Electronic Journal of Qualitative Theory of Differential Equations Spec. Ed. I, 2009 No. 11, 1–9; http://www.math.u-szeged.hu/ejqtde/

UNIQUENESS IMPLIES EXISTENCE OF SOLUTIONS FOR NONLINEAR (k; j) POINT BOUNDARY VALUE

PROBLEMS FOR NTH ORDER DIFFERENTIAL EQUATIONS

Jeffrey Ehme

Department of Mathematics, Spelman College Atlanta, GA 30314 USA e-mail: jehme@spelman.edu

Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday

Abstract Given appropriate growth conditions for f and a uniqueness assumption on y^{(n)} = 0 with respect to certain (k; j) point boundary value problems, it is shown that uniqueness of solutions to the nonlinear differential equation

y^{(n)} = f(t, y, y′, …, y^{(n−1)}),

subject to nonlinear (k; j) boundary conditions of the form

g_{ij}(y(t_{j}), …, y^{(n−1)}(t_{j})) = y_{ij},

implies existence of solutions.

Key words and phrases: (k; j) point boundary conditions, nonlinear, continuous dependence, uniqueness, existence. AMS (MOS) Subject Classifications: 34B10, 34B15

1 Introduction

In this paper, we will consider the differential equation, n ≥ 3,

y^{(n)} = f(t, y, y′, …, y^{(n−1)}), a < x < b.  (1)

We begin by defining an apppropriate linear point boundary condition. Given 1 ≤ j ≤ n − 1 and 1 ≤ k ≤ n − j, positive integers m_{1}, …, m_{k} such that m_{1} + ⋯ + m_{k} = n − j, points a < x_{1} < ⋯ < x_{k} < x_{k+1} < ⋯ < x_{k+j} < b, real numbers y_{il}, 1 ≤ i ≤ m_{l}, 1 ≤ l ≤ k, and real numbers y_{n−j+l}, 1 ≤ l ≤ j, a boundary condition of the form

\begin{matrix}y^{(i−1)}(x_{l}) = y_{il}, 1 ≤ i ≤ m, 1 ≤ l ≤ k, \\ y′(x_{k+l}) = y_{n−j+l}, 1 ≤ l ≤ j, \end{matrix}  (2)

will be referred to as a linear (k; j) point boundary condition, and a condition of the form

\begin{matrix}g_{il}(y(t_{l}), …, y^{(n−1)}(t_{l})) = y_{il}, 1 ≤ i ≤ m, 1 ≤ l ≤ k, \\ g_{kl}(y(t_{k+l}), …, y^{(n−1)}(t_{k+l})) = y_{n−j+l}, 1 ≤ l ≤ j, \end{matrix}  (3)

will be referred to as a nonlinear (k; j) point boundary condition. If

g_{il}(x_{0}, x_{1}, …, x_{n−1}) = x_{i−1}

for 1 ≤ i ≤ m, 1 ≤ l ≤ k, and if

g_{kl}(x_{0}, x_{1}, …, x_{n−1}) = x_{1}

1 ≤ l ≤ j, then the nonlinear (k; j) boundary conditions become linear (k; j) boundary conditions. Our interest in these boundary conditions is stimulated by recent work by Eloe and Henderson in [6].

Moreover, it will be assumed that for x_{p} ≥ 0, x_{p} ≤ g_{pl}(x_{0}, x_{1}, …, x_{n−1}), and for x_{p} < 0, g_{pl}(x_{0}, x_{1}, …, x_{n−1}) ≤ x_{p}, for all p and l. We will refer to this property by saying the nonlinear (k; j) condition dominates the linear (k; j) condition. Let (x_{0}, …, x_{n−1}) denote a vector in R^{n}. If h : R → R is an odd continuous function and k : R^{n} → R is any continuous positive function, then g_{il}(x_{0}, …, x_{n−1}) = x_{i−1}+h(x_{i−1})k(x_{0}, …, x_{n−1}), 1 ≤ i ≤ m, 1 ≤ l ≤ k and g_{kl}(x_{0}, …, x_{n−1}) = x_{1} + h(x_{1})k(x_{0}, …, x_{n−1}), 1 ≤ l ≤ j, is an example of such a boundary condition satisfying the property.

Our goal will be to establish uniqueness implies existence results for (1), (2) and (1), (3). Our standing assumptions for this paper are the following.

(A1) f : [a, b] × R^{n} → R is continuous.

(A2) Solutions of initial value problems for (1) are unique and extend across the interval [a, b].

Later, in the statements of our main theorems, growth conditions will be placed on the function f.

Results concerning uniqueness implies existence have been considered by authors for many types of boundary conditions. For several examples of these types of arguments as applied to conjugate, focal, or Lidstone problems, see [2]-[9] and the references cited within. Eloe and Henderson obtained existence and uniqueness results for non-local boundary value problems in [5]. Some papers in which nonlinear boundary conditions have also been studied include Abadi and Thompson [1] and Thompson’s studies of fully nonlinear problems in [14]-[16]. Schrader [17] and Ehme, Eloe, and Henderson [4] considered various problems with non-linear boundary conditions. In this paper, we will establish uniqueness implies existence results for nonlinear linear (k; j) problems. These new results will yield as special cases the linear (k; j) problems in [6].

Uniqueness Implies Existence of Solutions for (k; j) Boundary Value Problems 3

2 Preliminary Results

By ordering the linear (k; j) boundary conditions (2) in some order, we may denote these conditions by L_{i}(y), where 1 ≤ i ≤ n. Likewise, we can label the nonlinear (k; j) boundary conditions (3) as S_{i}(y) by using the same ordering as in the linear conditions. We obtain |L_{i}(y)| ≤ |S_{i}(y)|, for all i, because we have assumed the nonlinear condition dominates the linear condition.

Next, we define φ : R^{n} → R^{n} by

φ(c_{0}, c_{1}, …, c_{n−1}) = (S_{1}(y(t, c_{0}, …, c_{n−1})), , …, S_{n}(y(t, c_{0}, …, c_{n−1}))),  (4)

where y(t, c_{0}, …, c_{n−1}) denotes the unique solution of the initial value problem

\begin{matrix}y^{(n)} = f(t, y, y′, …, y^{(n−1)}), \\ y^{(i)}(t_{0}) = c_{i}, 0 ≤ i ≤ n − 1. \end{matrix}

and t_{0} is a fixed point in (a, b).

The following representation theorem will be indispensable. For completeness, we state and prove this result here.

Theorem 2.1 (Representation Theorem). Let u(t) ∈ C^{(n)}[a, b]. Assume solutions of y^{(n)} = 0 satisfying L_{i}(y) = 0, i = 1, …, n, are unique when they exist. Then

u(t) = L_{1}(u)p_{1}(t) + L_{2}(u)p_{2}(t) + ⋯ + L_{n}(u)p_{n}(t) + ∫_{a}^{b} G(t, s)u^{(n)}(s) ds,

where p_{j} is a polynomial of degree less than or equal to n − 1, L_{i}(p_{j}) = δ_{ij}, where δ_{ij} is the Kronecker delta, and G(t, s) is the Green’s function for y^{(n)} = 0 satisfying L_{i}(y) = 0.

Proof. Let p_{j} denote the unique solution of

\{ \begin{matrix}y^{(n)} = 0, \\ L_{i}(y) = δ_{ij}, for i = 1, …, n. \end{matrix}

The existence of the Green’s function implies such p_{j}’s exist. Clearly, p_{j} is a polynomial with a degree at most n − 1, and L_{i}(p_{j}) = δ_{ij}. Next, let

w(t) = u(t) − L_{1}(u)p_{1}(t) − L_{2}(u)p_{2}(t) − ⋯ − L_{n}(u)p_{n}(t) − ∫_{a}^{b} G(t, s)u^{(n)}(s) ds.

Then w^{(n)}(t) = u^{(n)}(t) − L_{1}(u) · 0 − ⋯ − L_{n}(u) · 0 − u^{(n)}(t) = 0. Thus, using the fact that

L_{i} (∫_{a}^{b} G(t, s)u^{(n)}(s)ds) = 0,

we obtain

L_{i}(w) = L_{i}(u) − L_{1}(u)L_{i}(p_{1}) − ⋯ − L_{n}(u)L_{i}(p_{n}) − 0
= L_{i}(u) − L_{i}(u)
= 0.

By uniqueness of of solutions of y^{(n)} = 0, L_{i}(y) = 0, it follows that w(t) = 0. This completes the proof.

We next establish that solutions depend continuously on the boundary values.

Theorem 2.2 (Continuous Dependence). Assume (A1) and (A2), and suppose solutions to (1), (3) are unique when they exist. Then, given a solution y_{0} of (1), (3), and ε > 0, there exists δ > 0 such that if |S_{i}(y_{0}) − r_{i}| < δ, i = 1, …, n, then there exists a solution z of (1) such that S_{i}(z) = r_{i}, for i = 1, …, n, and |y_{0}^{(i)}(t) − z^{(i)}(t)| < ε, i = 0, …, n − 1.

Proof. Let φ be defined as in (4). If φ(c⃗_{1}) = φ(c⃗_{2}), then this implies S_{i}(y(t, c⃗_{1})) = S_{i}(y(t, c⃗_{2})), i = 1, 2, …, n. Our uniqueness assumption on the nonlinear boundary value problems implies that y(t, c⃗_{1}) = y(t, c⃗_{2}), for all t, including t_{0}. But, this implies c⃗_{1} = c⃗_{2} because solutions of initial value problems are unique, and, hence, φ is one-to-one. The results now follows from the Brouwer Theorem on Invariance of Domain and the fact that solutions depend continuously on initial conditions.

We note that the previous theorem also establishes that φ is continuous. As the linear (k; j) problems are special cases of the nonlinear (k; j) problems, the above theorem also establishes continuous dependence for the linear (k; j) problems.

Lemma 2.1. If φ is onto, then solutions to (1), (3) exist for all choices of y_{il}.

The proof of this lemma follows immediately from the definition of ϕ.

3 Main Results

We now in a position to state our first main theorem.

Theorem 3.1. Assume (A1) and (A2), and solutions to (1), (3) are unique when they exist. Assume solutions to the linear problem y^{(n)} = 0, L_{i}(y) = 0, i = 1, …, n, are unique when they exist. Also assume

\frac{|f(t, y_{1}, …, y_{n})|}{[max\{|y_{1}|, …, |y_{n}|\}]^{p}} ≤ M,

for some M > 0 and for all (y_{1}, …, y_{n}) ∈ R^{n} such that |(y_{1}, …, y_{n})| > R, for some R > 0, 0 < p < 1. Then solutions to the nonlinear (k; j) problem (1), (3) exist for all choices of boundary values.

Proof. Assume φ : R^{n} → R^{n} is defined as in (4). Theorem 1.1 implies the image of φ is open. The image of φ is clearly non-empty. To complete the proof we will show the image of φ is both open and closed which will imply the image is R^{n}, and hence the mapping φ is onto. By Lemma 1.3, we will obtain existence of solutions to (1), (3).

First, suppose there exists a sequence c⃗_{k} ∈ R^{n} such that φ(c⃗_{k}) → y_{0}. If < c⃗_{k} > is bounded, then there exists a convergent subsequence < c⃗_{k_{l}} >. Suppose c⃗_{k_{l}} → c⃗_{0} for some c⃗_{0} ∈ R^{n}. Then, the continuity of φ implies φ(c⃗_{k_{l}}) → φ(c⃗_{0}), which implies y_{0} = φ(c⃗_{0}). And, hence, the image of φ is closed.

We may now assume φ(c⃗_{k}) → y_{0} and |c⃗_{k}| → ∞. Let c⃗_{k} = (c_{0_{k}}, c_{1_{k}}, …, c_{n−1_{k}}). Since y_{n}^{(i)}(t, c⃗_{k}) is continuous in t for 0 ≤ i ≤ n − 1, there exists t_{k} ∈ [a, b] such that |y^{(i_{k})}(t_{k}, c⃗_{k})| = max {∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥} for some i_{k} ∈ {0, …, n − 1}. Since 0 ≤ i_{k} ≤ n − 1 and each i_{k} is an integer, there exists a subsequence of the i_{k} that is constant. By choosing this subsequence, we may assume |y^{(j)}(t_{k}, c⃗_{k})| = max ∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥, for all k and some fixed j. Applying our Representation Theorem, we have

y^{(j)}(t_{k}, c⃗_{k}) = L_{1}(y(t, c⃗_{k}))p_{1}^{(j)}(t_{k}) + ⋯ + L_{n}(y(t, c⃗_{k}))p_{n}^{(j)}(t_{k})
+ ∫_{a}^{b} \frac{∂^{(j)}G}{∂t^{j}}(t_{k}, s)f(s, y(s, c⃗_{k}), …, y^{(n−1)}(s, c⃗_{k}))ds.

This will then give us

\frac{y^{(j)}(t_{k}, c⃗_{k})}{[max \{∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥\}]^{p}} = \frac{L_{1}(y(t, c⃗_{k}))p_{1}^{(j)}(t_{k})}{[max \{∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥\}]^{p}}
+  ⋮
+ \frac{L_{n}(y(t, c⃗_{k}))p_{n}^{(j)}(t_{k})}{[max \{∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥\}]^{p}}

+ ∫_{a}^{b} \frac{∂^{(j)}G}{∂t^{j}}(t_{k}, s)\frac{f(s, y(s, c⃗_{k}), …, y^{(n−1)}(s, c⃗_{k}))}{[max \{∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥\}]^{p}} ds.  (5)

As

|y^{(j)}(t_{k}, c⃗_{k})| = max {∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥}
≥ max{|c_{0_{k}}|, |c_{1_{k}}|, …, |c_{n_{k}−1}|} → ∞

and p < 1, we see that

\frac{|y^{(j)}(t_{k}, c⃗_{k})|}{[max \{∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥\}]^{p}} = |y^{(j)}(t_{k}, c⃗_{k})|^{1−p} → ∞

We will obtain a contradiction by showing the right hand side of (5) is bounded. By assumption φ(c⃗_{k}) → y_{0} implies < φ(c⃗_{k}) > is a bounded sequence. From the definition of φ(c⃗_{k}), we obtain each of its component functions S_{i}(y(t, c⃗_{k})) is bounded. By our dominance assumption, |L_{i}(y(t, c⃗_{k}))| ≤ |S_{i}(y(t, c⃗_{k}))|, we obtain that L_{i}(y(t, c⃗_{k})) are all bounded.

Moreover,

∫_{a}^{b} |\frac{∂^{(j)}G}{∂t^{j}}(t_{k}, s)| · |\frac{f(s, y(s, c⃗_{k}), …, y^{(n−1)}(s, c⃗_{k}))}{[max \{∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥\}]^{p}} | ds ≤ ∫_{a}^{b} |\frac{∂^{(j)}G}{∂t^{j}}(t_{k}, s)| Mds,

which is clearly bounded.This implies the right hand side of (5) is bounded, which is a contradiction. Hence, |c⃗_{k}| ↛ ∞. This yields the image of φ is closed and the proof is complete.

If the g_{il} are actually linear (k; j) boundary conditions, then we obtain the following special case of results recently proved by Eloe and Henderson in [6] using completely different techniques.

Theorem 3.2. Assume (A1) and (A2), and solutions of (1), (2) are unique when they exist. Assume solutions of the linear problem y^{(n)} = 0 satisfying L_{i}(y) = 0, i = 1, …, n, are unique when they exist. Also assume

\frac{|f(t, y_{1}, …, y_{n})|}{[max\{|y_{1}|, …, |y_{n}|\}]^{p}} ≤ M,

for some M > 0 and for all (y_{1}, …, y_{n}) ∈ R^{n} such that |(y_{1}, …, y_{n})| > R, for some R > 0, 0 < p < 1. Then solutions to the linear (k; j) boundary problem (1), (2) exist for all choices of boundary values.

The hypotheses on f in Theorem 3.1 are satisfied by any bounded function f. In addition, any unbounded function f such that

|f(t, y_{1}, …, y_{n})| ≤ α_{1}y_{1}^{p_{1}} + ⋯ + α_{n}y_{n}^{p_{n}},

where α_{i} ≥ 0, 0 < p_{i} < 1, i = 1, ..., n, for all (y_{1}, …, y_{n}) ∈ R^{n} such that |(y_{1}, …, y_{n})| > R, for some R > 0 will also satisfy the hypothesis of Theorem 3.1.

We will now consider the case when p_{i} = 1, i = 1, ..., n. This case is satisfied by functions f that are bounded by linear functions. That is,

|f(t, y_{1}, …, y_{n})| ≤ α_{1}|y_{1}| + ⋯ + α_{n}|y_{n}|,

for appropriate α_{i} ∈ R^{+} and for all (y_{1}, …, y_{n}) ∈ R^{n} such that |(y_{1}, …, y_{n})| > R, for some R > 0.

Theorem 3.3. Assume (A1) and (A2), and solutions to (1), (3) are unique when they exist. Assume solutions of y^{(n)} = 0 satisfying L_{i}(y) = 0, i = 1, …, n, are unique. Also assume there exists an α such that

\frac{|f(t, y_{1}, …, y_{n})|}{max\{|y_{1}|, …, |y_{n}|\}} < α < \frac{1}{max_{l∈\{0,1,...,n−1\}} \{max_{t∈[a,b]} \{∫_{a}^{b} \frac{∂^{(l)}G}{∂t^{l}}(t, s)ds\}\}}

EJQTDE Spec. Ed. I, 2009 No. 11

for all (y_{1}, …, y_{n}) ∈ R^{n} such that |(y_{1}, …, y_{n})| > R, for some R > 0. Then solutions to (1), (3) exist for all choices of boundary values.

Because the proof of Theorem 3.3 is similar to the proof of Theorem 3.1, we provide a sketch of the proof. Sketch of Proof. Building on our work in the proof of Theorem 3.1, it suffices to consider the equality

y^{(j)}(t_{k}, c⃗_{k}) = L_{1}(y(t, c⃗_{k}))p_{1}^{(j)}(t_{k}) + ⋯ + L_{n}(y(t, c⃗_{k}))p_{n}^{(j)}(t_{k})
+ ∫_{a}^{b} \frac{∂^{(j)}G}{∂t^{j}}(t_{k}, s)f(s, y(s, c⃗_{k}), …, y^{(n−1)}(s, c⃗_{k}))ds,

where |y^{(j)}(t_{k}, c⃗_{k})| = max {∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥}, for all k, as in the proof of Theorem 3.1. As before, |L_{i}(y(t, c⃗_{k}))| ≤ |S_{i}(y(t, c⃗_{k}))|, for all i and all k, and hence the |L_{i}(y(t, c⃗_{k}))| are bounded. Thus,

\frac{y^{(j)}(t_{k}, c⃗_{k})}{max \{∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥\}} = \frac{L_{1}(y(t, c⃗_{k}))p_{1}^{(j)}(t_{k})}{max \{∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥\}} + ⋯
+ \frac{L_{n}(y(t, c⃗_{k}))p_{n}^{(j)}(t_{k})}{max \{∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥\}}

+ ∫_{a}^{b} \frac{∂^{(j)}G}{∂t^{j}}(t_{k}, s)\frac{f(s, y(s, c⃗_{k}), …, y^{(n−1)}(s, c⃗_{k}))}{max \{∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥\}}ds.  (6)

Then, max {∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥} → ∞, as k → ∞, and as before,

\frac{L_{i}(y(t, c⃗_{k}))p_{i}^{(j)}(t_{k})}{max \{∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥\}} → 0.

By construction, |\frac{y^{(j)}(t_{k},c⃗_{k})}{max\{∥y(t,c⃗_{k})∥,...,∥y^{(n−1)}(t,c⃗_{k})∥\}}| = 1, for all k. However,

|∫_{a}^{b} \frac{∂^{(j)}G}{∂t^{j}}(t_{k}, s)\frac{f(s, y(s, c⃗_{k}), …, y^{(n−1)}(s, c⃗_{k}))}{max \{∥y(t, c⃗_{k})∥, …, ∥y^{(n−1)}(t, c⃗_{k})∥\}}ds|

≤ ∫_{a}^{b} |\frac{∂^{(j)}G}{∂t^{j}}(t_{k}, s)| · |\frac{f(s, y(s, c⃗_{k}), …, y^{(n−1)}(s, c⃗_{k}))}{max \{|y(t, c⃗_{k})|, …, |y^{(n−1)}(t, c⃗_{k})|\}}ds|
≤ ∫_{a}^{b} |\frac{∂^{(j)}G}{∂t^{j}}(t_{k}, s)| · α ds < 1,

from our assumption on α. We see the right hand side of (6) is eventually less than 1 in an absolute value, which is a contradiction.

Applying Theorem 3.3 to linear (k; j) boundary value problems yields the following theorem.

Theorem 3.4. Assume (A1) and (A2), and solutions to (1), (2) are unique when they exist. Assume solutions of y^{(n)} = 0 satisfying L_{i}(y) = 0, i = 1, …, n, are unique. Also assume there exists an α such that

\frac{|f(t, y_{1}, …, y_{n})|}{max\{|y_{1}|, …, |y_{n}|\}} < α < \frac{1}{max_{l∈\{0,1,...,n−1\}} \{max_{t∈[a,b]} \{∫_{a}^{b} \frac{∂^{(l)}G}{∂t^{l}}(t, s)ds\}\}}

for all (y_{1}, …, y_{n}) ∈ R^{n} such that |(y_{1}, …, y_{n})| > R, for some R > 0. Then solutions to (1), (2) exist for all choices of boundary values.

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