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,
The following result is a modification of the ideas in Theorem 2 and its proof.
Theorem 5. Let be continuous and let If then problem (1) and (2) has at least one solution such that for .
Proof. The proof is very similar to that of Theorem 2 and so is only outlined.
Consider the sequence of successive approximations defined by for . 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.
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 in the proofs of Theorems 2 and 5 was chosen to be a constant vector with components . With suitable modifications on (16) we may use any vector as our initial approximation provided for . For BVPs that have more than one solution, different choices in our initial approximation can lead to the generation of distinct limit functions . That is, through various choices of we can observe convergence of 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 , 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 ], 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 .
We require the following notation. Denote the sequence as ; let ; and let for . If problem (1) and (2) has a solution for and that we denote by then we construct the following sequence of continuous functions from (35) via linear interpolation to form for and . Note that for .
Lemma 9. Let be continuous and let be a constant. If problem (1) and (2) has a solution for and that we denote by with then problem (3) and (4) has a solution that is the limit of a subsequence of (36).
Proof. For consider the sequence of functions for in (36). We show that the sequence of functions is uniformly bounded and equicontinuous on . For and we have where Thus, is uniformly bounded on .
For and given , consider whenever . Thus, is equicontinuous on .
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 . That is, The continuity of ensures that the above limit function will be a solution to (3) and (4).
The next theorem, in the spirit of [7, Theorem ], will require the following notation. If problem (1) and (2) has a solution for then we define the continuous function by
Theorem 10. Let be continuous and let be a constant. Assume problem (1) and (2) has a solution for that we denote by with Given there exists such that if then problem (3) and (4) has a solution with
Proof. Suppose, for some , there is a sequence such that as and for problem (1) and (2) has a solution with every solution to (3) and (4) satisfying By assumption, for sufficiently large, there is such that the solution to (1) and (2) satisfies Thus, the conditions of Lemma 9 are satisfied and so we obtain a subsequence of that converges uniformly on to a solution 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 there is such that if then problem (3) and (4) has a solution that satisfies (44).
Proof. We show that the conditions of Theorem 10 are satisfied for . Assumption (13) ensures that the solution to (1) and (2) guaranteed to exist by Theorem 2 satisfies for and so (43) holds with .
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.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.