- Open Access
Solvability and positive solution of a system of second-order boundary value problem with integral boundary conditions
© Jebari and Boukricha; licensee Springer 2014
Received: 24 July 2014
Accepted: 11 November 2014
Published: 16 December 2014
The main purpose of this paper is to establish the existence, uniqueness and positive solution of a system of second-order boundary value problem with integral conditions. Using Banach’s fixed point theorem and the Leray-Schauder nonlinear alternative, we discuss the existence and uniqueness solution of this problem, and we apply Guo-Krasnoselskii’s fixed point theorem in cone to study the existence of positive solution. We also give some examples to illustrate our results.
For more knowledge about the boundary value problems, we refer the reader to –. Our aim is to use the Banach contraction principle and the Leray-Schauder nonlinear alternative to prove the existence and uniqueness solutions of our problem. For this, we formulate the boundary value problem as the fixed point problem. However, the Schauder fixed point theorem cannot ensure the solutions to be positive. Since only positive solutions are useful for many applications, motivated by the above works, the existence of positive solution is obtained by Guo-Krasnoselskii’s fixed point theorem. Our work is more general than –, , ; for example, (1.7) investigated in the case , , , and (1.3) in the case , , . To the best of our knowledge, no one has studied the existence and positiveness of solutions for system (1.1)-(1.2).
This paper is organized as follows. In Section 2, we present some preliminaries that will be used to prove our results. In Section 3, we discuss the existence and uniqueness of a solution for problem (1.1)-(1.2) by using the Leray-Schauder nonlinear alternative and Banach’s fixed point theorem. In Section 4, the study of the positivity of a solution is based on Guo-Krasnoselskii’s fixed point theorem in cone. Finally, we shall give three examples to illustrate our main results.
2 Preliminaries and lemmas
We define the set by .
The function is called a nonnegative solution of system (1.1)-(1.2) if and only if u satisfies (1.1)-(1.2) and for all , for .
The function is called a positive solution of system (1.1)-(1.2) if and only if u satisfies (1.1)-(1.2) and for all , for .
is the Green functions of the corresponding homogeneous problem.
We can write Lemma 2.2 in the following form: is a solution of (1.1)-(1.2) if and only if u is a fixed point of the operator T.
3 Existence and uniqueness results
We deduce that for all , for all and for all , . Since for all , and . We obtain for all , , , where is given by (3.2). □
Now, we prove the existence and uniqueness of solutions in the Banach space E. The uniqueness result is based on Banach’s contraction principle .
Then problem (1.1)-(1.2) has a unique solution u in E.
Since , then T is a contraction, hence it has a unique fixed point which is the unique solution of (1.1)–(1.2). The proof is completed. □
We establish an existence result using the nonlinear alternative of Leray-Schauder type.
(Leray-Schauder nonlinear alternative (see ))
Let F be a Banach space and Ω be a bounded open subset of F, . Letbe a completely continuous operator. Then there exists, such that, or there exists a fixed point.
From this theorem we have the following result.
Then the boundary value problem (1.1)-(1.2) has at least one nontrivial solution.
It is easy to see that for all , are continuous since , , , and are continuous. T maps bounded sets into bounded sets in E; to establish this step, we use the Arzelà-Ascoli theorem . Let be a bounded subset in E. We shall prove that is relatively compact.
- (ii)Now, we apply the Leray-Schauder nonlinear alternative to prove that T has at least a nontrivial solution in E. We define . Then, for such that , , we have
which is a contradiction to the fact that . Lemma 3.3 allows us to conclude that T has a fixed point , and then problem (1.1)-(1.2) has a nontrivial solution . This achieves the proof. □
4 Existence of a positive solution
In this section, we will give some preliminary considerations and some lemmas which are essential to establish sufficient conditions for the existence of at least one positive solution for our problem. We make the following additional assumption.
(H1) The functions , and are continuous.
(H2) For all , there exist and such that for all , .
(H3) For all , there exist , and such that , and for all .
Now, we need some properties of the Green functions for .
The proof of Lemma 4.1 is similar to that of Lemma 3.1.
, implies ;
, implies .
is a cone of E.
(Guo-Krasnoselskii’s fixed point theorem in cone )
, and , or
, and , .
Then A has a fixed point in.
We employ Guo-Krasnoselskii’s fixed point theorem in cone to prove the existence of a positive solution of our problem, we have the following theorem.
Assume that conditions (H1), (H2) and (H3) hold. Then equation (1.1)-(1.2) has at least one positive solution.
Remark 4.3 shows that C is a cone subset of E. Lemma 3.1 and (H1) show that . In addition, a standard argument involving the Arzelà-Ascoli theorem implies that T is a completely continuous operator.
Based on Theorem 4.4, we get from (4.1) and (4.2) that the operator T has at least one fixed point. Thus, it follows that (1.1)-(1.2) has at least one nonnegative solution and from (H1) and (H2), (1.1)-(1.2) has at least one positive solution. □
Finally, we give some examples to illustrate the results obtained in this paper.
Then from Theorem 3.2 we conclude that the system of boundary value problem has a unique solution .
Then from Theorem 3.4 we conclude that the system of boundary value problem has at least one solution .
We can easily show that conditions (H1), (H2) and (H3) are satisfied. Hence, by Theorem 4.5 this problem has at least one positive solution.
The authors thank the reviewers for their constructive remarks that led to the improvement of the original manuscript.
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