## Complex Boundary Value Problems of Nonlinear Differential Equations: Theory, Computational Methods, and Applications

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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

**Academic Editor:**Xinan Hao

#### 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

#### References

- S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
*Results in Mathematics*, vol. 18, no. 1-2, pp. 18–56, 1990. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Bohner and A. Peterson,
*Dynamic Equations on Time Scales: An Introduction with Applications*, Birkhäuser, Boston, Mass, USA, 2001. View at: Publisher Site | MathSciNet - M. Bohner and A. Peterson,
*Advances in Dynamic Equations on Time Scales*, Birkhäuser, Boston, Mass, USA, 2003. View at: Publisher Site | MathSciNet - F. A. Davidson and B. P. Rynne, “The formulation of second-order boundary value problems on time scales,”
*Advances in Difference Equations*, vol. 2006, Article ID 31430, 10 pages, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Erbe and A. Peterson, “Green's functions and comparison theorems for differential equations on measure chains,”
*Dynamics of Continuous, Discrete and Impulsive Systems*, vol. 6, no. 1, pp. 121–137, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. A. Khan, “The generalized method of quasilinearization and nonlinear boundary value problems with integral boundary conditions,”
*Electronic Journal of Qualitative Theory of Differential Equations*, vol. 10, pp. 1–9, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Z. Yang, “Existence and nonexistence results for positive solutions of an integral boundary value problem,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 65, no. 8, pp. 1489–1511, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Ahmad, A. Alsaedi, and B. S. Alghamdi, “Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions,”
*Nonlinear Analysis: Real World Applications*, vol. 9, no. 4, pp. 1727–1740, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. M. Atici and G. Sh. Guseinov, “On Green's functions and positive solutions for boundary value problems on time scales,”
*Journal of Computational and Applied Mathematics*, vol. 141, no. 1-2, pp. 75–99, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. W. Batten Jr., “Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations,”
*Mathematics of Computation*, vol. 17, pp. 405–413, 1963. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. R. Cannon, S. P. Esteva, and J. van der Hoek, “A Galerkin procedure for the diffusion equation subject to the specification of mass,”
*SIAM Journal on Numerical Analysis*, vol. 24, no. 3, pp. 499–515, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. I. Kamynin, “A boundary value problem in the theory of heat conduction with boundary condition,”
*USSR Computational Mathematics and Mathematical Physics*, vol. 4, no. 6, pp. 33–59, 1964. View at: Publisher Site | Google Scholar - A. Bouziani and N.-E. Benouar, “Mixed problem with integral conditions for a third order parabolic equation,”
*Kobe Journal of Mathematics*, vol. 15, no. 1, pp. 47–58, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Denche and A. L. Marhoune, “Mixed problem with integral boundary condition for a high order mixed type partial differential equation,”
*Journal of Applied Mathematics and Stochastic Analysis*, vol. 16, no. 1, pp. 69–79, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. M. Gallardo, “Second-order differential operators with integral boundary conditions and generation of analytic semigroups,”
*The Rocky Mountain Journal of Mathematics*, vol. 30, no. 4, pp. 1265–1292, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Jankowski, “Differential equations with integral boundary conditions,”
*Journal of Computational and Applied Mathematics*, vol. 147, no. 1, pp. 1–8, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. V. Kartynnik, “A three-point mixed problem with an integral condition with respect to the space variable for second-order parabolic equations,”
*Differential Equations*, vol. 26, no. 9, pp. 1160–1166, 1990. View at: Google Scholar | MathSciNet - L. Liu, X. Hao, and Y. Wu, “Positive solutions for singular second order differential equations with integral boundary conditions,”
*Mathematical and Computer Modelling*, vol. 57, no. 3-4, pp. 836–847, 2013. View at: Publisher Site | Google Scholar - X. Zhang and W. Ge, “Symmetric positive solutions of boundary value problems with integral boundary conditions,”
*Applied Mathematics and Computation*, vol. 219, no. 8, pp. 3553–3564, 2012. View at: Publisher Site | Google Scholar | MathSciNet - S. A. Brykalov, “A second-order nonlinear problem with two-point and integral boundary conditions,”
*Georgian Mathematical Journal*, vol. 1, no. 3, pp. 243–249, 1994. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. A. Krasnosel'skii,
*Positive Solutions of Operator Equations*, P. Noordhoff Ltd., Groningen, The Netherlands, 1964. View at: MathSciNet

#### Copyright

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.