A discrete Gronwall–Halanay-type inequality with infinite delay and its applications to difference equations

Semilinear functional difference equations with infinite delay

We obtain boundness and asymptotic behavior of solutions for semilinear functional difference equations with infinite delay. Applications on Volterra difference equations with infinite delay are shown.

Nonlinear Delay Discrete Inequalities and Their Applications to Volterra Type Difference Equations

Gronwall-Bellman inequalities and their various linear and nonlinear generalizations play very important roles in the discussion of existence, uniqueness, continuation, boundedness, and stability properties of solutions of differential equations and difference equations. The literature on such inequalities and their applications is vast. For example, see 1–12 for continuous cases, and 13–20 for...

On a Max-Type Difference Inequality and Its Applications

A Note on Discrete Maximal Regularity for Functional Difference Equations with Infinite Delay

The maximal regularity problem for the discrete time evolution equations has been recently considered by Blunck [4, 5]. Discrete maximal regularity properties appears not to be considered in the literature before the paper [5]. The continuous maximal regularity problem for time evolution equations is well-know (see [1, 4, 5, 19, 20] and the reference contained therein). In the present paper we ...

On a New Discrete Hilbert–type Inequality and Its Applications

In this paper, a theorem related to the Hilbert-type inequality is corrected. By introducing parameters, and using Euler-Maclaurin summation formula, we give a discrete form of the Hilbert-type inequality involving a non-homogeneous kernel. Furthermore, we prove that our result is a concise generalization of the corrected theorem and some known results. As applications, some particular new resu...