Monotone iterative method for first-order functional difference equations with nonlinear boundary value conditions

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Abstract

In this paper, we show that the method of monotone iterative coupled with the upper and lower solutions is valid to obtain constructive proofs to the existence of solutions for first-order functional difference equations with nonlinear boundary value conditions. An example is given to illustrate the results obtained.

Introduction

The method of upper and lower solutions coupled with the monotone iterative technique has been applied successfully to obtain the results of differential equations and difference equations [1], [2], [3]. In the last decades, the methods were extended to functional differential equations [4], [5], [6], [7], [8]. However, we notice that the results only considered linear boundary values that complied with the condition u(t)=u(0),t[-r,0] and these few results for functional difference equations with the nonlinear functional boundary values which include the usual boundary conditions (such as initial and periodic). Thus, we shall consider the following boundary value problems:Δy(k-1)=f(k,y(k),y(θ(k)),k{1,2,,T}=I1,B(y(0),y)=0,y(k)=φ(k),kZ[-τ,0]={-τ,τ+1,,0},where Δy(k-1)=y(k)-y(k-1), fC(I1×R2,R), θ(k)C(I1,Z[-τ,k]), andΔy(k)=f(k,y(k),y(θ(k)),k{0,1,,T-1}=I2,B(y(0),y)=0,y(k)=φ(k),kZ[-τ,0]={-τ,τ+1,,0},where Δy(k)=y(k+1)-y(k), fC(I2×R2,R), θ(k)C(I2,Z[-τ,k]), and in (1.1), (1.2), φC(Z[-τ,0),R+), R+=[0,+), BC(R×RT+1+τ,R), τ{0,1,}. A solution of problem (1.1), (1.2) is a real sequence x={x(-τ),x(-τ+1),,x(T)} satisfying (1.1), (1.2).

The assumptions we shall impose to function B will be adequate to cover the usual linear boundary conditions (for example, periodic condition x(0)=x(T)) in the formulation of problem (1.1). By the way, more general conditions as, for instance, u(0)=jJu(j) or u(0)=maxjJu(j) for some JZ[-τ,T] will also be included.

We shall divide the results of this paper into two sections. In Section 2, we establish some results about functional difference equations that we shall need later and prove a maximum principle. Section 3 is devoted to developing the monotone iterative method for problem (1.1), (1.2). Finally, to assert our results, we give an example.

Section snippets

Preliminaries

Firstly, we consider the following problem for linear functional difference equations:Δy(k-1)+My(k)+Ny(θ(k))=σ(k),kI1,y(k)=φ(k),kZ[-τ,0],where σC[I1,R], τ{0,1,}, andΔy(k)+My(k)+Ny(θ(k))=σ(k),kI2,y(k)=φ(k),kZ[-τ,0],where σC[I2,R], τ{0,1,}.

Next we give a maximum principle to develop the monotone method in Section 3.

Theorem 2.1

Let yC(Z[-τ,T],R) and M and N are positive constants such that

  • (i)

    Δy(k-1)+My(k)+Ny(θ(k))0, kI1;

  • (ii)

    0y(k)y(0), kZ[-τ,0];

  • (iii)

    N(1+M)TM+N<1.

Then y(k)0, kZ[-τ,T].

Proof

From (ii), we only

Monotone iterative method

In order to develop the monotone iterative technique for problem (1.1), (1.2), we shall first consider the following boundary value problem for linear equations:Δy(k-1)+My(k)+Ny(θ(k))=σ(k),B(y(0),y)=0,y(k)=φ(k),k[-τ,0]andΔy(k)+My(k)+Ny(θ(k))=σ(k),B(y(0),y)=0,y(k)=φ(k),k[-τ,0].We shall denote by E={uE:u(k)=φ(k),kZ[-τ,0]}.

A function αE is said to be a lower solution of (3.1) if it satisfiesΔα(k-1)+Mα(k)+Nα(θ(k))σ(k),B(α(0),α)0.and an upper solution of (3.1) is defined analogously by

Acknowledgement

The authors thank the reviewers and the editor for their valuable suggestions and comments.

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Supported by the Key Project of Chinese Ministry of Education (207014) and the Natural Science Foundation of Hebei Province of China (A2006000941).

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