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.
Keywords:measure of non-comparison; positive solution; boundary value problem; impulsive integro-differential equation
In this paper, we study the existence of positive solutions to second-order singular nonlinear impulsive integro-differential equation of the form:
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.
2 Preliminaries and lemmas
In this section, we shall state some necessary definitions and preliminaries results.
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 2.4 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.
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.
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 .
For convenience and simplicity in the following discussion, we denote
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:
Integrate again, we get
Substituting (2.12) and (2.13) into (2.9), we obtain
Thanks to (2.1), we know that
Lemma 2.2 ()
Lemma 2.3 ()
Lemma 2.4 ()
Lemma 2.5 ()
3 Main results
In this section, we establish the existence of positive solutions for problem (1.1) by making use of Lemma 2.8.
Theorem 3.1Suppose that (H1)-(H4) hold. Then problem (1.1) has at least one positive solution.
From (3.2) and (3.3), we get
Theorem 3.2Suppose that (H1)∼(H3) and (H5) are satisfied. Then problem (1.1) has at least one positive solution.
According to (3.6) and (3.7), we get
4 Concerned results and applications
In this section, we deal with a special case of the problem (1.1). The method is just similar to what we have done in Section 3, so we omit the proof of some main results of the section. Case is treated in the following theorem. Under the case, the problem (1.1) reduces to the following boundary value problems:
Theorem 4.1Assume that (H2) holds, and the following conditions are satisfied:
Theorem 4.2Assume that (H2) and (C1)∼(C2) hold, and the following condition is satisfied:
To illustrate how our main results can be used in practice, we present an example.
Example 4.1 Consider the following boundary value problem for scalar second-order impulsive integro-differential equation:
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 declares that she has no competing interests.
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).
Wu, CZ, Teo, KL, Zhou, Y, Yan, WY: An optimal control problem involving impulsive integro-differential systems. Optim. Methods Softw.. 22, 531–549 (2007). Publisher Full Text
Wu, CZ, Teo, KL, Zhou, Y, Yan, WY: Solving an identification problem as an impulsive optimal parameter selection problem. Comput. Math. Appl.. 50, 217–229 (2005). Publisher Full Text
Guo, D: Boundary value problems for impulsive integro-differential equations on unbounded domains in a Banach space. Appl. Math. Comput.. 99, 1–15 (1999). Publisher Full Text
Guo, D: Existence of positive solutions for nth order nonlinear impulsive singular integro-differential equations in a Banach space. Nonlinear Anal.. 68, 2727–2740 (2008). Publisher Full Text
Guo, D: Extremal solutions for nth order impulsive integro-differential equations on the half-line in Banach space. Nonlinear Anal.. 65, 677–696 (2006). Publisher Full Text
Guo, D: Multiple positive solutions for nth order impulsive integro-differential equations in Banach space. Nonlinear Anal.. 60, 955–976 (2005). Publisher Full Text
Liu, L, Xu, Y, Wu, Y: On unique solution of an initial value problem for nonlinear first-order impulsive integro-differential equations of Volterra type in Banach spaces. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal.. 13, 641–652 (2006)
Liu, L, Wu, Y, Zhang, X: On well-posedness of an initial value problem for nonlinear second order impulsive integro differential equations of Volterra type in Banach spaces. J. Math. Anal. Appl.. 317, 634–649 (2006). Publisher Full Text
Zhang, X, Liu, L: Initial value problem for nonlinear second order impulsive integro differential equations of mixed type in Banach spaces. Nonlinear Anal.. 64, 2562–2574 (2006). Publisher Full Text
Guo, D: Existence of solutions for nth-order impulsive integro-differential equations in a Banach space. Nonlinear Anal.. 47, 741–752 (2001). Publisher Full Text
Liu, L, Wu, C, Guo, F: A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces. J. Math. Anal. Appl.. 275, 369–385 (2002). Publisher Full Text
Liu, L: Iterative method for solution and coupled quasi-solutions of nonlinear integro differential equations of mixed type in Banach spaces. Nonlinear Anal.. 42, 583–598 (2000). Publisher Full Text
Zhang, X, Feng, M, Ge, W: Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces. J. Comput. Appl. Math.. 233, 1915–1926 (2010). Publisher Full Text
Feng, M, Pang, H: A class of three-point boundary-value problems for second-order impulsive integro-differential equations in Banach spaces. Nonlinear Anal.. 70, 64–82 (2009). Publisher Full Text