Behaviour of solutions of some linear functional equations at infinity

Relationship between the boundedness and the existence of a finite limit at the infinity of solutions of the linear functional equationx(φ(t))=αx(t)+f(t),where α∈C,φ:R+→R+ is a continuous increasing function satisfying the condition φ(t)>t,t∈R+, and f is a continuous function such that there is a finite limit limt→+∞f(t), is studied here. By using obtained results we study behaviour of bounded solutions of the functional equationx(φ[k](t))=∑i=0k-1αix(φ[i](t))+f(t),where αi,i=1,k¯ are real numbers such that ∑i=0k-1αi=1, functions φ and f satisfy above mentioned conditions and limt→+∞f(t)=0.