^{t}(s))ds,f(x(t))-f(y(t))\right)$
$\displaystyle\ \ \ \
-2\left(\int_{0}^{\infty}A(s)(\eta_{1}^{t}(s)-\eta_{2}^{t}(s))ds,\varepsilon_{1}\int_{0}^{\infty}M(s)\,dsx_{1}(t)-\varepsilon_{2}\int_{0}^{\infty}M(s)\,dsx_{2}(t)\right)$
$\displaystyle\ \ \ \ \ \
-2\left(\int_{0}^{\infty}A(s)(\eta_{1}^{t}(s)-\eta_{2}^{t}(s))ds,\varepsilon_{1}\int_{0}^{\infty}M(s)\eta_{1}^{t}(s)\,ds-\varepsilon_{2}\int_{0}^{\infty}M(s)\eta_{2}^{t}(s)ds\right)$
$\displaystyle\ \ \
=-C\|\eta_{1}^{t}-\eta_{2}^{t}\|^{2}-2\left(\int_{0}^{\infty}A(s)(\eta_{1}^{t}(s)-\eta_{2}^{t}(s))ds,f(x(t))-f(y(t))\right)$
$\displaystyle\ \ \ \
-2(\varepsilon_{1}-\varepsilon_{2})\left(\int_{0}^{\infty}A(s)(\eta_{1}^{t}(s)-\eta_{2}^{t}(s))ds,\int_{0}^{\infty}M(s)\,dsx_{1}(t)+\int_{0}^{\infty}M(s)\eta_{1}^{t}(s)\,ds\right)$
$\displaystyle\ \ \ \
-2\varepsilon_{2}\left(\int_{0}^{\infty}A(s)(\eta_{1}^{t}(s)-\eta_{2}^{t}(s))ds,\int_{0}^{\infty}M(s)\,ds(x_{1}(t)-x_{2}(t))+\int_{0}^{\infty}M(s)(\eta_{1}^{t}(s)-\eta_{2}^{t}(s))\,ds\right).$

It follows that

$\displaystyle\frac{d}{dt}\|\eta_{1}^{t}-\eta_{2}^{t}\|^{2}\leq-C\|\eta^{t}_{1}-\eta^{t}_{2}\|^{2}+2D\|\eta_{1}^{t}-\eta_{2}^{t}\|\cdot\|Df(N)\|\cdot|x_{1}(t)-x_{2}(t)|$
$\displaystyle\ \ \
+2|\varepsilon_{1}-\varepsilon_{2}|D\|\eta_{1}^{t}-\eta_{2}^{t}\|\overline{R}+2\Delta
C_{1}D\|\eta_{1}^{t}-\eta_{2}^{t}\|^{2}+2\Delta
D\|\eta_{1}^{t}-\eta_{2}^{t}\|\ |x_{1}(t)-x_{2}(t)|.$

Putting together the two estimates we obtain

$\displaystyle\frac{d}{dt}\left((x_{1}(t)-x_{2}(t))^{\top}Q(x_{1}(t)-x_{2}(t))-E\|\eta_{1}(t)-\eta_{2}(t)\|^{2}\right)\geq
G|x_{1}(t)-x_{2}(t)|^{2}+E(C-2\Delta C_{1}D)\|\eta^{t}_{1}-\eta^{t}_{2}\|^{2}$
$\displaystyle\ \ \ -2|x_{1}(t)-x_{2}(t)|(\Delta\|Q\|C_{1}+ED\|Df(N)\|+E\Delta
D)\|\eta_{1}^{t}-\eta_{2}^{t}\|$ $\displaystyle\ \ \
-2|\varepsilon_{1}-\varepsilon_{2}|\,|x_{1}(t)-x_{2}(t)|\,\|Q\|\overline{R}-2E|\varepsilon_{1}-\varepsilon_{2}|D\|\eta_{1}^{t}-\eta_{2}^{t}\|\overline{R}.$

We need the right-hand side of the last estimate to be bounded from below by
$0$ at $t=0$, on the boundary of the cone, i.e. for
${Q}(x^{0}_{1}-x^{0}_{2})-E\|\eta^{0}_{1}-\eta_{2}^{0}\|^{2}+L|\varepsilon_{1}-\varepsilon_{2}|^{2}=0$,
whereas we can estimate from above as follows

$|\varepsilon_{1}-\varepsilon_{2}|\leq\frac{1}{\sqrt{|L|}}\left(\sqrt{\|Q\|}\cdot|x^{0}_{1}-x^{0}_{2}|+\sqrt{E}\|\eta^{0}_{1}-\eta_{2}^{0}\|\right)$

We deduce that, at $t=0$ we have

$\displaystyle\frac{d}{dt}\left((x_{1}(t)-x_{2}(t))^{\top}Q(x_{1}(t)-x_{2}(t))-E\|\eta_{1}(t)-\eta_{2}(t)\|^{2}\right)|_{t=0}\geq\left(G-\frac{2\|Q\|^{\frac{3}{2}}\overline{R}}{\sqrt{|L|}}\right)|x_{1}^{0}-x_{2}^{0}|^{2}$
$\displaystyle\ \ \ \ +E\left(C-2\Delta
C_{1}D-\frac{2\sqrt{E}D\overline{R}}{\sqrt{|L|}}\right)\|\eta^{t}_{1}-\eta^{t}_{2}\|^{2}$
$\displaystyle\ \ \
-2|x_{1}^{0}-x_{2}^{0}|\left(\Delta\|Q\|C_{1}+ED\|Df(N)\|+E\Delta
D+\frac{E\sqrt{\|Q\|}D\overline{R}}{\sqrt{|L|}}+\frac{\|Q\|\overline{R}\sqrt{E}}{\sqrt{|L|}}\right)\|\eta_{1}^{t}-\eta_{2}^{t}\|.$

We are free to choose sufficiently large (positive or negative) $L$,
sufficiently small $\Delta$ and sufficiently small $E$. We already have the
upper bound on $E$ in Lemma 5.5 given by $E\leq E_{max}$.

Now suppose that $|L|$ is large enough and $\Delta$ is small enough such that

$\sqrt{|L|}\geq\max\left\\{\frac{4\|Q\|^{\frac{3}{2}}R}{G},\frac{8\sqrt{E_{max}}D\overline{R}}{C}\right\\}\qquad\textrm{and}\qquad\Delta\leq\frac{C}{8C_{1}D}.$

With these assumption the above estimate takes the form

$\displaystyle\frac{d}{dt}\left((x_{1}(t)-x_{2}(t))^{\top}Q(x_{1}(t)-x_{2}(t))-E\|\eta_{1}(t)-\eta_{2}(t)\|^{2}\right)|_{t=0}\geq\frac{G}{2}|x_{1}^{0}-x_{2}^{0}|^{2}$
$\displaystyle\ \ \ \
+\frac{EC}{2}\|\eta^{t}_{1}-\eta^{t}_{2}\|^{2}-2|x_{1}^{0}-x_{2}^{0}|\left(\Delta\|Q\|C_{1}+ED\|Df(N)\|+E\Delta
D+\frac{E\sqrt{\|Q\|}D\overline{R}}{\sqrt{|L|}}+\frac{\|Q\|\overline{R}\sqrt{E}}{\sqrt{|L|}}\right)\|\eta_{1}^{t}-\eta_{2}^{t}\|.$

The above quadratic form on $|x_{1}^{0}-x_{2}^{0}|$ and
$\|\eta^{t}_{1}-\eta^{t}_{2}\|$ is nonnegatively defined provided

$\frac{ECG}{4}\geq\left(\Delta\|Q\|C_{1}+ED\|Df(N)\|+E\Delta
D+\frac{E\sqrt{\|Q\|}D\overline{R}}{\sqrt{|L|}}+\frac{\|Q\|\overline{R}\sqrt{E}}{\sqrt{|L|}}\right)^{2}.$

But we know that

$\displaystyle\left(\Delta\|Q\|C_{1}+ED\|Df(N)\|+E\Delta
D+\frac{E\sqrt{\|Q\|}D\overline{R}}{\sqrt{|L|}}+\frac{\|Q\|\overline{R}\sqrt{E}}{\sqrt{|L|}}\right)^{2}$
$\displaystyle\ \ \leq
5\Delta^{2}\|Q\|^{2}C_{1}^{2}+5E^{2}D^{2}\|Df(N)\|^{2}+5E^{2}\Delta^{2}D^{2}+\frac{5E^{2}\|Q\|D^{2}\overline{R}^{2}}{|L|}+\frac{5\|Q\|^{2}\overline{R}^{2}{E}}{|L|}.$

Hence we need the following five inequalities

$\displaystyle 5\Delta^{2}\|Q\|^{2}C_{1}^{2}\leq\frac{ECG}{20},\ \
5E^{2}D^{2}\|Df(N)\|^{2}\leq\frac{ECG}{20},\ \
5E^{2}\Delta^{2}D^{2}\leq\frac{ECG}{20},$ $\displaystyle\ \ \
\frac{5E^{2}\|Q\|D^{2}\overline{R}^{2}}{|L|}\leq\frac{ECG}{20},\ \
\frac{5\|Q\|^{2}\overline{R}^{2}{E}}{|L|}\leq\frac{ECG}{20},$

or, after the simplification,

$\displaystyle 100\Delta^{2}\|Q\|^{2}C_{1}^{2}\leq ECG,\ \
100ED^{2}\|Df(N)\|^{2}\leq CG,\ \ 100E\Delta^{2}D^{2}\leq CG,$ $\displaystyle\
\ \ {100E\|Q\|D^{2}\overline{R}^{2}}\leq{CG|L|},\ \
{100\|Q\|^{2}\overline{R}^{2}}\leq{CG}{|L|}.$

We see that it is enough to choose

$|L|\geq\max\left\\{\frac{100E_{max}\|Q\|D^{2}\overline{R}^{2}}{CG},\frac{100\|Q\|^{2}\overline{R}^{2}}{CG}\right\\}$

and the last two inequalities hold. We are now free to pick $E$ which
satisfies

$0<E\leq\frac{CG}{100D^{2}\|Df(N)\|^{2}}\ \ \textrm{and}\ \ E<E_{max},$

and we finally need to pick $\Delta$ such that

$\Delta^{2}\leq\min\left\\{\frac{ECG}{100\|Q\|^{2}C_{1}^{2}},\frac{CG}{100ED^{2}}\right\\}.$

The proof is complete. ∎

##  6\. Equilibria and their nonexpansion in the limit

In this section we relate the equilibria of the unperturbed system (4) with
the equilibria of the problem (8)–(9). We show that if $\varepsilon$ is small,
then for every equilibrium $e$ of (4) there exists an equilibrium
$(0,e^{\varepsilon})$ of (8)–(9) in its vicinity and the perturbed system has
no other equilibria. Moreover, we show the upper semi-continuity result on the
connections between the equilibria, that is, if the two equilibria of (8)–(9)
are connected for a sequence of parameters $\varepsilon\to 0$, then the
connection also exists for $\varepsilon=0$. We remind that the limit equation
(4) has only a finite number of isolated and hyperbolic equilibria and the
system is Morse–Smale, i.e. the intersections of their stable and unstable
manifolds are always transversal.

######  Lemma 6.1.

There exists $\varepsilon_{0}>0$ and $R>0$ such that for every
$\varepsilon\in[0,\varepsilon_{0})$ if $e\in\mathbb{R}^{d}$ is an equilibrium
for (4) with an isolating block with cones $N^{x}$ then the problem governed
by (8)–(9) has an equilibrium $(0,e^{\varepsilon})$ which is unique in the set
$\widetilde{N}^{x}=\\{\|\eta\|\leq R\\}\times N^{x}$. Moreover
$(0,e^{\varepsilon})$ are the only equilibria for (8)–(9).

###### Proof.

Denote by $\mathcal{E}$ the set of equilibria of (4) and let
$e\in\mathcal{E}$. Take $R$ and $N^{x}$ from Theorem 5.4 (isolating set) and
take $E$ satisfying the constraints from Lemma 5.5 (the cone condition), Lemma
5.6 (the cone condition with parameters) and Lemma 5.1 (the Lyapunov
function). Now take $\varepsilon\in[0,\varepsilon_{0}]$, where
$\varepsilon_{0}$ satisfies all the constraints of the previous results: the
constraint of Lemma 5.5 (the cone condition), Lemma 5.6 (the cone condition
with parameters), Lemma 5.1 (the Lyapunov function) and Theorem 5.4 (isolating
set) and the constraints of Section 3. From Lemma 5.1 we deduce that the
equilibria of (8) must have the $\eta$ component equal to zero. Theorem 9.9
together with Lemma 5.5 imply that the problem governed by (8)–(9) has a
unique equilibrium in the set $\widetilde{N}^{x}$. We denote this equilibrium
by $(0,e^{\varepsilon})$. It must be $f^{\varepsilon}(e^{\varepsilon})=0$. We
must show that problem (8) does not have other equilibria than the ones which
lie in $\widetilde{N}^{x}$.

From (12) we deduce that if $f^{\varepsilon}(e^{\varepsilon})=0$ then
$|x^{\varepsilon}|\leq\overline{R}$. Assume that
$e^{\varepsilon}\notin\bigcup_{y\in\mathcal{E}}\textrm{int}\,N^{y}$. Now let

$\beta=\min\left\\{|f(x)|\,:\
|x|\leq\overline{R},x\notin\bigcup_{y\in\mathcal{E}}\textrm{int}\,N^{y}\right\\}.$

This is a positive constant. We have

$|f^{\varepsilon}(e^{\varepsilon})|\geq|f(e^{\varepsilon})|-\varepsilon|e^{\varepsilon}|\int_{0}^{\infty}\|M(s)\|\,ds\geq\beta-\varepsilon\overline{R}\int_{0}^{\infty}\|M(s)\|\,ds.$

Decreasing $\varepsilon$ is necessary we note that we must have
$|f^{\varepsilon}(e^{\varepsilon})|>0$, a contradiction. ∎

In the sequel we always assume that $E,\varepsilon_{0},R$ satisfy the
constraints which follow from the above Lemma.

######  Definition 6.2.

Let $\varepsilon\geq 0$. The function $(\eta,x):\mathbb{R}\to
L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}$ is a bounded complete
(eternal) solution for (8)–(9) if for every $t\in\mathbb{R}$ the function
$(\eta^{t+\cdot},x(t+\cdot)):\mathbb{R}\to
L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}$ is a solution for (8)–(9)
and moreover
$\sup_{t\in\mathbb{\mathbb{R}}}\left(|x(t)|+\|\eta^{t}\|_{L^{2}_{A}(\mathbb{R}^{+})^{d}}\right)$
is bounded.

The existence of the Liapunov function in Lemma 5.1 directly implies the
following result

######  Lemma 6.3.

The pair $t\mapsto(\eta^{t},x(t))$ is a bounded complete solution for (8)–(9)
if and only if there exists two points
$e^{\varepsilon}_{1},e^{\varepsilon}_{2}\in\mathbb{R}^{d}$ satisfying
$f^{\varepsilon}(e^{\varepsilon}_{1})=f^{\varepsilon}(e^{\varepsilon}_{2})=0$
such that

$\lim_{t\to-\infty}(\eta^{t},x(t))=(0,e^{\varepsilon}_{1}),\ \ \
\lim_{t\to\infty}(\eta^{t},x(t))=(0,e^{\varepsilon}_{2})$

In such case we say that the exists a connection between the equilibria
$e^{\varepsilon}_{1}$ and $e^{\varepsilon}_{2}$.

In the next lemma we prove that the existing connections are preserved in the
limit, see [2, Proposition 4].

######  Lemma 6.4.

If for a sequence $\varepsilon^{n}\to 0^{+}$ there exist connections between
equilibria $e^{\varepsilon^{n}}_{1}$ and $e^{\varepsilon^{n}}_{2}$ through the
system (8)-(9) where $\lim_{n\to\infty}e^{\varepsilon^{n}}_{1}=e_{1}$ and
$\lim_{n\to\infty}e^{\varepsilon^{n}}_{2}=e_{2}$ then $e_{1}$ and $e_{2}$ are
equilibria of (4) and there exists a sequence of equilibria
$e_{1}=g_{1},\ldots,g_{N}=e_{2}$ such that there exist complete trajectories
of (4) which connect $e_{i}\to e_{i+1}$ for $i\in\\{1,\ldots,N-1\\}$.

###### Proof.

The fact that $e_{1}$ and $e_{2}$ are equilibria of (4) follows from the
definition of $f^{\varepsilon}$. Denote by
$(\eta_{n}^{(\cdot)},x_{n}(\cdot)):\mathbb{R}\to
L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}$ the bounded complete
solutions for $\epsilon_{n}$ such that for each $n$

$\lim_{t\to-\infty}(\eta_{n}^{t},x_{n}(t))=(0,e_{1}^{\epsilon^{n}}),\ \ \
\lim_{t\to\infty}(\eta_{n}^{t},x_{n}(t))=(0,e_{2}^{\epsilon^{n}}).$

Now $x_{n}$ and $\eta_{n}$ are bounded uniformly with respect to $t$. The
Lyapunov function in Lemma 5.1 implies that they are also bounded uniformly
with respect to $n$. Hence $x_{n}^{\prime}(t)$ is also uniformly bounded with
respect to both $n$ and $t$. For every $\delta>0$ there exists $t_{n}^{1}$
such that if only $t\leq t_{n}^{1}$ then
$|x_{n}(t)-e_{1}^{\varepsilon_{n}}|\leq\delta$. From the Arzela–Ascoli lemma,
using the diagonal argument we can construct a function
$x^{1}:\mathbb{R}\to\mathbb{R}^{n}$ such that $x_{n}(t+t_{n}^{1})\to u^{1}(t)$
uniformly for $t$ on every bounded time interval. Since

$x_{n}(t_{n}^{1}+t)=x_{n}(t_{n}^{1})+\int_{t_{n}^{1}}^{t_{n}^{1}+t}\left(f(x_{n}(s))+\varepsilon_{n}\int_{0}^{\infty}M(r)\,drx_{n}(s)+\varepsilon_{n}\int_{0}^{\infty}M(r)\eta_{n}^{s}(r)\,dr\right)\,ds,$

we can pass to the limit with $n$ to infinity whence

$u_{1}(t)=u_{1}(0)+\int_{0}^{t}f(u_{1}(s))\,ds,$

i.e. $x_{1}$ solves (4). Now let $t\leq 0$ be fixed. We have

$|u_{1}(t)-e_{1}|\leq|u_{1}(t)-x_{n}(t+t_{n}^{1})|+|x_{n}(t+t_{n}^{1})-e_{1}^{\varepsilon_{n}}|+|e_{1}^{\varepsilon_{n}}-e_{1}|\leq|u_{1}(t)-x_{n}(t+t_{n}^{1})|+\delta+|e_{1}^{\varepsilon_{n}}-e_{1}|.$

Passing with $n$ to infinity we deduce that

$|u_{1}(t)-e_{1}|\leq\delta.$

Since $\lim_{t\to-\infty}u_{1}(t)=e$, an equilibrium of (4), we deduce by
taking $\delta$ small enough related to minimal distance between the
equilibria of the system, that it must be $e=e_{1}$. Now
$\lim_{t\to\infty}u_{1}(t)=g_{2}$, an equilibrium of (4). If $g_{2}=e_{2}$ the
proof is complete. Otherwise for every $n$ there exists $k(n)\to\infty$ as
$t\to\infty$ and $\tau^{2}_{n}$ such that
$|x_{k(n)}(\tau^{2}_{n})-g_{2}|\leq\frac{1}{n}$. Hence
$x_{k(n)}(\tau^{2}_{n}+t)$ converges to $e_{2}$ uniformly on bounded time
intervals. This means that for every sufficiently small $\delta$ and every $n$
there exists a maximal $t^{2}_{n}>\tau^{2}_{n}$ such that for
$t\in[\tau^{2}_{n},t^{2}_{n}]$ we have $|x_{n}(t)-g_{2}|\leq\delta$ and it
must be $t^{2}_{n}-\tau^{2}_{n}\to\infty$ as $n\to\infty$. Solutions
$x_{n}(t_{n}^{2}+t)$, again from the Arzela–Ascoli lemma converge to
$u_{2}(t)$, the solution of (4), uniformly on bounded time intervals. Moreover
for every $t\leq 0$ we are able to find $n_{0}$ such that for every $n\geq
n_{0}$ we have $\tau_{n}^{2}<t+t_{n}^{2}$. Then

$|u_{2}(t)-g_{2}|\leq|u_{2}(t)-x_{n}(t+t_{n}^{2})|+|x_{n}(t+t_{n}^{2})-g_{2}|\leq|u_{2}(t)-x_{n}(t+t_{n}^{2})|+\delta.$

Passing with $n\to\infty$ we deduce that $|u^{2}(t)-g_{2}|\leq\delta$ for
every $t\leq 0$ and it is enough to choose $\delta$ sufficiently small so that
$\lim_{t\to-\infty}u_{2}(t)=g_{2}$. Now, $\lim_{t\to\infty}u_{2}(t)=g_{3}$. If
$g_{3}=e_{2}$ the proof is complete. If not, we continue the procedure, which
is always possible if the equilibrium is not $e_{2}$. Since the number of
equilibria of (4) is finite and the system is gradient, the procedure must end
after finite number of steps, which concludes the proof. ∎

As the limit system (4) is Morse–Smale, the existence of the sequence of
connections $e_{1}=g_{1}\to g_{2}\to\ldots\to g_{N}=e_{2}$ implies the
existence of connection $e_{1}\to e_{2}$, whence we can formulate the
following lemma

######  Lemma 6.5.

If for a sequence $\varepsilon^{n}\to 0^{+}$ there exist connections between
equilibria $(0,e^{\varepsilon^{n}}_{1})$ and $(0,e^{\varepsilon^{n}}_{2})$
through the system (8)–(9) where
$\lim_{n\to\infty}e^{\varepsilon^{n}}_{1}=e_{1}$ and
$\lim_{n\to\infty}e^{\varepsilon^{n}}_{2}=e_{2}$ then $e_{1}$ and $e_{2}$ are
equilibria of (4) and there exists a connection $e_{1}\to e_{2}$ through the
system (4).

##  7\. Continuation of the intersection of manifolds.

We recall that for the system (4) there exists a finite number of equilibria
$\\{e_{1},\ldots,e_{N}\\}$ which are all hyperbolic, that is
$Df(e_{i})=D^{2}F(e_{i})$ is a nonsingular matrix for every $e_{i}$.

Using the Hadamard–Perron theorem, cf. for example [7, Theorem 3.2.1], and the
fact that $f\in C^{2}(\mathbb{R}^{d};\mathbb{R}^{2})$ we deduce that each
equilibrium has the stable and unstable manifold $W^{u}(e_{i})$ and
$W^{s}(e_{i})$ which is of class $C^{2}$. If, for the two equilibria $e_{i}$
and $e_{j}$, there exists the solution $\gamma$ that connects $e_{i}$ to
$e_{j}$ then this solution belong to both the unstable manifold of $e_{i}$ and
the stable manifold of $e_{j}$.

In the previous sections we have constructed the local stable and unstable
manifolds of all equilibria in such their neighbourhoods, that are isolating
h-sets with cones, which are moreover preserved when the problem is perturbed
by the delay term $\varepsilon$. So, if the equilibrium $e_{i}$ is connected
to $e_{j}$ if $W^{u}(e_{i})\cap W^{s}(e_{j})\neq\emptyset$, we can take as an
intersection point, a point $z\in W^{s}_{loc,N(e_{j})}$ where $N(e_{j})$ is an
h-set with cones for the equlibrium $e_{j}$. Then $z\in W^{u}(e_{i})$.

We assume that the intersection of $W^{u}(e_{i})$ and
$W^{s}_{loc,A(e_{j})}(e_{j})$ is transversal that is

$T_{z}W^{u}(e_{i})\oplus T_{z}W^{s}_{loc,A(e_{j})}(e_{j})=\mathbb{R}^{d}.$

Let $\textrm{dim}W^{u}(e_{i})=u_{i}$ and
$\textrm{dim}W^{s}_{loc,A(e_{j})}(e_{j})=s_{j}$. Then $u_{i}+s_{j}\geq d+1$.
Tangent space $T_{z}W^{u}(e_{i})$ is the $u_{i}$ dimensional subspace of
$\mathbb{R}^{d}$ and $T_{z}W^{s}_{loc,A(e_{j})}(e_{j})$ is its $s_{j}$
dimensional subspace. The intersection of both spaces is $c$ dimensional
subspace of $\mathbb{R}^{d}$ where $c=u_{i}+s_{j}-d$. Denote $u_{i}=k_{1}+c$
and $s_{j}=k_{2}+c$. There exists an invertible $d\times d$ matrix $M$ such
that
$M\cdot(\mathbb{R}^{c}\otimes\mathbb{R}^{k_{1}}\otimes(0)_{k_{2}})=T_{z}W^{u}(e_{i})$
and
$M\cdot(\mathbb{R}^{c}\otimes(0)_{k_{1}}\otimes\mathbb{R}^{k_{2}})=T_{z}W^{s}_{loc,A(e_{j})}(e_{j})$.
This matrix defines the linear change of coordinates in $\mathbb{R}^{d}$. We
will denote the new coordinates by
$(\overline{a},\overline{x},\overline{y})\in\mathbb{R}^{c}\otimes\mathbb{R}^{k_{1}}\otimes\mathbb{R}^{k_{2}}$.
The next lemma states that in that local systems of coordinates the local
stable and unstable manifolds constitute the horizontal and vertical disks
with the arbitrarily small Lipschitz constants.

######  Lemma 7.1.

There exist constants $\delta_{k_{1}},\delta_{k_{2}},\delta_{c}>0$ and
Lipschitz functions $\overline{x}^{0}:B(0,c)\times B(0,\delta_{k_{2}})\to
B(0,\delta_{k_{1}})$ and $\overline{y}^{0}:B(0,c)\times B(0,\delta_{k_{1}})\to
B(0,\delta_{k_{2}})$ such that

$M\cdot\\{(\overline{a},\overline{x}^{0}(\overline{a},\overline{y}),\overline{y})\,:\
(\overline{a},\overline{y})\in B(0,\delta_{c})\times
B(0,\delta_{k_{2}})\\}=(W^{s}(e_{j})-z)\cap M\cdot(B(0,\delta_{c})\times
B(0,\delta_{k_{1}})\times B(0,\delta_{k_{2}})).$

and

$M\cdot\\{(\overline{a},\overline{x},\overline{y}^{0}(\overline{a},\overline{x}))\,:\
(\overline{a},\overline{x})\in B(0,\delta_{c})\times
B(0,\delta_{k_{1}})\\}=(W^{u}(e_{i})-z)\cap M\cdot(B(0,\delta_{c})\times
B(0,\delta_{k_{1}})\times B(0,\delta_{k_{2}})).$

Moreover, with decreasing $\delta_{k_{1}},\delta_{k_{2}},\delta_{c}$, the
Lipschitz constants of both disks can be made arbitrarily small.

###### Proof.

We will denote by $S^{0}(t)$ the flow for $\varepsilon=0$. We first study the
local stable manifold of $e_{j}$. This manifold is a graph of the Lipschitz
function in the coordinates that we denote by $(x,y)$, where $x$ is in the
unstable space of $e_{j}$, and $y$ is in the stable one. In these coordinates
the point of intersection, $z$, can be represented as $x_{z}+y_{z}$. Then,
this manifold translated by $z$ is a graph of a function $x=x^{0}(y)$. Since
it is of class $C^{1}$, the points on it have the form

$x_{z}+x^{0}(y)+y_{z}+y=z+\frac{Dx^{0}(0)}{Dy}y+y+\Delta(y),$

where, $\frac{Dx^{0}(0)}{Dy}y+y\in T_{z}W^{s}(e_{j})$, and $\Delta(y)\in
o(|y|)$. Denoting $\Pi_{M\cdot(\overline{a},0,\overline{y})}$ the projection
on the tangent space $T_{z}W^{s}(e_{j})$ and by
$\Pi_{M\cdot(0,\overline{x},0)}$ the complementary projection, we can
represent the considered point on $W^{s}(e_{j})$ as

$z+\frac{Dx^{0}(0)}{Dy}y+y+\Pi_{M\cdot(\overline{a},0,\overline{y})}\Delta(y)+\Pi_{M\cdot(0,\overline{x},0)}\Delta(y).$

We need to prove that there exists $\delta_{0},\delta_{1}>0$ such that for
every $(\overline{a},\overline{y})\in B(0,\delta_{1})$ there exists $y\in
B(0,\delta_{0})$ such that

$(\overline{a},0,\overline{y})=M^{-1}\left(\frac{Dx^{0}(0)}{Dy}y+y+\Pi_{M\cdot(\overline{a},0,\overline{y})}\Delta(y)\right).$

The mapping

$y\mapsto M^{-1}\left(\frac{Dx^{0}(0)}{Dy}y+y\right)$

is a linear invertible mapping from $s_{j}$ dimensional space into $s_{j}$
dimensional space. Therefore for every $\delta_{0}$ we can find a ball
$B(0,\overline{\delta}(\delta_{0}))$ such that

$\\{(\overline{a},0,\overline{y})\,:\ (\overline{a},\overline{y})\in
B(0,\overline{\delta})\\}\subset\left\\{M^{-1}\left(\frac{Dx^{0}(0)}{Dy}y+y\right)\,:\
y\in B(0,\delta_{0})\right\\}.$

Consider the homotopy

$B(0,\delta_{0})\times[0,1]\ni(y,\theta)\mapsto
f(y,\theta)=M^{-1}\left(\frac{Dx^{0}(0)}{Dy}y+y+\theta\Pi_{M\cdot(\overline{a},0,\overline{y})}\Delta(y)\right).$

Now for some constants $C_{1},C_{2}>0$ we have

$|f(y,\theta)|\geq C_{1}|y|-C_{2}|\Delta_{1}(y)|\ \ \textrm{for}\ \
(y,\theta)\in B(0,\delta_{0})\times[0,1].$

Therefore if $|y|=\delta_{0}$, by taking the constant $\delta_{0}$ small
enough, we obtain $|f(y,\theta)|>\frac{C_{1}}{2}\delta_{0}$. Hence, by the
homotopy invariance of the Brouwer degree if only
$\delta_{1}<\min\left\\{\frac{C_{1}}{2}\delta_{0},\overline{\delta}(\delta_{0})\right\\}$
we obtain the existence of the pair $(\overline{a},\overline{y})$ such that
there exists $\overline{x}$ for which the point
$M(\overline{a},\overline{x},\overline{y})$ belongs to the local stable
manifold.

In the next step we show that for this pair the point $\overline{x}$ is unique
and that the dependence
$\overline{x}=\overline{x}^{0}(\overline{a},\overline{y})$ is Lipschitz with a
constant that can be made arbitrarily small by decreasing, if necessary the
radius $\delta_{1}$. We will use the notation
$\overline{F([a,b])}=\mathrm{conv}\\{F(x)\>x=\lambda a+(1-\lambda
b),\lambda\in[0,1]\\}$. Consider two points on the local stable manifold,
denote them by
$x_{1}+y_{1}=M\cdot(\overline{a}_{1},\overline{x}_{1},\overline{y}_{1})$ and
$x_{2}+y_{2}=M\cdot(\overline{a}_{2},\overline{x}_{2},\overline{y}_{2})$. Now

$M(\overline{a}_{1}-\overline{a}_{2},\overline{x}_{1}-\overline{x}_{2},\overline{y}_{1}-\overline{y}_{2})=(x^{0}_{1}(y_{1})-x_{2}^{0}(y_{2})+y_{1}-y_{2}).$

Now

$\displaystyle
M\cdot(\overline{a}_{1}-\overline{a}_{2},0,\overline{y}_{1}-\overline{y}_{2})\in\Pi_{M\cdot(\overline{a},0,\overline{y})}\left(\frac{Dx^{0}(0)}{Dy}(y_{1}-y_{2})+y_{1}-y_{2}\right)$
$\displaystyle\qquad+\Pi_{M\cdot(\overline{a},0,\overline{y})}\left(\overline{\frac{Dx^{0}([y_{1},y_{2}])}{Dy}}-\frac{Dx^{0}(0)}{Dy}\right)(y_{1}-y_{2}),$

and, as $\left(\frac{Dx^{0}(0)}{Dy}(y_{1}-y_{2})+y_{1}-y_{2}\right)$ belongs
to the tangent space,

$M\cdot(0,\overline{x}_{1}-\overline{x}_{2},0)\in\Pi_{M\cdot(0,\overline{x},0)}\left(\overline{\frac{Dx^{0}([y_{1},y_{2}])}{Dy}}-\frac{Dx^{0}(0)}{Dy}\right)(y_{1}-y_{2}).$

We use the fact that $x^{0}$ is of class $C^{1}$. The first of the above two
inclusions implies that there exist constants $C_{1},C_{2}>0$

$|(\overline{a}_{1}-\overline{a}_{2},0,\overline{y}_{1}-\overline{y}_{2})|\geq
C_{1}|y_{1}-y_{2}|-C_{2}|y_{1}-y_{2}|,$

where $C_{2}$ can be made as small as we need by taking sufficiently small
$\delta_{0}$. Moreover

$|(0,\overline{x}_{1}-\overline{x}_{2},0)|\leq C_{3}|y_{1}-y_{2}|,$

where $C_{3}$ again can be made as small as necessary by taking sufficiently
small $\delta_{0}$. Both above inequalities imply that
$|(0,\overline{x}_{1}-\overline{x}_{2},0)|\leq
C_{s}|(\overline{a}_{1}-\overline{a}_{2},0,\overline{y}_{1}-\overline{y}_{2})|$
with the Lipschitz constant $C_{s}$ being as small as we need, which can be
obtained by taking small $\delta_{0}$. Observe that this gives the restriction
on the radii $c,\delta_{k_{1}},\delta_{k_{2}}$ in order to guarantee that the
constructed disk remains in the box, namely that $\delta_{k_{1}}^{2}\geq
C_{s}^{2}(\delta_{c}^{2}+\delta_{k_{2}}^{2})$.

We pass to the analysis of the unstable manifold $W^{u}(e_{i})$. There exists
a point $p$ in the local unstable manifold of $e_{i}$ and time $t$ such that
$z=S^{0}(t)p$. Denote $p=x_{p}+y_{p}$ with $x$ being the $u_{i}$-dimensional
coordinates in the unstable space of $e_{i}$ and $y$ being the $d-u_{i}$
dimensional coordinates in its stable space. The local unstable manifold is a
graph of a $C^{1}$ function $y=y^{0}(x)$ (translated such that
$y_{p}=x^{0}(0)$). Take a ball $B(0,\delta_{0})$ such that the graph of the
unstable manifold over that ball is inside the $h$-set. For $x$ in this ball
we have

$\displaystyle
S(t)(x_{p}+x+y^{0}(x))=S(t)\left(x_{p}+x+y^{0}(0)+\frac{Dy^{0}(0)}{Dx}x+\Delta(x)\right)=S(t)\left(p+x+\frac{Dy^{0}(0)}{Dx}x+\Delta(x)\right),$

where $\Delta(x)\in o(|x|)$. Denote

$\Delta_{1}(x)=x+\frac{Dy^{0}(0)}{Dx}x+\Delta(x).$

From now on we will denote by $\frac{\partial}{\partial u_{0}}$ the derivative
with respect to the initial data. Then

$\displaystyle S^{0}(t)(x_{p}+x+y^{0}(x))=S^{0}(t)p+\frac{\partial
S^{0}(t)p}{\partial u_{0}}\Delta_{1}(x)+\Delta_{2}(\Delta_{1}(x))$
$\displaystyle\qquad=S^{0}(t)p+\frac{\partial S^{0}(t)p}{\partial
u_{0}}\left(x+\frac{Dy^{0}(0)}{Dx}x\right)+\frac{\partial S^{0}(t)p}{\partial
u_{0}}\Delta(x)+\Delta_{2}(\Delta_{1}(x)),$

with $\Delta_{2}(\Delta_{1}(x))\in o(|\Delta_{1}(x)|)$. Denote $\frac{\partial
S^{0}(t)p}{\partial u_{0}}\Delta(x)+\Delta_{2}(\Delta_{1}(x))=\Delta_{3}(x)$.
This quantity belongs to $o(|x|)$. Then we have

$S^{0}(t)(x_{p}+x+y^{0}(x))=z+\frac{\partial S^{0}(t)p}{\partial
u_{0}}\left(x+\frac{Dy^{0}(0)}{Dx}x\right)+\Pi_{M\cdot(\overline{a},\overline{x},0)}\Delta_{3}(x)+\Pi_{M\cdot(0,0,\overline{y})}\Delta_{3}(x),$

and the expression $\frac{\partial S^{0}(t)p}{\partial
u_{0}}\left(x+\frac{Dy^{0}(0)}{Dx}x\right)+\Pi_{M\cdot(\overline{a},\overline{x},0)}\Delta_{3}(x)$
belongs to the tangent space $T_{z}W^{u}(e_{i})$. We need to show that there
exists $\delta_{1}$ such that for every $(\overline{a},\overline{x})\in
B(0,\delta_{1})$ there exists $x\in B(0,\delta)$ such that

$M\cdot(\overline{a},\overline{x},0)=\frac{\partial S^{0}(t)p}{\partial
u_{0}}\left(x+\frac{Dy^{0}(0)}{Dx}x\right)+\Pi_{M\cdot(\overline{a},\overline{x},0)}\Delta_{3}(x),$

the argument follows by homotopy, analogously as for the stable manifold.
Indeed, the invertibility of the linear mapping $x\mapsto\frac{\partial
S^{0}(t)p}{\partial u_{0}}\left(x+\frac{Dy^{0}(0)}{Dx}x\right)$ implies that
there exists $\overline{\delta}(\delta_{0})$ such that

$\\{(\overline{a},\overline{x},0)\,:\ (\overline{a},\overline{x})\in
B(0,\overline{\delta})\\}\subset\left\\{M^{-1}\frac{\partial
S^{0}(t)p}{\partial u_{0}}\left(x+\frac{Dy^{0}(0)}{Dx}x\right)\,:\ y\in
B(0,\delta_{0})\right\\},$

and, decreasing $\overline{\delta}(\delta_{0})$ if necessary, as in the case
of the stable manifold, the result follows by considering the homotopy

$B(0,\delta_{0})\times[0,1]\ni(x,\theta)\mapsto M^{-1}\frac{\partial
S^{0}(t)p}{\partial
u_{0}}\left(x+\frac{Dy^{0}(0)}{Dx}x\right)+\theta\Pi_{M\cdot(\overline{a},\overline{x},0)}\Delta_{3}(x).$

To demonstrate that the point $\overline{y}$ is uniquely determined for a
given pair $(\overline{a},\overline{x})$ and the Lipschitz condition holds,
consider the two points on the local unstable manifold of $e_{i}$, denote them
by $p_{1}=x_{1}+y_{1}$ and $p_{2}=x_{2}+y_{2}$. We consider the difference

$\displaystyle S^{0}(t)(p_{2})-S^{0}(t)(p_{1})\in\overline{\frac{\partial
S^{0}(t)([p_{1},p_{2}])}{\partial x}}(x_{2}-x_{1})+\overline{\frac{\partial
S^{0}(t)([p_{1},p_{2}])}{\partial y}}(y_{2}-y_{1})$
$\displaystyle\qquad\subset\left(\overline{\frac{\partial
S^{0}(t)([p_{1},p_{2}])}{\partial x}}+\overline{\frac{\partial
S^{0}(t)([p_{1},p_{2}])}{\partial y}}\
\overline{\frac{Dy^{0}([x_{1},x_{2}])}{Dx}}\right)(x_{2}-x_{1})$
$\displaystyle\qquad\subset\frac{\partial S^{0}(t)(p)}{\partial
x}(x_{2}-x_{1})+\frac{\partial S^{0}(t)(p)}{\partial
y}\frac{Dy^{0}(0)}{Dx}(x_{2}-x_{1})+\frac{\partial S^{0}(t)(p)}{\partial
y}\left(\overline{\frac{Dy^{0}([x_{1},x_{2}])}{Dx}}-\frac{Dy^{0}(0)}{Dx}\right)(x_{2}-x_{1})$
$\displaystyle\qquad\qquad+\left(\overline{\frac{\partial
S^{0}(t)([p_{1},p_{2}])}{\partial x}}-\frac{\partial S^{0}(t)(p)}{\partial
x}+\left(\overline{\frac{\partial S^{0}(t)([p_{1},p_{2}])}{\partial
y}}-\frac{\partial S^{0}(t)(p)}{\partial
y}\right)\overline{\frac{Dy^{0}([x_{1},x_{2}])}{Dx}}\right)(x_{2}-x_{1}).$

Now denote
$S^{0}(t)p_{1}=M\cdot(\overline{a}_{1},\overline{x}_{1},\overline{y}_{1})$ and
$S^{0}(t)p_{2}=M\cdot(\overline{a}_{2},\overline{x}_{2},\overline{y}_{2})$.
Because the sum of the first two terms in the above expression belongs to the
tangent space $T_{z}W^{u}(e_{i})$, we have

$\displaystyle
M\cdot(\overline{a}_{1}-\overline{a}_{2},\overline{x}_{1}-\overline{x}_{2},0)\in\frac{\partial
S^{0}(t)(p)}{\partial x}(x_{2}-x_{1})+\frac{\partial S^{0}(t)(p)}{\partial
y}\frac{Dy^{0}(0)}{Dx}(x_{2}-x_{1})$ $\displaystyle\
+\Pi_{M\cdot(\overline{a},\overline{x},0)}\frac{\partial S^{0}(t)(p)}{\partial
y}\left(\overline{\frac{Dy^{0}([x_{1},x_{2}])}{Dx}}-\frac{Dy^{0}(0)}{Dx}\right)(x_{2}-x_{1})$
$\displaystyle\
+\Pi_{M\cdot(\overline{a},\overline{x},0)}\left(\overline{\frac{\partial
S^{0}(t)([p_{1},p_{2}])}{\partial x}}-\frac{\partial S^{0}(t)(p)}{\partial
x}+\left(\overline{\frac{\partial S^{0}(t)([p_{1},p_{2}])}{\partial
y}}-\frac{\partial S^{0}(t)(p)}{\partial
y}\right)\overline{\frac{Dy^{0}([x_{1},x_{2}])}{Dx}}\right)(x_{2}-x_{1}),$

and

$\displaystyle
M\cdot(0,0,\overline{y}_{1}-\overline{y}_{2})\in\Pi_{M\cdot(0,0\overline{y})}\frac{\partial
S^{0}(t)(p)}{\partial
y}\left(\overline{\frac{Dy^{0}([x_{1},x_{2}])}{Dx}}-\frac{Dy^{0}(0)}{Dx}\right)(x_{2}-x_{1})$
$\displaystyle\ +\Pi_{M\cdot(0,0,\overline{y})}\left(\overline{\frac{\partial
S^{0}(t)([p_{1},p_{2}])}{\partial x}}-\frac{\partial S^{0}(t)(p)}{\partial
x}+\left(\overline{\frac{\partial S^{0}(t)([p_{1},p_{2}])}{\partial
y}}-\frac{\partial S^{0}(t)(p)}{\partial
y}\right)\overline{\frac{Dy^{0}([x_{1},x_{2}])}{Dx}}\right)(x_{2}-x_{1}),$

Now, as $\frac{\partial S^{0}(t)(p)}{\partial x}(x_{2}-x_{1})+\frac{\partial
S^{0}(t)(p)}{\partial y}$ is a nonsingular matrix, by the $C^{1}$ continuity
of the flow $S^{0}$ and of the unstable manifold $y^{0}$ we deduce as in the
stable case that

$|(\overline{a}_{1}-\overline{a}_{2},\overline{x}_{1}-\overline{x}_{2},0)|\geq
C_{1}|x_{1}-x_{2}|-C_{2}|x_{1}-x_{2}|,$

and

$|(0,0,\overline{y}_{1}-\overline{y}_{2},0)|\leq C_{3}|x_{1}-x_{2}|,$

where $C_{1}$ is fixed and $C_{2},C_{3}$ can be made as small as we need by
taking sufficiently small $\delta_{0}$. This implies the required Lipschitz
condition $|(0,0,\overline{y}_{1}-\overline{y}_{2},0)|\leq
C_{u}|(\overline{a}_{1}-\overline{a}_{2},\overline{x}_{1}-\overline{x}_{2},0)|$
with the arbitrarily small constant $C_{u}$. We also have the restriction
$\delta_{k_{2}}^{2}\geq C_{u}^{2}(\delta_{c}^{2}+\delta_{k_{1}}^{2})$ that
needs to be satisfied together with $\delta_{k_{1}}^{2}\geq
C_{s}^{2}(\delta_{c}^{2}+\delta_{k_{2}}^{2})$ so that the graphs are vertical
and horizontal disks in the box. But these restrictions can be made to hold
together as $C_{u}\to 0$ as $(\delta_{c},\delta_{k_{1}})\to 0$ and $C_{s}\to
0$ as $(\delta_{c},\delta_{k_{2}})\to 0$, so we can rescale the three radii if
necessary. ∎

Now we consider the problem with $\varepsilon>0$. Then, the local stable
manifold of the corresponding equilibrium $(0,e_{j}^{\varepsilon})$ is a disk
over variables $(\eta,x)$ where $\eta\in
B_{L^{2}_{A}(\mathbb{R}_{+})^{d}}(0,R_{1})$ with values denoted by $y$ and
$(x,y)$ are the local variables in the $h$-set containing $e_{j}$. Likewise,
the local unstable manifold is the disk over the variable $y$ with values
being variables $(x,\eta)$ with $\eta\in
B_{L^{2}_{A}(\mathbb{R}_{+})^{d}}(0,R_{2})$ and $(x,y)$ are the local
coordinates in the $h$-set containing $e_{i}$. In the next lemma we prove that
for every $\eta\in B_{L^{2}_{A}(\mathbb{R}_{+})^{d}}(0,R_{1})$ the section of
stable manifold of $e_{j}$ is a vertical disk in the box constructed in Lemma
7.1 and the image by $S^{0}(t)$ of the local unstable manifold of $e_{i}$ is
the horizontal disk in the same box. We also calculate the Lipschitz constants
of these disks.

######  Lemma 7.2.

Consider the box $(\overline{a},\overline{x},\overline{y})\in
B(0,\delta_{c})\times B(0,\delta_{k_{1}})\times B(0,\delta_{k_{2}})$ from
Lemma 7.1. There exists $\varepsilon_{0}>0$ such that for every
$\varepsilon\in[0,\varepsilon_{0}]$, if $(0,e_{j}^{\varepsilon})$ is an
equilibrium for $\varepsilon$ that corresponds to $e^{j}$, then, decreasing
the size of the box, if necessary, the intersection of the local stable
manifold of $(0,e_{j}^{\varepsilon})$ with this box is a vertical disk
$\overline{x}^{\varepsilon}(\cdot,\eta)$ for every $\eta\in
B_{L^{2}_{A}(\mathbb{R}_{+})^{d}}(0,R_{1})$ with some $R_{1}>0$ and the
following Lipschitz condition holds

$|\overline{x}^{\varepsilon}(\overline{a}_{2},\overline{y}_{2},\eta_{2})-\overline{x}^{\varepsilon}(\overline{a}_{1},\overline{y}_{1},\eta_{1})|\leq
D_{1}|(\overline{a}_{1}-\overline{a}_{2},0,\overline{y}_{1}-\overline{y}_{2})|+D_{2}E\|\eta_{1}-\eta_{2}\|_{L^{2}_{A}(\mathbb{R}_{+})^{d}},$

for every $\eta_{1},\eta_{2}\in B_{L^{2}_{A}(\mathbb{R}_{+})^{d}}(0,R_{1})$
and with the constant $D_{1}$ that can be made arbitrarily small be scaling
down the size of the box and decreasing $\varepsilon$, and $E$ being the
Lipschitz constant of the $\eta$ variable in the local unstable manifolds of
$(0,e_{j}^{\varepsilon})$.

Moreover, the intersection of the unstable manifold of
$(0,e^{\varepsilon}_{i})$ with the box is the horizontal disk
$\overline{y}^{\varepsilon}$ satisfying

$|\overline{y}^{\varepsilon}(\overline{a}_{2},\overline{x}_{2})-\overline{y}^{\varepsilon}(\overline{a}_{1},\overline{x}_{1})|\leq
D_{3}|(\overline{a}_{2}-\overline{a}_{1},\overline{x}_{2}-\overline{x}_{1},0)|,$

where the constant $D_{3}$ can be made arbitrarily small by decreasing
$\varepsilon$ and the size of the box. The corresponding variables
$\overline{\eta}$ satisfy the Lipschitz condition

$\|\overline{\eta}^{\varepsilon}(\overline{a}_{2},\overline{x}_{2})-\overline{\eta}^{\varepsilon}(\overline{a}_{1},\overline{x}_{1})\|_{L^{2}_{A}(\mathbb{R}_{+})^{d}}\leq
D_{4}|(\overline{a}_{2}-\overline{a}_{1},\overline{x}_{2}-\overline{x}_{1},0)|,$

for some constant $D_{4}>0$.

###### Proof.

Consider first the local stable manifold of $(e^{\varepsilon}_{j},0)$, denote
its graph translated by $z$ by $x^{\varepsilon}(y,\eta)$. Note that
$\lim_{\varepsilon_{t}o0}x^{\varepsilon}(0,\eta)=0$. Now fix $\eta\in
B_{L^{2}_{A}(\mathbb{R}_{+})^{d}}(0,R_{1})$ and consider the point $z+x+y$ on
the local stable manifold of $e_{j}$ with this $\eta$. We have

$x=x^{\varepsilon}(y,\eta)\in x^{\varepsilon}(0,\eta)+\overline{\frac{\partial
x^{\varepsilon}([0,y],\eta)}{\partial y}}y,$

and hence the point can be written as

$z+y+\frac{\partial x^{0}(0)}{\partial
y}y+x^{\varepsilon}(0,\eta)+\overline{\frac{\partial
x^{\varepsilon}([0,y],\eta)}{\partial y}-\frac{\partial x^{0}([0,y])}{\partial
y}}y+\left(\overline{\frac{\partial x^{0}([0,y])}{\partial y}}-\frac{\partial
x^{0}(0)}{\partial y}\right)y.$

The proof now follows the same scheme using the Brouwer degree and the
homotopy argument as for the case $\varepsilon=0$ noting that
$y+\frac{\partial x^{0}(0)}{\partial y}y$ belongs to the tangent space of the
stable manifold at $z$ for $\varepsilon=0$, the term
$x^{\varepsilon}(0,\eta)+\overline{\frac{\partial
x^{\varepsilon}([0,y],\eta)}{\partial y}-\frac{\partial x^{0}([0,y])}{\partial
y}}y$ can be made arbitrarily small by decreasing $\varepsilon$, uniformly
with respect to $\eta$, and the term $\left(\overline{\frac{\partial
x^{0}([0,y])}{\partial y}}-\frac{\partial x^{0}(0)}{\partial y}\right)y$ can
be decreased by scaling down the size of the box, if necessary. Note that the
projection on the above point on $M\cdot(0,\overline{x},0)$ can be made to lie
in $B(0,\delta_{k_{1}})$ again by scaling down the size of the box, and
decreasing $\varepsilon$.

Now consider the two points in the stable manifold for $\varepsilon$, and
denote them by $(\eta_{1},x_{1}+y_{1})$ with
$x_{1}+y_{1}=M(\overline{a}_{1},\overline{x}_{1},\overline{y}_{1})$ and
$(\eta_{2},x_{2}+y_{2})$ with
$x_{2}+y_{2}=M(\overline{a}_{2},\overline{x}_{2},\overline{y}_{2})$. The
argument again follows the one for $\varepsilon=0$ with some extra terms that
become small for small $\varepsilon$. Indeed, we have

$\displaystyle
M(\overline{a}_{1}-\overline{a}_{2},\overline{x}_{1}-\overline{x}_{2},\overline{y}_{1}-\overline{y}_{2})=(x^{\varepsilon}_{1}(y_{1},\eta_{1})-x_{2}^{\varepsilon}(y_{2},\eta_{2})+y_{1}-y_{2})$
$\displaystyle\ =\overline{\frac{\partial
x^{\varepsilon}([(y_{2},\eta_{2})-(y_{1},\eta_{1})])}{\partial
y}}(y_{1}-y_{2})+\overline{\frac{\partial
x^{\varepsilon}([(y_{2},\eta_{2})-(y_{1},\eta_{1})])}{\partial\eta}}(\eta_{1}-\eta_{2})+y_{1}-y_{2}$
$\displaystyle\ =\overline{\frac{\partial
x^{\varepsilon}([(y_{2},\eta_{2})-(y_{1},\eta_{1})])}{\partial
y}-\frac{\partial x^{0}([y_{2},y_{1}])}{\partial
y}}(y_{1}-y_{2})+\left(\overline{\frac{\partial x^{0}([y_{2},y_{1}])}{\partial
y}}-\frac{\partial x^{0}(0)}{\partial y}\right)(y_{1}-y_{2})$
$\displaystyle\qquad\qquad+\overline{\frac{\partial
x^{\varepsilon}([(y_{2},\eta_{2})-(y_{1},\eta_{1})])}{\partial\eta}}(\eta_{1}-\eta_{2})+y_{1}-y_{2}+\frac{\partial
x^{0}(0)}{\partial y}(y_{1}-y_{2}).$

Like for $\varepsilon=0$ we project this formula on the tangent space
$T_{z}W^{s}(e_{j})$ and its complement. We obtain

$\displaystyle
M\cdot(\overline{a}_{1}-\overline{a}_{2},0,\overline{y}_{1}-\overline{y}_{2})\in\left(\frac{Dx^{0}(0)}{Dy}(y_{1}-y_{2})+y_{1}-y_{2}\right)$
$\displaystyle\qquad+\Pi_{M\cdot(\overline{a},0,\overline{y})}\overline{\frac{\partial
x^{\varepsilon}([(y_{2},\eta_{2})-(y_{1},\eta_{1})])}{\partial
y}-\frac{\partial x^{0}([y_{2},y_{1}])}{\partial y}}(y_{1}-y_{2})$
$\displaystyle\qquad+\Pi_{M\cdot(\overline{a},0,\overline{y})}\left(\overline{\frac{\partial
x^{0}([y_{2},y_{1}])}{\partial y}}-\frac{\partial x^{0}(0)}{\partial
y}\right)(y_{1}-y_{2})+\Pi_{M\cdot(\overline{a},0,\overline{y})}\overline{\frac{\partial
x^{\varepsilon}([(y_{2},\eta_{2})-(y_{1},\eta_{1})])}{\partial\eta}}(\eta_{1}-\eta_{2}).$

This means that

$|(\overline{a}_{1}-\overline{a}_{2},0,\overline{y}_{1}-\overline{y}_{2})|\geq
C_{1}|y_{1}-y_{2}|-(C_{2}(\varepsilon)+C_{3})|y_{1}-y_{2}|-C_{4}E|\eta_{1}-\eta_{2}|_{L^{2}_{A}(\mathbb{R}^{+})^{d}},$

where $C_{2}(\varepsilon)$ can be made arbitrarily small by taking small
$\varepsilon$, $C_{3}$ can be decreased by decreasing the size of the box, and
$E$ is the Lipschitz constant for $\eta$ in the graph of $x^{\varepsilon}$.
Now

$\displaystyle
M\cdot(0,\overline{x}_{1}-\overline{x}_{2},0)\in\Pi_{M\cdot(0,\overline{x},0)}\overline{\frac{\partial
x^{\varepsilon}([(y_{2},\eta_{2})-(y_{1},\eta_{1})])}{\partial
y}-\frac{\partial x^{0}([y_{2},y_{1}])}{\partial y}}(y_{1}-y_{2})$
$\displaystyle\qquad+\Pi_{M\cdot(0,\overline{x},0)}\left(\overline{\frac{\partial
x^{0}([y_{2},y_{1}])}{\partial y}}-\frac{\partial x^{0}(0)}{\partial
y}\right)(y_{1}-y_{2})+\Pi_{M\cdot(0,\overline{x},0)}\overline{\frac{\partial
x^{\varepsilon}([(y_{2},\eta_{2})-(y_{1},\eta_{1})])}{\partial\eta}}(\eta_{1}-\eta_{2}).$

It follows that

$|(0,\overline{x}_{1}-\overline{x}_{2},0)|\leq(C_{5}(\varepsilon)+C_{6})|y_{1}-y_{2}|+C_{7}E\|\eta_{1}-\eta_{2}\|_{L^{2}_{A}(\mathbb{R}^{+})^{d}}.$

Summarizing, we obtain

$|(0,\overline{x}_{1}-\overline{x}_{2},0)|\leq\frac{C_{5}(\varepsilon)+C_{6}}{C_{1}-C_{2}(\varepsilon)-C_{3}}|(\overline{a}_{1}-\overline{a}_{2},0,\overline{y}_{1}-\overline{y}_{2})|+\left(C_{7}E+\frac{C_{5}(\varepsilon)+C_{6}}{C_{1}-C_{2}(\varepsilon)-C_{3}}C_{4}E\right)\|\eta_{1}-\eta_{2}\|_{L^{2}_{A}(\mathbb{R}^{+})^{d}}.$

Note that decreasing $\varepsilon$ and the size of the box can make the
constant $\frac{C_{5}(\varepsilon)+C_{6}}{C_{1}-C_{2}(\varepsilon)-C_{3}}$
arbitrarily small, which ends the proof of the assertion for the stable
manifold.

We pass to the argument for the unstable manifold of
$(0,e_{i}^{\varepsilon})$. We denote the coordinates that follow from the
stable and unstable directions in the local unstable manifold of
$(0,e_{i}^{\varepsilon})$ by $(\eta,x+y,\eta)$. The point in this manifold is
denoted by

$\displaystyle(\eta^{\varepsilon}(x),x_{p}+x+y^{\varepsilon}(x))\in\left(\eta^{\varepsilon}(x_{p})+\frac{D\eta^{\varepsilon}(0)}{Dx}x+\Delta_{\eta}(x),x_{p}+x+y^{0}(0)+(y^{\varepsilon}(0)-y^{0}(0))+\frac{Dy^{\varepsilon}(0)}{Dx}x+\Delta_{y}(x)\right)$
$\displaystyle\ \
=(\eta^{\varepsilon}(x_{p}),p)+\left(\frac{D\eta^{\varepsilon}(0)}{Dx}x,x+\frac{Dy^{\varepsilon}(0)}{Dx}x\right)+(0,y^{\varepsilon}(0)-y^{0}(0))+\left(\Delta_{\eta}(x),\Delta_{y}(x)\right)$
$\displaystyle\ \
=(\eta^{\varepsilon}(x_{p}),p)+\left(\frac{D\eta^{\varepsilon}(0)}{Dx}x,x+\frac{Dy^{0}(0)}{Dx}x\right)+\left(0,\frac{Dy^{\varepsilon}(0)}{Dx}-\left(\frac{Dy^{0}(0)}{Dx}\right)x\right)+(0,y^{\varepsilon}(0)-y^{0}(0))+\left(\Delta_{\eta}(x),\Delta_{y}(x)\right)$
$\displaystyle\ \ =I+II+III+IV+V.$

Now denote the flow by
$S^{\varepsilon}(t)=\left(S^{\varepsilon}_{\eta}(t),S^{\varepsilon}_{(x,y)}(t)\right)$.
Hence

$\displaystyle
S^{\varepsilon}_{(x,y)}(t)(\eta^{\varepsilon}(x),x_{p}+x+y^{\varepsilon}(x))$
$\displaystyle\
=z+(S^{\varepsilon}_{(x,y)}(t)(\eta^{\varepsilon}(x_{p}),p)-S^{0}(t)p)+\frac{\partial
S^{\varepsilon}_{(x,y)}(t)(\eta^{\varepsilon}(x_{p}),p)}{\partial u_{0}}II$
$\displaystyle\qquad\qquad+\frac{\partial
S^{\varepsilon}_{(x,y)}(t)(\eta^{\varepsilon}(x_{p}),p)}{\partial
u_{0}}(III+IV+V)+o(|II+III+IV+V|)$ $\displaystyle\ =z+\frac{\partial
S^{0}(t)(p)}{\partial
u_{0}}\left(x+\frac{Dy^{0}(0)}{Dx}x\right)+(S^{\varepsilon}_{(x,y)}(t)(\eta^{\varepsilon}(x_{p}),p)-S^{0}(t)p)+\left(\frac{\partial
S^{\varepsilon}_{u}(t)(\eta^{\varepsilon}(x_{p}),p)}{\partial
u_{0}}-\frac{\partial S^{0}(t)(p)}{\partial u_{0}}\right)II+$
$\displaystyle\qquad\qquad+\frac{\partial
S^{\varepsilon}_{(x,y)}(t)(\eta^{\varepsilon}(x_{p}),p)}{\partial
u_{0}}(III+IV+V)+o(|II+III+IV+V|).$

The proof of the fact that for every $(\overline{a},\overline{x})$ there
exists $\overline{y}$ in the box in the image by $S^{\varepsilon}(t)$ of the
graph of the local unstable manifold follows again by the Brouwer degree and
the homotopy argument as in previous cases. Indeed, terms $III,IV$ can be made
as small as we need by taking small $\varepsilon$. Same thing can be done with
the terms $S^{\varepsilon}_{(x,y)}(t)(\eta^{\varepsilon}(x_{p}),p)-S^{0}(t)p$
and $\left(\frac{\partial
S^{\varepsilon}_{(x,y)}(t)(\eta^{\varepsilon}(x_{p}),p)}{\partial
u_{0}}-\frac{\partial S^{0}(t)(p)}{\partial u_{0}}\right)II$. The term $V$
satisfies the estimate $|V|\leq c(|x|)|x|$ with $c(|x|)\to 0$ as $|x|\to 0$.
This term can be decreased by decreasing the size of the ball in the variable
$x$. Finally in the term $o(|II+III+IV+V|)$ the dominating contribution comes
from $II$ but this, same as in case of $V$ has the form of $c(|x|)|x|$ with
$c(|x|)\to 0$ as $|x|\to 0$. We observe that the projection of the above point
on $M\cdot(0,0,\overline{y})$ can be made to belong to $B(0,\delta_{k_{2}})$
by scaling down the box and decreasing $\varepsilon$ if necessary.

Now let us consider two points in the local unstable manifold of
$(0,e^{\varepsilon}_{i})$. Denote them by
$(\eta_{1},p_{1})=(\eta_{1},x_{1}+y_{1})$ and
$(\eta_{2},p_{2})=(\eta_{2},x_{2}+y_{2})$. Now
$S^{\varepsilon}_{(x,y)}(t)(\eta_{1},p_{1})=M\cdot(\overline{a}_{1},\overline{x}_{1},\overline{y}_{1})$
and
$S^{\varepsilon}_{(x,y)}(t)(\eta_{2},p_{2})=M\cdot(\overline{a}_{2},\overline{x}_{2},\overline{y}_{2}).$
We calculate

$\displaystyle
M\cdot(\overline{a}_{2}-\overline{a}_{1},\overline{x}_{2}-\overline{x}_{1},\overline{y}_{2}-\overline{y}_{1})=S^{\varepsilon}_{(x,y)}(t)(\eta_{2},x_{2}+y_{2})-S^{\varepsilon}_{(x,y)}(t)(\eta_{1},x_{1}+y_{1})$
$\displaystyle\ \in\overline{\frac{\partial
S^{\varepsilon}_{(x,y)}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial
x}}(x_{2}-x_{1})$ $\displaystyle\qquad\qquad+\overline{\frac{\partial
S^{\varepsilon}_{(x,y)}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial
y}}(y_{2}-y_{1})+\overline{\frac{\partial
S^{\varepsilon}_{(x,y)}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial\eta}}(\eta_{2}-\eta_{1})$
$\displaystyle\ \in\overline{\frac{\partial
S^{\varepsilon}_{(x,y)}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial
x}}(x_{2}-x_{1})+\overline{\frac{\partial
S^{\varepsilon}_{(x,y)}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial y}}\
\overline{\frac{Dy^{\varepsilon}([x_{1},x_{2}])}{Dx}}(x_{2}-x_{1})$
$\displaystyle\qquad\qquad+\overline{\frac{\partial
S^{\varepsilon}_{(x,y)}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial\eta}}\
\overline{\frac{D\eta^{\varepsilon}([x_{1},x_{2}])}{Dx}}(x_{2}-x_{1}).$

Furthermore

$\displaystyle
M\cdot(\overline{a}_{2}-\overline{a}_{1},\overline{x}_{2}-\overline{x}_{1},\overline{y}_{2}-\overline{y}_{1})\in\frac{\partial
S^{0}(t)p}{\partial
x}\left(\frac{Dy^{0}(x_{p})}{Dx}(x_{2}-x_{1})+x_{2}-x_{1}\right)$
$\displaystyle\qquad+\overline{\frac{\partial
S^{\varepsilon}_{(x,y)}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial
x}-\frac{\partial S^{0}(t)[p_{1},p_{2}]}{\partial
x}}(x_{2}-x_{1})+\left(\overline{\frac{\partial
S^{0}(t)[p_{1},p_{2}]}{\partial x}}-\frac{\partial S^{0}(t)p}{\partial
x}\right)(x_{2}-x_{1})$ $\displaystyle\qquad+\overline{\frac{\partial
S^{\varepsilon}_{(x,y)}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial
y}-\frac{\partial S^{0}(t)[p_{1},p_{2}]}{\partial y}}\
\overline{\frac{Dy^{\varepsilon}([x_{1},x_{2}])}{Dx}}(x_{2}-x_{1})$
$\displaystyle\qquad+\left(\overline{\frac{\partial
S^{0}(t)[p_{1},p_{2}]}{\partial y}}-\frac{\partial S^{0}(t)p}{\partial
y}\right)\overline{\frac{Dy^{\varepsilon}([x_{1},x_{2}])}{Dx}}(x_{2}-x_{1})$
$\displaystyle\qquad+\frac{\partial S^{0}(t)p}{\partial
y}\overline{\frac{Dy^{\varepsilon}([x_{1},x_{2}])}{Dx}-\frac{Dy^{0}([x_{1},x_{2}])}{Dx}}(x_{2}-x_{1})+\frac{\partial
S^{0}(t)p}{\partial
y}\left(\overline{\frac{Dy^{0}([x_{1},x_{2}])}{Dx}}-\frac{Dy^{0}(x_{p})}{Dx}\right)(x_{2}-x_{1})$
$\displaystyle\qquad\qquad+\overline{\frac{\partial
S^{\varepsilon}_{(x,y)}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial\eta}}\
\overline{\frac{D\eta^{\varepsilon}([x_{1},x_{2}])}{Dx}}(x_{2}-x_{1})=I+II+III+IV+V+VI+VII+VIII.$

The first term in the last sum belong to the tangent space $T_{z}W^{u}(e_{i})$
for $\varepsilon=0$. Terms $II,IV,VI$ satisfy the estimate $|II+IV+VI|\leq
C_{2}(\varepsilon)|x_{1}-x_{2}|$ with $C_{2}(\varepsilon)\to 0$ as
$\varepsilon\to 0$. Terms $III,V,VII$ satisfy the bound $|III+V+VII|\leq
C_{3}|x_{2}-x_{1}|$, where $C_{2}$ may be made arbitrarily small by decreasing
the size of the box. As for term $VIII$ we have the bound $|VIII|\leq
C_{4}E|x_{2}-x_{1}|$, where $E$ is the Lipshitz constant in variable $\eta$ of
the local unstable manifold at $\varepsilon$ of the point
$(0,e^{\varepsilon}_{i})$. It follows that

$|(\overline{a}_{1}-\overline{a}_{2},\overline{x}_{1}-\overline{x}_{2},0)|\geq
C_{1}|x_{1}-x_{2}|-(C_{2}(\varepsilon)+C_{3}+C_{4}E)|x_{1}-x_{2}|,$

and

$|(0,0,\overline{y}_{1}-\overline{y}_{2})|\leq(C_{2}(\varepsilon)+C_{3}+C_{4}E)|x_{1}-x_{2}|.$

Now note that $E$ can be decreased to arbitrarily low value by decreasing
$\varepsilon$ (cf. Lemma 5.6). Hence

$|(0,0,\overline{y}_{1}-\overline{y}_{2})|\leq\frac{C_{2}(\varepsilon)+C_{3}+C_{4}E}{C_{1}-C_{2}(\varepsilon)-C_{3}-C_{4}E}|(\overline{a}_{1}-\overline{a}_{2},\overline{x}_{1}-\overline{x}_{2},0)|,$

and we have the required Lipschitz condition with the arbitrarily small
constant obtained by decreasing $\varepsilon$ and the size of the box, if
necessary.

Finally, let us estimate

$\displaystyle\overline{\eta}_{2}-\overline{\eta}_{1}=S^{\varepsilon}_{\eta}(t)(\eta_{2},x_{2}+y_{2})-S^{\varepsilon}_{\eta}(t)(\eta_{1},x_{1}+y_{1})$
$\displaystyle\ \in\overline{\frac{\partial
S^{\varepsilon}_{\eta}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial
x}}(x_{2}-x_{1})$ $\displaystyle\qquad\qquad+\overline{\frac{\partial
S^{\varepsilon}_{\eta}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial
y}}(y_{2}-y_{1})+\overline{\frac{\partial
S^{\varepsilon}_{\eta}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial\eta}}(\eta_{2}-\eta_{1})$
$\displaystyle\ \in\overline{\frac{\partial
S^{\varepsilon}_{\eta}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial
x}}(x_{2}-x_{1})+\overline{\frac{\partial
S^{\varepsilon}_{\eta}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial y}}\
\overline{\frac{Dy^{\varepsilon}([x_{1},x_{2}])}{Dx}}(x_{2}-x_{1})$
$\displaystyle\qquad\qquad+\overline{\frac{\partial
S^{\varepsilon}_{\eta}(t)[(\eta_{1},p_{1}),(\eta_{2},p_{2})]}{\partial\eta}}\
\overline{\frac{D\eta^{\varepsilon}([x_{1},x_{2}])}{Dx}}(x_{2}-x_{1}).$

This means that

$\|\overline{\eta}_{2}-\overline{\eta}_{1}\|_{L^{2}_{A}(\mathbb{R}^{+})^{d}}\leq
L_{1}|x_{1}-x_{2}|\leq
L_{2}|(\overline{a}_{1}-\overline{a}_{2},\overline{x}_{1}-\overline{x}_{2},0)|,$

which completes the proof. ∎

Now we prove that the manifolds for $\varepsilon>0$ intersect. We find the
intersection, in the constructed box, of the unstable manifold of
$(0,e^{\varepsilon}_{i})$ with the stable manifold of
$(0,e^{\varepsilon}_{j})$. To this end we consider first the mapping

$B(0,\delta_{k_{1}})\ni\overline{x}\mapsto(\overline{y}^{\varepsilon}(0,\overline{x}),\overline{\eta}^{\varepsilon}(0,\overline{x}))\in
B(0,\delta_{k_{2}})\times B_{L^{2}_{A}(\mathbb{R}^{+})^{d}}(0,R_{1}).$

We will compose it with the mapping

$B(0,\delta_{k_{1}})\times
B_{L^{2}_{A}(\mathbb{R}^{+})^{d}}(0,R_{1})\ni(\overline{y},\overline{\eta})\mapsto\overline{x}^{\varepsilon}(0,\overline{y},\overline{\eta})\in
B(0,\delta_{k_{1}}).$

If we are able to prove that the composition of the above mappings has a fixed
point then this fixed point corresponds to the intersection point of the
manifolds for $\varepsilon>0$.

######  Lemma 7.3.

There exists $\varepsilon_{0}>0$ such that for every
$\varepsilon\in[0,\varepsilon_{0}]$ the unstable manifold of
$(0,e^{\varepsilon}_{i})$ intersects with the stable manifold of
$(0,e^{\varepsilon}_{j})$.

###### Proof.

Take $\overline{x}_{1},\overline{x}_{2}\in B(0,\delta_{k_{1}})$. We have

$|\overline{y}^{\varepsilon}(0,\overline{x}_{2})-\overline{y}^{\varepsilon}(0,\overline{x}_{1})|\leq
D_{3}|\overline{x}_{2}-\overline{x}_{1}|,$

and

$\|\overline{\eta}^{\varepsilon}(0,\overline{x}_{2})-\overline{\eta}^{\varepsilon}(0,\overline{x}_{1})\|_{L^{2}_{A}(\mathbb{R}^{+})^{d}}\leq
D_{4}|\overline{x}_{2}-\overline{x}_{1}|.$

We already know that the $\overline{y}$ variable belongs to the ball
$B(0,\delta_{k_{2}})$. But we still need to show that
$\overline{\eta}^{\varepsilon}(0,\overline{x})\in
B_{L^{2}_{A}(\mathbb{R}^{+})^{d}}(0,R_{1})$ for $\overline{x}\in
B(0,\delta_{k_{1}})$. To this end observe that $S^{0}(t)p\to e_{j}$ as
$t\to\infty$. Let us estimate the norm of the delay at $\varepsilon=0$
corresponding to this solution as $t\to\infty$. We have

$\eta^{t}(r)=S^{0}(t-r)p-S^{0}(t)p.$

The norm of this delay is given by

$\displaystyle\|\eta^{t}\|^{2}=\int_{0}^{\infty}(A(r)(S^{0}(t-r)p-S^{0}(t)p),(S^{0}(t-r)p-S^{0}(t)p))\,dr$
$\displaystyle\qquad=\int_{0}^{r_{0}}(A(r)(S^{0}(t-r)p-S^{0}(t)p),(S^{0}(t-r)p-S^{0}(t)p))\,dr$
$\displaystyle\qquad\ \
+\int_{r_{0}}^{\infty}(A(r)(S^{0}(t-r)p-S^{0}(t)p),(S^{0}(t-r)p-S^{0}(t)p))\,dr.$

In the following calculation by $C$ we will denote a generic constant. Fix
$\gamma>0$ and $r_{0}$. For these values we can find $t$ such that for
$r\in[0,r_{0}]$ we have $t-r\geq t-r_{0}\geq t_{0}$ with $t_{0}$ sufficiently
large to guarantee that $S(s)p\in B(e_{j},\gamma)$ for $s\geq t_{0}$. We have

$\|\eta^{t}\|^{2}\leq
4\gamma^{2}\int_{0}^{r_{0}}\|A(r)\|dr+C\int_{r_{0}}^{\infty}\|A(r)\|dr\leq
C\gamma^{2}+Ce^{-Cr_{0}}.$

whence

$\|\eta^{t}\|\leq C\gamma+Ce^{-Cr_{0}}.$

Now $\overline{\eta}^{\varepsilon}(0,\overline{x})$ is an image by
$S^{\varepsilon}_{\eta}(t)$ of a certain point $(\xi,x+y)$ in a local unstable
manifold of $(0,e^{\varepsilon}_{i})$ such that the distance $|x-x_{p}|$ does
not exceed $\delta$. From (14) we deduce that

$\|\overline{\eta}^{\varepsilon}(0,\overline{x})-\eta^{t}\|\leq
Ce^{Ct}(\|\xi-\eta_{0}\|+|x-x_{p}|+|y-y_{p}|+\varepsilon),$

where $\eta^{0}$ is the delay term corresponding to the total solution passing
through $p$ with $\varepsilon=0$. By Theorem 10.2 the last quantity can be
estimated from above as follows

$\|\overline{\eta}^{\varepsilon}(0,\overline{x})-\eta^{t}\|\leq
Ce^{Ct}(|x-x_{p}|+\varepsilon)\leq Ce^{Ct}(\delta+\varepsilon).$

Now we estimate $\|\overline{\eta}^{\varepsilon}(0,\overline{x})\|$. We have

$\|\overline{\eta}^{\varepsilon}(0,\overline{x})\|\leq\|\overline{\eta}^{\varepsilon}(0,\overline{x})-\eta^{t}\|+\|\eta^{t}\|\leq
Ce^{Ct}\delta+Ce^{Ct}\varepsilon+C\gamma+Ce^{-Cr_{0}}.$

We need the last quantity to be less than $R_{1}$. We fix $r_{0}$ and $\gamma$
so that each of two two last terms is no larger than $\frac{R_{1}}{4}$. This
forces us to choose $t$. Now choose $\varepsilon$ such that the second term is
no larger than $\frac{R_{1}}{4}$. Finally, if necessary, decrease $\delta$ so
that the first term does not exceed $\frac{R_{1}}{4}$. We come back to the
calculation of the Lipschitz constant for the fixed point mapping. We have

$\displaystyle|\overline{x}^{\varepsilon}(0,\overline{y}^{\varepsilon}(0,\overline{x}_{2}),\overline{\eta}^{\varepsilon}(0,\overline{x}_{2}))-\overline{x}^{\varepsilon}(0,\overline{y}^{\varepsilon}(0,\overline{x}_{1}),\overline{\eta}^{\varepsilon}(0,\overline{x}_{1}))|$
$\displaystyle\qquad\leq
D_{1}|\overline{y}^{\varepsilon}(0,\overline{x}_{2})-\overline{y}^{\varepsilon}(0,\overline{x}_{1})|+D_{2}E\|\overline{\eta}^{\varepsilon}(0,\overline{x}_{2})-\overline{\eta}^{\varepsilon}(0,\overline{x}_{1})\|_{L^{2}_{A}(\mathbb{R}^{+})^{d}}$
$\displaystyle\qquad\leq(D_{1}D_{3}+D_{2}ED_{4})|\overline{x}_{2}-\overline{x}_{1}|.$

The constants $D_{1},D_{3}$ can be made arbitrarily small by decreasing
$\varepsilon$ and scaling down the size of the box. Moreover $E$ can be made
arbitrarily small by decreasing $\varepsilon$ (cf. Lemma 5.6). We decrease
these constants such that $D_{1}D_{3}+D_{2}ED_{4}<1$. Then the constructed
mapping is a contraction and hence it has a fixed point which is the sought
intersection of the manifolds. ∎

##  8\. Appendix 1: Asymptotic compactness

We consider the semigroup

$S^{\varepsilon}(t):L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}\times\to
L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d},$

given by the solutions of (8)–(9), namely as
$S^{\varepsilon}(t)(\eta^{0},x_{0})=(\eta^{t},x(t))$. We do not expect the
compactness of $S^{\varepsilon}(t)$ for a finite $t$. Instead we prove the
following lemma

######  Lemma 8.1.

Assume that we have the estimate $|x(t)|\leq C(|x_{0}|,\|\eta_{0}\|)$ for the
function $C$ nondecreasing with respect to both arguments (this a priori
estimate follows from the Lyapunov function in Lemma 5.1). Assume that
$\\{\eta^{0,n},x_{0}^{n}\\}$ is a sequence of initial data bounded in
$L^{2}_{A}(\mathbb{R}^{+})^{d}\times\mathbb{R}^{d}$ and $t_{n}\to\infty$. Then
$S^{\varepsilon}(t_{n})(\eta^{0,n},x_{0}^{n})$ is relatively compact.

###### Proof.

Denote $(\eta^{t,n},x^{n}(t))=S^{\varepsilon}(t)(\eta^{0,n},x_{0}^{n})$. Then
$|x^{n}(t_{n})|$ is bounded, so it has a convergent subsequence. We denote
this subsequence by the same index $n$, without renumbering. Then
$x^{n}(t_{n})\to\xi$ in $\mathbb{R}^{d}$. We need to show the relative
compactness of $\eta^{{t_{n}},n}$, that is of $x^{n}(t_{n}-s)-x^{n}(t_{n}).$
in $L^{2}_{A}(\mathbb{R}^{+})^{d}$. Observe that

$0\leq\int_{0}^{\infty}(A(s)(x^{n}(t_{n})-\xi),x^{n}(t_{n})-\xi)\,ds\leq\int_{0}^{\infty}\|A(s)\|\,ds|x^{n}(t_{n})-\xi|^{2}\to
0.$

It is enough to prove the relative compactness of $[0,\infty)\ni s\to
x^{n}(t_{n}-s)\in\mathbb{R}^{d}$ in the space $L^{2}_{A}(\mathbb{R}^{+})^{d}$.
We first demonstrate the relative compactness on $L^{2}_{A}(0,T)^{d}$ for
every $T$. Note that the continuity and positive definiteness of $[0,T]\ni
s\mapsto A(s)$ implies that the norms $L^{2}_{A}(0,T)^{d}$ and
$L^{2}(0,T)^{d}$ are equivalent. We are in position to use the Kolmogorov-
Riesz-Frechet theorem which states that the set $B\subset L^{2}(0,T)^{d}$ is
relatively compact if and only if it is bounded in that space and

$\lim_{h\to 0}\sup_{u\in B}\int_{0}^{T-h}|u(s+h)-u(s)|^{2}\,ds=0.$

In our case we need to show that

$\lim_{h\to
0}\sup_{n}\int_{0}^{T-h}|x^{n}(t_{n}-s-h)-x^{n}(t_{n}-s)|^{2}\,ds=0,$

or

$\lim_{h\to
0}\sup_{n}\int_{0}^{T-h}\int_{t_{n}-s-h}^{t_{n}-s}|(x^{n})^{\prime}(r)|^{2}\,dr\,ds=0.$

It is enough that the result is obtained for $n\geq n_{0}$ where $n_{0}$ may
depend on $T$. We chose $n_{0}$ sufficiently large such that $t_{n}\geq T$.
Then

$\int_{0}^{T-h}\int_{t_{n}-s-h}^{t_{n}-s}|(x^{n})^{\prime}(r)|^{2}\,dr\,ds\leq
h\int_{t_{n}-T}^{t_{n}}|(x^{n})^{\prime}(r)|^{2}\,dr.$

But, cf. (8) and Lemma 5.1,

$\displaystyle\int_{t_{n}-T}^{t_{n}}|(x^{n})^{\prime}(r)|^{2}\,dr\leq
3\int_{t_{n}-T}^{t_{n}}|f(x^{n}(s))|^{2}\,ds+\varepsilon
C\int_{t_{n}-T}^{t_{n}}|x^{n}(s)|^{2}\,ds+\varepsilon
C\int_{t_{n}-T}^{T_{n}}\|\eta^{s,n}\|^{2}\,ds$ $\displaystyle\ \ \leq
TC(\|\eta^{0,n}\|,|x_{0}|),$

and the assertion follows. By the diagonal argument we can construct a
subsequence, still denoted by $n$, which converges in $L^{2}_{A}(0,T)^{d}$ for
every $T$. We denote the limit by $\eta$. We also have that
$\eta^{t_{n},n}(r)\to\eta(r)$ for almost every $r\geq 0$. As
$\eta^{t_{n},n}(r)=x^{n}(t_{n}-r)-x^{n}(r)$, it follows that
$|\eta^{t_{n},n}(r)|\leq E$ for a constant $E>0$ and $r\in[0,t_{n}]$, whence
$|\eta(r)|\leq E$ for a.e. $r\geq 0$. We claim that this subsequence actually
converges in $L^{2}_{A}(\mathbb{R}^{+})^{d}$. To get this assertion we need to
show that for every $\delta>0$ there exists $n_{\delta}$ such that for every
$n\geq n_{\delta}$ there holds

$\|\eta^{t_{n},n}-\eta\|=\int_{0}^{\infty}(A(r)(\eta^{t_{n},n}(r)-\eta(r)),(\eta^{t_{n},n}(r)-\eta(r)))\,dr\leq\delta$

We choose $T_{\delta}$ such that

$\int_{T_{\delta}}^{\infty}\|A(r)\|\,dr\leq\frac{\delta}{12E^{2}}.$

For large $T_{\delta}$ we split the integral into three parts

$\displaystyle\int_{0}^{\infty}(A(r)(\eta^{t_{n},n}(r)-\eta(r)),\eta^{t_{n},n}(r)-\eta(r))\,dr$
$\displaystyle=\int_{0}^{T_{\delta}}(A(r)(\eta^{t_{n},n}(r)-\eta(r)),\eta^{t_{n},n}(r)-\eta(r))\,dr$
$\displaystyle+\int_{T_{\delta}}^{t_{n}}(A(r)(\eta^{t_{n},n}(r)-\eta(r)),\eta^{t_{n},n}(r)-\eta(r))\,dr+\int_{t_{n}}^{\infty}(A(r)(\eta^{t_{n},n}(r)-\eta(r)),\eta^{t_{n},n}(r)-\eta(r))\,dr.$

From convergence in $L^{2}_{A}(0,T)^{d}$ for every $T$ it follows that we can
find $n_{\delta}$ such that the first integral in no greater than $\delta/3$.
Norm of the second integral is majorized as follows

$\left|\int_{T_{\delta}}^{t_{n}}(A(r)(\eta^{t_{n},n}(r)-\eta(r)),\eta^{t_{n},n}(r)-\eta(r))\,dr\right|\leq
4E^{2}\int_{T_{\delta}}^{t_{n}}\|A(r)\|\,dr\leq
4E^{2}\int_{T_{\delta}}^{\infty}\|A(r)\|\,dr\leq\frac{\delta}{3}.$

To deal with the last integral let us compute

$\displaystyle\int_{t_{n}}^{\infty}(A(r)(\eta^{t_{n},n}(r)-\eta(r)),\eta^{t_{n},n}(r)-\eta(r))\,dr$
$\displaystyle\qquad\qquad\leq
2\int_{t_{n}}^{\infty}(A(r)\eta^{t_{n},n}(r),\eta^{t_{n},n}(r))\,dr+2\int_{t_{n}}^{\infty}(A(r)\eta(r),\eta(r))\,dr$
$\displaystyle\qquad\qquad\leq
2\int_{0}^{\infty}(A(r+t_{n})\eta^{0,n}(r),\eta^{0,n}(r))\,dr+2E^{2}\int_{t_{n}}^{\infty}\|A(r)\|\,dr.$

From 2.1 we deduce

$\displaystyle\int_{t_{n}}^{\infty}(A(r)(\eta^{t_{n},n}(r)-\eta(r)),\eta^{t_{n},n}(r)-\eta(r))\,dr$
$\displaystyle\qquad\qquad\leq
2e^{-Ct_{n}}\int_{0}^{\infty}(A(r)\eta^{0,n}(r),\eta^{0,n}(r))\,dr+\frac{\delta}{6}=2e^{-Ct_{n}}\|\eta^{0,n}\|^{2}+\frac{\delta}{6}.$

We can find $n_{\delta}$ large enough, that the right-hand side of the last
bound is no greater than $\delta/3$ and the proof is complete. ∎

##  9\. Appendix 2: Graph transform for existence of local stable and unstable
manifolds.

We begin with the definitions and properties of hyperbolic sets, isolation,
and cone conditions adapted for the problems with the distributed delay.

######  Definition 9.1.

A family of mappings $\\{S(t)\\}_{t\geq 0}$ will be called a $C^{0}$ semiflow
on $X$ if

  * •

$[0,\infty)\times X\ni(t,x)\to S(t)x$ is continuous,

  * •

$S(0)=I_{X}$, the identity,

  * •

$S(t+s)x=S(t)(S(s)x)$ for every $s,t\geq 0$ and $x\in X$

######  Definition 9.2.

A $C^{0}$ semiflow on $X$ $\\{S(t)\\}_{t\geq 0}$ is asymptotically compact if
for a bounded sequence $\\{x_{n}\\}\subset X$ and a sequence $t_{n}\to\infty$
the sequence $S(t_{n})x_{n}$ is relatively compact.

######  Definition 9.3.

For a bounded set $B$ we define its $\omega$-limit set as

$\omega(B)=\\{x\in X\,:\ x=\lim_{n\to\infty}S(t_{n})x_{n}\ \ \textrm{for
sequences}\ \ t_{n}\to\infty\ \ \textrm{and}\ \ \\{x_{n}\\}\subset B\\}.$

The following result is well known.

######  Lemma 9.4.

If a a $C^{0}$ semiflow is assymptotically compact, then for every nonempty
bounded set $B\subset X$, the set $\omega(B)$ is nonempty, compact, connected,
invariant, and

$\lim_{t\to\infty}\mathrm{dist}\,(S(t)B,\omega(B))=0.$

######  Definition 9.5.

Let $X$ be a Banach space. The set $A\subset X$ is called an h-set (hyperbolic
set) if there exist the linear closed subspaces $X_{1},X_{2}$ of $X$ with
$X=X_{1}\oplus X_{2}$ and $\textrm{dim}\,X_{1}<\infty$,
$\textrm{dim}\,X_{1}=s+u$, with $s,u\in\mathbb{N}$, $u=u_{1}+2u_{2}$ and
$s=s_{1}+2s_{2}$, the numbers $\\{a_{k}\\}_{k=1}^{s_{1}+s_{2}+u_{1}+u_{2}}$
with $a_{k}>0$ and an affine bijective mapping
$L:\mathbb{R}^{\textrm{dim}\,X_{1}}\to X_{1}$ such that

$A=L\left(N_{u}\times N_{s}\right)\oplus\overline{B}_{X_{2}}(0,r),$

where

$N_{u}=\prod_{k=1}^{u_{1}}[-a_{k},a_{k}]\times\prod_{k=u_{1}+1}^{u_{1}+u_{2}}\\{(x,y)\in\mathbb{R}^{2}\,:\
x^{2}+y^{2}\leq a_{k}^{2}\\},$

and

$N_{s}=\prod_{k=u_{1}+u_{2}+1}^{u_{1}+u_{2}+s_{1}}[-a_{k},a_{k}]\times\prod_{k=u_{1}+u_{2}+s_{1}+1}^{u_{1}+u_{2}+s_{1}+s_{2}}\\{(x,y)\in\mathbb{R}^{2}\,:\
x^{2}+y^{2}\leq a_{k}^{2}\\},$

We also define

$N_{u,\varepsilon}=\prod_{k=1}^{u_{1}}[-a_{k}-\varepsilon,a_{k}+\varepsilon]\times\prod_{k=u_{1}+1}^{u_{1}+u_{2}}\\{(x,y)\in\mathbb{R}^{2}\,:\
x^{2}+y^{2}\leq(a_{k}+\varepsilon)^{2}\\}.$

If an element $x$ belongs to an h-set $A$ we can represent it uniquely as

$x=L((x_{u},x_{s}))+y,$

where $y\in\overline{B}_{X_{2}}(0,r)$, $x_{u}\in N_{u}\subset\mathbb{R}^{u}$
and $x_{s}\in N_{s}\subset\mathbb{R}^{s}$. We will use the notation
$P_{u}x=x_{u}$, $P_{s}x=x_{s}$ and $P_{X_{2}}x=y$. For and h-set we define its
exit set as

$A_{exit}=L\left(\partial N_{u}\times
N_{s}\right)\oplus\overline{B}_{X_{2}}(0,r).$

and its $\varepsilon$ exit extension as

$A^{\varepsilon}=L\left(N_{u,\varepsilon}\times
N_{s}\right)\oplus\overline{B}_{X_{2}}(0,r).$

Note that $P_{u},P_{s}$ and $P_{X_{2}}$ make sense for elements of
$A^{\varepsilon}$. An equivalent norm on $X$ will be denoted by
$\|x\|_{X}=|P_{u}x|+|P_{s}x|+\|P_{X_{2}}x\|_{X_{2}}$, where by $|\cdot|$ we
denote an euclidean norm on $\mathbb{R}^{s}$ or $\mathbb{R}^{u}$.

######  Definition 9.6.

Let $X$ be a Banach space and let $\\{S(t)\\}_{t\geq 0}$ be a $C^{0}$ semiflow
of mappings $S(t):X\to X$. An h-set $A$ is isolating with respect to this
semiflow if there exixts $\varepsilon>0$ and $t(\varepsilon)>0$ such that for
every $s\in(0,t(\varepsilon)]$

  * (A1)

$S(s)A\subset A^{\varepsilon}$,

  * (A2)

$\left[S(s)\left(A_{exit}\right)\right]\cap A=\emptyset$.

Condition (A1) imples that if via the evolution $S(t)$ we leave an isolating
h-set, we have to stay in $A^{\varepsilon}$ within the short time interval,
while condition (A2) implies that if we are on the exit set of $A$, then,
although we stay in $A^{\varepsilon}$, we cannot reenter $A$ is a short time.

######  Definition 9.7.

The h-set $A\subset X$ is called an h-set with cones if there exist three
continuous quadratic forms $\alpha:\mathbb{R}^{u}\to\mathbb{R}$,