A Fourth-Order Boundary Value Problem with One-Sided Nagumo Condition

  • Wenjing Song1, 2Email author and

    Affiliated with

    • Wenjie Gao1

      Affiliated with

      Boundary Value Problems20112011:569191

      DOI: 10.1155/2011/569191

      Received: 10 January 2011

      Accepted: 9 March 2011

      Published: 14 March 2011

      Abstract

      The aim of this paper is to study a fourth-order separated boundary value problem with the right-hand side function satisfying one-sided Nagumo-type condition. By making a series of a priori estimates and applying lower and upper functions techniques and Leray-Schauder degree theory, the authors obtain the existence and location result of solutions to the problem.

      1. Introduction

      In this paper we apply the lower and upper functions method to study the fourth-order nonlinear equation
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ1_HTML.gif
      (1.1)

      with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq1_HTML.gif being a continuous function.

      This equation can be used to model the deformations of an elastic beam, and the type of boundary conditions considered depends on how the beam is supported at the two endpoints [1, 2]. We consider the separated boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ2_HTML.gif
      (1.2)

      with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq3_HTML.gif .

      For the fourth-order differential equation
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ3_HTML.gif
      (1.3)

      the authors in [3] obtained the existence of solutions with the assumption that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq4_HTML.gif satisfies the two-sided Nagumo-type conditions. For more related works, interested readers may refer to [114]. The one-sided Nagumo-type condition brings some difficulties in studying this kind of problem, as it can be seen in [1518].

      Motivated by the above works, we consider the existence of solutions when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq5_HTML.gif satisfies one-sided Nagumo-type conditions. This is a generalization of the above cases. We apply lower and upper functions technique and topological degree method to prove the existence of solutions by making a priori estimates for the third derivative of all solutions of problems (1.1) and (1.2). The estimates are essential for proving the existence of solutions.

      The outline of this paper is as follows. In Section 2, we give the definition of lower and upper functions to problems (1.1) and (1.2) and obtain some a priori estimates. Section 3 will be devoted to the study of the existence of solutions. In Section 4, we give an example to illustrate the conclusions.

      2. Definitions and A Priori Estimates

      Upper and lower functions will be an important tool to obtain a priori bounds on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq7_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq8_HTML.gif . For this problem we define them as follows.

      Definition 2.1.

      The functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq9_HTML.gif verifying
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ4_HTML.gif
      (2.1)

      define a pair of lower and upper functions of problems (1.1) and (1.2) if the following conditions are satisfied:

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq11_HTML.gif ,

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq12_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq13_HTML.gif ,

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq14_HTML.gif .

      Remark 2.2.

      By integration, from (iii) and (2.1), we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ5_HTML.gif
      (2.2)

      that is, lower and upper functions, and their first derivatives are also well ordered.

      To have an a priori estimate on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq15_HTML.gif , we need a one-sided Nagumo-type growth condition, which is defined as follows.

      Definition 2.3.

      Given a set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq16_HTML.gif , a continuous http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq17_HTML.gif is said to satisfy the one-sided Nagumo-type condition in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq18_HTML.gif if there exists a real continuous function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq19_HTML.gif , for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq20_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ6_HTML.gif
      (2.3)
      with
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ7_HTML.gif
      (2.4)

      Lemma 2.4.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq21_HTML.gif satisfy
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ8_HTML.gif
      (2.5)
      and consider the set
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ9_HTML.gif
      (2.6)

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq22_HTML.gif be a continuous function satisfying one-sided Nagumo-type condition in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq23_HTML.gif .

      Then, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq24_HTML.gif , there exists an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq25_HTML.gif such that for every solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq26_HTML.gif of problems (1.1) and (1.2) with
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ10_HTML.gif
      (2.7)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ11_HTML.gif
      (2.8)

      for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq27_HTML.gif and every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq28_HTML.gif , one has http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq29_HTML.gif .

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq30_HTML.gif be a solution of problems (1.1) and (1.2) such that (2.7) and (2.8) hold. Define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ12_HTML.gif
      (2.9)
      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq31_HTML.gif , and suppose, for contradiction, that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq32_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq33_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq34_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq35_HTML.gif , then we obtain the following contradiction:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ13_HTML.gif
      (2.10)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq36_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq37_HTML.gif , a similar contradiction can be derived. So there is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq38_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq39_HTML.gif . By (2.4) we can take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq40_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ14_HTML.gif
      (2.11)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq41_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq42_HTML.gif , then we have trivially http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq43_HTML.gif . If not, then we can take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq44_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq45_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq46_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq47_HTML.gif . Suppose that the first case holds. By (2.7) we can consider http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq48_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ15_HTML.gif
      (2.12)
      Applying a convenient change of variable, we have, by (2.3) and (2.11),
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ16_HTML.gif
      (2.13)

      Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq49_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq50_HTML.gif can be taken arbitrarily as long as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq51_HTML.gif , we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq52_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq53_HTML.gif provided that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq54_HTML.gif .

      In a similar way, it can be proved that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq55_HTML.gif , for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq56_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq57_HTML.gif . Therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ17_HTML.gif
      (2.14)
      Consider now the case http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq58_HTML.gif , and take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq59_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ18_HTML.gif
      (2.15)
      In a similar way, we may show that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ19_HTML.gif
      (2.16)

      Taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq60_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq61_HTML.gif .

      Remark 2.5.

      Observe that the estimation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq62_HTML.gif depends only on the functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq63_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq64_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq65_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq66_HTML.gif and it does not depend on the boundary conditions.

      3. Existence and Location Result

      In the presence of an ordered pair of lower and upper functions, the existence and location results for problems (1.1) and (1.2) can be obtained.

      Theorem 3.1.

      Suppose that there exist lower and upper functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq67_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq68_HTML.gif of problems (1.1) and (1.2), respectively. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq69_HTML.gif be a continuous function satisfying the one-sided Nagumo-type conditions (2.3) and (2.4) in
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ20_HTML.gif
      (3.1)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq70_HTML.gif verifies
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ21_HTML.gif
      (3.2)
      for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq71_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ22_HTML.gif
      (3.3)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq72_HTML.gif means http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq73_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq74_HTML.gif , then problems (1.1) and (1.2) has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq75_HTML.gif satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ23_HTML.gif
      (3.4)

      for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq76_HTML.gif .

      Proof.

      Define the auxiliary functions
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ24_HTML.gif
      (3.5)
      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq77_HTML.gif , consider the homotopic equation
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ25_HTML.gif
      (3.6)
      with the boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ26_HTML.gif
      (3.7)
      Take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq78_HTML.gif large enough such that, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq79_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ27_HTML.gif
      (3.8)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ28_HTML.gif
      (3.9)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ29_HTML.gif
      (3.10)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ30_HTML.gif
      (3.11)

      Step 1.

      Every solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq80_HTML.gif of problems (3.6) and (3.7) satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ31_HTML.gif
      (3.12)

      for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq81_HTML.gif , for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq82_HTML.gif independent of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq83_HTML.gif .

      Assume, for contradiction, that the above estimate does not hold for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq84_HTML.gif . So there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq85_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq86_HTML.gif , and a solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq87_HTML.gif of (3.6) and (3.7) such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq88_HTML.gif . In the case http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq89_HTML.gif define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ32_HTML.gif
      (3.13)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq90_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq91_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq92_HTML.gif . Then, by (3.2) and (3.10), for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq93_HTML.gif , the following contradiction is obtained:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ33_HTML.gif
      (3.14)
      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq94_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ34_HTML.gif
      (3.15)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq95_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ35_HTML.gif
      (3.16)
      and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq96_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq97_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq98_HTML.gif and so http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq99_HTML.gif . Therefore, the above computations with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq100_HTML.gif replaced by 0 yield a contradiction. For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq101_HTML.gif , by (3.11), we get the following contradiction:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ36_HTML.gif
      (3.17)

      The case http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq102_HTML.gif is analogous. Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq103_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq104_HTML.gif . In a similar way, we may prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq105_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq106_HTML.gif .

      By the boundary condition (3.7) there exists a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq107_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq108_HTML.gif . Then by integration we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ37_HTML.gif
      (3.18)

      Step 2.

      There is an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq109_HTML.gif such that for every solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq110_HTML.gif of problems (3.6) and (3.7)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ38_HTML.gif
      (3.19)

      with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq111_HTML.gif independent of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq112_HTML.gif .

      Consider the set
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ39_HTML.gif
      (3.20)
      and for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq113_HTML.gif the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq114_HTML.gif given by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ40_HTML.gif
      (3.21)
      In the following we will prove that the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq115_HTML.gif satisfies the one-sided Nagumo-type conditions (2.3) and (2.4) in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq116_HTML.gif independently of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq117_HTML.gif . Indeed, as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq118_HTML.gif verifies (2.3) in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq119_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ41_HTML.gif
      (3.22)
      So, defining http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq120_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq121_HTML.gif , we see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq122_HTML.gif verifies (2.3) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq123_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq124_HTML.gif replaced by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq125_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq126_HTML.gif , respectively. The condition (2.4) is also verified since
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ42_HTML.gif
      (3.23)

      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq127_HTML.gif satisfies the one-sided Nagumo-type condition in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq128_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq129_HTML.gif replaced by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq130_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq131_HTML.gif independent of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq132_HTML.gif .

      Moreover, for
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ43_HTML.gif
      (3.24)
      every solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq133_HTML.gif of (3.6) and (3.7) satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ44_HTML.gif
      (3.25)
      Define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ45_HTML.gif
      (3.26)

      The hypotheses of Lemma 2.4 are satisfied with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq134_HTML.gif replaced by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq135_HTML.gif . So there exists an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq136_HTML.gif , depending on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq137_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq138_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq139_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq140_HTML.gif . As http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq141_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq142_HTML.gif do not depend on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq143_HTML.gif , we see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq144_HTML.gif is maybe independent of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq145_HTML.gif .

      Step 3.

      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq146_HTML.gif , the problems (3.6) and (3.7) has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq147_HTML.gif .

      Define the operators
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ46_HTML.gif
      (3.27)
      by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ47_HTML.gif
      (3.28)
      and for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq148_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq149_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ48_HTML.gif
      (3.29)
      with
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ49_HTML.gif
      (3.30)
      Observe that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq150_HTML.gif has a compact inverse. Therefore, we can consider the completely continuous operator
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ50_HTML.gif
      (3.31)
      given by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ51_HTML.gif
      (3.32)
      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq151_HTML.gif given by Step 2, take the set
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ52_HTML.gif
      (3.33)
      By Steps 1 and 2, degree http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq152_HTML.gif is well defined for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq153_HTML.gif and by the invariance with respect to a homotopy
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ53_HTML.gif
      (3.34)
      The equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq154_HTML.gif is equivalent to the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ54_HTML.gif
      (3.35)
      and has only the trivial solution. Then, by the degree theory,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ55_HTML.gif
      (3.36)
      So the equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq155_HTML.gif has at least one solution, and therefore the equivalent problem
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ56_HTML.gif
      (3.37)

      has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq156_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq157_HTML.gif .

      Step 4.

      The function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq158_HTML.gif is a solution of the problems (1.1) and (1.2).

      The proof will be finished if the above function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq159_HTML.gif satisfies the inequalities
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ57_HTML.gif
      (3.38)
      Assume, for contradiction, that there is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq160_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq161_HTML.gif , and define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ58_HTML.gif
      (3.39)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq162_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq163_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq164_HTML.gif . Therefore, by (3.2) and Definition 2.1, we obtain the contradiction
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ59_HTML.gif
      (3.40)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq165_HTML.gif , then we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ60_HTML.gif
      (3.41)
      By Definition 2.1 this yields a contradiction
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ61_HTML.gif
      (3.42)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq166_HTML.gif and, by similar arguments, we prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq167_HTML.gif . Thus,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ62_HTML.gif
      (3.43)
      Using an analogous technique, it can be deduced that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq168_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq169_HTML.gif . So we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ63_HTML.gif
      (3.44)
      On the other hand, by (1.2),
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ64_HTML.gif
      (3.45)
      that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ65_HTML.gif
      (3.46)
      Applying the same technique, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ66_HTML.gif
      (3.47)
      and then by Definition 2.1 (iii), (3.44) and (3.46), we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ67_HTML.gif
      (3.48)
      that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ68_HTML.gif
      (3.49)
      Since, by (3.44), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq170_HTML.gif is nondecreasing, we have by (3.49)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ69_HTML.gif
      (3.50)
      and, therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq171_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq172_HTML.gif . By the monotonicity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq173_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ70_HTML.gif
      (3.51)

      and so http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq174_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq175_HTML.gif .

      The inequalities http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq176_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq177_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq178_HTML.gif can be proved in the same way. Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq179_HTML.gif is a solution of problems (1.1) and (1.2).

      4. An Example

      The following example shows the applicability of Theorem 3.1 when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq180_HTML.gif satisfies only the one-sided Nagumo-type condition.

      Example 4.1.

      Consider now the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ71_HTML.gif
      (4.1)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ72_HTML.gif
      (4.2)
      with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq181_HTML.gif . The nonlinear function
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ73_HTML.gif
      (4.3)
      is continuous in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq182_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq183_HTML.gif , then the functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq184_HTML.gif defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ74_HTML.gif
      (4.4)
      are, respectively, lower and upper functions of (4.1) and (4.2). Moreover, define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ75_HTML.gif
      (4.5)

      Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq185_HTML.gif satisfies condition (3.2) and the one-sided Nagumo-type condition with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq186_HTML.gif , in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq187_HTML.gif .

      Therefore, by Theorem 3.1, there is at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq188_HTML.gif of Problem (4.1) and (4.2) such that, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq189_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ76_HTML.gif
      (4.6)
      Notice that the function
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ77_HTML.gif
      (4.7)

      does not satisfy the two-sided Nagumo condition.

      Declarations

      Acknowledgments

      The authors would like to thank the referees for their valuable comments on and suggestions regarding the original manuscript. This work was supported by NSFC (10771085), by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education, and by the 985 Program of Jilin University.

      Authors’ Affiliations

      (1)
      Institute of Mathematics, Jilin University
      (2)
      Institute of Applied Mathematics, Jilin University of Finance and Economics

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      © W. Song and W. Gao. 2011

      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.