Article
Kyungpook Mathematical Journal 2020; 60(1): 163-175
Published online March 31, 2020
Copyright © Kyungpook Mathematical Journal.
Stability Criterion for Volterra Type Delay Difference Equations Including a Generalized Difference Operator
Murat Gevgesoglu∗ and Yasar Bolat
Department of Mathematics, Kastamonu University, Kastamonu, Turkey
e-mail : mgevgesoglu@kastamonu.edu.tr and ybolat@kastamonu.edu.tr
Received: January 17, 2018; Revised: May 10, 2019; Accepted: November 18, 2019
Abstract
The stability of a class of Volterra-type difference equations that include a generalized difference operator Δ
Keywords: stability, Volterra diff,erence equations.
1. Introduction
Difference equations are the discrete analogues of differential equations and they usually describe certain phenomena over the course of time. Difference equations have many applications in a wide variety of disciplines, such as economics, mathematical biology, social sciences and physics. We refer to [1, 2, 4, 6] for the basic theory and some applications of difference equations. Volterra difference equations are extensively used to model phenomena in engineering, economics, and in the natural and social sciences; their stability has been studied by many authors.
In [5], Khandaker and Raffoul considered a Volterra discrete system with nonlinear perturbation
and obtained necessary and sufficient conditions for stability properties of the zero solution employing the resolvent equation coupled with a variation of parameters formula.
In [7], Migda et al. investigated the boundedness and asymptotic stability of the zero solution of the discrete Volterra equation
using fixed point theory.
In [3], Islam and Yankson studied the stability and boundedness of the nonlinear difference equation
using fixed point theorems.
In [9], Yankson studied the asymptotic stability of the zero solution of the Volterra difference delay equation
using Krasnoselskii’s fixed point theorem.
In this paper, motivated by [9], we investigate the asymptotic stability of the zero solution of neutral and Volterra type difference equations which include a generalized difference operator of the form
using Krasnoselskii’s fixed point theorem. Here
The difference operator Δ and generalized difference operator Δ
and
respectively.
We assume that
for some positive constant
2. Basic Definitions, Theorems and Lemmas
For any integer
Definition 2.1
Definition 2.2
The zero solution of
Definition 2.3
The zero solution of
Lemma 2.1
It is obvious.
Now below we state Krasnoselskii’s theorem. For the proof we refer to [8].
Theorem 2.1
x ,y ∈M implies Ax +Qy ∈M ,A is continuous and AM is contained in a compact set ,Q is a contraction mapping.
Theorem 2.2.(Ascoli-Arzela Theorem)
3. Main Results
Lemma 3.1
From
Using the definition of the operator Δ
By summing both sides of
from this last equality, we write
Because
we can write
or
Now, using Lemma 2.1 in the second term on the right-hand side of
Hence, by putting this last equality in
Because in the last term on the right-hand side of
from
This completes the proof.
Now let
where
Then, we can see that (
and for
Lemma 3.2
Due to the condition
Given
Hence, for
This completes the proof.
To use Krasnoselskii’s theorem, we construct two mappings
where
and
respectively.
Lemma 3.3
First, we show that the mapping
which shows that the mapping
which shows that (
for some positive constant
Lemma 3.4
Take any two functions
which shows that Q is a contraction mapping.
Theorem 3.1
Given
Let
and take any
which shows that (
By the last result, Lemma 4 and Lemma 5 all conditions of Theorem 1 are satisfied on
Example 3.1
Consider the difference equation
Here,
We see that
so
The solution is of the form
References
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, Chapman and Hall, New York, 1991. - M. Migda, M. Ruzickova, and E. Schmeidel.
Boundedness and stability of discrete Volterra equations . Adv Difference Equ.,2015 (2015) 11 pp, 47. - DR. Smart. Fixed point theorems,
, Cambridge University Press, Cambridge, 1980. - E. Yankson.
Stability of Volterra difference delay equations . Electron J Qual Theory Differ Equ.,20 (2006), 14.