International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 465867, 7 pages

doi:10.1155/2012/465867

Research Article

Solution Matching for a Second Order Boundary Value Problem on Time Scales

Aprillya Lanz¹ and Ana Tameru²

¹Department of Mathematics and Computer Science, Virginia Military Institute, Lexington, VA 24450, USA

²Department of Mathematics and Computer Science, Alabama State University, 915 S. Jackson Street, Montgomery, AL 36101, USA

Received 7 June 2012; Revised 9 August 2012; Accepted 9 August 2012

Academic Editor: Palle E. Jorgensen

Copyright © 2012 Aprillya Lanz and Ana Tameru. 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.

Let be a time scale such that . We will show the existence and uniqueness of solutions for the second-order boundary value problem , by matching a solution of the first equation satisfying boundary conditions on with a solution of the first equation satisfying boundary conditions on , where .

1. Introduction

The result discussed in this paper was inspired by the solution matching technique that was first introduced by Bailey et al. [1]. In their work, they dealt with the existence and uniqueness of solutions for the second-order conjugate boundary value problems

As shown in their work, the uniqueness and existence of the solutions of (1.1), (1.2) were obtained by matching a solution of (1.1) satisfying the boundary condition

with a solution of (1.1) satisfying the boundary condition

where .

Since the initial work by Bailey et al., there have been many studies utilizing the solution matching technique on boundary value problems, see for example, Rao et al. [2], Henderson [3, 4], Henderson and Taunton [5].

Existence and Uniqueness for solutions of boundary value problems have quite a history for ordinary differential equations as well for difference equations, we mentioned papers of Barr and Sherman [6], Hartman [7], Henderson [8, 9], Henderson and Yin [10], Moorti and Garner [11], Rao et al. [12] and many others.

While many of the work mentioned above considered boundary value problems for differential and difference equations, our study applies the solution matching technique to obtain a solution to a similar boundary value problem (1.1), (1.2) on a time scale. The theory of time scales was first introduced by Hilger [13] in 1990 to unify results in differential and difference equations. Since then, there has been much activity focused on dynamic equations on time scales, with a good deal of this activity devoted to boundary value problems. Efforts have been made in the context of time scales, in establishing that some results for boundary value problems for ordinary differential equations and their discrete analogues are special cases of more general results on time scales. For the context of dynamic equations on time scales, we mention the results by Bohner and Peterson [14, 15], Chyan [16], Henderson [4], and Henderson and Yin [17].

In this work, is assumed to be a nonempty closed subset of with and . We shall also use the convention on notation that for each interval of ,

For readers' convenience, we state a few definitions which are basic to the calculus on the time scale . The forward jump operator is defined by

If , is said to be right-scattered, whereas, if , is said to be right-dense. The backward jump operator is defined by

If , is said to be left-scattered, and if , then is said to be left-dense. If and , then the delta derivative of g at t, , is defined to be the number (provided that it exists), with the property that, given any , there is a neighborhood of , such that

for all . In this definition, , where this set is derived from the time scale as follows: if has a left-scattered maximum , then . Otherwise, we define .

We say that the function has a generalized zero at if or if . In the latter case, we would say the generalized zero is in the real interval .

Theorem 1.1 (Mean Value Theorem). If is continuous and has a generalized zero at and , , then there exists a point such that has a generalized zero at .

Let be a time scale such that . In this paper, we are concerned with the existence and uniqueness of solutions of boundary value problems on the interval for the second-order delta derivative equation

satisfying the boundary conditions,

where and . Throughout this paper, we will assume

(A1) is a real-valued continuous function defined on .

We obtain solutions by matching a solution of (1.9) satisfying boundary conditions on to a solution of (1.9) satisfying boundary conditions on . In particular, we will give sufficient conditions such that if is a solution of (1.9) satisfying the boundary conditions and is a solution of (1.9) satisfying the boundary conditions , the solutions of (1.9) is

Moreover, we will assume the following conditions throughout this paper.

(A2) Solutions of initial value problems for (1.9) are unique and extend throughout . (A3) is right dense and is fixed.

And the uniqueness of solutions assumptions are stated in terms of generalized zeros as follows:

(A4) For any in , if and are solutions of (1.9) such that has a generalized zero at and has a generalized zero at , then on .

2. Uniqueness of Solutions

In this section, we establish that under conditions (A1) through (A4), solutions of the conjugate boundary value problems of this paper are unique, when they exist.

Theorem 2.1. Let be given and assume conditions through are satisfied. Then, given , each of boundary value problems of (1.9) satisfying any of the following boundary conditions has at most one solution.

Proof. Assume for some , there exists distinct solutions and of (1.9), (2.1), and set . Then, we have

Clearly, since and has a generalized zero at and has a generalized zero at , this contradicts condition (A4). Hence, the boundary value problems (1.9), (2.1) have unique solutions.

Next, we will look at a special boundary value problem of (1.9) satisfying the boundary condition

We will show the uniqueness of solutions of the boundary value problems (1.9), (2.4) and use it to obtain the uniqueness of solutions of the boundary value problems (1.9), (2.2).

Assume that for some there are two distinct solutions, and , of (1.9), (2.4). Let . Then, we have

By the uniqueness of solutions of initial value problems of (1.9), . Without loss of generality, we may assume . We consider the two cases of .

If is right-dense, , then

If is right scattered, , then

Regardless of whether is right dense or right scattered, we have , which is a contradiction to condition (A2). Hence, .

The uniqueness of solutions of boundary value problems of (1.9), (2.4) implies the uniqueness of solutions of boundary value problems of (1.9), (2.2) because the boundary conditions are defined at . This completes the proof.

Theorem 2.2. Let be given and assume conditions through are satisfied. Then the boundary value problems (1.9), (1.10) has at most one solution.

Proof. Again, we argue by contradiction. Assume for some values , there are two distinct solutions, and , of (1.9), (1.10). Let . Then, we have and . By the uniqueness of solutions of initial value problems of (1.9), and . We may assume, without loss of generality, and .

Then, there exists a point , , such that has a generalized zero at . That is, either or . But, by condition (A3), .

Since and , there exists a point such that has generalized zero at . Since we also obtain that has a generalized zero at , it implies that , and this contradicts condition (A4).

Similarly, since and , there exists a point such that has generalized zero at . Note that and we obtain that has a generalized zero at c and has a generalized zero at . This, again, implies that , and, hence, contradicts condition (A4).

3. Existence of Solutions

In this section, we establish monotonicity of the derivative as a function of , of solutions of (1.9) satisfying each of the boundary conditions (2.1), (2.2). We use these monotonicity properties then to obtain solutions of (1.9), (1.10).

Theorem 3.1. Suppose that conditions through are satisfied and that for each there exists solutions of (1.9), (2.1) and (1.9), (2.2). Then, and are both strictly increasing function of whose range is .

Proof. The strictness of the conclusion arises from Theorem 2.1. Let and let

Then, by Theorem 2.1,

Suppose to the contrary that . Then there exists a point such that has a generalized zero at . This contradicts condition (A4). Thus, and as a consequence, is a strictly increasing function of .

We now show that . Let and consider the solution of (1.9), (2.1), with as defined above. Consider also the solution of (1.9), (2.1). Hence, by Theorem 2.1, and the range of as a function of is the set of real numbers.

The argument for is quite similar. This completes the proof.

In a similar way, we also have a monotonicity result on the functions and .

Theorem 3.2. Assume the hypotheses of Theorem 3.1. Then, and are, respectively, strictly increasing and decreasing functions of with ranges all of .

We now provide our existence result.

Theorem 3.3. Assume the hypotheses of Theorem 3.1. Then, the boundary value problems (1.9), (1.10) has a unique solution.

Proof. The existence is immediate from Theorem 3.1 or Theorem 3.2. Making use of Theorem 3.1, there exists a unique such that . Then,

is a solution of (1.9), (1.10), and by Theorem 2.2, is the unique solution.

References

P. B. Bailey, L. F. Shampine, and P. E. Waltman, Nonlinear Two Point Boundary Value Problems, Academic Press, New York, NY, USA, 1968.

D. R. K. S. Rao, K. N. Murty, and A. S. Rao, “Existence and uniqueness theorems for four-point boundary value problems for fourth order differential equations,” Indian Journal of Pure and Applied Mathematics, vol. 13, no. 9, pp. 1006–1010, 1982.

J. Henderson, “Three-point boundary value problems for ordinary differential equations by matching solutions,” Nonlinear Analysis, vol. 7, no. 4, pp. 411–417, 1983.

J. Henderson, “Solution matching for boundary value problems for linear equations,” International Journal of Mathematics & Mathematical Sciences, vol. 12, no. 4, pp. 713–720, 1989.

J. Henderson and R. D. Taunton, “Solutions of boundary value problems by matching methods,” Applicable Analysis, vol. 49, no. 3-4, pp. 235–246, 1993.

D. Barr and T. Sherman, “Existence and uniqueness of solutions of three-point boundary value problems,” Journal of Differential Equations, vol. 13, pp. 197–212, 1973.

P. Hartman, “On -parameter families and interpolation problems for nonlinear ordinary differential equations,” Transactions of the American Mathematical Society, vol. 154, pp. 201–226, 1971.

J. Henderson, “Existence theorems for boundary value problems for th-order nonlinear difference equations,” SIAM Journal on Mathematical Analysis, vol. 20, no. 2, pp. 468–478, 1989.

J. Henderson, “Focal boundary value problems for nonlinear difference equations. I, II,” Journal of Mathematical Analysis and Applications, vol. 141, no. 2, pp. 559–579, 1989.

J. Henderson and W. K. C. Yin, “Two-point and three-point problems for fourth order dynamic equations,” Dynamic Systems and Applications, vol. 12, no. 1-2, pp. 159–169, 2003.

V. R. G. Moorti and J. B. Garner, “Existence-uniqueness theorems for three-point boundary value problems for th-order nonlinear differential equations,” Journal of Differential Equations, vol. 29, no. 2, pp. 205–213, 1978.

D. R. K. S. Rao, K. N. Murthy, and A. S. Rao, “On three-point boundary value problems associated with third order differential equations,” Nonlinear Analysis, vol. 5, no. 6, pp. 669–673, 1981.

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.

M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001.

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003.

C. J. Chyan, “Uniqueness implies existence on time scales,” Journal of Mathematical Analysis and Applications, vol. 258, no. 1, pp. 359–365, 2001.

J. Henderson and W. K. C. Yin, “Existence of solutions for fourth order boundary value problems on a time scale,” Journal of Difference Equations and Applications, vol. 9, no. 1, pp. 15–28, 2003.