Positive Solutions for a Class of Coupled System of Singular Three-Point Boundary Value Problems
© N. A. Asif and R. A. Khan. 2009
Received: 27 February 2009
Accepted: 15 May 2009
Published: 22 June 2009
Existence of positive solutions for a coupled system of nonlinear three-point boundary value problems of the type , , , , , , , is established. The nonlinearities , are continuous and may be singular at , and/or , while the parameters , satisfy . An example is also included to show the applicability of our result.
Multipoint boundary value problems (BVPs) arise in different areas of applied mathematics and physics. For example, the vibration of a guy wire composed of parts with a uniform cross-section and different densities in different parts can be modeled as a Multipoint boundary value problem . Many problems in the theory of elastic stability can also be modeled as Multipoint boundary value problem .
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 . Since then the study of nonlinear three-point BVPs has attracted much attention of many researchers, see [8–11] and references therein for boundary value problems with ordinary differential equations and also  for boundary value problems on time scales. Recently, the study of singular BVPs has attracted the attention of many authors, see for example, [13–18] and the recent monograph by Agarwal et al. .
where , and may be singular at , , and/or .
where , , . We allow and to be singular at , , and also and/or . We study the sufficient conditions for existence of positive solution for the singular system (1.3) under weaker hypothesis on and 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 and are allowed to be unbounded at , , , and/or . 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, 24–26]. Throughout this paper, we assume that are continuous and may be singular at , , , and/or . We also assume that the following conditions hold:
The main result of this paper is as follows.
Assume that hold. Then the system (1.3) has at least one positive solution.
For each , we write . Let . Clearly, is a Banach space and is a cone. Similarly, for each , we write . Clearly, is a Banach space and is a cone in . For any real constant , define . By a positive solution of (1.3), we mean a vector such that satisfies (1.3) and , on . 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 ).
Let be a cone of a real Banach space , , be bounded open subsets of and . Suppose that is completely continuous such that one of the following condition hold:
(i) for and for ;
(ii) for and for .
Then, has a fixed point in .
The following result can be easily verified.
Let such that . Let , and concave on . Then, for all .
Choose such that . For fixed and , the linear three-point BVP
Lemma 2.2 (see ).
We need the following properties of the Green's function in the sequel.
Lemma 2.3 (see ).
Following the idea in , we calculate upper bound for the Green's function in the following lemma.
For , we discuss various cases.
For , the maximum occurs at , hence
For , the maximum occurs at , so
Now, we consider the nonlinear nonsingular system of BVPs
Clearly, if is a fixed point of , then is a solution of the system (2.19).
Assume that holds. Then is completely continuous.
which implies that is equicontinuous. Similarly, using (2.22)–(2.27), we can show that is also equicontinuous. Thus, is equicontinuous. By Arzelà-Ascoli theorem, is relatively compact. Hence, is a compact operator.
that is, is continuous. Hence, is completely continuous.
3. Main Results
Proof of Theorem 1.1.
implies that for all . Similarly, for all . The proof of Theorem 1.1 is complete.
where the real constants satisfy , with and the real constants satisfy , with . Clearly, and satisfy the assumptions . Hence, by Theorem 1.1, the system (1.3) has a positive solution.
Research of R. A. Khan is supported by HEC, Pakistan, Project 2- 3(50)/PDFP/HEC/2008/1.
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