Open Access

Positive Solutions for a Class of Coupled System of Singular Three-Point Boundary Value Problems

Boundary Value Problems20092009:273063

DOI: 10.1155/2009/273063

Received: 27 February 2009

Accepted: 15 May 2009

Published: 22 June 2009

Abstract

Existence of positive solutions for a coupled system of nonlinear three-point boundary value problems of the type https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq7_HTML.gif , is established. The nonlinearities https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq9_HTML.gif are continuous and may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq10_HTML.gif , and/or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq11_HTML.gif , while the parameters https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq13_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq14_HTML.gif . An example is also included to show the applicability of our result.

1. Introduction

Multipoint boundary value problems (BVPs) arise in different areas of applied mathematics and physics. For example, the vibration of a guy wire composed of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq15_HTML.gif parts with a uniform cross-section and different densities in different parts can be modeled as a Multipoint boundary value problem [1]. Many problems in the theory of elastic stability can also be modeled as Multipoint boundary value problem [2].

The study of Multipoint boundary value problems for linear second order ordinary differential equations was initiated by Il'in and Moiseev, [3, 4], and extended to nonlocal linear elliptic boundary value problems by Bitsadze et al. [5, 6]. Existence theory for nonlinear three-point boundary value problems was initiated by Gupta [7]. Since then the study of nonlinear three-point BVPs has attracted much attention of many researchers, see [811] and references therein for boundary value problems with ordinary differential equations and also [12] for boundary value problems on time scales. Recently, the study of singular BVPs has attracted the attention of many authors, see for example, [1318] and the recent monograph by Agarwal et al. [19].

The study of system of BVPs has also fascinated many authors. System of BVPs with continuous nonlinearity can be seen in [2022] and the case of singular nonlinearity can be seen in [8, 21, 2326]. Wei [25], developed the upper and lower solutions method for the existence of positive solutions of the following coupled system of BVPs:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq16_HTML.gif , and may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq19_HTML.gif and/or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq20_HTML.gif .

By using fixed point theorem in cone, Yuan et al. [26] studied the following coupled system of nonlinear singular boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ2_HTML.gif
(1.2)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq21_HTML.gif are allowed to be superlinear and are singular at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq22_HTML.gif and/or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq23_HTML.gif . Similarly, results are studied in [8, 21, 23].

In this paper, we generalize the results studied in [25, 26] to the following more general singular system for three-point nonlocal BVPs:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq25_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq26_HTML.gif . We allow https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq27_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq28_HTML.gif to be singular at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq29_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq30_HTML.gif , and also https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq31_HTML.gif and/or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq32_HTML.gif . We study the sufficient conditions for existence of positive solution for the singular system (1.3) under weaker hypothesis on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq34_HTML.gif as compared to the previously studied results. We do not require the system (1.3) to have lower and upper solutions. Moreover, the cone we consider is more general than the cones considered in [20, 21, 26].

By singularity, we mean the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq36_HTML.gif are allowed to be unbounded at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq37_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq38_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq39_HTML.gif , and/or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq40_HTML.gif . To the best of our knowledge, existence of positive solutions for a system (1.3) with singularity with respect to dependent variable(s) has not been studied previously. Moreover, our conditions and results are different from those studied in [21, 2426]. Throughout this paper, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq41_HTML.gif are continuous and may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq42_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq43_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq44_HTML.gif , and/or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq45_HTML.gif . We also assume that the following conditions hold:

(A1) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq46_HTML.gif and satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ4_HTML.gif
(1.4)
(A2)There exist real constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq47_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq48_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq49_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq50_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq51_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq52_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ5_HTML.gif
(1.5)
(A3)There exist real constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq53_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq54_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq56_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq57_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq58_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ6_HTML.gif
(1.6)
for example, a function that satisfies the assumptions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq59_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq60_HTML.gif is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ7_HTML.gif
(1.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq61_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq62_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq63_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq64_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ8_HTML.gif
(1.8)

The main result of this paper is as follows.

Theorem 1.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq65_HTML.gif hold. Then the system (1.3) has at least one positive solution.

2. Preliminaries

For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq66_HTML.gif , we write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq67_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq68_HTML.gif . Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq69_HTML.gif is a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq70_HTML.gif is a cone. Similarly, for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq71_HTML.gif , we write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq72_HTML.gif . Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq73_HTML.gif is a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq74_HTML.gif is a cone in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq75_HTML.gif . For any real constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq76_HTML.gif , define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq77_HTML.gif . By a positive solution of (1.3), we mean a vector https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq78_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq79_HTML.gif satisfies (1.3) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq80_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq81_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq82_HTML.gif . The proofs of our main result (Theorem 1.1) is based on the Guo's fixed-point theorem.

Lemma 2.1 (Guo's Fixed-Point Theorem [27]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq83_HTML.gif be a cone of a real Banach space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq84_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq85_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq86_HTML.gif be bounded open subsets of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq87_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq88_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq89_HTML.gif is completely continuous such that one of the following condition hold:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq90_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq91_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq92_HTML.gif   for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq93_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq94_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq95_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq96_HTML.gif   for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq97_HTML.gif .

Then, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq98_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq99_HTML.gif .

The following result can be easily verified.

Result.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq100_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq101_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq102_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq103_HTML.gif and concave on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq104_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq105_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq106_HTML.gif .

Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq107_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq108_HTML.gif . For fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq109_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq110_HTML.gif , the linear three-point BVP

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ9_HTML.gif
(2.1)
has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ10_HTML.gif
(2.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq111_HTML.gif is the Green's function and is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ11_HTML.gif
(2.3)
We note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq112_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq113_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ12_HTML.gif
(2.4)
is the Green's function corresponding the boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ13_HTML.gif
(2.5)
whose integral representation is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ14_HTML.gif
(2.6)

Lemma 2.2 (see [9]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq114_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq115_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq116_HTML.gif , then then unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq117_HTML.gif of the problem (2.5) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ15_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq118_HTML.gif .

We need the following properties of the Green's function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq119_HTML.gif in the sequel.

Lemma 2.3 (see [11]).

The function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq120_HTML.gif can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ16_HTML.gif
(2.8)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ17_HTML.gif
(2.9)

Following the idea in [10], we calculate upper bound for the Green's function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq121_HTML.gif in the following lemma.

Lemma 2.4.

The function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq122_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ18_HTML.gif
(2.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq123_HTML.gif

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq124_HTML.gif , we discuss various cases.

Case 1.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq125_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq126_HTML.gif ; using (2.3), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ19_HTML.gif
(2.11)
If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq127_HTML.gif , the maximum occurs at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq128_HTML.gif , hence
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ20_HTML.gif
(2.12)
and if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq129_HTML.gif , the maximum occurs at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq130_HTML.gif , hence
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ21_HTML.gif
(2.13)

Case 2.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq131_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq132_HTML.gif ; using (2.3), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ22_HTML.gif
(2.14)

Case 3.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq133_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq134_HTML.gif ; using (2.3), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ23_HTML.gif
(2.15)

Case 4.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq135_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq136_HTML.gif ; using (2.3), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ24_HTML.gif
(2.16)

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq137_HTML.gif , the maximum occurs at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq138_HTML.gif , hence

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ25_HTML.gif
(2.17)

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq139_HTML.gif , the maximum occurs at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq140_HTML.gif , so

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ26_HTML.gif
(2.18)

Now, we consider the nonlinear nonsingular system of BVPs

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ27_HTML.gif
(2.19)
We write (2.19) as an equivalent system of integral equations
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ28_HTML.gif
(2.20)
By a solution of the system (2.19), we mean a solution of the corresponding system of integral equations (2.20). Define a retraction https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq141_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq142_HTML.gif and an operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq143_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ29_HTML.gif
(2.21)
where operators https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq144_HTML.gif are defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ30_HTML.gif
(2.22)

Clearly, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq145_HTML.gif is a fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq146_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq147_HTML.gif is a solution of the system (2.19).

Lemma 2.5.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq148_HTML.gif holds. Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq149_HTML.gif is completely continuous.

Proof.

Clearly, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq150_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq151_HTML.gif . We show that the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq152_HTML.gif is uniformly bounded. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq153_HTML.gif be fixed and consider
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ31_HTML.gif
(2.23)
Choose a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq154_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq155_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq156_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq157_HTML.gif . Then, for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq158_HTML.gif , using (2.22), Lemma 2.4, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq159_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq160_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ32_HTML.gif
(2.24)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ33_HTML.gif
(2.25)
that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq161_HTML.gif is uniformly bounded. Similarly, using (2.22), Lemma 2.4, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq162_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq163_HTML.gif , we can show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq164_HTML.gif is also uniformly bounded. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq165_HTML.gif is uniformly bounded. Now we show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq166_HTML.gif is equicontinuous. Define
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ34_HTML.gif
(2.26)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq167_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq168_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq169_HTML.gif is uniformly continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq170_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq171_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq172_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq173_HTML.gif implies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ35_HTML.gif
(2.27)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq174_HTML.gif , using (2.22)–(2.27), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ36_HTML.gif
(2.28)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ37_HTML.gif
(2.29)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq175_HTML.gif is equicontinuous. Similarly, using (2.22)–(2.27), we can show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq176_HTML.gif is also equicontinuous. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq177_HTML.gif is equicontinuous. By Arzelà-Ascoli theorem, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq178_HTML.gif is relatively compact. Hence, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq179_HTML.gif is a compact operator.

Now we show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq180_HTML.gif is continuous. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq181_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq182_HTML.gif Then by using (2.22) and Lemma 2.4, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ38_HTML.gif
(2.30)
Consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ39_HTML.gif
(2.31)
By Lebesgue dominated convergence theorem, it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ40_HTML.gif
(2.32)
Similarly, by using (2.22) and Lemma 2.4, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ41_HTML.gif
(2.33)
From (2.32) and (2.33), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ42_HTML.gif
(2.34)

that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq183_HTML.gif is continuous. Hence, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq184_HTML.gif is completely continuous.

3. Main Results

Proof of Theorem 1.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq185_HTML.gif . Choose a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq186_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ43_HTML.gif
(3.1)
Choose a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq187_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq188_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq189_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq190_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq191_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq192_HTML.gif , using (2.22), (3.1), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq193_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq194_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ44_HTML.gif
(3.2)
Since,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ45_HTML.gif
(3.3)
it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ46_HTML.gif
(3.4)
Similarly, using (2.22), (3.1), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq195_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq196_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ47_HTML.gif
(3.5)
From (3.4), and (3.5), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ48_HTML.gif
(3.6)
Choose a real constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq197_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ49_HTML.gif
(3.7)
Choose a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq198_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq199_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq200_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq201_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq202_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq203_HTML.gif , using (2.22), (3.7), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq204_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq205_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ50_HTML.gif
(3.8)
We used the fact that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ51_HTML.gif
(3.9)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ52_HTML.gif
(3.10)
Similarly, using (2.22), (3.7), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq206_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq207_HTML.gif , we have,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ53_HTML.gif
(3.11)
From (3.10) and (3.11), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ54_HTML.gif
(3.12)
Hence by Lemma 2.1, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq208_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq209_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ55_HTML.gif
(3.13)
Moreover, by (3.4), (3.5), (3.10) and (3.11), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ56_HTML.gif
(3.14)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq210_HTML.gif is a solution of the system (2.19), hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq211_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq212_HTML.gif are concave on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq213_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq214_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq215_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq216_HTML.gif , using result (2.2) and (3.14), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ57_HTML.gif
(3.15)
which implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq217_HTML.gif is uniformly bounded on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq218_HTML.gif . Now we show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq219_HTML.gif is equicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq220_HTML.gif . Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq221_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq222_HTML.gif and consider the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ58_HTML.gif
(3.16)
Using Lemma 2.3, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ59_HTML.gif
(3.17)
Differentiating with respect to t, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ60_HTML.gif
(3.18)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ61_HTML.gif
(3.19)
In view of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq223_HTML.gif and (3.15), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ62_HTML.gif
(3.20)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ63_HTML.gif
(3.21)
Similarly, consider the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ64_HTML.gif
(3.22)
using https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq224_HTML.gif and (3.15), we can show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ65_HTML.gif
(3.23)
In view of (3.21) and (3.23), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq225_HTML.gif is equicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq226_HTML.gif . Hence by Arzelà-Ascoli theorem, the sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq227_HTML.gif has a subsequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq228_HTML.gif converging uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq229_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq230_HTML.gif . Let us consider the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ66_HTML.gif
(3.24)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq231_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ67_HTML.gif
(3.25)
Differentiating twice with respect to t, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ68_HTML.gif
(3.26)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq232_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ69_HTML.gif
(3.27)
Similarly, consider the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ70_HTML.gif
(3.28)
we can show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ71_HTML.gif
(3.29)
Now, we show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq233_HTML.gif also satisfies the boundary conditions. Since,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ72_HTML.gif
(3.30)
Similarly, we can show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ73_HTML.gif
(3.31)
Equations (3.27)–(3.31) imply that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq234_HTML.gif is a solution of the system (1.3). Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq235_HTML.gif is positive. In fact, by (3.27) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq236_HTML.gif is concave and by Lemma 2.2
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ74_HTML.gif
(3.32)

implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq237_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq238_HTML.gif . Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq239_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq240_HTML.gif . The proof of Theorem 1.1 is complete.

Example.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ75_HTML.gif
(3.33)

where the real constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq241_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq242_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq243_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq244_HTML.gif and the real constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq245_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq246_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq247_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq248_HTML.gif . Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq249_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq250_HTML.gif satisfy the assumptions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq251_HTML.gif . Hence, by Theorem 1.1, the system (1.3) has a positive solution.

Declarations

Acknowledgment

Research of R. A. Khan is supported by HEC, Pakistan, Project 2- 3(50)/PDFP/HEC/2008/1.

Authors’ Affiliations

(1)
Centre for Advanced Mathematics and Physics, Campus of College of Electrical and Mechanical Engineering, National University of Sciences and Technology

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Copyright

© N. A. Asif and R. A. Khan. 2009

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