*Asymptotic Behavior by Krasnoselskii Fixed Point Theorem for Nonlinear Neutral… DOI: http://dx.doi.org/10.5772/intechopen.96040*

and

$$\begin{split} (\mathcal{B}\boldsymbol{\varrho})(t) &:= \left( \boldsymbol{\nu}(t\_{0}) - \frac{p(t\_{0} - \tau\_{1}(t\_{0}))}{p(t\_{0})} \frac{c(t\_{0})}{(1 - \tau\_{1}'(t\_{0}))} \boldsymbol{\nu}(t\_{0} - \tau\_{1}(t\_{0})) \right. \\ &\left. - \int\_{t\_{0} - \tau\_{1}(t\_{0})}^{t\_{0}} \left( h(u) - \frac{p'(u)}{p(u)} \right) \boldsymbol{\rho}(u) du \right) \boldsymbol{\varepsilon}^{-\int\_{t\_{0}}^{t} h(s) ds} \\ &+ \frac{p(t - \tau\_{1}(t))}{p(t)} \frac{c(t)}{1 - \tau\_{1}'(t)} \boldsymbol{\rho}(t - \tau\_{1}(t)) + \int\_{t - \tau\_{1}(t)}^{t} \left( h(u) - \frac{p'(u)}{p(u)} \right) \boldsymbol{\rho}(u) du \\ &+ \int\_{t\_{0}}^{t} e^{-\int\_{t}^{t} h(u) du} \left\{ -\overline{\boldsymbol{\mu}}(s) + \left( h(s - \tau\_{1}(s)) - \frac{p'(s - \tau\_{1}(s))}{p(s - \tau\_{1}(s))} \right) \left( 1 - \tau\_{1}'(s) \right) - \overline{\boldsymbol{\mu}}(s) \right\} du \\ &\times \boldsymbol{\rho}(s - \tau\_{1}(s)) ds \\ &- \int\_{t\_{0}}^{t} e^{-\int\_{t}^{t} h(u) du} h(s) \left( \int\_{s - \tau\_{1}(s)}^{t} \left( h(u) - \frac{p'(u)}{p(u)} \right) \boldsymbol{\rho}(u) du \right) ds. \end{split} \tag{23}$$

If we are able to prove that *H* possesses a fixed point *φ* on the set *Xψ* , then *y t*ð Þ¼ , *t*0, *ψ φ*ð Þ*t* for *t*≥*t*0, *y t*ð Þ¼ , *t*0, *ψ ψ*ð Þ*t* on ½ � *m t*ð Þ0 , *t*0 , *y t*ð Þ , *t*0, *ψ* satisfies (13) when its derivative exists and j j *y t*ð Þ , *t*0, *ψ* < 1 for *t*≥*t*0*:* That A maps *Xψ* into itself can be deduced from condition (21).

For *α* ∈ð Þ 0, 1 , we choose *δ*> 0 such that

$$\begin{aligned} \left| \mathbf{1} + \frac{\left| p(t\_0 - \tau\_1(t\_0))}{p(t\_0)} \frac{c(t\_0)}{\left(\mathbf{1} - \tau\_1'(t\_0)\right)} \right|} \right| \\ + \int\_{t\_0 - \tau\_1(t\_0)}^{t\_0} \left| h(u) - \frac{p'(u)}{p(u)} \right| du \Big| e^{-\int\_{t\_0}^t h(s) ds} \delta + a \le 1. \end{aligned} \tag{24}$$

Let *ψ* : ½ �! *m t*ð Þ0 , *t*0 be a given continuous initial function with k k *ψ* <*δ:* Let *g* : ½ Þ! *m t*ð Þ0 , ∞ ½ Þ 1, ∞ be any strictly increasing and continuous function with *gmt* ð Þ¼ ð Þ0 1, *g s*ðÞ! ∞ as *s* ! ∞, such that

$$\begin{split} & \left| \frac{p(t - \tau\_1(t))}{p(t)} \frac{c(t)}{1 - \tau\_1'(t)} \right| + \int\_{t - \tau\_1(t)}^t \left| h(u) - \frac{p'(u)}{p(u)} \right| \mathbf{g}(u) / \mathbf{g}(t) du \\ & + \int\_{t 0}^t e^{-\int\_u^t h(u) du} |h(s)| \left( \int\_{s - \tau\_1(s)}^s \left| h(u) - \frac{p'(u)}{p(u)} \right| \mathbf{g}(u) / \mathbf{g}(t) du \right) ds \\ & + \int\_{t 0}^t e^{-\int\_u^t h(u) du} \left| -\overline{\mu}(s) + \left( h(s - \tau\_1(s)) - \frac{p'(s - \tau\_1(s))}{p(s - \tau\_1(s))} \right) \left( 1 - \tau\_1'(s) \right) - \overline{\rho}(s) \right| \\ & \times g(s - \tau\_1(s)) / \mathbf{g}(t) ds < a. \end{split} \tag{25}$$

Now we split the rest of our proof into three steps.

**First step:** We now show that *φ*, *ϕ* ∈*Xψ* implies that A*φ* þ B*ϕ*∈*Xψ :* Now, let k k*:* be the supremum norm on ½ Þ *m t*ð Þ0 , ∞ of *φ*∈*Xψ* if *φ* is bounded. Note that if *φ*, *ϕ*∈*Xψ* then

$$|(\mathcal{A}\rho)(t) + (\mathcal{B}\phi)(t)| \le \frac{\epsilon}{2}$$

$$\leq 2\int\_{t\_1}^{t\_2} \left| b(u) \frac{p^{\sigma}(u - \tau\_2(u))}{p(u)} \right| du + a \left| \int\_{t\_1}^{t\_2} h(u) du \right| \leq (2k\_1 + ak\_2)|t\_2 - t\_1|,$$

$$\begin{split} &|(\mathcal{A}\rho)(t) - (\mathcal{A}\phi)(t)| / \mathcal{g}(t) \\ &= (\mathbbm{1}/\mathcal{g}(t)) \left| \int\_{t\_{0}}^{t} e^{-\int\_{t}^{s} h(u) du} \Big| b(s) \frac{p^{\sigma}(s - \tau\_{2}(s))}{p(s)} \Big| |\rho^{\sigma}(s - \tau\_{2}(s)) - \phi^{\sigma}(s - \tau\_{2}(s))| ds \right| \\ &\leq (\mathbbm{1}/\mathcal{g}(t)) \left\{ \int\_{t\_{0}}^{T} e^{-\int\_{s}^{s} h(u) du} \Big| b(s) \frac{p^{\sigma}(s - \tau\_{2}(s))}{p(s)} \Big| |\rho^{\sigma}(s - \tau\_{2}(s)) - \phi^{\sigma}(s - \tau\_{2}(s))| ds \right\} \\ &+ 2 \int\_{T}^{t} e^{-\int\_{s}^{t} h(u) du} \Big| b(s) \frac{p^{\sigma}(s - \tau\_{2}(s))}{p(s)} \Big| ds \Big| \\ &\leq \{ (ae/2\mathfrak{g}(t)) + (2\alpha/g(T)) \} \leq ae. \end{split}$$

Since *α* ∈ð Þ 0, 1 , we can conclude that B is a contraction on *Xψ* , j j*: g* � �*:*

The conditions of Krasnoselskii's theorem are satisfied with M ¼ *Xψ* . Hence, we deduce that *H* : *Xψ* ! *Xψ* has a fixed point *y t*ð Þ, which is a solution of (13) with *y s*ðÞ¼ *ψ*ð Þ*s* on *s*∈ *m t*ð Þ0 , *t*0� and j j *y t*ð Þ , *t*0, *ψ* ≤ 1 for *t*∈½ Þ *m t*ð Þ0 , ∞ . Since there exists a bounded function *p* : ½ ½! *m t*ð Þ0 , ∞ ð Þ 0, ∞ with *p t*ðÞ¼ 1 for *t*∈ ½ � *m t*ð Þ0 , *t*0 , by hypotheses (12) and from the above arguments we deduce that there exists a solution *x* of (1) with *x* ¼ *ψ* on ½ � *m t*ð Þ0 , *t*0 satisfies j j *x t*ð Þ , *t*0, *ψ* ≤1 for all *t*∈½ Þ *m t*ð Þ0 , ∞ . The proof is complete.

by (28), *K* is well defined. Suppose that (29) holds.

ð*t t*0 *e* � Ð*t s*

exists a solution *x*∈ *Xψ* of (1) with *x t*ð Þ! , *t*0, *ψ* 0 as *t* ! ∞*:*

for all *n* ≥1*:* To simplify the expression, we define

*ω*ð Þ*s* ≔ �*μ*ðÞþ*s h s*ð Þ� � *τ*1ð Þ*s p*0

� � � �

*s*�*τ*1ð Þ*s*

ð*tn t*0 *e* Ð*s* 0

subsequence. Without loss of generality, we can assume that

lim*n*!∞ ð*tn t*0 *e* Ð*s* 0

for some *θ* ∈ þ*:* Let *m* be an integer such that

*t*0 *e* Ð*s* 0 �*J* ≤ ð*tn t*0

*h u*ð Þ� *p*0

ð*tn t*0 *e* � Ð*tn*

ð Þ *u p u*ð Þ

*h u*ð Þ*duω*ð Þ*s ds* ≤*αe*

� � � �

asymptotically stable.

For

replaced by the supremum norm k k*: :*

*DOI: http://dx.doi.org/10.5772/intechopen.96040*

j j ð Þ A*φ* ð Þ*t* ≤

where *q t*ðÞ! 0 as *t* ! ∞ by (27).

with *tn* ! ∞ as *n* ! ∞ such that lim*n*!∞

choose a positive constant *J* satisfying

� � � �

þj j *h s*ð Þ ð*s*

for all *s*≥ 0*:* By (21), we have

This yields

**37**

The sequence Ð*tn*

Since *p* is bounded, it remains to prove that the zero solution of (1) is

*Asymptotic Behavior by Krasnoselskii Fixed Point Theorem for Nonlinear Neutral…*

All of the calculations in the proof of Theorem 3.1 hold with *g t*ðÞ¼ 1 when j j*: g* is

*φ*∈*Xψ* ,

� � � �

*h u*ð Þ*du b s*ð Þ *pσ*ð Þ *s* � *τ*2ð Þ*s*

Add to*Xψ* the condition that *φ* ∈*Xψ* implies that *φ*ðÞ!*t* 0 as*t* ! ∞.We can see that for *φ*∈ *Xψ* then ð Þ A*φ* ðÞ!*t* 0 as*t* ! ∞ by (31), and ð Þ B*φ* ðÞ!*t* 0 as*t* ! ∞ by (29). Since A*Xψ* has been shown to be equicontinuous, A maps *Xψ* into a compact subset of *Xψ* . By Krasnoselskii's theorem, there is *y*∈*Xψ* with A*y* þ B*y* ¼ *y*. As *y*∈*Xψ* , *y t*ð Þ! , *t*0, *ψ* 0 as *t* ! ∞*:* By condition (12), it is very easy to show that there

Conversely, we suppose that (29) fails. From (28) there exists a sequence f g *tn*

*h u*ð Þ*du*≤ þ *J*,

ð Þ *s* � *τ*1ð Þ*s p s*ð Þ � *τ*1ð Þ*s* � � 1 � *τ*0

> � � � �

*s h u*ð Þ*duω*ð Þ*s ds*≤*α:*

Ð*tn*

*h u*ð Þ*duω*ð Þ*s ds* ¼ *θ*,

0 *h u*ð Þ*du* ≤*e*

*h u*ð Þ*duω*ð Þ*s ds* � � is bounded, hence there exists a convergent

*J :*

*du* þ *b s*ð Þ *pσ*ð Þ *s* � *τ*2ð Þ*s*

*p s*ð Þ

Ð*tn*

*p s*ð Þ

� � � �

*t*0 *h u*ð Þ*du* ¼ *ξ* for some *ξ*∈ þ*:* We may also

1ð Þ*s* � � � *β*ð Þ*s*

� � � � , � � � �

*ds*≕*q t*ð Þ, (31)

Letting *σ* ¼ 1*=*3, and *c t*ðÞ¼ 0 in Theorem 3.1. Then we have the following corollary.

**Corollary 3.1.** *Let* (19) *and* (20) *hold, and* (21) *be replaced by*

$$\begin{aligned} &\int\_{t-\tau\_1(t)}^t \left| h(u) - \frac{p'(u)}{p(u)} \right| du \\ &+ \int\_{t0}^t e^{-\int\_0^t h(u) du} \left\{ \left| \left( h(s-\tau\_1(s)) - \frac{p'(s-\tau\_1(s))}{p(s-\tau\_1(s))} \right) \left( 1 - \tau\_1'(s) \right) - a(s) \frac{p(s-\tau(s))}{p(s)} \right| \right\} ds \\ &+ \int\_{t0}^t e^{-\int\_0^t h(u) du} |h(s)| \left( \int\_{s-\tau\_1(s)}^t \left| h(u) - \frac{p'(u)}{p(u)} \right| du \right) ds \le a. \end{aligned} \tag{26}$$

*Then there is a solution x t*ð Þ , *t*0, *ψ of* (4) *on* þ *with x t* j j ð Þ , *t*0, *ψ* ≤1*:*

**Remark 3.2:** When *p t*ðÞ¼ 1, then Corollary 3.1 reduces to Theorem A, which was recently stated in Jin and Luo [14]. Therefore, the paper (Jin and Luo [14]) is a particular case of ours.

For the next Theorem, we manipulate function spaces defined on infinite *t* -intervals. So, for compactness, we need an extension of the Arzelà-Ascoli theorem. This extension is taken from ([3], Theorem 1.2.2 p. 20).

**Theorem 3.2**. *Let* (19)–(21) *hold and assume that*

$$\int\_{t\_0}^t e^{-\int\_{s}^t h(u) du} \left| b(s) \frac{p^\sigma(s - \tau\_2(s))}{p(s)} \right| ds \to 0 \text{ as } t \to \infty,\tag{27}$$

and

$$\lim\_{t \to \infty} \inf \int\_{t\_0}^t h(s)ds > -\infty. \tag{28}$$

*If ψ is given continuous initial function which is sufficiently small, then* (1) *has a solution x t*ð Þ! , *t*0, *ψ* 0 *as t* ! ∞ *if and only if*

$$\int\_{t\_0}^{t} h(s)ds \to \infty \text{ as } t \to \infty. \tag{29}$$

**Proof.** We set

$$K = \sup\_{t \ge t\_0} \left\{ e^{-\int\_{t\_0}^t h(s)ds} \right\},\tag{30}$$

*Asymptotic Behavior by Krasnoselskii Fixed Point Theorem for Nonlinear Neutral… DOI: http://dx.doi.org/10.5772/intechopen.96040*

by (28), *K* is well defined. Suppose that (29) holds.

Since *p* is bounded, it remains to prove that the zero solution of (1) is asymptotically stable.

All of the calculations in the proof of Theorem 3.1 hold with *g t*ðÞ¼ 1 when j j*: g* is replaced by the supremum norm k k*: :*

For

$$\wp \in X\_{\psi},$$

$$|(\mathcal{A}\wp)(t)| \le \int\_{t\_0}^t e^{-\int\_u^t h(u)du} \left| b(s) \frac{p^{\sigma}(s - \tau\_2(s))}{p(s)} \right| ds =: q(t),\tag{31}$$

where *q t*ðÞ! 0 as *t* ! ∞ by (27).

Add to*Xψ* the condition that *φ* ∈*Xψ* implies that *φ*ðÞ!*t* 0 as*t* ! ∞.We can see that for *φ*∈ *Xψ* then ð Þ A*φ* ðÞ!*t* 0 as*t* ! ∞ by (31), and ð Þ B*φ* ðÞ!*t* 0 as*t* ! ∞ by (29).

Since A*Xψ* has been shown to be equicontinuous, A maps *Xψ* into a compact subset of *Xψ* . By Krasnoselskii's theorem, there is *y*∈*Xψ* with A*y* þ B*y* ¼ *y*. As *y*∈*Xψ* , *y t*ð Þ! , *t*0, *ψ* 0 as *t* ! ∞*:* By condition (12), it is very easy to show that there exists a solution *x*∈ *Xψ* of (1) with *x t*ð Þ! , *t*0, *ψ* 0 as *t* ! ∞*:*

Conversely, we suppose that (29) fails. From (28) there exists a sequence f g *tn* with *tn* ! ∞ as *n* ! ∞ such that lim*n*!∞ Ð*tn t*0 *h u*ð Þ*du* ¼ *ξ* for some *ξ*∈ þ*:* We may also choose a positive constant *J* satisfying

$$-J \le \int\_{t\_0}^{t\_u} h(u) du \le \, +J,$$

for all *n* ≥1*:* To simplify the expression, we define

$$\begin{aligned} \rho(s) &:= \left| -\overline{\mu}(s) + \left( h(s - \tau\_1(s)) - \frac{p'(s - \tau\_1(s))}{p(s - \tau\_1(s))} \right) \left( 1 - \tau\_1'(s) \right) - \overline{\rho}(s) \right|, \\ &+ |h(s)| \int\_{s - \tau\_1(s)}^s \left| h(u) - \frac{p'(u)}{p(u)} \right| du + \left| b(s) \frac{p^{\sigma}(s - \tau\_2(s))}{p(s)} \right|, \end{aligned}$$

for all *s*≥ 0*:* By (21), we have

$$\int\_{t\_0}^{t\_u} e^{-\int\_{t}^{t\_u} h(u) du} o(s) ds \le a.$$

This yields

$$\int\_{t\_0}^{t\_n} e^{\int\_0^{\int\_0^{\tilde{h}(u)} du} o(s)ds} \le \alpha e^{\int\_0^{\epsilon\_n} h(u)du} \le e^{I} \dots$$

The sequence Ð*tn t*0 *e* Ð*s* 0 *h u*ð Þ*duω*ð Þ*s ds* � � is bounded, hence there exists a convergent subsequence. Without loss of generality, we can assume that

$$\lim\_{n \to \infty} \int\_{t\_0}^{t\_n} e^{\int\_0^t h(u) du} o(s) ds = \theta,$$

for some *θ* ∈ þ*:* Let *m* be an integer such that

$$\int\_{t\_m}^{t\_n} e^{\int\_0^t h(u) du} \alpha(s) ds \le \frac{\delta\_0}{4K},$$

ð*t t*0

*Asymptotic Behavior by Krasnoselskii Fixed Point Theorem for Nonlinear Neutral…*

order with two variable delays, which is a special case of (1):

0*:*6 0ð Þ *:*95*t* þ 1 1

for *t*≥0 where *τ*2ðÞ¼ *t* 0*:*5*t*, *τ*1ðÞ¼ *t* 0*:*05*t*, and *a t*ð Þ satisfies

ðÞ¼� *t a t*ð Þ*x t*ð Þþ � *τ*1ð Þ*t* ln 0*:*95*t* þ 1

ð Þ *t* þ 1 2 *x*

1 4 � 0*:*95

*h u*ð Þ� *p*0

*ds* ≤0*:*4, and since Ð*t*

However, for the asymptotic stable of the zero solution of (33), the

*α* ¼ 0*:*263 þ 0*:*026 þ 0*:*026 þ 0*:*2 þ 0*:*4. It is easy to see that all the conditions of Theorem 3.1 hold for *α*≃0*:*915<1*:* Thus, Theorem 3.1 implies that the zero solution

corresponding conditions used by the fixed point theory in Ardjouni and Djoudi [6]

3

1

ð Þ *t* � *τ*1ð Þ*t p t*ð Þ � *τ*1ð Þ*t* � � 1 � *τ*0

where *μ*ð Þ*t* and *β*ð Þ*t* are defined in (15) and (16), respectively. Choosing *h t*ðÞ¼

� � � �

ð Þ *u p u*ð Þ

� � � � *du* !*ds* ≤0*:*026,

> ð Þ *s* � *τ*1ð Þ*s p s*ð Þ � *τ*1ð Þ*s* � � 1 � *τ*0

. By straightforward computations, we can check that condition

≤0*:*263,

neutral differential equations than Eq. (1).

*DOI: http://dx.doi.org/10.5772/intechopen.96040*

*x*0

*t*þ1

*h u*ð Þ� *p*0

*h u*ð Þ*du*j j *h s*ð Þ ð*s*

� � � �

*h u*ð Þ*du*j j *b s*ð Þ *pσ* ð Þ *s*�*τ*2ð Þ*s*

of (33) is asymptotic stable.

� � �

*p s*ð Þ

*p t*ð Þ � *τ*1ð Þ*t p t*ð Þ

> � � � �

� � � �

*t*þ1 and *p t*ðÞ¼ 1

� � � �

ð*t*

ð*t* 0 *e* � Ð*t s*

ð*t* 0 *e* � Ð*t s*

≤ ð*t* 0 *e* � Ð*t s* 1*:*5 *u*þ1 *du* 0*:*3 *s* þ 1

and Ð*t* 0*e* � Ð*t s*

are

**39**

*t*�*τ*1ð Þ*t*

1*:*5

þ

�*μ*ðÞþ*t h t*ð Þ� � *τ*1ð Þ*t p*0

(21) in Theorem 3.1 holds true. As *t* ! ∞, we have

� � � � ≤

*s*�*τ*1ð Þ*s*

*h u*ð Þ*du* �*μ*ðÞþ*s h s*ð Þ� � *τ*1ð Þ*s p*0

*ds*≤ 0*:*2,

� � � � � � �

*du*≤0*:*026,

� � � �

*c t*ð Þ 1 � *τ*0 1ð Þ*t*

> ð Þ *u p u*ð Þ

� � � �

some *tρ* ∈ þ.

**4. Example**

*h s*ð Þ*ds* ! ∞ as *t* ! ∞*:*

**Remark 3.3.** The method in this paper can be applied to more general nonlinear

**Remark 3.4.** Theorem 3.1 is still true if condition (21) is satisfied for *t*≥*tρ* with

In this section, we now give an example to show the applicability of Theorem 3.1. **Example.** Let us consider the following neutral differential equation of first

> 4ð Þ *t* þ 1 � �*x*0

ð Þ *t* � *τ*1ð Þ*t*

� � � � ≤ 0*:*03 *t* þ 1 ,

1ð Þ*s* � � � *β*ð Þ*s*

0*h s*ð Þ*ds* ! ∞ as *t* ! ∞, *p t*ð Þ≤ 1*:* Let

� � � � *ds*

3ð Þ *t* � *τ*2ð Þ*t* , (33)

1ð Þ*t* � � � *β*ð Þ*t*

for all *n* ≥ *m*, where *δ*0 >0 satisfies 2*δ*0*KeJ* þ *α*≤1*:*

We now consider the solution *y t*ðÞ¼ *y t*ð Þ , *tm*, *ψ* of (1) with *ψ*ð Þ¼ *tm δ*0 and j j *ψ*ð Þ*s* ≤*δ*0 for *s*≤*tm:* We may choose *ψ* so that j j *y t*ð Þ ≤1 for *t* ≥*tm* and

$$\begin{aligned} \,\_2\psi(t\_m) - \frac{p(t\_m - \tau\_1(t\_m))}{p(t\_m)} \frac{c(t\_m)}{(1 - \tau\_1'(t\_m))} \,\_2\psi(t\_m - \tau\_1(t\_m)) \\ - \int\_{t\_m - \tau\_1(t\_m)}^{t\_m} \left( h(u) - \frac{p'(u)}{p(u)} \right) z(u) du \geq \frac{1}{2} \delta\_0. \end{aligned}$$

In follows from (22) and (23) with *y t*ðÞ¼ Að Þ*y* ð Þþ B *t* ð Þ*y* ð Þ*t* that for *n*≥ *m*

$$\begin{split} & \left| y(t\_n) - \frac{p(t\_n - \tau\_1(t\_n))}{p(t\_n)} \frac{c(t\_n)}{\left(1 - \tau\_1'(t\_n)\right)} y(t\_n - \tau\_1(t\_n)) - \int\_{t\_n - \tau\_1(t\_n)}^{t\_n} \left(h(s) - \frac{p'(s)}{p(s)}\right) y(s) ds \right| \\ & \geq \frac{1}{2} \delta\_0 e^{-\int\_{t\_n}^{t\_n} h(u) du} - \int\_{t\_n}^{t\_n} e^{-\int\_{t\_n}^{u\_n} h(u) du} a(s) ds \\ & = e^{-\int\_{t\_n}^{u\_n} h(u) du} \left(\frac{1}{2} \delta\_0 - e^{-\int\_{t\_n}^{u\_n} h(u) du} \int\_{t\_n}^{t\_n} e^{\int\_0^u (h(u) du} a(s) ds} \right) \\ & \geq e^{-\int\_{t\_n}^{u\_n} h(u) du} \left(\frac{1}{2} \delta\_0 - K \int\_{t\_n}^{t\_n} e^{\int\_0^u (h(u) du} a(s) ds} \right) \\ & \geq \frac{1}{4} \delta\_0 e^{-\int\_{t\_n}^{u\_n} h(u) du} \geq \frac{1}{4} \delta\_0 e^{-2\delta} > 0. \tag{32} \end{split} \tag{33}$$

On the other hand, if the zero solution of (13) *y t*ðÞ¼ *y t*ð Þ! , *tm*, *ψ* 0 as *t* ! ∞, since *tn* � *τi*ð Þ! *tn* ∞ as *t* ! ∞, *i* ¼ 1, 2, and (21) holds, we have

$$y(t\_n) - \frac{p(t\_n - \tau\_1(t\_n))}{p(t\_n)} \frac{c(t\_n)}{\left(1 - \tau\_1'(t\_n)\right)} y(t\_n - \tau\_1(t\_n)) - \int\_{t\_n - \tau\_1(t\_n)}^{t\_n} \left(h(s) - \frac{p'(s)}{p(s)}\right) y(s)ds \to 0$$

as *t* ! ∞, which contradicts (32). Hence condition (29) is necessary for the asymptotic stability of the zero solution of (13), and hence the zero solution of (1) is asymptotically stable. The proof is complete.

For the special case *c t*ðÞ¼ 0 and *σ* ¼ 1 3 , we can get. **Corollary 3.2.** *Let* (19), (20) *and* (27) *hold and* (21) *be replaced by*

$$\begin{split} &\int\_{t-\tau\_{1}(\varepsilon)}^{t} \left| h(u) - \frac{p'(u)}{p(u)} \right| du \\ &+ \int\_{t0}^{t} e^{-\int\_{u}^{t} h(u) du} \left\{ \left| \left( h(s-\tau\_{1}(s)) - \frac{p'(s-\tau\_{1}(s))}{p(s-\tau\_{1}(s))} \right) (1-\tau\_{1}'(s)) - a(s) \frac{p(s-\tau(s))}{p(s)} \right| \right\} ds \\ &+ \int\_{t0}^{t} e^{-\int\_{u}^{t} h(u) du} |h(s)| \left( \int\_{s-\tau\_{1}(s)}^{s} \left| h(u) - \frac{p'(u)}{p(u)} \right| du \right) ds \leq a. \end{split}$$

*Then the zero solution x t*ð Þ , *t*0, *ψ of* (4) *with a small continuous function ψ*ð Þ*t is asymptotically stable if only if*

*Asymptotic Behavior by Krasnoselskii Fixed Point Theorem for Nonlinear Neutral… DOI: http://dx.doi.org/10.5772/intechopen.96040*

$$\int\_{t\_0}^{t} h(s)ds \to \infty \text{ as } t \to \infty.$$

**Remark 3.3.** The method in this paper can be applied to more general nonlinear neutral differential equations than Eq. (1).

**Remark 3.4.** Theorem 3.1 is still true if condition (21) is satisfied for *t*≥*tρ* with some *tρ* ∈ þ.

#### **4. Example**

In this section, we now give an example to show the applicability of Theorem 3.1. **Example.** Let us consider the following neutral differential equation of first order with two variable delays, which is a special case of (1):

$$\begin{split} \mathbf{x}'(t) &= -a(t)\mathbf{x}(t - \tau\_1(t)) + \ln \left( \frac{\mathbf{0.95t} + 1}{4(t+1)} \right) \mathbf{x}'(t - \tau\_1(t)) \\ &+ \frac{\mathbf{0.6}(\mathbf{0.95t} + 1)^{\frac{1}{3}}}{\left(t + 1\right)^2} \mathbf{x}^{\frac{1}{3}}(t - \tau\_2(t)), \end{split} \tag{33}$$

for *t*≥0 where *τ*2ðÞ¼ *t* 0*:*5*t*, *τ*1ðÞ¼ *t* 0*:*05*t*, and *a t*ð Þ satisfies

$$\left| -\overline{\mu}(t) + \left( h(t - \tau\_1(t)) - \frac{p'(t - \tau\_1(t))}{p(t - \tau\_1(t))} \right) \left( 1 - \tau\_1'(t) \right) - \overline{\beta}(t) \right| \le \frac{0.03}{t + 1},$$

where *μ*ð Þ*t* and *β*ð Þ*t* are defined in (15) and (16), respectively. Choosing *h t*ðÞ¼ 1*:*5 *t*þ1 and *p t*ðÞ¼ 1 *t*þ1 . By straightforward computations, we can check that condition (21) in Theorem 3.1 holds true. As *t* ! ∞, we have

$$\begin{aligned} &\left|\frac{p(t-\tau\_1(t))}{p(t)}\frac{c(t)}{1-\tau\_1'(t)}\right| \leq \left|\frac{1}{4\times0.95}\right| \leq 0.263, \\ &\int\_{t-\tau\_1(t)}^t \left|h(u) - \frac{p'(u)}{p(u)}\right| du \leq 0.026, \\ &\int\_0^t e^{-\int\_u^t h(u)du} |h(s)| \left(\int\_{s-\tau\_1(s)}^t \left|h(u) - \frac{p'(u)}{p(u)}\right| du\right) ds \leq 0.026, \\ &\int\_0^t e^{-\int\_u^t h(u)du} \left|-\overline{\mu}(s) + \left(h(s-\tau\_1(s)) - \frac{p'(s-\tau\_1(s))}{p(s-\tau\_1(s))}\right) \left(1-\tau\_1'(s)\right) - \overline{\beta}(s)\right| ds \\ \leq \int\_0^t e^{-\int\_{s+\tau}^{1.5} du} \frac{0.3}{s+1} ds \leq 0.2, \end{aligned}$$

and

Ð*t* 0*e* � Ð*t s h u*ð Þ*du*j j *b s*ð Þ *pσ* ð Þ *s*�*τ*2ð Þ*s p s*ð Þ � � � � � � *ds* ≤0*:*4, and since Ð*t* 0*h s*ð Þ*ds* ! ∞ as *t* ! ∞, *p t*ð Þ≤ 1*:* Let *α* ¼ 0*:*263 þ 0*:*026 þ 0*:*026 þ 0*:*2 þ 0*:*4. It is easy to see that all the conditions of Theorem 3.1 hold for *α*≃0*:*915<1*:* Thus, Theorem 3.1 implies that the zero solution of (33) is asymptotic stable.

However, for the asymptotic stable of the zero solution of (33), the corresponding conditions used by the fixed point theory in Ardjouni and Djoudi [6] are

$$\lim \left| \frac{c(t)}{1 - \tau\_1'(t)} \right| = \lim \left| \frac{1}{0.95} \ln \left( \frac{0.95t + 1}{4(t+1)} \right) \right| = 1.513 \text{ast} \to \infty.$$

**References**

(2003), 15–32.

York, (2006).

17–30.

328–339.

**41**

(2012), no. 1, 5–19.

[1] T. A. Burton, Stability by fixed point

*DOI: http://dx.doi.org/10.5772/intechopen.96040*

[10] T. A. Burton, T. Furumochi, Krasnoselskii's fixed point theorem and stability, Nonlinear Anal. 49 (2002),

[11] G. Chen, D. Li, O. Gaans, S. Lunel, Stability of nonlinear neutral delay differential equations with variable delays, Electronic Journal of Differential Equations, Vol. 2017 (2017), no. 118,

[12] Y. M. Dib, M. R. Maroun, Y. N. Raffoul, Periodicity a stability in neutral nonlinear differential equations with functional delay. Electronic Journal of Differential Equations, Vol. 2005 (2005), no. 142, 1072–6691.

[13] M. N. Islam and Y. N. Raffoul, Periodic solutions of neutral nonlinear system of differentiale quations with functional delay, J. Math. Anal. Appl,

[14] C. H. Jin, J.W. Luo, Stability in functional differential equations established using fixed point theory, Nonlinear Anal. 68 (2008), 3307–3315.

[15] E. R. Kaufman and Y. N. Raffoul, Periodic solutions for a neutral

nonlinear dynamical equation on a time scale, J. Math. Anal. Appl, 319 (2006),

[16] J. W. Luo, Fixed points and stability of neutral stochastic delay differential equations, J. Math. Anal. Appl., 334

[17] J. W. Luo and T. Taniguchi, Fixed points and stability of stochastic neutral partial differential equations with infinite delays, Stoch. Anal. Appl., 27

[18] B. Lisena, Global attractivity in nonautonomous logistic equations with delay. Nonlinear Analysis: RWA, 9

331 (2007), 1175–1186.

315–325.

(2007), 431–440.

(2009), 1163–1173.

(2008), 53–63.

445–454.

*Asymptotic Behavior by Krasnoselskii Fixed Point Theorem for Nonlinear Neutral…*

1–14.

[2] T. A. Burton, Liapunov functionals,

[3] T. A. Burton, Stability by fixed point theory for functional differential equations, Dover Publications, New

[4] Ardjouni A, Djoudi A. Fixed points and stability in linear neutral differential equations with variable delays. Nonlinear

[5] A. Ardjouni, A. Djoudi; Stability by Krasnoselskii fixed point theorem for neutral differential equations with variable delays, Acta Universitatis Apulensis Mathematica, (2012), no. 32,

[6] A. Ardjouni, A. Djoudi; Fixed points and stability in neutral nonlinear differential equations with variable delays, Opuscula Mathematica 32

[7] M. Ad var, H. C. Koyuncuoğlu and Y.

[8] S. Althubiti, H.A. Makhzoum, Y. N. Raffoul, Periodic solution and stability in nonlinear neutral system with infinite delay, Applied Mathematical Sciences, Vol. 7, (2013), no. 136, 6749–6764.

[9] T. A. Burton, T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynam. Systems

Appl. 11 (2002), 499–519.

N. Raffoul, Existence of periodic solutions in shifts *δ* for neutral nonlinear dynamic systems on time scales, Appl. Math. Comput. 242 (2014),

theory or Liapunov's theory: A comparison, Fixed Point Theory 4

fixed points, and stability by Krasnoselskii's theorem, Nonlinear

Studies 9 (2001), 181–190.

Analysis. 2011;**74**:2062-2070

This implies that condition (11) does not hold. So it is clear that the reduction of the conservatism by our method is quite significant when compared to Ardjouni and Djoudi [6].

**Remark 4.1**. It is an open problem whether the zero solution of (1) is uniform asymptotically stable, perseverance, and so on.

#### **5. Conclusion**

This work is a new attempt at applying the fixed point theory to the stability analysis of neutral differential equations with variable delays, several special cases of which have been studied in [9, 14]. Some of the results, like Theorem B, is mainly dependent on the constraint

$$\left| \frac{c(t)}{1 - \tau\_1'(t)} \right| < 1.$$

But in many environments, the constraint does not hold. So by employing two auxiliary continuous functions *g* and *p* to define an appropriate mapping, and present new criteria for asymptotic stability of Eq. (1) which makes stability conditions more feasible and the results in [14] are improved and generalized. From what has been discussed above, we see that Krasnoselskii's fixed point theorem is effective for not only the investigation of the existence of solution but also for the boundedness and the stability analysis of trivial equilibrium. We introduce an example to verify the applicability of the results established. In the future, we will continue to explore the application of other kinds of fixed point theorems to the stability research of fractional neutral systems with variable delays.

#### **Additional classifications**

**AMS Subject Classifications**: 34K20, 34K30, 34B40

#### **Author details**

Benhadri Mimia LAMAHIS Lab, Faculty of Sciences, Department of Mathematics, University of Skikda, Skikda, Algeria

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Asymptotic Behavior by Krasnoselskii Fixed Point Theorem for Nonlinear Neutral… DOI: http://dx.doi.org/10.5772/intechopen.96040*

#### **References**

[1] T. A. Burton, Stability by fixed point theory or Liapunov's theory: A comparison, Fixed Point Theory 4 (2003), 15–32.

[2] T. A. Burton, Liapunov functionals, fixed points, and stability by Krasnoselskii's theorem, Nonlinear Studies 9 (2001), 181–190.

[3] T. A. Burton, Stability by fixed point theory for functional differential equations, Dover Publications, New York, (2006).

[4] Ardjouni A, Djoudi A. Fixed points and stability in linear neutral differential equations with variable delays. Nonlinear Analysis. 2011;**74**:2062-2070

[5] A. Ardjouni, A. Djoudi; Stability by Krasnoselskii fixed point theorem for neutral differential equations with variable delays, Acta Universitatis Apulensis Mathematica, (2012), no. 32, 17–30.

[6] A. Ardjouni, A. Djoudi; Fixed points and stability in neutral nonlinear differential equations with variable delays, Opuscula Mathematica 32 (2012), no. 1, 5–19.

[7] M. Ad var, H. C. Koyuncuoğlu and Y. N. Raffoul, Existence of periodic solutions in shifts *δ* for neutral nonlinear dynamic systems on time scales, Appl. Math. Comput. 242 (2014), 328–339.

[8] S. Althubiti, H.A. Makhzoum, Y. N. Raffoul, Periodic solution and stability in nonlinear neutral system with infinite delay, Applied Mathematical Sciences, Vol. 7, (2013), no. 136, 6749–6764.

[9] T. A. Burton, T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynam. Systems Appl. 11 (2002), 499–519.

[10] T. A. Burton, T. Furumochi, Krasnoselskii's fixed point theorem and stability, Nonlinear Anal. 49 (2002), 445–454.

[11] G. Chen, D. Li, O. Gaans, S. Lunel, Stability of nonlinear neutral delay differential equations with variable delays, Electronic Journal of Differential Equations, Vol. 2017 (2017), no. 118, 1–14.

[12] Y. M. Dib, M. R. Maroun, Y. N. Raffoul, Periodicity a stability in neutral nonlinear differential equations with functional delay. Electronic Journal of Differential Equations, Vol. 2005 (2005), no. 142, 1072–6691.

[13] M. N. Islam and Y. N. Raffoul, Periodic solutions of neutral nonlinear system of differentiale quations with functional delay, J. Math. Anal. Appl, 331 (2007), 1175–1186.

[14] C. H. Jin, J.W. Luo, Stability in functional differential equations established using fixed point theory, Nonlinear Anal. 68 (2008), 3307–3315.

[15] E. R. Kaufman and Y. N. Raffoul, Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. Math. Anal. Appl, 319 (2006), 315–325.

[16] J. W. Luo, Fixed points and stability of neutral stochastic delay differential equations, J. Math. Anal. Appl., 334 (2007), 431–440.

[17] J. W. Luo and T. Taniguchi, Fixed points and stability of stochastic neutral partial differential equations with infinite delays, Stoch. Anal. Appl., 27 (2009), 1163–1173.

[18] B. Lisena, Global attractivity in nonautonomous logistic equations with delay. Nonlinear Analysis: RWA, 9 (2008), 53–63.

[19] Y. N. Raffoul, Existence of positive periodic solutions in neutral nonlinear equations with functional delay. Rocky Mountain J. Math. 42 (2012), no. 6, 1983–1993.

[20] D. Zhao, New criteria for stability of neutral differential equations with variable delays by fixed points method, Zhao Advances in Difference Equations (2011), 2011:48.

[21] D. Zhao, The boundedness and stability of neutral delay differential equations with damped stochastic perturbations via fixed point theory, Applied Mathematics and Computation 219 (2013), 6792–6803.

[22] B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal. 63 (2005) e233–e242.

[23] Y. Guo, X. Chao, W. Jun, Stability analysis of neutral stochastic delay di ¤erential equations by a generalization of Banach.s contraction principle. International Journal of Control, 90 (2017), no. 8, 1555-1560.

[24] R. K. Brayton, Bifurcation of periodic solutions in a nonlinear difference-differential equation of neutral type, Quart. Appl. Math. 24 (1966), 215–224.

[25] J. K. Hale and S.M.V. Lunel , Introduction to functional differential equations, Ser. Applied Mathematical Sciences. New York: Springer-Verlag, Vol. 99, (1993).

[26] J. K. Hale and K.R. Meyer, A class of functional equations of neutral type, Mem. Am. Math. Soc. 76(1967), 1–65.

[27] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego, (1993).

[28] V. B. Kolmanovskii, A.D. Myshkis, Applied Theory of Functional

Differential Equations, Kluwer Academic, Dordrecht, 1992.

[29] M. Liu, I. Dassios, and F. Milano, "On the stability analysis of systems of neutral delay differential equations, " Circuits, Systems, and Signal Processing, Vol. 38, (2019), Issue 4, 1639–1653.

**Chapter 3**

**Abstract**

**1. Introduction**

earlier.

**43**

*Aleksandr Kholkin*

Spectral Properties of a

Operator Coefficients

having spectral singularities is constructed.

operator coefficients, Green's function, resolvent

Non-Self-Adjoint Differential

Operator with Block-Triangular

In this chapter, the Sturm-Liouville equation with block-triangular, increasing at

infinity operator potential is considered. A fundamental system of solutions is constructed, one of which decreases at infinity, and the second increases. The asymptotic behavior at infinity was found out. The Green's function and the resolvent for a non-self-adjoint differential operator are constructed. This allows to obtain sufficient conditions under which the spectrum of this non-self-adjoint differential operator is real and discrete. For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator

**Keywords:** differential operators, spectrum, non-self-adjoint, block-triangular

The question of the generalization of the oscillatory Sturm theorem for scalar equations of higher orders and for equations with matrix coefficients for a long time remained open. Only in recent joint papers by F. Rofe-Beketov and A. Kholkin (see [1]) a connection was established between spectral and oscillatory properties for self-adjoint operators generated by equations of arbitrary even order with operator coefficients and boundary conditions of general form. Later, a Sturm-type oscillation theorem was proved [2] for a problem on finite and infinite intervals for a second-order equation with block-triangular matrix coefficients. In the case of nonself-adjoint differential operators, oscillation theorems have not been considered

Results turning out in self-adjoint and non-self-adjoint cases differentiate substantially. The theory of non-self-adjoint singular differential operators, generated by scalar differential expressions, has been well studied. An overview on the theory of non-self-adjoint singular ordinary differential operators is provided in V.E. Lyantse's Appendix I to the monograph [3]. In the study of the connection between spectral and oscillation properties of non-self-adjoint differential operators with block-triangular operator coefficients [2, 4] the question arises of the structure of the spectrum of such operators. For scalar non- self-adjoint differential operators

[30] Y. N. Raffoul, Stability in neutral nonlinear differential equations with functional delays using fixed-point theory, Math. Comput. Modelling 40 (2004), 691–700.

[31] D. R. Smart, Fixed points theorems. Cambridge University Press, Cambridge, (1980).

#### **Chapter 3**

## Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular Operator Coefficients

*Aleksandr Kholkin*

### **Abstract**

In this chapter, the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential is considered. A fundamental system of solutions is constructed, one of which decreases at infinity, and the second increases. The asymptotic behavior at infinity was found out. The Green's function and the resolvent for a non-self-adjoint differential operator are constructed. This allows to obtain sufficient conditions under which the spectrum of this non-self-adjoint differential operator is real and discrete. For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator having spectral singularities is constructed.

**Keywords:** differential operators, spectrum, non-self-adjoint, block-triangular operator coefficients, Green's function, resolvent

#### **1. Introduction**

The question of the generalization of the oscillatory Sturm theorem for scalar equations of higher orders and for equations with matrix coefficients for a long time remained open. Only in recent joint papers by F. Rofe-Beketov and A. Kholkin (see [1]) a connection was established between spectral and oscillatory properties for self-adjoint operators generated by equations of arbitrary even order with operator coefficients and boundary conditions of general form. Later, a Sturm-type oscillation theorem was proved [2] for a problem on finite and infinite intervals for a second-order equation with block-triangular matrix coefficients. In the case of nonself-adjoint differential operators, oscillation theorems have not been considered earlier.

Results turning out in self-adjoint and non-self-adjoint cases differentiate substantially. The theory of non-self-adjoint singular differential operators, generated by scalar differential expressions, has been well studied. An overview on the theory of non-self-adjoint singular ordinary differential operators is provided in V.E. Lyantse's Appendix I to the monograph [3]. In the study of the connection between spectral and oscillation properties of non-self-adjoint differential operators with block-triangular operator coefficients [2, 4] the question arises of the structure of the spectrum of such operators. For scalar non- self-adjoint differential operators

these questions were studied in the papers [5–8]. The theory of singular non-selfadjoint differential operators with matrix and operator coefficients is relatively new. In the context of the inverse scattering problem, for an operator with a triangular matrix potential decreasing at infinity, the first moment of which is bounded, the structure of the spectrum was established in [9, 10]. The theory of equations with block - triangular operator coefficients the first results were published in 2012 in the works of the author [11–13].

ð ∞

j j *U t*ð Þ � *v*�1

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular…*

∞ð

*v*00ðÞ� *t v*�3

�*y*00 þ ð Þ *v x*ð Þþ *q x*ð Þ *y* ¼ ð Þ ð Þ *λ* þ *q x*ð Þ *I* � *U x*ð Þ *y*, (7)

� 1 4 *v*00ð Þ *x*

A, *γ*∞ð Þ¼ *x*, *λ* 1

ffiffiffiffiffiffiffiffiffiffiffi 4*v x*ð Þ p4 � exp ð

�*z*00 þ ð Þ *v x*ð Þþ *q x*ð Þ *z* ¼ 0, (10)

Φ0 ð Þ *x*, *λ*

*γ*0

0

In case of *v x*ð Þ¼ *x*2*α*, 0 <*α*≤ 1, we suppose that the coefficients of the Eq. (1)

2ð Þ*t dt* < ∞, (4)

�*αdt* < ∞, *a*>0*:* (6)

2ð Þ*t dt*< ∞*:* (5)

*v x*ð Þ *:* (8)

*x*

0 @

ffiffiffiffiffiffiffiffiffi *v u*ð Þ p *du*

1 A*:* (9)

0

0ð Þ� *x*, *λ γ*∞ð Þ¼ *x*, *λ* 1*:* (11)

0ð Þ *x*, *λ* ¼ *I:* (12)

0

2ð Þ*t dt*< ∞,

j j *U t*ð Þ � *t*

**2.1 Construction of the fundamental system of solutions for an operator differential equation with a rapidly increasing at infinity potential**

Condition (3) is performed, for example, quickly increasing functions

where *q x*ð Þ determined by a formula (cf. with the monograph [15])

*v*0 ð Þ *x v x*ð Þ � �2

1

It is easy to see that *γ*0ð Þ! *x*, *λ* 0, *γ*∞ð Þ! *x*, *λ* ∞ as *x* ! ∞. These solutions constitute a fundamental system of solutions of the scalar differential equation

∞ð Þ� *x*, *λ γ*0

*Also, there exists increasing at infinity operator solution* Ψð Þ *x*, *λ* ∈*B*ð Þ **H** *satisfying the*

**Theorem 2.1** *Under conditions (3), (4), (5) Eq. (1) has a unique decreasing at*

*γ*0ð Þ *x*, *λ* ¼ *I and* lim*x*!∞

ð ∞

*DOI: http://dx.doi.org/10.5772/intechopen.95820*

*v*02

ð Þ� *t v*�5

∞ð

*a*

Consider first the case where *v x*ð Þ≥*Cx*2*α*,*C*> 0, *α* >1.

*q x*ð Þ¼ 5 16

> ffiffiffiffiffiffiffiffiffi *v u*ð Þ p *du*

ð *x*

0 @

in such a way that for all *x*∈½ Þ 0, ∞ one has.

lim*x*!∞

*W γ*0 ð Þ , *γ*∞ ≔ *γ*0ð Þ� *x*, *λ γ*0

*infinity operator solution* Φð Þ *x*, *λ* ∈*B*ð Þ **H** *, satisfying the conditions*

Φð Þ *x*, *λ*

0

0

Rewrite the Eq. (1) in the form

satisfy the relation

*ex*, exp *ex* f g etc.

Now let us denote.

ffiffiffiffiffiffiffiffiffiffiffi 4*v x*ð Þ p4 � exp �

*γ*0ð Þ¼ *x*, *λ* 1

*conditions*

**45**

In this works we construct the fundamental system of solutions of differential equation with block-triangular operator potential that increases at infinity, one of that is decreasing at infinity, and the second growing. The asymptotics of the fundamental system of solutions of this equation is established. The Green's function is constructed for a non-self-adjoint system with a block-triangular potential, the diagonal blocks of which are self-adjoint operators. We obtained a resolvent for a non-self-adjoint differential operator, using which the structure of the operator spectrum is set. Sufficient conditions at which a spectrum of such non-self-adjoint differential operator is real and discrete are obtained. Here the rate of growth elements, not on the main diagonal, is subordinated to the rate of growth of the diagonal elements. In case of infringement of this condition, the operator can have spectral singularities [14].

#### **2. The fundamental solutions for an non-self-adjoint differential operator with block – triangular operator coefficients.**

Let us designate *Hk*, *k* ¼ 1,*r* as a finite-dimensional or infinite-dimensional separable Hilbert space with inner product ð Þ �, � and norm j j� . Denote by **H** ¼ *H*1 ⊕ *H*2 ⊕ … ⊕ *Hr:* Element *h*∈ **H** will be written in the form of *h* ¼ col *h*1, *h*2, … , *hr* � �, where *hk* ∈ *Hk*, *k* ¼ 1, 2, … ,*r*, *Ik*,*I*- are identity operators in *Hk* and **H** accordingly.

We denote by *L*2ð Þ **H**, 0, ð Þ ∞ the Hilbert space of vector-valued functions *y x*ð Þ with values in **H** with inner product h i *y*, *z* ¼ Ð ∞ 0 ð Þ *y x*ð Þ, *z x*ð Þ *dx* and the norm k k� .

Now let us consider the equation with block-triangular operator potential in *B*ð Þ **H**

$$d[\overline{\eta}] = -\overline{\eta}'' + V(\mathfrak{x})\overline{\eta} = \lambda \overline{\eta}, \quad 0 \le \mathfrak{x} < \infty,\tag{1}$$

where

$$V(\mathbf{x}) = v(\mathbf{x}) \cdot I + U(\mathbf{x}), \quad U(\mathbf{x}) = \begin{pmatrix} U\_{11}(\mathbf{x}) & U\_{12}(\mathbf{x}) & \dots & U\_{1r}(\mathbf{x}) \\ 0 & U\_{22}(\mathbf{x}) & \dots & U\_{2r}(\mathbf{x}) \\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & U\_{rr}(\mathbf{x}) \end{pmatrix},\tag{2}$$

*v x*ð Þ is a real scalar function such that 0<*v x*ð Þ! ∞ monotonically, as *x* ! ∞, and it has monotone absolutely continuous derivative. Also, *U x*ð Þ is a relatively small perturbation, e. g. as *x* ! ∞ j j *U x*ð Þ � *v*�1ð Þ! *x* 0 or j j *U v*�1 ∈*L*∞ð Þ þ . The diagonal blocks *Ukk*ð Þ *x* , *k* ¼ 1,*r* are assumed to be bounded self-adjoint operators in *Hk*.

In case where

$$\nu(\mathfrak{x}) \ge \mathbb{C}\mathfrak{x}^{2a}, \mathbb{C} > 0, a > 1,\tag{3}$$

we suppose that coefficients of the Eq. (1) satisfy relations:

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular… DOI: http://dx.doi.org/10.5772/intechopen.95820*

$$\int\_0^\infty |U(t)| \cdot v^{-\frac{1}{2}}(t)dt < \infty,\tag{4}$$

$$\int\_0^\infty v'^2(t) \cdot v^{-\frac{5}{2}}(t)dt < \infty, \quad \int\_0^\infty v''(t) \cdot v^{-\frac{3}{2}}(t)dt < \infty. \tag{5}$$

In case of *v x*ð Þ¼ *x*2*α*, 0 <*α*≤ 1, we suppose that the coefficients of the Eq. (1) satisfy the relation

$$\int\_{a}^{\infty} |U(t)| \cdot t^{-a} dt < \infty, \ a > 0. \tag{6}$$

#### **2.1 Construction of the fundamental system of solutions for an operator differential equation with a rapidly increasing at infinity potential**

Consider first the case where *v x*ð Þ≥*Cx*2*α*,*C*> 0, *α* >1.

Condition (3) is performed, for example, quickly increasing functions *ex*, exp *ex* f g etc.

Rewrite the Eq. (1) in the form

$$-\overline{\mathfrak{y}}'' + (\upsilon(\mathfrak{x}) + q(\mathfrak{x})) \overline{\mathfrak{y}} = ((\lambda + q(\mathfrak{x}))I - U(\mathfrak{x})) \overline{\mathfrak{y}},\tag{7}$$

where *q x*ð Þ determined by a formula (cf. with the monograph [15])

$$q(\boldsymbol{x}) = \frac{5}{16} \left( \frac{v'(\boldsymbol{x})}{v(\boldsymbol{x})} \right)^2 - \frac{1}{4} \frac{v''(\boldsymbol{x})}{v(\boldsymbol{x})}.\tag{8}$$

Now let us denote.

these questions were studied in the papers [5–8]. The theory of singular non-selfadjoint differential operators with matrix and operator coefficients is relatively new. In the context of the inverse scattering problem, for an operator with a triangular matrix potential decreasing at infinity, the first moment of which is bounded, the structure of the spectrum was established in [9, 10]. The theory of equations with block - triangular operator coefficients the first results were

In this works we construct the fundamental system of solutions of differential equation with block-triangular operator potential that increases at infinity, one of that is decreasing at infinity, and the second growing. The asymptotics of the fundamental system of solutions of this equation is established. The Green's function is constructed for a non-self-adjoint system with a block-triangular potential, the diagonal blocks of which are self-adjoint operators. We obtained a resolvent for a non-self-adjoint differential operator, using which the structure of the operator spectrum is set. Sufficient conditions at which a spectrum of such non-self-adjoint differential operator is real and discrete are obtained. Here the rate of growth elements, not on the main diagonal, is subordinated to the rate of growth of the diagonal elements. In case of infringement of this condition, the operator can have

**2. The fundamental solutions for an non-self-adjoint differential operator with block – triangular operator coefficients.**

*H*1 ⊕ *H*2 ⊕ … ⊕ *Hr:* Element *h*∈ **H** will be written in the form of *h* ¼

with values in **H** with inner product h i *y*, *z* ¼ Ð ∞

*V x*ð Þ¼ *v x*ð Þ� *I* þ *U x*ð Þ, *U x*ð Þ¼

Let us designate *Hk*, *k* ¼ 1,*r* as a finite-dimensional or infinite-dimensional separable Hilbert space with inner product ð Þ �, � and norm j j� . Denote by **H** ¼

� �, where *hk* ∈ *Hk*, *k* ¼ 1, 2, … ,*r*, *Ik*,*I*- are identity operators in *Hk*

We denote by *L*2ð Þ **H**, 0, ð Þ ∞ the Hilbert space of vector-valued functions *y x*ð Þ

0

BBB@

we suppose that coefficients of the Eq. (1) satisfy relations:

*v x*ð Þ is a real scalar function such that 0<*v x*ð Þ! ∞ monotonically, as *x* ! ∞, and it has monotone absolutely continuous derivative. Also, *U x*ð Þ is a relatively small perturbation, e. g. as *x* ! ∞ j j *U x*ð Þ � *v*�1ð Þ! *x* 0 or j j *U v*�1 ∈*L*∞ð Þ þ . The diagonal blocks *Ukk*ð Þ *x* , *k* ¼ 1,*r* are assumed to be bounded self-adjoint operators in *Hk*.

Now let us consider the equation with block-triangular operator potential in *B*ð Þ **H**

*l*½ �¼� *y y*00 þ *V x*ð Þ*y* ¼ *λy*, 0≤ *x*< ∞, (1)

*U*11ð Þ *x U*12ð Þ *x* … *U*1*r*ð Þ *x* 0 *U*22ð Þ *x* … *U*2*r*ð Þ *x* … ……… 0 0 … *Urr*ð Þ *x*

*v x*ð Þ≥*Cx*2*α*,*C*>0, *α*>1, (3)

0 ð Þ *y x*ð Þ, *z x*ð Þ *dx* and the norm k k� .

1

CCCA

, (2)

published in 2012 in the works of the author [11–13].

*Recent Developments in the Solution of Nonlinear Differential Equations*

spectral singularities [14].

col *h*1, *h*2, … , *hr*

where

In case where

**44**

and **H** accordingly.

$$\gamma\_0(\mathbf{x}, \boldsymbol{\lambda}) = \frac{1}{\sqrt[4]{4\nu(\mathbf{x})}} \cdot \exp\left(-\int\_0^\mathbf{x} \sqrt{v(u)} du\right), \\ \gamma\_{\ast\ast}(\mathbf{x}, \boldsymbol{\lambda}) = \frac{1}{\sqrt[4]{4\nu(\boldsymbol{\lambda})}} \cdot \exp\left(\int\_0^\mathbf{x} \sqrt{v(u)} du\right). \tag{9}$$

It is easy to see that *γ*0ð Þ! *x*, *λ* 0, *γ*∞ð Þ! *x*, *λ* ∞ as *x* ! ∞. These solutions constitute a fundamental system of solutions of the scalar differential equation

$$-z'' + (v(\varkappa) + q(\varkappa))z = 0,\tag{10}$$

in such a way that for all *x*∈½ Þ 0, ∞ one has.

$$W(\boldsymbol{\chi}\_{0}, \boldsymbol{\chi}\_{\circ \circ}) \coloneqq \boldsymbol{\chi}\_{0}(\boldsymbol{\mathfrak{x}}, \boldsymbol{\lambda}) \cdot \boldsymbol{\chi}\_{\circ \circ}^{\prime}(\boldsymbol{\mathfrak{x}}, \boldsymbol{\lambda}) - \boldsymbol{\chi}\_{0}^{\prime}(\boldsymbol{\mathfrak{x}}, \boldsymbol{\lambda}) \cdot \boldsymbol{\chi}\_{\circ \circ}(\boldsymbol{\mathfrak{x}}, \boldsymbol{\lambda}) = \mathbf{1}.\tag{11}$$

**Theorem 2.1** *Under conditions (3), (4), (5) Eq. (1) has a unique decreasing at infinity operator solution* Φð Þ *x*, *λ* ∈*B*ð Þ **H** *, satisfying the conditions*

$$\lim\_{\chi \to \infty} \frac{\Phi(\mathbf{x}, \boldsymbol{\lambda})}{\chi\_0(\mathbf{x}, \boldsymbol{\lambda})} = I \quad \text{and} \quad \lim\_{\mathbf{x} \to \infty} \frac{\Phi'(\mathbf{x}, \boldsymbol{\lambda})}{\mathbf{y}\_0'(\mathbf{x}, \boldsymbol{\lambda})} = I. \tag{12}$$

*Also, there exists increasing at infinity operator solution* Ψð Þ *x*, *λ* ∈*B*ð Þ **H** *satisfying the conditions*

$$\lim\_{\lambda \to \infty} \frac{\Psi(\varkappa, \lambda)}{\chi\_{\infty}(\varkappa, \lambda)} = I \text{ and } \lim\_{\varkappa \to \infty} \frac{\Psi'(\varkappa, \lambda)}{\chi\_{\infty}'(\varkappa, \lambda)} = I. \tag{13}$$

and therefore integral equation has a unique solution *χ*ð Þ *x*, *λ* and j j *χ*ð Þ *x*, *λ* ≤const. By (17), one has that lim *x*!∞*χ*ð Þ¼ *x*, *λ I*, where the first part of formula (12) follows

b. Denote by Ψ^ð Þ *x*, *λ* ∈*B*ð Þ **H** block-triangular operator solution of Eq. (1) that increases at infinity, Ψ*kk*ð Þ *x*, *λ* ∈*B H*ð Þ *k*, *Hk* , *k* ¼ 1,*r* -its diagonal blocks. Now

> ð *x*

0

where, just as in (14), the kernel *K x*ð Þ , *t*, *λ* is given by (15). Now set *χ*ð Þ¼ *x*, *λ*

ð *x*

0

Ψ^ð Þ *x*, *λ*

*kk* ð Þ *t*, *λ* Φ∗

Ψ~ *kk*ð Þ *x*, *λ*

that the integral Eq. (23) has a unique solution *χ*ð Þ *x*, *λ* and j j *χ*ð Þ *x*, *λ* ≤const. Pass in (23) to a limit as *x* ! ∞ to get lim *x*!∞*χ*ð Þ¼ *x*, *λ I* þ *C*~ð Þ*λ* where *C*~ð Þ*λ* is block-

Now consider another block-triangular operator solution Ψ~ð Þ *x*, *λ* that increases at

*kk*ð Þ *t*, *λ* � ��1

Φ*kk*ð Þ *x*, *λ* are the diagonal blocks of operator solution Φð Þ *x*, *λ* as in Section a). In view (16) and the definition of the functions *γ*0ð Þ *x* , *γ*∞ð Þ *x* can be proved that

Since Ψ^ð Þ *x*, *λ* and Ψ~ð Þ *x*, *λ* are the operator solutions of Eq. (1) that increase at

*x S x*ð Þ , *t*, *λ χ*ð Þ *t*, *λ dt*, where *S x*ð Þ¼ , *t*, *λ*

*v t*ð Þ p � ½j j *λ* þ j jþ *q t*ð Þ

*K x*ð Þ� , *t*, *λ* Ψ^ð Þ *t*, *λ dt*, (22)

*R x*ð Þ� , *t*, *λ χ*ð Þ *t*, *λ dt*, (23)

*γ*∞ð Þ *x*, *λ* � ½ � ð Þ� *q t*ðÞþ *λ I* � *U t*ð Þ . Similarly we can prove

*γ*∞ð Þ *x*, *λ* ¼ *I* þ *C*~ð Þ*λ :* (24)

*γ*∞ð Þ *x*, *λ* ¼ *Ik*, *k* ¼ 1,*r:* (26)

Ψ^ð Þ¼ *x*, *λ* Ψ~ð Þþ *x*, *λ* Φð Þ� *x*, *λ C*0ð Þ*λ* , (27)

*dt*, *k* ¼ 1,*r*,ð Þ *a*≥0 , (25)

0ð Þ *x*, *λ* � ½ � ð Þ *λ* þ *q t*ð Þ *I* � *U t*ð Þ . We have similarly (18), that

ffiffiffiffiffi

ð Þ *x*, *λ γ*0

*x*ð Þ� *x*, *t γ*0ð Þ *t*, *λ γ*0

Eq. (7) is equivalent to the integral equation

Ψ^ð Þ¼ *x*, *λ γ*∞ð Þ� *x*, *λ I* �

*χ*ð Þ¼ *x*, *λ I* �

lim*x*!∞

ð *x*

Φ�1

*a*

lim*x*!∞

0ð Þ *x* ¼ *I* þ Ð ∞

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular…*

*v t*ð Þ p , and therefore j j *S x*ð Þ , *t*, *λ* ≤ 1

j j� *U t*ð Þ ∈*L*ð Þ 0, ∞ , where the second part of formula (12) follows from.

from.

*x*ð Þ *x*, *t*, *λ γ*0ð Þ *t*, *λ γ*0

*x*ð Þ� *x*, *t γ*0ð Þ *t*, *λ γ*0 0ð Þ *x*, *λ*

*K*0

� � �

*C*0

Ψ^ð Þ *x*, *λ*

infinity,

**47**

Differentiable (14) to get Φ0

*DOI: http://dx.doi.org/10.5772/intechopen.95820*

0ð Þ *x*, *λ* ¼ *C*0

� � �≤ 1 ffiffiffiffiffi

*γ*∞ð Þ *x*, *λ* to rewrite Eq. (22) in form

triangular operator in **H**, that is

Ψ~ *kk*ð Þ¼ *x*, *λ* Φ*kk*ð Þ *x*, *λ*

where *R x*ð Þ¼ , *t*, *λ C x*ð Þ� , *t*, *λ γ*∞ð Þ *t*, *λ*

infinity diagonal blocks which are defined by.

#### **Proof**

a. Eq. (7) equivalently to integral equation

$$\Phi(\mathbf{x}, \boldsymbol{\lambda}) = \chi\_0(\mathbf{x}, \boldsymbol{\lambda}) I + \int\_{\mathbf{x}}^{\mathbf{x}} K(\mathbf{x}, t, \boldsymbol{\lambda}) \cdot \Phi(t, \boldsymbol{\lambda}) dt,\tag{14}$$

where

$$K(\mathbf{x}, t, \lambda) = \mathbf{C}(\mathbf{x}, t, \lambda) \cdot [(\lambda + q(t))I - U(t)],\tag{15}$$

$$\mathbf{C}(\mathbf{x}, \mathbf{t}, \boldsymbol{\lambda}) = \boldsymbol{\chi}\_{\boldsymbol{\omega}}(\mathbf{x}, \boldsymbol{\lambda}) \cdot \boldsymbol{\chi}\_{0}(\mathbf{t}, \boldsymbol{\lambda}) - \boldsymbol{\chi}\_{\boldsymbol{\omega}}(\mathbf{t}, \boldsymbol{\lambda}) \cdot \boldsymbol{\chi}\_{0}(\mathbf{x}, \boldsymbol{\lambda}), \tag{16}$$

with *C x*ð Þ , *t*, *λ* being the Cauchy function that in each variable satisfies Eq. (10) and the initial conditions *C x*ð Þ , *t*, *λ x*¼*t* j ¼ 0, *C*0 *x*ð Þ *x*, *t*, *λ* j*x*¼*t* ¼ 1, *C*0 *t* ð Þ *x*, *t*, *λ* j*x*¼*t* ¼ �1. Set *χ*ð Þ¼ *x*, *λ* Φð Þ *x*, *λ γ*0ð Þ *x*, *λ* to rewrite Eq. (14) in form

$$\chi(\mathbf{x}, \boldsymbol{\lambda}) = \boldsymbol{I} + \int\_{\mathbf{x}}^{\mathbf{\hat{o}}} \boldsymbol{R}(\mathbf{x}, \mathbf{t}, \boldsymbol{\lambda}) \chi(\mathbf{t}, \boldsymbol{\lambda}) d\mathbf{t}, \tag{17}$$

where *R x*ð Þ¼ , *t*, *λ K x*ð Þ� , *t*, *λ γ*0ð Þ *t*, *λ γ*0ð Þ *x*, *λ* . Thus

$$\begin{split} \left| C(\mathbf{x}, t) \cdot \frac{\boldsymbol{\gamma}\_{0}(t, \lambda)}{\boldsymbol{\gamma}\_{0}(\mathbf{x}, \lambda)} \right| &= \left| \boldsymbol{\gamma}\_{0}^{2}(t, \mathbf{y}) \cdot \frac{\boldsymbol{\gamma}\_{\alpha}(\mathbf{x}, \lambda)}{\boldsymbol{\gamma}\_{0}(\mathbf{x}, \lambda)} - \boldsymbol{\gamma}\_{0}(\mathbf{t}, \lambda) \cdot \boldsymbol{\gamma}\_{\alpha}(\mathbf{t}, \lambda) \right| = \\ &= \left| \frac{1}{2\sqrt{\boldsymbol{\nu}(t)}} \cdot \exp\left( -2\int\_{0}^{t} \sqrt{\boldsymbol{\nu}(u)} du \right) \cdot \exp\left( 2\int\_{0}^{x} \sqrt{\boldsymbol{\nu}(u)} du \right) - \frac{1}{2\sqrt{\boldsymbol{\nu}(t)}} \right| = \\ &= \frac{1}{2\sqrt{\boldsymbol{\nu}(t)}} \cdot \left| \exp\left( -2\int\_{x}^{t} \sqrt{\boldsymbol{\nu}(u)} du \right) - 1 \right| \end{split} \tag{18}$$

and since with *x*≤*t* one has exp �2 Ð*t x* ffiffiffiffiffiffiffiffiffi *v u*ð Þ p *du* � � ≤ 1, we deduce that

$$\left| \mathbf{C}(\mathbf{x}, t) \cdot \frac{\boldsymbol{\gamma}\_0(t, \lambda)}{\boldsymbol{\gamma}\_0(\mathbf{x}, \lambda)} \right| \leq \frac{1}{\sqrt{\boldsymbol{\nu}(t)}}.\tag{19}$$

Hence.

$$|R(\mathbf{x}, t, \lambda)| = \left| C(\mathbf{x}, t) \cdot \frac{\chi\_0(t, \lambda)}{\chi\_0(\mathbf{x}, \gamma)} \cdot [(\lambda + q(t))I - U(t)] \right| \le \frac{1}{\sqrt{\nu(t)}} (|\lambda| + |q(t)| + |U(t)|). \tag{20}$$

By virtue of (3)–(5), (8),

$$\frac{1}{\sqrt{v(t)}}(|\lambda| + |q(t)| + |U(t)|) \in L(\mathbf{0}, \infty),\tag{21}$$

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular… DOI: http://dx.doi.org/10.5772/intechopen.95820*

and therefore integral equation has a unique solution *χ*ð Þ *x*, *λ* and j j *χ*ð Þ *x*, *λ* ≤const. By (17), one has that lim *x*!∞*χ*ð Þ¼ *x*, *λ I*, where the first part of formula (12) follows from.

Differentiable (14) to get Φ0 ð Þ *x*, *λ γ*0 0ð Þ *x* ¼ *I* þ Ð ∞ *x S x*ð Þ , *t*, *λ χ*ð Þ *t*, *λ dt*, where *S x*ð Þ¼ , *t*, *λ K*0 *x*ð Þ *x*, *t*, *λ γ*0ð Þ *t*, *λ γ*0 0ð Þ *x*, *λ* ¼ *C*0 *x*ð Þ� *x*, *t γ*0ð Þ *t*, *λ γ*0 0ð Þ *x*, *λ* � ½ � ð Þ *λ* þ *q t*ð Þ *I* � *U t*ð Þ . We have similarly (18), that *C*0 *x*ð Þ� *x*, *t γ*0ð Þ *t*, *λ γ*0 0ð Þ *x*, *λ* � � � � � �≤ 1 ffiffiffiffiffi *v t*ð Þ p , and therefore j j *S x*ð Þ , *t*, *λ* ≤ 1 ffiffiffiffiffi *v t*ð Þ p � ½j j *λ* þ j jþ *q t*ð Þ j j� *U t*ð Þ ∈*L*ð Þ 0, ∞ , where the second part of formula (12) follows from.

b. Denote by Ψ^ð Þ *x*, *λ* ∈*B*ð Þ **H** block-triangular operator solution of Eq. (1) that increases at infinity, Ψ*kk*ð Þ *x*, *λ* ∈*B H*ð Þ *k*, *Hk* , *k* ¼ 1,*r* -its diagonal blocks. Now Eq. (7) is equivalent to the integral equation

$$
\hat{\Psi}(\mathbf{x},\boldsymbol{\lambda}) = \boldsymbol{\chi}\_{\rm os}(\mathbf{x},\boldsymbol{\lambda}) \cdot \boldsymbol{I} - \int\_{0}^{\boldsymbol{\lambda}} \boldsymbol{K}(\mathbf{x},\boldsymbol{t},\boldsymbol{\lambda}) \cdot \hat{\Psi}(\boldsymbol{t},\boldsymbol{\lambda}) d\boldsymbol{t},\tag{22}
$$

where, just as in (14), the kernel *K x*ð Þ , *t*, *λ* is given by (15). Now set *χ*ð Þ¼ *x*, *λ* Ψ^ð Þ *x*, *λ γ*∞ð Þ *x*, *λ* to rewrite Eq. (22) in form

$$\chi(\mathbf{x}, \boldsymbol{\lambda}) = I - \int\_0^\mathbf{x} \mathbf{R}(\mathbf{x}, t, \boldsymbol{\lambda}) \cdot \chi(t, \boldsymbol{\lambda}) dt,\tag{23}$$

where *R x*ð Þ¼ , *t*, *λ C x*ð Þ� , *t*, *λ γ*∞ð Þ *t*, *λ γ*∞ð Þ *x*, *λ* � ½ � ð Þ� *q t*ðÞþ *λ I* � *U t*ð Þ . Similarly we can prove that the integral Eq. (23) has a unique solution *χ*ð Þ *x*, *λ* and j j *χ*ð Þ *x*, *λ* ≤const. Pass in (23) to a limit as *x* ! ∞ to get lim *x*!∞*χ*ð Þ¼ *x*, *λ I* þ *C*~ð Þ*λ* where *C*~ð Þ*λ* is blocktriangular operator in **H**, that is

$$\lim\_{\chi \to \infty} \frac{\hat{\Psi}(\varkappa, \lambda)}{\chi\_{\infty}(\varkappa, \lambda)} = I + \breve{\mathcal{C}}(\lambda). \tag{24}$$

Now consider another block-triangular operator solution Ψ~ð Þ *x*, *λ* that increases at infinity diagonal blocks which are defined by.

$$\tilde{\Psi}\_{kk}(\mathbf{x},\boldsymbol{\lambda}) = \Phi\_{kk}(\mathbf{x},\boldsymbol{\lambda}) \int\_{a}^{\mathbf{x}} \boldsymbol{\Phi}\_{kk}^{-1}(t,\boldsymbol{\lambda}) \left(\boldsymbol{\Phi}\_{kk}^{\*}(t,\boldsymbol{\lambda})\right)^{-1} dt,\\ k = \overline{1,r}, (a \ge 0), \tag{25}$$

Φ*kk*ð Þ *x*, *λ* are the diagonal blocks of operator solution Φð Þ *x*, *λ* as in Section a). In view (16) and the definition of the functions *γ*0ð Þ *x* , *γ*∞ð Þ *x* can be proved that

$$\lim\_{\chi \to \infty} \frac{\tilde{\Psi}\_{kk}(\mathbf{x}, \boldsymbol{\lambda})}{\chi\_{\infty}(\mathbf{x}, \boldsymbol{\lambda})} = I\_k, \ \boldsymbol{k} = \overline{\mathbf{1}, r}. \tag{26}$$

Since Ψ^ð Þ *x*, *λ* and Ψ~ð Þ *x*, *λ* are the operator solutions of Eq. (1) that increase at infinity,

$$
\hat{\Psi}(\mathbf{x},\boldsymbol{\lambda}) = \tilde{\Psi}(\mathbf{x},\boldsymbol{\lambda}) + \Phi(\mathbf{x},\boldsymbol{\lambda}) \cdot \mathbf{C}\_0(\boldsymbol{\lambda}),\tag{27}
$$

lim*x*!∞

a. Eq. (7) equivalently to integral equation

and the initial conditions *C x*ð Þ , *t*, *λ x*¼*t* j ¼ 0, *C*0

where *R x*ð Þ¼ , *t*, *λ K x*ð Þ� , *t*, *λ γ*0ð Þ *t*, *λ*

� � � �

¼ 1 2 ffiffiffiffiffiffiffiffi

¼ 1 2 ffiffiffiffiffiffiffiffi

and since with *x*≤*t* one has exp �2

j j *R x*ð Þ , *t*, *λ* ¼ *C x*ð Þ� , *t γ*0ð Þ *t*, *λ*

By virtue of (3)–(5), (8),

� � � � � � � � � � 0ð Þ� *t*, *γ*

� � � � ¼ *γ*2

*γ*0ð Þ *x*, *λ* to rewrite Eq. (14) in form

*χ*ð Þ¼ *x*, *λ I* þ

*γ*∞ð Þ *x*, *λ*

0 @

0 @

*v t*ð Þ p � exp �2

*v t*ð Þ p � exp �2

� � � �

1 ffiffiffiffiffiffiffiffi

� � � � � �

**Proof**

where

Set *χ*ð Þ¼ *x*, *λ* Φð Þ *x*, *λ*

*C x*ð Þ� , *t γ*0ð Þ *t*, *λ γ*0ð Þ *x*, *λ*

Hence.

**46**

� � � � Ψð Þ *x*, *λ*

*Recent Developments in the Solution of Nonlinear Differential Equations*

Φð Þ¼ *x*, *λ γ*0ð Þ *x*, *λ I* þ

*γ*∞ð Þ *x*, *λ* ¼ *I and* lim*x*!∞

∞ð

*x*

with *C x*ð Þ , *t*, *λ* being the Cauchy function that in each variable satisfies Eq. (10)

∞ð

*x*

*γ*0ð Þ *x*, *λ* � *γ*0ð Þ� *t*, *λ γ*∞ð Þ *t*, *λ*

ffiffiffiffiffiffiffiffiffi *v u*ð Þ p *du*

ffiffiffiffiffiffiffiffiffi *v u*ð Þ p *du*

1

1 A � 1

ffiffiffiffiffiffiffiffiffi *v u*ð Þ p *du* � �

> 1 ffiffiffiffiffiffiffiffi

> > � � � � ≤

*v t*ð Þ p ð Þ j j *λ* þ j j *q t*ð Þ þ j j *U t*ð Þ ∈*L*ð Þ 0, ∞ , (21)

1 ffiffiffiffiffiffiffiffi

� � � � ≤

*γ*0ð Þ *x*, *λ* .  Thus lim*x*!∞ Ψ^ð Þ *x*, *λ γ*∞ð Þ *x* ¼ lim*x*!∞ Ψ~ð Þ *x*, *λ γ*∞ð Þ *x* , hence, by virtue (26), lim*x*!∞ Ψ*kk*ð Þ *x*, *λ γ*∞ð Þ *x* ¼ *Ik*, *k* ¼ 1,*r* and in (24) has

$$
\tilde{\mathbf{C}}(\boldsymbol{\lambda}) = \begin{pmatrix}
\mathbf{0} & \mathbf{C}\_{12}(\boldsymbol{\lambda}) & \dots & \mathbf{C}\_{1r}(\boldsymbol{\lambda}) \\
\mathbf{0} & \mathbf{0} & \dots & \mathbf{C}\_{2r}(\boldsymbol{\lambda}) \\
\dots & \dots & \dots & \dots \\
\mathbf{0} & \mathbf{0} & \dots & \mathbf{0}
\end{pmatrix}.\tag{28}
$$

After expanding here the integral, we obtain the exponential as follows

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular…*

1 � 3 � … � ð Þ 2*k* � 3

1 � 3 � … � ð Þ 2*k* � 3

2 � �*n*

!

1 � 3 � … � ð Þ 2*k* � 3

� �

!

� �*k* !*du*

*k*! � 2*k* � *λ*

*k*! � *λ*

*u*2*α*

, this expression after integration acquires the

2 � �*k*

> 2 � �*k*

*k*! � *λ*

� ð Þ 1 þ *o*ð Þ1 *:* (35)

2 � �*k*

ð Þ 1 þ *o*ð Þ1 *:* (37)

2 � �*k*

� ln *x* þ *o*ð Þ1

*k*! � *λ*

1 � 3 � … � ð Þ 2*k* � 3

!

2 � ð Þ 1 þ *o*ð Þ1 *:*

2 � �

� � þ 1, with ½ � *β* being the integral part of *β*, to

*k*! � *λ*

� 1 þ *o x*�*α* ð Þ ð Þ

� 1 þ *o x*�1

2 � � � � *:* (39)

1 � 3 � … � ð Þ 2*k* � 3

!

1

� *x*1�ð Þ 2*k*�1 *α* 1 � ð Þ 2*k* � 1 *α*

� *x*1�ð Þ 2*k*�1 *α* 1 � ð Þ 2*k* � 1 *α*

> � *x*1�ð Þ 2*k*�1 *α* 1 � ð Þ 2*k* � 1 *α*

� *x*1�ð Þ 2*k*�1 *α* 1 � ð Þ 2*k* � 1 *α*

�

(38)

(36)

�

¼

A*:* (34)

exp �

In case *α*þ1

form:

0 @ ð *x*

*uα* � 1 � 1

*DOI: http://dx.doi.org/10.5772/intechopen.95820*

2*α* ¼ *n* ∈ *N*, i.e. *α* ¼ 1

*λ* 2 � *x*1�*α*

> *λ* 2 � *x*1�*α*

1 þ *α* þ

1 þ *α* þ

� exp

1 þ *α* þ

2 � *λ*

*u*2*α* �X∞ *k*¼2

2*n*�1

1 � *α* þX*n*�1

*k*¼2

*k*¼2

1 � *α* þX*n*�1

*λ*�1

*n*! � *λ*

!

A similar procedure allows to establish the asymptotics of *γ*∞ð Þ *x* as *x* ! ∞*:* If *α*þ1

� �

2 ð Þ *n* ¼ 2 one has

3 *x* 3 2 þ *λx* 1 2 � *λ* 2 � �2 *x*�1 2

2 � �*n* � *x*�*α α*

2*α*

*λ* 2 � *x*1�*α* 1 � *α* þX*n*�1 *k*¼2

*k*¼2

In particular, for *α* ¼ 1 ð Þ *n* ¼ 1 , *γ*0ð Þ *x*, *λ* has the following asymptotics at infinity:

2 � exp � *x*2

1�3� … �ð Þ 2*n*�3 *n*! � *λ* ð Þ2 *n*

*n*! � *λ*

1 � 3 � … � ð Þ 2*n* � 3

1 � *α* þX*n*�1

� *x*

The asymptotics of *γ*0ð Þ *x*, *λ* as *x* ! ∞ is as follows:

*λ* 2 � *x*1�*α*

�*x* 1�3� … �ð Þ 2*n*�3 *n*! � *λ* ð Þ2 *n* �*α*

*γ*0ð Þ¼ *x*, *λ c* � *x*

obtain the following asymptotics for *γ*0ð Þ *x*, *λ* at infinity:

1 þ *α* þ

4 � exp � 2

, then

2*n*�1

� exp � 1 � 3 � … � ð Þ 2*n* � 3

2*α* ∉ *N* we set *n* ¼ *α*þ1

2 exp � *x*1þ*α*

In particular, with *α* ¼ 1

*γ*0ð Þ¼ *x*, *λ cx*�1

2*α* ¼ *n*∈ *N*, i.e. *α* ¼ 1

**49**

*a*

*c* � exp � *x*1þ*α*

¼ *c* � exp � *x*1þ*α*

*γ*0ð Þ¼ *x; λ c* � exp � *x*1þ*α*

In case *α*þ1

*γ*0ð Þ¼ *x*, *λ c* � *x*�*α*

The solution Ψð Þ *x*, *λ* given by Ψð Þ¼ *x*, *λ* Ψ^ð Þ *x*, *λ I* þ *C*~ð Þ*λ* � ��1 is subject to first from condition (13). Use (12) to differentiate (27), then find the asymptotes of ~ Ψ0 ð Þ *x*, *λ* as *x* ! ∞ similarly to (21) to obtain the second part of formula (13). Theorem is proved. □

In this section, the fundamental system of solution is constructed for an operator differential equation with a rapidly increasing at infinity potential.

#### **2.2 Asymptotic of the fundamental system solutions of equation with block-triangular potential**

Now consider the case when *v x*ð Þ¼ *x*2*α*, 0 <*α* ≤1 and coefficients of Eq. (1) satisfy the condition (6). Rewrite Eq. (1) in the form

$$-\overline{\mathfrak{y}}'' + \left(\mathfrak{x}^{2a} - \lambda + q(\mathfrak{x}, \lambda)\right)\overline{\mathfrak{y}} = (q(\mathfrak{x}, \lambda) \cdot I - U(\mathfrak{x}))\overline{\mathfrak{y}},\tag{29}$$

where *q x*ð Þ , *λ* determined by a formula

$$q(\mathbf{x}, \boldsymbol{\lambda}) = \frac{5\alpha^2}{4} \left( \frac{\mathbf{x}^{2a-1}}{\mathbf{x}^{2a} - \boldsymbol{\lambda}} \right)^2 - \frac{a(2a-1)\mathbf{x}^{2a-2}}{2(\mathbf{x}^{2a} - \boldsymbol{\lambda})}.\tag{30}$$

Denote

$$\gamma\_0(\mathbf{x}, \boldsymbol{\lambda}) = \frac{1}{\sqrt[4]{4(\mathbf{x}^{2a} - \boldsymbol{\lambda})}} \cdot \exp\left(-\int\_a^x \sqrt{u^{2a} - \boldsymbol{\lambda}} du\right),\tag{31}$$

$$\chi\_{\infty}(\mathbf{x}, \boldsymbol{\lambda}) = \frac{1}{\sqrt[4]{4(\boldsymbol{\omega}^{2a} - \boldsymbol{\lambda})}} \cdot \exp\left(\int\_{a}^{\boldsymbol{\chi}} \sqrt{u^{2a} - \boldsymbol{\lambda}} du\right). \tag{32}$$

There solutions constitute a fundamental system of solutions of the scalar differential equation �*z*00 þ *x*ð Þ 2*α* � *λ* þ *q x*ð Þ , *λ z* ¼ 0, in such a way that for all *x*∈½ Þ 0, ∞ one has *W γ*0 ð Þ , *γ*∞ ≔ *γ*0ð Þ� *x*, *λ γ*0 ∞ð Þ� *x*, *λ γ*0 0ð Þ� *x*, *λ γ*∞ð Þ¼ *x*, *λ* 1.

We are about to establish the asymptotics1 of *γ*0ð Þ *x*, *λ* as *x* ! ∞:

$$\chi\_0(\mathbf{x}, \boldsymbol{\lambda}) = (2\mathbf{x}^a)^{-\frac{1}{2}} \cdot \left(\mathbf{1} - \frac{\boldsymbol{\lambda}}{\mathbf{x}^{2a}}\right)^{-\frac{1}{4}} \cdot \exp\left(-\int\_a^x \mathbf{u}^a \left(\mathbf{1} - \frac{\boldsymbol{\lambda}}{u^{2a}}\right)^{\frac{1}{2}} du\right). \tag{33}$$

1 For *α* ¼ 1 and *α* ¼ 1 2 , i.e., for *v x*ð Þ¼ *x*2 and *v x*ð Þ¼ *x*, the asymptotics of the functions *γ*0ð Þ *x*, *λ* and *γ*∞ð Þ *x*, *λ* is known.

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular… DOI: http://dx.doi.org/10.5772/intechopen.95820*

After expanding here the integral, we obtain the exponential as follows

$$\exp\left(-\int\_{a}^{\mathbf{x}} u^{a} \cdot \left(\mathbf{1} - \frac{1}{2} \cdot \frac{\lambda}{u^{2a}} - \sum\_{k=2}^{\infty} \frac{\mathbf{1} \cdot \mathbf{3} \cdot \ldots \cdot (2k-3)}{k! \cdot 2^{k}} \cdot \left(\frac{\lambda}{u^{2a}}\right)^{k}\right) du\right). \tag{34}$$

In case *α*þ1 2*α* ¼ *n* ∈ *N*, i.e. *α* ¼ 1 2*n*�1 , this expression after integration acquires the form:

$$c \cdot \exp\left(-\frac{\mathbf{x}^{1+a}}{1+a} + \frac{\lambda}{2} \cdot \frac{\mathbf{x}^{1-a}}{1-a} + \sum\_{k=2}^{n-1} \frac{\mathbf{1} \cdot 3 \cdot \dots \cdot (2k-3)}{k!} \cdot \binom{\lambda}{2}^k \cdot \frac{\mathbf{x}^{1-(2k-1)a}}{1-(2k-1)a}\right)$$

$$\cdot \exp\left(\frac{\mathbf{1} \cdot 3 \cdot \dots \cdot (2n-3)}{n!} \cdot \left(\frac{\lambda}{2}\right)^n \cdot \ln x + o(1)\right) =$$

$$= c \cdot \exp\left(-\frac{\mathbf{x}^{1+a}}{1+a} + \frac{\lambda}{2} \cdot \frac{\mathbf{x}^{1-a}}{1-a} + \sum\_{k=2}^{n-1} \frac{\mathbf{1} \cdot 3 \cdot \dots \cdot (2k-3)}{k!} \cdot \binom{\lambda}{2}^k \cdot \frac{\mathbf{x}^{1-(2k-1)a}}{1-(2k-1)a}\right).$$

$$\cdot \mathbf{x}^{\frac{\mathbf{1}\cdot3 \dots (2n-3)}{n!} \left(\frac{\lambda}{2}\right)^n} \cdot (1+o(1)).\tag{35}$$

The asymptotics of *γ*0ð Þ *x*, *λ* as *x* ! ∞ is as follows:

$$\gamma\_0(\mathbf{x}, \boldsymbol{\lambda}) = \boldsymbol{c} \cdot \exp\left(-\frac{\mathbf{x}^{1+a}}{\mathbf{1}+a} + \frac{\boldsymbol{\lambda}}{2} \cdot \frac{\mathbf{x}^{1-a}}{\mathbf{1}-a} + \sum\_{k=2}^{n-1} \frac{\mathbf{1} \cdot \mathbf{3} \cdot \dots \cdot (2k-3)}{k!} \cdot \binom{\boldsymbol{\lambda}}{2}^k \cdot \frac{\mathbf{x}^{1-(2k-1)a}}{\mathbf{1}-(2k-1)a}\right). \tag{36}$$

$$\propto \frac{\mathbf{x}^{\frac{1\boldsymbol{3}\cdot\dots(2k-3)}{n}}(\frac{\mathbf{1}}{2})^n \cdot \frac{\mathbf{-}}{2} \cdot (\mathbf{1} + o(\mathbf{1})). \tag{37}$$

In particular, for *α* ¼ 1 ð Þ *n* ¼ 1 , *γ*0ð Þ *x*, *λ* has the following asymptotics at infinity:

$$\gamma\_0(\varkappa, \lambda) = c \cdot \varkappa^{\frac{\lambda - 1}{2}} \cdot \exp\left(-\frac{\varkappa^2}{2}\right) (1 + o(1)). \tag{37}$$

In case *α*þ1 2*α* ∉ *N* we set *n* ¼ *α*þ1 2*α* � � þ 1, with ½ � *β* being the integral part of *β*, to obtain the following asymptotics for *γ*0ð Þ *x*, *λ* at infinity:

$$\begin{split} \gamma\_{0}(\mathbf{x},\boldsymbol{\lambda}) = & c \cdot \mathbf{x}^{-\frac{\sigma}{2}} \exp\left(-\frac{\mathbf{x}^{1+a}}{\mathbf{1}+a} + \frac{\boldsymbol{\lambda}}{2} \cdot \frac{\mathbf{x}^{1-a}}{\mathbf{1}-a} + \sum\_{k=2}^{n-1} \frac{\mathbf{1} \cdot \mathbf{3} \cdot \dots \cdot (2k-3)}{k!} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{k} \cdot \frac{\mathbf{x}^{1-(2k-1)a}}{\mathbf{1} - (2k-1)a}\right) \cdot \mathbf{1} \\ & \cdot \exp\left(-\frac{\mathbf{1} \cdot \mathbf{3} \cdot \dots \cdot (2n-3)}{n!} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} \cdot \frac{\mathbf{x}^{-a}}{a}\right) \cdot (\mathbf{1} + o(\mathbf{x}^{-a})) \end{split} \tag{38}$$

In particular, with *α* ¼ 1 2 ð Þ *n* ¼ 2 one has

$$\chi\_0(\mathbf{x}, \boldsymbol{\lambda}) = c\boldsymbol{\lambda}^{-\frac{1}{4}} \cdot \exp\left(-\frac{2}{3}\boldsymbol{\lambda}^{\frac{1}{2}} + \boldsymbol{\lambda}\boldsymbol{\tilde{x}}^{\frac{1}{2}} - \left(\frac{\boldsymbol{\lambda}}{2}\right)^2 \boldsymbol{\lambda}^{-\frac{1}{2}}\right) \cdot \left(\mathbf{1} + o\left(\boldsymbol{\lambda}^{-\frac{1}{2}}\right)\right). \tag{39}$$

A similar procedure allows to establish the asymptotics of *γ*∞ð Þ *x* as *x* ! ∞*:* If *α*þ1 2*α* ¼ *n*∈ *N*, i.e. *α* ¼ 1 2*n*�1 , then

where *C*0ð Þ*λ* is some block-triangular operator. Thus lim*x*!∞

*Recent Developments in the Solution of Nonlinear Differential Equations*

0

BBB@

The solution Ψð Þ *x*, *λ* given by Ψð Þ¼ *x*, *λ* Ψ^ð Þ *x*, *λ I* þ *C*~ð Þ*λ* � ��1

differential equation with a rapidly increasing at infinity potential.

**2.2 Asymptotic of the fundamental system solutions of equation with**

from condition (13). Use (12) to differentiate (27), then find the asymptotes of

ð Þ *x*, *λ* as *x* ! ∞ similarly to (21) to obtain the second part of formula (13).

In this section, the fundamental system of solution is constructed for an operator

Now consider the case when *v x*ð Þ¼ *x*2*α*, 0 <*α* ≤1 and coefficients of Eq. (1)

*x*2*α*�1 *x*2*α* � *λ* � �2

4 *x*ð Þ 2*α* � *λ* p4 � exp �

There solutions constitute a fundamental system of solutions of the scalar differential equation �*z*00 þ *x*ð Þ 2*α* � *λ* þ *q x*ð Þ , *λ z* ¼ 0, in such a way that for all *x*∈½ Þ 0, ∞

� exp �

0 @

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 *x*ð Þ 2*α* � *λ* p4 � exp

∞ð Þ� *x*, *λ γ*0

4

We are about to establish the asymptotics1 of *γ*0ð Þ *x*, *λ* as *x* ! ∞:

*x*2*α* � ��1

2 � 1 � *λ*

�*y*00 þ *x*2*α* � *λ* þ *q x*ð Þ , *λ* � �*y* ¼ ð Þ *q x*ð Þ� , *λ I* � *U x*ð Þ *y*, (29)

� *α*ð Þ 2*α* � 1 *x*2*α*�2

ð *x*

0 @

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *u*2*α* � *λ* p *du*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *u*2*α* � *λ* p *du*

*uα* 1 � *λ*

*u*2*α* � �1

2 *du*

1

A*:* (33)

*a*

ð *x*

0 @

*a*

0ð Þ� *x*, *λ γ*∞ð Þ¼ *x*, *λ* 1.

ð *x*

*a*

, i.e., for *v x*ð Þ¼ *x*2 and *v x*ð Þ¼ *x*, the asymptotics of the functions *γ*0ð Þ *x*, *λ* and

2 *x*ð Þ 2*α* � *λ :* (30)

1

1

A, (31)

A*:* (32)

*γ*∞ð Þ *x* ¼ *Ik*, *k* ¼ 1,*r* and in (24) has

0 *C*12ð Þ*λ* … *C*1*r*ð Þ*λ* 0 0 … *C*2*r*ð Þ*λ* ………… 0 0 … 0

Ψ*kk*ð Þ *x*, *λ*

*C*~ð Þ¼ *λ*

satisfy the condition (6). Rewrite Eq. (1) in the form

*q x*ð Þ¼ , *λ* 5*α*2

*γ*0ð Þ¼ *x*, *λ* 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*γ*∞ð Þ¼ *x*, *λ*

one has *W γ*0 ð Þ , *γ*∞ ≔ *γ*0ð Þ� *x*, *λ γ*0

*γ*0ð Þ¼ *x*, *λ* 2*xα* ð Þ�1

2

1 For *α* ¼ 1 and *α* ¼ 1

*γ*∞ð Þ *x*, *λ* is known.

**48**

4

where *q x*ð Þ , *λ* determined by a formula

hence, by virtue (26), lim*x*!∞

Theorem is proved. □

Denote

**block-triangular potential**

~ Ψ0 Ψ^ð Þ *x*, *λ γ*∞ð Þ *x* ¼ lim*x*!∞

1

Ψ~ð Þ *x*, *λ γ*∞ð Þ *x* ,

CCCA*:* (28)

is subject to first

$$\begin{split} \boldsymbol{\chi}\_{\boldsymbol{\omega}}(\boldsymbol{x},\boldsymbol{\lambda}) = & c \cdot \exp\left(\frac{\mathbf{x}^{1+a}}{\mathbf{1}+a} - \frac{\boldsymbol{\lambda}}{2} \cdot \frac{\mathbf{x}^{1-a}}{\mathbf{1}-a} - \sum\_{k=2}^{n-1} \frac{\mathbf{1} \cdot \mathbf{3} \cdot \dots \cdot (2k-3)}{k!} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{k} \cdot \frac{\mathbf{x}^{1-(2k-1)a}}{\mathbf{1} - (2k-1)a}\right) \cdot \\ & \cdot \boldsymbol{\chi}^{-\left(\frac{3\mathbf{3}\dots(2k-3)}{n!} \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} + \frac{\mathbf{1}}{2}\right)} \cdot (\mathbf{1} + o(\mathbf{1})). \end{split} \tag{40}$$

With *α* ¼ 1 ð Þ *n* ¼ 1 , this becomes

$$\gamma\_{\infty}(\mathbf{x}, \boldsymbol{\lambda}) = \boldsymbol{\sigma} \cdot \boldsymbol{\pi}^{-\frac{l+1}{2}} \cdot \exp\left(\frac{\boldsymbol{\pi}^2}{2}\right) (\mathbf{1} + o(\mathbf{1})). \tag{41}$$

The remaining statements of Theorem 2.1 are proved similarly. □

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular…*

2

*γ*∞ð Þ *x*, *λ you can take the following functions.*

*DOI: http://dx.doi.org/10.5772/intechopen.95820*

*λ*�1

4 � exp � 2

2 � exp � *x*2

3 *x* 3 2 þ *λx* 1 2 , *γ*∞ð Þ¼ *x*, *λ x*�1

for *λ* ¼ 2*n* þ 1 has the solution *φn*ð Þ¼ *x Hn*ð Þ� *x exp* � *x*2

following asymptotics at infinity: *φn*ð Þ¼ *x* ð Þ 2*x n* � *exp* � *x*2

2 � exp � *x*2

yields the solution proportional to *φn*ð Þ *x* .

**block – triangular coefficients**

which satisfy the conditions

**51**

*lk yk*

¼ �*yk*

2

**3. Green's function for an operator differential equation with**

Let us suppose that at the *x* ¼ 0 given boundary conditions

� *π* 2

cos *A* � *y*0

**Remark 2.1** It is known that scalar equation

*γ*0ð Þ¼ *x*, *λ x*

*If α* ¼ 1 2

ð Þ 2*x n*

*γ*0ð Þ¼ *x*, *λ x*�1

infinity: *φ*ð Þ¼ *x*, *λ* ð Þ *x λ*�1

one increasing.

From Theorem 2.2 and the asymptotic formulas (37), (39), (41), (43) follows. **Corollary 2.1** *If α* ¼ 1*, i.e. v x*ð Þ¼ *x*2*, then, under condition (6), the solutions* Φð Þ *x*, *λ and* Ψð Þ *x*, *λ have common (known) asymptotics, as in the quality γ*0ð Þ *x*, *λ and*

, *::γ*∞ð Þ¼ *x*, *λ x*�*λ*þ1

*, i.e. the coefficient v x*ð Þ¼ *x, and the condition (6) holds, then.*

Chebyshev – Hermitre polynomial, that at *x* ! ∞ has next asymptotics *Hn*ð Þ¼ *x*

ð Þ 1 þ *o*ð Þ1 . Hence the solution *φn*ð Þ *x* of the Eq. (49) at *x* ! ∞ will have the

In the case of *U x*ð Þ¼ 0, *v x*ð Þ¼ *x*2 in (2), the Eq. (1) is splitting into infinity system scalar equations of the form (49). The operator solution Φð Þ *x*, *λ* will be diagonal in this case. Denote by *φ*ð Þ *x*, *λ* the diagonal elements of the operator Φð Þ *x*, *λ* . Then, by Corollary 2.1, the solution *φ*ð Þ *x*, *λ* will have the following asymptotics at

In this section, the asymptotics of the fundamental system of solutions for the Sturm-Liouville equation with block-triangular operator potential, increasing at infinity is established. One of the solutions is found decreasing at infinity, the other

where *A*- the block-triangular operator of the same structure as the coefficients

*Ik* < < *Akk* ≤ *π*

2

00 þ ð Þ *v x*ð Þ*Ik* þ *Ukk*ð Þ *x yk* ¼ *λyk*, *k* ¼ 1,*r* (52)

of the differential equation, *Akk*, *k* ¼ 1,*r*- bounded self-adjoint operators in *Hk*,

Together with the problem (1), (50) we consider the separated system

2 � exp

4 � exp

�*φ*00 þ *x*2 � *φ* ¼ *λφ* (49)

2

2

ð Þ� 0 sin *A* � *y*ð Þ¼ 0 0, (50)

*Ik:* (51)

ð Þ 1 þ *o*ð Þ1 . In particular, for *λ* ¼ 2*n* þ 1, this

� ð Þ 1 þ *o*ð Þ1 .

, where *Hn*ð Þ *x* is the

*x*2 2

2 3 *x* 3 2 � *λx* 1 2 *:* (48)

*:* (47)

In case *α*þ1 2*α* ∉ *N*, we set *n* ¼ *α*þ1 2*α* � � þ 1 to get the asymptotics.

$$\gamma\_{\boldsymbol{\alpha}}(\boldsymbol{x},\boldsymbol{\lambda}) = \boldsymbol{c} \cdot \boldsymbol{x}^{-\frac{a}{2}} \exp\left(-\frac{\mathbf{x}^{1+a}}{\mathbf{1}+a} + \frac{\boldsymbol{\lambda}}{2} \cdot \frac{\mathbf{x}^{1-a}}{\mathbf{1}-a} + \sum\_{k=2}^{n-1} \frac{\mathbf{1} \cdot \mathbf{3} \cdot \dots \cdot (2k-3)}{k!} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{k} \cdot \frac{\mathbf{x}^{1-(2k-1)a}}{\mathbf{1} - (2k-1)a}\right) \cdot \boldsymbol{\lambda} \cdot \mathbf{x}^{-a} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} \cdot \boldsymbol{\lambda}^{-a} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} \cdot \boldsymbol{\lambda}^{-a} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} \cdot \boldsymbol{\lambda}^{-a} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} \cdot \boldsymbol{\lambda}^{-a} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} \cdot \boldsymbol{\lambda}^{-a} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} \cdot \boldsymbol{\lambda}^{-a} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} \cdot \boldsymbol{\lambda}^{-a} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} \cdot \boldsymbol{\lambda}^{-a} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} \cdot \boldsymbol{\lambda}^{-a} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} \cdot \boldsymbol{\lambda}^{-a} \cdot \left(\frac{\boldsymbol{\lambda}}{2}\right)^{n} \cdot \boldsymbol{\lambda}^{-a} \cdot \left(\$$

In case *α* ¼ 1 2 ð Þ *n* ¼ 2 , one has

$$\gamma\_{\infty}(\mathbf{x}, \boldsymbol{\lambda}) = c\mathbf{x}^{-\frac{1}{4}} \cdot \exp\left(\frac{2}{3}\mathbf{x}^{\frac{3}{2}} - \lambda \mathbf{x}^{\frac{1}{2}} + \left(\frac{\boldsymbol{\lambda}}{2}\right)^{2}\mathbf{x}^{-\frac{1}{2}}\right) \cdot \left(1 + o\left(\mathbf{x}^{-\frac{1}{2}}\right)\right). \tag{43}$$

**Theorem 2.2** *Under* 0<*α* ≤1 *and condition (6), the statement of Theorem 2.1 is also valid for Eq. (1).*

*Proof* is similar to Theorem 2.1. Moreover, note that

$$\begin{split} & \left| C(\mathbf{x}, t, \boldsymbol{\lambda}) \cdot \frac{\boldsymbol{\chi}\_{0}(t, \boldsymbol{\lambda})}{\boldsymbol{\chi}\_{0}(\mathbf{x}, \boldsymbol{\lambda})} \right| = \left| \nu\_{0}^{2}(t, \boldsymbol{\lambda}) \cdot \frac{\boldsymbol{\chi}\_{\text{oc}}(\mathbf{x}, \boldsymbol{\lambda})}{\boldsymbol{\chi}\_{0}(\mathbf{x}, \boldsymbol{\lambda})} - \boldsymbol{\chi}\_{0}(t, \boldsymbol{\lambda}) \cdot \boldsymbol{\chi}\_{\text{oc}}(t, \boldsymbol{\lambda}) \right| = \\ &= \left| \frac{1}{2\sqrt{t^{2\alpha} - \boldsymbol{\lambda}}} \cdot \exp\left( -2\int\_{\boldsymbol{u}}^{t} \sqrt{u^{2\alpha} - \boldsymbol{\lambda}} du \right) \cdot \exp\left( 2\int\_{\boldsymbol{u}}^{\mathbf{x}} \sqrt{u^{2\alpha} - \boldsymbol{\lambda}} du \right) - \frac{1}{2\sqrt{t^{2\alpha} - \boldsymbol{\lambda}}} \right| \\ &= \frac{1}{2\sqrt{t^{2\alpha} - \boldsymbol{\lambda}}} \cdot \left| \exp\left( -2\int\_{\boldsymbol{x}}^{t} \sqrt{u^{2\alpha} - \boldsymbol{\lambda}} du - 1 \right) \right| . \end{split} \tag{44}$$

As *x*≤ *t*, one has exp �2 Ð*t x* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *u*2*α* � *λ* p *du* � � ≤ 1, and that is why

$$\left| \mathbf{C}(\mathbf{x}, t, \lambda) \cdot \frac{\chi\_0(t, \lambda)}{\chi\_0(\mathbf{x}, \lambda)} \right| \le \frac{1}{\sqrt{t^{2a} - \lambda}}.\tag{45}$$

Hence

$$|R(\mathbf{x}, \mathbf{t}, \boldsymbol{\lambda})| = \left| \mathbf{C}(\mathbf{x}, \mathbf{t}, \boldsymbol{\lambda}) \cdot \frac{\boldsymbol{\chi}\_0(\mathbf{t}, \boldsymbol{\lambda})}{\boldsymbol{\chi}\_0(\mathbf{x}, \boldsymbol{\lambda})} \cdot [q(\mathbf{t}, \boldsymbol{\lambda}) \cdot \boldsymbol{I} - U(\mathbf{t})] \right| \leq \frac{1}{\sqrt{t^{2a} - \lambda}} (|q(\mathbf{t}, \boldsymbol{\lambda})| + |U(\mathbf{t})|). \tag{46}$$

By virtue of (6) and (30), 1ffiffiffiffiffiffiffiffi *t*2*α*�*λ* p ð Þ j j *q t*ð Þ , *λ* þ j j *U t*ð Þ ∈ *L a*ð Þ , ∞ and therefore integral equation has a unique solution *χ*ð Þ *x*, *λ* and j j *χ*ð Þ *x*, *λ* ≤ const. By (17), one has that lim *x*!∞*χ*ð Þ¼ *x*, *λ I*, where the first part of formula (12) follows from.

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular… DOI: http://dx.doi.org/10.5772/intechopen.95820*

The remaining statements of Theorem 2.1 are proved similarly. □

From Theorem 2.2 and the asymptotic formulas (37), (39), (41), (43) follows. **Corollary 2.1** *If α* ¼ 1*, i.e. v x*ð Þ¼ *x*2*, then, under condition (6), the solutions* Φð Þ *x*, *λ and* Ψð Þ *x*, *λ have common (known) asymptotics, as in the quality γ*0ð Þ *x*, *λ and*

$$\boldsymbol{\gamma}\_{0}(\mathbf{x},\boldsymbol{\lambda}) = \boldsymbol{\pi}^{\frac{\boldsymbol{\omega}-1}{2}} \cdot \exp\left(-\frac{\boldsymbol{\pi}^{2}}{2}\right), \boldsymbol{\dots}\_{\boldsymbol{\omega}}(\mathbf{x},\boldsymbol{\lambda}) = \boldsymbol{\pi}^{-\frac{\boldsymbol{\lambda}+1}{2}} \cdot \exp\left(\frac{\boldsymbol{\pi}^{2}}{2}\right). \tag{47}$$

*If α* ¼ 1 2 *, i.e. the coefficient v x*ð Þ¼ *x, and the condition (6) holds, then.*

$$\gamma\_0(\mathbf{x}, \boldsymbol{\lambda}) = \mathbf{x}^{-\frac{1}{4}} \cdot \exp\left(-\frac{2}{3}\mathbf{x}^{\frac{1}{4}} + \boldsymbol{\lambda}\mathbf{x}^{\frac{1}{4}}\right), \\ \gamma\_\infty(\mathbf{x}, \boldsymbol{\lambda}) = \mathbf{x}^{-\frac{1}{4}} \cdot \exp\left(\frac{2}{3}\mathbf{x}^{\frac{1}{4}} - \boldsymbol{\lambda}\mathbf{x}^{\frac{1}{4}}\right). \tag{48}$$

**Remark 2.1** It is known that scalar equation

*γ*∞ð Þ *x*, *λ you can take the following functions.*

$$-\varphi'' + \mathfrak{x}^2 \cdot \varphi = \lambda \varphi \tag{49}$$

for *λ* ¼ 2*n* þ 1 has the solution *φn*ð Þ¼ *x Hn*ð Þ� *x exp* � *x*2 2 , where *Hn*ð Þ *x* is the Chebyshev – Hermitre polynomial, that at *x* ! ∞ has next asymptotics *Hn*ð Þ¼ *x* ð Þ 2*x n* ð Þ 1 þ *o*ð Þ1 . Hence the solution *φn*ð Þ *x* of the Eq. (49) at *x* ! ∞ will have the following asymptotics at infinity: *φn*ð Þ¼ *x* ð Þ 2*x n* � *exp* � *x*2 2 � ð Þ 1 þ *o*ð Þ1 .

In the case of *U x*ð Þ¼ 0, *v x*ð Þ¼ *x*2 in (2), the Eq. (1) is splitting into infinity system scalar equations of the form (49). The operator solution Φð Þ *x*, *λ* will be diagonal in this case. Denote by *φ*ð Þ *x*, *λ* the diagonal elements of the operator Φð Þ *x*, *λ* . Then, by Corollary 2.1, the solution *φ*ð Þ *x*, *λ* will have the following asymptotics at infinity: *φ*ð Þ¼ *x*, *λ* ð Þ *x λ*�1 2 � exp � *x*2 2 ð Þ 1 þ *o*ð Þ1 . In particular, for *λ* ¼ 2*n* þ 1, this yields the solution proportional to *φn*ð Þ *x* .

In this section, the asymptotics of the fundamental system of solutions for the Sturm-Liouville equation with block-triangular operator potential, increasing at infinity is established. One of the solutions is found decreasing at infinity, the other one increasing.

#### **3. Green's function for an operator differential equation with block – triangular coefficients**

Let us suppose that at the *x* ¼ 0 given boundary conditions

$$
\cos A \cdot \overline{\mathfrak{y}}'(\mathbf{0}) - \sin A \cdot \overline{\mathfrak{y}}(\mathbf{0}) = \mathbf{0},\tag{50}
$$

where *A*- the block-triangular operator of the same structure as the coefficients of the differential equation, *Akk*, *k* ¼ 1,*r*- bounded self-adjoint operators in *Hk*, which satisfy the conditions

$$-\frac{\pi}{2}I\_k < < A\_{kk} \le \frac{\pi}{2}I\_k. \tag{51}$$

Together with the problem (1), (50) we consider the separated system

$$\mathcal{I}\_k \left[ \mathcal{Y}\_k \right] = -\mathcal{Y}\_k'' + (v(\mathfrak{x})I\_k + U\_{kk}(\mathfrak{x})) \mathcal{Y}\_k = \lambda \mathcal{Y}\_k, \ \ k = \overline{1, r} \tag{52}$$

*γ*∞ð Þ¼ *x*, *λ c* � exp *x*1þ*α*

In case *α*þ1

*γ*∞ð Þ¼ *x*, *λ c* � *x*�*α*

In case *α* ¼ 1

*valid for Eq. (1).*

¼ 1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi

> ¼ 1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi

� � � � � �

Hence

**50**

� � � �

*γ*∞ð Þ¼ *x*, *λ cx*�1

1 þ *α* � *λ*

With *α* ¼ 1 ð Þ *n* ¼ 1 , this becomes

2*α* ∉ *N*, we set *n* ¼ *α*þ1

2 ð Þ *n* ¼ 2 , one has

4 � exp

*C x*ð Þ� , *t*, *λ γ*0ð Þ *t*, *λ*

0 @

*t*2*α* � *λ* p � exp �2

As *x*≤ *t*, one has exp �2

j j *R x*ð Þ , *t*, *λ* ¼ *C x*ð Þ� , *t*, *λ γ*0ð Þ *t*, *λ*

By virtue of (6) and (30), 1ffiffiffiffiffiffiffiffi

� � � �

*t*2*α* � *λ* p � exp �2

� � � � � � 1 þ *α* þ

2 exp � *x*1þ*α*

� exp

2 � *x*1�*α* 1 � *α* �X*n*�1

*Recent Developments in the Solution of Nonlinear Differential Equations*

*γ*∞ð Þ¼ *x*, *λ c* � *x*�*λ*þ1

*λ* 2 � *x*1�*α* 1 � *α* þX*n*�1 *k*¼2

1 � 3 � … � ð Þ 2*n* � 3

2 3 *x* 3 2 � *λx* 1 2 þ

*Proof* is similar to Theorem 2.1. Moreover, note that

� � � � ¼ *γ*2

� � � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *u*2*α* � *λ* p *du*

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *u*2*α* � *λ* p *du* � 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *u*2*α* � *λ* p *du* � �

*C x*ð Þ� , *t*, *λ γ*0ð Þ *t*, *λ*

*γ*0ð Þ *x*, *λ* � ½ � *q t*ð Þ� , *λ I* � *U t*ð Þ

lim *x*!∞*χ*ð Þ¼ *x*, *λ I*, where the first part of formula (12) follows from.

*γ*0ð Þ *x*, *λ*

gral equation has a unique solution *χ*ð Þ *x*, *λ* and j j *χ*ð Þ *x*, *λ* ≤ const.  (54) decreasing and increasing at infinity will be denoted by Φ1ð Þ *x*, *λ* , Ψ1ð Þ *x*, *λ* , and the corresponding solutions of the Eq. (59) denote by Φ~ ð Þ *x*, *λ* and Ψ~ð Þ *x*, *λ* . For the system operator solutions *Y x*ð Þ , *λ* , Φ~ ð Þ *x*, *λ* ∈*B*ð Þ **H** of the Eqs. (1) and (59), respectively, will take the form of Wronskian *W* Φ~ , *Y* � � ¼

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular…*

ð Þ *x*, *λ* and do not depend on *x*.

In the following theorem it is proved that the operator function *G x*ð Þ , *t*, *λ*

**Theorem 3.2** *The operator function G x*ð Þ , *t*, *λ is the Green's function of the*

*on each of the intervals* ½ Þ 0, *t and t*ð Þ , ∞ *, and at x* ¼ *t it has the jump*

2.*For any fixed t, the function G x*ð Þ , *t*, *λ has a continuous derivative with respect to x*

0

3.*For a fixed t, the function G x*ð Þ , *t*, *λ of the variable x is an operator solution of Eq. (1) on each of the intervals* ½ Þ 0, *t , t*ð Þ , ∞ *, and it satisfies the boundary condition (50), and at a fixed x function G x*ð Þ , *t*, *λ of the variable t is an operator solution of the Eq. (59) on each of the intervals* ½ Þ 0, *x* , ð Þ *x*, ∞ *, and it satisfies the boundary*

**Proof** The function *G x*ð Þ , *t*, *λ* is continuous with respect to *x* at each of the intervals 0, ½ Þ*t* and ð Þ *t*, ∞ . Similarly to the variable *t*. To prove the continuity of the

Φ~ ð Þþ *x*, *λ* Φð Þ *x*, *λ W Y*~, Φ � � � � �1

ð Þ *x*, *λ Y x*ð Þ , *λ*

ð Þ *x*, *λ*

ð Þ *x*, *λ Y*~0

ð Þ *x*, *λ Y*~0

ð Þ *x*, *λ* Φ~ 0

� *Y*~�1

ð Þ� *x*, *λ Y*~�1

ð Þ� *x*, *λ* Φ~ �1

function *G x*ð Þ , *t*, *λ* for all *x*, *t*≥0, it is sufficient that the identity shown as

ð Þ� *x*, *λ* Φ~ 0

� ��1

ð Þ *x*, *λ* Φð Þ� *x*, *λ Y x* ~ð Þ , *λ* Φ0

ð Þ *x*, *λ*

ð Þ *x*, *λ* Φ~ 0

� ��1

Φ~ ð Þ *t*, *λ* 0≤*x*≤ *t*

*Y t* ~ ð Þ , *λ x*≥*t*

ð Þ¼� *x* � 0, *x*, *λ I:* (61)

*Y x* ~ ð Þ� , *λ* 0*:* (62)

*Y x* ~ ð Þ� , *λ* 0 (63)

ð Þ *x*, *λ* Φ�1

ð Þ *x*, *λ* Φ�1

ð Þ *x*, *λ* Φ�1

ð Þ *x*, *λ*

ð Þ *x*, *λ* , (64)

ð Þ *x*, *λ* (65)

,

Φ~ ð Þ� *x*, *λ*

ð Þ� *x*, *λ* Φ0

ð Þ� *x*, *λ* Φ0

ð Þ� *x*, *λ* Φ0

� ��1

: *:* (60)

*Y x*ð Þ , *λ W* Φ~ , *Y* � � � � �1

�Φð Þ *x*, *λ W Y*~, Φ � � � � �1

1.*The function G x*ð Þ , *t*, *λ is continuous for all values x*, *t*∈½ Þ 0, ∞ *;*

ð Þ� *x* þ 0, *x*, *λ Gx*

ð Þ� 0 cos *A* � ~*y*ð Þ� 0 sin *A* ¼ 0*:*

is satisfied for all *x*≥0*:* This identity shown as

ð Þ *x*, *λ* Φ~ 0

ð Þ *x*, *λ Y*~0

*Y x*ð Þ , *λ* Φ~ ð Þ *x*, *λ Y*0

possesses all the classical properties of the Green's function.

Φ~ 0

ð Þ *x*, *λ Y x*ð Þ� , *λ* Φ~ ð Þ *x*, *λ Y*0

*G x*ð Þ¼ , *t*, *λ*

*differential operator L, i.e.:*

*condition* ~*y*0

*Y x*ð Þ , *λ W* Φ~ , *Y* � � � � �1

�Φð Þ *x*, *λ Y*~0

ð Þ� *x*, *λ* Φ~ �1

ð Þ� *x*, *λ Y*~�1

� ��1

ð Þ� *x*, *λ* Φ~ �1

or

*Y*0

ð Þ *x*, *λ Y*�1

ð Þ *x*, *λ Y*�1

ð Þ *x*, *λ Y*�1

which is equivalent to

*Y*0

*Y*0

**53**

8 <

*DOI: http://dx.doi.org/10.5772/intechopen.95820*

*Gx* 0

Let us designate

Let *L*0 denote the minimal differential operator generated by differential expression *l*½ � *y* (1) and the boundary condition (50), and let *Lk* 0 , *k* ¼ 1,*r* denote the minimal differential operator on *L*2ð Þ *Hk*, 0, ð Þ ∞ generated by differential expression *lk yk* and the boundary conditions (53). Taking into account the conditions on coefficients, as well as sufficient smallness of perturbations *Ukk*ð Þ *x* and conditions (51), we conclude that, for every symmetric operator *Lk* 0 , there is a case of limit point at infinity. Hence their self-adjoint extensions *Lk* are the closures of operators *Lk* 0 respectively. The operators *Lk* are semi-bounded below, and their spectra are discrete.

Let *L* denote the operator extensions *L*0 , by requiring that *L*2ð Þ **H**, 0, ð Þ ∞ be the domain of operator *L*.

The following theorem is proved in [4].

**Theorem 3.1** *Suppose that, for Eq. (1) conditions (3)-(5) are satisfied for α* >1 *or condition (6) for* 0<*α* ≤1*. Then the discrete spectrum of the operator L is real and coincides with the union of spectra of the self-adjoint operators Lk*, *k* ¼ 1,*r, i.e., σd*ð Þ¼ *L* ∪*r k*¼1*σ*ð Þ *Lk* .

**Comment 3.1** Note that this theorem contains a statement of the discrete spectrum of the non-self-adjoint operator *L* only and no allegations of its continuous and residual spectrum.

Along with the Eq. (1) we consider the equation

$$l\_1[\overline{\mathfrak{y}}] = -\overline{\mathfrak{y}}\,' + V^\*\left(\mathfrak{x}\right)\overline{\mathfrak{y}} = \lambda \overline{\mathfrak{y}} \tag{54}$$

(*V* ∗ ð Þ *x* is adjoint to the operator *V x*ð Þ). If the space **H** is finite-dimensional, then the Eq. (54) can be rewritten as

$$
\tilde{I}[\tilde{\mathbf{y}}] = -\tilde{\mathbf{y}}'' + \tilde{\mathbf{y}}V(\mathbf{x}) = \lambda \tilde{\mathbf{y}},\tag{55}
$$

where ~*y* ¼ ~*y*1 ~*y*2 …~*yr* and the equation is called the left. For operator -functions *Y x*ð Þ , *λ* , *Z x*ð Þ , *λ* ∈*B*ð Þ **H** let

$$W\{Z^\*, Y\} = Z^{\*\prime\prime}(\mathbf{x}, \overline{\lambda})Y(\mathbf{x}, \lambda) - Z^\*\left(\mathbf{x}, \overline{\lambda}\right)Y^\prime(\mathbf{x}, \lambda). \tag{56}$$

If *Y x*ð Þ , *λ* - operator solution of the Eq. (1), and *Z x*ð Þ , *λ* - operator solution of Eq. (54), the Wronskian does not depend on *x*.

Now we denote *Y x*ð Þ , *λ* and *Y*1ð Þ *x*, *λ* the solutions of the Eqs. (1) and (54), respectively, satisfying the initial conditions

$$Y(0,\lambda) = \cos A,\\ Y(0,\lambda) = \sin A,\\ Y\_1(0,\lambda) = \left(\cos A\right)^\*,\\ Y\_1(0,\lambda) = \left(\sin A\right)^\*, \ \lambda \in \mathbb{C}.\tag{57}$$

Because the operator function *Y* ∗ 1 *x*, *λ* satisfies equation

$$-Y\_1^{\*\prime\prime}(\mathbf{x}, \overline{\boldsymbol{\lambda}}) + Y\_1^\*\left(\boldsymbol{\x}, \overline{\boldsymbol{\lambda}}\right) \cdot V(\boldsymbol{x}) = \lambda Y\_1^\*\left(\boldsymbol{x}, \overline{\boldsymbol{\lambda}}\right),\tag{58}$$

the operator function *Y x* ~ð Þ , *λ* ≕*Y* ∗ 1 *x*, *λ* is a solution to the left of the equation

$$-\tilde{Y}''(\mathbf{x}, \boldsymbol{\lambda}) + \tilde{Y}(\mathbf{x}, \boldsymbol{\lambda}) \cdot V(\mathbf{x}) = \boldsymbol{\lambda} \tilde{Y}(\mathbf{x}, \boldsymbol{\lambda}) \tag{59}$$

and satisfies the initial conditions *Y*~ð Þ¼ 0, *λ* cos *A*, *Y*~0 ð Þ¼ 0, *λ* sin *A*, *λ*∈ . *Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular… DOI: http://dx.doi.org/10.5772/intechopen.95820*

Operator solutions of Eq. (54) decreasing and increasing at infinity will be denoted by Φ1ð Þ *x*, *λ* , Ψ1ð Þ *x*, *λ* , and the corresponding solutions of the Eq. (59) denote by Φ~ ð Þ *x*, *λ* and Ψ~ð Þ *x*, *λ* . For the system operator solutions *Y x*ð Þ , *λ* , Φ~ ð Þ *x*, *λ* ∈*B*ð Þ **H** of the Eqs. (1) and (59), respectively, will take the form of Wronskian *W* Φ~ , *Y* � � ¼ Φ~ 0 ð Þ *x*, *λ Y x*ð Þ� , *λ* Φ~ ð Þ *x*, *λ Y*0 ð Þ *x*, *λ* and do not depend on *x*.

Let us designate

with the boundary conditions

that, for every symmetric operator *Lk*

domain of operator *L*.

*k*¼1*σ*ð Þ *Lk* .

the Eq. (54) can be rewritten as

where ~*y* ¼ ~*y*1 ~*y*2 …~*yr*

*Y*ð Þ¼ 0, *λ* cos *A*, *Y*0

**52**

*σd*ð Þ¼ *L* ∪*r*

residual spectrum.

Let *L* denote the operator extensions *L*0

The following theorem is proved in [4].

Along with the Eq. (1) we consider the equation

For operator -functions *Y x*ð Þ , *λ* , *Z x*ð Þ , *λ* ∈*B*ð Þ **H** let

Eq. (54), the Wronskian does not depend on *x*.

respectively, satisfying the initial conditions

Because the operator function *Y* ∗

the operator function *Y x* ~ð Þ , *λ* ≕*Y* ∗

�*Y* ∗ 1

�*Y*~00

00 *x*, *λ* þ *Y* ∗

and satisfies the initial conditions *Y*~ð Þ¼ 0, *λ* cos *A*, *Y*~0

cos *Akk* � *yk*

0

*Recent Developments in the Solution of Nonlinear Differential Equations*

their self-adjoint extensions *Lk* are the closures of operators *Lk*

operators *Lk* are semi-bounded below, and their spectra are discrete.

sion *l*½ � *y* (1) and the boundary condition (50), and let *Lk*

ð Þ� 0 sin *Akk* � *yk*ð Þ¼ 0 0, *k* ¼ 1,*r:* (53)

0

*l*1½ �¼� *y y*00 þ *V* ∗ ð Þ *x y* ¼ *λy* (54)

~*l*½ �¼� ~*y* ~*y*00 þ ~*yV x*ð Þ¼ *λ*~*y*, (55)

0

1 *x*, *λ* is a solution to the left of the equation

ð Þþ *x*, *λ Y x* ~ð Þ� , *λ V x*ð Þ¼ *λY x* ~ð Þ , *λ* (59)

, there is a case of limit point at infinity. Hence

, by requiring that *L*2ð Þ **H**, 0, ð Þ ∞ be the

, *k* ¼ 1,*r* denote the minimal

0 respectively. The

ð Þ *x*, *λ :* (56)

ð Þ¼ 0, *λ* ð Þ sin *A* ∗ , *λ*∈ *:*

1 *x*, *λ* , (58)

ð Þ¼ 0, *λ* sin *A*, *λ*∈ .

(57)

and

Let *L*0 denote the minimal differential operator generated by differential expres-

differential operator on *L*2ð Þ *Hk*, 0, ð Þ ∞ generated by differential expression *lk yk*

0

the boundary conditions (53). Taking into account the conditions on coefficients, as well as sufficient smallness of perturbations *Ukk*ð Þ *x* and conditions (51), we conclude

**Theorem 3.1** *Suppose that, for Eq. (1) conditions (3)-(5) are satisfied for α* >1 *or condition (6) for* 0<*α* ≤1*. Then the discrete spectrum of the operator L is real and coincides with the union of spectra of the self-adjoint operators Lk*, *k* ¼ 1,*r, i.e.,*

**Comment 3.1** Note that this theorem contains a statement of the discrete spectrum of the non-self-adjoint operator *L* only and no allegations of its continuous and

(*V* ∗ ð Þ *x* is adjoint to the operator *V x*ð Þ). If the space **H** is finite-dimensional, then

and the equation is called the left.

*W Z* ∗ f g , *Y* ¼ *Z* ∗ 0 *x*, *λ Y x*ð Þ� , *λ Z* ∗ *x*, *λ Y*0

If *Y x*ð Þ , *λ* - operator solution of the Eq. (1), and *Z x*ð Þ , *λ* - operator solution of

1 *x*, *λ* satisfies equation

1 *x*, *λ* � *V x*ð Þ¼ *λY* ∗

Now we denote *Y x*ð Þ , *λ* and *Y*1ð Þ *x*, *λ* the solutions of the Eqs. (1) and (54),

ð Þ¼ 0, *λ* sin *A*, *Y*1ð Þ¼ 0, *λ* ð Þ cos *A* ∗ , *Y*1

$$G(\mathbf{x},t,\lambda) = \begin{cases} \begin{array}{ll} Y(\mathbf{x},\lambda) \left(W\{\tilde{\Phi},Y\}\right)^{-1} \tilde{\Phi}(t,\lambda) & 0 \le \mathbf{x} \le t \\\\ -\Phi(\mathbf{x},\lambda) \left(W\{\tilde{Y},\Phi\}\right)^{-1} \tilde{Y}(t,\lambda) & \mathbf{x} \ge t \end{array} \end{cases} \tag{60}$$

In the following theorem it is proved that the operator function *G x*ð Þ , *t*, *λ* possesses all the classical properties of the Green's function.

**Theorem 3.2** *The operator function G x*ð Þ , *t*, *λ is the Green's function of the differential operator L, i.e.:*

$$G\_{\mathbf{x}}{}^{\prime}(\mathbf{x}+\mathbf{0},\mathbf{x},\boldsymbol{\lambda}) - G\_{\mathbf{x}}{}^{\prime}(\mathbf{x}-\mathbf{0},\mathbf{x},\boldsymbol{\lambda}) = -I. \tag{61}$$

3.*For a fixed t, the function G x*ð Þ , *t*, *λ of the variable x is an operator solution of Eq. (1) on each of the intervals* ½ Þ 0, *t , t*ð Þ , ∞ *, and it satisfies the boundary condition (50), and at a fixed x function G x*ð Þ , *t*, *λ of the variable t is an operator solution of the Eq. (59) on each of the intervals* ½ Þ 0, *x* , ð Þ *x*, ∞ *, and it satisfies the boundary condition* ~*y*0 ð Þ� 0 cos *A* � ~*y*ð Þ� 0 sin *A* ¼ 0*:*

**Proof** The function *G x*ð Þ , *t*, *λ* is continuous with respect to *x* at each of the intervals 0, ½ Þ*t* and ð Þ *t*, ∞ . Similarly to the variable *t*. To prove the continuity of the function *G x*ð Þ , *t*, *λ* for all *x*, *t*≥0, it is sufficient that the identity shown as

$$\left(Y(\mathbf{x},\boldsymbol{\lambda})\left(\mathcal{W}\{\tilde{\Phi},Y\}\right)^{-1}\tilde{\Phi}(\mathbf{x},\boldsymbol{\lambda})+\Phi(\mathbf{x},\boldsymbol{\lambda})\left(\mathcal{W}\{\tilde{Y},\boldsymbol{\Phi}\}\right)^{-1}\tilde{Y}\left(\mathbf{x},\boldsymbol{\lambda}\right)\equiv\mathbf{0}.\tag{62}$$

is satisfied for all *x*≥0*:* This identity shown as

$$Y(\mathbf{x},\boldsymbol{\lambda})\left(\tilde{\Phi}(\mathbf{x},\boldsymbol{\lambda})Y(\mathbf{x},\boldsymbol{\lambda})-\tilde{\Phi}'(\mathbf{x},\boldsymbol{\lambda})Y(\mathbf{x},\boldsymbol{\lambda})\right)^{-1}\tilde{\Phi}(\mathbf{x},\boldsymbol{\lambda})-$$

$$-\Phi(\mathbf{x},\boldsymbol{\lambda})\left(\tilde{Y}'(\mathbf{x},\boldsymbol{\lambda})\Phi(\mathbf{x},\boldsymbol{\lambda})-\tilde{Y}(\mathbf{x},\boldsymbol{\lambda})\Phi'(\mathbf{x},\boldsymbol{\lambda})\right)^{-1}\tilde{Y}\left(\mathbf{x},\boldsymbol{\lambda}\right)\equiv\mathbf{0}\tag{63}$$

or

$$\left(Y'(\mathbf{x},\boldsymbol{\lambda})Y^{-1}(\mathbf{x},\boldsymbol{\lambda})-\tilde{\Phi}^{-1}(\mathbf{x},\boldsymbol{\lambda})\tilde{\Phi}'(\mathbf{x},\boldsymbol{\lambda})\right)^{-1} \equiv \left(\tilde{Y}^{-1}(\mathbf{x},\boldsymbol{\lambda})\tilde{Y}'(\mathbf{x},\boldsymbol{\lambda})-\Phi'(\mathbf{x},\boldsymbol{\lambda})\Phi^{-1}(\mathbf{x},\boldsymbol{\lambda})\right)^{-1},\tag{64}$$

$$Y'(\mathbf{x},\boldsymbol{\lambda})Y^{-1}(\mathbf{x},\boldsymbol{\lambda})-\tilde{\Phi}^{-1}(\mathbf{x},\boldsymbol{\lambda})\tilde{\Phi}'(\mathbf{x},\boldsymbol{\lambda}) \equiv \tilde{Y}^{-1}(\mathbf{x},\boldsymbol{\lambda})\tilde{Y}'(\mathbf{x},\boldsymbol{\lambda})-\Phi'(\mathbf{x},\boldsymbol{\lambda})\Phi^{-1}(\mathbf{x},\boldsymbol{\lambda}),\tag{65}$$

which is equivalent to

$$Y^{\prime}(\mathbf{x},\boldsymbol{\lambda})Y^{-1}(\mathbf{x},\boldsymbol{\lambda}) - \bar{Y}^{-1}(\mathbf{x},\boldsymbol{\lambda})\bar{Y}^{\prime}(\mathbf{x},\boldsymbol{\lambda}) \equiv \bar{\boldsymbol{\Phi}}^{-1}(\mathbf{x},\boldsymbol{\lambda})\bar{\boldsymbol{\Phi}}^{\prime}(\mathbf{x},\boldsymbol{\lambda}) - \boldsymbol{\Phi}^{\prime}(\mathbf{x},\boldsymbol{\lambda})\boldsymbol{\Phi}^{-1}(\mathbf{x},\boldsymbol{\lambda}) \tag{65}$$

or to.

$$\begin{split} &\tilde{Y}^{-1}(\mathbf{x},\boldsymbol{\lambda}) \Big( \tilde{Y}(\mathbf{x},\boldsymbol{\lambda})Y'(\mathbf{x},\boldsymbol{\lambda}) - \tilde{Y}'(\mathbf{x},\boldsymbol{\lambda})Y(\mathbf{x},\boldsymbol{\lambda}) \Big) Y^{-1}(\mathbf{x},\boldsymbol{\lambda}) \equiv \\ & \equiv -\tilde{\Phi}^{-1}(\mathbf{x},\boldsymbol{\lambda}) \Big( \tilde{\Phi}(\mathbf{x},\boldsymbol{\lambda})\Phi'(\mathbf{x},\boldsymbol{\lambda}) - \tilde{\Phi}'(\mathbf{x},\boldsymbol{\lambda})\Phi^{-1}(\mathbf{x},\boldsymbol{\lambda}) \Big) \Phi^{-1}(\mathbf{x},\boldsymbol{\lambda}). \end{split} \tag{66}$$

Ω1ð Þ¼ *λ* lim*x*!∞ *γ*0ð Þ *x*, *λ γ*∞

¼ � ð *x*

0

following form

ð Þ *Rλf* ð Þ¼� *x*

**55**

ð *a*

Φð Þ *x*, *λ W Y*~, Φ � � � � �1

*χ*1ð Þ¼ *x*, *λ* Φð Þ *x*, *λ W Y*~, Φ � � � � �1

*χ*2ð Þ¼ *x*, *λ* Φð Þ *x*, *λ*

0

where *a*>0 and

**block – triangular coefficients**

*DOI: http://dx.doi.org/10.5772/intechopen.95820*

Φð Þ *x*, *λ W Y*~, Φ � � � � �1

vector-function *y x*ð Þ¼ , *λ Rλf*

*A*ð Þ¼ *λ W* Ψ~, *Y* � �, *B*ð Þ¼� *λ W* Φ~ , *Y* � �,

0

ð Þ� *x*, *λ γ*0

*parameter λ with the poles coincide with the eigenvalues of the operator L.*

0 ½ ð Þ *x*, *λ γ*∞ð Þ *x*, *λ* � � *I* ¼ *W γ*0 f g� , *γ*∞ *I* ¼ *I:* (75)

*G x*ð Þ , *t*, *λ f t*ð Þ*dt* ¼ (76)

Φ~ ð Þ *t*, *λ f t*ð Þ*dt:*

(79)

*Y x*ð Þ , *λ W* Φ~ , *Y* � � � � �1

� �ð Þ *x* is a solution of the equation *l*½ �� *y λy* ¼ *f* when-

*Y x*ð Þ¼ , *λ* Φð Þ *x*, *λ W* Ψ~, *Y* � � � Ψð Þ *x*, *λ W* Φ~ , *Y* � �*:* (77)

*Y x* ~ð Þ¼ , *λ W Y*~, Φ � �Ψ~ð Þ� *x*, *λ W Y*~, Ψ � �Φ~ ð Þ *x*, *λ :* (78)

*W Y*~, Ψ � � ð

ð *x*

*a*

*x*

*a*

*Y t* ~ ð Þ , *λ f t*ð Þ*dt* þ *χ*1ð Þ� *x*, *λ χ*2ð Þþ *x*, *λ χ*3ð Þ� *x*, *λ χ*4ð Þ *x*, *λ* ,

Φ~ ð Þ *t*, *λ f t*ð Þ*dt*, (80)

Ψ~ ð Þ *t*, *λ f t*ð Þ*dt* , (81)

This completes the proof of the formula (61), and with it the theorem 3.1. □ **Corollary***. By the definition (60), function G x*ð Þ , *t*, *λ is meromorphic of the*

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular…*

We constructed Green's function for the non-self-adjoint differential operator.

**4. Resolvent for an non-self-adjoint operator differential equation with**

We consider the operator *Rλ* defined in *L*2ð Þ **H**, 0, ð Þ ∞ by the relation

∞ð

0

**Proof** One can directly verify that, for any function *f x*ð Þ ∈*L*2ð Þ **H**, 0, ð Þ ∞ , the

Since operator solutions Φð Þ *x*, *λ* and Ψð Þ *x*, *λ* form a fundamental system of solutions of Eq. (1), the operator solution *Y x*ð Þ , *λ* of Eq. (1) satisfying the initial conditions (57), can be written as *Y x*ð Þ¼ , *λ* Φð Þ *x*, *λ A*ð Þþ *λ* Ψð Þ *x*, *λ B*ð Þ*λ* , where

Similarly, the operator solution *Y x* ~ð Þ , *λ* of Eq. (59) can be represented in the

By using formulas (77) and (78), we can rewrite the relation (76) as follows:

∞ð

*x*

*Y t* ~ ð Þ , *λ f t*ð Þ*dt* þ

**Theorem 4.1** *The operator Rλ is the resolvent of the operatorL*.

*Rλf* � �ð Þ¼ *x*

ever *λ* ∉ *σ*ð Þ *L* . We will prove that *y x*ð Þ , *λ* ∈*L*2ð Þ **H**, 0, ð Þ ∞ .

This follows from the fact that *W Y*~, *Y* ¼ *W* Φ~ , Φ ¼ 0.

To make sure that the jump in the first derivative at *t* ¼ *x* is equal to ð Þ �*I* , i.e., that the equality (61) holds, it is sufficient to prove the identity

$$\left(Y'(\mathbf{x},\boldsymbol{\lambda})\left(\mathcal{W}\{\bar{\Phi},Y\}\right)^{-1}\bar{\Phi}(\mathbf{x},\boldsymbol{\lambda})+\Phi'(\mathbf{x},\boldsymbol{\lambda})\left(\mathcal{W}\{\bar{Y},\boldsymbol{\Phi}\}\right)^{-1}\bar{Y}\left(\mathbf{x},\boldsymbol{\lambda}\right)\equiv I.\tag{67}$$

Now we consider the function

$$\mathbf{C}(\mathbf{x},t,\lambda) = \mathbf{Y}(\mathbf{x},\lambda) \left(\mathcal{W}\{\tilde{\Phi},\mathbf{Y}\}\right)^{-1} \tilde{\Phi}(t,\lambda) + \Phi(\mathbf{x},\lambda) \left(\mathcal{W}\{\tilde{\mathbf{Y}},\Phi\}\right)^{-1} \tilde{\mathbf{Y}}(t,\lambda), \tag{68}$$

which is an analogue of the Cauchy function. This function is the solution of Eq. (1) of the variable *x*, and it is the solution of Eq. (59) of the variable *t*. By (62), we have *C x*ð Þ� , *x*, *λ* 0. But in this case *Cxx*00j*t*¼*x* ¼ ð Þ *V x*ð Þ� *λI C*j *t*¼*x* � 0, and, therefore, *Cx* 0 ð Þj *x*, *t*, *λ t*¼*x* � Ω1ð Þ*λ* , i.e.,

$$\left(Y(\mathbf{x},\boldsymbol{\lambda})\left(\mathcal{W}\{\tilde{\Phi},Y\}\right)\right)^{-1}\tilde{\Phi}(\mathbf{x},\boldsymbol{\lambda}) + \Phi'(\mathbf{x},\boldsymbol{\lambda})\left(\mathcal{W}\{\tilde{Y},\boldsymbol{\Phi}\}\right)^{-1}\tilde{Y}\left(\mathbf{x},\boldsymbol{\lambda}\right) \equiv \boldsymbol{\Omega}\_1(\boldsymbol{\lambda}).\tag{69}$$

It shows that Ω1ð Þ¼ *λ I*, we obtain (61).

Since operator solutions Φð Þ *x*, *λ* and Ψð Þ *x*, *λ* form a fundamental system of solutions of Eq. (1), the operator solution *Y x*ð Þ , *λ* of Eq. (1) satisfying the initial conditions (57), can be written as *Y x*ð Þ¼ , *λ* Φð Þ *x*, *λ A*ð Þþ *λ* Ψð Þ *x*, *λ B*ð Þ*λ* , where *A*ð Þ¼� *λ W* Ψ~, *Y* , *B*ð Þ¼ *λ W* Φ~ , *Y* ,

$$Y(\mathbf{x}, \boldsymbol{\lambda}) = \Psi(\mathbf{x}, \boldsymbol{\lambda}) W\{\tilde{\Phi}, Y\} - \Phi(\mathbf{x}, \boldsymbol{\lambda}) W\{\tilde{\Psi}, Y\}. \tag{70}$$

Similarly, operator solution *Y x* ~ð Þ , *λ* of Eq. (59) can be represented in the form

$$
\tilde{Y}(\mathbf{x},\boldsymbol{\lambda}) = \tilde{W}\{\tilde{\Phi}, Y\}\tilde{\Psi}(\mathbf{x},\boldsymbol{\lambda}) - \tilde{W}\{\tilde{\Psi}, Y\}\tilde{\Phi}(\mathbf{x},\boldsymbol{\lambda}),
\tag{71}
$$

where

$$\tilde{W}\{\tilde{\Phi}, Y\} = \sin A \cdot \Phi(0, \lambda) - \cos A \cdot \Phi'(0, \lambda) = -\Omega(0, \lambda) = -W\{\tilde{Y}, \Phi\}.\tag{72}$$

Similarly we get *W*~ Ψ~, *Y* ¼ �*W Y*~, Ψ . Thus,

$$
\tilde{Y}(\mathbf{x}, \boldsymbol{\lambda}) = \mathcal{W}\{\tilde{Y}, \boldsymbol{\Psi}\} \tilde{\Phi}(\mathbf{x}, \boldsymbol{\lambda}) - \mathcal{W}\{\tilde{Y}, \boldsymbol{\Phi}\} \tilde{\Psi}(\mathbf{x}, \boldsymbol{\lambda}).\tag{73}
$$

Substituting (70) and (73) into the formula (69), using the fact that the equality (69) is performed on *x* identically, we obtain

$$\Omega\_1(\lambda) = \lim\_{\mathbf{x} \to \mathbf{s}} \left[ \Psi'(\mathbf{x}, \lambda) \tilde{\Phi}(\mathbf{x}, \lambda) - \Phi'(\mathbf{x}, \lambda) \tilde{\Psi}(\mathbf{x}, \lambda) \right]. \tag{74}$$

By Theorem 2.1, on the asymptotic behavior of functions Φð Þ *x*, *λ* and Ψð Þ *x*, *λ* at infinity, we have

*Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular… DOI: http://dx.doi.org/10.5772/intechopen.95820*

$$\Omega\_1(\lambda) = \lim\_{\mathbf{x} \to \mathbf{os}} \left[ \chi\_0(\mathbf{x}, \lambda) \chi\_{\sf ss}^{\prime}(\mathbf{x}, \lambda) - \chi\_0^{\prime}(\mathbf{x}, \lambda) \chi\_{\sf ss}(\mathbf{x}, \lambda) \right] \cdot I = \mathcal{W}\{\chi\_0, \chi\_{\sf ss}\} \cdot I = I. \tag{75}$$

This completes the proof of the formula (61), and with it the theorem 3.1. □ **Corollary***. By the definition (60), function G x*ð Þ , *t*, *λ is meromorphic of the parameter λ with the poles coincide with the eigenvalues of the operator L.*

We constructed Green's function for the non-self-adjoint differential operator.

#### **4. Resolvent for an non-self-adjoint operator differential equation with block – triangular coefficients**

We consider the operator *Rλ* defined in *L*2ð Þ **H**, 0, ð Þ ∞ by the relation

$$\left(R\_{\vec{\ell}}\overline{f}\right)(\mathbf{x}) = \bigcap\_{\ell=1}^{\infty} G(\mathbf{x}, t, \lambda)\overline{f}(t)dt = \tag{76}$$

$$=-\int\_0^\infty \Phi(\mathbf{x},\boldsymbol{\lambda}) \left(\mathcal{W}\{\boldsymbol{\tilde{Y}},\boldsymbol{\Phi}\}\right)^{-1} \boldsymbol{\tilde{Y}}(t,\boldsymbol{\lambda}) \boldsymbol{\tilde{f}}(t) dt + \int\_\mathbf{x}^\infty \mathbf{y}(\boldsymbol{x},\boldsymbol{\lambda}) \left(\mathcal{W}\{\boldsymbol{\tilde{\Phi}},\boldsymbol{Y}\}\right)^{-1} \boldsymbol{\tilde{\Phi}}(t,\boldsymbol{\lambda}) \boldsymbol{\tilde{f}}(t) dt.$$

**Theorem 4.1** *The operator Rλ is the resolvent of the operatorL*.

**Proof** One can directly verify that, for any function *f x*ð Þ ∈*L*2ð Þ **H**, 0, ð Þ ∞ , the vector-function *y x*ð Þ¼ , *λ Rλf* � �ð Þ *x* is a solution of the equation *l*½ �� *y λy* ¼ *f* whenever *λ* ∉ *σ*ð Þ *L* . We will prove that *y x*ð Þ , *λ* ∈*L*2ð Þ **H**, 0, ð Þ ∞ .

Since operator solutions Φð Þ *x*, *λ* and Ψð Þ *x*, *λ* form a fundamental system of solutions of Eq. (1), the operator solution *Y x*ð Þ , *λ* of Eq. (1) satisfying the initial conditions (57), can be written as *Y x*ð Þ¼ , *λ* Φð Þ *x*, *λ A*ð Þþ *λ* Ψð Þ *x*, *λ B*ð Þ*λ* , where *A*ð Þ¼ *λ W* Ψ~, *Y* � �, *B*ð Þ¼� *λ W* Φ~ , *Y* � �,

$$Y(\mathbf{x}, \boldsymbol{\lambda}) = \Phi(\mathbf{x}, \boldsymbol{\lambda}) \mathcal{W}\{\tilde{\Psi}, Y\} - \Psi(\mathbf{x}, \boldsymbol{\lambda}) \mathcal{W}\{\tilde{\Phi}, Y\}. \tag{77}$$

Similarly, the operator solution *Y x* ~ð Þ , *λ* of Eq. (59) can be represented in the following form

$$
\tilde{Y}(\mathbf{x},\boldsymbol{\lambda}) = \mathcal{W}\{\tilde{Y}, \boldsymbol{\Phi}\} \tilde{\boldsymbol{\Psi}}(\mathbf{x}, \boldsymbol{\lambda}) - \mathcal{W}\{\tilde{Y}, \boldsymbol{\Psi}\} \tilde{\boldsymbol{\Phi}}(\mathbf{x}, \boldsymbol{\lambda}).\tag{78}
$$

By using formulas (77) and (78), we can rewrite the relation (76) as follows:

$$(R\_{\bar{f}}f)(\mathbf{x}) = -\int\_0^t \Phi(\mathbf{x}, \boldsymbol{\lambda}) \left(\mathcal{W}\{\bar{Y}, \boldsymbol{\Phi}\}\right)^{-1} \bar{Y}(t, \boldsymbol{\lambda}) \boldsymbol{f}(t) dt + \underline{\chi}\_1(\mathbf{x}, \boldsymbol{\lambda}) - \underline{\chi}\_2(\mathbf{x}, \boldsymbol{\lambda}) + \underline{\chi}\_3(\mathbf{x}, \boldsymbol{\lambda}) - \underline{\chi}\_4(\mathbf{x}, \boldsymbol{\lambda}), \tag{79}$$

where *a*>0 and

$$
\overline{\chi}\_1(\mathbf{x}, \boldsymbol{\lambda}) = \Phi(\mathbf{x}, \boldsymbol{\lambda}) \left( \mathcal{W} \{ \boldsymbol{\tilde{Y}}, \boldsymbol{\Phi} \} \right)^{-1} \mathcal{W} \{ \boldsymbol{\tilde{Y}}, \boldsymbol{\Psi} \} \int\_{\boldsymbol{a}}^{\boldsymbol{\mathfrak{x}}} \boldsymbol{\Phi} \,(t, \boldsymbol{\lambda}) \overline{\boldsymbol{\tilde{f}}}(t) dt,\tag{80}
$$

$$
\overline{\chi}\_2(\mathbf{x}, \boldsymbol{\lambda}) = \Phi(\mathbf{x}, \boldsymbol{\lambda}) \int\_{\boldsymbol{a}}^{\mathbf{x}} \tilde{\Psi}\left(t, \boldsymbol{\lambda}\right) \overline{f}(t) \, dt \; \tag{81}
$$

or to.

*Y*0

therefore, *Cx*

where

infinity, we have

**54**

*Y*0

0

ð Þ *x*, *λ W* Φ~ , *Y* �1

*Y*~�1

� �Φ~ �1

ð Þ *x*, *λ Y x* ~ð Þ , *λ Y*0

ð Þ *x*, *λ W* Φ~ , *Y* �1

Now we consider the function

*C x*ð Þ¼ , *t*, *λ Y x*ð Þ , *λ W* Φ~ , *Y* �1

ð Þ� *x*, *λ Y*~0

*Recent Developments in the Solution of Nonlinear Differential Equations*