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

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.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Douglas R. Anderson 1 and Christopher C. Tisdell 2

Academic Editor:Athanassios G. Bratsos

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; 24 August 2016

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.

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 f : [ 0,1 ] × D ⊆ [ 0,1 ] × R [arrow right] R be continuous and consider the discrete boundary value problem [figure omitted; refer to PDF] where 0 < h = 1 / n < 1 ; the grid points are denoted by _{t i} = i h for i = 0 , ... , n ; Δ _{x i} [: =] _{x i + 1} - _{x i} for i = 0 , ... , n - 1 ; and u , v , and w are constants.

Secondly, we study BVPs involving nonlinear ordinary differential equations (the following "continuous" BVP): [figure omitted; refer to PDF] where ^{[variant prime]} [: =] d / d t .

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. 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 [7-11]. 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. 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 vector x ~ [: =] ( _{x 0} , ... , _{x n} ) ∈ ^{R n + 1} having components _{x i} that

(a) satisfy ( _{t i} , _{x i} ) ∈ [ 0,1 ] × D for i = 0 , ... , n ,

(b) satisfy (1) for i = 0 , ... , n - 1 and also satisfy (2).

A solution to the continuous BVP (3) and (4) is differentiable function x = x ( t ) that

(a) satisfies ( t , x ( t ) ) ∈ [ 0,1 ] × D for t ∈ [ 0,1 ] ,

(b) satisfies (3) for t ∈ [ 0,1 ] 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 [figure omitted; refer to PDF] are equivalent, with [figure omitted; refer to PDF]

Proof.

For completeness we provide a proof. Let x ~ be a solution to (1) and (2). If we sum (1) from 0 to i - 1 then we obtain [figure omitted; refer to PDF] and so for i = n we obtain [figure omitted; refer to PDF] Using boundary conditions (2) we can eliminate _{x n} in (8) to obtain [figure omitted; refer to PDF] Thus, substitution of (9) into (7) yields [figure omitted; refer to PDF] which can then be recast into form (5) by splitting the second term to sum from j = 0 to i - 1 and from j = i to n - 1 .

Now let x ~ 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 [ 0,1 ] × D of f will be the rectangle [figure omitted; refer to PDF] for some positive number b .

Since f is continuous on the compact set _{R b} we may define a number M ≥ 0 such that [figure omitted; refer to PDF]

The main result of this section is the following.

Theorem 2.

Let f : _{R b} [arrow right] R be continuous and let [figure omitted; refer to PDF] If [figure omitted; refer to PDF] then problem (1) and (2) has at least one solution x ~ ∈ ^{R n + 1} for each h ∈ ( 0,1 ) such that ( _{t i} , _{x i} ) ∈ _{R b} for i = 0 , ... , n .

Proof.

Consider summation equation (5) that, by Lemma 1, is equivalent to (1) and (2) and define the sequence of vectors ^{[varphi] ~ ( k )} [: =] ( ^{[varphi] 0 ( k )} , ... , ^{[varphi] n ( k )} ) for k = 0,1 , 2 , ... recursively by [figure omitted; refer to PDF] Firstly we show that our sequence of vectors ^{[varphi] ~ ( k )} is well defined for k = 0,1 , ... by showing that each | ^{[varphi] i ( k )} - w / ( u + v ) | <= b for i = 0 , ... , n and so ( _{t i} , ^{[varphi] i ( k )} ) ∈ _{R b} for each i = 0 , ... , n and k = 0,1 , ... . We use proof by induction.

From the definition of ^{[varphi] i ( 0 )} it is easy to see that | ^{[varphi] i ( 0 )} - w / ( u + v ) | <= b for i = 0 , ... , n . Now assume for some _{k 1} ≥ 0 we have | ^{[varphi] i ( _{k 1} )} - w / ( u + v ) | <= b for i = 0 , ... , n . From (18) we have for i = 0 , ... , n [figure omitted; refer to PDF] from (13). Thus, by induction, we have ( _{t i} , ^{[varphi] i ( k )} ) ∈ _{R b} for each i = 0 , ... , n and k = 0,1 , ... and so our sequence of vectors ^{[varphi] ~ ( k )} is well defined for each k = 0,1 , ... .

Furthermore, the above has shown that the sequence of vectors ^{[varphi] ~ ( k )} is uniformly bounded for k = 0,1 , ... .

We now show that ^{[varphi] ~ ( k + 1 )} ≥ ^{[varphi] ~ ( k )} for k = 0,1 , ... , where the inequality holds in a componentwise fashion. Once again, we use induction. For i = 0 , ... , n consider [figure omitted; refer to PDF] where we have used (14) and (16). Thus, ^{[varphi] ~ ( 1 )} ≥ ^{[varphi] ~ ( 0 )} .

Now assume that ^{[varphi] ~ ( _{k 1} )} ≥ ^{[varphi] ~ ( _{k 1} - 1 )} for some _{k 1} ≥ 1 ; that is, assume ^{[varphi] i ( _{k 1} )} ≥ ^{[varphi] i ( _{k 1} - 1 )} for i = 0 , ... , n . For each i = 0 , ... , n we have [figure omitted; refer to PDF] where we have used assumptions (14) and (15). Thus, ^{[varphi] ~ ( k + 1 )} ≥ ^{[varphi] ~ ( k )} for k = 0,1 , ... .

From the above we conclude that ^{[varphi] ~ ( k )} is a uniformly bounded and nondecreasing sequence of vectors and so must converge to a vector [varphi] ~ ; that is, [figure omitted; refer to PDF] for some [varphi] ~ ∈ ^{R n + 1} .

We finally show that the above [varphi] ~ = ( _{[varphi] 0} , ... , _{[varphi] n} ) ∈ ^{R n + 1} is actually a solution to (1) and (2). Since each | ^{[varphi] i ( k )} - w / ( u + v ) | <= b we must have each | _{[varphi] i} - w / ( u + v ) | <= b and so ( _{t i} , _{[varphi] i} ) ∈ _{R b} for i = 0 , ... , n . Furthermore, the continuity of f on _{R b} ensures that [figure omitted; refer to PDF] for each i = 0 , ... , n .

If we now take limits in (18) as k [arrow right] ∞ then we obtain [figure omitted; refer to PDF] so that our limit vector [varphi] ~ is a solution to (1) and (2).

Example 3.

Consider the following discrete BVP: [figure omitted; refer to PDF] so that we have a special case of (1) and (2) with [figure omitted; refer to PDF] We claim that problem (25) and (26) has at least one solution x ~ such that ( _{x i} ) <= 1 for i = 0 , ... , n .

Proof.

We show that all of the conditions of Theorem 2 hold. Firstly, we see that the inequalities in (14) hold. If we choose b = 1 to form _{R b} then M = 1 / 5 and so (13) holds. Furthermore, f 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 n = 4 , pick ^{[varphi] i ( 0 )} = 0 for i = 0 , ... , 4 and construct the approximating iterates ^{[varphi] i ( k )} as in (18). The numbers in Table 1 signify the error [figure omitted; refer to PDF] that results upon substituting the generated ^{[varphi] i ( k )} into (25).

We notice that the error in terms of ^{[varphi] i ( k )} at each i decreases for this example as ^{[varphi] i ( k )} converge upward to a solution in the rectangle, so that ^{[varphi] i ( 12 )} , for example, is a good approximation to a solution _{x i} of (25). The actual values of ^{[varphi] i ( 12 )} [approximate] _{x i} are given by [figure omitted; refer to PDF] Note that by construction, ^{[varphi] i ( k )} satisfies boundary condition (26); namely, [figure omitted; refer to PDF]

Table 1

i	0	1	2	3	4
[...] i ( 2 )	0.00206246	0.00244859	0.00262248	0.00264573	0.00259853
[...] i ( 4 )	0.0000898399	0.000100995	0.000108355	0.000112198	0.000113191
[...] i ( 6 )	3.5115 × 1 0 - 6	3.9453 × 1 0 - 6	4.2343 × 1 0 - 6	4.3843 × 1 0 - 6	4.4242 × 1 0 - 6
[...] i ( 8 )	1.3681 × 1 0 - 7	1.5372 × 1 0 - 7	1.6497 × 1 0 - 7	1.7082 × 1 0 - 7	1.7238 × 1 0 - 7
[...] i ( 10 )	5.3299 × 1 0 - 9	5.9883 × 1 0 - 9	6.4269 × 1 0 - 9	6.6545 × 1 0 - 9	6.7152 × 1 0 - 9
[...] i ( 12 )	2.0764 × 1 0 - 10	2.3329 × 1 0 - 10	2.5037 × 1 0 - 10	2.5924 × 1 0 - 10	2.6161 × 1 0 - 10

The following result is a modification of the ideas in Theorem 2 and its proof.

Theorem 5.

Let f : _{R b} [arrow right] R be continuous and let [figure omitted; refer to PDF] If [figure omitted; refer to PDF] then problem (1) and (2) has at least one solution x ~ ∈ ^{R n + 1} such that ( _{t i} , _{x i} ) ∈ _{R b} for i = 0 , ... , n .

Proof.

The proof is very similar to that of Theorem 2 and so is only outlined.

Consider the sequence of successive approximations defined by [figure omitted; refer to PDF] for k = 0,1 , 2 , ... . The continuity of f 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.

Remark 6.

Note that (31) is a stronger assumption than (13), while (32) is weaker than (14).

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 approximation _{[varphi] ~ 0} in the proofs of Theorems 2 and 5 was chosen to be a constant vector with components w / ( u + v ) . With suitable modifications on (16) we may use any vector _{[varphi] ~ 0} as our initial approximation provided ( _{t i} , ^{[varphi] i ( 0 )} ) ∈ _{R b} for i = 0 , ... , n . For BVPs that have more than one solution, different choices in our initial approximation _{[varphi] ~ 0} can lead to the generation of distinct limit functions [varphi] ~ . That is, through various choices of _{[varphi] ~ 0} we can observe convergence of ^{[varphi] ~ ( k )} to various solutions of (1) and (2).

3. 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 [arrow right] 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 _{n m} [arrow right] ∞ as m [arrow right] ∞ ; let 0 < _{h m} = 1 / _{n m} < 1 ; and let ^{t i m} = i _{h m} for i = 0 , ... , n . If problem (1) and (2) has a solution for h = _{h m} and m ≥ _{m 0} that we denote by [figure omitted; refer to PDF] then we construct the following sequence of continuous functions from (35) via linear interpolation to form [figure omitted; refer to PDF] for m ≥ _{m 0} and t ∈ [ 0,1 ] . Note that ^{x m} ( ^{t i m} ) = ^{x i m} for i = 0 , ... , n .

Lemma 9.

Let f : [ 0,1 ] × D ⊆ [ 0,1 ] × R [arrow right] R be continuous and let R ≥ 0 be a constant. If problem (1) and (2) has a solution for h <= _{h m} and m ≥ _{m 0} that we denote by ^{x ~ m} with [figure omitted; refer to PDF] then problem (3) and (4) has a solution x = x ( t ) that is the limit of a subsequence of (36).

Proof.

For m ≥ _{m 0} consider the sequence of functions ^{x m} ( t ) for t ∈ [ 0,1 ] in (36). We show that the sequence of functions ^{x m} is uniformly bounded and equicontinuous on [ 0,1 ] . For t ∈ [ ^{t i m} , ^{t i + 1 m} ] and m ≥ _{m 0} we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Thus, ^{x m} is uniformly bounded on [ 0,1 ] .

For β , γ ∈ [ 0,1 ] and given [straight epsilon] > 0 , consider [figure omitted; refer to PDF] whenever ( β - γ ) < δ ( [straight epsilon] ) [: =] [straight epsilon] / _{M 1} . Thus, ^{x m} is equicontinuous on [ 0,1 ] .

The convergence Arzela-Ascoli theorem [12, p. 527] guarantees that the sequence of continuous functions ^{x m} = ^{x m} ( t ) has a subsequence ^{x k ( m )} ( t ) that converges uniformly to a continuous function x = x ( t ) for t ∈ [ 0,1 ] . That is, [figure omitted; refer to PDF]

The continuity of f ensures that the above limit function will be a solution to (3) and (4).

The next theorem, in the spirit of [7, Theorem 2.5 ], will require the following notation. If problem (1) and (2) has a solution x ~ for 0 < h <= _{h 0} then we define the continuous function x ( t , x ~ ) by [figure omitted; refer to PDF]

Theorem 10.

Let f : [ 0,1 ] × D ⊆ [ 0,1 ] × R [arrow right] R be continuous and let R ≥ 0 be a constant. Assume problem (1) and (2) has a solution for h <= _{h 0} that we denote by x ~ with [figure omitted; refer to PDF] Given [straight epsilon] > 0 there exists δ = δ ( [straight epsilon] ) such that if h <= δ then problem (3) and (4) has a solution x = x ( t ) with [figure omitted; refer to PDF]

Proof.

Suppose, for some [straight epsilon] > 0 , there is a sequence _{h m} such that _{h m} [arrow right] 0 as m [arrow right] ∞ and for h = _{h m} = 1 / _{n m} problem (1) and (2) has a solution ^{x ~ m} with every solution x = x ( t ) to (3) and (4) satisfying [figure omitted; refer to PDF] By assumption, for m sufficiently large, there is R ≥ 0 such that the solution ^{x ~ m} to (1) and (2) satisfies [figure omitted; refer to PDF] Thus, the conditions of Lemma 9 are satisfied and so we obtain a subsequence ^{x k ( m )} ( t ) of ^{x m} ( t ) that converges uniformly on [ 0,1 ] to a solution x of (3) and (4). Thus, (45) cannot hold.

We now relate the above abstract results to the ideas from earlier sections.

Theorem 11.

Let the conditions of Theorem 2 hold. Given any [straight epsilon] > 0 there is δ = δ ( [straight epsilon] ) such that if h <= δ then problem (3) and (4) has a solution x that satisfies (44).

Proof.

We show that the conditions of Theorem 10 are satisfied for _{R b} = [ 0,1 ] × D . Assumption (13) ensures that the solution x ~ to (1) and (2) guaranteed to exist by Theorem 2 satisfies | _{x i} | <= b for i = 0 , ... , n and so (43) holds with R = b .

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.

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