We present an existence result for fourthorder fourpoint boundary value problem on time scales. Our analysis is based on a fixed point theorem due to Krasnoselskii and Zabreiko.
1. Introduction
Very recently, Karaca [1] investigated the following fourthorder fourpoint boundary value problem on time scales:
for and And the author made the following assumptions:
() and
() If then
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 fourthorder fourpoint 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 timescale 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 [4–6]. 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 selfcontained. 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 timescale intervals
where with
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
(i) and
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
The following wellknown 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
where And we make the following assumptions:
() and
() and
()
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 and are satisfied. If then 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 are given as (2.12), respectively.
Proof.
If is a solution of (2.13), setting
then it follows from the first equation of (2.13) that
Multiplying (2.17) by and integrating from to we get
Similarly, by (2.18), we have
Then substituting (2.18) into (2.19), we get for each that
Substituting this expression for 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 satisfies (2.14), then direct differentiation of (2.14) yields
And it is easy to know that and satisfies (2.13).
Corollary 2.6.
If 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:
where
Proof.
If is a solution of (2.25), take then Hence, from (2.20) we have
Substituting this expression for into the boundary conditions of (2.25). By some calculations, we obtain
where is given as (2.28). Then substituting (2.31) into (2.30), we get
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:
has a unique solution
where the Green's function of (2.33) is as follows:
where
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 ()–() are satisfied. If then boundary value problem
has a unique solution
where
and is given in Lemma 2.5.
Proof.
Consider the following boundary value problem:
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
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 ).
Let then
The right hand side tends to uniformly zero as Consequently, Steps 1–3 together with the ArzelaAscoli 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
which yields
On the other hand, we have
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
By then for any there exist a such that
Set and select such that
Denote
Thus for any and when it follows that
In a similar way, we also conclude that for any
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 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 fourthorder fourpint 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
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

Karaca, IY: Fourthorder fourpoint boundary value problem on time scales. Applied Mathematics Letters. 21(10), 1057–1063 (2008). Publisher Full Text

Bai, C, Yang, D, Zhu, H: Existence of solutions for fourth order differential equation with fourpoint boundary conditions. Applied Mathematics Letters. 20(11), 1131–1136 (2007). Publisher Full Text

Hilger, S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics. 18(12), 18–56 (1990)

Atici, FM, Guseinov, GSh: On Green's functions and positive solutions for boundary value problems on time scales. Journal of Computational and Applied Mathematics. 141(12), 75–99 (2002). Publisher Full Text

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. 296(1), 65–73 (2004). Publisher Full Text

Boey, KL, Wong, PJY: Positive solutions of twopoint right focal boundary value problems on time scales. Computers & Mathematics with Applications. 52(34), 555–576 (2006). PubMed Abstract  Publisher Full Text

Stehlík, P: Periodic boundary value problems on time scales. Advances in Difference Equations.(1), 81–92 (2005)

Sun, JP: Existence of solution and positive solution of BVP for nonlinear thirdorder dynamic equation. Nonlinear Analysis: Theory, Methods & Applications. 64(3), 629–636 (2006). PubMed Abstract  Publisher Full Text

Sun, HR, Li, WT: Existence theory for positive solutions to onedimensional Laplacian boundary value problems on time scales. Journal of Differential Equations. 240(2), 217–248 (2007). Publisher Full Text

Su, H, Zhang, M: Solutions for higherorder dynamic equations on time scales. Applied Mathematics and Computation. 200(1), 413–428 (2008). Publisher Full Text

Wang, DB, Sun, JP: Existence of a solution and a positive solution of a boundary value problem for a nonlinear fourthorder dynamic equation. Nonlinear Analysis: Theory, Methods & Applications. 69(56), 1817–1823 (2008). PubMed Abstract  Publisher Full Text

Yaslan, İ: Existence results for an evenorder boundary value problem on time scales. Nonlinear Analysis: Theory, Methods & Applications. 70(1), 483–491 (2009). PubMed Abstract  Publisher Full Text

Su, YH: Multiple positive pseudosymmetric solutions of Laplacian dynamic equations on time scales. Mathematical and Computer Modelling. 49(78), 1664–1681 (2009). Publisher Full Text

Henderson, J, Tisdell, CC, Yin, WKC: Uniqueness implies existence for threepoint boundary value problems for dynamic equations. Applied Mathematics Letters. 17(12), 1391–1395 (2004). Publisher Full Text

Tian, Y, Ge, W: Existence and uniqueness results for nonlinear firstorder threepoint boundary value problems on time scales. Nonlinear Analysis: Theory, Methods & Applications. 69(9), 2833–2842 (2008). PubMed Abstract  Publisher Full Text

Anderson, DR, Smyrlis, G: Solvability for a thirdorder threepoint BVP on time scales. Mathematical and Computer Modelling. 49(910), 1994–2001 (2009). Publisher Full Text

Agarwal, R, Bohner, M, O'Regan, D, Peterson, A: Dynamic equations on time scales: a survey. Journal of Computational and Applied Mathematics. 141(12), 1–26 (2002). Publisher Full Text

Bohner, M, Peterson, A: Dynamic Equations on Time Scales: An Introduction with Applications,p. x+358. Birkhäuser, Boston, Mass, USA (2001)

Bohner, M, Peterson, A: Advances in Dynamic Equations on Time Scales,p. xii+348. Birkhäuser, Boston, Mass, USA (2003)

Krasnoselskiĭ, MA, Zabreĭko, PP: Geometrical Methods of Nonlinear Analysis, Grundlehren der Mathematischen Wissenschaften,p. xix+409. Springer, Berlin, Germany (1984)

Chai, G: Existence of positive solutions for secondorder boundary value problem with one parameter. Journal of Mathematical Analysis and Applications. 330(1), 541–549 (2007). Publisher Full Text