Approximation of Solution of Some m-Point Boundary Value Problems on Time Scales
© R. A. Khan and M. Rafique. 2010
Received: 24 August 2009
Accepted: 2 June 2010
Published: 24 June 2010
The method of upper and lower solutions and the generalized quasilinearization technique for second-order nonlinear m-point dynamic equations on time scales of the type , , , , , , are developed. A monotone sequence of solutions of linear problems converging uniformly and quadratically to a solution of the problem is obtained.
Many dynamical processes contain both continuous and discrete elements simultaneously. Thus, traditional mathematical modeling techniques, such as differential equations or difference equations, provide a limited understanding of these types of models. A simple example of this hybrid continuous-discrete behavior appears in many natural populations: for example, insects that lay their eggs at the end of the season just before the generation dies out, with the eggs laying dormant, hatching at the start of the next season giving rise to a new generation. For more examples of species which follow this type of behavior, we refer the readers to .
Hilger  introduced the notion of time scales in order to unify the theory of continuous and discrete calculus. The field of dynamical equations on time scales contain, links and extends the classical theory of differential and difference equations, besides many others. There are more time scales than just (corresponding to the continuous case) and (corresponding to the discrete case) and hence many more classes of dynamic equations. An excellent resource with an extensive bibliography on time scales was produced by Bohner and Peterson [3, 4].
Recently, existence theory for positive solutions of boundary value problems (BVPs) on time scales has attracted the attention of many authors; see, for example, [5–12] and the references therein for the existence theory of some two-point BVPs, and [13–16] for three-point BVPs on time scales. For the existence of solutions of -point BVPs on time scales, we refer the readers to .
However, the method of upper and lower solutions and the quasilinearization technique for BVPs on time scales are still in the developing stage and few papers are devoted to the results on upper and lower solutions technique and the method of quasilinearization on time scales [18–21]. The pioneering paper on multipoint BVPs on time scales has been the one in  where lower and upper solutions were combined with degree theory to obtain very wide-ranging existence results. Further, the authors of  studied existence results for more general three-point boundary conditions which involve first delta derivatives and they also developed some compatibility conditions. We are very grateful to the reviewer for directing us towards this important work.
where , and is from a so-called time scale (which is an arbitrary closed subset of ). Existence of at least one solution for (1.1) has already been studied in  by the Krasnosel'skii and Zabreiko fixed point theorems. We obtain existence and uniqueness results and develop a method to approximate the solutions.
Assume that has a topology that it inherits from the standard topology on and define the time scale interval . For , define the forward jump operator by and the backward jump operator by . If , is said to be right scattered, and if , is said to be right dense. If , is said to be left scattered, and if , is said to be left dense.
A solution of (1.1) is a function which satisfies (1.1) for each .
The purpose of this paper is to develop the method of upper and lower solutions and the method of quasilinearization [22–26]. Under suitable conditions on , we obtain a monotone sequence of solutions of linear problems. We show that the sequence of approximants converges uniformly and quadratically to a unique solution of the problem.
2. Upper and Lower Solutions Method
where , .
where is the identity. If and is bounded on , then by Arzela-Ascoli theorem is compact and Schauder's fixed point theorem yields a fixed point of . We discuss the case when is not necessarily bounded on .
(comparison result) Assume that , are lower and upper solutions of the boundary value problem (1.1). If and is strictly increasing in for each , then on .
a contradiction. Hence has no positive local maximum.
Hence, , which leads to , a contradiction.
Hence . Thus, on .
Under the hypotheses of Theorem 2.2, the solutions of the BVP (1.1), if they exist, are unique.
The following theorem establishes existence of solutions to the BVP (1.1) in the presence of well-ordered lower and upper solutions.
The proof essentially is a minor modification of the ideas in  and so is omitted.
3. Generalized Approximations Technique
are lower and upper solutions of the BVP (1.1) such that on ,
and is increasing in for each .
Then, there exists a monotone sequence of solutions of linear problems converging uniformly and quadratically to a unique solution of the BVP (1.1).
By standard arguments as in , the sequence converges to a solution of (1.1).
which shows the quadratic convergence.
The authors are thankful to reviewers for their valuable comments and suggestions.
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