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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012, Article IDΒ 465867, 7 pages
http://dx.doi.org/10.1155/2012/465867
Research Article

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

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.Β Jorgensen

Copyright Β© 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 𝕋 be a time scale such that π‘Ž<𝑏;π‘Ž,π‘βˆˆπ•‹. We will show the existence and uniqueness of solutions for the second-order boundary value problem 𝑦ΔΔ(𝑑)=𝑓(𝑑,𝑦(𝑑),𝑦Δ(𝑑)),π‘‘βˆˆ[π‘Ž,𝑏]𝕋,𝑦(π‘Ž)=𝐴,𝑦(𝑏)=𝐡, by matching a solution of the first equation satisfying boundary conditions on [π‘Ž,𝑐]𝕋 with a solution of the first equation satisfying boundary conditions on [𝑐,𝑏]𝕋, where π‘βˆˆ(π‘Ž,𝑏)𝕋.

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 π‘¦ξ…žξ…žξ€·(𝑑)=𝑓𝑑,𝑦,π‘¦ξ…žξ€Έ,(1.1)𝑦(π‘Ž)=𝑦1,𝑦(𝑏)=𝑦2.(1.2) As shown in their work, the uniqueness and existence of the solutions of (1.1), (1.2) were obtained by matching a solution 𝑦1 of (1.1) satisfying the boundary condition 𝑦(π‘Ž)=𝑦1,π‘¦ξ…ž(𝑐)=π‘š(1.3) with a solution 𝑦2 of (1.1) satisfying the boundary condition π‘¦ξ…ž(𝑐)=π‘š,𝑦(𝑏)=𝑦2,(1.4) where π‘š=π‘¦ξ…ž1(𝑐)=π‘¦ξ…ž2(𝑐).

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, 𝕋 is assumed to be a nonempty closed subset of ℝ with inf𝕋=βˆ’βˆž and sup𝕋=+∞. We shall also use the convention on notation that for each interval 𝐼 of ℝ, 𝐼𝕋=πΌβˆ©π•‹.(1.5)

For readers' convenience, we state a few definitions which are basic to the calculus on the time scale 𝕋. The forward jump operator πœŽβˆΆπ•‹β†’β„ is defined by 𝜎(𝑑)=inf{𝑠>π‘‘βˆ£π‘ βˆˆπ•‹}βˆˆπ•‹.(1.6) If 𝜎(𝑑)>𝑑, 𝑑 is said to be right-scattered, whereas, if 𝜎(𝑑)=𝑑, 𝑑 is said to be right-dense. The backward jump operator πœŒβˆΆπ•‹β†’β„ is defined by 𝜌(𝑑)=sup{𝑠<π‘‘βˆ£π‘ βˆˆπ•‹}βˆˆπ•‹.(1.7) If 𝜌(𝑑)<𝑑, 𝑑 is said to be left-scattered, and if 𝜌(𝑑)=𝑑, then 𝑑 is said to be left-dense. If π‘”βˆΆπ•‹β†’β„ and π‘‘βˆˆπ•‹, then the delta derivative of g at t, 𝑔Δ(𝑑), is defined to be the number (provided that it exists), with the property that, given any πœ€>0, there is a neighborhood π‘ˆ of 𝑑, such that ||[]𝑔(𝜎(𝑑))βˆ’π‘”(𝑠)βˆ’π‘”Ξ”([]||||||,𝑑)𝜎(𝑑)βˆ’π‘ β‰€πœ€πœŽ(𝑑)βˆ’π‘ (1.8) for all π‘ βˆˆπ‘ˆ. In this definition, π‘‘βˆˆπ•‹π’¦, where this set is derived from the time scale 𝕋 as follows: if 𝕋 has a left-scattered maximum π‘š, then 𝕋𝒦=𝕋⧡{π‘š}. Otherwise, we define 𝕋𝒦=𝕋.

We say that the function 𝑦 has a generalized zero at 𝑑 if 𝑦(𝑑)=0 or if 𝑦(𝑑)⋅𝑦(𝜎(𝑑))<0. In the latter case, we would say the generalized zero is in the real interval (𝑑,𝜎(𝑑)).

Theorem 1.1 (Mean Value Theorem). If π‘¦βˆΆπ•‹β†’β„ is continuous and 𝑦(𝑑) has a generalized zero at π‘Ž and 𝑏, π‘Ž,π‘βˆˆπ•‹, then there exists a point π‘Ÿβˆˆ[π‘Ž,𝑏]𝕋 such that 𝑦Δ has a generalized zero at π‘Ÿ.

Let 𝕋 be a time scale such that π‘Ž,π‘βˆˆπ•‹. In this paper, we are concerned with the existence and uniqueness of solutions of boundary value problems on the interval [π‘Ž,𝑏]𝕋 for the second-order delta derivative equation 𝑦ΔΔ=𝑓𝑑,𝑦,𝑦Δ,(1.9) satisfying the boundary conditions, 𝑦(π‘Ž)=𝑦1,𝑦(𝜎(𝑏))=𝑦2,(1.10) where π‘Ž<𝑏 and 𝑦1,𝑦2βˆˆβ„. Throughout this paper, we will assume (A1)𝑓(𝑑,π‘Ÿ1,π‘Ÿ2) is a real-valued continuous function defined on 𝕋×ℝ2.

We obtain solutions by matching a solution of (1.9) satisfying boundary conditions on [π‘Ž,𝑐]𝕋 to a solution of (1.9) satisfying boundary conditions on [𝑐,𝑏]𝕋. In particular, we will give sufficient conditions such that if 𝑦1(𝑑) is a solution of (1.9) satisfying the boundary conditions 𝑦(π‘Ž)=𝑦1,𝑦Δ(𝑐)=π‘š and 𝑦2(𝑑) is a solution of (1.9) satisfying the boundary conditions 𝑦(𝜎(𝑏))=𝑦2,𝑦Δ(𝑐)=π‘š, the solutions of (1.9) is 𝑦𝑦(𝑑)=1[](𝑑),π‘‘βˆˆπ‘Ž,𝑐𝕋,𝑦2([]𝑑),π‘‘βˆˆπ‘,𝑏𝕋.(1.11)

Moreover, we will assume the following conditions throughout this paper. (A2) Solutions of initial value problems for (1.9) are unique and extend throughout 𝕋. (A3)π‘βˆˆπ•‹ is right dense and is fixed.

And the uniqueness of solutions assumptions are stated in terms of generalized zeros as follows: (A4) For any 𝑑1<𝑑2 in 𝕋, if 𝑒 and 𝑣 are solutions of (1.9) such that (π‘’βˆ’π‘£) has a generalized zero at 𝑑1 and (π‘’βˆ’π‘£)Ξ” has a generalized zero at 𝑑2, then 𝑒≑𝑣 on 𝕋.

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 𝐴,π΅βˆˆβ„ be given and assume conditions (A1) through (A4) are satisfied. Then, given π‘šβˆˆβ„, each of boundary value problems of (1.9) satisfying any of the following boundary conditions has at most one solution. 𝑦(π‘Ž;π‘š)=𝐴,𝑦Δ(𝑐;π‘š)=π‘š,π‘Ž<𝑐,whereπ‘Ž,π‘βˆˆπ•‹,(2.1)𝑦(𝑏;π‘š)=𝐡,𝑦Δ(𝑐;π‘š)=π‘š,𝑐<𝑏,where𝑐,π‘βˆˆπ•‹.(2.2)

Proof. Assume for some π‘šβˆˆβ„, there exists distinct solutions 𝛼 and 𝛽 of (1.9), (2.1), and set 𝑀=π›Όβˆ’π›½. Then, we have 𝑀(π‘Ž)=0,𝑀Δ(𝑐)=0.(2.3) Clearly, since π‘Ž<𝑐 and 𝑀 has a generalized zero at π‘Ž and 𝑀Δ has a generalized zero at 𝑐, 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 𝑦(𝜎(𝑏);π‘š)=𝐢,𝑦Δ(𝑏;π‘š)=π‘š.(2.4) 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 π‘šβˆˆβ„ there are two distinct solutions, 𝛼 and 𝛽, of (1.9), (2.4). Let 𝑀=π›Όβˆ’π›½. Then, we have 𝑀(𝜎(𝑏);π‘š)=0,𝑀Δ(𝑏;π‘š)=0.(2.5) By the uniqueness of solutions of initial value problems of (1.9), 𝑀(𝑏)β‰ 0. Without loss of generality, we may assume 𝑀(𝑏)>0. We consider the two cases of 𝑏.
If 𝑏 is right-dense, 𝜎(𝑏)=𝑏, then 𝑀Δ(𝑏;π‘š)=lim𝑑→𝑏𝑀(𝑑;π‘š)βˆ’π‘€(𝑏;π‘š)π‘‘βˆ’π‘=0.(2.6)
If 𝑏 is right scattered, 𝜎(𝑏)>𝑏, then 𝑀Δ(𝑏;π‘š)=𝑀(𝜎(𝑏);π‘š)βˆ’π‘€(𝑏;π‘š)𝜎(𝑏)βˆ’π‘=0.(2.7)
Regardless of whether 𝑏 is right dense or right scattered, we have 𝑀(𝜎(𝑏);π‘š)=𝑀(𝑏;π‘š)=0, which is a contradiction to condition (A2). Hence, 𝛼≑𝛽.
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 𝑐<𝑏. This completes the proof.

Theorem 2.2. Let 𝐴,π΅βˆˆβ„ be given and assume conditions (A1) through (A4) 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 𝐴,π΅βˆˆβ„, there are two distinct solutions, 𝛼 and 𝛽, of (1.9), (1.10). Let 𝑀=π›Όβˆ’π›½. Then, we have 𝑀(π‘Ž)=0 and 𝑀(𝑏)=0. By the uniqueness of solutions of initial value problems of (1.9), 𝑀Δ(π‘Ž)β‰ 0 and 𝑀Δ(𝑏)β‰ 0. We may assume, without loss of generality, 𝑀Δ(π‘Ž)>0 and 𝑀Δ(𝑏)>0.
Then, there exists a point 𝑐, π‘Ž<𝑐<𝑏, such that 𝑀 has a generalized zero at 𝑐. That is, either 𝑀(𝑐)=0 or 𝑀(𝑐)⋅𝑀(𝜎(𝑐))<0. But, by condition (A3), 𝑀(𝑐)=0.
Since 𝑀(π‘Ž)=0 and 𝑀(𝑐)=0, there exists a point π‘Ÿ1∈(π‘Ž,𝑐)𝕋 such that 𝑀Δ has generalized zero at π‘Ÿ1. Since we also obtain that 𝑀 has a generalized zero at π‘Ž, it implies that 𝛼≑𝛽, and this contradicts condition (A4).
Similarly, since 𝑀(𝑐)=0 and 𝑀(𝑏)=0, there exists a point π‘Ÿ2∈(𝑐,𝑏)𝕋 such that 𝑀Δ has generalized zero at π‘Ÿ2. Note that 𝑐<π‘Ÿ2 and we obtain that 𝑀 has a generalized zero at c and 𝑀Δ has a generalized zero at π‘Ÿ2. This, again, implies that 𝛼≑𝛽, and, hence, contradicts condition (A4).

3. Existence of Solutions

In this section, we establish monotonicity of the derivative as a function of π‘š, 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 (A1) through (A4) are satisfied and that for each π‘šβˆˆβ„ there exists solutions of (1.9), (2.1) and (1.9), (2.2). Then, 𝛼Δ(𝑐;π‘š) and 𝛽Δ(𝑐;π‘š) are both strictly increasing function of π‘š whose range is ℝ.

Proof. The strictness of the conclusion arises from Theorem 2.1. Let π‘š1>π‘š2 and let 𝑀(π‘₯)=𝛼π‘₯;π‘š1ξ€Έξ€·βˆ’π›Όπ‘₯;π‘š2ξ€Έ.(3.1)
Then, by Theorem 2.1, 𝑀(π‘Ž)=π›Όπ‘Ž;π‘š1ξ€Έξ€·βˆ’π›Όπ‘Ž;π‘š2ξ€Έξ€·=0,𝑀(𝑐)=𝛼𝑐;π‘š1ξ€Έξ€·βˆ’π›Όπ‘;π‘š2𝑀=0,Ξ”(𝑐)=𝛼Δ𝑐;π‘š1ξ€Έβˆ’π›ΌΞ”ξ€·π‘;π‘š2ξ€Έβ‰ 0.(3.2)
Suppose to the contrary that 𝑀Δ(𝑐)<0. Then there exists a point π‘Ÿ1∈(π‘Ž,𝑐)𝕋 such that 𝑀Δ has a generalized zero at π‘Ÿ1. This contradicts condition (A4). Thus, 𝑀Δ(𝑐)>0 and as a consequence, 𝑀Δ(𝑐;π‘š) is a strictly increasing function of π‘š.
We now show that {𝛼Δ(𝑐;π‘š)βˆ£π‘šβˆˆβ„}=ℝ. Let π‘˜βˆˆβ„ and consider the solution 𝑒(π‘₯;π‘˜) of (1.9), (2.1), with 𝑒 as defined above. Consider also the solution 𝛼(π‘₯;𝑒Δ(𝑐;π‘˜)) of (1.9), (2.1). Hence, by Theorem 2.1, 𝛼(π‘₯;𝑒Δ(𝑐;π‘˜))≑𝑒Δ(𝑐;π‘˜) and the range of 𝛼Δ(𝑐;π‘š) as a function of π‘š is the set of real numbers.
The argument for 𝛽Δ is quite similar. This completes the proof.

In a similar way, we also have a monotonicity result on the functions 𝑒(𝑑;π‘š) and 𝑣(𝑑;π‘š).

Theorem 3.2. Assume the hypotheses of Theorem 3.1. Then, 𝑒(𝑑;π‘š) and 𝑣(𝑑;π‘š) are, respectively, strictly increasing and decreasing functions of π‘š with ranges all of ℝ.

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 π‘š0βˆˆβ„ such that 𝑒Δ(𝑐;π‘š0)=𝑣Δ(𝑐;π‘š0)=π‘š0. Then, 𝑒𝑦(𝑑)=𝑑;π‘š0𝑣,π‘Žβ‰€π‘‘β‰€π‘,𝑑;π‘š0ξ€Έ,𝑐≀𝑑≀𝑏(3.3) is a solution of (1.9), (1.10), and by Theorem 2.2, 𝑦(𝑑) is the unique solution.

References

  1. P. B. Bailey, L. F. Shampine, and P. E. Waltman, Nonlinear Two Point Boundary Value Problems, Academic Press, New York, NY, USA, 1968. View at Zentralblatt MATH
  2. D. R. K. S. Rao, K. N. Murty, and A. S. Rao, β€œExistence 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. View at Google Scholar Β· View at Zentralblatt MATH
  3. J. Henderson, β€œThree-point boundary value problems for ordinary differential equations by matching solutions,” Nonlinear Analysis, vol. 7, no. 4, pp. 411–417, 1983. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. J. Henderson, β€œSolution matching for boundary value problems for linear equations,” International Journal of Mathematics & Mathematical Sciences, vol. 12, no. 4, pp. 713–720, 1989. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. J. Henderson and R. D. Taunton, β€œSolutions of boundary value problems by matching methods,” Applicable Analysis, vol. 49, no. 3-4, pp. 235–246, 1993. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  6. D. Barr and T. Sherman, β€œExistence and uniqueness of solutions of three-point boundary value problems,” Journal of Differential Equations, vol. 13, pp. 197–212, 1973. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. P. Hartman, β€œOn N-parameter families and interpolation problems for nonlinear ordinary differential equations,” Transactions of the American Mathematical Society, vol. 154, pp. 201–226, 1971. View at Google Scholar
  8. J. Henderson, β€œExistence theorems for boundary value problems for nth-order nonlinear difference equations,” SIAM Journal on Mathematical Analysis, vol. 20, no. 2, pp. 468–478, 1989. View at Publisher Β· View at Google Scholar
  9. J. Henderson, β€œFocal boundary value problems for nonlinear difference equations. I, II,” Journal of Mathematical Analysis and Applications, vol. 141, no. 2, pp. 559–579, 1989. View at Publisher Β· View at Google Scholar
  10. J. Henderson and W. K. C. Yin, β€œTwo-point and three-point problems for fourth order dynamic equations,” Dynamic Systems and Applications, vol. 12, no. 1-2, pp. 159–169, 2003. View at Google Scholar Β· View at Zentralblatt MATH
  11. V. R. G. Moorti and J. B. Garner, β€œExistence-uniqueness theorems for three-point boundary value problems for nth-order nonlinear differential equations,” Journal of Differential Equations, vol. 29, no. 2, pp. 205–213, 1978. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. D. R. K. S. Rao, K. N. Murthy, and A. S. Rao, β€œOn three-point boundary value problems associated with third order differential equations,” Nonlinear Analysis, vol. 5, no. 6, pp. 669–673, 1981. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. S. Hilger, β€œAnalysis 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 Β· View at Zentralblatt MATH
  14. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001. View at Publisher Β· View at Google Scholar
  15. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003.
  16. C. J. Chyan, β€œUniqueness implies existence on time scales,” Journal of Mathematical Analysis and Applications, vol. 258, no. 1, pp. 359–365, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  17. J. Henderson and W. K. C. Yin, β€œExistence 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. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH