Existence and Uniqueness of Positive Solution for Discrete Multipoint Boundary Value Problems.

Huili Ma,Huifang Ma

doi:10.1155/2014/531978

Abstract

It is expected in this paper to investigate the existence and uniqueness of positive solution for the following difference equation: −Δ2 u(t − 1) = f(t, u(t)) + g(t, u(t)), t ∈ ℤ 1, T, subject to boundary conditions either u(0) − βΔu(0) = 0, u(T + 1) = αu(η) or Δu(0) = 0, u(T + 1) = αu(η), where 0 < α < 1, β > 0, and η ∈ ℤ 2,T−1. The proof of the main result is based upon a fixed point theorem of a sum operator. It is expected in this paper not only to establish existence and uniqueness of positive solution, but also to show a way to construct a series to approximate it by iteration.

Highlights

Let T > 1 be an integer; Za,b := {a, a + 1, . . . , b}, where a, b are positive integers
It is expected in this paper to investigate the existence and uniqueness of positive solution for the following difference equation: −Δ2u(t − 1) = f(t, u(t)) + g(t, u(t)), t ∈ Z1, T, subject to boundary conditions either u(0) − βΔu(0) = 0, u(T + 1) = αu(η) or Δu(0) = 0, u(T + 1) = αu(η), where 0 < α < 1, β > 0, and η ∈ Z2,T−1
It is expected in this paper to establish existence and uniqueness of positive solution, and to show a way to construct a series to approximate it by iteration

Summary

Huili Ma and Huifang Ma

Received 15 May 2014; Revised 9 August 2014; Accepted 10 August 2014; Published 29 October 2014. It is expected in this paper to investigate the existence and uniqueness of positive solution for the following difference equation: −Δ2u(t − 1) = f(t, u(t)) + g(t, u(t)), t ∈ Z1, T, subject to boundary conditions either u(0) − βΔu(0) = 0, u(T + 1) = αu(η) or Δu(0) = 0, u(T + 1) = αu(η), where 0 < α < 1, β > 0, and η ∈ Z2,T−1. The proof of the main result is based upon a fixed point theorem of a sum operator It is expected in this paper to establish existence and uniqueness of positive solution, and to show a way to construct a series to approximate it by iteration

Introduction

International Scholarly Research Notices

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