Differential Equations and Computational Simulations III J. Graef, R. Shivaji, B. Soni, & J. Zhu (Editors) Electronic Journal of Differential Equations, Conference 01, 1997, pp 129–136. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp 147.26.103.110 or 129.120.3.113 (login: ftp)

Multiple Solutions to a Boundary Value Problem for an n-th Order Nonlinear Difference Equation ^{∗}

Susan D. Lauer

Abstract We seek multiple solutions to the n-th order nonlinear difference equa-

tion

Δ^{n}x(t) = (−1)^{n−k}f(t, x(t)), t ∈ [0, T]

satisfying the boundary conditions

x(0) = x(1) = ⋯ = x(k − 1) = x(T + k + 1) = ⋯ = x(T + n) = 0 .

Guo’s fixed point theorem is applied multiple times to an operator defined on annular regions in a cone. In addition, the hypotheses invoked to obtain multiple solutions to this problem involves the condition (A) f : [0, T] × ℝ^{+} → ℝ^{+} is continuous in x, as well as one of the following: (B) f is sublinear at 0 and superlinear at ∞, or (C) f is superlinear at 0 and sublinear at ∞.

1 Introduction

Define the operator Δ to be the forward difference

Δu(t) = u(t + 1) − u(t),

and then for i ≥ 1 define

Δ^{i}u(t) = Δ(Δ^{i−1}u(t)).

For a ≤ b integers define [a, b] = {a, a+1, …, b−1, b}. Let the integers n, T ≥ 2 be given, and choose k ∈ {1, 2, …, n − 1}. Consider the nth order nonlinear difference equation

Δ^{n}x(t) = (−1)^{n−k}f(t, x(t)), t ∈ [0, T],  (1)

satisfying the boundary conditions

x(0) = x(1) = ⋯ = x(k − 1) = x(T + k + 1) = ⋯ = x(T + n) = 0. (2)

∗1991 Mathematics Subject Classifications: 39A10, 34B15.

Key words and phrases: n-th order difference equation, boundary value problem, superlinear, sublinear, fixed point theorem, Green’s function, discrete, nonlinear. \begin{matrix}c \\ ◯ \end{matrix}1998 Southwest Texas State University and University of North Texas. Published November 12, 1998.

To simplify the discussion of the desired properties of the function f define the following four functions:

f_{0,m} = lim_{u→0^{+}} min_{t∈[k,T +k]} \frac{f(t,u)}{u},  f_{∞,m} = lim_{u→+∞} min_{t∈[k,T +k]} \frac{f(t,u)}{u},
f_{0,M} = lim_{u→0^{+}} max_{t∈[k,T +k]} \frac{f(t,u)}{u}, and f_{∞,M} = lim_{u→+∞} max_{t∈[k,T +k]} \frac{f(t,u)}{u}.

We seek to prove the existence of multiple positive solutions to (1) and (2) where

(A) f : [0, T]×ℝ^{+} → ℝ^{+} is continuous in x, where ℝ^{+} denotes the nonnegative reals.

We also require that one of the following sublinearity and superlinearity conditions on the function f holds:

(B) f_{0,m} = +∞ and f_{∞,m} = +∞, or

(C) f_{0,M} = 0 and f_{∞,M} = 0.

We apply Guo’s Fixed point theorem, Guo and Lakshmikantham [5], using cone methods to accomplish this. This technique was first applied to differential equations in the landmark paper by Erbe and Wang [4] using Krasnosel’skiĭ’s fixed point theorem, Krasnosel’skiĭ [9]. A key to applying this fixed point theorem involves discrete concavity of solutions of the boundary value problem in conjunction with a lower bound on an appropriate Green’s function.

This work constitutes a complete generalization of the paper by Eloe, Henderson and Kaufmann [3] which we use extensively. We also utilize techniques from Hartman [6], Merdivenci [11], and Peterson [12]. Extensive use of the results by Eloe [2] concerning a lower bound for the Green’s function will be made.

2 Preliminaries

Let G(t, s) be the Green’s function for the disconjugate boundary value problem

Lx(t) ≡ Δ^{n}x(t) = 0, t ∈ [0, T]  (3)

and satisfying (2). The characterization of the Green’s function can be found in Kelley and Peterson [8]. We will use G(t, s) as the kernel of an integral operator preserving a cone in a Banach space, the setting for our fixed point theorem.

A closed, non-empty subset P of a Banach space B is said to be a cone provided (i) au + bv ∈ P for all u, v ∈ P and for all a, b ≥ 0, and (ii) u, −u ∈ P implies u = 0.

Repeated application of the following fixed point theorem from Guo, Guo and Lakshmikantham [5], will yield two solutions to (1) and (2).

Theorem 2.1 Let B be a Banach space and P ⊂ B be a cone. Let Ω_{1} and Ω_{2} be two bounded open sets in B such that 0 ∈ Ω_{1} ⊂ \overline{Ω}_{1} ⊂ Ω_{2}. Let

H : P ∩ (\overline{Ω}_{2} \ Ω_{1}) → P

be a completely continuous operator satisfying either

(i) ∥Hx∥ ≤ ∥x∥, x ∈ P ∩ ∂Ω_{1}, and ∥Hx∥ ≥ ∥x∥, x ∈ P ∩ ∂Ω_{2}, or

(ii) ∥Hx∥ ≥ ∥x∥, x ∈ P ∩ ∂Ω_{1}, and ∥Hx∥ ≤ ∥x∥, x ∈ P ∩ ∂Ω_{2}.

Then H has a fixed point in P ∩ (\overline{Ω}_{2} \ Ω_{1}).

Two applications of 2.1 to the problem (1) and (2) following along the lines of methods incorporated by Eloe, Henderson and Kaufmann [3] will be performed.

Note that x(t) is a solution of (1) and (2) if and only if

x(t) = (−1)^{n−k} ∑_{s=0}^{T} G(t, s)f(s, x(s)), t ∈ [0, T + n] .

Hartman [6] extensively studied the boundary value problem (1) and (2) with (−1)^{n−k}f(t, u) ≥ 0. Eloe [2] employed lemmas from Hartman to arrive at the following theorem that gives a lower bound for the solution to the class of boundary value problems studied by Hartman.

Theorem 2.2 Assume that u satisfies the difference inequality (−1)^{n−k}Δ^{n}u(t) ≥ 0, t ∈ [0, T], and the homogeneous boundary conditions, (2). Then for t ∈ [k, T + k],

(−1)^{n−k}Δ^{n}u(t) ≥ \frac{T! ν!}{(T + ν)!}∥u∥,

where ∥u∥ = max_{t∈[k,T +k]} |u(t)| and ν = max{k, n − k}.

We remark that Agarwal and Wong [1] have recently sharpened the inequality of Theorem 2.2. This sharper inequality is of little consequence for this work.

Eloe also contributed the following corollary.

Corollary 2.3 Let G(t, s) denote the Green’s function for the boundary value problem, (3) and (2). Then for all s ∈ [0, T], t ∈ [k, T + k],

(−1)^{n−k}Δ^{n}G(t, s) ≥ \frac{T! ν!}{(T + ν)!}∥G(·, s)∥,

where ∥G(·, s)∥ = max_{t∈[k,T +k]} |G(t, s)| and ν = max{k, n − k}.

To fulfill the hypotheses of Theorem 2.1 let B = {u : [0, T + n] → ℝ| u(0) = u(1) = ⋯ = u(k − 1) = u(T + k + 1) = ⋯ = u(T + n) = 0} with ∥u∥ = max_{t∈[k,T +k]} |u(t)|. Now (B, ∥ · ∥) is a Banach space.

Let

σ = \frac{T! ν!}{(T + ν)!}  (4)

with ν = max{k, n − k} and define a cone

P = {u ∈ B | u(t) ≥ 0 on [0, T + n] and min_{t∈[k,T +k]} u(t) ≥ σ∥u∥}.

3 Main Results

We first seek two solutions to the case when f is sublinear at 0 and superlinear at ∞. Define

η = (∑_{s=0}^{T} ∥G(·, s∥)^{−1} .  (5)

Theorem 3.1 Assume f(t, x) satisfies conditions (A) and (B). Suppose there exists p > 0 such that if 0 ≤ u(t) ≤ p, t ∈ [0, T], then f(t, u) ≤ ηp. Then the boundary value problem (1) and (2) has at least two positive solutions u_{1}, u_{2} ∈ P satisfying 0 ≤ ∥u_{1}∥ ≤ p ≤ ∥u_{2}∥.

proof Define a summation operator H : P → B by

Hx(t) = (−1)^{n−k} ∑_{s=0}^{T} G(t, s)f(s, x(s)),  x ∈ P  (6)

Now H : P → P and is completely continuous.

Choose α > 0 such that

ασ^{2} ∑_{s=k}^{T} ∥G(·, s)∥ ≥ 1.  (7)

By the sublinearity of f at 0 there exists 0 < r < p such that f(t, u) ≥ αu for all 0 ≤ u ≤ r, t ∈ [0, T + n]. For x ∈ P with ∥x∥ = r

Hx(t) = (−1)^{n−k} ∑_{s=0}^{T} G(t, s)f(s, x(s))

≥ σ ∑_{s=0}^{T} ∥G(·, s)∥f(s, x(s))

≥ ασ ∑_{s=0}^{T} ∥G(·, s)∥x(s)

≥ ασ^{2} ∑_{s=k}^{T} ∥G(·, s)∥∥x∥
≥ ∥x∥,  t ∈ [k, T + k].

Therefore ∥Hx∥ ≥ ∥x∥. Hence if we set

Ω_{1} = {u ∈ B | ∥u∥ < r}

then

∥Hx∥ ≥ ∥x∥, for all x ∈ P ∩ ∂Ω_{1}.  (8)

Now for x ∈ P with ∥x∥ = p,

Hx(t) = (−1)^{n−k} ∑_{s=0}^{T} G(t, s)f(s, x(s))

≤ ∑_{s=0}^{T} ∥G(·, s)∥f(s, x(s))

≤ ∑_{s=0}^{T} ∥G(·, s)∥ηp ≤ p = ∥x∥,  t ∈ [0, T + k].

Now if we take

Ω_{2} = {u ∈ B | ∥u∥ < p}

then

∥Hx∥ ≤ ∥x∥, for all x ∈ P ∩ ∂Ω_{2}.  (9)

Thus with (8) and (9), we have shown that H satisfies the hypotheses to Theorem 2.1(ii). This yields a fixed point u_{1} of H belonging to P ∩ (\overline{Ω}_{2} \ Ω_{1}). This fixed point is a solution of (1) and (2) satisfying r ≤ ∥u_{1}∥ ≤ p.

Next, choose ω > 0 such that

ωσ^{2} ∑_{s=k}^{T} ∥G(·, s)∥ ≥ 1.  (10)

By the superlinearity of f at infinity there exists R_{1} > 0 such that f(t, u) ≥ ωu for all u ≥ R_{1}, t ∈ [0, T + n]. Let R = max{2p, R_{1}}. Now for x ∈ P with ∥x∥ = R

Hx(t) = (−1)^{n−k} ∑_{s=0}^{T} G(t, s)f(s, x(s))

≥ σ ∑_{s=0}^{T} ∥G(·, s)∥f(s, x(s))

≥ ωσ ∑_{s=0}^{T} ∥G(·, s)∥x(s)

≥ ωσ^{2} ∑_{s=k}^{T} ∥G(·, s)∥∥x∥
≥ ∥x∥,  t ∈ [k, T + k].

Therefore ∥Hx∥ ≥ ∥x∥. Hence if we set

Ω_{3} = {u ∈ B | ∥u∥ < R}

then

∥Hx∥ ≥ ∥x∥, for all x ∈ P ∩ ∂Ω_{3}.  (11)

Thus with (9) and (11), we have shown that H satisfies the hypotheses to Theorem 2.1(i). This yields a fixed point u_{2} of H belonging to P ∩ (\overline{Ω}_{3} \ Ω_{2}). This fixed point is a solution of (1) and (2) satisfying p ≤ ∥u_{2}∥ ≤ R.

Therefore, the boundary value problem (1) and (2) has at least two positive solutions u_{1}, u_{2} ∈ P such that 0 ≤ ∥u_{1}∥ ≤ p ≤ ∥u_{2}∥. ♢

We now seek two solutions for the case when f is superlinear at 0 and sublinear at ∞.

Theorem 3.2 Assume f(t, x) satisfies conditions (A) and (C). Suppose there exists q > 0 such that if σq ≤ u(t) ≤ q, t ∈ [k, T + k], then f(t, u) ≥ τq, where

τ = (σ ∑_{s=k}^{T} ∥G(·, s)∥)^{−1} .  (12)

Then the boundary value problem (1) and (2) has at least two positive solutions u_{1}, u_{2} ∈ P such that 0 ≤ ∥u_{1}∥ ≤ q ≤ ∥u_{2}∥.

proof Define the summation operator as in (6) and define η as in (5). By the superlinearity of f at 0 there exists 0 < r < q such that f(t, u) ≤ ηu for all 0 ≤ u ≤ r, t ∈ [0, T]. For x ∈ P with ∥x∥ = r,

Hx(t) = (−1)^{n−k} ∑_{s=0}^{T} G(t, s)f(s, x(s))

≤ ∑_{s=0}^{T} ∥G(·, s)∥ηx(s)

≤ (∑_{s=0}^{T} ∥G(·, s)∥) η∥x∥ = ∥x∥,  t ∈ [0, T + k].

Therefore ∥Hx∥ ≤ ∥x∥. Hence if we set

Ω_{1} = {u ∈ B | ∥u∥ < r}

then

∥Hx∥ ≤ ∥x∥, for all x ∈ P ∩ ∂Ω_{1}.  (13)

Next, for x ∈ P with ∥x∥ = q

Hx(t) = (−1)^{n−k} ∑_{s=0}^{T} G(t, s)f(s, x(s))

≥ σ ∑_{s=k}^{T} ∥G(·, s)∥τq ≥ q = ∥x∥  t ∈ [k, T + k].

Therefore ∥Hx∥ ≥ ∥x∥. Hence if we set

Ω_{2} = {u ∈ B | ∥u∥ < q}

then

∥Hx∥ ≥ ∥x∥, for all x ∈ P ∩ ∂Ω_{2}.  (14)

Thus with (13) and (14), we have shown that H satisfies the hypotheses to Theorem 2.1(i) which yields a fixed point u_{1} of H belonging to P ∩ (\overline{Ω}_{2} \ Ω_{1}). This fixed point is a solution of (1) and (2) satisfying r ≤ ∥u_{1}∥ ≤ q.

Next, by condition (C), for every ε > 0, there exists a ξ > 0 such that for all u ≥ 0, t ∈ [0, T + k], f(t, u) ≤ ξ + εu. Let ε = \frac{η}{2} , where η is defined by (5) and select a corresponding ξ. Let R = max\{2q, 2\frac{ξ}{η} \}. Then for x ∈ P with ∥x∥ = R

Hx(t) = (−1)^{n−k} ∑_{s=0}^{T} G(t, s)f(s, x(s))

≤ ∑_{s=0}^{T} ∥G(·, s)∥[ξ + εx(s)]

≤ ξ ∑_{s=0}^{T} ∥G(·, s)∥ + ε ∑_{s=0}^{T} ∥G(·, s)∥x(s)

≤ \frac{ξ}{η} + ε ∑_{s=0}^{T} ∥G(·, s)∥∥x∥

≤ \frac{R}{2} + \frac{∥x∥}{2} = ∥x∥,  t ∈ [0, T + k].

Therefore ∥Hx∥ ≤ ∥x∥. Hence if we set

Ω_{3} = {u ∈ B | ∥u∥ < R}

then

∥Hx∥ ≤ ∥x∥, for all x ∈ P ∩ ∂Ω_{3}.  (15)

Thus with (14) and (15), we have shown that H satisfies the hypotheses to Theorem 2.1(i) which yields a fixed point of H belonging to P ∩ (\overline{Ω}_{3} \ Ω_{2}).

This fixed point, u_{2}, is a solution of (1) and (2) satisfying q ≤ ∥u_{2}∥ ≤ R. Therefore, the boundary value problem (1) and (2) has at least two positive solutions u_{1}, u_{2} ∈ P such that 0 ≤ ∥u_{1}∥ ≤ q ≤ ∥u_{2}∥.

References

[1] R. P. Agarwal and P. J. Y. Wong, Extension of continuous and discrete inequalities due to Eloe and Henderson, Nonlinear Analysis (in press).

[2] P. W. Eloe, A generalization of concavity for finite differences, Computers and Mathematics with Applications (in press).

[3] P. W. Eloe, J. L. Henderson, and E. R. Kaufmann, Multiple positive solutions for difference equations, Journal of Difference Equations and Applications (in press).

[4] L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proceedings of the American Mathematical Society 120 (1994), 743-748.

[5] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Inc., San Diego, 1988.

[6] P. Hartman, Difference equations: Disconjugacy, principal solutions, Green’s functions, complete monotonicity, Transactions of the American Mathematical Society 2465 (1978), 1-30.

[7] J. L. Henderson and S. D. Lauer, Existence of a positive solution for an nth order boundary value problem for nonlinear difference equations, Abstract and Applied Analysis 2, Nos. 3-4 (1997), 87-95.

[8] W. G. Kelley and A. C. Peterson, Difference Equations, An Introduction with Applications, Academic Press, Inc., San Diego, 1991.

[9] M. A. Krasnosel’skiĭ, Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen, The Netherlands.

[10] S. D. Lauer, Positive solutions of a boundary value problem for second order nonlinear difference equations, Communications on Applied Nonlinear Analysis 4 (1997), Number 3.

[11] F. Merdivenci, Two positive solutions of a boundary value problem for difference equations, Journal of Difference Equations and Applications 1 (1995), 263-270.

[12] A. C. Peterson, Boundary value problems for an nth order linear difference equation, SIAM Journal on Mathematical Analysis 15 (1984), 124-132.

Susan D. Lauer Department of Mathematics, Tuskegee University Tuskegee, Alabama 36088 USA E-mail address: lauersd@auburn.campus.mci.net