Multiple solutions of difference and partial difference equations with Lidstone conditions

Abstract

This paper considers the following boundary value problem: (-1)mΔ2my = F(k, y, Δy,..., Δ2m-1y), m ≥ 1, 0 ≤ k ≤ N, Δ(2i)y(0) = Δ(2i)y(N + 2m - 2i) = 0, 0 ≤ i ≤ m - 1. We offer criteria for the existence of double and triple positive solutions of the boundary value problem. In addition, for special cases upper and lower bounds are established for these positive solutions. As an application, we also investigate the existence of radial solutions of certain partial difference equation. Several examples are included to dwell upon the importance of the results obtained. (C) 2000 Elsevier Science Ltd.