Positive Solutions of Singular Multipoint Boundary Value Problems for Systems of Nonlinear Second-Order Differential Equations on Infinite Intervals in Banach Spaces
© Xingqiu Zhang. 2009
Received: 27 April 2009
Accepted: 12 June 2009
Published: 14 July 2009
The cone theory together with Mönch fixed point theorem and a monotone iterative technique is used to investigate the positive solutions for some boundary problems for systems of nonlinear second-order differential equations with multipoint boundary value conditions on infinite intervals in Banach spaces. The conditions for the existence of positive solutions are established. In addition, an explicit iterative approximation of the solution for the boundary value problem is also derived.
Very recently, by using Shauder fixed point theorem, Guo  obtained the existence of positive solutions for a class of th-order nonlinear impulsive singular integro-differential equations in a Banach space. Motivated by Guo's work, in this paper, we will use the cone theory and the Mönch fixed point theorem combined with a monotone iterative technique to investigate the positive solutions of BVP (1.2). The main features of the present paper are as follows. Firstly, compared with , the problem we discussed here is systems of multipoint boundary value problem and nonlinear term permits singularity not only at but also at . Secondly, compared with , the relative compact conditions we used are weaker. Furthermore, an iterative sequence for the solution under some normal type conditions is established which makes it very important and convenient in applications.
The rest of the paper is organized as follows. In Section 2, we give some preliminaries and establish several lemmas. The main theorems are formulated and proved in Section 3. Then, in Section 4, an example is worked out to illustrate the main results.
2. Preliminaries and Several Lemmas
Let be a normal cone in with normal constant which defines a partial ordering in by . If and , we write . Let . So, if and only if . For details on cone theory, see .
In what follows, we always assume that . Let . Obviously, for any . When , we write , that is, . Let and . It is clear, are cones in and , respectively. A map is called a positive solution of BVP (1.2) if and satisfies (1.2).
Let us list some conditions for convenience.
for all as The rest part of the proof is very similar to Lemma in , we omit the details.
The proof is similar to that of Lemma in , we omit it.
Lemma 2.7 (see ).
Lemma 2.8 (see ).
3. Main Results
On the other hand, . Then, (3.5), (3.6), , and Lemma 2.7 imply that is, is relatively compact in Hence, the Mönch fixed point theorem guarantees that has a fixed point in . Thus, Theorem 3.1 is proved.
The proof is almost the same as that of Theorem 3.2. The only difference is that, instead of using condition , the conclusion is implied directly by (3.15) and (3.16), the full regularity of and Lemma 2.4.
4. An Example
The project is supported financially by the National Natural Science Foundation of China (10671167) and the Natural Science Foundation of Liaocheng University (31805).
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