Behaviour at infinity of solutions of some linear functional equations in normed spaces

Let K is an element of{R, C}, I = (d, infinity), phi : I -> I be unbounded continuous and increasing, X be a normed space over K, F := {f is an element of X-I : lim(t ->) (infinity) f(t) exists in X},(a) over cap is an element of K, A (a) over cap:= {alpha is an element of K-I : lim(t -> infinity) a(t) =(a) over cap}, and X := {x is an element of X-I : lim sup(t -> infinity) parallel to x( t) parallel to < infinity}. We prove that the limit lim(t -> infinity) x(t) exists for every f is an element of F, alpha. is an element of A <(a)over cap> and every solution x is an element of X of the functional equation x(phi(t) = alpha(t)x(t) + (t) if and only if vertical bar(a) over cap vertical bar not equal 1. Using this result we study the behaviour of bounded at infinity solutions of the functional equation x(phi([k]) (t)) = k - 1 Sigma J - O alpha j(t)x(phi([j]) (t)) + f (t), under some conditions posed on functions alpha j (t), j = 0,1, ... , k - 1, phi and f.