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 equation xðn þ 1Þ ¼ aðnÞ þ bðnÞxðnÞ þ P n i¼0 Kðn; iÞxðiÞ In the paper, sufficient conditions are derived for the validity of a property of solutions that, for every admissible constant c 2 R, there exists a solution x ¼ xðnÞ such that xðnÞ $ c þ P nÀ1 i¼0 aðiÞ bði þ 1Þ bðnÞ; where bðnÞ ¼ Q nÀ1 j¼0 bðjÞ, for n ! 1 and inequalities for solutions are derived. Relevant comparisons and illustrative examples are given as well.