Abstract and Applied Analysis

Volume 2013 (2013), Article ID 708734, 7 pages

http://dx.doi.org/10.1155/2013/708734

Research Article

Existence Results for Solutions of Integral Boundary Value Problems on Time Scales

Erbil Cetin and F. Serap Topal

Department of Mathematics, Ege University, Bornova, 35100 Izmir, Turkey

Received 27 November 2012; Accepted 12 March 2013

Academic Editor: Xinan Hao

Copyright © 2013 Erbil Cetin and F. Serap Topal. 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.

Abstract.

This paper deals with the existence of solutions for integral boundary value problems (IBVPs) on time scales. We provide sufficient conditions for the existence of solutions by using Schauder fixed point theorem in a cone. Existence result for this problem is also given by the method of upper and lower solutions.

1. Introduction

The study of dynamic equations on time scales goes back to its founder Hilger [1]. The main motive of the subject of dynamic equations on time scales is that they build bridges between continuous and discrete cases. We begin by presenting some basic definitions on time scale calculus.

A time scale is a nonempty closed subset of . It follows that the jump operators ,

are well defined. The point is left-dense, left-scattered, right-dense, and right-scattered if , , , and , respectively. If has a right-scattered minimum , define ; otherwise, set . If has a left-scattered maximum , define ; otherwise, set . A function is ld-continuous provided it is continuous at left dense points in , and its right-sided limit exists at right dense points in . For and , the delta derivative of at , denoted by , is the number with the property that given any , there is a neighborhood of such that

for all . For and , the nabla derivative of at , denoted by , is the number with the property that given any , there is a neighborhood of such that

for all .

A function is called a nabla antiderivative of provided that holds for all . We then define the nabla integral of by

For the details of basic notions connected to time scales, we refer the readers to the books [2, 3] and the papers [4, 5], which are useful references for calculus on time scales. Hereafter, we use the notation to indicate the time scale interval . The intervals , and are similarly defined.

Let be a time scale such that and . We are concerned with existence of solutions of the following integral boundary value problem (IBVP):

where and are continuous, , , , and , and .

We would like to mention some results of Khan [6], Yang [7], Ahmad et al. [8], and Atici and Guseinov [9] which motivate us to consider the problem (5)–(7). In [6], Khan considered the method of quasilinearization for the nonlinear boundary value problem with integral boundary conditions

where and    are continuous functions and are nonnegative constants. He obtained some results for the existence of solutions in an ordered interval generated by the lower and upper solutions of the boundary value problem. Our work will extend some known results which Khan obtained in [6] for integral boundary value problems to any time scales.

Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. Various problems in heat conduction, chemical engineering, underground water flow, thermoelasticity, population dynamics, and plasmaphysics [8, 10–12] can be reduced to the problems with integral boundary conditions. For more details of boundary value problems involving integral boundary conditions, see, for instance, [6, 13–19] and references therein. Also this type of problems includes two-point, three-point, and multipoint boundary value problems as special cases [4, 5, 7, 20] and the references therein.

In this section, we obtain some inequalities needed later for certain Green's function. In Section 2, the main tool used in the proof of existence of solutions for the IBVP (5)–(7), is a fixed point theorem in a cone, result due to Schauder [21]. Besides this, in this section, we prove the existence of solutions which will lie between the lower and upper solutions when the lower solution is under the upper solution.

To obtain a solution for the IBVP (5)–(7), we need a mapping whose kernel is the Green's function of the equation

with the integral boundary conditions (6)-(7).

In [9], Atici and Guseinov have shown that the solution of the nonhomogeneous equation (9) with the nonhomogeneous boundary condition

is given by

where

Since the Wronskian of two solutions of the corresponding homogeneous equation

under the initial conditions

is independent of , taking and in (13), we find , where and are the linearly independent solutions of (14) subject to conditions (15).

In our problem, we can easily see that the solution of the IBVP (5)–(7) is

where

The Green's function in this formula is

Lemma 1. Let and be the solutions of (14) under conditions (15). Then, is strictly increasing and positive on , and is strictly decreasing and positive on .

Proof. In [9], it is shown by Lemma 5.1 that the solutions and of the BVP (14)-(15) possess the following properties:

Suppose that there exists at least one such . From (19) and (14), we obtain

and integrating over , we get

Since , , , and , we obtain . Thus, we determine . This contradiction shows that the solution is strictly increasing and positive on as desired. Similar arguments can be applied for the proof of and on .

Lemma 2. The Green's function defined by (18) satisfies the inequality , for all and .

Lemma 3. Let be defined by (14). Then, there results are

where

Proof. For this purpose, we have four cases which are , ,  , and . We consider only two cases; the others can be shown similarly.

Case  1. Let . By using Lemma 1, we get

Case  2. Let and . By using Lemma 1, we get

From these cases, we hold .

Also, we get

2. Existence of Solutions

We will consider the Banach space , with the norm .

Theorem 4. Assume that the function is continuous with respect to . If satisfies

where satisfies

then the IBVP (5)–(7) has a solution .

Proof. Let . Note that is closed, bounded, and convex subset of to which the Schauder fixed point theorem is applicable. Define by

for . Obviously, the solutions of problem (5)–(7) are the fixed points of operator . In view of the continuity of the function , it follows that is continuous.

Now, we show that . Let . Consider

for every . This implies that .

Thus, all functions which belong to are equi-bounded and . The uniform continuity of the and implies that all functions in are equi-continuous. So, by Arzela-Ascoli theorem, the operator is compact. Hence, has a fixed point in by Schauder fixed point theorem

Corollary 5. If is continuous and bounded on , then the IBVP (5)–(7) has a solution.

Let us define the set by

For any , we define the sector by

Definition 6. A real valued function on is a lower solution for IBVP (5)–(7) if

Similarly, real valued function on is an upper solution for IBVP (5)–(7) if the inequalities in (34) are satisfied in the reverse direction for .

Theorem 7. Assume that and are, respectively, lower and upper solutions of (5)–(7). If is continuous, () are continuously differentiable, is decreasing in for , and , then

Proof. Define . For the sake of contradiction, assume that the result is not true on . Then, the function has a positive maximum at and for . So, we have ,  , and ; hence,

a contradiction. If , then and . Using the boundary conditions, we have

On the other hand, using the mean value theorem and the assumption on , we obtain

where , a contradiction. Hence,

Now, we state and prove the existence and uniqueness of solutions in an ordered interval generated by the lower and upper solutions of the boundary value problem.

Theorem 8. Assume that and are, respectively, lower and upper solutions of (5)–(7) such that , . If and () are continuously differentiable and , then there exists a solution of (5)–(7) such that

Proof. Define the following modifications of and ,  :

for , and

Consider the modified problem

As and () are continuous and bounded, it follows that the boundary value problem (43) has a solution. Further, note that

which imply that is a lower solution of (43). Similarly, is an upper solution of (43). We need to show that any solution of (43) is such that , . Assume that is not true on . Then, the function has a positive maximum at and for . So, we have , and by Lemma  6.17 in [3]; hence,

a contradiction. If , then and , but then the boundary conditions and the nondecreasing property of give

If , then , and, hence, , a contradiction. If , then , which implies , a contradiction. Hence, and and , another contradiction.

We can illustrate our results in the following examples.

Example 9. Let be any time scales such that and . We consider the following IBVP:

From (15), we have and .

We calculate easily

For or , .

We have

where . Since for , there exists a positive real number which satisfies for the positive Q, and then all condition in Theorem 4 are satisfied. Therefore, the IBVP (5)–(7) has a solution .

Example 10. Let . We consider the following IBVP:

For , we get

Thus, is the lower solution.

For , we get

Thus, is the upper solution.

Theorem 8 implies that IBVP has a solution such that

Example 11. Let be any time scales such that and . We consider the following IBVP:

Let . We can easily see that satisfies all conditions of a lower solution. Hence, is the lower solution.

Let , and

We get . From

we get , and from

we get . Thus, is the upper solution.

Theorem 8 implies that IBVP has a solution such that

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