Existence of positive solutions for a coupled system of nonlinear threepoint 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.
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 parts with a uniform crosssection 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 threepoint boundary value problems was initiated by Gupta [7]. Since then the study of nonlinear threepoint BVPs has attracted much attention of many researchers, see [8–11] 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, [13–18] 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 [20–22] and the case of singular nonlinearity can be seen in [8, 21, 23–26]. Wei [25], developed the upper and lower solutions method for the existence of positive solutions of the following coupled system of BVPs:
where , and may be singular at , , and/or .
By using fixed point theorem in cone, Yuan et al. [26] studied the following coupled system of nonlinear singular boundary value problem:
are allowed to be superlinear and are singular at and/or . 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 threepoint nonlocal BVPs:
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:
(A_{1}) and satisfy
(A_{2})There exist real constants such that , , and for all , ,
(A_{3})There exist real constants such that , , and for all , ,
for example, a function that satisfies the assumptions and is
where , , ; such that
The main result of this paper is as follows.
Theorem 1.1.
Assume that hold. Then the system (1.3) has at least one positive solution.
2. Preliminaries
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 fixedpoint theorem.
Lemma 2.1 (Guo's FixedPoint Theorem [27]).
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.
Result.
Let such that . Let , and concave on . Then, for all .
Choose such that . For fixed and , the linear threepoint BVP
has a unique solution
where is the Green's function and is given by
We note that as , where
is the Green's function corresponding the boundary value problem
whose integral representation is given by
Lemma 2.2 (see [9]).
Let . If and , then then unique solution of the problem (2.5) satisfies
where .
We need the following properties of the Green's function in the sequel.
Lemma 2.3 (see [11]).
The function can be written as
where
Following the idea in [10], we calculate upper bound for the Green's function in the following lemma.
Lemma 2.4.
The function satisfies
where
Proof.
For , we discuss various cases.
Case 1.
, ; using (2.3), we obtain
If , the maximum occurs at , hence
and if , the maximum occurs at , hence
Case 2.
, ; using (2.3), we have
Case 3.
, ; using (2.3), we have
Case 4.
, ; using (2.3), we have
For , the maximum occurs at , hence
For , the maximum occurs at , so
Now, we consider the nonlinear nonsingular system of BVPs
We write (2.19) as an equivalent system of integral equations
By a solution of the system (2.19), we mean a solution of the corresponding system of integral equations (2.20). Define a retraction by and an operator by
where operators are defined by
Clearly, if is a fixed point of , then is a solution of the system (2.19).
Lemma 2.5.
Assume that holds. Then is completely continuous.
Proof.
Clearly, for any , . We show that the operator is uniformly bounded. Let be fixed and consider
Choose a constant such that , , . Then, for every , using (2.22), Lemma 2.4, and , we have
which implies that
that is, is uniformly bounded. Similarly, using (2.22), Lemma 2.4, and , we can show that is also uniformly bounded. Thus, is uniformly bounded. Now we show that is equicontinuous. Define
Let such that . Since is uniformly continuous on , for any , there exist such that implies
For , using (2.22)–(2.27), we have
Hence,
which implies that is equicontinuous. Similarly, using (2.22)–(2.27), we can show that is also equicontinuous. Thus, is equicontinuous. By ArzelàAscoli theorem, is relatively compact. Hence, is a compact operator.
Now we show that is continuous. Let such that Then by using (2.22) and Lemma 2.4, we have
Consequently,
By Lebesgue dominated convergence theorem, it follows that
Similarly, by using (2.22) and Lemma 2.4, we have
From (2.32) and (2.33), it follows that
that is, is continuous. Hence, is completely continuous.
3. Main Results
Proof of Theorem 1.1.
Let . Choose a constant such that
Choose a constant such that , , , . For any , using (2.22), (3.1), , and , we have
Since,
it follows that
Similarly, using (2.22), (3.1), , and , we have
From (3.4), and (3.5), it follows that
Choose a real constant such that
Choose a constant such that , , , . For any , using (2.22), (3.7), , and , we have
We used the fact that
Thus,
Similarly, using (2.22), (3.7), and , we have,
From (3.10) and (3.11), it follows that
Hence by Lemma 2.1, has a fixed point , that is,
Moreover, by (3.4), (3.5), (3.10) and (3.11), we have
Since is a solution of the system (2.19), hence and are concave on . Moreover, and . For , using result (2.2) and (3.14), we have
which implies that is uniformly bounded on . Now we show that is equicontinuous on . Choose and and consider the integral equation
Using Lemma 2.3, we have
Differentiating with respect to t, we obtain
which implies that
In view of and (3.15), we have
which implies that
Similarly, consider the integral equation
using and (3.15), we can show that
In view of (3.21) and (3.23), is equicontinuous on . Hence by ArzelàAscoli theorem, the sequence has a subsequence converging uniformly on to . Let us consider the integral equation
Letting , we have
Differentiating twice with respect to t, we have
Letting , we have
Similarly, consider the integral equation
we can show that
Now, we show that also satisfies the boundary conditions. Since,
Similarly, we can show that
Equations (3.27)–(3.31) imply that is a solution of the system (1.3). Moreover, is positive. In fact, by (3.27) is concave and by Lemma 2.2
implies that for all . Similarly, for all . The proof of Theorem 1.1 is complete.
Example.
Let
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.
Acknowledgment
Research of R. A. Khan is supported by HEC, Pakistan, Project 2 3(50)/PDFP/HEC/2008/1.
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