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

and Applied Analysis 3 Theorem 2. Let f : R b → R be continuous and let M |u + V| max {|u| , |V|} ≤ b; (13)


Introduction
In this paper we investigate two types of first-order, two-point boundary value problems (BVPs).
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. 2]; 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][8][9][10][11]. In particular, in [9][10][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. 2] 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 vector̃fl ( 0 , . . . , ) ∈ R +1 having components that 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
Proof. For completeness we provide a proof. Let̃be a solution to (1) and (2). If we sum (1) from 0 to − 1 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 = 0 to − 1 and from = to − 1. Now let̃be a solution to (10). It can be directly verified that (1) and (2) hold.

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 [0, 1] × of will be the rectangle for some positive number .
Since is continuous on the compact set we may define a number ≥ 0 such that The main result of this section is the following. If then problem (1) and (2) has at least one solutioñ∈ R +1 for each ℎ ∈ (0, 1) such that ( , ) ∈ for = 0, . . . , .
Proof. We show that all of the conditions of Theorem 2 hold. Firstly, we see that the inequalities in (14) hold. If we choose = 1 to form then = 1/5 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.  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 (12) , for example, is a good approximation to a solution of (25). The actual values of (12) ≈ are given by Note that by construction, ( ) satisfies boundary condition (26); namely, The following result is a modification of the ideas in Theorem 2 and its proof.
Proof. The proof is very similar to that of Theorem 2 and so is only outlined. Consider the sequence of successive approximations defined by Abstract and Applied Analysis 5 for = 0, 1, 2, . . .. The continuity of and (31) ensure that the successive approximations are well defined and uniformly bounded. The assumptions (32)-(33) ensure that the successive approximations are a nondecreasing sequence with the convergence and existence following in the same way as in the proof of Theorem 2.
There are a number of interesting variations of Theorems 2 and 5 that we now discuss.
Remark 7. The proofs of Theorems 2 and 5 essentially rest on generating a bounded, nondecreasing sequence of vectors. The statement of each theorem can be suitably modified so as to produce a bounded, nonincreasing sequence of vectors that converge to a solution of (1) and (2). All that is required is to reverse the differential inequalities in, for example, (15) and (16).

Remark 8.
For simplicity, the initial approximatioñ0 in the proofs of Theorems 2 and 5 was chosen to be a constant vector with components /( + V). With suitable modifications on (16) we may use any vector̃0 as our initial approximation provided ( , (0) ) ∈ for = 0, . . . , . For BVPs that have more than one solution, different choices in our initial approximatioñ0 can lead to the generation of distinct limit functions̃. That is, through various choices of̃0 we can observe convergence of̃( ) to various solutions of (1) and (2).

A Discrete Approach to Differential Equations
In this section we form a relationship between solutions to the discrete BVP (1) and (2) and solutions to the continuous BVP (3) and (4). We generate a sequence of functions that are based on the solutions to (1) and (2) guaranteed to exist from earlier sections and present some conditions under which they will converge to a function as ℎ → 0, with the function being a solution to (3) and (4). Thus, our approach uses the discrete problem to generate new existence results for the continuous problem in a constructive manner. Our first general convergence result is in the spirit of [7,Lemma 2.4], where Gaines applies the ideas to second-order BVPs. Our result involves a bound on the solutions to (1) and (2), with the bound being independent of ℎ. We require the following notation. Denote the sequence → ∞ as → ∞; let 0 < ℎ = 1/ < 1; and let = ℎ for = 0, . . . , . If problem (1) and (2) has a solution for ℎ = ℎ and ≥ 0 that we denote bỹ then we construct the following sequence of continuous functions from (35) via linear interpolation to form then problem (3) and (4) Thus, is uniformly bounded on [0, 1]. For , ∈ [0, 1] and given > 0, consider whenever | − | < ( ) fl / 1 . Thus, is equicontinuous on [0, 1].
The convergence Arzela-Ascoli theorem [12, p. 527] guarantees that the sequence of continuous functions = ( ) has a subsequence ( ) ( ) that converges uniformly to a continuous function = ( ) for ∈ [0, 1]. That is, The continuity of ensures that the above limit function will be a solution to (3) and (4).
We now relate the above abstract results to the ideas from earlier sections.

Theorem 11. Let the conditions of Theorem 2 hold. Given any
> 0 there is = ( ) such that if ℎ ≤ then problem (3) and (4) has a solution that satisfies (44).
Thus, all of the conditions of Theorem 10 hold and the result follows.
Remark 12. Similar results to that of Theorem 11 hold under the assumptions of Theorem 5 or Remark 7.

Disclosure
Douglas R. Anderson stated that research was carried out while being a visiting fellow at School of Mathematics and Statistics, UNSW, Sydney, NSW 2052, Australia.