Open Access

Homoclinic Solutions of Singular Nonautonomous Second-Order Differential Equations

Boundary Value Problems20092009:959636

DOI: 10.1155/2009/959636

Received: 27 April 2009

Accepted: 15 September 2009

Published: 13 October 2009

Abstract

This paper investigates the singular differential equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq1_HTML.gif , having a singularity at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq2_HTML.gif . The existence of a strictly increasing solution (a homoclinic solution) satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq4_HTML.gif is proved provided that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq5_HTML.gif has two zeros and a linear behaviour near https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq6_HTML.gif .

1. Introduction

Having a positive parameter https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq7_HTML.gif we consider the problem

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ2_HTML.gif
(1.2)

under the following basic assumptions for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq8_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq9_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ3_HTML.gif
(1.3)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ4_HTML.gif
(1.4)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ5_HTML.gif
(1.5)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ6_HTML.gif
(1.6)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ7_HTML.gif
(1.7)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ8_HTML.gif
(1.8)

Then problem (1.1), (1.2) generalizes some models arising in hydrodynamics or in the nonlinear field theory (see [15]). However (1.1) is singular at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq10_HTML.gif because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq11_HTML.gif .

Definition 1.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq12_HTML.gif , thena solution of (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq13_HTML.gif is a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq14_HTML.gif satisfying (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq15_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq16_HTML.gif is a solution of (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq17_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq18_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq19_HTML.gif is a solution of (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq20_HTML.gif .

Definition 1.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq21_HTML.gif be a solution of (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq22_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq23_HTML.gif moreover fulfils conditions (1.2), it is called a solution of problem(1.1), (1.2).

Clearly, the constant function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq24_HTML.gif is a solution of problem (1.1), (1.2). An important question is the existence of a strictly increasing solution of (1.1), (1.2) because if such a solution exists, many important physical properties of corresponding models can be obtained. Note that if we extend the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq25_HTML.gif in (1.1) from the half–line onto https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq26_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%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq27_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq28_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq29_HTML.gif . Therefore we will use the following definition.

Definition 1.3.

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

Numerical investigation of problem (1.1), (1.2), where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq31_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq32_HTML.gif , can be found in [1, 46]. Problem (1.1), (1.2) can be also transformed onto a problem about the existence of a positive solution on the half-line. For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq34_HTML.gif and for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq35_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq36_HTML.gif , such transformed problem was solved by variational methods in [7, 8], respectively. Some additional assumptions imposed on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq37_HTML.gif were needed there. Related problems were solved, for example, in [9, 10].

Here, we deal directly with problem (1.1), (1.2) and continue our earlier considerations of papers [11, 12], where we looked for additional conditions which together with (1.3)–(1.8) would guarantee the existence of a homoclinic solution.

Let us characterize some results reached in [11, 12] in more details. Both these papers assume (1.3)–(1.8). In [11] we study the case that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq38_HTML.gif has at least three zeros https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq39_HTML.gif . More precisely, the conditions,

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ9_HTML.gif
(1.9)

are moreover assumed. Then there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq40_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq41_HTML.gif and a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq42_HTML.gif of (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq43_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ10_HTML.gif
(1.10)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ11_HTML.gif
(1.11)

We call such solution an escape solution. The main result of [11] is that (under (1.3)–(1.8), (1.9)) the set of solutions of (1.1), (1.10) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq44_HTML.gif consists of escape solutions and of oscillatory solutions (having values in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq45_HTML.gif ) and of at least one homoclinic solution. In [12] we omit assumptions (1.9) and prove that assumptions (1.3)–(1.8) are sufficient for the existence of an escape solution and also for the existence of a homoclinic solution provided the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq46_HTML.gif fulfils

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ12_HTML.gif
(1.12)

If (1.12) is not valid, then the existence of both an escape solution and a homoclinic solution is proved in [12], provided that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq47_HTML.gif satisfies moreover

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ13_HTML.gif
(1.13)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ14_HTML.gif
(1.14)

Assumption (1.13) characterizes the case that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq48_HTML.gif has just two zeros https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq50_HTML.gif in the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq51_HTML.gif . Further, we see that if (1.14) holds, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq52_HTML.gif is either bounded on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq53_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq54_HTML.gif is unbounded earlier and has a sublinear behaviour near https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq55_HTML.gif .

This paper also deals with the case that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq56_HTML.gif satisfies (1.13) and is unbounded above on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq57_HTML.gif . In contrast to [12], here we prove the existence of a homoclinic solution for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq58_HTML.gif having a linear behaviour near https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq59_HTML.gif . The proof is based on a full description of the set of all solutions of problem (1.1), (1.10) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq60_HTML.gif and on the existence of an escape solutions in this set.

Finally, we want to mention the paper [13], where the problem

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ15_HTML.gif
(1.15)

is investigated under the assumptions that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq61_HTML.gif is continuous, it has three distinct zeros and satisfies the sign conditions similar to those in [11, (3.4)]. In [13], an approach quite different from [11, 12] is used. In particular, by means of properties of the associated vector field https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq62_HTML.gif together with the Kneser's property of the cross sections of the solutions' funnel, the authors provide conditions which guarantee the existence of a strictly increasing solution of (1.15). The authors apply this general result to problem

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ16_HTML.gif
(1.16)

and get a strictly increasing solution of (1.16) for a sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq63_HTML.gif . This corresponds to the results of [11], where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq64_HTML.gif may be arbitrary.

2. Initial Value Problem

In this section, under the assumptions (1.3)–(1.8) and (1.13) we prove some basic properties of solutions of the initial value problem (1.1), (1.10), where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq65_HTML.gif .

Lemma 2.1.

For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq66_HTML.gif there exists a maximal https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq67_HTML.gif such that problem (1.1), (1.10) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq68_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq69_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ17_HTML.gif
(2.1)
Further, for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq70_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq71_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ18_HTML.gif
(2.2)

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq72_HTML.gif be a solution of problem (1.1), (1.10) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq73_HTML.gif . By (1.1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ19_HTML.gif
(2.3)
and multiplying by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq74_HTML.gif and integrating between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq75_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq76_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ20_HTML.gif
(2.4)

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq77_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq78_HTML.gif . Then (2.4) yields https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq79_HTML.gif , which is not possible, because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq80_HTML.gif is decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq81_HTML.gif . Therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq82_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq83_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq84_HTML.gif . Consider the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq85_HTML.gif (with the maximum norm) and an operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq86_HTML.gif defined by

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ21_HTML.gif
(2.5)
A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq87_HTML.gif is a solution of problem (1.1), (1.2) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq88_HTML.gif if and only if it is a fixed point of the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq89_HTML.gif . Using the Lipschitz property of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq90_HTML.gif we can prove that the operator is contractive for each sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq91_HTML.gif and from the Banach Fixed Point Theorem we conclude that there exists exactly one solution of problem (1.1), (1.2) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq92_HTML.gif . This solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq93_HTML.gif has the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ22_HTML.gif
(2.6)

for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq94_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq95_HTML.gif can be extended onto each interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq96_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq97_HTML.gif is bounded. So, we can put https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq98_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq99_HTML.gif . Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq100_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq101_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq102_HTML.gif . So, (2.6) yields

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ23_HTML.gif
(2.7)
Put
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ24_HTML.gif
(2.8)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ25_HTML.gif
(2.9)
and, by "per partes" integration we derive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq103_HTML.gif . Multiplying (2.7) by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq104_HTML.gif and integrating it over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq105_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ26_HTML.gif
(2.10)
Estimates (2.2) follow from (2.7)–(2.10) for
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ27_HTML.gif
(2.11)

Remark 2.2.

The proof of Lemma 2.1 yields that if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq106_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq107_HTML.gif .

Let us put

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ28_HTML.gif
(2.12)
and consider an auxiliary equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ29_HTML.gif
(2.13)

Similarly as in the proof of Lemma 2.1 we deduce that problem (2.13), (1.10) has a unique solution on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq108_HTML.gif . Moreover the following lemma is true.

Lemma 2.3 ([12]).

For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq109_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq110_HTML.gif and each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq111_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq112_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq113_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq114_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ30_HTML.gif
(2.14)

Here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq115_HTML.gif is a solution of problem (2.13), (1.10) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq116_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq117_HTML.gif .

Proof.

Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq118_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq119_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq120_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq121_HTML.gif be the Lipschitz constant for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq122_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq123_HTML.gif . By (2.6) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq124_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq125_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq126_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq127_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ31_HTML.gif
(2.15)
From the Gronwall inequality, we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ32_HTML.gif
(2.16)
Similarly, by (2.6), (2.9), and (2.16),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ33_HTML.gif
(2.17)
If we choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq128_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ34_HTML.gif
(2.18)

we get (2.14).

Remark 2.4.

Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq129_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq130_HTML.gif , and consider the initial conditions
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ35_HTML.gif
(2.19)

Arguing as in the proof of Lemma 2.1, we get that problem (2.13), (2.19) has a unique solution on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq131_HTML.gif . In particular, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq132_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq133_HTML.gif , the unique solution of problem (2.13), (2.19) (and also of problem (1.1), (2.19)) is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq134_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq135_HTML.gif , respectively.

Lemma 2.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq136_HTML.gif be a solution of problem (1.1), (1.10). Assume that there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq137_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ36_HTML.gif
(2.20)
Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq138_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq139_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ37_HTML.gif
(2.21)

Proof.

By (1.13) and (2.20), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq140_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq141_HTML.gif and thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq142_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq143_HTML.gif are positive on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq144_HTML.gif . Consequently, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq145_HTML.gif . Further, by (1.1),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ38_HTML.gif
(2.22)
and, by multiplication and integration over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq146_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ39_HTML.gif
(2.23)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ40_HTML.gif
(2.24)
and hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq147_HTML.gif exists. Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq148_HTML.gif is bounded on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq149_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ41_HTML.gif
(2.25)

By (1.3), (1.8), and (2.22), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq150_HTML.gif exists and, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq151_HTML.gif is bounded on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq152_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq153_HTML.gif . Hence, letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq154_HTML.gif in (2.22), we obtain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq155_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq156_HTML.gif and (2.21) is proved.

Lemma 2.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq157_HTML.gif be a solution of problem (1.1), (1.10). Assume that there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq158_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq159_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ42_HTML.gif
(2.26)

Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq160_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq161_HTML.gif and (2.21) holds.

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq162_HTML.gif fulfils (2.26), we can find a maximal https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq163_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq164_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq165_HTML.gif and consequently https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq166_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq167_HTML.gif . By (4.23) and (2.26), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq168_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq169_HTML.gif and thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq170_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq171_HTML.gif are negative on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq172_HTML.gif . So, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq173_HTML.gif is positive and decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq174_HTML.gif which yields https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq175_HTML.gif (otherwise, we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq176_HTML.gif , contrary to (2.26)). Consequently there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq177_HTML.gif . By multiplication and integration (2.22) over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq178_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ43_HTML.gif
(2.27)

By similar argument as in the proof of Lemma 2.5 we get that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq179_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq180_HTML.gif . Therefore (2.21) is proved.

3. Damped Solutions

In this section, under assumptions (1.3)–(1.8) and (1.13) we describe a set of all damped solutions which are defined in the following way.

Definition 3.1.

A solution of problem (1.1), (1.10) (or of problem (2.13), (1.10)) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq181_HTML.gif is calleddamped if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ44_HTML.gif
(3.1)

Remark 3.2.

We see, by (2.12), that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq182_HTML.gif is a damped solution of problem (1.1), (1.10) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq183_HTML.gif is a damped solution of problem (2.13), (1.10). Therefore, we can borrow the arguments of [12] in the proofs of this section.

Theorem 3.3.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq184_HTML.gif is a damped solution of problem (1.1), (1.10), then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq185_HTML.gif has a finite number of isolated zeros and satisfies (2.21); or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq186_HTML.gif is oscillatory (it has an unbounded set of isolated zeros).

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq187_HTML.gif be a damped solution of problem (1.1), (1.10). By Remark 2.2, we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq188_HTML.gif in Lemma 2.1 and hence
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ45_HTML.gif
(3.2)

Step 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq189_HTML.gif has no zero in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq190_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq191_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq192_HTML.gif and, by Lemma 2.5, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq193_HTML.gif fulfils (2.21).

Step 2.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq194_HTML.gif is the first zero of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq195_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq196_HTML.gif . Then, due to Remark 2.4, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq197_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq198_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq199_HTML.gif . By virtue of (1.4), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq200_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq201_HTML.gif and thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq202_HTML.gif is decreasing. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq203_HTML.gif be positive on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq204_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq205_HTML.gif is also decreasing, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq206_HTML.gif is increasing and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq207_HTML.gif , due to (3.1). Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq208_HTML.gif . Letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq209_HTML.gif in (2.22), we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq210_HTML.gif , which is impossible because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq211_HTML.gif is bounded below. Therefore there are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq212_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq213_HTML.gif satisfying (2.26) and, by Lemma 2.6, either https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq214_HTML.gif fulfils (2.21) or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq215_HTML.gif has the second zero https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq216_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq217_HTML.gif . So https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq218_HTML.gif is positive on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq219_HTML.gif and has just one local maximum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq220_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq221_HTML.gif . Moreover, putting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq222_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq223_HTML.gif in (2.23), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ46_HTML.gif
(3.3)
and hence
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ47_HTML.gif
(3.4)

Step 3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq224_HTML.gif have no other zeros. Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq225_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq226_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq227_HTML.gif is negative on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq228_HTML.gif . Then, due to (2.1), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq229_HTML.gif . Putting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq230_HTML.gif in (2.23) and letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq231_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ48_HTML.gif
(3.5)
Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq232_HTML.gif exists and, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq233_HTML.gif is bounded, we deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ49_HTML.gif
(3.6)

Letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq234_HTML.gif in (2.22), we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq235_HTML.gif , which contradicts the fact that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq236_HTML.gif is bounded above. Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq237_HTML.gif cannot be negative on the whole interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq238_HTML.gif and there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq239_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq240_HTML.gif . Moreover, according to (3.2), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq241_HTML.gif .

Then, Lemma 2.5 yields that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq242_HTML.gif fulfils (2.21). Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq243_HTML.gif is positive on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq244_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq245_HTML.gif has just one minimum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq246_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq247_HTML.gif . Moreover, putting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq248_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq249_HTML.gif in (2.23), we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ50_HTML.gif
(3.7)
which together with (3.4) yields
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ51_HTML.gif
(3.8)

Step 4.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq250_HTML.gif has its third zero https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq251_HTML.gif . Then we prove as in Step 2 that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq252_HTML.gif has just one negative minimum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq253_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq254_HTML.gif and (3.8) is valid. Further, as in Step 2, we deduce that either https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq255_HTML.gif fulfils (2.21) or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq256_HTML.gif has the fourth zero https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq257_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq258_HTML.gif is positive on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq259_HTML.gif with just one local maximum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq260_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq261_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq262_HTML.gif . This together with (3.8) yields
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ52_HTML.gif
(3.9)

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq263_HTML.gif has no other zeros, we deduce as in Step 3 that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq264_HTML.gif has just one negative minimum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq265_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq266_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq267_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq268_HTML.gif fulfils (2.21).

Step 5.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq269_HTML.gif has other zeros, we use the previous arguments and get that either https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq270_HTML.gif has a finite number of zeros and then fulfils (2.21) or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq271_HTML.gif is oscillatory.

Remark 3.4.

According to the proof of Theorem 3.3, we see that if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq272_HTML.gif is oscillatory, it has just one positive local maximum between the first and the second zero, then just one negative local minimum between the second and the third zero, and so on. By (3.8), (3.9), (1.4)–(1.6) and (1.13), these maxima are decreasing (minima are increasing) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq273_HTML.gif increasing.

Lemma 3.5.

A solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq274_HTML.gif of problem (1.1), (1.10) fulfils the condition
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ53_HTML.gif
(3.10)
if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq275_HTML.gif fulfils the condition
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ54_HTML.gif
(3.11)

Proof.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq276_HTML.gif fulfils (3.10). Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq277_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq278_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq279_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq280_HTML.gif . Otherwise https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq281_HTML.gif , due to Lemma 2.5. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq282_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq283_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq284_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq285_HTML.gif . By Remark 2.4 and (3.10), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq286_HTML.gif . Integrating (1.1) over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq287_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ55_HTML.gif
(3.12)

Due to (1.4), we see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq288_HTML.gif is strictly decreasing for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq289_HTML.gif as long as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq290_HTML.gif . Thus, there are two possibilities. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq291_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq292_HTML.gif , then from Lemma 2.6 we get (2.21), which contradicts (3.10). If there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq293_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq294_HTML.gif , then in view Remark 2.4 we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq295_HTML.gif . Using the arguments of Steps 3–5 of the proof of Theorem 3.3, we get that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq296_HTML.gif is damped, contrary to (3.10). Therefore, such https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq297_HTML.gif cannot exist and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq298_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq299_HTML.gif . Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq300_HTML.gif . So, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq301_HTML.gif fulfils (3.11). The inverse implication is evident.

Remark 3.6.

According to Definition 1.3 and Lemma 3.5, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq302_HTML.gif is a homoclinic solution of problem (1.1), (1.10) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq303_HTML.gif is a homoclinic solution of problem (2.13), (1.10).

Theorem 3.7 (on damped solutions).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq304_HTML.gif satisfy (1.5) and (1.6). Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq305_HTML.gif is a solution of problem (1.1), (1.10) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq306_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq307_HTML.gif is damped.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq308_HTML.gif be a solution of (1.1), (1.10) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq309_HTML.gif . Then, by (1.4)–(1.6),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ56_HTML.gif
(3.13)
Assume on the contrary that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq310_HTML.gif is not damped. Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq311_HTML.gif is defined on the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq312_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq313_HTML.gif or there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq314_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq315_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq316_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq317_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq318_HTML.gif . If the latter possibility occurs, (2.22) and (3.13) give by integration
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ57_HTML.gif
(3.14)
a contradiction. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq319_HTML.gif , then, by Lemma 3.5, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq320_HTML.gif fulfils (3.11). So https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq321_HTML.gif has a unique zero https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq322_HTML.gif . Integrating (2.22) over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq323_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ58_HTML.gif
(3.15)
and so
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ59_HTML.gif
(3.16)
Integrating (2.22) over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq324_HTML.gif , we obtain for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq325_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ60_HTML.gif
(3.17)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq326_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq327_HTML.gif , and letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq328_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq329_HTML.gif . This together with (3.16) contradicts (3.13). We have proved that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq330_HTML.gif is damped.

Theorem 3.8.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq331_HTML.gif be the set of all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq332_HTML.gif such that corresponding solutions of problem (1.1), (1.10) are damped. Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq333_HTML.gif is open in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq334_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq335_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq336_HTML.gif be a solution of (1.1), (1.10) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq337_HTML.gif . So, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq338_HTML.gif is damped and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq339_HTML.gif is also a solution of (2.13).
  1. (a)

    Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq340_HTML.gif be oscillatory. Then its first local maximum belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq341_HTML.gif . Lemma 2.3 guarantees that if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq342_HTML.gif is sufficiently close to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq343_HTML.gif , the corresponding solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq344_HTML.gif of (2.13), (1.10) has also its first local maximum in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq345_HTML.gif . This means that there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq346_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq347_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq348_HTML.gif satisfies (2.26). Now, we can continue as in the proof of Theorem 3.3 using the arguments of Steps 2–5 and Remark 3.2 to get that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq349_HTML.gif is damped.

     
  2. (b)
    Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq350_HTML.gif have at most a finite number of zeros. Then, by Theorem 3.3, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq351_HTML.gif fulfils (2.21). Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq352_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq353_HTML.gif fulfils (2.22), we get by integration over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq354_HTML.gif
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ61_HTML.gif
    (3.18)
     
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq355_HTML.gif we get, by (2.21),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ62_HTML.gif
(3.19)
Therefore, we can find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq356_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ63_HTML.gif
(3.20)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq357_HTML.gif be the constant of Lemma 2.1. Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq358_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq359_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq360_HTML.gif is a corresponding solution of problem (2.13), (1.10). Using Lemma 2.1, Lemma 2.3 and the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq361_HTML.gif , we can find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq362_HTML.gif such that if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq363_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ64_HTML.gif
(3.21)
moreover https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq364_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq365_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ65_HTML.gif
(3.22)
Therefore, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ66_HTML.gif
(3.23)

Consequently, integrating (2.13) over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq366_HTML.gif and using (3.19)–(3.23), we get for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq367_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ67_HTML.gif
(3.24)
We get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq368_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq369_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq370_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq371_HTML.gif and, due to (1.4)–(1.6),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ68_HTML.gif
(3.25)

Assume that there is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq372_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq373_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq374_HTML.gif . Then, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq375_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq376_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq377_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq378_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq379_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq380_HTML.gif , contrary to (3.25). Hence we get that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq381_HTML.gif fulfils (3.1).

4. Escape Solutions

During the whole section, we assume (1.3)–(1.8) and (1.13). We prove that problem (1.1), (1.10) has at least one escape solution. According to Section 1 and Remark 2.2, we work with the following definitions.

Definition 4.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq382_HTML.gif . A solution of problem (1.1), (1.10) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq383_HTML.gif is called an escape solution if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ69_HTML.gif
(4.1)

Definition 4.2.

A solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq384_HTML.gif of problem (2.13), (1.10) is called an escape solution, if there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq385_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ70_HTML.gif
(4.2)

Remark 4.3.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq386_HTML.gif is an escape solution of problem (2.13), (1.10), then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq387_HTML.gif is an escape solution of problem (1.1), (1.10) on some interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq388_HTML.gif .

Theorem 4.4 (on three types of solutions.).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq389_HTML.gif be a solution of problem (1.1), (1.10). Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq390_HTML.gif is just one of the following three types

(I) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq391_HTML.gif is damped;

(II) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq392_HTML.gif is homoclinic;

(III) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq393_HTML.gif is escape.

Proof.

By Definition 3.1, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq394_HTML.gif is damped if and only if (3.1) holds. By Lemma 3.5 and Definition 1.3, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq395_HTML.gif is homoclinic if and only if (3.10) holds. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq396_HTML.gif be neither damped nor homoclinic. Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq397_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq398_HTML.gif is bounded on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq399_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq400_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq401_HTML.gif . So, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq402_HTML.gif has its first zero https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq403_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq404_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq405_HTML.gif . Assume that there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq406_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq407_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq408_HTML.gif . Then, by Lemma 2.6, either https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq409_HTML.gif fulfils (2.21) or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq410_HTML.gif has its second zero and, arguing as in Steps 2–5 of the proof of Theorem 3.3, we deduce that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq411_HTML.gif is a damped solution. This contradiction implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq412_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq413_HTML.gif . Therefore, by Definition 4.1, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq414_HTML.gif is an escape solution.

Theorem 4.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq415_HTML.gif be the set of all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq416_HTML.gif such that the corresponding solutions of (1.1), (1.10) are escape solutions. The set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq417_HTML.gif is open in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq418_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq419_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq420_HTML.gif be a solution of problem (1.1), (1.10) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq421_HTML.gif . So, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq422_HTML.gif fulfils (4.1) for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq423_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq424_HTML.gif be a solution of problem (2.13), (1.10) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq425_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq426_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq427_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq428_HTML.gif is increasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq429_HTML.gif . There exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq430_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq431_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq432_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq433_HTML.gif be a solution of problem (2.13), (1.10) for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq434_HTML.gif . Lemma 2.3 yields https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq435_HTML.gif such that if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq436_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq437_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq438_HTML.gif is an escape solution of problem (2.13), (1.10). By Remark 4.3, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq439_HTML.gif is also an escape solution of problem (1.1), (1.10) on some interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq440_HTML.gif .

To prove that the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq441_HTML.gif of Theorem 4.5 is nonempty we will need the following two lemmas.

Lemma 4.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq442_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq443_HTML.gif is a solution of problem (1.1), (1.10) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq444_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq445_HTML.gif is a maximal interval where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq446_HTML.gif is increasing and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq447_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq448_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ71_HTML.gif
(4.3)

Proof.

Step 1.

We show that the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq449_HTML.gif is nonempty. Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq450_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq451_HTML.gif satisfies (1.3), (1.13), we can find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq452_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ72_HTML.gif
(4.4)
Integrating (1.1) over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq453_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ73_HTML.gif
(4.5)

So, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq454_HTML.gif is an increasing solution of problem (1.1), (1.10) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq455_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq456_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq457_HTML.gif . Therefore the nonempty interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq458_HTML.gif exists.

Step 2.

By multiplication of (1.1) by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq459_HTML.gif and integration over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq460_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ74_HTML.gif
(4.6)
Using the "per partes" integration, we get for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq461_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ75_HTML.gif
(4.7)

This relation together with (4.6) implies (4.3).

Remark 4.7.

Consider a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq462_HTML.gif of Lemma 4.6. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq463_HTML.gif is an escape solution, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq464_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq465_HTML.gif is not an escape solution. Then both possibilities https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq466_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq467_HTML.gif can occur. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq468_HTML.gif . By Theorem 4.4 and Lemma 2.5, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq469_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq470_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq471_HTML.gif . We write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq472_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq473_HTML.gif . Using Lemmas 3.5 and 2.5 and Theorem 4.4, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq474_HTML.gif and either https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq475_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq476_HTML.gif .

Lemma 4.8.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq477_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq478_HTML.gif . Then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq479_HTML.gif

(i)there exists a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq480_HTML.gif of problem (1.1), (1.10) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq481_HTML.gif ,

(ii)there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq482_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq483_HTML.gif is the maximal interval on which the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq484_HTML.gif is increasing and its values in this interval are contained in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq485_HTML.gif ,

(iii)there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq486_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq487_HTML.gif .

If the sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq488_HTML.gif is unbounded, then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq489_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq490_HTML.gif is an escape solution.

Proof.

Similar arugmets can be found in [12]. By Lemma 2.1, the assertion (i) holds. The arguments in Step 1 of the proof of Lemma 4.6 imply (ii). The strict monotonicity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq491_HTML.gif and Remark 4.7 yields a unique https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq492_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq493_HTML.gif is unbounded. Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ76_HTML.gif
(4.8)
(otherwise, we take a subsequence). Assume on the contrary that for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq494_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq495_HTML.gif is not an escape solution. Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq496_HTML.gif . Then, by Remark 4.7,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ77_HTML.gif
(4.9)
Due to (4.9), (1.2) and (ii) there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq497_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ78_HTML.gif
(4.10)
By (i) and (ii), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq498_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ79_HTML.gif
(4.11)
Integrating it over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq499_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ80_HTML.gif
(4.12)
Put
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ81_HTML.gif
(4.13)
Then, by (4.12),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ82_HTML.gif
(4.14)
We see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq500_HTML.gif is decreasing. From (1.4) and (1.6) we get that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq501_HTML.gif is increasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq502_HTML.gif and consequently by (4.9) and (4.13), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ83_HTML.gif
(4.15)
Integrating (4.14) over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq503_HTML.gif and using (4.10), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ84_HTML.gif
(4.16)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ85_HTML.gif
(4.17)
Further, by (4.15),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ86_HTML.gif
(4.18)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ87_HTML.gif
(4.19)
Conditions (1.8) and (4.8) yield https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq504_HTML.gif , which implies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ88_HTML.gif
(4.20)
By (4.13) and (4.18),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ89_HTML.gif
(4.21)
and consequently
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ90_HTML.gif
(4.22)

which contradicts (4.20). Therefore, at least one escape solution of (1.1), (1.10) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq505_HTML.gif must exist.

Theorem 4.9 (on escape solution).

Assume that (1.3)–(1.8) and (1.13) hold and let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ91_HTML.gif
(4.23)

Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq506_HTML.gif such that the corresponding solution of problem (1.1), (1.10) is an escape solution.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq507_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq508_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq509_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq510_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq511_HTML.gif be sequences from Lemma 4.8. Moreover, let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ92_HTML.gif
(4.24)

By (4.24) we can find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq512_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq513_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq514_HTML.gif . We assume that for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq515_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq516_HTML.gif is not an escape solution and we construct a contradiction.

Step 1.

We derive some inequality for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq517_HTML.gif . By Remark 4.7, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ93_HTML.gif
(4.25)
and, by Lemma 4.8, the sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq518_HTML.gif is bounded. Therefore there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq519_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ94_HTML.gif
(4.26)
Choose an arbitrary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq520_HTML.gif . According to Lemma 4.6, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq521_HTML.gif satisfies equality (4.3), that is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ95_HTML.gif
(4.27)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq522_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq523_HTML.gif is increasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq524_HTML.gif , there exists a unique https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq525_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ96_HTML.gif
(4.28)
Having in mind, due to (1.4)–(1.8), that the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ97_HTML.gif
(4.29)
holds, we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ98_HTML.gif
(4.30)
By virtue of (1.6) and (1.13), we see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq526_HTML.gif is decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq527_HTML.gif , which yields
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ99_HTML.gif
(4.31)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ100_HTML.gif
(4.32)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq528_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq529_HTML.gif , the monotonicity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq530_HTML.gif yields https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq531_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq532_HTML.gif , and consequently
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ101_HTML.gif
(4.33)
Therefore (4.27) and (4.32) give
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ102_HTML.gif
(4.34)

Step 2.

We prove that the sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq533_HTML.gif is bounded below by some positive number. Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq534_HTML.gif is a solution of (1.1) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq535_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ103_HTML.gif
(4.35)
Integrating it, we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ104_HTML.gif
(4.36)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq536_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq537_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq538_HTML.gif . Having in mind (1.8), we see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq539_HTML.gif is increasing and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq540_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq541_HTML.gif . Consequently
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ105_HTML.gif
(4.37)
Integrating (4.36) over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq542_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ106_HTML.gif
(4.38)
and hence
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ107_HTML.gif
(4.39)
By (4.23) we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ108_HTML.gif
(4.40)
which, due to (4.39), yields
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ109_HTML.gif
(4.41)

So, by virtue of (4.37), there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq543_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq544_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq545_HTML.gif .

Step 3.

We construct a contradiction. Putting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq546_HTML.gif in (4.34), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ110_HTML.gif
(4.42)
Due to (4.23), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq547_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq548_HTML.gif , and consequently, by (4.24),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ111_HTML.gif
(4.43)

In order to get a contradiction, we distinguish two cases.

Case 1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq549_HTML.gif , that is, we can find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq550_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq551_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq552_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ112_HTML.gif
(4.44)
Then, by (4.43), for each sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq553_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ113_HTML.gif
(4.45)
Putting it to (4.42), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ114_HTML.gif
(4.46)

Therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq554_HTML.gif , contrary to (4.25).

Case 2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq555_HTML.gif . We may assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq556_HTML.gif (otherwise we take a subsequence). Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq557_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq558_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ115_HTML.gif
(4.47)
Due to (4.43), for each sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq559_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ116_HTML.gif
(4.48)
Putting it to (4.42), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ117_HTML.gif
(4.49)
Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq560_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq561_HTML.gif . Integrating it over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq562_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ118_HTML.gif
(4.50)

which yields, by (4.26), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq563_HTML.gif and also https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq564_HTML.gif , contrary to (4.25). These contradictions obtained in both cases imply that there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq565_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq566_HTML.gif is an escape solution.

5. Homoclinic Solution

The following theorem provides the existence of a homoclinic solution under the assumption that the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq567_HTML.gif in (1.1) has a linear behaviour near https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq568_HTML.gif . According to Definition 1.2, a homoclinic solution is a strictly increasing solution of problem (1.1), (1.2).

Theorem 5.1 (On homoclinic solution.).

Let the assumptions of Theorem 4.9 be satisfied. Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq569_HTML.gif such that the corresponding solution of problem (1.1), (1.10) is a homoclinic solution.

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq570_HTML.gif denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq571_HTML.gif the corresponding solution of problem (1.1), (1.10). Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq572_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq573_HTML.gif be the set of all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq574_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq575_HTML.gif is a damped solution and an escape solution, respectively. By Theorems 3.7, 3.8, 4.5, and 4.9, the sets https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq576_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq577_HTML.gif are nonempty and open in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq578_HTML.gif . Therefore, the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq579_HTML.gif is nonempty. Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq580_HTML.gif . Then, by Theorem 4.4, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq581_HTML.gif is a homoclinic solution. Moreover, due to Theorem 3.7, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq582_HTML.gif .

Example 5.2.

The function
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ119_HTML.gif
(5.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq583_HTML.gif is a negative constant, satisfies the conditions (1.3)–(1.6), (1.13), and (4.23).

Declarations

Acknowledgments

The authors thank the referee for valuable comments. This work 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

References

  1. Dell'Isola F, Gouin H, Rotoli G: Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations. European Journal of Mechanics. B 1996, 15(4):545-568.MATHGoogle Scholar
  2. Derrick GH: Comments on nonlinear wave equations as models for elementary particles. Journal of Mathematical Physics 1964, 5: 1252-1254. 10.1063/1.1704233MathSciNetView ArticleGoogle Scholar
  3. Gouin H, Rotoli G: An analytical approximation of density profile and surface tension of microscopic bubbles for Van Der Waals fluids. Mechanics Research Communications 1997, 24(3):255-260. 10.1016/S0093-6413(97)00022-0MATHView ArticleGoogle Scholar
  4. Kitzhofer G, Koch O, Lima P, Weinmüller E: Efficient numerical solution of the density profile equation in hydrodynamics. Journal of Scientific Computing 2007, 32(3):411-424. 10.1007/s10915-007-9141-0MATHMathSciNetView ArticleGoogle Scholar
  5. Lima PM, Konyukhova NB, Sukov AI, Chemetov NV: Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems. Journal of Computational and Applied Mathematics 2006, 189(1-2):260-273. 10.1016/j.cam.2005.05.004MATHMathSciNetView ArticleGoogle Scholar
  6. Koch O, Kofler P, Weinmüller EB: Initial value problems for systems of ordinary first and second order differential equations with a singularity of the first kind. Analysis 2001, 21(4):373-389.MATHMathSciNetView ArticleGoogle Scholar
  7. Bonheure D, Gomes JM, Sanchez L: Positive solutions of a second-order singular ordinary differential equation. Nonlinear Analysis: Theory, Methods & Applications 2005, 61(8):1383-1399. 10.1016/j.na.2005.02.029MATHMathSciNetView ArticleGoogle Scholar
  8. Conti M, Merizzi L, Terracini S:Radial solutions of superlinear equations on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq584_HTML.gif . I. A global variational approach. Archive for Rational Mechanics and Analysis 2000, 153(4):291-316. 10.1007/s002050050015MATHMathSciNetView ArticleGoogle Scholar
  9. Berestycki H, Lions P-L, Peletier LA:An ODE approach to the existence of positive solutions for semilinear problems in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq585_HTML.gif . Indiana University Mathematics Journal 1981, 30(1):141-157. 10.1512/iumj.1981.30.30012MATHMathSciNetView ArticleGoogle Scholar
  10. Maatoug L: On the existence of positive solutions of a singular nonlinear eigenvalue problem. Journal of Mathematical Analysis and Applications 2001, 261(1):192-204. 10.1006/jmaa.2001.7491MATHMathSciNetView ArticleGoogle Scholar
  11. Rachůnková I, Tomeček J: Singular nonlinear problem for ordinary differential equation of the second-order on the half-line. In Mathematical Models in Engineering, Biology and Medicine: International Conference on Boundary Value Problems Edited by: Cabada A, Liz E, Nieto JJ. 2009, 294-303.Google Scholar
  12. Rachůnková I, Tomeček J: Bubble-type solutions of nonlinear singular problem. submitted
  13. Palamides AP, Yannopoulos TG: Terminal value problem for singular ordinary differential equations: theoretical analysis and numerical simulations of ground states. Boundary Value Problems 2006, 2006:-28.Google Scholar

Copyright

© I. Rachůnková and J. Tomeček. 2009

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