On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence

Schauder's fixed point technique is applied to asymptotical analysis of solutions of a linear Volterra difference equationx(n+1)=a(n)+b(n)x(n)+∑i=0nK(n,i)x(i)where n∈N0, x:N0→R, a:N0→R, K:N0×N0→R, and b:N0→R⧹{0} is ω-periodic. In the paper, sufficient conditions are derived for the validity of a property of solutions that, for every admissible constant c∈R, there exists a solution x=x(n) such thatx(n)∼c+∑i=0n-1a(i)β(i+1)β(n),where β(n)=∏j=0n-1b(j), for n→∞ and inequalities for solutions are derived. Relevant comparisons and illustrative examples are given as well.