In this paper, we discuss the existence of positive solutions for second-order differential equations subject to nonlinear impulsive conditions and non-separated periodic boundary value conditions. Our criteria for the existence of positive solutions will be expressed in terms of the first eigenvalue of the corresponding nonimpulsive problem. The main tool of study is a fixed point theorem in a cone.
MSC: 34B37, 34B18.
Keywords:impulsive differential equation; positive solution; fixed point theorem; non-separated periodic boundary value condition
Let ω be a fixed positive number. In this paper, we are concerned with the existence of positive solutions for the following boundary value problem (BVP) with impulses:
We assume throughout, and with further mention, that the following conditions hold.
A function defined on is called a solution of BVP (1.1) ((1.1a)-(1.1c)) if its first derivative exists for each , is absolutely continuous on each close subinterval of , there exist finite values , the impulse conditions (1.1b) and the boundary conditions (1.1c) are satisfied, and the equation (1.1a) is satisfied almost everywhere on .
For the case of (), the problem (1.1) is related to a non-separated periodic boundary value problem of ODE. Atici and Guseinov  have proved the existence of a positive and twin positive solutions to BVP (1.1) by applying a fixed point theorem for the completely continuous operators in cones. In , Graef and Kong studied the following periodic boundary value problem:
where . Based upon the properties of Green’s function obtained in , the authors extended and improved the work of  by using topological degree theory. They derived new criteria for the existence of non-trivial solutions, positive solutions and negative solutions of the problem (1.2) when f is a sign-changing function and not necessarily bounded from below even over . Very recently, He et al. studied BVP (1.1) without impulses and generalized the results of [1,4] via the fixed point index theory. The problem (1.2) in the case of , the usual periodic boundary value problem, has been extensively investigated; see [4-7] for some results.
On the other hand, impulsive differential equations are a basic tool to study processes that are subjected to abrupt changes in their state. There has been a significant development in the last two decades. Boundary problems of second-order differential equations with impulse have received considerable attention and much literature has been published; see, for instance, [8-17] and their references. However, there are fewer results about positive solutions for second-order impulsive differential equations. To our best knowledge, there is no result about nonlinear impulsive differential equations with non-separated periodic boundary conditions.
Motivated by the work above, in this paper we study the existence of positive solutions for the boundary value problem (1.1). By using fixed point theorems in a cone, criteria are established under some conditions on concerning the first eigenvalue corresponding to the relevant linear operator. More important, the impulsive terms are different from those of papers [8,9].
under the initial boundary conditions
Put , then by [, Lemma 2.3], .
Definition 2.1 For two differential functions y and z, we define their Wronskian by
Proof Suppose that y and z are two solutions of (2.1), then
Consider the following equation:
From Theorem 2.5 in , equation (2.3) has a Green function for all , which has the following properties:
Combining with Theorem 2.1, we can also prove that
Remark 1 From paper , we can get when () and ,
Define an operator
then it is easy to check that is a completely continuous operator. By virtue of the Krein-Rutman theorem, the authors in  got the following result.
Lemma 2.2The fixed point of the mapping Φ is a solution of (1.1).
which implies that the fixed point of Φ is the solution of (1.1). The proof is complete. □
The proofs of the main theorems of this paper are based on fixed point theory. The following two well-known lemmas in  are needed in our argument.
is a completely continuous operator such that
is a completely continuous operator such that
3 Main results
On the other hand,
Next, we show that
Multiplying the first equation of (3.8) by ϕ and integrating from 0 to ω, we obtain that
One can find that
Substituting (3.10) into (3.9), we get
which implies that
Finally, we show that
Multiplying the first equation of (3.11) by ϕ and integrating from 0 to ω, we obtain that
One can get that
Substituting (3.13) into (3.12), we get
Example 1 Consider the equation
Example 2 Consider the equation
the first eigenvalue is 0 (see, for example, [, p.428]). It follows that the first eigenvalue is for the problem consisting of the equation
and the boundary condition (3.16). Meanwhile, we can obtain the positive eigenfunction corresponding to . It is also easy to check that , , and (here ). So, the right-hand side of the inequality in Corollary 3.2 is obviously satisfied. Considering the monotonicity of and , we can choose a sufficiently small positive constant α such that the left-hand side of the inequality is true. Therefore, by a direct application of Corollary 3.2, there exists one open interval such that (3.15) has at least two positive solutions for .
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
The authors would like to thank anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of work. This research was partially supported by the NNSF of China (No. 11001274, 11171085) and the Postdoctoral Science Foundation of Central South University and China (No. 2011M501280).
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