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 Δ3u(t) + a(t)f(u(t)) = 0, 2 ≤ t ≤ 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 λ for which there exist positive solutions of Δ3u(t) + λa(t)f(u(t)) = 0, 2 ≤ t ≤ T+ 2, satisfying the boundary conditions u(0) = u(1) = u(T + 3) = 0, when both limx→0+(f(x)x) and limx→∞(f(x)x) exist as positive real numbers.