Multiple positive solutions of a discrete difference system

doi:10.1016/S0096-3003(02)00354-5

Applied Mathematics and Computation

Volume 143, Issues 2–3, 10 November 2003, Pages 213-221

https://doi.org/10.1016/S0096-3003(02)00354-5 Get rights and content

Abstract

This paper is concerned with the following system $Δ^{2} u_{1} (k)+f_{1} (k,u_{1} (k),u_{2} (k))=0, k∈[0,T],$ $Δ^{2} u_{2} (k)+f_{2} (k,u_{1} (k),u_{2} (k))=0, k∈[0,T],$ $u_{1} (0)=u_{1} (T+2)=0=u_{2} (0)=u_{2} (T+2).$ By using Leggett–Williams fixed point theorem, sufficient conditions are obtained for the existence of three positive solutions to the above system.

Introduction

Many problems in applied mathematics lead to the study of the difference system (see [4], [8] and the references therein). Recently, many attentions have been paid to the existence of positive solutions of scalar difference equations (see [2], [3], [10], [11]). But very few work has been done to the existence of positive solutions of the discrete difference systems.

The purpose of this paper is to study the following discrete system $Δ^{2} u_{1} (k)+f_{1} (k,u_{1} (k),u_{2} (k))=0, k∈[0,T],$ $Δ^{2} u_{2} (k)+f_{2} (k,u_{1} (k),u_{2} (k))=0, k∈[0,T],$ with the Dirichlet boundary condition $u_{1} (0)=u_{1} (T+2)=0, u_{2} (0)=u_{2} (T+2)=0,$ where T>1 is a fixed positive integer, Δu(k)=u(k+1)−u(k), Δ²u(k)=Δ(Δu(k)), and $[a,b] ≜ {a,a+1,…,b}⊂Z$ the set of all integers. By using Leggett–Williams fixed point theorem [5], [7], we give some sufficient conditions for existence of at least three positive solutions to problem , .

We note that Agarwal and O’Reagn [4] or Wong [9] established a criteria for the existence of at least two positive solutions to the discrete system (1) with the general two-point boundary value conditions. The main tool of their proof is fixed point theorem in cone.

The rest of the paper is organized as follows. In Section 2, we shall state the well-known fixed point theorem due to Leggett–Williams, and prove some inequalities for Green function which are needed later. Criteria for the existence of three positive solutions of problem , is established in Section 3.

Section snippets

Several lemmas

In this section we will give some lemmas which are useful in proving our main results.

We denote by G(k,l) the Green’s function of the boundary value problem $− Δ^{2} u(k)=0, k∈[0,T],$ $u(0)=u(T+2)=0,$ which is explicitly given by [1] $G(k,l)= 1 T+2 (T+2−k)(l+1), 0⩽l⩽k−2, k(T+1−l), k−1⩽l⩽T.$ Then, for 0⩽l⩽k−2, $G(k,l) G(l+1,l) = (T+2−k)(l+1) (l+1)(T+1−l) = T+2−k T+1−l ⩽ T−l T+1−l ⩽1, 0⩽k⩽T+2,$ $G(k,l) G(l+1,l) = (T+2−k)(l+1) (l+1)(T+1−l) = T+2−k T+1−l ⩾ 1 T+1, 1⩽k⩽T+1$ and for k−1⩽l⩽T, $G(k,l) G(l+1,l) = k(T+1−l) (l+1)(T+1−l) = k l+1 ⩽ l+1 l+1 =1, 0⩽k⩽T+2,$ $G(k,l)$

Main result

In this section we state and prove our theorem concerning the existence of three positive solutions for the boundary value problem , . By a positive solution of , we mean a solution of , which is nonnegative and nontrivial.

Theorem 1

Suppose that f_i:[0,T]×[0,∞)×[0,∞)→[0,∞) (i=1,2) is continuous and that there exist numbers a and d with 0<d<a such that the following conditions are satisfied:

(i)
if k∈[0,T],u₁,u₂⩾0 and u₁+u₂⩽d, then $f_{i} (k,u_{1},u_{2})< d 2D, i=1,2,$ where $D= max_{k∈[0,T+2]} ∑_{l=0}^{T} G(k,l);$
(ii)
there exists i₀∈{1,2},

Acknowledgements

This work is supported by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011-A25-007-Z), the Foundation for University Key Teacher by the Ministry of Education of China, the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China, and the Key Research and Development Program for Outstanding Groups of Gansu University of Technology.

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Multiple positive solutions of a discrete difference system

Abstract

Introduction

Section snippets

Several lemmas

Main result

Acknowledgements

J. Math. Anal. Appl.

J. Math. Anal. Appl.

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Discrete polynomial interpolation, Green’s functions, maximum principle, error bounds and boundary value problem

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