Multiple solutions for higher-order difference equations

Abstract

The nth (n≥2) order discrete conjugate problem (-1)n-pΔny(k) = f(k,y(k)), k∈I0, Δiy(0) = 0, 0≤i≤p-1 (here 1≤p≤n-1), Δi(T+n-i) = 0, 0≤i≤n-p-1, and the nth (n≥2) order discrete (n,p) problem Δny(k)+f(k,y(k)) = 0, k∈I0, Δiy(0) = 0, 0≤i≤n-2, Δpy(T+n-p) = 0, 0≤p≤n-1 is fixed, are discussed. Let T∈{1,2, ... }, I0 = {0,1, ..., T}, and y:In = {0,1, ..., T+n}→R. Let C(In) denote the class of maps w continuous on In (discrete topology) with norm |m|0 = maxi∈I(n) |w(i)|.