Asymptotically periodic solutions of Volterra system of difference equations

Abstract

a b s t r a c tA Volterra system of difference equations of the formwhere n ∈ N, a s , b s , x s : N → R and K sp : N × N → R, s = 1, 2, . . . , r is studied. Sufficient conditions for the existence of asymptotically periodic solutions of this system are derived.Crown

Bounded Solutions to Nonhomogeneous Linear Second-Order Difference Equations

“…Using (22) into (18) we get (11). That (11) represents the general solution to (8), follows from the facts that the sequence…”

“…According to Lemma 1 we know that in the case p 2 = 4q, the general solution to difference Equation (8) is given by Formula (9), while if p 2 = 4q, the general solution is given by (11). Assume first that p 2 = 4q.…”

“…As in [30], we first use some solvability methods and then the contraction mapping principle in showing the existence of a unique bounded solution to Equation (5) under some conditions posed on the coefficients of the equation. For some other applications of fixed-point theorems in studying difference equations and systems, see, for example, [10][11][12]17,18,25] and the related references therein. Note that beside the contraction mapping principle, very frequent situation is application of a variant of the Schauder fixed-point theorem ( [10][11][12]17,18]), and since recently the Darbo fixed-point theorem ( [38]) which uses the notion of measure of non-compactness ( [25]).…”

Asymptotic behavior of solutions of discrete Volterra equations

“…In particular, the boundedness of solutions was studied by, i.e., Crisci et al [2], Diblík and Schmeidel [6], Gronek and Schmeidel [7], Győri and Horvath [10], Győri and Awwad [8], Kolmanovskii and Shaikhet [12], Migda and Migda [19], Migda and Morchało [20] or Morchało [21]. Asymptotically periodic solutions were studied, for example, by Baker and Song [1], Diblík et al [4,5] or Győri and Reynolds [11]. To the best of our knowledge, there are a few papers dealing with the asymptotic behavior of solutions of higher order Volterra difference equations, for example, the second order difference equation of Volterra type was studied in Medina [13].…”

Explicit asymptotic limits for a class of discrete Volterra equations

“…One motivation for this work was to explore the connection between the approaches used in Refs [1,2,5] for studying asymptotically periodic solutions of discrete Volterra equations.…”