Abstract
In this paper, we discuss the existence of positive solutions for secondorder differential equations subject to nonlinear impulsive conditions and nonseparated 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; nonseparated periodic boundary value condition1 Introduction
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:
Here,
We assume throughout, and with further mention, that the following conditions hold.
(H1) Let
(H)
A function
For the case of
where
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 secondorder differential equations with impulse have received considerable attention and much literature has been published; see, for instance, [817] and their references. However, there are fewer results about positive solutions for secondorder impulsive differential equations. To our best knowledge, there is no result about nonlinear impulsive differential equations with nonseparated 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
2 Preliminaries
In this section, we collect some preliminary results that will be used in the subsequent
section. We denote by
under the initial boundary conditions
Put
Definition 2.1 For two differential functions y and z, we define their Wronskian by
Theorem 2.1The Wronskian of any two solutions for equations (2.1) is constant. Especially,
Proof Suppose that y and z are two solutions of (2.1), then
therefore, the Wronskian is constant. Further, from the initial conditions (2.2),
we have
Consider the following equation:
From Theorem 2.5 in [1], equation (2.3) has a Green function
(
(
(
Combining with Theorem 2.1, we can also prove that
(
Remark 1 From paper [1], we can get
Especially, in the case of
Define an operator
then it is easy to check that
Lemma 2.1The spectral radius
In what follows, we denote the positive eigenfunction corresponding to
where
Lemma 2.2The fixed point of the mapping Φ is a solution of (1.1).
Proof Clearly, Φu is continuous in t. For
Using (
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 wellknown lemmas in [18] are needed in our argument.
Lemma 2.3[18]
LetXbe a Banach space andKbe a cone inX. Suppose
is a completely continuous operator such that
•
•
Then Φ has a fixed point in
Lemma 2.4[18]
LetXbe a Banach space andKbe a cone inX. Suppose
is a completely continuous operator such that
• There exists
• There exists
Then Φ has a fixed point in
3 Main results
Recalling that δ was defined after Lemma 2.1, for convenience, we introduce the following notations.
Assume that the constant
Theorem 3.1Assume that there exist positive constantsα, βsuch that
Then (1.1) has at least one positive solutionusuch that
Proof Clearly,
Then
and
Let
If not, there exist
which implies that
On the other hand, for
From Lemma 2.4 it follows that Φ has a fixed point
In the next theorem, we make use of the eigenvalue
Theorem 3.2Assume that there exist positive constantsα, βsuch that
here
Proof Obviously,
At first, we show that
On the other hand,
It is easy to check that
Next, we show that
If not, there exist
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
Noting that
which implies that
a contradiction.
Finally, we show that
Since
Suppose that there exist
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
Noting that
It is impossible for
a contradiction.
From Lemma 2.3 it follows that Φ has a fixed point
Corollary 3.1Assume that
or
here
Corollary 3.2Assume that there exists a constantαsuch that
here
Example 1 Consider the equation
where
here
Example 2 Consider the equation
where
It is well known that, for the problem consisting of the equation
the first eigenvalue is 0 (see, for example, [[19], p.428]). It follows that the first eigenvalue is
and the boundary condition (3.16). Meanwhile, we can obtain the positive eigenfunction
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
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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