Boundary Value Problems

, 2009:491952

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

Open Access
Research Article

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 and And the author made the following assumptions:

The following key lemma is provided in [1].

Lemma 1.1 (see [1, Lemma  2.5]).

Assume that conditions () and () are satisfied. If then the boundary value problem
has a unique solution
where
Here , and are given as follows:
Unfortunately, this lemma is wrong. Without considering the whole interval the author only considers in the Green's function Thus, the expression of (1.3) which is a solution to BVP (1.2) is incorrect. In fact, if one takes 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 , here 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 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

Definition 2.1.

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

Then is called derivative of

Definition 2.2.

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

The generalized function has then the following properties.

Lemma 2.3 (see [18]).

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 [20]).

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

Then has a fixed point in

Throughout this paper, let be endowed with the norm by
(2.10)

where And we make the following assumptions:

Set

(2.11)

For convenience, we denote

(2.12)

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

Lemma 2.5.

Assume that and are satisfied. If then is a solution of the following boundary value problem (BVP):
(2.13)
if and only if
(2.14)
where the Green's function of (2.13) is as follows:
(2.15)

where are given as (2.12), respectively.

Proof.

If is a solution of (2.13), setting
(2.16)
then it follows from the first equation of (2.13) that
(2.17)
Multiplying (2.17) by and integrating from to we get
(2.18)
Similarly, by (2.18), we have
(2.19)
Then substituting (2.18) into (2.19), we get for each that
(2.20)
Substituting this expression for into the boundary conditions of (2.13). By some calculations, we get
(2.21)
Then substituting (2.21) into (2.20), we get
(2.22)
By interchanging the order of integration and some rearrangement of (2.22), we obtain
(2.23)

Thus, we obtain (2.14) consequently.

On the other hand, if satisfies (2.14), then direct differentiation of (2.14) yields
(2.24)

And it is easy to know that and satisfies (2.13).

Corollary 2.6.

If then BVP (2.13) reduces to the following problem:
(2.25)
From Lemma 2.5, BVP (2.25) has a unique solution
(2.26)
where the Green's function of (2.25) is as follows:
(2.27)
where
(2.28)
(2.29)

Proof.

If is a solution of (2.25), take then Hence, from (2.20) we have
(2.30)
Substituting this expression for into the boundary conditions of (2.25). By some calculations, we obtain
(2.31)
where is given as (2.28). Then substituting (2.31) into (2.30), we get
(2.32)

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.

Corollary 2.7.

The following boundary value problem:
(2.33)
has a unique solution
(2.34)
where the Green's function of (2.33) is as follows:
(2.35)
where
(2.36)
After some rearrangement of (2.35), one obtains
(2.37)

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 ()–() are satisfied. If then boundary value problem
(2.38)
has a unique solution
(2.39)
where
(2.40)

and is given in Lemma 2.5.

Proof.

Consider the following boundary value problem:
(2.41)

The Green's function associated with the BVP (2.41) is . This completes the proof.

Remark 2.10.

In [1, Lemma  2.5], the solution of (1.2) is defined as
(2.42)

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

Theorem 3.1.

Assume ()–() are satisfied. Moreover, suppose that the following condition is satisfied:

where are continuous, with
and there exists a continuous nonnegative function such that If
where

then BVP (1.8) has a solution .

Proof.

Define an operator by

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).

Let be a sequence such that then we have

Since are continuous, we have which yields That is, is continuous.

Step 2 ( maps bounded sets into bounded sets in ).

Let be a bounded set. Then, for and any we have

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.

Now we consider the following boundary value problem:
Define

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.

If or suppose that the BVP (3.8) has a nontrivial solution and then we have
(3.10)
which yields
(3.11)
On the other hand, we have
(3.12)

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

At last, we show that
(3.13)
By then for any there exist a such that
(3.14)

Set and select such that

Denote
(3.15)
Thus for any and when it follows that
(3.16)
In a similar way, we also conclude that for any
(3.17)
Therefore,
(3.18)
On the other hand, we get
(3.19)
Combining (3.18) with (3.19), we have
(3.20)

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.

4. Application

We give an example to illustrate our result.

Example 4.1.

Consider the fourth-order four-pint boundary value problem
Notice that To show that (4.1) has at least one nontrivial solution we apply Theorem 3.1 with and Obviously, ()–() are satisfied. And

Since for each we have the following.

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

That is, is satisfied. Thus, Theorem 3.1 guarantees that (4.1) has at least one nontrivial solution

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).

References

1. 1.
Karaca IY: Fourth-order four-point boundary value problem on time scales. Applied Mathematics Letters 2008, 21(10):1057–1063. 10.1016/j.aml.2008.01.001
2. 2.
Bai C, Yang D, Zhu H: Existence of solutions for fourth order differential equation with four-point boundary conditions. Applied Mathematics Letters 2007, 20(11):1131–1136. 10.1016/j.aml.2006.11.013
3. 3.
Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990, 18(1–2):18–56.
4. 4.
Atici FM, Guseinov GSh: On Green's functions and positive solutions for boundary value problems on time scales. Journal of Computational and Applied Mathematics 2002, 141(1–2):75–99. 10.1016/S0377-0427(01)00437-X
5. 5.
Benchohra M, Ntouyas SK, Ouahab A: Existence results for second order boundary value problem of impulsive dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2004, 296(1):65–73. 10.1016/j.jmaa.2004.02.057
6. 6.
Boey KL, Wong PJY: Positive solutions of two-point right focal boundary value problems on time scales. Computers & Mathematics with Applications 2006, 52(3–4):555–576. 10.1016/j.camwa.2006.08.025
7. 7.
Stehlík P: Periodic boundary value problems on time scales. Advances in Difference Equations 2005, (1):81–92.Google Scholar
8. 8.
Sun J-P: Existence of solution and positive solution of BVP for nonlinear third-order dynamic equation. Nonlinear Analysis: Theory, Methods & Applications 2006, 64(3):629–636. 10.1016/j.na.2005.04.046
9. 9.
Sun H-R, Li W-T: Existence theory for positive solutions to one-dimensional Open image in new window-Laplacian boundary value problems on time scales. Journal of Differential Equations 2007, 240(2):217–248. 10.1016/j.jde.2007.06.004
10. 10.
Su H, Zhang M: Solutions for higher-order dynamic equations on time scales. Applied Mathematics and Computation 2008, 200(1):413–428. 10.1016/j.amc.2007.11.022
11. 11.
Wang D-B, Sun J-P: Existence of a solution and a positive solution of a boundary value problem for a nonlinear fourth-order dynamic equation. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(5–6):1817–1823. 10.1016/j.na.2007.07.028
12. 12.
Yaslan İ: Existence results for an even-order boundary value problem on time scales. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(1):483–491. 10.1016/j.na.2007.12.019
13. 13.
Su Y-H: Multiple positive pseudo-symmetric solutions of Open image in new window-Laplacian dynamic equations on time scales. Mathematical and Computer Modelling 2009, 49(7–8):1664–1681. 10.1016/j.mcm.2008.10.010
14. 14.
Henderson J, Tisdell CC, Yin WKC: Uniqueness implies existence for three-point boundary value problems for dynamic equations. Applied Mathematics Letters 2004, 17(12):1391–1395. 10.1016/j.am1.2003.08.015
15. 15.
Tian Y, Ge W: Existence and uniqueness results for nonlinear first-order three-point boundary value problems on time scales. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(9):2833–2842. 10.1016/j.na.2007.08.054
16. 16.
Anderson DR, Smyrlis G: Solvability for a third-order three-point BVP on time scales. Mathematical and Computer Modelling 2009, 49(9–10):1994–2001. 10.1016/j.mcm.2008.11.009
17. 17.
Agarwal R, Bohner M, O'Regan D, Peterson A: Dynamic equations on time scales: a survey. Journal of Computational and Applied Mathematics 2002, 141(1–2):1–26. 10.1016/S0377-0427(01)00432-0
18. 18.
Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.
19. 19.
Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.
20. 20.
Krasnoselskiĭ MA, Zabreĭko PP: Geometrical Methods of Nonlinear Analysis, Grundlehren der Mathematischen Wissenschaften. Volume 263. Springer, Berlin, Germany; 1984:xix+409.Google Scholar
21. 21.
Chai G: Existence of positive solutions for second-order boundary value problem with one parameter. Journal of Mathematical Analysis and Applications 2007, 330(1):541–549. 10.1016/j.jmaa.2006.07.092