On the existence and uniqueness of (N,lambda )-periodic solutions to a class of Volterra difference equations

Edgardo Alvarez,Stiven Díaz,Carlos Lizama

doi:10.1186/s13662-019-2053-0

Edgardo Alvarez, Stiven Díaz + Show 1 more

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https://doi.org/10.1186/s13662-019-2053-0

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Abstract

In this paper we introduce the class of (N,lambda )-periodic vector-valued sequences and show several notable properties of this new class. This class includes periodic, anti-periodic, Bloch and unbounded sequences. Furthermore, we show the existence and uniqueness of (N, lambda )-periodic solutions to the following class of Volterra difference equations with infinite delay: \t\t\tu(n+1)=α∑j=−∞na(n−j)u(j)+f(n,u(n)),n∈Z,α∈C,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u(n+1)=\\alpha \\sum_{j=-\\infty }^{n}a(n-j)u(j)+f \\bigl(n,u(n) \\bigr), \\quad n \\in \\mathbb{Z}, \\alpha \\in \\mathbb{C}, $$\\end{document} where the kernel a and the nonlinear term f satisfy suitable conditions.

Highlights

In this paper, we define and investigate a new class of vector-valued functions defined on Z called (N, λ)-periodic discrete functions
We establish a criterion of the existence and uniqueness of (N, λ)-periodic discrete solutions to the linear and nonlinear Volterra difference equations in Banach spaces
In order to obtain our results, first we show a characterization of the (N, λ)-periodic discrete functions, which says that f : Z → X is an (N, λ)-periodic discrete function if and only if there exists an N -periodic function u such that f (n) = λn/N u(n) for all n ∈ Z

Summary

Introduction

We define and investigate a new class of vector-valued functions defined on Z called (N, λ)-periodic discrete functions. We establish a criterion of the existence and uniqueness of (N, λ)-periodic discrete solutions to the linear and nonlinear Volterra difference equations in Banach spaces. Araya et al in [6] studied the existence and uniqueness of almost automorphic discrete solutions for a class of nonlinear Volterra difference equations of convolution type on a Banach space X with norm · X, namely n u(n + 1) = α a(n – j)u(j) + f n, u(u) , (1.1)

Results

Conclusion