Abstract and Applied Analysis

Volume 2016, Article ID 3910972, 6 pages

http://dx.doi.org/10.1155/2016/3910972

## Discrete Approaches to Continuous Boundary Value Problems: Existence and Convergence of Solutions

^{1}Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562, USA^{2}School of Mathematics and Statistics, The University of New South Wales (UNSW), Sydney, NSW 2052, Australia

Received 8 June 2016; Accepted 10 July 2016

Academic Editor: Athanassios G. Bratsos

Copyright © 2016 Douglas R. Anderson and Christopher C. Tisdell. 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

We investigate two types of first-order, two-point boundary value problems (BVPs). Firstly, we study BVPs that involve nonlinear difference equations (the “discrete” BVP); and secondly, we study BVPs involving nonlinear ordinary differential equations (the “continuous” BVP). We formulate some sufficient conditions under which the discrete BVP will admit solutions. For this, our choice of methods involves a monotone iterative technique and the method of successive approximations (a.k.a. Picard iterations) in the absence of Lipschitz conditions. Our existence results for the discrete BVP are of a constructive nature and are of independent interest in their own right. We then turn our attention to applying our existence results for the discrete BVP to the continuous BVP. We form new existence results for solutions to the continuous BVP with our methods involving linear interpolation of the data from the discrete BVP, combined with* a priori* bounds and the convergence Arzela-Ascoli theorem. Thus, our use of discrete BVPs to yield results for the continuous BVP may be considered as a discrete approach to continuous BVPs.

#### 1. Introduction

In this paper we investigate two types of first-order, two-point boundary value problems (BVPs).

Firstly, we study BVPs that involve nonlinear difference equations (the following “discrete” BVP). Let be continuous and consider the discrete boundary value problemwhere ; the grid points are denoted by for ; for ; and , , and are constants.

Secondly, we study BVPs involving nonlinear ordinary differential equations (the following “continuous” BVP): where .

Problem (1) and (2) may be considered as a discrete analogue of (3) and (4).

The study of discrete BVP (1) and (2) is significant for two main reasons, as these types of equations(a)naturally arise when modelling phenomena, for example, in oscillation and control theory [1, p. 1],(b)are of importance in the approximation of solutions to ordinary differential equations.

In this paper we discuss the existence and approximation of solutions of both sets of BVPs: (1) and (2); (3) and (4).

We formulate some sufficient conditions under which the discrete BVP (1) and (2) will admit solutions. For this, our choice of methods involves monotone iterative techniques and the method of successive approximations (a.k.a. Picard iterations). The classical method of successive approximations is powerful and constructive in nature and thus it is surprising to find that it has been significantly underutilized in the environment of discrete BVPs of the first order. Our existence results for the discrete BVP are of a constructive nature and, furthermore, some of our results bound solutions independently of the step size. These results are of independent interest of the continuous BVP (3) and (4).

We then turn our attention to applying our existence results for the discrete BVP (1) and (2) to the continuous BVP (3) and (4). We form new existence results for solutions to the continuous BVP with our methods involving linear interpolation of the data from the discrete BVP, combined with* a priori* bounds and the convergence Arzela-Ascoli theorem. Thus, our use of discrete BVPs to yield results for the continuous BVP may be considered as a discrete approach to continuous BVPs.

Several other authors have studied the existence of solutions to (1) and (2) via the method of lower and upper solutions [2, 3], [4, Sec. ]; and by employing* a priori* bounds on solutions and Brouwer degree [5]. Mohamed et al. [6] have recently studied variations of (1) and (2) via discrete approaches.

Several authors have used the discrete approach to continuous BVPs for second-order problems, such as [7–11]. In particular, in [9–11] the boundary conditions were separated; however, in this work our boundary conditions under consideration are not separated. In addition, we employ different assumptions and different methods. For example, we use the idea of a monotonic and bounded sequence herein, rather than the maximum principles of [9] or the growth conditions and* a priori* bounds of [10, 11].

Our ideas complement those of [2, 3, 5] and [4, Sec. ] and appear to be of a more constructive nature as solutions to (1) and (2) obtained by the theorems herein may be computed (or approximated) via an iterative process. Our results herein improve some of the results in [6] and our techniques and methods contrast with theirs; for example, we do not rely on Lipschitz conditions in our theorems.

Our results are innovative for two main reasons: (i) they are new for the discrete BVP; (ii) they form new connections to the continuous BVP. Furthermore, we believe that the discrete approach to continuous BVPs that we present open up several lines of inquiry for first-order BVPs.

A solution to the discrete BVP (1) and (2) is a vector having components that(a)satisfy for ,(b)satisfy (1) for and also satisfy (2).

A solution to the continuous BVP (3) and (4) is differentiable function that(a)satisfies for ,(b)satisfies (3) for and also satisfies (4).

We now present a simple result showing the equivalence between (1) and (2) and a particular summation equation that will be used throughout this work.

Lemma 1. *The discrete BVP (1) and (2) and the summation equation are equivalent, with *

*Proof. *For completeness we provide a proof. Let be a solution to (1) and (2). If we sum (1) from 0 to then we obtain and so for we obtain Using boundary conditions (2) we can eliminate in (8) to obtain Thus, substitution of (9) into (7) yields which can then be recast into form (5) by splitting the second term to sum from to and from to .

Now let be a solution to (10). It can be directly verified that (1) and (2) hold.

#### 2. Monotone Sequential Approach

In this section we formulate some existence results for solutions to (1) and (2) by generating a monotone and bounded sequence of vectors whose limit will be a solution to (1) and (2).

Throughout this section the domain of will be the rectangle for some positive number .

Since is continuous on the compact set we may define a number such that

The main result of this section is the following.

Theorem 2. *Let be continuous and let If then problem (1) and (2) has at least one solution for each such that for .*

*Proof. *Consider summation equation (5) that, by Lemma 1, is equivalent to (1) and (2) and define the sequence of vectors for recursively by Firstly we show that our sequence of vectors is well defined for by showing that each for and so for each and . We use proof by induction.

From the definition of it is easy to see that for . Now assume for some we have for . From (18) we have for from (13). Thus, by induction, we have for each and and so our sequence of vectors is well defined for each .

Furthermore, the above has shown that the sequence of vectors is uniformly bounded for .

We now show that for , where the inequality holds in a componentwise fashion. Once again, we use induction. For consider where we have used (14) and (16). Thus, .

Now assume that for some ; that is, assume for . For each we have where we have used assumptions (14) and (15). Thus, for .

From the above we conclude that is a uniformly bounded and nondecreasing sequence of vectors and so must converge to a vector ; that is, for some .

We finally show that the above is actually a solution to (1) and (2). Since each we must have each and so for . Furthermore, the continuity of on ensures that for each .

If we now take limits in (18) as then we obtain so that our limit vector is a solution to (1) and (2).

*Example 3. *Consider the following discrete BVP: so that we have a special case of (1) and (2) with We claim that problem (25) and (26) has at least one solution such that for .

*Proof. *We show that all of the conditions of Theorem 2 hold. Firstly, we see that the inequalities in (14) hold. If we choose to form then and so (13) holds. Furthermore, is nondecreasing in the second variable and so (15) is satisfied. Finally, (16) holds. Thus, all of the conditions of Theorem 2 hold and the result follows.

*Remark 4. *In Example 3 above, letting , pick for and construct the approximating iterates as in (18). The numbers in Table 1 signify the error that results upon substituting the generated into (25).

We notice that the error in terms of at each decreases for this example as converge upward to a solution in the rectangle, so that , for example, is a good approximation to a solution of (25). The actual values of are given by Note that by construction, satisfies boundary condition (26); namely,