- Open Access
Existence results for p-Laplacian boundary value problems of impulsive dynamic equations on time scales
© Ozen et al.; licensee Springer. 2013
- Received: 26 March 2013
- Accepted: 25 September 2013
- Published: 20 November 2013
In this paper, Bai-Ge’s fixed point theorem is used to investigate the existence of positive solutions for second-order boundary value problems of p-Laplacian impulsive dynamic equations on time scales. As an application, we give an example to demonstrate our results.
MSC:34B18, 34N05, 39B37.
- impulsive dynamic equation
- positive solutions
- fixed point theorems
- time scales
Impulse differential equations describe processes which experience a sudden change of state at certain moments; see the monographs of Lakshmikantham et al.  and Samoilenko and Perestyuk . Impulsive differential equations can be used to describe a lot of natural phenomena such as the dynamics of populations subject to abrupt changes (harvesting, diseases, etc.), which cannot be described using classical differential equations. That is why in recent years they have attracted much attention of investigators (cf., e.g., [3, 4]).
The study of dynamic equations on time scales goes back to Stefan Hilger . Now it is still a new area of fairly theoretical exploration in mathematics. We refer to the books by Bohner and Peterson [6, 7]. There are a lot of works concerning the p-Laplacian problems on time scales; see, for example, [8–10]. Few works have been done on the existence of solutions to boundary value problems (BVP) for p-Laplacian impulsive dynamic equations on time scales; see [11–14]. Moreover, there is not much work on m-point boundary value problems for the p-Laplacian impulsive dynamic equations on time scales except for that in  by Li et al. Our aim in this paper is to fill the gap.
where is a time scale, , , , , with , () with , and , , . is a p-Laplacian operator, i.e., for , , where and , , and represent the right-hand limit and left-hand limit of the function at , .
In this paper we assume that
(A2) , , , with and ,
(A3) is nonnegative and there exists an integer , such that ,
(A4) is a bounded function, , ,
(A5) , where and is a constant which is given by Theorem 3.1.
In this study, by employing Bai-Ge’s fixed point theorem , we get the existence of at least three positive solutions for boundary value problem (1.1). In fact, our result is also new when (the differential case) and ℤ (the discrete case). Therefore, the result can be considered as a contribution to this field.
This paper is organized as follows. In Section 2, we give some definitions and preliminary lemmas which are key tools for our proof. The main results are given in Section 3. Finally, in Section 4, we give an example to demonstrate our results.
In this section, we give some lemmas which are useful for our main results.
Throughout the rest of this paper, we always assume that the points of impulse are right dense for each . Let , .
respectively. A function is called a solution to (1.1) if it satisfies all the equations of (1.1).
Substituting (2.7) into (2.6), we get (2.1), which completes the proof of sufficiency.
The proof is complete. □
Lemma 2.4 Assume that (A1)-(A4) hold. Then is a completely continuous operator.
Proof From the definition of T, it is clear that . On the other hand, by conditions (A1)-(A4) and the definition of , it is clear that is continuous.
So, Tu and are bounded on J and equicontinuous on each (). This implies that T Ω is relatively compact. Therefore, the operator is completely continuous. □
The following assumptions as regards the nonnegative continuous convex functions φ, ω are used:
(B1) there exists such that for all ;
(B2) for any and .
To prove our main result, we need the following fixed point theorem due to Bai and Ge in .
Lemma 3.1 
Let P be a cone in a real Banach space , and let , . Assume that φ and ω are nonnegative continuous convex functions satisfying (B1) and (B2), ψ is a nonnegative continuous concave function on P such that for all and is a completely continuous operator. Suppose that
(B3) , for ,
(B4) , for all ,
(B5) for all with .
Then, on the cone P, ψ is a concave functional, φ and ω are convex functionals satisfying (B1) and (B2).
Theorem 3.1 Assume that (A1)-(A4) hold. There exist constants , and the following assumptions are satisfied:
(A5) , for , ;
(A6) for ;
(A7) , for , .
Proof Problem (1.1) has a solution if and only if u solves the operator equation . We have shown is completely continuous by Lemma 2.4. We now verify that all the conditions of Lemma 3.1 are satisfied. The proof is divided into four steps.
So, (3.1) holds.
Step 3. We now show that (B4) in Lemma 3.1 is satisfied. If , by condition (A5), in the same way as in Step 1, we can obtain that . Hence, condition (B4) in Lemma 3.1 is satisfied.
Thus, condition (B5) in Lemma 3.1 is satisfied.
The proof is complete. □
From the proof of Theorem 3.1, it is easy to see that if conditions like (A5)-(A7) are appropriately combined, we can obtain an arbitrary number of positive solutions of problem (1.1).
Corollary 3.1 Assume that (A1)-(A4) hold. There exist constants , , , and the following conditions are satisfied:
(A8) , for , , ;
(A9) for , .
Then problem (1.1) possesses at least positive solutions.
Here , , , , , , , , .
Choose , , , , , .
We would like to thank the reviewers for their valuable comments and suggestions on the manuscript.
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