Positive solutions and nonlinear eigenvalue problems for third-order difference equations

First, solutions that lie in a cone intersected with an annular type region are obtained for the third-order difference equation Delta(3)u(t) + a(t)f(u(t)) = 0, 2 less than or equal to t less than or equal to T + 2, satisfying the boundary conditions u(0) = u(1) = u(T + 3) = 0, in the cases that f is either superlinear or sublinear. The second part is devoted to determining values of lambda for which there exist positive solutions of Delta(3)u(t) + lambda a(t)f(u(t)) = 0, 2 less than or equal to t less than or equal to T + 2, satisfying the boundary conditions u(0) = u(1) = u(T + 3) = 0, when both lim(x-->0+)(f(x)/x) and lim(x-->infinity)(f(x)/x) exist as positive real numbers. (C) 1998 Elsevier Science Ltd. All rights reserved.