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Erbil Cetin, F. Serap Topal, "Existence Results for Solutions of Integral Boundary Value Problems on Time Scales", Abstract and Applied Analysis, vol. 2013, Article ID 708734, 7 pages, 2013. https://doi.org/10.1155/2013/708734
Existence Results for Solutions of Integral Boundary Value Problems on Time Scales
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.
The study of dynamic equations on time scales goes back to its founder Hilger . 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 , Yang , Ahmad et al. , and Atici and Guseinov  which motivate us to consider the problem (5)–(7). In , 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  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 . 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.
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).
Proof. In , 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 .
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
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 .
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.
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 ; 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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