Homoclinic Solutions of Singular Nonautonomous Second-Order Differential Equations

  • Irena Rachůnková1Email author and

    Affiliated with

    • Jan Tomeček1

      Affiliated with

      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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq1_HTML.gif , having a singularity at http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq4_HTML.gif is proved provided that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq6_HTML.gif .

      1. Introduction

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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ1_HTML.gif
      (1.1)
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq9_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ3_HTML.gif
      (1.3)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ4_HTML.gif
      (1.4)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ5_HTML.gif
      (1.5)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ6_HTML.gif
      (1.6)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ7_HTML.gif
      (1.7)
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq10_HTML.gif because http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq11_HTML.gif .

      Definition 1.1.

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

      Definition 1.2.

      Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq22_HTML.gif . If http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq27_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq28_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq31_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq34_HTML.gif and for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq35_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq38_HTML.gif has at least three zeros http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq39_HTML.gif . More precisely, the conditions,

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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ10_HTML.gif
      (1.10)
      http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq46_HTML.gif fulfils

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq47_HTML.gif satisfies moreover

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ13_HTML.gif
      (1.13)
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq48_HTML.gif has just two zeros http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq49_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq50_HTML.gif in the interval http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq52_HTML.gif is either bounded on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq53_HTML.gif or http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq58_HTML.gif having a linear behaviour near http://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 http://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

      http://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 http://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 http://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

      http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq65_HTML.gif .

      Lemma 2.1.

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

      Proof.

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

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

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq84_HTML.gif . Consider the Banach space http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq86_HTML.gif defined by

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ21_HTML.gif
      (2.5)
      A function http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq89_HTML.gif . Using the Lipschitz property of http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq92_HTML.gif . This solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq93_HTML.gif has the form
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ22_HTML.gif
      (2.6)

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

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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ23_HTML.gif
      (2.7)
      Put
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ24_HTML.gif
      (2.8)
      Then
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq103_HTML.gif . Multiplying (2.7) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq104_HTML.gif and integrating it over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq105_HTML.gif , we get
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq106_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq107_HTML.gif .

      Let us put

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

      Here http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq116_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq117_HTML.gif .

      Proof.

      Choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq118_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq119_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq120_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq121_HTML.gif be the Lipschitz constant for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq122_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq123_HTML.gif . By (2.6) for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq124_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq125_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq127_HTML.gif ,
      http://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
      http://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),
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ33_HTML.gif
      (2.17)
      If we choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq128_HTML.gif such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq129_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq130_HTML.gif , and consider the initial conditions
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq131_HTML.gif . In particular, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq132_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq135_HTML.gif , respectively.

      Lemma 2.5.

      Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq137_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ36_HTML.gif
      (2.20)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq138_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq139_HTML.gif and
      http://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), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq140_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq141_HTML.gif and thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq142_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq143_HTML.gif are positive on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq144_HTML.gif . Consequently, there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq145_HTML.gif . Further, by (1.1),
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq146_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ39_HTML.gif
      (2.23)
      Therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ40_HTML.gif
      (2.24)
      and hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq147_HTML.gif exists. Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq148_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq149_HTML.gif , we get
      http://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), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq150_HTML.gif exists and, since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq151_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq152_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq153_HTML.gif . Hence, letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq154_HTML.gif in (2.22), we obtain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq155_HTML.gif . Therefore, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq158_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq159_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ42_HTML.gif
      (2.26)

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

      Proof.

      Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq163_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq164_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq165_HTML.gif and consequently http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq166_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq167_HTML.gif . By (4.23) and (2.26), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq168_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq169_HTML.gif and thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq170_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq171_HTML.gif are negative on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq172_HTML.gif . So, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq173_HTML.gif is positive and decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq174_HTML.gif which yields http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq175_HTML.gif (otherwise, we get http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq178_HTML.gif , we obtain
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq179_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq181_HTML.gif is calleddamped if
      http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq188_HTML.gif in Lemma 2.1 and hence
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ45_HTML.gif
      (3.2)

      Step 1.

      If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq189_HTML.gif has no zero in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq190_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq191_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq192_HTML.gif and, by Lemma 2.5, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq194_HTML.gif is the first zero of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq195_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq196_HTML.gif . Then, due to Remark 2.4, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq197_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq198_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq199_HTML.gif . By virtue of (1.4), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq200_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq201_HTML.gif and thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq202_HTML.gif is decreasing. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq203_HTML.gif be positive on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq204_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq205_HTML.gif is also decreasing, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq206_HTML.gif is increasing and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq207_HTML.gif , due to (3.1). Consequently, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq208_HTML.gif . Letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq209_HTML.gif in (2.22), we get http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq210_HTML.gif , which is impossible because http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq211_HTML.gif is bounded below. Therefore there are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq212_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq214_HTML.gif fulfils (2.21) or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq215_HTML.gif has the second zero http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq216_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq217_HTML.gif . So http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq218_HTML.gif is positive on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq219_HTML.gif and has just one local maximum http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq220_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq221_HTML.gif . Moreover, putting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq222_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq223_HTML.gif in (2.23), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ46_HTML.gif
      (3.3)
      and hence
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ47_HTML.gif
      (3.4)

      Step 3.

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

      Letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq234_HTML.gif in (2.22), we get http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq235_HTML.gif , which contradicts the fact that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq236_HTML.gif is bounded above. Therefore, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq238_HTML.gif and there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq239_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq240_HTML.gif . Moreover, according to (3.2), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq241_HTML.gif .

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

      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ51_HTML.gif
      (3.8)

      Step 4.

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

      If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq264_HTML.gif has just one negative minimum http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq265_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq266_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq267_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq268_HTML.gif fulfils (2.21).

      Step 5.

      If http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq273_HTML.gif increasing.

      Lemma 3.5.

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

      Proof.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq276_HTML.gif fulfils (3.10). Then there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq277_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq278_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq279_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq280_HTML.gif . Otherwise http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq281_HTML.gif , due to Lemma 2.5. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq282_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq283_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq284_HTML.gif , http://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), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq286_HTML.gif . Integrating (1.1) over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq287_HTML.gif , we get
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq288_HTML.gif is strictly decreasing for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq289_HTML.gif as long as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq290_HTML.gif . Thus, there are two possibilities. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq291_HTML.gif for all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq293_HTML.gif such that http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq297_HTML.gif cannot exist and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq298_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq299_HTML.gif . Consequently, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq300_HTML.gif . So, http://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, http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq306_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq307_HTML.gif is damped.

      Proof.

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

      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq326_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq327_HTML.gif , and letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq328_HTML.gif , we get http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq330_HTML.gif is damped.

      Theorem 3.8.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq331_HTML.gif be the set of all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq333_HTML.gif is open in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq334_HTML.gif .

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq335_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq337_HTML.gif . So, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq338_HTML.gif is damped and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq341_HTML.gif . Lemma 2.3 guarantees that if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq342_HTML.gif is sufficiently close to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq343_HTML.gif , the corresponding solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq345_HTML.gif . This means that there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq346_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq347_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq349_HTML.gif is damped.

         
      2. (b)
        Let http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq351_HTML.gif fulfils (2.21). Choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq352_HTML.gif . Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq354_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ61_HTML.gif
        (3.18)
         
      For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq355_HTML.gif we get, by (2.21),
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ62_HTML.gif
      (3.19)
      Therefore, we can find http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq356_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ63_HTML.gif
      (3.20)
      Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq358_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq359_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq361_HTML.gif , we can find http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq362_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq363_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ64_HTML.gif
      (3.21)
      moreover http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq364_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq365_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ65_HTML.gif
      (3.22)
      Therefore, we have
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq367_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ67_HTML.gif
      (3.24)
      We get http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq368_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq369_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq370_HTML.gif for http://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),
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ68_HTML.gif
      (3.25)

      Assume that there is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq372_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq373_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq374_HTML.gif . Then, since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq375_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq376_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq377_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq378_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq379_HTML.gif for http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq383_HTML.gif is called an escape solution if
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq385_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ70_HTML.gif
      (4.2)

      Remark 4.3.

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

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

      (III) http://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, http://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, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq397_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq398_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq399_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq400_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq401_HTML.gif . So, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq402_HTML.gif has its first zero http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq403_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq404_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq405_HTML.gif . Assume that there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq406_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq407_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq408_HTML.gif . Then, by Lemma 2.6, either http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq409_HTML.gif fulfils (2.21) or http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq412_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq413_HTML.gif . Therefore, by Definition 4.1, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq415_HTML.gif be the set of all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq417_HTML.gif is open in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq418_HTML.gif .

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq419_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq421_HTML.gif . So, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq422_HTML.gif fulfils (4.1) for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq423_HTML.gif . Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq425_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq426_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq427_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq428_HTML.gif is increasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq429_HTML.gif . There exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq430_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq431_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq432_HTML.gif . Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq434_HTML.gif . Lemma 2.3 yields http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq435_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq436_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq437_HTML.gif . Therefore, http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq440_HTML.gif .

      To prove that the set http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq442_HTML.gif . Assume that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq444_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq445_HTML.gif is a maximal interval where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq446_HTML.gif is increasing and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq447_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq448_HTML.gif . Then
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq449_HTML.gif is nonempty. Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq450_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq452_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ72_HTML.gif
      (4.4)
      Integrating (1.1) over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq453_HTML.gif we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ73_HTML.gif
      (4.5)

      So, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq455_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq456_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq457_HTML.gif . Therefore the nonempty interval http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq459_HTML.gif and integration over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq460_HTML.gif we obtain
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq461_HTML.gif
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq462_HTML.gif of Lemma 4.6. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq463_HTML.gif is an escape solution, then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq464_HTML.gif . Assume that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq466_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq467_HTML.gif can occur. Let http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq469_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq470_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq471_HTML.gif . We write http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq472_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq474_HTML.gif and either http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq475_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq476_HTML.gif .

      Lemma 4.8.

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

      (i)there exists a solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq481_HTML.gif ,

      (ii)there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq482_HTML.gif such that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq485_HTML.gif ,

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

      If the sequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq488_HTML.gif is unbounded, then there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq489_HTML.gif such that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq492_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq493_HTML.gif is unbounded. Then
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq494_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq495_HTML.gif is not an escape solution. Choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq496_HTML.gif . Then, by Remark 4.7,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq497_HTML.gif satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ78_HTML.gif
      (4.10)
      By (i) and (ii), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq498_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ79_HTML.gif
      (4.11)
      Integrating it over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq499_HTML.gif we get
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ80_HTML.gif
      (4.12)
      Put
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ81_HTML.gif
      (4.13)
      Then, by (4.12),
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ82_HTML.gif
      (4.14)
      We see that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq501_HTML.gif is increasing on http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ83_HTML.gif
      (4.15)
      Integrating (4.14) over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq503_HTML.gif and using (4.10), we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ84_HTML.gif
      (4.16)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ85_HTML.gif
      (4.17)
      Further, by (4.15),
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ86_HTML.gif
      (4.18)
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq504_HTML.gif , which implies
      http://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),
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ89_HTML.gif
      (4.21)
      and consequently
      http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ91_HTML.gif
      (4.23)

      Then there exists http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq507_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq508_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq509_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq510_HTML.gif and http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq512_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq513_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq514_HTML.gif . We assume that for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq515_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq517_HTML.gif . By Remark 4.7, we have
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq518_HTML.gif is bounded. Therefore there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq519_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ94_HTML.gif
      (4.26)
      Choose an arbitrary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq520_HTML.gif . According to Lemma 4.6, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq521_HTML.gif satisfies equality (4.3), that is
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ95_HTML.gif
      (4.27)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq522_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq523_HTML.gif is increasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq524_HTML.gif , there exists a unique http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq525_HTML.gif such that
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ97_HTML.gif
      (4.29)
      holds, we get
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq526_HTML.gif is decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq527_HTML.gif , which yields
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ99_HTML.gif
      (4.31)
      Hence,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ100_HTML.gif
      (4.32)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq528_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq529_HTML.gif , the monotonicity of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq530_HTML.gif yields http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq531_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq532_HTML.gif , and consequently
      http://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
      http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq535_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ103_HTML.gif
      (4.35)
      Integrating it, we get
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ104_HTML.gif
      (4.36)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq536_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq537_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq539_HTML.gif is increasing and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq540_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq541_HTML.gif . Consequently
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ105_HTML.gif
      (4.37)
      Integrating (4.36) over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq542_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ106_HTML.gif
      (4.38)
      and hence
      http://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
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq543_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq544_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq546_HTML.gif in (4.34), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ110_HTML.gif
      (4.42)
      Due to (4.23), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq547_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq548_HTML.gif , and consequently, by (4.24),
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq549_HTML.gif , that is, we can find http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq550_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq551_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq552_HTML.gif , such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq553_HTML.gif , we get
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ114_HTML.gif
      (4.46)

      Therefore http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq555_HTML.gif . We may assume http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq557_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq558_HTML.gif , such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq559_HTML.gif , we get
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_Equ117_HTML.gif
      (4.49)
      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq560_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq561_HTML.gif . Integrating it over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq562_HTML.gif , we obtain
      http://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), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq563_HTML.gif and also http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq565_HTML.gif such that http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq570_HTML.gif denote by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq572_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq573_HTML.gif be the set of all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq574_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq576_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq577_HTML.gif are nonempty and open in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq578_HTML.gif . Therefore, the set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq579_HTML.gif is nonempty. Choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq580_HTML.gif . Then, by Theorem 4.4, http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F959636/MediaObjects/13661_2009_Article_892_IEq582_HTML.gif .

      Example 5.2.

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

      where http://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

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      Copyright

      © I. Rachůnková and J. Tomeček. 2009

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.