Open Access

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

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: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq3_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq4_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq5_HTML.gif and https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq8_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq11_HTML.gif defined by
https://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 https://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, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq13_HTML.gif maybe singular at the end points https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq14_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq15_HTML.gif , or at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq16_HTML.gif . We will introduce the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq17_HTML.gif -class of functions and another space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq18_HTML.gif (see [18, 19]) as follows:
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq20_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ5_HTML.gif
(2.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq22_HTML.gif are given as (1.2) and
https://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 https://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
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ7_HTML.gif
    (2.3)
     
 (ii) The function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq25_HTML.gif , we have
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ8_HTML.gif
    (2.4)
     

Lemma 2.2.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq26_HTML.gif , then the problem
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq27_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ10_HTML.gif
(2.6)

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

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

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

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

We in this section assume that https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq48_HTML.gif is called a https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq50_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq51_HTML.gif , and

(ii)for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq52_HTML.gif , either https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq53_HTML.gif , or there exists an open interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq54_HTML.gif containing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq55_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq56_HTML.gif and, for almost every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq57_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq58_HTML.gif is called a https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq60_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq61_HTML.gif , and

(ii)for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq62_HTML.gif , either https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq63_HTML.gif , or there exists an open interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq64_HTML.gif containing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq65_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq66_HTML.gif and, for almost every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq67_HTML.gif , we have
https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq68_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ14_HTML.gif
(3.4)

Proposition 3.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq69_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq70_HTML.gif be respective https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq72_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq73_HTML.gif . If https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq75_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq76_HTML.gif .

Proof.

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

Case 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq78_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq79_HTML.gif , which implies https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq81_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq82_HTML.gif , there exist an open interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq83_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq84_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq85_HTML.gif and a neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq86_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq87_HTML.gif contained in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq88_HTML.gif such that for almost every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq89_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ16_HTML.gif
(3.6)
Furthermore, it follows from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq90_HTML.gif that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq91_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq92_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq93_HTML.gif cannot occur at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq94_HTML.gif , a contradiction.

Case 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq95_HTML.gif , by the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq96_HTML.gif -lower solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq97_HTML.gif , we then have
https://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 https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq99_HTML.gif on https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ20_HTML.gif
(3.10)

Theorem 3.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq101_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq102_HTML.gif be https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq104_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq105_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq106_HTML.gif be a Carathéodory function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq107_HTML.gif , where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq108_HTML.gif such that, for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq109_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq110_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq111_HTML.gif . Consider the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq112_HTML.gif with supremum and the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq113_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ23_HTML.gif
(3.13)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq114_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq115_HTML.gif is defined as (2.1). Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq116_HTML.gif is a Carathéodory function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq117_HTML.gif , for almost all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq118_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq119_HTML.gif , there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq120_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ24_HTML.gif
(3.14)
Define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ25_HTML.gif
(3.15)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ26_HTML.gif
(3.16)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ27_HTML.gif
(3.17)

It is clear that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq122_HTML.gif and one can show that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq124_HTML.gif has a fixed point in https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq126_HTML.gif is a Carathéodory function satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq127_HTML.gif is essentially bounded for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq128_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq130_HTML.gif and there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq131_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq132_HTML.gif small enough such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq133_HTML.gif and for almost all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq134_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq135_HTML.gif large enough, we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq136_HTML.gif of the form
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ32_HTML.gif
(3.22)
Clearly, one can choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq137_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ33_HTML.gif
(3.23)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ34_HTML.gif
(3.24)
and choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq138_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq139_HTML.gif , which is a positive solution of
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ35_HTML.gif
(3.25)

Hence, if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq142_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq143_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq144_HTML.gif is a positive https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq146_HTML.gif -lower solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq147_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq149_HTML.gif is a strict https://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), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq151_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq152_HTML.gif and for any https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq154_HTML.gif ;

(ii)there exist an interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq155_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq156_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq157_HTML.gif int https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq158_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq159_HTML.gif and for almost every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq160_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq161_HTML.gif we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq162_HTML.gif is a strict https://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), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq164_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq165_HTML.gif and for any https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq167_HTML.gif ,

(ii)there exist an interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq169_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq170_HTML.gif int https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq171_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq172_HTML.gif and for almost every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq173_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq174_HTML.gif we have
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq177_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq178_HTML.gif be respective strict https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq180_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq181_HTML.gif . If https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq183_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq184_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq185_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq186_HTML.gif .

Proof.

As https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq187_HTML.gif is not a solution, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq188_HTML.gif is not identical to https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ38_HTML.gif
(3.28)

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

Case 1.

Set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq193_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq194_HTML.gif has minimum at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq195_HTML.gif , we have https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq197_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq198_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq199_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq200_HTML.gif such that, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq201_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq202_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq203_HTML.gif and for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq204_HTML.gif
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ40_HTML.gif
(3.30)

Case 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq205_HTML.gif , by the definition of strict https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq206_HTML.gif -lower solution that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq207_HTML.gif , we then have
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq209_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq210_HTML.gif . The inequality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq211_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq213_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq214_HTML.gif be strict https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq216_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq217_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq218_HTML.gif be a Carathéodory function, where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq219_HTML.gif such that, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq220_HTML.gif ,
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq222_HTML.gif is singular at the end point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq223_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq224_HTML.gif .

Theorem 4.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq225_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq226_HTML.gif be https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq228_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq229_HTML.gif and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq231_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq232_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq233_HTML.gif ;

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

(iii)there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq237_HTML.gif such that, for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq238_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ44_HTML.gif
(4.1)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq239_HTML.gif such that, for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq240_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq241_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq242_HTML.gif and define https://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, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq244_HTML.gif is well defined. Define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ47_HTML.gif
(4.4)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ48_HTML.gif
(4.5)

and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ49_HTML.gif
(4.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq246_HTML.gif is a constant and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq247_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq249_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq250_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq251_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ50_HTML.gif
(4.7)
Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq252_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq253_HTML.gif -lower solution of (4.7) and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ51_HTML.gif
(4.8)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq254_HTML.gif and define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq255_HTML.gif . Since, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq256_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq257_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ53_HTML.gif
(4.10)
that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq258_HTML.gif , we have, from Lemma 2.2, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq259_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq260_HTML.gif exists. Let
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq261_HTML.gif such that
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq263_HTML.gif . Next, let https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ56_HTML.gif
(4.13)
Hence, such https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq265_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq266_HTML.gif -upper solution of (4.7) and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq267_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq268_HTML.gif . Clearly, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq270_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ57_HTML.gif
(4.14)
and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq271_HTML.gif . Therefore, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq273_HTML.gif such that, for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq274_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq275_HTML.gif is singular at end points https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq276_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq277_HTML.gif . However, when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq278_HTML.gif is singular at https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq281_HTML.gif is continuous;

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq292_HTML.gif , there is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq293_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq294_HTML.gif has at least one solution
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq295_HTML.gif ]).

Assumption https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq297_HTML.gif and a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq298_HTML.gif such that:

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

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

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

where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq306_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq307_HTML.gif and the function
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ64_HTML.gif
(4.21)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq308_HTML.gif is chosen small enough so that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ65_HTML.gif
(4.22)
Next, we choose https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ66_HTML.gif
(4.23)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq311_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq312_HTML.gif , we have:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ67_HTML.gif
(4.24)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq313_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ69_HTML.gif
(4.26)
we have:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ70_HTML.gif
(4.27)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq314_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq315_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ72_HTML.gif
(4.29)
and set
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq316_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq317_HTML.gif is continuous and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ74_HTML.gif
(4.31)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq318_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq319_HTML.gif uniformly on any set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq320_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq321_HTML.gif is an arbitrary compact subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq322_HTML.gif . Next we define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ76_HTML.gif
(4.33)
Each of the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq323_HTML.gif is a continuous function defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq324_HTML.gif , moreover
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ77_HTML.gif
(4.34)
and the sequence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq325_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq326_HTML.gif uniformly on the compact subsets of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq327_HTML.gif since
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq328_HTML.gif such that
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ80_HTML.gif
(Pn)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq331_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ81_HTML.gif
(4.37)
As the sequence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq332_HTML.gif is decreasing, we also have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ82_HTML.gif
(4.38)
Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq333_HTML.gif satisfies
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq336_HTML.gif of (4.7) such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ84_HTML.gif
(4.40)
From assumption https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq337_HTML.gif , we can find https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq338_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq339_HTML.gif such that, for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq340_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq341_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ85_HTML.gif
(4.41)
Also, one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ86_HTML.gif
(4.42)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq343_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq344_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ87_HTML.gif
(4.43)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq345_HTML.gif be a constant such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ88_HTML.gif
(4.44)
Choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq346_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ89_HTML.gif
(4.45)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ90_HTML.gif
(4.46)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq348_HTML.gif is well-defined and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq349_HTML.gif since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq350_HTML.gif . It is easy to see that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq351_HTML.gif of (4.7) such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq353_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ93_HTML.gif
(4.49)
Notice that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq354_HTML.gif is an upper solution of ( ), since
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq356_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ96_HTML.gif
(4.52)
and therefore https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq357_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq358_HTML.gif . Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq360_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq361_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq362_HTML.gif and therefore for these https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq363_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ97_HTML.gif
(4.53)
Moreover, we have
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq365_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq366_HTML.gif is arbitrary, we find that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq367_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq368_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ99_HTML.gif
(4.55)
Since
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq369_HTML.gif at https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq371_HTML.gif and the fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq372_HTML.gif as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq374_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq375_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq376_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq377_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ101_HTML.gif
(4.57)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq378_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq379_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq380_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq381_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq382_HTML.gif . Moreover, for any given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq383_HTML.gif and for any compact set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq384_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq385_HTML.gif small enough, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ102_HTML.gif
(4.58)
Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq386_HTML.gif holds. Furthermore, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq387_HTML.gif large enough, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq389_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq390_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_Equ103_HTML.gif
(4.59)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq391_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq392_HTML.gif holds. By Theorem 4.4, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F901796/MediaObjects/13661_2010_Article_66_IEq393_HTML.gif has at least one solution
https://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.