Dynamic Systems and Applications 15 (2006) 111-118

SOLUTIONS OF MULTIPOINT BOUNDARY VALUE PROBLEMS FOR SECOND ORDER EQUATIONS

JOHNNY HENDERSON

Baylor University, Department of Mathematics, Waco, Texas 76798-7328, USA E-mail: Johnny Henderson@baylor.edu

Dedicated to John Baxley on the occasion of his retirement from Wake Forest University.

ABSTRACT. Shooting methods are employed to obtain solutions of the second order differential equation, y′′ = f(x, y, y′), satisfying the multipoint nonlocal boundary conditions, y(x_{0}) = y_{1}, y(x_{n+1}) − ∑_{i=1}^{n} y(x_{i}) = y_{2}, a < x_{0} < x_{1} < ⋯ < x_{n+1} < b, and y_{1}, y_{2} ∈ ℝ, where f : (a, b) × ℝ^{2} → ℝ is continuous. It is assumed that solutions of such problems are unique, when they exist. 2000 MSC: 34B10, 34B15.

1. INTRODUCTION

Let n ≥ 1 be fixed. We are concerned with the question of uniqueness of solutions implying their existence for (n + 2)-point nonlocal boundary value problems for the second order ordinary differential equation,

(1.1)  y′′ = f(x, y, y′), a < x < b.

In particular, given points a < x_{0} < x_{1} < ⋯ < x_{n+1} < b, and y_{1}, y_{2} ∈ ℝ, we shall consider uniqueness implies existence results for solutions of (1.1) satisfying the boundary conditions,

(1.2)  y(x_{0}) = y_{1}, y(x_{n+1}) − ∑_{i=1}^{n} y(x_{i}) = y_{2}.

There has been much interest in multipoint boundary value problems for second order differential equations with the papers by Bai and Fang [1], Gupta and Trofimchuk [15], Ma [26, 27] and Yang [30] among some of the recent principal works.

We assume throughout:

(A) f(x, r_{1}, r_{2}) : (a, b) × ℝ^{2} → ℝ is continuous, and (B) Solutions of initial value problems for (1.1) are unique and extend to (a, b).

Received November 1, 2004               1056-2176 $03.50 \begin{matrix}c \\ ◯ \end{matrix}Dynamic Publishers, Inc.

112 JOHNNY HENDERSON

Partial motivation for this paper is a couple of recent papers by Henderson [20, 21] that were devoted to certain nonlocal 3-point and 4-point boundary value problems for (1.1). There have been several papers written in which uniqueness conditions on solutions have led to the existence of solutions. For ordinary differential equations, the papers by Baxley et al. [2, 3, 5, 6, 10], Hartman [16], Henderson et al. [17, 22], Klaasen [24], Lasota and Luczynski [25] and Peterson [28] constitute some of the main contributions, and for finite difference equations, Davis and Henderson [14] and Henderson [18, 19] obtained some uniqueness implies existence results for conjugate and focal boundary value problems. In the context of boundary value problems for dynamic equations on time scales, Bohner and Peterson [12], Chyan [13] and Henderson and Yin [23] have also obtained uniqueness implies existence results. Key to each of these papers have been uses made of continuous dependence of solutions on boundary conditions along with extensive use of modified shooting methods. Baxley has exploited these continuity properties and coupled them with computational shooting to produce numerical solutions in some of the above scenarios; see, for example, [4, 7, 8, 9, 11].

Our uniqueness condition on solutions of the boundary value problem (1.1), (1.2) takes the following form:

(C) Given points a < x_{0} < x_{1} < ⋯ < x_{n+1} < b, if y and z are solutions of (1.1) such that

y(x_{0}) = z(x_{0}),
y(x_{n+1}) − ∑_{i=1}^{n} y(x_{i}) = z(x_{n+1}) − ∑_{i=1}^{n} z(x_{i}),

then y(x) = z(x), for all a < x < b.

Remark 1 If n = 1, so that (1.2) is given by

(1.3)  y(x_{0}) = y_{1}, y(x_{2}) − y(x_{1}) = y_{2},

and if (A) - (C) are satisfied, then for any points a < r_{1} < r_{2} < b, the 2-point conjugate boundary value problem for (1.1) satisfying

(1.4)  y(r_{1}) = s_{1}, y(r_{2}) = s_{2},

s_{1}, s_{2} ∈ ℝ, has at most one solution on (a, b). It follows in turn from [25] that, for any a < r_{1} < r_{2} < b and any s_{1}, s_{2} ∈ ℝ, there exists a unique solution of the 2-point conjugate boundary value problem (1.1), (1.4). Remark 2 Again, if n = 1, and if (A) - (C) are satisfied, then Henderson [20] proved there exists a unique solution of the 3-point problem (1.1), (1.3).

In outlining the development of this paper, it will first be shown that, if (C) holds, then for each 1 ≤ k ≤ n, solutions of (1.1) satisfying

(1.5)  y(x_{0}) = y_{1}, y(x_{k+1}) − ∑_{i=1}^{k} y(x_{i}) = y_{2},

a < x_{0} < ⋯ < x_{k+1} < b and y_{1}, y_{2} ∈ ℝ, are unique, when they exist. After that, an inductive argument will be given showing conditions (A) - (C) imply the existence of solutions of (1.1), (1.2).

Fundamental to the uniqueness implies existence arguments will be our use of a precompactness condition on uniformly bounded sequences of solutions of (1.1). We now state the condition as a theorem, the proof of which is straight-forward.

Theorem 1.1. Assume that with respect to (1.1), conditions (A) and (B) hold. If {y_{k}(x)} is a sequence of solutions of (1.1) for which there exists an interval [c, d] ⊂ (a, b) and there exists an M > 0 such that |y_{k}(x)| ≤ M, for all x ∈ [c, d] and for all k ∈ ℕ, then there exists a subsequence {y_{k_{j}}(x)} such that, for i = 0, 1, {y_{k_{j}}^{(i)}(x)} converges uniformly on each compact subinterval of (a, b).

2. UNIQUENESS OF (1.1), (1.2) IMPLIES UNIQUENESS OF (1.1), (1.5)

In this section, we prove that conditions (A) - (C) yield the uniqueness of solutions of (1.1), (1.5), for each 1 ≤ k ≤ n. After having established that, we then prove a theorem concerning continuous dependence of solutions of (1.1), (1.5) on boundary conditions.

Theorem 2.1. Assume that conditions (A), (B) and (C) hold. Then, given 1 ≤ k ≤ n, solutions of (1.1), (1.5) are unique, when they exist.

Proof. Of course from assumption (C), the condition is true when k = n. So we assume for the purpose of contradiction that, for some 1 ≤ k < n, there exist distinct solutions y and z of (1.1) and points a < x_{0} < ⋯ < x_{k+1} < b, so that

y(x_{0}) = z(x_{0}),
y(x_{k+1}) − ∑_{i=1}^{k} y(x_{i}) = z(x_{k+1}) − ∑_{i=1}^{k} z(x_{i}).

Set w = y − z. Then,

w(x_{0}) = 0, w(x_{k+1}) − ∑_{i=1}^{k} w(x_{i}) = 0.

114 JOHNNY HENDERSON

By (B), we may assume w′(x_{0}) > 0, and so there exists x_{0} < τ < b so that w′(x) > 0 on [x_{0}, τ]. Then there exist x_{0} < τ_{1} < ⋯ < τ_{n} < τ such that ∑_{i=1}^{n} w(τ_{i}) < w(τ). By the monotonicity of w on [x_{0}, τ], there exists τ_{n} < τ_{n+1} < τ such that

w(τ_{n+1}) = ∑_{i=1}^{n} w(τ_{i}).

Since w(x_{0}) = 0, we conclude that

y(x_{0}) = z(x_{0}),
y(τ_{n+1}) − ∑_{i=1}^{n} y(τ_{i}) = z(τ_{n+1}) − ∑_{i=1}^{n} z(τ_{i}),

which is a contradiction to assumption (C). The proof is complete.

We complete this section by stating a theorem showing that under the hypotheses of Theorem 2.1, solutions of (1.1) satisfying (1.5), for each 1 ≤ k ≤ n, depend continuously on boundary conditions. The key tool for proving the theorem involves the Brouwer theorem on invariance of domain [29]. The continuous dependence result is fairly standard, and so we refer the readers to [20] for a typical argument.

Theorem 2.2. Assume that conditions (A), (B) and (C) hold. Also, let 1 ≤ k ≤ n be given. Given a solution y(x) of (1.1) on (a, b), an interval [c, d] ⊂ (a, b), points c < x_{0} < ⋯ < x_{k+1} < d, and an ϵ > 0, there exists a δ(ϵ, [c, d]) > 0 such that, if |x_{i} − t_{i}| < δ, 0 ≤ i ≤ k + 1, and c < t_{0} < ⋯ < t_{k+1} < d, and if |y(x_{0}) − z_{1}| < δ, and |y(x_{k+1}) − ∑_{i=1}^{k} y(x_{i}) − z_{2}| < δ, then there exists a solution z(x) of (1.1) satisfying z(t_{0}) = z_{1}, z(t_{k+1}) − ∑_{i=1}^{k} z(t_{i}) = z_{2}, and |y^{(i)}(x) − z^{(i)}(x)| < ϵ on [c, d], i = 0, 1.

3. EXISTENCE OF SOLUTIONS

In this section, we show that condition (C) leads to solutions of (1.1), (1.5), for each 1 ≤ k ≤ n. The arguments rely on continuous dependence, the precompactness condition, and a shooting technique.

Theorem 3.1. Assume conditions (A) - (C) hold. Then, for each 1 ≤ k ≤ n, there exists a unique solution of (1.1), (1.5).

Proof. First, solutions of (1.1), (1.5), for each 1 ≤ k ≤ n, are unique by Theorem 2.1. In addition, as mentioned in Remark 2, when k = 1, the statement of this theorem is true.

So assume 1 < k ≤ n and that, for each 1 ≤ h < k, given points a < t_{0} < ⋯ < t_{h+1} < b, and given z_{1}, z_{2} ∈ ℝ, there exists a unique solution z of (1.1) satisfying,

z(t_{0}) = z_{1}, z(t_{h+1}) − ∑_{i=1}^{h} z(t_{i}) = z_{2}.

Now choose points a < x_{0} < ⋯ < x_{k+1} < b and y_{1}, y_{2} ∈ ℝ. By the inductive hypotheses, when h = k − 1, there exists a unique solution z of (1.1) satisfying

z(x_{1}) = 0, z(x_{k+1}) − ∑_{i=2}^{k} z(x_{i}) = y_{2}.

Next, we define a set

S = {u(x_{0}) | u is a solution of (1.1) satisfying
u(x_{k+1}) − ∑_{i=1}^{k} u(x_{i}) = z(x_{k+1}) − ∑_{i=1}^{k} z(x_{i})}.

We observe that S is nonempty, since z(x_{0}) ∈ S.

To see that S is an open subset, choose ξ ∈ S. Then, there exists a solution u_{ξ} of (1.1) such that

u_{ξ}(x_{0}) = ξ, u_{ξ}(x_{k+1}) − ∑_{i=1}^{k} u_{ξ}(x_{i}) = z(x_{k+1}) − ∑_{i=1}^{k} z(x_{i}).

By Theorem 2.2, there exists a δ > 0 such that, for each |η − ξ| < δ, there exists a solutions u_{η} of (1.1) such that

u_{η}(x_{0}) = η, u_{η}(x_{k+1}) − ∑_{i=1}^{k} u_{η}(x_{i}) = u_{ξ}(x_{k+1}) − ∑_{i=1}^{k} u_{ξ}(x_{i}) = z(x_{k+1}) − ∑_{i=1}^{k} z(x_{i}),

or in other words, η ∈ S; in particular, (ξ − δ, ξ + δ) ⊂ S, and S is an open subset of ℝ.

The remainder of the argument is devoted to showing that S is also a closed subset of ℝ. To that end, we assume for the purpose of contradiction that S is not closed. Then there exists an r_{0} ∈ \overline{S} \ S and a strictly monotone sequence \{r_{j}\} ⊂ S such that lim_{j→∞} r_{j} = r_{0}.

We may assume, without loss of generality, that r_{j} ↑ r_{0}. By the definition of S, we denote, for each j ∈ ℕ, by u_{j}(x) the solution of (1.1) satisfying

u_{j}(x_{0}) = r_{j}, u_{j}(x_{k+1}) − ∑_{i=1}^{k} u_{j}(x_{i}) = z(x_{k+1}) − ∑_{i=1}^{k} z(x_{i}).

By uniqueness of solutions of (1.1), (1.2), we have

u_{j}(x) < u_{j+1}(x) on (a, x_{2}).

Consequently, from Theorem 1.1 and the fact that r_{0} ∉ S, we may conclude that {u_{j}(x)} is not uniformly bounded above on each compact subinterval of each of (a, x_{1}) and (x_{1}, x_{2}).

Now let w(x) be the solution of the initial value problem for (1.1) satisfying

w(x_{0}) = r_{0}, w′(x_{0}) = 0.

It follows that, for some J large, there exist points a < τ_{1} < x_{0} < τ_{2} < x_{2} so that

u_{J}(τ_{1}) = w(τ_{1}) and u_{J}(τ_{2}) = w(τ_{2}).

But this contradicts the part of the statement in Remark 1 concerning the uniqueness of solutions of the 2-point conjugate boundary value problem (1.1), (1.4). Thus, S is also a closed subset of ℝ.

In summary, S is a nonempty subset of ℝ that is both open and closed. We have S ≡ ℝ. By choosing y_{1} ∈ S, there is a corresponding solution y(x) of (1.1) such that

y(x_{0}) = y_{1}, y(x_{k+1}) − ∑_{i=1}^{k} y(x_{i}) = z(x_{k+1}) − ∑_{i=1}^{k} z(x_{i}) = y_{2}.

This completes the proof.

For completeness of the paper, we state as a corollary to Theorem 3.1 in the case when k = n, a uniqueness implies existence result with respect to (1.1), (1.2).

Corollary 3.2. Assume conditions (A) - (C) hold. Then, given a < x_{0} < ⋯ < x_{n+1} < b and y_{1}, y_{2} ∈ ℝ, there exists a unique solution of (1.1), (1.2).

REFERENCES

[1] C. Bai and J. Fang, Existence of multiple positive solutions for m-point boundary value problems, J. Math. Anal. Appl. 281 (2003), 76–85.

[2] J. V. Baxley, Global existence and uniqueness for second-order ordinary differential equations, J. Diff. Eqs., 23 (1977), 315–334.

[3] J. V. Baxley, Nonliner second-order boundary value problems: intervals of existence, uniqueness and continuous dependence. J. Diff. Eqs. 45 (1982) 389–407.

[4] J. V. Baxley, A singular nonlinear boundary value problem: membrane response to a spherical cap, SIAM J. Appl. Math., 48 (1988), 497–505.

[5] J. V. Baxley, Existence theorems for nonlinear second order boundary value problems, J. Diff. Eqs. 85 (1990), 125–150.

[6] J. V. Baxley and S. E. Brown, Existence and uniqueness for two-point boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 219–234.

[7] J. V. Baxley and L. J. Haywood, Nonlinear boundary value problems with multiple solutions, Nonlinear Anal. 47 (2001), 1187–1198.

[8] J. V. Baxley and L. J. Haywood, Multiple positive solutions of nonlinear boundary value problems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10 (2003), 157–167.

[9] J. V. Baxley and C. R. Houmand, Nonlinear higher order boundary value problems with multiple positive solutions, J. Math. Anal. Appl. 286 (2003), 682–691.

[10] J. V. Baxley and H. B. Thompson, Nonlinear boundary value problems and competition in the chemostat, Nonlinear Anal. 22 (1994), 1329–1344.

[11] J. V. Baxley and H. B. Thompson, Boundary behavior and computation of solutions of singular nonlinear boundary value problems, Commun. Appl. Anal. 4 (2000), 207–226.

[12] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkauser, Boston, 2001.

[13] C. J. Chyan, Uniqueness implies existence for second order differential equations on measure chains, J. Math. Anal. Appl. 258 (2001), 359–365.

[14] J. M. Davis and J. Henderson, Uniqueness implies existence for fourth-order Lidstone boundary value problems, Panamer. Math. J. 8 (1998), 23–35.

[15] C. P. Gupta and S. I. Trofimchuk, Solvability of a multi-point boundary value problem and related a priori estimates, Canad. Appl. Math. Quart. 6 (1998), 45–60.

[16] P. Hartman, On n-parameter families and interpolation problems for nonlinear ordinary differential equations, Trans. Amer. Math. Soc. 154 (1971), 201-226.

[17] J. Henderson, Existence of solutions of right focal point boundary value problems for ordinary differential equations, Nonlinear Anal. 5 (1981), 989–1002.

[18] J. Henderson,. Focal boundary value problems for nonlinear difference equations, II, J. Math. Anal. Appl. 141 (1989), 559–567.

[19] J. Henderson, Existence theorems for boundary value problems for nth-order nonlinear difference equations, SIAM J. Math Anal. 20 (1989), 468-478.

[20] J. Henderson, Uniqueness implies existence for three-point boundary value problems for second order differential equations, Appl. Math. Lett. 17 (2004), 1391–1395.

[21] J. Henderson, Uniqueness implies existence for four-point nonlocal boundary value problems for second order equations, Int. J. Appl. Math. Sci., in press.

[22] J. Henderson, B. Karna and C. C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations, Proc. Amer. Math. Soc., in press.

[23] J. Henderson and W. K. C. Yin, Two-point and three-point problems for fourth order dynamic equations, Dynam. Systems Appl. 12 (2003), 159–169.

[24] G. Klaasen, Existence theorems for boundary value problems for nth order ordinary differential equations, Rocky Mtn. J. Math. 3: (1973), 457–472.

[25] A. Lasota and M. Luczynski, A note on the uniqueness of two point boundary value problems I, Zeszyty Naukowe UJ, Prace matematyezne 12 (1968), 27–29.

[26] R. Ma, Existence theorems for a second-order three-point boundary value problem, J. Math. Anal. Appl. 212 (1997), 430–442.

[27] R. Ma, Existence and uniqueness of solutions to first-order three-point boundary value problems, Appl. Math. Lett., 15 (2002), 211–216.

[28] A. C. Peterson, Existence-uniqueness for focal-point boundary value problems, SIAM J. Math. Anal. 12 (1982), 173–185.

[29] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.

[30] B. Yang, Boundary Value Problems for Ordinary Differential Equations, Ph.D. Dissertation, Mississippi State University, Mississippi State, MS, 2002.