Mathematics Faculty Articles

Title

A Difference Equation with Anti-Periodic Boundary Conditions

Document Type

Article

Publication Date

1-1-2015

Publication Title

Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis

Keywords

Fixed Point Theorem, Difference Equation, Antiperiodic, Antisymmetric, Functional

ISSN

1201-3390

Volume

22

Issue/No.

1

First Page

47

Last Page

60

Abstract

In this paper, we apply an extension of the Leggett-Williams fixed point theorem to the second order difference equation ∆2u(k)+f(u(k+1)) = 0, k ∈ {0, 1, . . . , N}, satisfying the anti-periodic boundary conditions u(0) + u(N + 2) = 0, ∆u(0) + ∆u(N +1) = 0. Two important results of this paper involve providing the Green’s function for −∆2u(k) = 0 satisfying u(0) + u(N + 2) = 0, ∆u(0) + ∆u(N + 1) = 0 and showing this Green’s function satisfies a concavity like property. An example is also given.

Comments

AMS (MOS) subject classification: 39A10.

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