Solvability of second-order m-point difference equation boundary value problems on infinite intervals

In this paper, we study second-order m-point difference boundary value problems on infinite intervals { ∆²x(k− 1) + f(k, x(k),∆x(k− 1)) = 0, k ∈ N, x(0) = m−2 ∑ i=1 αix(ηi), lim k→∞∆x(k) = 0, where N = {1, 2, ...}, f : N× R2 → R is continuous, αi ∈ R, m−2 ∑ i=1 αi 6= 1, ηi ∈ N, 0 η1 η2 ... ∞ and ∆x(k) = x(k+ 1) − x(k), the nonlinear term is dependent in a difference of lower order on infinite intervals. By using Leray-Schauder continuation theorem, the existence of solutions are investigated. Finally, we give one example to demonstrate the use of the main result.