Boundary Value Problems

, 2009:491952 | Cite as

Existence of Solutions for Fourth-Order Four-Point Boundary Value Problem on Time Scales

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Abstract

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.

Keywords

Unique Solution Boundary Value Problem Dynamic Equation Green Function Fixed Point Theorem 

1. Introduction

Very recently, Karaca [1] investigated the following fourth-order four-point boundary value problem on time scales:

for Open image in new window Open image in new window and Open image in new window And the author made the following assumptions:

(A1) Open image in new window and Open image in new window

(A2) Open image in new window If Open image in new window then Open image in new window

The following key lemma is provided in [1].

Lemma 1.1 (see [1, Lemma  2.5]).

Assume that conditions ( Open image in new window ) and ( Open image in new window ) are satisfied. If Open image in new window then the boundary value problem
has a unique solution
Unfortunately, this lemma is wrong. Without considering the whole interval Open image in new window the author only considers Open image in new window in the Green's function Open image in new window Thus, the expression of Open image in new window (1.3) which is a solution to BVP (1.2) is incorrect. In fact, if one takes Open image in new window Open image in new window Open image in new window Open image in new window Open image in new window Open image in new window Open image in new window then (1.1) reduces to the following boundary value problem:
The counterexample is given by [2], from which one can see clearly that [1, Lemma  2.5] is wrong. If one takes Open image in new window , here Open image in new window is a constant, then (1.1) reduces to the following fourth-order four-point boundary value problem on time scales:

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.

2. Preliminaries

The study of dynamic equations on time scales goes back to its founder Hilger [3] 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, 5, 6]. And an increasing interest in studying the existence of solutions to dynamic equations on time scales is observed, for example, see [7, 8, 9, 10, 11, 12, 13, 14, 15, 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, 18, 19] which are excellent works for the calculus of time scales.

A time scale Open image in new window is an arbitrary nonempty closed subset of real numbers Open image in new window . The operators Open image in new window and Open image in new window from Open image in new window to Open image in new window

are called the forward jump operator and the backward jump operator, respectively.

For all Open image in new window we assume throughout that Open image in new window has the topology that it inherits from the standard topology on Open image in new window The notations Open image in new window Open image in new window and so on, will denote time-scale intervals

where Open image in new window with Open image in new window

Definition 2.1.

Fix Open image in new window Let Open image in new window Then we define Open image in new window to be the number (if it exists) with the property that given Open image in new window there is a neighborhood Open image in new window of Open image in new window with

Then Open image in new window is called derivative of Open image in new window

Definition 2.2.

If Open image in new window then we define the integral by
We say that a function Open image in new window is regressive provided
where Open image in new window which is called graininess function. If Open image in new window is a regressive function, then the generalized exponential function Open image in new window is defined by
for Open image in new window Open image in new window is the cylinder transformation, which is defined by
Let Open image in new window be two regressive functions, then define

The generalized function Open image in new window has then the following properties.

Lemma 2.3 (see [18]).

Assume that Open image in new window are two regressive functions, then

(i) Open image in new window and Open image in new window

(ii) Open image in new window

(iii) Open image in new window

(iv) Open image in new window

(v) Open image in new window

(vi) Open image in new window

(vii) Open image in new window

The following well-known fixed point theorem will play a very important role in proving our main result.

Theorem 2.4 (see [20]).

Let Open image in new window be a Banach space, and let Open image in new window be completely continuous. Assume that Open image in new window is a bounded linear operator such that Open image in new window is not an eigenvalue of Open image in new window and

Then Open image in new window has a fixed point in Open image in new window

Throughout this paper, let Open image in new window be endowed with the norm by

where Open image in new window And we make the following assumptions:

() Open image in new window and Open image in new window

() Open image in new window and Open image in new window Open image in new window

() Open image in new window

Set

For convenience, we denote

First, we present two lemmas about the calculus on Green functions which are crucial in our main results.

Lemma 2.5.

Assume that Open image in new window and Open image in new window are satisfied. If Open image in new window then Open image in new window is a solution of the following boundary value problem (BVP):
if and only if
where the Green's function of (2.13) is as follows:

where Open image in new window are given as (2.12), respectively.

Proof.

If Open image in new window is a solution of (2.13), setting
then it follows from the first equation of (2.13) that
Similarly, by (2.18), we have
Then substituting (2.18) into (2.19), we get for each Open image in new window that
Substituting this expression for Open image in new window into the boundary conditions of (2.13). By some calculations, we get
Then substituting (2.21) into (2.20), we get
By interchanging the order of integration and some rearrangement of (2.22), we obtain

Thus, we obtain (2.14) consequently.

On the other hand, if Open image in new window satisfies (2.14), then direct differentiation of (2.14) yields

And it is easy to know that Open image in new window and Open image in new window satisfies (2.13).

Corollary 2.6.

If Open image in new window then BVP (2.13) reduces to the following problem:
From Lemma 2.5, BVP (2.25) has a unique solution
where the Green's function of (2.25) is as follows:

Proof.

Substituting this expression for Open image in new window into the boundary conditions of (2.25). By some calculations, we obtain
where Open image in new window is given as (2.28). Then substituting (2.31) into (2.30), we get

where Open image in new window 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 Open image in new window take Open image in new window Open image in new window Open image in new window Open image in new window we get the following result.

Corollary 2.7.

The following boundary value problem:
has a unique solution
where the Green's function of (2.33) is as follows:
After some rearrangement of (2.35), one obtains

Remark 2.8.

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].

Lemma 2.9.

Assume that conditions ( Open image in new window )–( Open image in new window ) are satisfied. If Open image in new window then boundary value problem
has a unique solution

and Open image in new window is given in Lemma 2.5.

Proof.

Consider the following boundary value problem:

The Green's function associated with the BVP (2.41) is Open image in new window . This completes the proof.

Remark 2.10.

In [1, Lemma  2.5], the solution of (1.2) is defined as

where Open image in new window and Open image in new window are given as (1.4) and (1.5), respectively. In fact, Open image in new window is incorrect. Thus, we give the right form of Open image in new window as the special case Open image in new window in our Lemma 2.9.

3. Main Results

Theorem 3.1.

Assume ( Open image in new window )–( Open image in new window ) are satisfied. Moreover, suppose that the following condition is satisfied:

and there exists a continuous nonnegative function Open image in new window such that Open image in new window If

then BVP (1.8) has a solution Open image in new window .

Proof.

where Open image in new window is given by (2.40). Then by Lemmas 2.5 and 2.9, it is clear that the fixed points of Open image in new window are the solutions to the boundary value problem (1.8). First of all, we claim that Open image in new window is a completely continuous operator, which is divided into 3 steps.

Step 1 ( Open image in new window is continuous).

Let Open image in new window be a sequence such that Open image in new window then we have

Since Open image in new window are continuous, we have Open image in new window which yields Open image in new window That is, Open image in new window is continuous.

Step 2 ( Open image in new window maps bounded sets into bounded sets in Open image in new window ).

By virtue of the continuity of Open image in new window and Open image in new window , we conclude that Open image in new window is bounded uniformly, and so Open image in new window is a bounded set.

Step 3 ( Open image in new window maps bounded sets into equicontinuous sets of Open image in new window ).

The right hand side tends to uniformly zero as Open image in new window Consequently, Steps 1–3 together with the Arzela-Ascoli theorem show that Open image in new window is completely continuous.

Now we consider the following boundary value problem:

Obviously, Open image in new window is a completely continuous bounded linear operator. Moreover, the fixed point of Open image in new window is a solution of the BVP (3.8) and conversely.

We are now in the position to claim that Open image in new window is not an eigenvalue of Open image in new window

If Open image in new window and Open image in new window then (3.8) has no nontrivial solution.

If Open image in new window or Open image in new window suppose that the BVP (3.8) has a nontrivial solution Open image in new window and Open image in new window then we have
which yields
On the other hand, we have

From the above discussion (3.11) and (3.12), we have Open image in new window . This contradiction implies that boundary value problem (3.8) has no trivial solution. Hence, Open image in new window is not an eigenvalue of Open image in new window

At last, we show that

Set Open image in new window and select Open image in new window such that Open image in new window

In a similar way, we also conclude that for any Open image in new window
Therefore,
On the other hand, we get
Combining (3.18) with (3.19), we have

Theorem 2.4 guarantees that boundary value problem (1.8) has a solution Open image in new window It is obvious that Open image in new window when Open image in new window for some Open image in new window In fact, if Open image in new window then Open image in new window will lead to a contradiction, which completes the proof.

4. Application

We give an example to illustrate our result.

Example 4.1.

Consider the fourth-order four-pint boundary value problem

Since Open image in new window for each Open image in new window we have the following.

By simple calculation we have
On the other hand, we notice that

That is, Open image in new window is satisfied. Thus, Theorem 3.1 guarantees that (4.1) has at least one nontrivial solution Open image in new window

Notes

Acknowledgments

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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Copyright information

© Dandan Yang et al. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.School of Mathematical ScienceYangzhou UniversityYangzhouChina
  2. 2.Department of MathematicsHuaiyin Normal UniversityHuaianChina

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