An application of topological transversality to non-positive higher order difference equations

摘要

In this paper we shall provide sufficient conditions on the non-positive function ƒ(i, u1, …, un−1) so that the boundary value problem (BVP) given by (E) Δnu(i) + μƒ(i,u(i), Δu(i), …, Δn−2(i)) = 0, and (BC) Δmu(0) = 0, 0 ⩽ m ⩽ n − 3, αΔn−2(0) −βΔn−1u(0) = 0, γΔn−2u(T + 1) + δΔn−1u(T + 1) = 0, has at least one positive solution.