- Open Access
Positive solutions of Sturm-Liouville boundary value problems for singular nonlinear second-order impulsive integro-differential equation in Banach spaces
© Sun; licensee Springer 2012
- Received: 5 June 2012
- Accepted: 23 July 2012
- Published: 6 August 2012
In this work, we investigate the existence of positive solutions of Sturm-Liouville boundary value problems for singular nonlinear second-order impulsive integro differential equation in a real Banach space. Some new existence results of positive solutions are established by applying fixed-point index theory together with comparison theorem. Some discussions and an example are given to demonstrate the applications of our main results.
MSC:34B15, 34B25, 45J05.
- measure of non-comparison
- positive solution
- boundary value problem
- impulsive integro-differential equation
where , and , represent the right-hand limit and left-hand limit of and at , respectively. and may be singular at and/or .
Boundary value problems for impulsive differential equations arise from many nonlinear problems in sciences, such as physics, population dynamics, biotechnology, and economics etc. (see [1, 2, 4–14, 16–18]). As it is well known that impulsive differential equations contain jumps and/or impulses which are main characteristic feature in computational biology. Over the past 15 years, a significant advance has been achieved in theory of impulsive differential equations. However, the corresponding theory of impulsive integro-differential equations in Banach spaces does not develop rapidly. Recently, Guo [5–8] established the existence of a solution, multiple solutions and extremal solutions for nonlinear impulsive integro-differential equations with nonsingular argument in Banach spaces. The main tools of Guo [5–8] are the Schauder fixed-point theorem, fixed-point index theory, upper and lower solutions together with the monotone iterative technique, respectively. The conditions of the Kuratowski measure of non-compactness in Guo [5–8] play an important role in the proof of the results. But all kinds of compactness type conditions is difficult to verify in abstract spaces. As a result, it is an interesting and important problem to remove or weak compactness type conditions.
Inspired and motivated greatly by the above works, the aim of the paper is to consider the existence of positive solutions for the boundary value problem (1.1) under simpler conditions. The main results of problem (1.1) are obtained by making use of fixed-point index theory and fixed-point theorem. More specifically, in the proof of these theorems, we construct a special cone for strict set contraction operator. Our main results in essence improve and generalize the corresponding results of Guo [5–8]. Moreover, our method is different from those in Guo [5–8].
The rest of the paper is organized as follows: In Section 2, we present some known results and introduce conditions to be used in the next section. The main theorem formulated and proved in Section 3. Finally, in Section 4, some discussions and an example for singular nonlinear integro-differential equations are presented to demonstrate the application of the main results.
In this section, we shall state some necessary definitions and preliminaries results.
, implies ;
, implies .
A cone is said solid if it contains interior points, . A cone P is called to be generating if , i.e., every element can be represented in the form , where . A cone P in E induces a partial ordering in E given by if . If and , we write ; if cone P is solid and , we write .
Definition 2.2 A cone is said to be normal if there exists a positive constant N such that , , , .
Definition 2.4 An operator is said to be completely continuous if it is continuous and compact. B is called a k-set-contraction () if it is continuous, bounded and for any bounded set , where denotes the measure of noncompactness of S.
A k-set-contraction is called a strict-set contraction if . An operator B is said to be condensing if it is continuous, bounded, and for any bounded set with .
Obviously, if B is a strict-set contraction, then B is a condensing mapping, and if operator B is completely continuous, then B is a strict-set contraction.
where . In what follows, we write , (), . By making use of (2.1), we can prove that has the following properties.
Proposition 2.1, .
Let . It is easy to verify is a Banach space with norm . Obviously, is a cone in Banach space .
Let , . It is easy to see that is a Banach space with the norm . Evidently, and is a cone in Banach space . For any , by making use of the mean value theorem (), obviously we see that exists and .
Let . For any , let , , .
A map is called a nonnegative solution of problem (1.1) if , for and satisfies problem (1.1). An operator is called a positive solution of problem (1.1) if y is a nonnegative solution of problem (1.1) and .
where ν denote 0 or ∞.
To establish the existence of multiple positive solutions in E of problem (1.1), let us list the following assumptions, which will stand throughout the paper:
(H3) for any bounded set , , and together with are relatively compact sets,
i.e., y is a fixed point of operator A defined by (2.6) in.
and . So and (). It is easy to verify that and . The proof is complete. □
In the following, let . For , we denote , and ().
Lemma 2.2 ()
where ϒ anddenote the Kuratowski measures of noncompactness of bounded sets in E and, respectively.
Lemma 2.3 ()
Lemma 2.4 ()
is relatively compact if and only if each elementandare uniformly bounded and equicontinuous on each ().
Lemma 2.5 ()
Hence, . □
Lemma 2.7 Suppose that (H1) and (H3) hold. Thenis completely continuous.
Hence, is continuous.
From (2.17), (2.18), and the absolutely continuity of integral function, we see that is equicontinuous.
So, . Therefore, A is compact. To sum up, the conclusion of Lemma 2.7 follows. □
If for , then .
If for , then .
In this section, we establish the existence of positive solutions for problem (1.1) by making use of Lemma 2.8.
Theorem 3.1 Suppose that (H1)-(H4) hold. Then problem (1.1) has at least one positive solution.
Therefore, A has at least one fixed point on . Consequently, problem (1.1) has at least one positive solution. □
Theorem 3.2 Suppose that (H1)∼(H3) and (H5) are satisfied. Then problem (1.1) has at least one positive solution.
Therefore, A has at least one fixed point on . Consequently, problem (1.1) has at least one positive solution. The proof is complete. □
where , .
Theorem 4.1 Assume that (H2) holds, and the following conditions are satisfied:
(C2) for any bounded set (), andtogether withare relatively compact sets.
(C3) , where m is defined by (2.4). Then the problem (4.1) has at least one positive solution.
Theorem 4.2 Assume that (H2) and (C1)∼(C2) hold, and the following condition is satisfied:
(C4) , where m is defined by (2.4). Then the problem (4.1) has at least one positive solution.
To illustrate how our main results can be used in practice, we present an example.
The problem (4.2) has at least one positive solution .
Then conditions (H1)∼(H4) are satisfied. Therefore, by Theorem 3.1, the problem (4.2) has at least one positive solution.
Remark 5.1 In , by requiring that f satisfies some noncompact measure conditions and P is a normal cone, Guo established the existence of positive solutions for initial value problem. In the paper, we impose some weaker condition on f, we obtain the positive solution of the problem (1.1).
The author is very grateful to Professor Lishan Liu and Professor R. P. Agarwal for their making many valuable comments. The author would like to express her thanks to the editor of the journal and the anonymous referees for their carefully reading of the first draft of the manuscript and making many helpful comments and suggestions which improved the presentation of the paper. The author was supported financially by the Foundation of Shanghai Municipal Education Commission (Grant Nos. DYL201105).
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