POSITIVE SOLUTIONS OF q-DIFFERENCE EQUATION

In this paper we investigate the existence of positive solutions of the q-difference equation −D2 qu(t) = a(t) f(u(t)) with some boundary conditions by applying a fixed point theorem in cones. 1. Preliminaries In many of the mathematical models in science such as models of chemical problems, population or concentration in biology, and many problems in physics and economics, we need to investigate the existence of nonnegative solutions. What we understand by nonnegativity can be described by cones, that is, a closed convex set K of a Banach space X such that λK ⊂ K for all λ ≥ 0 and K ⋂ (−K) = {0}. Recently there have been many results for positive solutions of different types of boundary value problems. The main task of these results is based on the work of Krasnoselskii [6]. He worked on nonlinear operator equations by using the theory of cones in Banach spaces. In [2], the authors studied the existence of positive solutions of the second order boundary value problem, − u′′ = a(t) f(u), 0 t 1, (1) αu(0)− βu′(0) = 0 γu(1) + δu′(1) = 0, (2) with some conditions imposed on f(·), a(·), and the constants of (2). It is shown that there is a positive solutions in both the superlinear and sublinear cases; see the section below. The authors used a Fixed Point Theorem of Krasnoselskii; see [7]. More precisely, they use a modified version of Krasnoselskii due to Guo [4, p. 94]; it reads: Theorem 1. Let ∆1 and ∆2 be two bounded open sets in a Banach space E such that 0 ∈ ∆1,∆1 ⊂ ∆2. Let A : K ⋂( ∆2\∆1 ) −→ K be completely continuous and let one of the conditions (1) ‖Ax‖ ≤ ‖x‖, ∀x ∈ K ⋂ ∂∆1, and ‖Ax‖ ≥ ‖x‖, ∀x ∈ K ⋂ ∂∆2; Received by the editors April 27, 2009, and, in revised form, August 27, 2009. 2010 Mathematics Subject Classification. Primary 39A13, 45M20, 34B18, 34B27.