Discrete First-Order Three-Point Boundary Value Problem

Abstract We study difference equations which arise as discrete approximations to three-point boundary value problems for systems of first-order ordinary differential equations. We obtain new results of the existence of solutions to the discrete problem by employing Euler’s method. The existence of solutions are proven by the contraction mapping theorem and the Brouwer fixed point theorem in Euclidean space. We apply our results to show that solutions to the discrete problem converge to solutions of the continuous problem in an aggregate sense. We also give some examples to illustrate the existence of a unique solution of the contraction mapping theorem.


Introduction
In this paper, we study the three-point boundary value problem (1) which arises as a discrete approximation to the continuous problem (3) Here f is a continuous, vector-valued and possibly nonlinear function, the step size h = (c−a)/n and grid points t k = a+kh for k = 0, • • • , n. M, N and R are given d × d matrices, and α ∈ R d .Let s ∈ {0, 1, • • • , n − 1} be such that s < b < s + 1. Choose a θ ∈ [0, 1] so that x ŝ = θx s+1 + (1 − θ)x s .Thus we approximate x ŝ by linear interpolation (see McCormick, 1964, p. 50-51).
¢ www.ccsenet.org/jmrISSN: 1916-9795 Numerical solutions to (3), (4) involve discretization.The numerical methods of interest provide solutions that closely approximate the exact solutions for sufficiently small step size (see Keller, 1991).Since the Euler method is the simplest numerical scheme for solving initial value problems, we employ this method for approximating the solution of (3), (4), requiring an extremely small step size.
The discretized boundary value problems and the 'effect' that this discretization may have on possible solutions when compared with solutions to the original continuous boundary value problem, have been researched by (Agarwal, 1985), (Gaines, 1974) and (Lasota, 1968).For example, (Agarwal, 1985) provides some examples showing that even though the continuous boundary value problem may have a solution, its discretization may have no solution.Thus we formulate a convergence theorem which is a generalization of Theorem 2.5, (Gaines, 1974) showing that if solutions to the continuous problem (3), (4) are unique, then solutions to the discrete problem (1), (2) converge to solutions of the continuous problem.
The primary motivation here for the research in this paper is the work by (Ma, 2002) who studied the existence and uniqueness of solutions of three-point boundary value problems using the Leray-Schauder continuation theorem when f is a Carathéodory function.In this work, by employing Euler's method, we obtain new results of the existence of solutions to (1), ( 2) with uniqueness as well as the existence of solutions to (1), (2) without uniqueness.We prove existence and uniqueness results for nonlinear boundary value problems using the contraction mapping theorem and the Brouwer fixed point theorem in Euclidean space.We also give some examples to illustrate the existence of a unique solution of the contraction mapping theorem.

Denote
Let B be a d × d matrix with elements b i j , with the norm   3) for all t ∈ [a, c] and (4).By a solution to (1), we mean a vector (2).The value of the k th component x k of a solution x of (1) is expected to approximate x(t k ), for some solution x of (3).We assume the following: Assumption (A1).M, N and R are constant square matrices of order d such that Then where, and 0 j=1 he j = 0 by definition.Proof We have x s = x 0 + s j=1 he j , x s+1 = x 0 + s j=1 he j + he s+1 , x ŝ = x 0 + θhe s+1 + s j=1 he j .Combining this with (2), we conclude that where 0 j=1 he j = 0 by definition.Thus where and .∞ is given in (5).For t 0 ≤ t s ≤ t s+1 ≤ t k ≤ t n , we have where 0 j=s+1 he j = 0 by definition.Thus where and .∞ is given in (5).
Combining ( 11) with (13), we obtain where and w be defined by ( 16).Then the problem has a unique solution x = u + w if and only if u is the only solution of Proof Suppose u is the only solution of ( 19), (20 and Conversely, suppose x = u + w is the only solution of ( 17), ( 18).Then it is clear that Thus the proof is complete.

Existence Results
In this section, first to obtain an existence theorem without uniqueness of the solution, we will apply the Brouwer Fixed Point Theorem which is given in (Keller, 1991, p. 382).Then we shall use the contraction mapping theorem which is given in (Keller, 1991, p. 372) to establish the existence of a unique solution to the boundary value problem (1), (2).
then the three-point boundary value problem (1), (2) has at least one solution.
Proof In view of Lemma 4, to prove that (1), (2) has a solution x = u + w, it suffices to prove the following problem has a solution in u, where w is defined by ( 16).The general solution of ( 23), ( 24) is ¢ www.ccsenet.org/jmrwhere 0 j=1 h f (t j , u j + w) = 0 by definition.By the continuity of f , T is continuous.We now show that T : Ω −→ Ω.Then by ( 21) we have For t 0 ≤ t k ≤ t s ≤ t s+1 ≤ t n we have where 0 j=1 h f (t j , u j + w) = 0 by definition.Thus where Γ 1,0 is given in ( 12).Also, for t 0 ≤ t s ≤ t s+1 ≤ t k ≤ t n we have where 0 j=s+1 h f (t j , u j + w) = 0 by definition.Thus where Γ 2,0 is given in ( 14).Combining ( 28) with ( 29), we obtain where Γ 0 is given in (9) and Hence T u ∈ Ω, and the conclusion follows from the Brouwer Fixed Point Theorem that there exists at least one solution to the boundary value problem ( 1), ( 2 then the three point boundary value problem (1), (2) has a unique solution.
Proof The proof is similar to the proof of Theorem 1.Let l > 0, and set By the continuity of f , T is continuous.We have already shown that T : Ω −→ Ω in the proof of Theorem 1. Then by (31) we have We obtained where Γ 0 is given in ( 9) and Hence T u ∈ Ω.
We shall prove that T : Also, for t 0 ≤ t s ≤ t s+1 ≤ t k ≤ t n we have Combining ( 35) with (36), we obtain ¢ www.ccsenet.org/jmrwhere Γ 0 is given in (9).It follows that for Γ 0 h(n + θ) p < 1, T u = u has a unique solution u in Ω ⊂ X = R (n+1)d .This fixed point is the unique solution of the boundary value problem (1), (2).This completes the proof of the theorem.
The following two examples give the existence of a unique solution of Theorem 2.

Example 1
Consider the following discrete boundary value problem where The boundary value problem ( 37), ( 38) has a unique solution.
Example 2 Consider the following discrete boundary value problem

Convergence of Solutions
In this section, the previous results are applied to formulate a convergence theorem.The following is a generalization of Theorem 2.5, (Gaines, 1974).
Proof The proof is similar to that of (Gaines, 1974) and so is omitted.
Remark 1 It follows from Theorem 3 that if the solutions to the continuous problem (3), (4) are unique, then solutions to (1), (2) converge to solutions of the continuous problem in the sense of Theorem 3.
with norms e j = max 1≤i≤d | e ji |, where | .| denotes the modulus of e ji ∈ R. By abuse of notation we let e ∞ = max 1≤ j≤n e j .Let p k