Periodic solutions of discrete Volterra equations

Abstract

In this paper, we investigate periodic solutions of linear and nonlinear discrete Volterra equations of convolution or non-convolution type with unbounded memory.For linear discrete Volterra equations of convolution type, we establish Fredholm's alternative theorem and for equations of non-convolution type, and we prove that a unique periodic solution exists for a particular bounded initial function under appropriate conditions. Further, this unique periodic solution attracts all other solutions with bounded initial function. All solutions of linear discrete Volterra equations with bounded initial functions are asymptotically periodic under certain conditions. A condition for periodic solutions in the nonlinear case is established.

Asymptotic periodicity of nonlinear discrete Volterra equations and applications

“…For the linear case, in Refs. [1,3,9], sufficient conditions for the DVEs to have an asymptotically periodic solution are given. In Refs.…”

“…In Refs. [1,9], some results on the asymptotic periodicity are provided for a particular case of nonlinear DVEs.…”

Periodic and Almost Periodic Solutions of Functional Difference Equations with Finite Delay

“…The aim of this paper is to extend results in [1] to (1.1). Delay difference equations or functional difference equations (no matter with finite or infinite delay), inspired by the development of the study of delay differential equations, have been studied extensively in the past few decades (see, [2][3][4][5][6][7][8][9][10][11], to mention a few, and references therein). Recently, several papers [12][13][14][15][16][17] are devoted to study almost periodic solutions of difference equations.…”

Almost periodic solutions of discrete Volterra equations

“…To the best of our knowledge, relatively few papers (in particular, [3,4,8,9,12,[14][15][16] and [18][19][20]23]) deal with discrete Volterra equations. Much of the general qualitative theory and asymptotic properties of solutions remains to be developed.…”