Superlinear Singular Problems on the Half Line

  • Irena Rachunková1Email author and

    Affiliated with

    • Jan Tomecek1

      Affiliated with

      Boundary Value Problems20102010:429813

      DOI: 10.1155/2010/429813

      Received: 19 October 2010

      Accepted: 7 December 2010

      Published: 15 December 2010

      Abstract

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

      1. Introduction

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

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

      Definition 1.1.

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

      Definition 1.2.

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

      Definition 1.3.

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

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

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

      Definition 1.4.

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

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

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

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

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

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

      2. Escape Solutions

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

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

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

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

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

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

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

      Lemma 2.3.

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

      Proof.

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

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

      Lemma 2.4.

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

      Proof.

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

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

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

      Lemma 2.5.

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

      Proof.

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

      We have proved that (2.15) is valid.

      Lemma 2.6.

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

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

      Proof.

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

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

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

      Lemma 2.7.

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

      Proof.

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

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

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

      We will consider two cases.

      Case 1.

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

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

      Case 2.

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

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

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

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

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

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

      Lemma 2.8.

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

      Proof.

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

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

      Lemma 2.9.

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

      Proof.

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

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

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

      Theorem 2.10.

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

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

      Proof.

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

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

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

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

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

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

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

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

      Step 4 (final contradictions).

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

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

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

      contrary to (2.53).

      Remark 2.11.

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

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

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

      Example 2.12.

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

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

      Example 2.13.

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

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

      3. Homoclinic Solutions

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

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

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

      Definition 3.1.

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

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

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

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

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

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

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

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

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

      Theorem 3.5 (On a homoclinic solution).

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

      Proof.

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

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

      Corollary 3.6.

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

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

      Example 3.7.

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

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

      Declarations

      Acknowledgment

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

      Authors’ Affiliations

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

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      Copyright

      © Rachunková and J. Tomecek. 2010

      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.