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

  • NaseerAhmad Asif1 and

    Affiliated with

    • RahmatAli Khan1Email author

      Affiliated with

      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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq7_HTML.gif , is established. The nonlinearities http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq8_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq10_HTML.gif , and/or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq11_HTML.gif , while the parameters http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq12_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq13_HTML.gif satisfy http://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 http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ1_HTML.gif
      (1.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq16_HTML.gif , and may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq17_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq19_HTML.gif and/or http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ2_HTML.gif
      (1.2)

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq22_HTML.gif and/or http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ3_HTML.gif
      (1.3)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq24_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq25_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq26_HTML.gif . We allow http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq27_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq28_HTML.gif to be singular at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq29_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq30_HTML.gif , and also http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq31_HTML.gif and/or http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq33_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq36_HTML.gif are allowed to be unbounded at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq37_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq38_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq39_HTML.gif , and/or http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq42_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq43_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq44_HTML.gif , and/or http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq46_HTML.gif and satisfy
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq47_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq48_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq49_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq50_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq51_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq52_HTML.gif ,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq53_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq54_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq55_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq56_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq57_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq58_HTML.gif ,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq60_HTML.gif is
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ7_HTML.gif
      (1.7)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq62_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq63_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq64_HTML.gif such that
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq66_HTML.gif , we write http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq67_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq68_HTML.gif . Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq69_HTML.gif is a Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq70_HTML.gif is a cone. Similarly, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq71_HTML.gif , we write http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq72_HTML.gif . Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq73_HTML.gif is a Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq74_HTML.gif is a cone in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq75_HTML.gif . For any real constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq76_HTML.gif , define http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq78_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq79_HTML.gif satisfies (1.3) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq80_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq81_HTML.gif on http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq84_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq85_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq86_HTML.gif be bounded open subsets of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq88_HTML.gif . Suppose that http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq90_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq91_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq92_HTML.gif   for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq93_HTML.gif ;

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

      Then, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq98_HTML.gif has a fixed point in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq100_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq101_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq102_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq103_HTML.gif and concave on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq104_HTML.gif . Then, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq105_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq106_HTML.gif .

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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ9_HTML.gif
      (2.1)
      has a unique solution
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ10_HTML.gif
      (2.2)
      where http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ11_HTML.gif
      (2.3)
      We note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq112_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq113_HTML.gif , where
      http://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
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq114_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq115_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq116_HTML.gif , then then unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq117_HTML.gif of the problem (2.5) satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ15_HTML.gif
      (2.7)

      where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq120_HTML.gif can be written as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ16_HTML.gif
      (2.8)
      where
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq122_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ18_HTML.gif
      (2.10)

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

      Proof.

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

      Case 1.

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

      Case 2.

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

      Case 3.

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

      Case 4.

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

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

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

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

      http://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

      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq141_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq142_HTML.gif and an operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq143_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ29_HTML.gif
      (2.21)
      where operators http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq144_HTML.gif are defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ30_HTML.gif
      (2.22)

      Clearly, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq145_HTML.gif is a fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq146_HTML.gif , then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq148_HTML.gif holds. Then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq150_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq151_HTML.gif . We show that the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq152_HTML.gif is uniformly bounded. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq153_HTML.gif be fixed and consider
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ31_HTML.gif
      (2.23)
      Choose a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq154_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq155_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq156_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq157_HTML.gif . Then, for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq158_HTML.gif , using (2.22), Lemma 2.4, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq159_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq160_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ32_HTML.gif
      (2.24)
      which implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ33_HTML.gif
      (2.25)
      that is, http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq162_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq163_HTML.gif , we can show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq164_HTML.gif is also uniformly bounded. Thus, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq166_HTML.gif is equicontinuous. Define
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ34_HTML.gif
      (2.26)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq167_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq168_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq169_HTML.gif is uniformly continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq170_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq171_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq172_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq173_HTML.gif implies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ35_HTML.gif
      (2.27)
      For http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ36_HTML.gif
      (2.28)
      Hence,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ37_HTML.gif
      (2.29)

      which implies that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq176_HTML.gif is also equicontinuous. Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq177_HTML.gif is equicontinuous. By Arzelà-Ascoli theorem, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq178_HTML.gif is relatively compact. Hence, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq180_HTML.gif is continuous. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq181_HTML.gif such that http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ38_HTML.gif
      (2.30)
      Consequently,
      http://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
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ42_HTML.gif
      (2.34)

      that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq183_HTML.gif is continuous. Hence, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq185_HTML.gif . Choose a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq186_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ43_HTML.gif
      (3.1)
      Choose a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq187_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq188_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq189_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq190_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq191_HTML.gif . For any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq192_HTML.gif , using (2.22), (3.1), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq193_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq194_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ44_HTML.gif
      (3.2)
      Since,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ45_HTML.gif
      (3.3)
      it follows that
      http://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), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq195_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq196_HTML.gif , we have
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ48_HTML.gif
      (3.6)
      Choose a real constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq197_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ49_HTML.gif
      (3.7)
      Choose a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq198_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq199_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq200_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq201_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq202_HTML.gif . For any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq203_HTML.gif , using (2.22), (3.7), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq204_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq205_HTML.gif , we have
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ51_HTML.gif
      (3.9)
      Thus,
      http://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), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq206_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq207_HTML.gif , we have,
      http://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
      http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq208_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq209_HTML.gif , that is,
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ56_HTML.gif
      (3.14)
      Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq211_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq212_HTML.gif are concave on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq213_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq214_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq215_HTML.gif . For http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ57_HTML.gif
      (3.15)
      which implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq217_HTML.gif is uniformly bounded on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq218_HTML.gif . Now we show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq219_HTML.gif is equicontinuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq220_HTML.gif . Choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq221_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq222_HTML.gif and consider the integral equation
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ60_HTML.gif
      (3.18)
      which implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ61_HTML.gif
      (3.19)
      In view of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq223_HTML.gif and (3.15), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ62_HTML.gif
      (3.20)
      which implies that
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ64_HTML.gif
      (3.22)
      using http://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
      http://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), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq225_HTML.gif is equicontinuous on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq227_HTML.gif has a subsequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq228_HTML.gif converging uniformly on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq229_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq230_HTML.gif . Let us consider the integral equation
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ66_HTML.gif
      (3.24)
      Letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq231_HTML.gif , we have
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ68_HTML.gif
      (3.26)
      Letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq232_HTML.gif , we have
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ70_HTML.gif
      (3.28)
      we can show that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ71_HTML.gif
      (3.29)
      Now, we show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq233_HTML.gif also satisfies the boundary conditions. Since,
      http://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
      http://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 http://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, http://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) http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ74_HTML.gif
      (3.32)

      implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq237_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq238_HTML.gif . Similarly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq239_HTML.gif for all http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_Equ75_HTML.gif
      (3.33)

      where the real constants http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq241_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq242_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq243_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq244_HTML.gif and the real constants http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq245_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq246_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq247_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq248_HTML.gif . Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq249_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F273063/MediaObjects/13661_2009_Article_836_IEq250_HTML.gif satisfy the assumptions http://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

      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.