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:
for and And the author made the following assumptions:
(A2) If then
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.
A time scale is an arbitrary nonempty closed subset of real numbers . The operators and from to
are called the forward jump operator and the backward jump operator, respectively.
For all we assume throughout that has the topology that it inherits from the standard topology on The notations and so on, will denote time-scale intervals
Then is called derivative of
The generalized function has then the following properties.
Lemma 2.3 (see ).
Assume that are two regressive functions, then
The following well-known fixed point theorem will play a very important role in proving our main result.
Theorem 2.4 (see ).
Then has a fixed point in
where And we make the following assumptions:
For convenience, we denote
First, we present two lemmas about the calculus on Green functions which are crucial in our main results.
where are given as (2.12), respectively.
Thus, we obtain (2.14) consequently.
And it is easy to know that and satisfies (2.13).
where are as in (2.29), respectively. By some rearrangement of (2.32), we obtain (2.26) consequently.
From the proof of Corollary 2.6, if take we get the following result.
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].
and is given in Lemma 2.5.
The Green's function associated with the BVP (2.41) is . This completes the proof.
where and are given as (1.4) and (1.5), respectively. In fact, is incorrect. Thus, we give the right form of as the special case in our Lemma 2.9.
3. Main Results
Assume ( )–( ) are satisfied. Moreover, suppose that the following condition is satisfied:
then BVP (1.8) has a solution .
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.
Step 1 ( is continuous).
Since are continuous, we have which yields That is, is continuous.
Step 2 ( maps bounded sets into bounded sets in ).
By virtue of the continuity of and , we conclude that is bounded uniformly, and so is a bounded set.
Step 3 ( maps bounded sets into equicontinuous sets of ).
The right hand side tends to uniformly zero as Consequently, Steps 1–3 together with the Arzela-Ascoli theorem show that is completely continuous.
Obviously, is a completely continuous bounded linear operator. Moreover, the fixed point of is a solution of the BVP (3.8) and conversely.
We are now in the position to claim that is not an eigenvalue of
If and then (3.8) has no nontrivial solution.
From the above discussion (3.11) and (3.12), we have . This contradiction implies that boundary value problem (3.8) has no trivial solution. Hence, is not an eigenvalue of
Set and select such that
Theorem 2.4 guarantees that boundary value problem (1.8) has a solution It is obvious that when for some In fact, if then will lead to a contradiction, which completes the proof.
We give an example to illustrate our result.
Since for each we have the following.
That is, is satisfied. Thus, Theorem 3.1 guarantees that (4.1) has at least one nontrivial solution
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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