Existence of Solutions for Fourth-Order Four-Point Boundary Value Problem on Time Scales
© Dandan Yang et al. 2009
Received: 11 April 2009
Accepted: 28 July 2009
Published: 19 August 2009
We present an existence result for fourth-order four-point boundary value problem on time scales. Our analysis is based on a fixed point theorem due to Krasnoselskii and Zabreiko.
Very recently, Karaca  investigated the following fourth-order four-point boundary value problem on time scales:
The following key lemma is provided in .
Lemma 1.1 (see [1, Lemma 2.5]).
The purpose of this paper is to establish some existence criteria of solution for BVP (1.8) which is a special case of (1.1). The paper is organized as follows. In Section 2, some basic time-scale definitions are presented and several preliminary results are given. In Section 3, by employing a fixed point theorem due to Krasnoselskii and Zabreiko, we establish existence of solutions criteria for BVP (1.8). Section 4 is devoted to an example illustrating our main result.
The study of dynamic equations on time scales goes back to its founder Hilger  and it is a new area of still fairly theoretical exploration in mathematics. In the recent years boundary value problem on time scales has received considerable attention [4–6]. And an increasing interest in studying the existence of solutions to dynamic equations on time scales is observed, for example, see [7–16].
For convenience, we first recall some definitions and calculus on time scales, so that the paper is self-contained. For the further details concerning the time scales, please see [17–19] which are excellent works for the calculus of time scales.
are called the forward jump operator and the backward jump operator, respectively.
Lemma 2.3 (see ).
The following well-known fixed point theorem will play a very important role in proving our main result.
Theorem 2.4 (see ).
For convenience, we denote
First, we present two lemmas about the calculus on Green functions which are crucial in our main results.
Thus, we obtain (2.14) consequently.
Green function (2.37) associated with BVP (2.33) which is a special case of (2.13) is coincident with part of [21, Lemma 1].
3. Main Results
where is given by (2.40). Then by Lemmas 2.5 and 2.9, it is clear that the fixed points of are the solutions to the boundary value problem (1.8). First of all, we claim that is a completely continuous operator, which is divided into 3 steps.
We give an example to illustrate our result.
The authors would like to thank the referees for their valuable suggestions and comments. This work is supported by the National Natural Science Foundation of China (10771212) and the Natural Science Foundation of Jiangsu Education Office (06KJB110010).
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