Existence Results of Three-Point Boundary Value Problems for Second-order Ordinary Differential Equations

  • Sheng-Ping Wang1Email author and

    Affiliated with

    • Long-Yi Tsai2

      Affiliated with

      Boundary Value Problems20102011:901796

      DOI: 10.1155/2011/901796

      Received: 19 May 2010

      Accepted: 24 September 2010

      Published: 28 September 2010

      Abstract

      We establish existence results of the following three-point boundary value problems: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq3_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq4_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq6_HTML.gif . The approach applied in this paper is upper and lower solution method associated with basic degree theory or Schauder's fixed point theorem. We deal with this problem with the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq7_HTML.gif which is Carathéodory or singular on its domain.

      1. Introduction

      In this paper, we consider three-point boundary value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ1_HTML.gif
      (1.1)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ2_HTML.gif
      (1.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq9_HTML.gif .

      In the mathematical literature, a number of works have appeared on nonlocal boundary value problems, and one of the first of these was [1]. Il'in and Moiseev initiated the research of multipoint boundary value problems for second-order linear ordinary differential equations, see [2, 3], motivated by the study [46] of Bitsadze and Samarskii.

      Recently, nonlinear multipoint boundary value problems have been receiving considerable attention, and have been studied extensively by using iteration scheme (e.g., [7]), fixed point theorems in cones (e.g., [8]), and the Leray-Schauder continuation theorem (e.g., [9]). We refer more detailed treatment to more interesting research [10, 11] and the references therein.

      The theory of upper and lower solutions is also a powerful tool in studying boundary value problems. For the existence results of two-point boundary value problem, there already are lots of interesting works by applying this essential technique (see [12, 13]). Recently, it is shown that this method plays an important role in proving the existence of solutions for three-point boundary value problems (see [1416]).

      Last but not least, as the singular source term appearing in two-point problems, singular three-point boundary value problems also attract more attention (e.g., [17]).

      In this paper, we will discuss the existence of solutions of some general types on three-point boundary value problems by using upper and lower solution method associated with basic degree theory or Schauder's fixed point theorem.

      This paper is organized as follows. In Section 2, we give two lemmas which will be extensively used later. In Section 3, when the source term http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq10_HTML.gif is a Carathéodory function, we consider the Sobolev space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq11_HTML.gif defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ3_HTML.gif
      (1.3)
      and obtain the existence of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq12_HTML.gif -solution in Theorems 3.5 and 3.11. In Section 4, we discuss the singular case, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq13_HTML.gif maybe singular at the end points http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq14_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq15_HTML.gif , or at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq16_HTML.gif . We will introduce the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq17_HTML.gif -class of functions and another space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq18_HTML.gif (see [18, 19]) as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ4_HTML.gif
      (1.4)

      and prove the existence of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq19_HTML.gif -solution in Theorems 4.1 and 4.4. Some sufficient conditions for constructing upper and lower solutions are given in each section for applications.

      2. Preliminaries

      Define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq20_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ5_HTML.gif
      (2.1)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq21_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq22_HTML.gif are given as (1.2) and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ6_HTML.gif
      (2.2)

      By direct computations, we get the following results.

      Lemma 2.1.
      1. (i)
        The function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq23_HTML.gif defined by (2.1), is the Green function corresponding for the problem
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ7_HTML.gif
        (2.3)
         
       (ii) The function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq24_HTML.gif defined by (2.1), is continuous.
      1. (iii)
        In the case http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq25_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ8_HTML.gif
        (2.4)
         

      Lemma 2.2.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq26_HTML.gif , then the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ9_HTML.gif
      (2.5)
      with boundary condition (1.2) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq27_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ10_HTML.gif
      (2.6)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq28_HTML.gif is defined by (2.1).

      3. Carathéodory Case

      In this section we first introduce the Carathéodory function as follows.

      Definition 3.1.

      A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq29_HTML.gif defined on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq30_HTML.gif is called a Carathéodory function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq31_HTML.gif if

      (i)for almost every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq32_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq33_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq34_HTML.gif ;

      (ii)for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq35_HTML.gif the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq36_HTML.gif is measurable on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq37_HTML.gif ;

      (iii)for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq38_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq39_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq40_HTML.gif and for almost every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq41_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq42_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq43_HTML.gif .

      We in this section assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq44_HTML.gif is a Carathéodory function and discuss the existence of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq45_HTML.gif -solution by assuming the existence of upper and lower solutions.

      3.1. Existence of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq46_HTML.gif -Solutions

      We first introduce the definitions of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq47_HTML.gif -upper and lower solutions as below.

      Definition 3.2.

      A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq48_HTML.gif is called a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq49_HTML.gif -lower solution of problem (1.1) and (1.2) if it satisfies

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq50_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq51_HTML.gif , and

      (ii)for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq52_HTML.gif , either http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq53_HTML.gif , or there exists an open interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq54_HTML.gif containing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq55_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq56_HTML.gif and, for almost every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq57_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ11_HTML.gif
      (3.1)

      Definition 3.3.

      A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq58_HTML.gif is called a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq59_HTML.gif -upper solution of problem (1.1) and (1.2) if it satisfies

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq60_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq61_HTML.gif , and

      (ii)for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq62_HTML.gif , either http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq63_HTML.gif , or there exists an open interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq64_HTML.gif containing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq65_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq66_HTML.gif and, for almost every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq67_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ12_HTML.gif
      (3.2)
      Before proving our main results, we first consider such a modified problem given as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ13_HTML.gif
      (3.3)
      with boundary condition (1.2), where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq68_HTML.gif is defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ14_HTML.gif
      (3.4)

      Proposition 3.4.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq70_HTML.gif be respective http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq71_HTML.gif -lower and upper solutions of problem (1.1) and (1.2) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq72_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq73_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq74_HTML.gif is a solution of problem (3.3) and (1.2), then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq75_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq76_HTML.gif .

      Proof.

      Suppose there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq77_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ15_HTML.gif
      (3.5)

      Case 1.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq78_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq79_HTML.gif , which implies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq80_HTML.gif Hence, by Definition 3.2 and the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq81_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq82_HTML.gif , there exist an open interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq83_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq84_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq85_HTML.gif and a neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq86_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq87_HTML.gif contained in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq88_HTML.gif such that for almost every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq89_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ16_HTML.gif
      (3.6)
      Furthermore, it follows from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq90_HTML.gif that for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq91_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq92_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ17_HTML.gif
      (3.7)

      This implies that the minimum of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq93_HTML.gif cannot occur at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq94_HTML.gif , a contradiction.

      Case 2.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq95_HTML.gif , by the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq96_HTML.gif -lower solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq97_HTML.gif , we then have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ18_HTML.gif
      (3.8)

      And we get a contradiction.

      Case 3.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq98_HTML.gif , it follows from the conclusion of Case 1 that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ19_HTML.gif
      (3.9)

      which is impossible.

      Consequently, we obtain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq99_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq100_HTML.gif . By the similar arguments as above, we also have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ20_HTML.gif
      (3.10)

      Theorem 3.5.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq101_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq102_HTML.gif be http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq103_HTML.gif -lower and upper solutions of problem (1.1) and (1.2) such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq104_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq105_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq106_HTML.gif be a Carathéodory function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq107_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ21_HTML.gif
      (3.11)
      Then problem (1.1) and (1.2) has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq108_HTML.gif such that, for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq109_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ22_HTML.gif
      (3.12)

      Proof.

      We consider the modified problem (3.3) and (1.2) with respect to the given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq110_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq111_HTML.gif . Consider the Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq112_HTML.gif with supremum and the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq113_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ23_HTML.gif
      (3.13)
      for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq114_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq115_HTML.gif is defined as (2.1). Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq116_HTML.gif is a Carathéodory function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq117_HTML.gif , for almost all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq118_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq119_HTML.gif , there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq120_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ24_HTML.gif
      (3.14)
      Define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ25_HTML.gif
      (3.15)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ26_HTML.gif
      (3.16)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ27_HTML.gif
      (3.17)

      It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq121_HTML.gif is a closed, bounded and convex set in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq122_HTML.gif and one can show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq123_HTML.gif is a completely continuous mapping by Arzelà-Ascoli theorem and Lebesgue dominated convergence theorem. By applying Schauder's fixed point theorem, we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq124_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq125_HTML.gif which is a solution of problem (3.3) and (1.2). From Proposition 3.4, this fixed point is also a solution of problem (1.1) and (1.2). Hence, we complete the proof.

      We further illustrate the use of Theorem 3.11 in the following second-order differential equation:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ28_HTML.gif
      (3.18)

      with the boundary condition (1.2).

      Corollary 3.6.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq126_HTML.gif is a Carathéodory function satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq127_HTML.gif is essentially bounded for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq128_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq129_HTML.gif is a constant large enough. Assume further that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq130_HTML.gif and there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq131_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ29_HTML.gif
      (3.19)

      Then, problem (3.18) and (1.2) has at least one solution.

      Proof.

      By hypothesis, for any given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq132_HTML.gif small enough such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq133_HTML.gif and for almost all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq134_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq135_HTML.gif large enough, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ30_HTML.gif
      (3.20)
      We now choose an upper solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq136_HTML.gif of the form
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ31_HTML.gif
      (3.21)
      To this end, we compute
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ32_HTML.gif
      (3.22)
      Clearly, one can choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq137_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ33_HTML.gif
      (3.23)
      that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ34_HTML.gif
      (3.24)
      and choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq138_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq139_HTML.gif , which is a positive solution of
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ35_HTML.gif
      (3.25)

      Hence, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq140_HTML.gif is large enough, we can show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq141_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq142_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq143_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq144_HTML.gif is a positive http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq145_HTML.gif -upper solution. In the same way we construct a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq146_HTML.gif -lower solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq147_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq148_HTML.gif .

      3.2. Nontangency Solution

      In this subsection, we afford another stronger lower and upper solutions to get a strict inequality of the solution between them.

      Definition 3.7.

      A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq149_HTML.gif is a strict http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq150_HTML.gif -lower solution of problem (1.1) and (1.2), if it is not a solution of problem (1.1) and (1.2), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq151_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq152_HTML.gif and for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq153_HTML.gif , one of the following is satisfied:

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq154_HTML.gif ;

      (ii)there exist an interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq155_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq156_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq157_HTML.gif int http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq158_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq159_HTML.gif and for almost every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq160_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq161_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ36_HTML.gif
      (3.26)

      Definition 3.8.

      A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq162_HTML.gif is a strict http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq163_HTML.gif -upper solution of problem (1.1) and (1.2), if it is not a solution of problem (1.1) and (1.2), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq164_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq165_HTML.gif and for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq166_HTML.gif , one of the following is satisfied:

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq167_HTML.gif ,

      (ii)there exist an interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq168_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq169_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq170_HTML.gif int http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq171_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq172_HTML.gif and for almost every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq173_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq174_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ37_HTML.gif
      (3.27)

      Remark 3.9.

      Every strict http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq175_HTML.gif -lower(upper) solution of problem (1.1) and (1.2) is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq176_HTML.gif -lower(upper) solution.

      Now we are going to show that the solution curve of problem (1.1) and (1.2) cannot be tangent to upper or lower solutions from below or above.

      Proposition 3.10.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq177_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq178_HTML.gif be respective strict http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq179_HTML.gif -lower and upper solutions of problem (1.1) and (1.2) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq180_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq181_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq182_HTML.gif is a solution of problem (1.1) and (1.2) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq183_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq184_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq185_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq186_HTML.gif .

      Proof.

      As http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq187_HTML.gif is not a solution, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq188_HTML.gif is not identical to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq189_HTML.gif . Assume, the conclusion does not hold, then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ38_HTML.gif
      (3.28)

      exists. Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq190_HTML.gif has minimum at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq191_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq192_HTML.gif .

      Case 1.

      Set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq193_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq194_HTML.gif has minimum at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq195_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq196_HTML.gif . According to the Definition 3.7, there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq197_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq198_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq199_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq200_HTML.gif such that, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq201_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq202_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq203_HTML.gif and for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq204_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ39_HTML.gif
      (3.29)
      Hence, we have the contradiction since
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ40_HTML.gif
      (3.30)

      Case 2.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq205_HTML.gif , by the definition of strict http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq206_HTML.gif -lower solution that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq207_HTML.gif , we then have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ41_HTML.gif
      (3.31)

      And we get a contradiction.

      Case 3.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq208_HTML.gif , repeat the same arguments in Case 3 of the proof of Proposition 3.4. Therefore, we obtain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq209_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq210_HTML.gif . The inequality http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq211_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq212_HTML.gif can be proved by the similar arguments as above.

      Theorem 3.11.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq213_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq214_HTML.gif be strict http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq215_HTML.gif -lower and upper solutions of problem (1.1) and (1.2) such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq216_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq217_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq218_HTML.gif be a Carathéodory function, where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ42_HTML.gif
      (3.32)
      Then, problem (1.1) and (1.2) has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq219_HTML.gif such that, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq220_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ43_HTML.gif
      (3.33)

      Proof.

      This is a consequence of Theorem 3.5 and Proposition 3.10 and hence, we omits this proof.

      4. Singular Case

      In this section we give a more general existence result than Theorem 3.11 by assuming the existence of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq221_HTML.gif -lower and upper solutions. This makes us to deal with problem (1.1) and (1.2), where the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq222_HTML.gif is singular at the end point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq223_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq224_HTML.gif .

      Theorem 4.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq225_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq226_HTML.gif be http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq227_HTML.gif -lower and upper solutions of problem (1.1) and (1.2) such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq228_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq229_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq230_HTML.gif satisfy the following conditions:

      (i)for almost every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq231_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq232_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq233_HTML.gif ;

      (ii)for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq234_HTML.gif the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq235_HTML.gif is measurable on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq236_HTML.gif ;

      (iii)there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq237_HTML.gif such that, for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq238_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ44_HTML.gif
      (4.1)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ45_HTML.gif
      (4.2)
      Then problem (1.1) and (1.2) has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq239_HTML.gif such that, for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq240_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ46_HTML.gif
      (4.3)

      Proof.

      Consider the modified problem (3.3) and (1.2) with respect to the given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq241_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq242_HTML.gif and define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq243_HTML.gif by (3.13). Note that by Lemma 2.2, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq244_HTML.gif is well defined. Define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ47_HTML.gif
      (4.4)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ48_HTML.gif
      (4.5)

      and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq245_HTML.gif is defined by (3.17). The rest arguments are similar to the proof of Theorem 3.5.

      Remark 4.2.

      We have similar results of Theorems 3.5–4.1, respectively, for (1.1) equipped with
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ49_HTML.gif
      (4.6)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq246_HTML.gif is a constant and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq247_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq248_HTML.gif are given as (1.2).

      Example 4.3.

      Consider the problem (4.7), for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq249_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq250_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq251_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ50_HTML.gif
      (4.7)
      Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq252_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq253_HTML.gif -lower solution of (4.7) and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ51_HTML.gif
      (4.8)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ52_HTML.gif
      (4.9)
      From Lemma 2.1, we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq254_HTML.gif and define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq255_HTML.gif . Since, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq256_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq257_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ53_HTML.gif
      (4.10)
      that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq258_HTML.gif , we have, from Lemma 2.2, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq259_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq260_HTML.gif exists. Let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ54_HTML.gif
      (4.11)
      and, by Lemma 2.2 again, choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq261_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ55_HTML.gif
      (4.12)
      Note that according to the direct computation, we see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq262_HTML.gif is well-defined and is bounded by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq263_HTML.gif . Next, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq264_HTML.gif . By Young's inequality, it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ56_HTML.gif
      (4.13)
      Hence, such http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq265_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq266_HTML.gif -upper solution of (4.7) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq267_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq268_HTML.gif . Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq269_HTML.gif satisfies (i), (ii) of Theorem 4.1. By using Young's inequality again, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq270_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ57_HTML.gif
      (4.14)
      and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq271_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq272_HTML.gif satisfies the assumption (iii) of Theorem 4.1. Consequently, we conclude that this problem has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq273_HTML.gif such that, for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq274_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ58_HTML.gif
      (4.15)

      Notice that in Theorem 4.1, one can only deal with the case that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq275_HTML.gif is singular at end points http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq276_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq277_HTML.gif . However, when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq278_HTML.gif is singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq279_HTML.gif , there is no hope to obtain the solutions directly from Theorem 4.1. We will establish the following theorem to deal with this case by constructing upper and lower solutions to solve this problem.

      Theorem 4.4.

      Assume

      the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq281_HTML.gif is continuous;

      there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq283_HTML.gif and for any compact set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq284_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq285_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ59_HTML.gif
      (4.16)
      for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq287_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq288_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq289_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ60_HTML.gif
      (4.17)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq290_HTML.gif is defined as in Lemma 2.1.

      for any compact set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq292_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq293_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ61_HTML.gif
      (4.18)
      Then problem (1.1) and (1.2) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq294_HTML.gif has at least one solution
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ62_HTML.gif
      (4.19)

      Remark 4.5 (see [12, Remark http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq295_HTML.gif ]).

      Assumption http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq296_HTML.gif is equivalent to the assumption that there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq297_HTML.gif and a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq298_HTML.gif such that:

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq299_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq300_HTML.gif ,

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq301_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq302_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq303_HTML.gif ,

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq304_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq305_HTML.gif ,

      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ63_HTML.gif
      (4.20)

      Proof.

      Step 1.

      Construction of lower solutions. Consider http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq306_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq307_HTML.gif and the function
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ64_HTML.gif
      (4.21)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq308_HTML.gif is chosen small enough so that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ65_HTML.gif
      (4.22)
      Next, we choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq309_HTML.gif from the Remark 4.5, and let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ66_HTML.gif
      (4.23)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq310_HTML.gif is small enough so that for some points http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq311_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq312_HTML.gif , we have:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ67_HTML.gif
      (4.24)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ68_HTML.gif
      (4.25)
      Notice that by (4.24) and (4.25), for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq313_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ69_HTML.gif
      (4.26)
      we have:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ70_HTML.gif
      (4.27)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ71_HTML.gif
      (4.28)

      Step 2.

      Approximation problems. We define for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq314_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq315_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ72_HTML.gif
      (4.29)
      and set
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ73_HTML.gif
      (4.30)
      We have that, for each index http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq316_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq317_HTML.gif is continuous and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ74_HTML.gif
      (4.31)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ75_HTML.gif
      (4.32)
      Hence, the sequence of functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq318_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq319_HTML.gif uniformly on any set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq320_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq321_HTML.gif is an arbitrary compact subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq322_HTML.gif . Next we define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ76_HTML.gif
      (4.33)
      Each of the functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq323_HTML.gif is a continuous function defined on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq324_HTML.gif , moreover
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ77_HTML.gif
      (4.34)
      and the sequence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq325_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq326_HTML.gif uniformly on the compact subsets of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq327_HTML.gif since
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ78_HTML.gif
      (4.35)
      Define now a decreasing sequence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq328_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ79_HTML.gif
      (4.36)
      and consider a sequence of the following approximation problems:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ80_HTML.gif
      (Pn)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq329_HTML.gif .

      Step 3.

      A lower solution of ( ). It is clear that for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq331_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ81_HTML.gif
      (4.37)
      As the sequence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq332_HTML.gif is decreasing, we also have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ82_HTML.gif
      (4.38)
      Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq333_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ83_HTML.gif
      (4.39)

      It follows from (4.25) and (4.27) that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq334_HTML.gif is a lower solution of ( ).

      Step 4.

      Existence of a solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq336_HTML.gif of (4.7) such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ84_HTML.gif
      (4.40)
      From assumption http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq337_HTML.gif , we can find http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq338_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq339_HTML.gif such that, for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq340_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq341_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ85_HTML.gif
      (4.41)
      Also, one has
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ86_HTML.gif
      (4.42)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq342_HTML.gif is a suitable constant. Hence, we obtain, for such http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq343_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq344_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ87_HTML.gif
      (4.43)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq345_HTML.gif be a constant such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ88_HTML.gif
      (4.44)
      Choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq346_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ89_HTML.gif
      (4.45)
      that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ90_HTML.gif
      (4.46)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq347_HTML.gif is defined by (2.1). Note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq348_HTML.gif is well-defined and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq349_HTML.gif since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq350_HTML.gif . It is easy to see that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ91_HTML.gif
      (4.47)
      So by Remark 4.2, there is a solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq351_HTML.gif of (4.7) such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ92_HTML.gif
      (4.48)

      Step 5.

      The problem ( ) has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq353_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ93_HTML.gif
      (4.49)
      Notice that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq354_HTML.gif is an upper solution of ( ), since
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ94_HTML.gif
      (4.50)

      Step 6.

      Existence of a solution. Consider the pointwise limit
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ95_HTML.gif
      (4.51)
      It is clear that, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq356_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ96_HTML.gif
      (4.52)
      and therefore http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq357_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq358_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq359_HTML.gif be a compact interval. There is an index http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq360_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq361_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq362_HTML.gif and therefore for these http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq363_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ97_HTML.gif
      (4.53)
      Moreover, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ98_HTML.gif
      (4.54)
      By Arzelá-Ascoli theorem it is standard to conclude that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq364_HTML.gif is a solution of problem (1.1) and (1.2) on the interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq365_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq366_HTML.gif is arbitrary, we find that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq367_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq368_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ99_HTML.gif
      (4.55)
      Since
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ100_HTML.gif
      (4.56)

      it remains only to check the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq369_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq370_HTML.gif . This can be deduced from the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq371_HTML.gif and the fact that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq372_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq373_HTML.gif .

      Example 4.6.

      Consider the following problem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq374_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq375_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq376_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq377_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ101_HTML.gif
      (4.57)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq378_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq379_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq380_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq381_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq382_HTML.gif . Moreover, for any given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq383_HTML.gif and for any compact set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq384_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq385_HTML.gif small enough, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ102_HTML.gif
      (4.58)
      Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq386_HTML.gif holds. Furthermore, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq387_HTML.gif large enough, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq388_HTML.gif , we have, from Young's inequality by choosing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq389_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq390_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ103_HTML.gif
      (4.59)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq391_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq392_HTML.gif holds. By Theorem 4.4, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq393_HTML.gif has at least one solution
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ104_HTML.gif
      (4.60)

      Authors’ Affiliations

      (1)
      Holistic Education Center, Cardinal Tien College of Healthcare and Management
      (2)
      Department of Mathematical Sciences, National ChengChi University

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      Copyright

      © S.-P.Wang and L.-Y. Tsai. 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.