A survey: stability and boundedness of Volterra difference equations

On the boundedness of the solutions in nonlinear discrete Volterra difference equations

“…For some recent literature on the boundedness of the solutions of linear Volterra difference equations, we refer the readers to [10][11][12]. We give some applications of our main result for sub-linear, linear, and super-linear Volterra difference equations.…”

“…It is worth to note that in this case our Theorem 4.5 is applicable for any x 0 ℝ + , but the results in [2,[8][9][10][11][12]14] are not applicable in this case.…”

“…We consider the nonlinear system of Volterra difference equations In recent years, there has been an increasing interest in the study of the asymptotic behavior of the solutions of both convolution and non-convolution-type linear and nonlinear Volterra difference equations (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and references therein). Appleby et al [2], under appropriate assumptions, have proved that the solutions of the discrete linear Volterra equation converge to a finite limit, which in general is non-trivial.…”

Maximal Regularity of the Discrete Harmonic Oscillator Equation

“…In the recent work 15 , the authors have found a way around the "short time interval" problem to treat semilinear problems for certain evolution equations of second order. One more case merits mentioning here is Volterra difference equations which describe processes whose current state is determined by their entire prehistory see, e.g., 16,17 , and the references given there . These processes are encountered, for example, in mathematical models in population dynamics as well as in models of propagation of perturbation in matter with memory.…”

A note on discrete maximal regularity for functional difference equations with infinite delay

“…In the third section, we study the maximal regularity problem for (1.1); while in the fourth section, we present applications to discrete Volterra difference equations with infinite delay. During the last few years discrete Volterra equations have emerged vigorously in several applied fields and there is much interest in developing the qualitative theory for such equations (see [15] for discussion and references).…”