tailieunhanh - On the asymptotic behavior of solution of certain systems of Volterra equations

This paper is concerned with the asymptotic property of the solution of a system of the linear Volterra difference equations. The criterion for the existence of a solution of the considered system that is asymptotically equivalent to a given sequence is established. | Turk J Math (2018) 42: 2994 – 3001 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article On the asymptotic behavior of solution of certain systems of Volterra equations Ewa SCHMEIDEL∗,, Małgorzata ZDANOWICZ, Institute of Mathematics, University of Bialystok, Białystok, Poland Received: • Accepted/Published Online: • Final Version: Abstract: This paper is concerned with the asymptotic property of the solution of a system of the linear Volterra difference equations. The criterion for the existence of a solution of the considered system that is asymptotically equivalent to a given sequence is established. The results presented here improve and generalize the results published by Diblik et al. Unlike in those works, here periodicity of the nonhomogeneous term of the equation is not assumed. Examples illustrate the obtained results. Key words: Linear Volterra difference equation, asymptotic equivalence 1. Introduction and notation We consider a Volterra system of difference equations X (n + 1) = A (n) + B (n) X (n) + n ∑ K (n, i) X (i) (1) i=0 where n ∈ N := {0, 1, 2, . . . }. Let R denote the set of real numbers. Here A = (a1 , a2 , . . . , ar )T , B = diag(b1 , . . . , br ), and X = (x1 , x2 , . . . , xr )T , where as , xs : N → R and bs : N → R \ {0}, s = 1, . . . , r . Moreover K = (Ksp )s,p=1,.,r , where Ksp : N × N → R for s, p ∈ {1, . . . , r}. By a solution of system (1) we mean a sequence X = (x1 , x2 , . . . , xr )T whose terms satisfy (1) for every n ∈ N . In the last years, there has been an interest among many authors to study the asymptotic behavior of solutions of Volterra difference equations. The results were published, ., by Appleby et al. [1], by Appleby and Patterson [2], by Berezansky et al. [3], by Gajda et al. [7], by Gil and Medina [8], by Gronek and Schmeidel [9], by Györi and Horváth [10], by Györi and Reynolds [11], by .