Báo Cáo Báo cáo hóa học CONSTRUCTION OF UPPER AND LOWER SOLUTIONS FOR SINGULAR DISCRETE INITIAL AND BOUNDARY

CONSTRUCTION OF UPPER AND LOWER SOLUTIONS FOR SINGULAR DISCRETE INITIAL AND BOUNDARY VALUE PROBLEMS VIA INEQUALITY THEORY ¨ HAISHEN LU AND DONAL O’REGAN Received 25 May 2004

We present new existence results for singular discrete initial and boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign.

1. Introduction

An upper- and lower-solution theory is presented for the singular discrete boundary value problem ư∆ ϕ p ∆u(k ư 1) = q(k) f k,u(k) , k ∈ N = {1, .,T }, u(0) = u(T + 1) = 0, and the singular discrete initial value problem ∆u(k ư 1) = q(k) f k,u(k) , u(0) = 0, k ∈ N = {1, . ,T }, (1.1) (1.2) where ϕ p (s) = |s| pư2 s, p 1, ∆u(k ư 1) = u(k) ư u(k ư 1), T ∈ {1,2, . }, N + = {0,1, .,T }, and u : N + → R. Throughout this paper, we will assume f : N × (0, ∞) → R is continuous. As a result, our nonlinearity f (k,u) may be singular at u = 0 and may change sign