On the solvability of some systems of integro-

differential equations with and without a drift

Messoud Efendiev1,2, Vitali Vougalter${}^{3\ *}$

1 Helmholtz Zentrum München, Institut für Computational Biology, Ingolstädter
Landstrasse 1

Neuherberg, 85764, Germany

e-mail<EMAIL_ADDRESS>
2 Department of Mathematics, Marmara University, Istanbul, Türkiye

e-mail<EMAIL_ADDRESS>

${}^{3\ *}$ Department of Mathematics, University of Toronto, Toronto,
Ontario, M5S 2E4, Canada

e-mail<EMAIL_ADDRESS>

Abstract. We prove the existence of solutions for some integro-differential
systems containing equations with and without the drift terms in the $H^{2}$
spaces by virtue of the fixed point technique when the elliptic equations
contain second order differential operators with and without the Fredholm
property, on the whole real line or on a finite interval with periodic
boundary conditions. Let us emphasize that the study of the system case is
more complicated than of the scalar situation and requires to overcome more
cumbersome technicalities.

Keywords: solvability conditions, non-Fredholm operators, integro-differential
systems, drift terms

AMS subject classification: 35J61, 35R09, 35K57

##  1 Introduction

We recall that a linear operator $L$ acting from a Banach space $E$ into
another Banach space $F$ satisfies the Fredholm property if its image is
closed, the dimension of its kernel and the codimension of its image are
finite. Consequently, the problem $Lu=f$ is solvable if and only if
$\phi_{i}(f)=0$ for a finite number of functionals $\phi_{i}$ from the dual
space $F^{*}$. These properties of the Fredholm operators are broadly used in
many methods of the linear and nonlinear analysis.

Elliptic equations in bounded domains with a sufficiently smooth boundary
satisfy the Fredholm property if the ellipticity condition, proper ellipticity
and Lopatinskii conditions are fulfilled (see e.g. [3], [10], [20], [22]).
This is the main result of the theory of the linear elliptic problems. In the
case of the unbounded domains, these conditions may not be sufficient and the
Fredholm property may not be satisfied. For example, the Laplace operator,
$Lu=\Delta u$, in $\mathbb{R}^{d}$ fails to satisfy the Fredholm property when
considered in Hölder spaces, $L:C^{2+\alpha}(\mathbb{R}^{d})\to
C^{\alpha}(\mathbb{R}^{d})$, or in Sobolev spaces, $L:H^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})$.

Linear elliptic problems in unbounded domains satisfy the Fredholm property if
and only if, in addition to the assumptions stated above, the limiting
operators are invertible (see [23]). In some simple cases, the limiting
operators can be constructed explicitly. For example, if

$Lu=a(x)u^{\prime\prime}+b(x)u^{\prime}+c(x)u,\;\;\;x\in\mathbb{R},$

where the coefficients of the operator have limits at the infinities,

$a_{\pm}=\lim_{x\to\pm\infty}a(x),\;\;\;b_{\pm}=\lim_{x\to\pm\infty}b(x),\;\;\;c_{\pm}=\lim_{x\to\pm\infty}c(x),$

the limiting operators are given by:

$L_{\pm}u=a_{\pm}u^{\prime\prime}+b_{\pm}u^{\prime}+c_{\pm}u.$

Since the coefficients here are the constants, the essential spectrum of the
operator, that is the set of the complex numbers $\lambda$ for which the
operator $L-\lambda$ does not satisfy the Fredholm property, can be found
explicitly by means of the Fourier transform:

$\lambda_{\pm}(\xi)=-a_{\pm}\xi^{2}+b_{\pm}i\xi+c_{\pm},\;\;\;\xi\in\mathbb{R}.$

The invertibility of the limiting operators is equivalent to the condition
that the essential spectrum does not contain the origin.

In the cases of the general elliptic equations, the same assertions hold true.
The Fredholm property is satisfied if the essential spectrum does not contain
the origin or if the limiting operators are invertible. However, these
conditions may not be explicitly written.

In the situations with non-Fredholm operators, the usual solvability
conditions may not be applicable and the solvability relations are, in
general, not known. There are some classes of operators for which the
solvability conditions are obtained. We illustrate that with the following
example. Consider the equation

$Lu\equiv\Delta u+au=f$ (1.1)

in $\mathbb{R}^{d}$, where $a$ is a positive constant. Clearly, the operator
$L$ coincides with its limiting operators. The homogeneous problem admits a
nonzero bounded solution. Thus the Fredholm property is not satisfied.
However, since the operator has constant coefficients, we can apply the
Fourier transform and find the solution explicitly. Solvability relations can
be formulated as follows. If $f\in L^{2}(\mathbb{R}^{d})$ and $xf\in
L^{1}(\mathbb{R}^{d})$, then there exists a unique solution of this problem in
$H^{2}(\mathbb{R}^{d})$ if and only if

$\Bigg{(}f(x),\frac{e^{ipx}}{(2\pi)^{\frac{d}{2}}}\Bigg{)}_{L^{2}(\mathbb{R}^{d})}=0,\quad
p\in S_{\sqrt{a}}^{d}\quad a.e.$

(see [29]). Here $S_{\sqrt{a}}^{d}$ denotes the sphere in $\mathbb{R}^{d}$ of
radius $\sqrt{a}$ centered at the origin. Thus, though the operator fails to
satisfy the Fredholm property, the solvability conditions are formulated
similarly. However, this similarity is only formal since the range of the
operator is not closed.

In the case of the operator with a potential,

$Lu\equiv\Delta u+a(x)u=f,$

the Fourier transform is not directly applicable. Nevertheless, the
solvability conditions in ${\mathbb{R}}^{3}$ can be obtained by a rather
sophisticated application of the theory of self-adjoint operators (see [26]).
As before, the solvability relations are formulated in terms of the
orthogonality to the solutions of the homogeneous adjoint problem. There are
several other examples of the linear elliptic non-Fredholm operators for which
the solvability conditions can be obtained (see [14], [23], [24], [26], [28],
[29]).

Solvability conditions play a crucial role in the analysis of the nonlinear
elliptic equations. In the case of the operators without Fredholm property, in
spite of some progress in the understanding of the linear problems, there
exist only few examples where nonlinear non-Fredholm operators are analyzed
(see [8], [9], [13], [14], [29]).

In the present article we study another class of stationary nonlinear systems
of equations, for which the Fredholm property may or may not be satisfied:

$\frac{d^{2}u_{k}}{dx^{2}}+b_{k}\frac{du_{k}}{dx}+a_{k}u_{k}+\int_{\Omega}G_{k}(x-y)F_{k}(u_{1}(y),u_{2}(y),...,u_{N}(y),y)dy=0,\quad
1\leq k\leq K,$ (1.2)
$\frac{d^{2}u_{k}}{dx^{2}}+a_{k}u_{k}+\int_{\Omega}G_{k}(x-y)F_{k}(u_{1}(y),u_{2}(y),...,u_{N}(y),y)dy=0,\quad
K+1\leq k\leq N,$ (1.3)

with the constants $a_{k}\geq 0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ and
$K\geq 2$. Here $x\in\Omega\subseteq{\mathbb{R}}$. In the first case we
consider the situation when $\Omega={\mathbb{R}}$, such that $N\geq 4$. In the
second part of the article we discuss the case of the finite interval
$\Omega=I:=[0,2\pi]$ with periodic boundary conditions, so that $N\geq 5$.
Throughout the work the vector function

$u:=(u_{1},u_{2},...,u_{N})^{T}\in{\mathbb{R}}^{N}.$ (1.4)

For the simplicity of the presentation we restrict ourselves to the one
dimensional case (the multidimensional case will be considered in our
forthcoming paper). Article [13] is devoted to the studies of a single
integro-differential equation with a drift term and the case without a
transport term was covered in [27]. In the population dynamics the integro-
differential equations describe the models with the intra-specific competition
and the nonlocal consumption of resources (see e.g. [4], [5], [17]). The
studies of the systems of integro- differential equations are of interest to
us in the context of the complicated biological systems, where $u_{k}(x,t),\
k=1,...,N$ stand for the cell densities for various groups of cells in the
organism. Let us use the explicit form of the solvability relations and study
the existence of solutions of such nonlinear systems. We would like to
emphasize especially that the solutions of the integro-differential equations
with the drift terms are relevant to the understanding of the emergence and
propagation of patterns in the theory of speciation (see [25]). The
solvability of the linear problems involving the Laplace operator with the
drift term was treated in [28], see also [6]. Standing lattice solitons in the
discrete NLS equation with saturation were discussed in [2]. Fredholm
structures, topological invariants and applications were covered in [10]. The
work [11] deals with the finite and infinite dimensional attractors for the
evolution equations of mathematical physics. The large time behavior of
solutions of fourth order parabolic equations and $\varepsilon$-entropy of
their attractors were analyzed in [12]. The articles [15] and [21] are crucial
for the understanding of the Fredholm and properness properties of the
quasilinear elliptic systems of the second order and of the operators of this
kind on ${\mathbb{R}}^{N}$. The work [16] is devoted to the exponential decay
and Fredholm properties in the second-order quasilinear elliptic systems of
equations.

##  2 Formulation of the results

Our technical conditions are analogous to the ones of the [13], adapted to the
work with vector functions. It is also more difficult to work in the Sobolev
space $H^{2}(\Omega,{\mathbb{R}}^{N})$, especially when $\Omega$ is a finite
interval with periodic boundary conditions with the constraints applied. The
nonlinear parts of system (1.2), (1.3) will satisfy the following regularity
conditions.

Assumption 2.1. Let $1\leq k\leq N$. Functions
$F_{k}(u,x):{\mathbb{R}}^{N}\times\Omega\to{\mathbb{R}}$ are satisfying the
Caratheodory condition (see [19]), such that

$\sqrt{\sum_{k=1}^{N}F_{k}^{2}(u,x)}\leq{\cal
K}|u|_{{\mathbb{R}}^{N}}+h(x)\quad for\quad u\in{\mathbb{R}}^{N},\ x\in\Omega$
(2.1)

with a constant ${\cal K}>0$ and $h(x):\Omega\to{\mathbb{R}}^{+},\quad h(x)\in
L^{2}(\Omega)$. Furthermore, they are Lipschitz continuous functions, such
that for any $u^{(1),(2)}\in{\mathbb{R}}^{N},\ x\in\Omega:$

$\sqrt{\sum_{k=1}^{N}(F_{k}(u^{(1)},x)-F_{k}(u^{(2)},x))^{2}}\leq
L|u^{(1)}-u^{(2)}|_{{\mathbb{R}}^{N}},$ (2.2)

with a constant $L>0$.

In the case of $\Omega=I$ we assume that $F_{k}(u,0)=F_{k}(u,2\pi)$ for
$u\in{\mathbb{R}}^{N}$ and all $1\leq k\leq N$.

Here and further down the norm of a vector function given by (1.4) is

$|u|_{{\mathbb{R}}^{N}}:=\sqrt{\sum_{k=1}^{N}u_{k}^{2}}.$

Note that the solvability of a local elliptic equation in a bounded domain in
${\mathbb{R}}^{N}$ was considered in [7], where the nonlinear function was
allowed to have a sublinear growth.

In order to study the existence of solutions of (1.2), (1.3), we introduce the
auxiliary system of equations as

$-\frac{d^{2}u_{k}}{dx^{2}}-b_{k}\frac{du_{k}}{dx}-a_{k}u_{k}=\int_{\Omega}G_{k}(x-y)F_{k}(v_{1}(y),v_{2}(y),...,v_{N}(y),y)dy,\quad
1\leq k\leq K,$ (2.3)
$-\frac{d^{2}u_{k}}{dx^{2}}-a_{k}u_{k}=\int_{\Omega}G_{k}(x-y)F_{k}(v_{1}(y),v_{2}(y),...,v_{N}(y),y)dy,\quad
K+1\leq k\leq N,$ (2.4)

where $a_{k}\geq 0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ are the constants and
$K\geq 2$.

Let us designate

$(f_{1}(x),f_{2}(x))_{L^{2}(\Omega)}:=\int_{\Omega}f_{1}(x)\bar{f_{2}}(x)dx,$
(2.5)

with a slight abuse of notations when these functions are not square
integrable, like for instance those involved in orthogonality relation (4.8)
below. Indeed, if $f_{1}(x)\in L^{1}(\Omega)$ and $f_{2}(x)$ is bounded, the
integral in the right side of (2.5) makes sense.

Let us first consider the situation of the whole real line, such that
$\Omega={\mathbb{R}}$. The appropriate Sobolev space is equipped with the norm

$\|\phi\|_{H^{2}({\mathbb{R}})}^{2}:=\|\phi\|_{L^{2}({\mathbb{R}})}^{2}+\Bigg{\|}\frac{d^{2}\phi}{dx^{2}}\Bigg{\|}_{L^{2}({\mathbb{R}})}^{2}.$
(2.6)

For a vector function with real valued components given by (1.4), we have

$\|u\|_{H^{2}({\mathbb{R}},{\mathbb{R}}^{N})}^{2}:=\sum_{k=1}^{N}\|u_{k}\|_{H^{2}({\mathbb{R}})}^{2}=\sum_{k=1}^{N}\Bigg{\\{}\|u_{k}\|_{L^{2}({\mathbb{R}})}^{2}+\Bigg{\|}\frac{d^{2}u_{k}}{dx^{2}}\Bigg{\|}_{L^{2}({\mathbb{R}})}^{2}\Bigg{\\}}.$
(2.7)

We also use the norm

$\|u\|_{L^{2}({\mathbb{R}},{\mathbb{R}}^{N})}^{2}:=\sum_{k=1}^{N}\|u_{k}\|_{L^{2}({\mathbb{R}})}^{2}.$

By means of Assumption 2.1 above, we are not allowed to consider the higher
powers of the nonlinearities, than the first one, which is restrictive from
the point of view of the applications. But this guarantees that our nonlinear
vector function is a bounded and continuous map from
$L^{2}(\Omega,{\mathbb{R}}^{N})$ to $L^{2}(\Omega,{\mathbb{R}}^{N})$.

The main issue for the system above is that in the absence of the drift terms
we are dealing with the self-adjoint, non-Fredholm operators

$-\frac{d^{2}}{dx^{2}}-a_{k}:H^{2}({\mathbb{R}})\to L^{2}({\mathbb{R}}),\
a_{k}\geq 0,$

which is the obstacle to solve our problem. The similar situations but in
linear problems, both self- adjoint and non-self-adjoint involving the
differential operators without the Fredholm property have been studied
extensively in recent years (see [14], [23], [24], [26], [28], [29]). However,
the situation differs when the constants in the drift terms $b_{k}\neq 0$. For
$1\leq k\leq K$, the operators

$L_{a,\ b,\ k}:=-\frac{d^{2}}{dx^{2}}-b_{k}\frac{d}{dx}-a_{k}:\quad
H^{2}({\mathbb{R}})\to L^{2}({\mathbb{R}})$ (2.8)

with $a_{k}\geq 0$ and $b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ involved in the
left side of (2.3) are non-self-adjoint.

By means of the standard Fourier transform, it can be easily verified that the
essential spectra of the operators $L_{a,\ b,\ k}$ are given by

$\lambda_{a,\ b,\ k}(p)=p^{2}-a_{k}-ib_{k}p,\quad p\in{\mathbb{R}}.$

Obviously, when $a_{k}>0$ the operators $L_{a,\ b,\ k}$ are Fredholm, because
their essential spectra stay away from the origin. But when $a_{k}=0$ our
operators $L_{a,\ b,\ k}$ fail to satisfy the Fredholm property since the
origin belongs to their essential spectra.

We manage to establish that under the reasonable technical assumptions system
(2.3), (2.4) defines a map $T_{a,b}:H^{2}({\mathbb{R}},{\mathbb{R}}^{N})\to
H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$, which is a strict contraction.

Theorem 2.2. Let $\Omega={\mathbb{R}},\ N\geq 4,\ K\geq 2,\ 1\leq l\leq K-1,\
K+1\leq r\leq N-1$, the integral kernels
$G_{k}(x):{\mathbb{R}}\to{\mathbb{R}},\ G_{k}(x)\in L^{1}({\mathbb{R}})$ for
all $1\leq k\leq N$ and Assumption 2.1 holds.

a) Let $a_{k}>0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ for $1\leq k\leq l$.

b) Let $a_{k}=0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ for $l+1\leq k\leq K$,
additionally $xG_{k}(x)\in L^{1}({\mathbb{R}})$ and orthogonality conditions
(4.8) hold.

c) Let $a_{k}>0,\ xG_{k}(x)\in L^{1}({\mathbb{R}})$ for $K+1\leq k\leq r$ and
orthogonality relations (4.9) hold.

d) Let $a_{k}=0,\ x^{2}G_{k}(x)\in L^{1}({\mathbb{R}})$ for $r+1\leq k\leq N$,
orthogonality conditions (4.10) hold and $2\sqrt{\pi}QL<1$ with $Q$ defined in
(4.7) below. Then the map $v\mapsto T_{a,b}v=u$ on
$H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ defined by problem (2.3), (2.4) has a
unique fixed point $v^{(a,b)}$, which is the only solution of the system of
equations (1.2), (1.3) in $H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$.

The fixed point $v^{(a,b)}$ is nontrivial provided that for some $1\leq k\leq
N$ the intersection of supports of the Fourier transforms of functions
$\hbox{supp}\widehat{F_{k}(0,x)}\cap\hbox{supp}\widehat{G_{k}}$ is a set of
nonzero Lebesgue measure in ${\mathbb{R}}$.

Let us note that in the case a) of the theorem above, when $a_{k}>0,\
b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$, the orthogonality relations are not
needed.

In the second part of the article we study the analogous system on the finite
interval $I=[0,2\pi]$ with periodic boundary conditions with $a_{k}\geq 0,\
b_{k}\in{\mathbb{R}},\ b_{k}\neq 0,\ K\geq 2,\ N\geq 5$, namely for $1\leq
k\leq K$

$\frac{d^{2}u_{k}}{dx^{2}}+b_{k}\frac{du_{k}}{dx}+a_{k}u_{k}+\int_{0}^{2\pi}G_{k}(x-y)F_{k}(u_{1}(y),u_{2}(y),...,u_{N}(y),y)dy=0,$
(2.9)

and for $K+1\leq k\leq N$

$\frac{d^{2}u_{k}}{dx^{2}}+a_{k}u_{k}+\int_{0}^{2\pi}G_{k}(x-y)F_{k}(u_{1}(y),u_{2}(y),...,u_{N}(y),y)dy=0.$
(2.10)

Let us use the function space

$H^{2}(I):=\\{v(x):I\to{\mathbb{R}}\ |\ v(x),v^{\prime\prime}(x)\in L^{2}(I),\
v(0)=v(2\pi),\ v^{\prime}(0)=v^{\prime}(2\pi)\\}$ (2.11)

aiming at $u_{k}(x)\in H^{2}(I)$ for $1\leq k\leq l$ and $K+1\leq k\leq r$
with $1\leq l\leq K-1$ and $K+1\leq r\leq q-1$ (see Theorem 2.3 and Lemma A2
below).

For the technical purposes we introduce the auxiliary constrained subspaces

$H_{k}^{2}(I):=\Big{\\{}v\in H^{2}(I)\ \Big{|}\ \Big{(}v(x),\frac{e^{\pm
in_{k}x}}{\sqrt{2\pi}}\Big{)}_{L^{2}(I)}=0\Big{\\}},\quad
n_{k}\in{\mathbb{N}},\quad r+1\leq k\leq q.$ (2.12)

Our goal is to have $u_{k}(x)\in H_{k}^{2}(I),\ r+1\leq k\leq q$, where
$r+1\leq q\leq N-1$. Also,

$H_{0}^{2}(I):=\\{v\in H^{2}(I)\ |\ (v(x),1)_{L^{2}(I)}=0\\}.$ (2.13)

We plan to have $u_{k}(x)\in H_{0}^{2}(I)$ for $l+1\leq k\leq K$ and $q+1\leq
k\leq N$ (see Theorem 2.3 and Lemma A2). The constrained subspaces (2.12) and
(2.13) are Hilbert spaces as well (see e.g. Chapter 2.1 of [18]).

The resulting space used for demonstrating the existence of the solution
$u(x):I\to{\mathbb{R}}^{N}$ of the system of equations (2.9), (2.10) will be
the direct sum of the spaces introduced above, namely

$H_{c}^{2}(I,{\mathbb{R}}^{N}):=\oplus_{k=1}^{l}H^{2}(I)\oplus_{k=l+1}^{K}H_{0}^{2}(I)\oplus_{k=K+1}^{r}H^{2}(I)\oplus_{k=r+1}^{q}H_{k}^{2}(I)\oplus_{k=q+1}^{N}H_{0}^{2}(I).$
(2.14)

The corresponding Sobolev norm will be

$\|u\|_{H_{c}^{2}(I,{\mathbb{R}}^{N})}^{2}:=\sum_{k=1}^{N}\\{\|u_{k}\|_{L^{2}(I)}^{2}+\|u_{k}^{\prime\prime}\|_{L^{2}(I)}^{2}\\},$
(2.15)

where $u(x):I\to{\mathbb{R}}^{N}$.

Let us demonstrate that under the reasonable technical conditions system
(2.3), (2.4) with $\Omega=I$ defines a map
$\tau_{a,b}:H_{c}^{2}(I,{\mathbb{R}}^{N})\to H_{c}^{2}(I,{\mathbb{R}}^{N})$,
which is a strict contraction.

Theorem 2.3. Let $\Omega=I,\ N\geq 5,\ K\geq 2,\ 1\leq l\leq K-1,\ K+1\leq
r\leq q-1,\ r+1\leq q\leq N-1$, the integral kernels
$G_{k}(x):I\to{\mathbb{R}},\ G_{k}(x)\in C(I),\ G_{k}(0)=G_{k}(2\pi)$ for all
$1\leq k\leq N$ and Assumption 2.1 is valid.

a) Let $a_{k}>0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ for $1\leq k\leq l$.

b) Let $a_{k}=0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ for $l+1\leq k\leq K$
and orthogonality condition (4.28) holds.

c) Let $a_{k}>0,\ a_{k}\neq n^{2},\ n\in{\mathbb{Z}}$ for $K+1\leq k\leq r$.

d) Let $a_{k}=n_{k}^{2},\ n_{k}\in{\mathbb{N}}$ for $r+1\leq k\leq q$ and
orthogonality relations (4.29) are valid.

e) Let $a_{k}=0$ for $q+1\leq k\leq N$, orthogonality condition (4.28) holds
and $2\sqrt{\pi}{\cal Q}L<1$, where ${\cal Q}$ is introduced in (4.27). Then
the map $v\mapsto\tau_{a,b}v=u$ on $H_{c}^{2}(I,{\mathbb{R}}^{N})$ defined by
the system of equations (2.3), (2.4) has a unique fixed point $v^{(a,b)}$,
which is the only solution of system (2.9), (2.10) in
$H_{c}^{2}(I,{\mathbb{R}}^{N})$.

The fixed point $v^{(a,b)}$ does not vanish identically in $I$ provided that
for some $1\leq k\leq N$ and a certain $n\in{\mathbb{Z}}$ the Fourier
coefficients $G_{k,n}F_{k}(0,x)_{n}\neq 0$.

Remark 2.4. Note that in the present work we deal with real valued vector
functions by means of the assumptions on $F_{k}(u,x)$ and $G_{k}(x)$ involved
in the nonlocal terms of the systems of equations considered above.

##  3 The existence of solutions for the integro-differential systems

Proof of Theorem 2.2. First we suppose that for a certain $v\in
H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ there exist two solutions
$u^{(1),(2)}\in H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ of problem (2.3), (2.4).
Then their difference $w(x):=u^{(1)}(x)-u^{(2)}(x)\in
H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ will be a solution of the homogeneous
system of equations

$-\frac{d^{2}w_{k}}{dx^{2}}-b_{k}\frac{dw_{k}}{dx}-a_{k}w_{k}=0,\quad 1\leq
k\leq K,$ $-\frac{d^{2}w_{k}}{dx^{2}}-a_{k}w_{k}=0,\quad K+1\leq k\leq N.$

But the operators $\displaystyle{-\frac{d^{2}}{dx^{2}}-a_{k},\ L_{a,\ b,\
k}:H^{2}({\mathbb{R}})\to L^{2}({\mathbb{R}})}$, where $L_{a,\ b,\ k}$ is
defined in (2.8) do not have any nontrivial zero modes. Therefore, $w(x)$
vanishes on ${\mathbb{R}}$.

Let us choose an arbitrary $v(x)\in H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$. We
apply the standard Fourier transform (4.1) to both sides of system (2.3),
(2.4). This gives us

$\widehat{u_{k}}(p)=\sqrt{2\pi}{\widehat{G_{k}}(p)\widehat{f_{k}}(p)\over
p^{2}-a_{k}-ib_{k}p},\quad
p^{2}\widehat{u_{k}}(p)=\sqrt{2\pi}{p^{2}\widehat{G_{k}}(p)\widehat{f_{k}}(p)\over
p^{2}-a_{k}-ib_{k}p},\quad 1\leq k\leq K,$ (3.1)
$\widehat{u_{k}}(p)=\sqrt{2\pi}{\widehat{G_{k}}(p)\widehat{f_{k}}(p)\over
p^{2}-a_{k}},\quad
p^{2}\widehat{u_{k}}(p)=\sqrt{2\pi}{p^{2}\widehat{G_{k}}(p)\widehat{f_{k}}(p)\over
p^{2}-a_{k}},\quad K+1\leq k\leq N.$ (3.2)

Here $\widehat{f_{k}}(p)$ stands for the Fourier image of $F_{k}(v(x),x)$.

Clearly, for $1\leq k\leq K$, we have the upper bounds with $N_{a,\ b,\ k}$
defined in (4.3), namely

$|\widehat{u_{k}}(p)|\leq\sqrt{2\pi}N_{a,\ b,\ k}|\widehat{f_{k}}(p)|\quad
and\quad|p^{2}\widehat{u_{k}}(p)|\leq\sqrt{2\pi}N_{a,\ b,\
k}|\widehat{f_{k}}(p)|.$

For $K+1\leq k\leq N$, we obtain

$|\widehat{u_{k}}(p)|\leq\sqrt{2\pi}M_{a,\ k}|\widehat{f_{k}}(p)|\quad
and\quad|p^{2}\widehat{u_{k}}(p)|\leq\sqrt{2\pi}M_{a,\
k}|\widehat{f_{k}}(p)|,$

with $M_{a,\ k}$ introduced in (4.4). Note that $N_{a,\ b,\ k}<\infty$ by
virtue of Lemma A1 of the Appendix without any orthogonality conditions for
$a_{k}>0,\ 1\leq k\leq l$ and under orthogonality relation (4.8) when
$a_{k}=0,\ l+1\leq k\leq K$. Also, $M_{a,\ k}<\infty$ under orthogonality
conditions (4.9) when $a_{k}>0,\ K+1\leq k\leq r$ and under orthogonality
relations (4.10) for $a_{k}=0,\ r+1\leq k\leq N$.

This allows us to obtain the upper bound on the norm as

$\|u\|_{H^{2}({\mathbb{R}},{\mathbb{R}}^{N})}^{2}=\sum_{k=1}^{N}\\{\|\widehat{u_{k}}(p)\|_{L^{2}({\mathbb{R}})}^{2}+\|p^{2}\widehat{u_{k}}(p)\|_{L^{2}({\mathbb{R}})}^{2}\\}\leq$
$\leq 4\pi\sum_{k=1}^{K}N_{a,\ b,\
k}^{2}\|F_{k}(v(x),x)\|_{L^{2}({\mathbb{R}})}^{2}+4\pi\sum_{k=K+1}^{N}M_{a,\
k}^{2}\|F_{k}(v(x),x)\|_{L^{2}({\mathbb{R}})}^{2}.$ (3.3)

The right side of (3.3) is finite via (2.1) of Assumption 2.1 since
$|v(x)|_{{\mathbb{R}}^{N}}\in L^{2}({\mathbb{R}})$. Thus, for an arbitrary
$v(x)\in H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ there exists a unique solution
$u(x)\in H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ of problem (2.3), (2.4). Its
Fourier image is given by (3.1), (3.2). Therefore, the map
$T_{a,b}:H^{2}({\mathbb{R}},{\mathbb{R}}^{N})\to
H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ is well defined.

This enables us to choose arbitrarily $v^{(1),(2)}(x)\in
H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$, such that under the given conditions
their images $u^{(1),(2)}:=T_{a,b}v^{(1),(2)}\in
H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$. By means of (2.3), (2.4) along with
(4.1),

$\widehat{u_{k}^{(1)}}(p)=\sqrt{2\pi}{\widehat{G_{k}}(p)\widehat{f_{k}^{(1)}}(p)\over
p^{2}-a_{k}-ib_{k}p},\quad\widehat{u_{k}^{(2)}}(p)=\sqrt{2\pi}{\widehat{G_{k}}(p)\widehat{f_{k}^{(2)}}(p)\over
p^{2}-a_{k}-ib_{k}p},\quad 1\leq k\leq K,$
$\widehat{u_{k}^{(1)}}(p)=\sqrt{2\pi}{\widehat{G_{k}}(p)\widehat{f_{k}^{(1)}}(p)\over
p^{2}-a_{k}},\quad\widehat{u_{k}^{(2)}}(p)=\sqrt{2\pi}{\widehat{G_{k}}(p)\widehat{f_{k}^{(2)}}(p)\over
p^{2}-a_{k}},\quad K+1\leq k\leq N.$

Here $\widehat{f_{k}^{(1)}}(p)$ and $\widehat{f_{k}^{(2)}}(p)$ denote the
Fourier transforms of $F_{k}(v^{(1)}(x),x)$ and $F_{k}(v^{(2)}(x),x)$
respectively.

Hence, for $1\leq k\leq K$, we easily derive

$\Bigg{|}\widehat{u_{k}^{(1)}}(p)-\widehat{u_{k}^{(2)}}(p)\Bigg{|}\leq\sqrt{2\pi}N_{a,\
b,\ k}\Bigg{|}\widehat{f_{k}^{(1)}}(p)-\widehat{f_{k}^{(2)}}(p)\Bigg{|},$
$\Bigg{|}p^{2}\widehat{u_{k}^{(1)}}(p)-p^{2}\widehat{u_{k}^{(2)}}(p)\Bigg{|}\leq\sqrt{2\pi}N_{a,\
b,\ k}\Bigg{|}\widehat{f_{k}^{(1)}}(p)-\widehat{f_{k}^{(2)}}(p)\Bigg{|}$

and for $K+1\leq k\leq N$

$\Bigg{|}\widehat{u_{k}^{(1)}}(p)-\widehat{u_{k}^{(2)}}(p)\Bigg{|}\leq\sqrt{2\pi}M_{a,\
k}\Bigg{|}\widehat{f_{k}^{(1)}}(p)-\widehat{f_{k}^{(2)}}(p)\Bigg{|},$
$\Bigg{|}p^{2}\widehat{u_{k}^{(1)}}(p)-p^{2}\widehat{u_{k}^{(2)}}(p)\Bigg{|}\leq\sqrt{2\pi}M_{a,\
k}\Bigg{|}\widehat{f_{k}^{(1)}}(p)-\widehat{f_{k}^{(2)}}(p)\Bigg{|}.$

Then for the appropriate norm of the difference of vector functions we obtain

$\|u^{(1)}-u^{(2)}\|_{H^{2}({\mathbb{R}},{\mathbb{R}}^{N})}^{2}=\sum_{k=1}^{N}\Big{\\{}\Big{\|}\widehat{u_{k}^{(1)}}(p)-\widehat{u_{k}^{(2)}}(p)\Big{\|}_{L^{2}({\mathbb{R}})}^{2}+\Big{\|}p^{2}\widehat{u_{k}^{(1)}}(p)-p^{2}\widehat{u_{k}^{(2)}}(p)\Big{\|}_{L^{2}({\mathbb{R}})}^{2}\Big{\\}}\leq$
$\leq 4\pi
Q^{2}\sum_{k=1}^{N}\|F_{k}(v^{(1)}(x),x)-F_{k}(v^{(2)}(x),x)\|_{L^{2}({\mathbb{R}})}^{2},$

where $Q$ is defined in (4.7). Clearly, all $v_{k}^{(1),(2)}(x)\in
H^{2}({\mathbb{R}})\subset L^{\infty}({\mathbb{R}})$ due to the Sobolev
embedding.

Condition (2.2) gives us

$\sum_{k=1}^{N}\|F_{k}(v^{(1)}(x),x)-F_{k}(v^{(2)}(x),x)\|_{L^{2}({\mathbb{R}})}^{2}\leq
L^{2}\|v^{(1)}-v^{(2)}\|_{L^{2}({\mathbb{R}},{\mathbb{R}}^{N})}^{2}.$

Thus,

$\|T_{a,b}v^{(1)}-T_{a,b}v^{(2)}\|_{H^{2}({\mathbb{R}},{\mathbb{R}}^{N})}\leq
2\sqrt{\pi}QL\|v^{(1)}-v^{(2)}\|_{H^{2}({\mathbb{R}},{\mathbb{R}}^{N})}.$
(3.4)

The constant in the right side of (3.4) is less than one via the one of our
assumptions. Therefore, by virtue of the Fixed Point Theorem, there exists a
unique vector function $v^{(a,b)}\in H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$,
such that $T_{a,b}v^{(a,b)}=v^{(a,b)}$, which is the only solution of problem
(1.2), (1.3) in $H^{2}({\mathbb{R}},{\mathbb{R}}^{N})$. Suppose $v^{(a,b)}(x)$
vanishes identically on the real line. This will contradict to our assumption
that for a certain $1\leq k\leq N$, the Fourier transforms of $G_{k}(x)$ and
$F_{k}(0,x)$ are nontrivial on a set of nonzero Lebesgue measure in
${\mathbb{R}}$.

Then we turn our attention to establishing the existence of solutions for our
system of integro-differential equations on the finite interval with periodic
boundary conditions.

Proof of Theorem 2.3. Let us first suppose that for some $v\in
H_{c}^{2}(I,{\mathbb{R}}^{N})$ there exist two solutions $u^{(1),(2)}\in
H_{c}^{2}(I,{\mathbb{R}}^{N})$ of system (2.3), (2.4) with $\Omega=I$. Then
the difference ${\tilde{w}}(x):=u^{(1)}(x)-u^{(2)}(x)\in
H_{c}^{2}(I,{\mathbb{R}}^{N})$ will satisfy the homogeneous system of
equations

$-\frac{d^{2}{\tilde{w}}_{k}}{dx^{2}}-b_{k}\frac{d{\tilde{w}}_{k}}{dx}-a_{k}{\tilde{w}}_{k}=0,\quad
1\leq k\leq K,$ (3.5)
$-\frac{d^{2}{\tilde{w}}_{k}}{dx^{2}}-a_{k}{\tilde{w}}_{k}=0,\quad K+1\leq
k\leq N.$ (3.6)

Evidently, each operator contained in the left side of system (3.5)

${\cal L}_{a,\ b,\ k}:=-\frac{d^{2}}{dx^{2}}-b_{k}\frac{d}{dx}-a_{k}:\quad
H^{2}(I)\to L^{2}(I),$ (3.7)

where $1\leq k\leq l,\ a_{k}>0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ is
Fredholm, non-self-adjoint.

Its set of eigenvalues is

$\lambda_{a,\ b,\ k}(n)=n^{2}-a_{k}-ib_{k}n,\quad n\in{\mathbb{Z}}.$ (3.8)

The corresponding eigenfunctions are the standard Fourier harmonics

$\frac{e^{inx}}{\sqrt{2\pi}},\quad n\in{\mathbb{Z}}.$ (3.9)

When $l+1\leq k\leq K$, we have $a_{k}=0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq
0$. Let us deal with the operators

${\cal L}_{0,\ b,\ k}:=-\frac{d^{2}}{dx^{2}}-b_{k}\frac{d}{dx}:\quad
H_{0}^{2}(I)\to L^{2}(I)$ (3.10)

involved in system (3.5).

Obviously, each operator (3.10) has the eigenvalues given by formula (3.8)
with $a_{k}=0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0,\ n\in{\mathbb{Z}},\ n\neq
0$.

The corresponding eigenfunctions in this case are (3.9) with
$n\in{\mathbb{Z}},\ n\neq 0$.

Clearly, every operator contained in the left side of the system of equations
(3.6)

${\cal L}_{a,\ 0,\ k}:=-\frac{d^{2}}{dx^{2}}-a_{k}:\quad H^{2}(I)\to L^{2}(I)$
(3.11)

with $K+1\leq k\leq r,\ a_{k}>0,\ a_{k}\neq n^{2},\ n\in{\mathbb{Z}}$ is
Fredholm, self-adjoint.

Its set of eigenvalues is

$\lambda_{a,\ 0,\ k}(n)=n^{2}-a_{k},\quad n\in{\mathbb{Z}}.$ (3.12)

The corresponding eigenfunctions are given by (3.9).

For $r+1\leq k\leq q$, we have $a_{k}=n_{k}^{2},\ n_{k}\in{\mathbb{N}}$. Let
us consider the operators

${\cal L}_{n_{k}^{2},\ 0,\ k}:=-\frac{d^{2}}{dx^{2}}-n_{k}^{2}:\quad
H_{k}^{2}(I)\to L^{2}(I)$ (3.13)

involved in the left side of system (3.6).

The eigenvalues of each operator (3.13) are written in (3.12) with
$a_{k}=n_{k}^{2},\ n\in{\mathbb{Z}},\ n\neq\pm n_{k}$.

The corresponding eigenfunctions are given by formula (3.9) with
$n\in{\mathbb{Z}},\ n\neq\pm n_{k}$.

When $q+1\leq k\leq N$, all the constants $a_{k}$ are trivial. Then the
operator contained in the left side of the system of equations (3.6) in this
situation is

${\cal L}_{0,\ 0,\ k}:=-\frac{d^{2}}{dx^{2}}:\quad H_{0}^{2}(I)\to L^{2}(I).$
(3.14)

Its eigenvalues are

$\lambda_{0,\ 0,\ k}(n)=n^{2},\quad n\in{\mathbb{Z}},\quad n\neq 0.$ (3.15)

We have the corresponding eigenfunctions written in (3.9) with
$n\in{\mathbb{Z}},\ n\neq 0$.

Note that all the operators mentioned above, which are involved in the left
side of the homogeneous system (3.5), (3.6) have the trivial kernels. Thus,
the vector function ${\tilde{w}}(x)$ vanishes identically on the interval $I$.

Let us choose arbitrarily $v(x)\in H_{c}^{2}(I,{\mathbb{R}}^{N})$ and apply
the Fourier transform (4.21) to both sides of the system of equations (2.3),
(2.4) with $\Omega=I$. This yields

$u_{k,n}=\sqrt{2\pi}\frac{G_{k,n}f_{k,n}}{n^{2}-a_{k}-ib_{k}n},\quad
n^{2}u_{k,n}=\sqrt{2\pi}\frac{n^{2}G_{k,n}f_{k,n}}{n^{2}-a_{k}-ib_{k}n},\quad
1\leq k\leq K,\quad n\in{\mathbb{Z}},$ (3.16)
$u_{k,n}=\sqrt{2\pi}\frac{G_{k,n}f_{k,n}}{n^{2}-a_{k}},\quad
n^{2}u_{k,n}=\sqrt{2\pi}\frac{n^{2}G_{k,n}f_{k,n}}{n^{2}-a_{k}},\quad K+1\leq
k\leq N,\quad n\in{\mathbb{Z}},$ (3.17)

where $f_{k,n}:=F_{k}(v(x),x)_{n}$.

From (3.16) and (3.17) we easily obtain that

$|u_{k,n}|\leq\sqrt{2\pi}{\cal N}_{a,\ b,\
k}|f_{k,n}|,\quad|n^{2}u_{k,n}|\leq\sqrt{2\pi}{\cal N}_{a,\ b,\
k}|f_{k,n}|,\quad 1\leq k\leq K,\quad n\in{\mathbb{Z}},$ (3.18)
$|u_{k,n}|\leq\sqrt{2\pi}{\cal M}_{a,\
k}|f_{k,n}|,\quad|n^{2}u_{k,n}|\leq\sqrt{2\pi}{\cal M}_{a,\ k}|f_{k,n}|,\quad
K+1\leq k\leq N,\quad n\in{\mathbb{Z}}.$ (3.19)

In the estimates above we have ${\cal N}_{a,\ b,\ k}$ introduced in (4.23) and
${\cal M}_{a,\ k}$ was defined in (4.24). Clearly, ${\cal N}_{a,\ b,\
k}<\infty$ by means of Lemma A2 of the Appendix without any orthogonality
relations for $a_{k}>0,\ 1\leq k\leq l$ and under orthogonality condition
(4.28) when $a_{k}=0,\ l+1\leq k\leq K$. Similarly, ${\cal M}_{a,\ k}<\infty$
for $a_{k}>0,\ a_{k}\neq n^{2},\ n\in{\mathbb{Z}},\ K+1\leq k\leq r$, under
orthogonality relations (4.29) when $a_{k}=n_{k}^{2},\ n_{k}\in{\mathbb{N}},\
r+1\leq k\leq q$ and under orthogonality condition (4.28) for $a_{k}=0,\
q+1\leq k\leq N$ by virtue of Lemma A2.

By means of (3.18) and (3.19), we derive

$\|u\|_{H_{c}^{2}(I,{\mathbb{R}}^{N})}^{2}=\sum_{k=1}^{K}\Big{[}\sum_{n=-\infty}^{\infty}|u_{k,n}|^{2}+\sum_{n=-\infty}^{\infty}|n^{2}u_{k,n}|^{2}\Big{]}+\sum_{k=K+1}^{N}\Big{[}\sum_{n=-\infty}^{\infty}|u_{k,n}|^{2}+\sum_{n=-\infty}^{\infty}|n^{2}u_{k,n}|^{2}\Big{]}\leq$
$\leq 4\pi\sum_{k=1}^{K}{\cal N}_{a,\ b,\
k}^{2}\|F_{k}(v(x),x)\|_{L^{2}(I)}^{2}+4\pi\sum_{k=K+1}^{N}{\cal M}_{a,\
k}^{2}\|F_{k}(v(x),x)\|_{L^{2}(I)}^{2}.$ (3.20)

Let us recall inequality (2.1) of Assumption 2.1. We have here
$|v(x)|_{{\mathbb{R}}^{N}}\in L^{2}(I)$, such that all $F_{k}(v(x),x)\in
L^{2}(I)$ and the right side of (3.20) is finite. Hence, for any $v(x)\in
H_{c}^{2}(I,{\mathbb{R}}^{N})$ there exists a unique $u(x)\in
H_{c}^{2}(I,{\mathbb{R}}^{N})$, which solves system (2.3), (2.4) with
$\Omega=I$. Its Fourier transform is given by formulas (3.16) and (3.17).
Thus, the map $\tau_{a,b}:H_{c}^{2}(I,{\mathbb{R}}^{N})\to
H_{c}^{2}(I,{\mathbb{R}}^{N})$ is well defined.

Let us choose arbitrarily $v^{(1),(2)}(x)\in H_{c}^{2}(I,{\mathbb{R}}^{N})$.
Under the stated assumptions, their images under the map discussed above are
$u^{(1),(2)}:=\tau_{a,b}v^{(1),(2)}\in H_{c}^{2}(I,{\mathbb{R}}^{N})$. By
virtue of (2.3) and (2.4) with $\Omega=I$ along with (4.21), we arrive at

$u_{k,n}^{(1)}=\sqrt{2\pi}\frac{G_{k,n}f_{k,n}^{(1)}}{n^{2}-a_{k}-ib_{k}n},\quad
u_{k,n}^{(2)}=\sqrt{2\pi}\frac{G_{k,n}f_{k,n}^{(2)}}{n^{2}-a_{k}-ib_{k}n},\quad
1\leq k\leq K,\quad n\in{\mathbb{Z}},$
$u_{k,n}^{(1)}=\sqrt{2\pi}\frac{G_{k,n}f_{k,n}^{(1)}}{n^{2}-a_{k}},\quad
u_{k,n}^{(2)}=\sqrt{2\pi}\frac{G_{k,n}f_{k,n}^{(2)}}{n^{2}-a_{k}},\quad
K+1\leq k\leq N,\quad n\in{\mathbb{Z}}.$

Here $f_{k,n}^{(1)}$ and $f_{k,n}^{(2)}$ stand for the Fourier images of
$F_{k}(v^{(1)}(x),x)$ and $F_{k}(v^{(2)}(x),x)$ respectively under transform
(4.21).

Thus, for $1\leq k\leq K,\ n\in{\mathbb{Z}}$, we have

$|u_{k,n}^{(1)}-u_{k,n}^{(2)}|\leq\sqrt{2\pi}{\cal N}_{a,\ b,\
k}|f_{k,n}^{(1)}-f_{k,n}^{(2)}|,\quad|n^{2}u_{k,n}^{(1)}-n^{2}u_{k,n}^{(2)}|\leq\sqrt{2\pi}{\cal
N}_{a,\ b,\ k}|f_{k,n}^{(1)}-f_{k,n}^{(2)}|.$

Similarly, for $K+1\leq k\leq N,\ n\in{\mathbb{Z}}$, we obtain

$|u_{k,n}^{(1)}-u_{k,n}^{(2)}|\leq\sqrt{2\pi}{\cal M}_{a,\
k}|f_{k,n}^{(1)}-f_{k,n}^{(2)}|,\quad|n^{2}u_{k,n}^{(1)}-n^{2}u_{k,n}^{(2)}|\leq\sqrt{2\pi}{\cal
M}_{a,\ k}|f_{k,n}^{(1)}-f_{k,n}^{(2)}|.$

Let us estimate the appropriate norm of the difference of the vector functions
as

$\|u^{(1)}-u^{(2)}\|_{H_{c}^{2}(I,{\mathbb{R}}^{N})}^{2}=\sum_{k=1}^{K}\Big{[}\sum_{n=-\infty}^{\infty}|u_{k,n}^{(1)}-u_{k,n}^{(2)}|^{2}+\sum_{n=-\infty}^{\infty}|n^{2}u_{k,n}^{(1)}-n^{2}u_{k,n}^{(2)}|^{2}\Big{]}+$
$+\sum_{k=K+1}^{N}\Big{[}\sum_{n=-\infty}^{\infty}|u_{k,n}^{(1)}-u_{k,n}^{(2)}|^{2}+\sum_{n=-\infty}^{\infty}|n^{2}u_{k,n}^{(1)}-n^{2}u_{k,n}^{(2)}|^{2}\Big{]}\leq$
$\leq 4\pi{\cal
Q}^{2}\sum_{k=1}^{N}\|F_{k}(v^{(1)}(x),x)-F_{k}(v^{(2)}(x),x)\|_{L^{2}(I)}^{2},$

with ${\cal Q}$ introduced in (4.27). Evidently, all $v_{k}^{(1),(2)}(x)\in
H^{2}(I)\subset L^{\infty}(I)$ via the Sobolev embedding.

We recall condition (2.2) of Assumption 2.1, such that

$\sum_{k=1}^{N}\|F_{k}(v^{(1)}(x),x)-F_{k}(v^{(2)}(x),x)\|_{L^{2}(I)}^{2}\leq
L^{2}\|v^{(1)}-v^{(2)}\|_{H_{c}^{2}(I,{\mathbb{R}}^{N})}^{2}.$

Hence,

$\|\tau_{a,b}v^{(1)}-\tau_{a,b}v^{(2)}\|_{H_{c}^{2}(I,{\mathbb{R}}^{N})}\leq
2\sqrt{\pi}{\cal Q}L\|v^{(1)}-v^{(2)}\|_{H_{c}^{2}(I,{\mathbb{R}}^{N})}.$
(3.21)

The constant in the right side of bound (3.21) is less than one as assumed. By
means of the Fixed Point Theorem, there exists a unique vector function
$v^{(a,b)}\in H_{c}^{2}(I,{\mathbb{R}}^{N})$, so that
$\tau_{a,b}v^{(a,b)}=v^{(a,b)}$. This is the only solution of the system of
equations (2.9), (2.10) in $H_{c}^{2}(I,{\mathbb{R}}^{N})$. Let us suppose
that $v^{(a,b)}(x)$ vanishes identically in I. This will contradict to the
given condition that for a certain $1\leq k\leq N$ and some $n\in{\mathbb{Z}}$
the Fourier coefficients $G_{k,n}F_{k}(0,x)_{n}\neq 0$.

##  4 Appendix

Let $G_{k}(x)$ be a function, $G_{k}(x):{\mathbb{R}}\to{\mathbb{R}}$, for
which we designate its standard Fourier transform using the hat symbol as

$\widehat{G_{k}}(p):={1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}G_{k}(x)e^{-ipx}dx,\quad
p\in{\mathbb{R}}.$ (4.1)

Clearly

$\|\widehat{G_{k}}(p)\|_{L^{\infty}({\mathbb{R}})}\leq{1\over\sqrt{2\pi}}\|G_{k}\|_{L^{1}({\mathbb{R}})}$
(4.2)

and
$\displaystyle{G_{k}(x)={1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}\widehat{G_{k}}(q)e^{iqx}dq,\
x\in{\mathbb{R}}.}$ For the technical purposes we introduce the auxiliary
quantities

$N_{a,\ b,\ k}:=\hbox{max}\Big{\\{}\Big{\|}{\widehat{G_{k}}(p)\over
p^{2}-a_{k}-ib_{k}p}\Big{\|}_{L^{\infty}({\mathbb{R}})},\quad\Big{\|}{p^{2}\widehat{G_{k}}(p)\over
p^{2}-a_{k}-ib_{k}p}\Big{\|}_{L^{\infty}({\mathbb{R}})}\Big{\\}},\quad 1\leq
k\leq K,$ (4.3) $M_{a,\
k}:=\hbox{max}\Big{\\{}\Big{\|}{\widehat{G_{k}}(p)\over
p^{2}-a_{k}}\Big{\|}_{L^{\infty}({\mathbb{R}})},\quad\Big{\|}{p^{2}\widehat{G_{k}}(p)\over
p^{2}-a_{k}}\Big{\|}_{L^{\infty}({\mathbb{R}})}\Big{\\}},\quad K+1\leq k\leq
N,$ (4.4)

where $a_{k}\geq 0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ are the constants and
$K\geq 2,\ N\geq 4$. Let $N_{0,\ b,\ k}$ denote (4.3) when $a_{k}$ vanishes
and $M_{0,\ k}$ stands for (4.4) when $a_{k}=0$. Under the assumptions of
Lemma A1 below, quantities (4.3) and (4.4) will be finite. This will allow us
to define

$N_{a,\ b}:=\hbox{max}_{1\leq k\leq K}N_{a,\ b,\ k}<\infty,$ (4.5)
$M_{a}:=\hbox{max}_{K+1\leq k\leq N}M_{a,\ k}<\infty$ (4.6)

and

$Q:=\hbox{max}\\{N_{a,\ b},\ M_{a}\\}.$ (4.7)

The auxiliary lemma below is the adaptation of the ones established in [13]
and [27] for the studies of the single integro-differential equation with and
without a drift. Let us present it for the convenience of the readers.

Lemma A1. Let $N\geq 4,\ K\geq 2,\ 1\leq l\leq K-1,\ K+1\leq r\leq N-1$, the
integral kernels $G_{k}(x):{\mathbb{R}}\to{\mathbb{R}},\ G_{k}(x)\in
L^{1}({\mathbb{R}})$ for all $1\leq k\leq N$.

a) Let $a_{k}>0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ for $1\leq k\leq l$.
Then $N_{a,\ b,\ k}<\infty$.

b) Let $a_{k}=0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ for $l+1\leq k\leq K$
and additionally $xG_{k}(x)\in L^{1}({\mathbb{R}})$. Then $N_{0,\ b,\
k}<\infty$ if and only if

$(G_{k}(x),1)_{L^{2}({\mathbb{R}})}=0$ (4.8)

holds.

c) Let $a_{k}>0$ and $xG_{k}(x)\in L^{1}({\mathbb{R}})$ for $K+1\leq k\leq r$.
Then $M_{a,\ k}<\infty$ if and only if

$\Bigg{(}G_{k}(x),\frac{e^{\pm
i\sqrt{a_{k}}x}}{\sqrt{2\pi}}\Bigg{)}_{L^{2}({\mathbb{R}})}=0$ (4.9)

is valid.

d) Let $a_{k}=0$ and $x^{2}G_{k}(x)\in L^{1}({\mathbb{R}})$ for $r+1\leq k\leq
N$. Then $M_{0,\ k}<\infty$ if and only if

$(G_{k}(x),1)_{L^{2}({\mathbb{R}})}=0\quad
and\quad(G_{k}(x),x)_{L^{2}({\mathbb{R}})}=0$ (4.10)

holds.

Proof. Note that in both cases a) and b) of our lemma the boundedness of
$\displaystyle{\frac{\widehat{G_{k}}(p)}{p^{2}-a_{k}-ib_{k}p}}$ implies the
boundedness of
$\displaystyle{\frac{p^{2}\widehat{G_{k}}(p)}{p^{2}-a_{k}-ib_{k}p}}$. Indeed,
we can express
$\displaystyle{\frac{p^{2}\widehat{G_{k}}(p)}{p^{2}-a_{k}-ib_{k}p}}$ as the
following sum

$\widehat{G_{k}}(p)+a_{k}\frac{\widehat{G_{k}}(p)}{p^{2}-a_{k}-ib_{k}p}+ib_{k}\frac{p\widehat{G_{k}}(p)}{p^{2}-a_{k}-ib_{k}p}.$
(4.11)

Obviously, the first term in (4.11) is bounded by virtue of (4.2) because
$G_{k}(x)\in L^{1}({\mathbb{R}})$ due to the one of our assumptions. The third
term in (4.11) can be trivially estimated from above in the absolute value by
means of (4.2) as

$\frac{|b_{k}||p||\widehat{G_{k}}(p)|}{\sqrt{(p^{2}-a_{k})^{2}+b_{k}^{2}p^{2}}}\leq\frac{1}{\sqrt{2\pi}}\|G_{k}(x)\|_{L^{1}({\mathbb{R}})}<\infty.$

Therefore, $\displaystyle{\frac{\widehat{G_{k}}(p)}{p^{2}-a_{k}-ib_{k}p}\in
L^{\infty}({\mathbb{R}})}$ yields
$\displaystyle{\frac{p^{2}\widehat{G_{k}}(p)}{p^{2}-a_{k}-ib_{k}p}\in
L^{\infty}({\mathbb{R}})}$.

To establish the result of the part a) of our lemma, we need to estimate

$\frac{|\widehat{G_{k}}(p)|}{\sqrt{(p^{2}-a_{k})^{2}+b_{k}^{2}p^{2}}}.$ (4.12)

Evidently, the numerator of (4.12) can be easily estimated from above by means
of (4.2) and the denominator in (4.12) can be trivially bounded below by a
finite, positive constant, such that

$\Bigg{|}\frac{\widehat{G_{k}}(p)}{p^{2}-a_{k}-ib_{k}p}\Bigg{|}\leq
C\|G_{k}(x)\|_{L^{1}({\mathbb{R}})}<\infty.$

Here and further down $C$ will stand for a finite, positive constant. This
implies that under our assumptions, when $a_{k}>0$ we have $N_{a,\ b,\
k}<\infty$.

In the case b) of the lemma when $a_{k}=0$, we use the identity

$\widehat{G_{k}}(p)=\widehat{G_{k}}(0)+\int_{0}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds.$

Hence

$\frac{\widehat{G_{k}}(p)}{p^{2}-ib_{k}p}=\frac{\widehat{G_{k}}(0)}{p(p-ib_{k})}+\frac{\int_{0}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds}{p(p-ib_{k})}.$
(4.13)

By virtue of definition (4.1) of the standard Fourier transform, we easily
obtain the upper bound

$\Bigg{|}\frac{d\widehat{G_{k}}(p)}{dp}\Bigg{|}\leq\frac{1}{\sqrt{2\pi}}\|xG_{k}(x)\|_{L^{1}({\mathbb{R}})}.$
(4.14)

We easily arrive at

$\Bigg{|}\frac{\int_{0}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds}{p(p-ib_{k})}\Bigg{|}\leq\frac{\|xG_{k}(x)\|_{L^{1}({\mathbb{R}})}}{\sqrt{2\pi}|b_{k}|}<\infty$

via our assumptions. Thus, the expression in the left side of (4.13) is
bounded if and only if $\widehat{G_{k}}(0)$ vanishes, which is equivalent to
orthogonality relation (4.8).

In the cases c) and d) of the lemma, we can write

$\frac{p^{2}\widehat{G_{k}}(p)}{p^{2}-a_{k}}=\widehat{G_{k}}(p)+a_{k}\frac{\widehat{G_{k}}(p)}{p^{2}-a_{k}},$

so that $\displaystyle{\frac{\widehat{G_{k}}(p)}{p^{2}-a_{k}}\in
L^{\infty}({\mathbb{R}})}$ implies that
$\displaystyle{\frac{p^{2}\widehat{G_{k}}(p)}{p^{2}-a_{k}}\in
L^{\infty}({\mathbb{R}})}$ as well.

To demonstrate the validity of the result of the part c) of our lemma, we
express

$\frac{\widehat{G_{k}}(p)}{p^{2}-a_{k}}=\frac{\widehat{G_{k}}(p)}{p^{2}-a_{k}}\chi_{I_{\delta_{k}}^{+}}+\frac{\widehat{G_{k}}(p)}{p^{2}-a_{k}}\chi_{I_{\delta_{k}}^{-}}+\frac{\widehat{G_{k}}(p)}{p^{2}-a_{k}}\chi_{I_{\delta_{k}}^{c}}.$
(4.15)

Here and below $\chi_{A}$ will denote the characteristic function of a set
$A\subseteq{\mathbb{R}}$ and $A^{c}$ will stand for its complement. The sets

$I_{\delta_{k}}^{+}:=\\{p\in{\mathbb{R}}\ |\
\sqrt{a_{k}}-\delta_{k}<p<\sqrt{a_{K}}+\delta_{k}\\},\quad
I_{\delta_{k}}^{-}:=\\{p\in{\mathbb{R}}\ |\
-\sqrt{a_{k}}-\delta_{k}<p<-\sqrt{a_{k}}+\delta_{k}\\}$

with $0<\delta_{k}<\sqrt{a_{k}}$ and $I_{\delta_{k}}:=I_{\delta_{k}}^{+}\cup
I_{\delta_{k}}^{-}$.

The third term in the right side of (4.15) can be trivially bounded from above
in the absolute value by means of (4.2) by
$\displaystyle{{1\over\sqrt{2\pi}\delta_{k}^{2}}\|G_{k}\|_{L^{1}({\mathbb{R}})}<\infty}$.

Clearly, we can write

$\widehat{G_{k}}(p)=\widehat{G_{k}}(\sqrt{a_{k}})+\int_{\sqrt{a_{k}}}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds,\quad\widehat{G_{k}}(p)=\widehat{G_{k}}(-\sqrt{a_{k}})+\int_{-\sqrt{a_{k}}}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds.$

Thus, the sum of the first two terms in the right side of (4.15) is given by

$\frac{\widehat{G_{k}}(\sqrt{a_{k}})}{p^{2}-a_{k}}\chi_{I_{\delta_{k}}^{+}}+\frac{\widehat{G_{k}}(-\sqrt{a_{k}})}{p^{2}-a_{k}}\chi_{I_{\delta_{k}}^{-}}+\frac{\int_{\sqrt{a_{k}}}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds}{p^{2}-a_{k}}\chi_{I_{\delta_{k}}^{+}}+\frac{\int_{-\sqrt{a_{k}}}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds}{p^{2}-a_{k}}\chi_{I_{\delta_{k}}^{-}}.$

By virtue of (4.14), we derive

$\Bigg{|}\frac{\int_{\sqrt{a_{k}}}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds}{p^{2}-a_{k}}\chi_{I_{\delta_{k}}^{+}}\Bigg{|}\leq\frac{\|xG_{k}(x)\|_{L^{1}({\mathbb{R}})}}{\sqrt{2\pi}(2\sqrt{a_{k}}-\delta_{k})}<\infty,$
$\Bigg{|}\frac{\int_{-\sqrt{a_{k}}}^{p}\frac{d\widehat{G_{k}}(s)}{ds}ds}{p^{2}-a_{k}}\chi_{I_{\delta_{k}}^{-}}\Bigg{|}\leq\frac{\|xG_{k}(x)\|_{L^{1}({\mathbb{R}})}}{\sqrt{2\pi}(2\sqrt{a_{k}}-\delta_{k})}<\infty.$

Therefore, it remains to consider

$\frac{\widehat{G_{k}}(\sqrt{a_{k}})}{p^{2}-a_{k}}\chi_{I_{\delta_{k}}^{+}}+\frac{\widehat{G_{k}}(-\sqrt{a_{k}})}{p^{2}-a_{k}}\chi_{I_{\delta_{k}}^{-}}.$
(4.16)

Evidently, (4.16) is bounded if and only if
$\widehat{G_{k}}(\pm\sqrt{a_{k}})=0$. This is equivalent to the orthogonality
conditions (4.9).

Finally, we turn our attention to the case d) of the lemma when $a_{k}=0$, so
that

$\frac{\widehat{G_{k}}(p)}{p^{2}}=\frac{\widehat{G_{k}}(p)}{p^{2}}\chi_{\\{|p|\leq
1\\}}+\frac{\widehat{G_{k}}(p)}{p^{2}}\chi_{\\{|p|>1\\}}.$ (4.17)

The second term in the right side of (4.17) can be easily estimated from above
in the absolute value as

$\Bigg{|}\frac{\widehat{G_{k}}(p)}{p^{2}}\chi_{\\{|p|>1\\}}\Bigg{|}\leq\|\widehat{G_{k}}(p)\|_{L^{\infty}({\mathbb{R}})}<\infty$

via (4.2). Obviously,

$\widehat{G_{k}}(p)=\widehat{G_{k}}(0)+p\frac{d\widehat{G_{k}}}{dp}(0)+\int_{0}^{p}\Bigg{(}\int_{0}^{s}\frac{d^{2}\widehat{G_{k}}(q)}{dq^{2}}dq\Bigg{)}ds,$

such that the first term in right side of (4.17) equals to

$\Bigg{[}\frac{\widehat{G_{k}}(0)}{p^{2}}+\frac{\frac{d\widehat{G_{k}}}{dp}(0)}{p}+\frac{\int_{0}^{p}\Bigg{(}\int_{0}^{s}\frac{d^{2}\widehat{G_{k}}(q)}{dq^{2}}dq\Bigg{)}ds}{p^{2}}\Bigg{]}\chi_{\\{|p|\leq
1\\}}.$ (4.18)

Using definition (4.1) of the standard Fourier transform, we derive

$\Bigg{|}\frac{d^{2}\widehat{G_{k}}(p)}{dp^{2}}\Bigg{|}\leq\frac{1}{\sqrt{2\pi}}\|x^{2}G_{k}(x)\|_{L^{1}({\mathbb{R}})},$

so that

$\Bigg{|}\frac{\int_{0}^{p}\Bigg{(}\int_{0}^{s}\frac{d^{2}\widehat{G_{k}}(q)}{dq^{2}}dq\Bigg{)}ds}{p^{2}}\chi_{\\{|p|\leq
1\\}}\Bigg{|}\leq\frac{\|x^{2}G_{k}(x)\|_{L^{1}({\mathbb{R}})}}{2\sqrt{2\pi}}<\infty$

as assumed. Hence, it remains to analyze

$\Bigg{[}\frac{\widehat{G_{k}}(0)}{p^{2}}+\frac{\frac{d\widehat{G_{k}}}{dp}(0)}{p}\Bigg{]}\chi_{\\{|p|\leq
1\\}}.$ (4.19)

From definition (4.1) of the standard Fourier transform we deduce that

$\widehat{G_{k}}(0)=\frac{1}{\sqrt{2\pi}}(G_{k}(x),1)_{L^{2}({\mathbb{R}})},\quad\frac{d\widehat{G_{k}}}{dp}(0)=-\frac{i}{\sqrt{2\pi}}(G_{k}(x),x)_{L^{2}({\mathbb{R}})},$

such that (4.19) is equal to

$\frac{1}{\sqrt{2\pi}}\Bigg{[}\frac{(G_{k}(x),1)_{L^{2}({\mathbb{R}})}}{p^{2}}-i\frac{(G_{k}(x),x)_{L^{2}({\mathbb{R}})}}{p}\Bigg{]}\chi_{\\{|p|\leq
1\\}}.$ (4.20)

Obviously, (4.20) belongs to $L^{\infty}({\mathbb{R}})$ if and only if
orthogonality relations (4.10) are valid.

Let the continuous function $G_{k}(x):I\to{\mathbb{R}},\
G_{k}(0)=G_{k}(2\pi)$. Its Fourier transform on the finite interval is given
by

$G_{k,n}:=\int_{0}^{2\pi}G_{k}(x)\frac{e^{-inx}}{\sqrt{2\pi}}dx,\quad
n\in{\mathbb{Z}},$ (4.21)

so that
$\displaystyle{G_{k}(x)=\sum_{n=-\infty}^{\infty}G_{k,n}\frac{e^{inx}}{\sqrt{2\pi}}}$.
Obviously, the inequalities

$\|G_{k,n}\|_{l^{\infty}}\leq\frac{1}{\sqrt{2\pi}}\|G_{k}(x)\|_{L^{1}(I)},\quad\|G_{k}(x)\|_{L^{1}(I)}\leq
2\pi\|G_{k}(x)\|_{C(I)}$ (4.22)

hold.

Similarly to the whole real line case, we define

${\cal N}_{a,\ b,\
k}:=\hbox{max}\Big{\\{}\Big{\|}\frac{G_{k,n}}{n^{2}-a_{k}-ib_{k}n}\Big{\|}_{l^{\infty}},\quad\Big{\|}\frac{n^{2}G_{k,n}}{n^{2}-a_{k}-ib_{k}n}\Big{\|}_{l^{\infty}}\Big{\\}},\quad
1\leq k\leq K,$ (4.23) ${\cal M}_{a,\
k}:=\hbox{max}\Big{\\{}\Big{\|}\frac{G_{k,n}}{n^{2}-a_{k}}\Big{\|}_{l^{\infty}},\quad\Big{\|}\frac{n^{2}G_{k,n}}{n^{2}-a_{k}}\Big{\|}_{l^{\infty}}\Big{\\}},\quad
K+1\leq k\leq N$ (4.24)

with the constants $a_{k}\geq 0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ and
$K\geq 2,\ N\geq 5$. Let ${\cal N}_{0,\ b,\ k}$ stand for (4.23) when
$a_{k}=0$ and ${\cal M}_{0,\ k}$ denote (4.24) when $a_{k}$ is trivial. Under
the conditions of Lemma A2 below, expressions (4.23), (4.24) will be finite.
This will enable us to introduce

${\cal N}_{a,\ b}:=\hbox{max}_{1\leq k\leq K}{\cal N}_{a,\ b,\ k}<\infty,$
(4.25) ${\cal M}_{a}:=\hbox{max}_{K+1\leq k\leq N}{\cal M}_{a,\ k}<\infty$
(4.26)

and

${\cal Q}:=\hbox{max}\\{{\cal N}_{a,\ b},\ {\cal M}_{a}\\}.$ (4.27)

Our final technical proposition is as follows.

Lemma A2. Let $N\geq 5,\ K\geq 2,\ 1\leq k\leq N,\ 1\leq l\leq K-1,\ K+1\leq
r\leq q-1,\ r+1\leq q\leq N-1$, the integral kernels
$G_{k}(x):I\to{\mathbb{R}},\ G_{k}(x)\in C(I)$ and $G_{k}(0)=G_{k}(2\pi)$ for
all $1\leq k\leq N$.

a) Let $a_{k}>0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ for $1\leq k\leq l$.
Then ${\cal N}_{a,\ b,\ k}<\infty$.

b) Let $a_{k}=0,\ b_{k}\in{\mathbb{R}},\ b_{k}\neq 0$ for $l+1\leq k\leq K$.
Then ${\cal N}_{0,\ b,\ k}<\infty$ if and only if

$(G_{k}(x),1)_{L^{2}(I)}=0.$ (4.28)

c) Let $a_{k}>0$ and $a_{k}\neq n^{2},\ n\in{\mathbb{Z}}$ for $K+1\leq k\leq
r$. Then ${\cal M}_{a,\ k}<\infty$.

d) Let $a_{k}=n_{k}^{2},\ n_{k}\in{\mathbb{N}}$ for $r+1\leq k\leq q$. Then
${\cal M}_{a,\ k}<\infty$ if and only if

$\Big{(}G_{k}(x),\frac{e^{\pm in_{k}x}}{\sqrt{2\pi}}\Big{)}_{L^{2}(I)}=0$
(4.29)

holds.

e) Let $a_{k}=0$ for $q+1\leq k\leq N$. Then ${\cal M}_{0,\ k}<\infty$ if and
only if orthogonality condition (4.28) is valid.

Proof. In both cases a) and b) of the lemma
$\displaystyle{\frac{G_{k,n}}{n^{2}-a_{k}-ib_{k}n}\in l^{\infty}}$ yields
$\displaystyle{\frac{n^{2}G_{k,n}}{n^{2}-a_{k}-ib_{k}n}\in l^{\infty}}$.
Indeed, $\displaystyle{\frac{n^{2}G_{k,n}}{n^{2}-a_{k}-ib_{k}n}}$ can be
written as

$G_{k,n}+a_{k}\frac{G_{k,n}}{n^{2}-a_{k}-ib_{k}n}+ib_{k}\frac{nG_{k,n}}{n^{2}-a_{k}-ib_{k}n}.$
(4.30)

The first term in (4.30) is bounded by means of (4.22) under the given
conditions. The third term in (4.30) can be easily estimated from above in the
absolute value using (4.22) as well as

$|b_{k}|\frac{|n||G_{k,n}|}{\sqrt{(n^{2}-a_{k})^{2}+b_{k}^{2}n^{2}}}\leq|G_{k,n}|\leq\sqrt{2\pi}\|G_{k}(x)\|_{C(I)}<\infty.$

Thus, $\displaystyle{\frac{G_{k,n}}{n^{2}-a_{k}-ib_{k}n}\in l^{\infty}}$
implies the boundedness of
$\displaystyle{\frac{n^{2}G_{k,n}}{n^{2}-a_{k}-ib_{k}n}}$.

To treat the case a) of the lemma, we need to consider

$\frac{|G_{k,n}|}{\sqrt{(n^{2}-a_{k})^{2}+b_{k}^{2}n^{2}}}.$ (4.31)

Obviously, the denominator in fraction (4.31) can be estimated from below by a
positive constant. Let us apply (4.22) to the numerator in (4.31). Hence
${\cal N}_{a,\ b,\ k}<\infty$ in the situation a) of our lemma when $a_{k}>0$.

Let us establish the validity of the result of the part b) when $a_{k}$
vanishes. Evidently, $\displaystyle{\frac{G_{k,n}}{n^{2}-ib_{k}n}\in
l^{\infty}}$ if and only if $G_{k,0}=0$. This is equivalent to orthogonality
relation (4.28). In this case we easily obtain the upper bound

$\Bigg{|}\frac{G_{k,n}}{n^{2}-ib_{k}n}\Bigg{|}=\frac{|G_{k,n}|}{|n|\sqrt{n^{2}+b_{k}^{2}}}\leq\sqrt{2\pi}\frac{\|G_{k}(x)\|_{C(I)}}{|b_{k}|}<\infty$

using (4.22) along with our assumptions.

In the cases c), d) and e) of the lemma we can express

$\frac{n^{2}G_{k,n}}{n^{2}-a_{k}}=G_{k,n}+a_{k}\frac{G_{k,n}}{n^{2}-a_{k}}.$
(4.32)

By virtue of (4.22), the boundedness of
$\displaystyle{\frac{G_{k,n}}{n^{2}-a_{k}}}$ yields
$\displaystyle{\frac{n^{2}G_{k,n}}{n^{2}-a_{k}}\in l^{\infty}}$.

In the situation c) we have $\displaystyle{\frac{G_{k,n}}{n^{2}-a_{k}}\in
l^{\infty}}$ since such expression does not contain any singularities and the
result of our lemma is obvious.

In the part d) the quantity $\displaystyle{\frac{G_{k,n}}{n^{2}-n_{k}^{2}}\in
l^{\infty}}$ if and only if $G_{k,\pm n_{k}}=0$. This is equivalent to
orthogonality conditions (4.29).

In the case e) the expression $\displaystyle{\frac{G_{k,n}}{n^{2}}\in
l^{\infty}}$ if and only if $G_{k,0}=0$, which is equivalent to orthogonality
relation (4.28).

## Acknowledgement

V. V. is grateful to Israel Michael Sigal for the partial support by the NSERC
grant NA 7901.

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# On the apparent molar volume of methanol in water-methanol mixtures.
Composition and temperature effects from molecular dynamics study

M. Cruz Sanchez, M. Aguilar, O. Pizio Corresponding author<EMAIL_ADDRESS>
(Received August 4, 2020, in final form August 21, 2020)

###### Abstract

We explore the effects of composition and temperature on the apparent molar
volumes of species of water-methanol mixtures. Isothermal-isobaric molecular
dynamics simulations are used with this purpose. Several combinations of
models for water and for methanol are explored. Validity of predictions
concerned with a puzzling minimum of apparent molar volume of methanol in
water-rich solutions is tested against experimental results.

Key words: water-methanol mixtures, excess mixing volume, apparent molar
volume, molecular dynamics simulations

###### Abstract

Ìè äîñëäæóìî âïëèâ ñêëàäó òåìïåðàòóðè íà âèäèì ìîëÿðí îá’ìè ñîðòâ ñóìøåé âîäà-
ìåòàíîë. Äëÿ öüîãî âèêîðèñòàíî ñèìóëÿö¿ ìåòîäîì ìîëåêóëÿðíî¿ äèíàìêè â
çîòåðìíî-çîáàðèíîìó àíñàìáë. Äîñëäæåíî äåêëüêà êîìáíàöé ìîäåëåé âîäè ìåòàíîëó.
ßêñòü ïåðåäáàåíü, ïîâ’ÿçàíèõ ç çàãàäêîþ ìíìóìó âèäèìîãî ìîëÿðíîãî îá’ìó
ìåòàíîëó ó âîäíèõ ðîçèíàõ, ïåðåâðÿòüñÿ ïîðâíÿííÿì ç åêñïåðèìåíòàëüíèìè
ðåçóëüòàòàìè.

Ключов слова: ñóìø âîäà-ìåòàíîë, íàäëèøêîâèé îá’ì çìøóâàííÿ, âèäèìèé ìîëÿðíèé
îá’ì, ñèìóëÿö¿ ìîëåêóëÿðíî¿ äèíàìêè

Experimental studies of aqueous solutions of alcohols and other co-solvents
have long history and continue nowadays. Early research is frequently
attributed to Mendeleev [1]. Concerning particular systems, the water-methanol
liquid mixtures seemingly are the most studied in various aspects. One of the
principal properties measured and discussed in several works is the behaviour
of density on composition, temperature and pressure. Namely, the density of
water-methanol mixtures in the entire composition interval at normal pressure
and various temperatures (e.g. 278.15 K, 288.15 K, 298.15 K, 313.15 K and
323.15 K) has been reported in [2, 3, 4, 5, 6]. Equivalently, the mixing
properties of two components, methanol and water, can be expressed in terms of
the excess mixing volume, $\Delta
V_{\text{mix}}=V_{\text{mix}}-(1-X_{\text{m}})V_{\text{w}}-X_{\text{m}}V_{\text{m}}$,
where $V_{\text{mix}}$, $V_{\text{w}}$ and $V_{\text{m}}$ refer to the molar
volumes of liquid solution and its components, $X_{\text{m}}$ denotes the
molar fraction of methanol. This definition can be rewritten straightforwardly
in terms of the respective densities, see e.g. equation (2) of [7]. It is
observed that the excess mixing volume is negative in the entire composition
interval and exhibits maximum deviation from ideality around equimolar
composition, at $X_{\text{m}}\approx$ 0.5.

Moreover, the experimental data can be elaborated in terms of apparent molar
volume for each species [7], $V_{\phi}^{(\text{w})}=V_{\text{w}}-\Delta
V_{\text{mix}}/(1-X_{\text{m}})$ and $V_{\phi}^{(m)}=V_{m}-\Delta
V_{\text{mix}}/X_{\text{m}}$. While, $V_{\phi}^{(\text{w})}$ takes linearly
descending values with increasing methanol fraction, the behaviour of
$V_{\phi}^{(\text{m})}$ is peculiar. Namely, if $X_{\text{m}}$ increases
starting from very small values (i.e. from the limit of water-rich solution),
$V_{\phi}^{(\text{m})}$ falls down and passes through a minimum value, next it
grows up with further growth of $X_{\text{m}}$. The minimum value of
$V_{\phi}^{(\text{m})}$ occurs at $X_{\text{m}}\approx$ 0.12 for the mixture
at room temperature, 298.15 K. Similar observations in terms of contraction of
the volume of water-methanol mixtures on composition have been discussed
recently in [8, 9].

Peculiar points of the behaviour of properties of water-methanol mixtures
along composition axes and at different temperatures are also drawn as a
logical conclusion from the analyses of light scattering experimental data,
see e.g. [10, 11]. A qualitatively similar picture involving the apparent
molar volume behaviour with trends of light scattering experimental data has
been discussed for water-DMSO solutions in [12]. It is quite clear that the
peculiarities of mixing of species in which one component consists of
molecules with hydrophilic and hydrophobic parts (besides, both species are
able to form hydrogen bonds) result from changes of intermolecular structure
upon changing composition and temperature.

After these preliminaries, we would like to establish the principal objective
of the present brief report. Specifically, our focus is to make clear if
computer simulations of models for water and methanol are able to describe
fine details of some properties observable from experimental data of this type
of mixtures. Previously, there have been very many computer simulation study
of water-methanol mixtures. For the sake of convenience of the reader we refer
to the list of references in our recent works on these mixtures [13, 14].

Common statement of the problem includes the choice of the model for water and
for methanol complemented by using certain combination rule for the crossed
interactions. Usually, the Lorentz-Berthelot (LB) combination rules (CR2 in
the Gromacs nomenclature) or the CR3 geometric rules are applied.

Alternative strategy for a single-component fluid or a mixture does not rely
on the combination rules. In this framework, it is assumed that the parameters
for the cross-interactions of the given functional form follow from the
fitting procedure to yield the target properties known for example, from the
experiment. The choice of target properties is not simply the question of
taste, actually it depends on the focus of specific research and determines
range of applicability of the designed model. Different realizations of this
type of procedure have been presented in [15, 16, 17, 18, 19, 20, 21, 22].

Figure 1: (Colour online) Panel a: Excess mixing volume of water-methanol
mixtures on methanol molar fraction for different combinations of models
together with the experimental data (blue squares). The nomenclature of
symbols is the following: TIP4P-2005-L1 (black circles), TIP4P-2005-TraPPE
(red circles), TIP4P-2005-OPLS (magenta circles), SPC-E-TraPPE (red
triangles), TIP4P-2005-OPLS-2016 (green hollow circles) and
TIP4P-2005-OPLS-2016-F (fitting with parameters from [19], green diamonds).
Panel b: Excess mixing enthalpy of water-methanol mixtures on methanol molar
fraction for different combinations of models together with the experimental
data (blue squares). The nomenclature of lines and symbols as in panel a.

The protocol of simulation of the systems with three thousand of particles
using Gromacs software (version 5.1) under the isobaric-isothermal conditions
of the present work is the same as in our previous study [13]. We restrict our
attention to nonpolarizable models only. The results of simulations refer to
the TIP4P-2005 [23] and the SPC-E [24] water models. It is known that the the
TIP4P-2005 version provides reasonable performance of the microscopic
structure of water [25] and a quite good description of various properties, in
comparison to other water models, see e.g. table 2 of [26].

On the other hand, a set of united type methanol models has been explored.
Specifically, we consider three models conveniently described in table I of
[18]: L1 [27], OPLS [28] and OPLS-2016 [18]. Besides, the TraPPE methanol
model [29] is incorporated. For all these cases standard strategy is applied.
If the OPLS-type modelling is involved, then the CR3 rules are used.
Otherwise, the LB rules determining the cross interactions are applied. On the
other hand, it appeared that the parametrized version of water-methanol
mixture [19] is helpful for the purposes of our study, as explained below.

A set of results concerning the excess mixing volume of the models of the
present study at normal pressure, 1 bar, and at room temperature, 298.15 K, is
given in figure 1 a. Formally, all the models under exploration yield a
qualitatively correct result describing the presence of a minimum of $\Delta
V_{\text{mix}}$, in accordance with the experimental data. However, the
TIP4P-2005 water combined with L1, TraPPE and OPLS methanol models all provide
the minimum at a correct composition value, $X_{\text{m}}\approx$ 0.5, and
predict $\Delta V_{\text{mix}}$ value at minimum reasonably well. The best
predictions are in fact provided by the TIP4P-2005/L1 and TIP4P-2005/TraPPE
models. In contrast, the SPC-E/TraPPE potential set underestimates the
magnitude of the $\Delta V_{\text{mix}}$ at minimum. The curve coming from the
TIP4P-2005/OPLS-2016 model exhibits very pronounced deviation from the
experimental values in the entire composition interval, in spite of the fact
that the models for each individual species apparently are the best. In our
opinion, the reason of such discrepancy lies in the apparent “conflict”
between the combination rules and the methodology of the design of the
OPLS-2016 model for methanol. Namely, the parametrization of the OPLS-2016
model has been performed taking the fluid-solid equilibrium as a target [18].
It has been commented that better global performance of the methanol modelling
is expected from the inclusion of the so-called polarization term [18],
similar observation has been advertised in [30] while considering water-
methanol mixture.

Having in mind unsatisfactory predictions of the TIP4P-2005/OPLS-2016
modelling for $\Delta V_{\text{mix}}$ within standard procedure, it has been
proposed to apply fitting, according to the alternative procedure with the
excess mixing enthalpy and the excess mixing volume as target properties. In
the case of the excess mixing enthalpy, $\Delta H_{\text{mix}}$ (defined as
$\Delta
H_{\text{mix}}=H_{\text{mix}}-(1-X_{\text{m}})H_{\text{w}}-X_{\text{m}}H_{\text{m}}$),
fitting leads to a curve perfectly coinciding with experimental data in the
entire interval of $X_{\text{m}}$, figure 1 b, in contrast to wrong
predictions within standard procedure. On the other hand, the excess mixing
volume, $\Delta V_{\text{mix}}$, after fitting, see the green line with solid
diamonds in figure 1 a, still deviates from the experimental data predicting
minimum at a lower $X_{\text{m}}$ than required. Rather accurate description
of $\Delta V_{\text{mix}}(X_{\text{m}})$ has been obtained within different,
microscopic structure demanding, fitting procedure in [17]. Unfortunately, the
excess mixing enthalpy has not been reported. Final comment concerning figure
1 b is that the TIP4P-2005, if combined with either L1, TraPPE or OPLS within
standard procedure, is of superior quality comparing to SPC-E/TraPPE, but does
not describe well the magnitude of $\Delta H_{\text{mix}}$ at minimum and its
position along composition axes. In close similarity to these developments,
Nezbeda and co-workers [15, 16] have explored $\Delta
V_{\text{mix}}(X_{\text{m}})$ and $\Delta H_{\text{mix}}(X_{\text{m}})$ at
room temperature applying adjustment of the parameters for the cross
interactions and obtained certain improvement of the results, in comparison to
standard modelling with LB combination rules. However, in summary one can
conclude that the energetic and geometric aspects of mixing of species are
intrinsically coupled and it is difficult to describe both of them perfectly
well simultaneously within nonpolarizable modelling.

Figure 2: (Colour online) Apparent molar volumes of methanol and of water
(panels a and b, respectively) for water-methanol mixtures on methanol molar
fraction for different combinations of models together with the experimental
data (blue squares). The nomenclature of lines and symbols as in figure 1.

It is worth examining the available results for $\Delta V_{\text{mix}}$ more
rigorously expecting to find its possible peculiar behaviour on $X_{\text{m}}$
at a low methanol mole fraction. For this purpose, we have evaluated the
apparent molar volume for each component, defined at the beginning of this
brief report, of the water-methanol mixture in the entire composition
interval. A set of results following from the application of different models
is given in two panels of figure 2. Namely, in figure 2 a, the results for
$V_{\phi}^{(\text{m})}$ are shown for the models in which TIP4P-2005 water is
combined with L1, TraPPE and OPLS methanol. The experimental data are given as
well. We observe that all three combined models capture the presence of a
minimum of the apparent molar volume of methanol at a low value of
$X_{\text{m}}$ in agreement with experimental data. Concerning the position of
the minimum along $X_{\text{m}}$ axes, the simulations predict slightly lower
value for $X_{\text{m}}$ comparing to the experiment. The value for
$V_{\phi}^{(\text{m})}$ at minimum is predicted by the TIP4P-2005/TraPPE model
best. For this particular model, the entire curve describing
$V_{\phi}^{(\text{m})}(X_{\text{m}})$ is in very good agreement with
experimental results. The shape of the curves coming from TIP4P-2005/L1 and
from TIP4P-2005/OPLS is satisfactory as well, but the absolute values for
$V_{\phi}^{(\text{m})}(X_{\text{m}})$ are less accurate because the results
for pure methanol within these two models slightly deviate from the
experiment, in contrast to TIP4P-2005/TraPPE. Interestingly, the combined
model TIP4P-2005/OPLS-2016 does not yield correct behaviour of
$V_{\phi}^{(\text{m})}(X_{\text{m}})$ at low values for $X_{\text{m}}$ in the
framework of modelling using the combination rules. If one applies the
parametrization according to [19], then the minimum at low $X_{\text{m}}$ is
captured, but the absolute values for $V_{\phi}^{(\text{m})}$ are less
accurate, in comparison to other models mentioned above. According to the
authors of recent [17] concerned with the parametrized version of the original
BJH and PHH flexible models for water and methanol, respectively [31, 32], the
minimum of the partial volume of methanol at low concentration has been
reproduced in that work for the first time. To be fair, it is worth mentioning
that this minimum has been obtained in [15], by using a particular set of
parameters for the cross interactions in the TIP4P/OPLS model. Nevertheless,
the $X_{\text{m}}$ value, and $V_{\phi}^{(\text{m})}$ at minimum in both
contributions [15, 17], are less accurately predicted, in comparison to the
TIP4P-2005/TraPPE modelling of the present report.

Water, in contrast to methanol, does not exhibit peculiar behaviour on the
methanol mole fraction in water-methanol mixtures in terms of
$V_{\phi}^{(\text{w})}$. The relevant results are shown in figure 2 b. Here,
it is important to note that the TIP4P-2005, combined with either L1, TraPPE
or OPLS with combination rules describes the $V_{\phi}^{(\text{w})}$ quite
accurately in the entire composition interval. On the other hand, the
TIP4P-2005/OPLS-2016 model is not adequate in this aspect. Solely
parametrization of the cross interactions leads to certain improvement of the
results. The curves for $V_{\phi}^{(\text{w})}(X_{\text{m}})$, reported in
[16, 17], are satisfactory with and without respective parametrizations.

Figure 3: (Colour online) Temperature dependence of the apparent molar volume
of methanol for water-methanol mixtures (panel a: TIP4P-2005-TraPPE model;
panel b: SPC-E-TraPPE model) on methanol molar fraction for different
temperatures, in comparison to the experimental data (blue squares). The
nomenclature of lines is given in the figure 1.

Finally, we would like to comment on the temperature trends for the observed
peculiar behaviour. The experimental data for density of water-methanol
mixtures have been reported in the temperature interval from 278.15 K to
323.15 K. The methanol molecule contains hydrophobic group, CH3, besides the
OH entity able to participate in the methanol-methanol and methanol-water
hydrogen bonding. Thus, one should expect changes in the balance between
hydrophilic and albeit weak hydrophobic effects. The former would be expected
to weaken as temperature increases while the latter formally could possibly
drive hydrophobic aggregation, as it has been discussed, for example, for
water-DMSO mixtures [33]. As it follows, from the experimental data for
density, the minimum of the apparent molar volume of methanol,
$V_{\phi}^{(\text{m})}$, becomes less pronounced with increasing temperature
from 278.15 K to 323.15 K, just a flat fragment is observed at low values of
$X_{\text{m}}$ at the highest temperature studied, figure 3 a. The data from
simulations of TIP4P-2005/TraPPE model follow experimental trends. Namely, the
magnitude of $V_{\phi}^{(\text{m})}$ increases at each fixed composition, if
the temperature increases, i.e., if the system becomes less non-ideal. This
behaviour is characteristic for the entire composition interval. The minimum
for the $V_{\phi}^{(\text{m})}$ from simulations is less pronounced at 323.15
K comparing to the lowest temperature in question, 278.15 K. Moreover, the
minimum slightly shifts to a lower value of $X_{\text{m}}$ upon increasing
temperature, similar trend can be deduced from the available experimental
data. On the other hand, accuracy of computer simulation predictions
deteriorates, if the temperature increases. Still, the predictions from the
TIP4P-2005/TraPPE model are quite satisfactory. General trends of behaviour of
$V_{\phi}^{(\text{m})}$ with temperature are predicted by the SPC-E/TraPPE
water-methanol model, figure 3 b. However, this kind of modelling has serious
deficiency — peculiarity of the $V_{\phi}^{(\text{m})}$ at low $X_{\text{m}}$
is not captured at all, at two temperatures studied. Possibly, one needs to
resort to a parametrization of crossed interactions similar to [19] as we
discussed above. The dependencies of $V_{\phi}^{(\text{w})}$ on $X_{\text{m}}$
are weakly sensitive to temperature, all of them behave as declining lines for
different models, cf. figure 2 b.

To conclude, we have performed a wide set of computer simulations for
different models of water-methanol mixtures and analyzed their feasibility to
capture peculiarity of mixing of species in water-rich region. We have
resorted to a standard procedure of modelling by picking up model for each
component and apply the combination rules. It has been established that the
TIP4P-2005 water model combined with either L1 or TraPPE or OPLS models of
methanol reproduce experimentally observed minimum for the apparent molar
volume of methanol at a low methanol mole fraction. The best performance is
observed while using the TIP4P-2005/TraPPE model. Almost perfect description
of this peculiar behaviour is due to a very good description of the slope of
the mixing volume following from correct evolution of the density of the
system. Moreover, analyses of the evolution of model structure, in terms of
the pair distribution functions, coordination numbers and the numbers of
hydrogen bonds, evidence a progressive mixing of components with increasing
$X_{\text{m}}$ and do not indicate any segregation trends similar to the model
studied in [17] (the corresponding figures are not presented due to the format
of this communication). Interestingly, one can attempt to investigate partial
properties related to the enthalpy of mixing. The results of the present study
permit to choose adequate models with the aim of exploration of properties
coming from vibrational spectroscopy experiments and of refractive index, for
example.

## Acknowledgements

M. Cruz is grateful to CONACyT for support of the Ph.D. studies.

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Ukrainian 0

[Water-methanol mixtures] Âèäèìèé ìîëÿðíèé îá’ì ìåòàíîëó â ñóìøàõ âîäà-
ìåòàíîë. Äîñëäæåííÿ âïëèâó ñêëàäó òåìïåðàòóðè ìåòîäîì ìîëåêóëÿðíî¿ äèíàìêè

M. Êðóç Ñàíåñ, M. Àãëÿð, O. Ïçî

Iíñòèòóò õiìi¿, Íàöiîíàëüíèé àâòîíîìíèé óíiâåðñèòåò ì. Ìåõiêî, Ìåõiêî, Ìåêñèêà