Open Access

Superlinear Singular Problems on the Half Line

Boundary Value Problems20102010:429813

DOI: 10.1155/2010/429813

Received: 19 October 2010

Accepted: 7 December 2010

Published: 15 December 2010

Abstract

The paper studies the singular differential equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq1_HTML.gif , which has a singularity at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq2_HTML.gif . Here the existence of strictly increasing solutions satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq3_HTML.gif is proved under the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq4_HTML.gif has two zeros 0 and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq5_HTML.gif and a superlinear behaviour near https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq6_HTML.gif . The problem generalizes some models arising in hydrodynamics or in the nonlinear field theory.

1. Introduction

Let us consider the problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq7_HTML.gif is a positive real parameter.

Definition 1.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq8_HTML.gif . A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq9_HTML.gif satisfying (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq10_HTML.gif is called a solution of ( 1.1 ) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq11_HTML.gif .

Definition 1.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq12_HTML.gif be a solution of (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq13_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq14_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq15_HTML.gif is called a solution of ( 1.1 ) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq16_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq17_HTML.gif moreover fulfils conditions (1.2), it is called a solution of problem ( 1.1 ), ( 1.2 ).

Definition 1.3.

A strictly increasing solution of problem (1.1), (1.2) is called a homoclinic solution.

In this paper we are interested in the existence of strictly increasing solutions and, in particular, of homoclinic solutions. In what follows we assume
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ3_HTML.gif
(1.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ4_HTML.gif
(1.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ5_HTML.gif
(1.5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ6_HTML.gif
(1.6)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ7_HTML.gif
(1.7)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ8_HTML.gif
(1.8)
Under assumptions (1.3)–(1.8) problem (1.1), (1.2) generalizes some models arising in hydrodynamics or in the nonlinear field theory (see [15]). If a homoclinic solution exists, many important properties of corresponding models can be obtained. Note that if we extend the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq18_HTML.gif in (1.1) from the half-line onto https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq19_HTML.gif (as an even function), then any solution of (1.1), (1.2) has the same limit https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq20_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq22_HTML.gif . This is a motivation for Definition 1.3. Equation (1.1) is singular at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq23_HTML.gif because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq24_HTML.gif . In [6, 7] we have proved that assumptions (1.3)–(1.8) are sufficient for the existence of strictly increasing solutions and homoclinic solutions provided
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ9_HTML.gif
(1.9)
Here we assume that (1.9) is not valid. Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ10_HTML.gif
(1.10)
and the papers [6, 8] provide existence theorems for problem (1.1), (1.2) if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq25_HTML.gif has a sublinear or linear behaviour near https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq26_HTML.gif . The case that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq27_HTML.gif has a superlinear behaviour near https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq28_HTML.gif is studied in this paper. To this aim we consider the initial conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ11_HTML.gif
(1.11)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq29_HTML.gif , and introduce the following definition.

Definition 1.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq30_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq31_HTML.gif be a solution of (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq32_HTML.gif satisfying (1.11). Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq33_HTML.gif is called a solution of problem ( 1.1 ), ( 1.11 ) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq34_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq35_HTML.gif moreover fulfils
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ12_HTML.gif
(1.12)

then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq36_HTML.gif is called an escape solution of problem ( 1.1 ), ( 1.11 ).

We have proved in [6, 8] that for sublinear or linear https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq37_HTML.gif the existence of a homoclinic solution follows from the existence of an escape solution of problem (1.1), (1.11). Therefore our first task here is to prove that at least one escape solution of (1.1), (1.11) exists, provided (1.3)–(1.8), (1.10), and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ13_HTML.gif
(1.13)

hold, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq38_HTML.gif has a superlinear behaviour near https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq39_HTML.gif . This is done in Section 2. Using the results of Section 2 "Theorem 2.10", and of [6, Theroms 13, 14 and 20] we get the existence of a homoclinic solution in Section 3.

Note that by Definitions 1.3 and 1.4 just the values of a solution which are less than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq40_HTML.gif are important for a decision whether the solution is homoclinic or escape one. Therefore condition (1.13) can be assumed without any loss of generality.

Close problems about the existence of positive solutions have been studied in [911].

2. Escape Solutions

In this section we assume that (1.3)–(1.8), (1.10), and (1.13) hold. We will need some lemmas.

Lemma 2.1 (see [6, Lemma 3]).

For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq41_HTML.gif , problem (1.1), (1.11) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq42_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq43_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ14_HTML.gif
(2.1)

In what follows by a solution of (1.1), (1.11) we mean a solution on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq44_HTML.gif .

Remark 2.2 (see [6, Remark 4]).

Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq46_HTML.gif , and consider the initial conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ15_HTML.gif
(2.2)

Problem (1.1), (2.2) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq47_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq48_HTML.gif . In particular, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq50_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq52_HTML.gif , respectively. Clearly, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq53_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq54_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq55_HTML.gif are solutions of (1.1) on the whole interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq56_HTML.gif .

Lemma 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq57_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq58_HTML.gif be a solution of problem (1.1), (1.11) which is not an escape solution. Let us denote
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ16_HTML.gif
(2.3)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq59_HTML.gif holds and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq60_HTML.gif is increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq61_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq62_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq63_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ17_HTML.gif
(2.4)

Proof.

The inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq64_HTML.gif yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq65_HTML.gif . By (1.1) and (1.10), we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq66_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq67_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq68_HTML.gif is increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq69_HTML.gif . As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq70_HTML.gif , one has https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq71_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq72_HTML.gif and consequently https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq73_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq74_HTML.gif . Therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq75_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq76_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq77_HTML.gif is the first zero of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq79_HTML.gif . Remark 2.2 yields that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq80_HTML.gif is not possible. This implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq81_HTML.gif . As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq82_HTML.gif is strictly increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq84_HTML.gif is not an escape solution, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq85_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq86_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq87_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq88_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq89_HTML.gif is decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq90_HTML.gif . This gives (2.4).

Lemma 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq91_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq92_HTML.gif be a solution of problem (1.1), (1.11) which is not an escape solution. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq93_HTML.gif is given by Lemma 2.3. Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ18_HTML.gif
(2.5)

Proof.

From (1.1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ19_HTML.gif
(2.6)
and, by multiplication and integration over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq94_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ20_HTML.gif
(2.7)
?(1) Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq95_HTML.gif . The definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq96_HTML.gif yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq97_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq98_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq99_HTML.gif is not an escape solution, it is bounded above and there exists
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ21_HTML.gif
(2.8)
Therefore the following integral is bounded and, since it is increasing, it has a limit
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ22_HTML.gif
(2.9)
So, by (2.7), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq100_HTML.gif exists. By virtue of (2.8), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ23_HTML.gif
(2.10)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq101_HTML.gif , then by (1.4), (1.10) and (2.6) we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq102_HTML.gif , which contradicts (2.10). Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq103_HTML.gif . In particular, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq104_HTML.gif is defined as in Lemma 2.3, then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ24_HTML.gif
(2.11)

? (2) Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq105_HTML.gif . Then the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq106_HTML.gif gives https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq108_HTML.gif of Lemma 2.3 fulfils https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq109_HTML.gif . We deduce that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq110_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq111_HTML.gif as in the proof of Lemma 2.3. Remark 2.2 yields that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq112_HTML.gif , then neither https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq113_HTML.gif nor https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq114_HTML.gif can occur. Therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq115_HTML.gif .

Denote
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ25_HTML.gif
(2.12)

Lemma 2.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq116_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq117_HTML.gif be a solution of problem (1.1), (1.11). Further assume maximal https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq118_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq119_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq120_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq121_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ26_HTML.gif
(2.13)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq122_HTML.gif , let us denote
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ27_HTML.gif
(2.14)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ28_HTML.gif
(2.15)

Proof.

For equality (2.13) see Lemma 4.6 in [8]. Let us prove (2.15). Using the per partes integration, we get for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq123_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ29_HTML.gif
(2.16)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ30_HTML.gif
(2.17)
By multiplication and integration of (1.1) we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ31_HTML.gif
(2.18)
and by the per partes integration,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ32_HTML.gif
(2.19)
To compute https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq124_HTML.gif , we use (1.1) and get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ33_HTML.gif
(2.20)
By the per partes integration we derive
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ34_HTML.gif
(2.21)

We have proved that (2.15) is valid.

Lemma 2.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq125_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq126_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq127_HTML.gif be solutions of problem (1.1), (1.11) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq128_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq129_HTML.gif . Let us denote
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ35_HTML.gif
(2.22)
Then for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq130_HTML.gif there exists a unique https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq131_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ36_HTML.gif
(2.23)

If the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq132_HTML.gif is unbounded, then there exists an escape solution in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq133_HTML.gif .

Proof.

Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq134_HTML.gif . The monotonicity and continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq135_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq136_HTML.gif give a unique https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq137_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq138_HTML.gif is unbounded we argue as in the proof of Lemma 4.8 in [8].

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq139_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq140_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq141_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq142_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq143_HTML.gif be sequences from Lemma 2.6. Assume that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq144_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq145_HTML.gif is not an escape solution of problem (1.1), (1.11). Lemma 2.6 implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ37_HTML.gif
(2.24)
We can assume that that either there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq146_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ38_HTML.gif
(2.25)
or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ39_HTML.gif
(2.26)

Otherwise we take a subsequence. Some additional properties of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq147_HTML.gif are given in the next two lemmas.

Lemma 2.7.

Denote
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ40_HTML.gif
(2.27)
and assume that the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq148_HTML.gif is bounded above. Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq149_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ41_HTML.gif
(2.28)

Proof.

By Lemma 2.4 we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ42_HTML.gif
(2.29)

Step 1 (sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq150_HTML.gif is bounded).

Assume on the contrary that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq151_HTML.gif is unbounded. We may write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ43_HTML.gif
(2.30)
(otherwise we take a subsequence). Equality (2.13) yields for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq152_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq153_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ44_HTML.gif
(2.31)
Using (1.4), (1.6), (1.10), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq154_HTML.gif and the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq155_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq156_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ45_HTML.gif
(2.32)
Consequently, inequality in (2.31) leads to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ46_HTML.gif
(2.33)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq157_HTML.gif . Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ47_HTML.gif
(2.34)

We will consider two cases.

Case 1.

If (2.25) holds, then (2.34) gives for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq158_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ48_HTML.gif
(2.35)
By (2.30), for each sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq159_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ49_HTML.gif
(2.36)

Putting it to (2.35), we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq160_HTML.gif , contrary to (2.29).

Case 2.

If (2.26) holds, then (2.34) gives for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq161_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ50_HTML.gif
(2.37)
Due to (2.30), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ51_HTML.gif
(2.38)

for each sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq162_HTML.gif . Putting it to (2.37), we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq163_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq164_HTML.gif . Integrating it over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq165_HTML.gif , we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq166_HTML.gif . Equation (1.1) and condition (1.13) yield https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq167_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq168_HTML.gif , and so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq169_HTML.gif , contrary to (2.29).

We have proved that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq170_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ52_HTML.gif
(2.39)

Step 2 (estimate for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq171_HTML.gif ).

Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq172_HTML.gif . By (2.32) we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ53_HTML.gif
(2.40)
This together with (2.31) and (2.39) imply
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ54_HTML.gif
(2.41)
According to (2.27) and Lemma 2.3 we see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq173_HTML.gif is the first zero of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq174_HTML.gif . Since the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq175_HTML.gif is bounded above, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq176_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq177_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq178_HTML.gif . Then (1.8) and (2.41) give
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ55_HTML.gif
(2.42)
Put
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ56_HTML.gif
(2.43)

Then, by virtue of (2.4), inequality (2.28) is valid.

Lemma 2.8.

Consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq179_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq180_HTML.gif satisfying (2.23) and (2.24). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq181_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq182_HTML.gif be given by (2.27). Assume that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ57_HTML.gif
(2.44)
Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq183_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ58_HTML.gif
(2.45)

Proof.

Assume on the contrary that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ59_HTML.gif
(2.46)
By Lemma 2.3, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq184_HTML.gif is increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq185_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq186_HTML.gif . Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ60_HTML.gif
(2.47)
and therefore there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq187_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ61_HTML.gif
(2.48)
Moreover (2.23), (2.24), (2.27), (2.44), and the monotonicity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq188_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq189_HTML.gif yield
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ62_HTML.gif
(2.49)

Integrating the last inequality over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq190_HTML.gif , we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq191_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq192_HTML.gif , a contradiction.

Lemma 2.9.

Let real sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq193_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq194_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq195_HTML.gif be given and assume that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ63_HTML.gif
(2.50)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq196_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ64_HTML.gif
(2.51)
(for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq197_HTML.gif we assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq198_HTML.gif ) be such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ65_HTML.gif
(2.52)
Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq199_HTML.gif is given by (2.14) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq200_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ66_HTML.gif
(2.53)

Proof.

By (2.50), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq201_HTML.gif . Condition (2.52) yields that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq202_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ67_HTML.gif
(2.54)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ68_HTML.gif
(2.55)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ69_HTML.gif
(2.56)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq203_HTML.gif . Consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ70_HTML.gif
(2.57)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq204_HTML.gif , because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq205_HTML.gif is less than the critical value https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq206_HTML.gif . We have proved (2.53).

Now we are ready to prove the following main result of this paper.

Theorem 2.10.

Assume that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ71_HTML.gif
(2.58)

for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq207_HTML.gif . Further, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq208_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq209_HTML.gif be such that (2.51) and (2.52) are valid. Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq210_HTML.gif such that the corresponding solution of problem (1.1), (1.11) is an escape solution.

Proof.

Assumption (2.51) implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq211_HTML.gif , and hence we can choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq212_HTML.gif and define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq213_HTML.gif by (2.14). According to (1.4), (1.10), and (2.56), there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq214_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq215_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq216_HTML.gif . Consequently, we can find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq217_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ72_HTML.gif
(2.59)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq218_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq219_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq220_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq221_HTML.gif be sequences defined in Lemma 2.6. Moreover, let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ73_HTML.gif
(2.60)
Assume that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq222_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq223_HTML.gif is not an escape solution of problem (1.1), (1.11). By Lemma 2.4 we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ74_HTML.gif
(2.61)
Condition (2.60) gives https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq224_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ75_HTML.gif
(2.62)

Choose an arbitrary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq225_HTML.gif . We will construct a contradiction.

Step 1 (inequality for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq226_HTML.gif ).

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq227_HTML.gif is increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq228_HTML.gif , (2.62) gives a unique https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq229_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ76_HTML.gif
(2.63)
By (2.59) we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ77_HTML.gif
(2.64)
because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq230_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq231_HTML.gif . Further, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq232_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ78_HTML.gif
(2.65)
Therefore, according to (2.12),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ79_HTML.gif
(2.66)
Let us put
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ80_HTML.gif
(2.67)
Then inequalities (2.15) and (2.66) imply
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ81_HTML.gif
(2.68)

Step 2 (estimate of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq233_HTML.gif from below).

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq234_HTML.gif is a solution of (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq235_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ82_HTML.gif
(2.69)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ83_HTML.gif
(2.70)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq236_HTML.gif , are such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ84_HTML.gif
(2.71)
Integrating (2.70) over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq237_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ85_HTML.gif
(2.72)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ86_HTML.gif
(2.73)
By (2.52), (2.60) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq238_HTML.gif , we deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ87_HTML.gif
(2.74)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq239_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq240_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ88_HTML.gif
(2.75)
Due to (2.58), there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq241_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq242_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq243_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq244_HTML.gif . Hence for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq245_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq246_HTML.gif such that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq247_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ89_HTML.gif
(2.76)
Consequently
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ90_HTML.gif
(2.77)
Having in mind (2.75), we can choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq248_HTML.gif in (2.62) such that for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq249_HTML.gif the inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq250_HTML.gif holds. Hence (2.72) and the first inequality in (2.77) yield
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ91_HTML.gif
(2.78)
Put https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq251_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq252_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ92_HTML.gif
(2.79)
On the other hand, by (2.76),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ93_HTML.gif
(2.80)
By (2.79), this yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ94_HTML.gif
(2.81)

Step 3 (estimate of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq253_HTML.gif ).

The inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq254_HTML.gif gives https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq255_HTML.gif . Hence there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq256_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ95_HTML.gif
(2.82)
Having in mind (2.76), we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq257_HTML.gif to this https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq258_HTML.gif and then, by the second inequality in (2.77), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ96_HTML.gif
(2.83)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ97_HTML.gif
(2.84)
By Lemmas 2.7 and 2.8 there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq259_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ98_HTML.gif
(2.85)
Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq260_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq261_HTML.gif , if (2.25) holds and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq262_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq263_HTML.gif , if (2.26) holds. In addition there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq264_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq265_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq266_HTML.gif . (Note that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq267_HTML.gif in Lemma 2.8 is not bounded but does not fulfil (2.44), we work with a proper subsequence fulfilling (2.44).) By virtue of (2.84) and (2.85) we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ99_HTML.gif
(2.86)
Inequalities (2.84) and (2.86) yield
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ100_HTML.gif
(2.87)

Step 4 (final contradictions).

Putting (2.81) and (2.87) to (2.68) and using (1.6), (1.10) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq268_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ101_HTML.gif
(2.88)
First, let us assume that (2.26) holds and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq269_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq270_HTML.gif . So, conditions (2.85), and (2.88) yield
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ102_HTML.gif
(2.89)

Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq271_HTML.gif we get a contradiction to (2.53).

Finally, let us assume that (2.25) holds and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq272_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq273_HTML.gif . Then (2.61), (2.88), and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq274_HTML.gif yield
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ103_HTML.gif
(2.90)

contrary to (2.53).

Remark 2.11.

We assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq275_HTML.gif in Theorem 2.10. In particular for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq276_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq277_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq278_HTML.gif , the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq279_HTML.gif can behave in neighbourhood of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq280_HTML.gif as a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq281_HTML.gif for arbitrary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq282_HTML.gif .

Now, let (2.58) hold for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq283_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq284_HTML.gif and therefore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ104_HTML.gif
(2.91)

which is the first condition in (1.9). We have proved in [6, 7] that, in this case, assumptions (1.3)–(1.8) are sufficient for the existence of an escape solution.

Example 2.12.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq285_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq286_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq287_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq288_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ105_HTML.gif
(2.92)

Hence, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq289_HTML.gif condition (2.58) is satisfied. The critical value https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq290_HTML.gif is equal to 3. By Theorem 2.10, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq291_HTML.gif fulfils (2.52) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq292_HTML.gif , problem (1.1), (1.11) has an escape solution.

Example 2.13.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq293_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq294_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq295_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ106_HTML.gif
(2.93)

Hence, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq296_HTML.gif condition (2.58) is satisfied. The critical value https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq297_HTML.gif is equal to 5. By Theorem 2.10, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq298_HTML.gif fulfils (2.52) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq299_HTML.gif , problem (1.1), (1.11) has an escape solution.

3. Homoclinic Solutions

Having an escape solution we can deduce the existence of a homoclinic solution by the same arguments as in [6]. For completeness we bring here the main ideas. Remember that our basic assumptions (1.3)–(1.8), (1.10) and (1.13) are fulfilled in this section.

By Lemma 11 in [6], a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq300_HTML.gif of problem (1.1), (1.11) is homoclinic if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ107_HTML.gif
(3.1)
By Theorem 16 in [6], a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq301_HTML.gif of problem (1.1), (1.11) is an escape solution if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ108_HTML.gif
(3.2)

The third type of solutions of problem (1.1), (1.11) is characterized in the next definition.

Definition 3.1.

A solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq302_HTML.gif of problem (1.1), (1.11) is called damped, if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ109_HTML.gif
(3.3)

The following properties of damped and escape solutions are important for the existence of homoclinic solutions.

Theorem 3.2 (see [6, Theorem 13] (on damped solutions)).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq303_HTML.gif be of (1.5) and (1.6). Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq304_HTML.gif is a solution of problem (1.1), (1.11) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq305_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq306_HTML.gif is damped.

Theorem 3.3 (see [6, Theorem 14]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq307_HTML.gif be the set of all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq308_HTML.gif such that corresponding solutions of problem (1.1), (1.11) are damped. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq309_HTML.gif is open in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq310_HTML.gif .

Theorem 3.4 (see [6, Theorem 20]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq311_HTML.gif be the set of all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq312_HTML.gif such that corresponding solutions of problem (1.1), (1.11) are escape ones. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq313_HTML.gif is open in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq314_HTML.gif .

Having these theorems we get the main result of this section.

Theorem 3.5 (On a homoclinic solution).

Assume that the assumptions of Theorem 2.10 are satisfied. Then problem (1.1), (1.2) has a homoclinic solution.

Proof.

By Theorems 3.2 and 3.3, the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq315_HTML.gif is nonempty and open in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq316_HTML.gif . By Theorem 3.4, the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq317_HTML.gif is open in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq318_HTML.gif . Using Theorem 2.10, we get that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq319_HTML.gif is nonempty. Therefore the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq320_HTML.gif is nonempty and if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq321_HTML.gif , then the corresponding solution of problem (1.1), (1.11) is neither damped nor an escape solution. Therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq322_HTML.gif , and by Lemma 11 in [6], such solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq323_HTML.gif is homoclinic.

The proof of Theorem 3.5 implies that if problem (1.1), (1.11) has an escape solution, then it has also a homoclinic solution. Hence the following corollary is true.

Corollary 3.6.

Assume that the assumptions of Theorem 2.10 are satisfied. Let problem (1.1), (1.11) have no homoclinic solution. Then it has no escape solution.

If we assume (2.51) and (2.52), then the growth of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq324_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq325_HTML.gif is less than the critical value https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq326_HTML.gif . This is necessary for the existence of homoclinic solutions of some types of (1.1). See the next example.

Example 3.7.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq327_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq328_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq329_HTML.gif . Consider (1.1), where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq330_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq331_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq332_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq333_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq334_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq335_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq336_HTML.gif satisfy conditions (1.3)–(1.8), (1.10), (1.13), (2.52) and (2.58) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_IEq337_HTML.gif . By Theorem 3.5, if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ110_HTML.gif
(3.4)
then problem (1.1), (1.11) has a homoclinic solution. But if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F429813/MediaObjects/13661_2010_Article_924_Equ111_HTML.gif
(3.5)

then we have proved in [12] that problem (1.1), (1.11) has no homoclinic solution and consequently no escape solution.

Declarations

Acknowledgment

This paper was supported by the Council of Czech Government MSM 6198959214.

Authors’ Affiliations

(1)
Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University

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Copyright

© Rachunková and J. Tomecek. 2010

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