In this paper we study nonlinear second order difference equations subject to separated linear boundary conditions. Sign properties of the associated Green’s functions are investigated and existence results for positive solutions of the nonlinear boundary value problem are established. Upper and lower bounds for these positive solutions also are given. | Turk J Math 27 (2003) , 481 – 507. ¨ ITAK ˙ c TUB On Positive Solutions of Boundary Value Problems for Nonlinear Second Order Difference Equations N. Aykut and G. Sh. Guseinov Abstract In this paper we study nonlinear second order difference equations subject to separated linear boundary conditions. Sign properties of the associated Green’s functions are investigated and existence results for positive solutions of the nonlinear boundary value problem are established. Upper and lower bounds for these positive solutions also are given. Key Words: Difference equations, boundary value problems, positive solutions, Green’s function, fixed point theorem in cones. 1. Introduction Positive solutions of operator equations in Banach spaces are investigated in the monographs by Krasnosel’skii [11] and Guo and Lakshmikantham [9] making use of the theory of operators acting in Banach spaces with a cone and leaving this cone invariant. The significance of this investigation is due to the fact that in analysing nonlinear phenomena many mathematical models give rise to problems for which only nonnegative solutions make sense. In [11] and subsequent studies (see, [9, 2]) the idea of the method was used to prove the existence of positive solutions of nonlinear ordinary and partial differential equations, integral and integro-differential equations, and difference equations. At the beginning, a main example investigated by the cone method was the nonlinear boundary value problem (BVP) AMS No.: 39A10. 481 AYKUT, GUSEINOV 00 −y = f(x, y), x ∈ [a, b], () y(a) = y(b) = 0. () Later in [7, 8] instead of the simple boundary conditions of (), general separated linear boundary conditions 0 αy(a) − βy (a) = 0, 0 γy(b) + δy (b) = 0 () were explored and the existence of positive solutions for the BVP (), () was studied by a similar method. In the sequel, a discrete analogue of the BVP (), () was considered. To formulate it, let a, b (a < b) be integers
