Approximation of Solution of Some m-Point Boundary Value Problems on Time Scales

Abstract

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.

1. Introduction

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 [1].

Hilger [2] 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 [17].

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 [21] where lower and upper solutions were combined with degree theory to obtain very wide-ranging existence results. Further, the authors of [21] 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.

Recently, existence results via upper and lower solutions method and approximation of solutions via generalized quasilinearization method for some three-point boundary value problems on time scales have been studied in [16]. Motivated by the work in [16, 17], in this paper, we extend the results studied in [16] to a class of -point BVPs of the type

(1.1)

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 [17] 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 function is said to be rd-continuous provided it is continuous at all right-dense points of and its left-sided limit exists at left-dense points of . A function is said to be ld-continuous provided it is continuous at all left-dense points of and its right-sided limit exists at right-dense points of . Define if has a left-scattered maximum at ; otherwise . For and , the delta derivative of at (if exists) is defined by the following Given that , there exists a neighborhood of such that

(1.2)

If there exists a function such that , is said to be the delta antiderivative of and the delta integral is defined by

(1.3)

Definition 1.1.

Define to be the set of all functions such that

(1.4)

A solution of (1.1) is a function which satisfies (1.1) for each .

Let us denote

(1.5)

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

We write the BVP (1.1) as an equivalent -integral equation

(2.1)

where is a Green's function for the problem

(2.2)

and it is given by [17]

(2.3)

where , .

Notice that on and is rd-continuous. Define an operator by

(2.4)

By a solution of (2.1), we mean a solution of the operator equation

(2.5)

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 .

Definition 2.1.

We say that is a lower solution of the BVP (1.1), if

(2.6)

Similarly, is an upper solution of the BVP (1.1) if

(2.7)

Theorem 2.2.

(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 .

Proof.

Define . Then and the BCs imply that

(2.8)

Assume that the conclusion of the theorem is not true. Then, has a positive maximum at some . Clearly, . If , then, the point is not simultaneously left dense and right scattered; see, for example, [12]. Hence by Lemma of [12],

(2.9)

On the other hand, using the definitions of lower and upper solutions, we obtain

(2.10)

Since is not simultaneously left dense and right scattered, it is left scattered and right scattered, left dense and right dense, or left scattered and right dense. In either case . Using the increasing property of in , we obtain

(2.11)

a contradiction. Hence has no positive local maximum.

If , then . If any one of the is such that , then has a positive local maximum, a contradiction. Hence

(2.12)

Moreover, if for each , then, from the BCs

(2.13)

we have , a contradiction. Hence, for some , and consequently, in view of (2.12) and the BCs, it follows that

(2.14)

Hence, , which leads to , a contradiction.

Hence . Thus, on .

Corollary 2.3.

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.

Theorem 2.4.

Assume that , are lower and upper solutions of the BVP (1.1) such that . If , then the BVP (1.1) has a solution such that

(2.15)

The proof essentially is a minor modification of the ideas in [21] and so is omitted.

3. Generalized Approximations Technique

We develop the approximation technique and show that, under suitable conditions on , there exists a bounded monotone sequence of solutions of linear problems that converges uniformly and quadratically to a solution of the nonlinear original problem. If and is bounded on , where

(3.1)

there always exists a function such that

(3.2)

where , and it is such that . For example, let  : , then we choose . Clearly,

(3.3)

Define by . Note that and

(3.4)

Theorem 3.1.

Assume that

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).

Proof.

Conditions and ensure the existence of a unique solution of the BVP (1.1) such that

(3.5)

In view of (3.4), we have

(3.6)

The mean value theorem and the fact that is nonincreasing in on for each yield

(3.7)

where such that . Substituting in (3.6), we have

(3.8)

on . Define by

(3.9)

We note that is continuous for each and is rd-continuous for each . Moreover, satisfies the following relations on :

(3.10)

(3.11)

Now, we develop the iterative scheme to approximate the solution. As an initial approximation, we choose and consider the linear problem

(3.12)

Using (3.11) and the definition of lower and upper solutions, we get

(3.13)

which imply that and are lower and upper solutions of (3.12), respectively. Hence by Theorem 2.4 and Corollary 2.3, there exists a unique solution of (3.12) such that

(3.14)

Using (3.11) and the fact that is a solution of (3.12), we obtain

(3.15)

which implies that is a lower solution of the problem (1.1). Similarly, in view of (3.11), and (3.15), we can show that and are lower and upper solutions of the problem

(3.16)

Hence by Theorem 2.4 and Corollary 2.3, there exists a unique solution of the problem (3.16) such that

(3.17)

Continuing in the above fashion, we obtain a bounded monotone sequence of solutions of linear problems satisfying

(3.18)

where the element of the sequence is a solution of the linear problem

(3.19)

and is given by

(3.20)

By standard arguments as in [19], the sequence converges to a solution of (1.1).

Now, we show that the convergence is quadratic. Set , where is a solution of (1.1). Then, on and the boundary conditions imply that

(3.21)

Now, in view of the definitions of and , we obtain

(3.22)

Using the mean value theorem repeatedly and the fact that on , we obtain

(3.23)

where ,  : , and  : . Hence (3.22) can be rewritten as

(3.24)

where ,  : , and we used the fact that on . Choose such that

(3.25)

Therefore, we obtain

(3.26)

where .

By comparison result, , where is the unique solution of the linear BVP

(3.27)

Hence,

(3.28)

where . Taking the maximum over , we obtain

(3.29)

which shows the quadratic convergence.