scholarly journals Oscillation Results for a Class of Nonlinear Fractional Order Difference Equations with Damping Term

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
Mary Jacintha ◽  
Abdullah Özbekler
Keyword(s):  
Fractional Order ◽  
Difference Equations ◽  
Difference Operator ◽  
Fractional Operators ◽  
Damping Term ◽  
Fractional Difference ◽  
Riccati Technique ◽  
Order Difference ◽  
Numerical Examples ◽  
Positive Integers

The paper studies the oscillation of a class of nonlinear fractional order difference equations with damping term of the form Δψλzηλ+pλzηλ+qλF∑s=λ0λ−1+μ λ−s−1−μys=0, where zλ=aλ+bλΔμyλ, Δμ stands for the fractional difference operator in Riemann-Liouville settings and of order μ, 0<μ≤1, and η≥1 is a quotient of odd positive integers and λ∈ℕλ0+1−μ. New oscillation results are established by the help of certain inequalities, features of fractional operators, and the generalized Riccati technique. We verify the theoretical outcomes by presenting two numerical examples.

Oscillation criteria for a class of nonlinear discrete fractional order equations with damping term

2020 ◽  
Vol 70 (5) ◽  
pp. 1165-1182
Author(s):  
George E. Chatzarakis ◽  
George M. Selvam ◽  
Rajendran Janagaraj ◽  
George N. Miliaras
Keyword(s):  
Fractional Order ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Damping Term ◽  
Transformation Technique ◽  
Order Difference ◽  
Numerical Examples ◽  
Oscillation Criteria ◽  
The Difference ◽  
Theoretical Results

AbstractThe aim in this work is to investigate oscillation criteria for a class of nonlinear discrete fractional order equations with damping term of the form$$\begin{array}{} \displaystyle \Delta\left[a(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta\right]+p(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta+F(t,G(t))=0, t\in N_{t_0}. \end{array}$$In the above equation α (0 < α ≤ 1) is the fractional order, $\begin{array}{} \displaystyle G(t)=\sum\limits_{s=t_0}^{t-1+\alpha}\left(t-s-1\right)^{(-\alpha)}x(s) \end{array}$ and Δα is the difference operator of the Riemann-Liouville (R-L) derivative of order α. We establish some new sufficient conditions for the oscillation of fractional order difference equations with damping term based on a Riccati transformation technique and some inequalities. We provide numerical examples to illustrate the validity of the theoretical results.

scholarly journals Multiparameter Fractional Difference Linear Control Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Dorota Mozyrska
Keyword(s):  
Control Systems ◽  
Fractional Order ◽  
Initial Conditions ◽  
Linear Control Systems ◽  
Difference Operators ◽  
Fractional Operators ◽  
Fractional Difference ◽  
Order Difference ◽  
Time Control ◽  
Controllability And Observability

The Riemann-Liouville-, Caputo-, and Grünwald-Letnikov-type fractional order difference operators are discussed and used to state and solve the controllability and observability problems of linear fractional order discrete-time control systems with multiorder and multistep. It is shown that the obtained results do not depend on the type of fractional operators and steps. The comparison of systems is made under the number of steps needed, firstly to achieve a final point, and secondly to distinguish initial conditions for particular operator.

scholarly journals Fractional Order Difference Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
J. Jagan Mohan ◽  
G. V. S. R. Deekshitulu
Keyword(s):  
Difference Equation ◽  
Fractional Order ◽  
Difference Equations ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Fractional Difference ◽  
Order Difference ◽  
Fractional Difference Equations ◽  
Independent Variable

A difference equation is a relation between the differences of a function at one or more general values of the independent variable. These equations usually describe the evolution of certain phenomena over the course of time. The present paper deals with the existence and uniqueness of solutions of fractional difference equations.

scholarly journals Ulam stability results for boundary value problem of fractional difference equations

2020 ◽  
pp. 219-230
Author(s):  
A. George Maria Selvam ◽  
R. Dhineshbabu
Keyword(s):  
Stability Analysis ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Difference Equations ◽  
Difference Operator ◽  
Boundary Value ◽  
Ulam Stability ◽  
Fractional Difference ◽  
Stability Results

Boundary value problems have wide applications in science and technology. This paper is concerned with various kinds of Ulam stability analysis for the nonlinear discrete boundary value problem of fractional order $\sigma\in(2,3]$ with Riemann-Liouville fractional difference operator. Finally, some examples are presented to illustrate the main results.

On the spectral analysis of self adjoint operators generated by second order difference equations

1991 ◽  
Vol 118 (1-2) ◽  
pp. 139-151 ◽  
Author(s):  
Dale T. Smith
Keyword(s):  
Essential Spectrum ◽  
Difference Equations ◽  
Difference Operator ◽  
Simple Criterion ◽  
Order Difference ◽  
Forward Difference ◽  
Other Information ◽  
Image Position ◽  
Adjoint Operators ◽  
Oscillation Constant

SynopsisIn this paper, I shall consider operators generated by difference equations of the formwhere Δ is the forward difference operator, and a, p, and r are sequences of real numbers. The connection between the oscillation constant of this equation and the bottom of the essential spectrum of self-adjoint extensions of the operator generated by the equation is given, as well as various other information about the spectrum of such extensions. In particular, I derive conditions for the spectrum to have only countably many eigenvalues below zero, and a simple criterion for the invariance of the essential spectrum.

scholarly journals Oscillation Tests for Fractional Difference Equations

2018 ◽  
Vol 71 (1) ◽  
pp. 53-64 ◽  
Author(s):  
George E. Chatzarakis ◽  
Palaniyappan Gokulraj ◽  
Thirunavukarasu Kalaimani
Keyword(s):  
Difference Equation ◽  
Difference Operator ◽  
Oscillatory Behavior ◽  
Riccati Transformation ◽  
Fractional Difference ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Fractional Difference Equation ◽  
Positive Integers ◽  
Theoretical Results

Abstract In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the form $$\Delta \left( {r\left( t \right)g\left( {{\Delta ^\alpha }x(t)} \right)} \right) + p(t)f\left( {\sum\limits_{s = {t_0}}^{t - 1 + \alpha } {{{(t - s - 1)}^{( - \alpha )}}x(s)} } \right) = 0, & t \in {_{{t_0} + 1 - \alpha }},$$ where Δα denotes the Riemann-Liouville fractional difference operator of order α, 0 < α ≤ 1, ℕt0+1−α={t0+1−αt0+2−α…}, t0 > 0 and γ > 0 is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation, using the Riccati transformation and Hardy type inequalities. Examples are provided to illustrate the theoretical results.

scholarly journals Oscillation theorems for second order difference equations woth negative neutral term

2015 ◽  
Vol 46 (4) ◽  
pp. 441-451 ◽  
Author(s):  
Ethiraju Thandapani ◽  
Devarajulu Seghar ◽  
Sandra Pinelas
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Second Order ◽  
Neutral Difference Equation ◽  
Order Difference ◽  
Oscillation Criteria ◽  
Neutral Term ◽  
Oscillation Theorems ◽  
Positive Integers

In this paper we obtain some new oscillation criteria for the neutral difference equation \begin{equation*} \Delta \Big(a_n (\Delta (x_n-p_n x_{n-k}))\Big)+q_n f(x_{n-l})=0 \end{equation*} where $0\leq p_n\leq p0$ and $l$ and $k$ are positive integers. Examples are presented to illustrate the main results. The results obtained in this paper improve and complement to the existing results.

On difference sequence spaces of fractional-order involving Padovan numbers

2020 ◽  
pp. 2150095
Author(s):  
Taja Yaying ◽  
Bipan Hazarika ◽  
Syed Abdul Mohiuddine
Keyword(s):  
Fractional Order ◽  
Measure Of Noncompactness ◽  
Difference Operator ◽  
Sequence Spaces ◽  
Difference Sequence ◽  
Topological Properties ◽  
Hausdorff Measure Of Noncompactness ◽  
Order Difference ◽  
Matrix Formula ◽  
Difference Sequence Spaces

In this paper, we introduce Padovan difference sequence spaces of fractional-order [Formula: see text] [Formula: see text] [Formula: see text] by the composition of the fractional-order difference operator [Formula: see text] and the Padovan matrix [Formula: see text] defined by [Formula: see text] and [Formula: see text] respectively, where the sequence [Formula: see text] is the Padovan sequence. We give some topological properties, Schauder basis and [Formula: see text]-, [Formula: see text]- and [Formula: see text]-duals of the newly defined spaces. We characterize certain matrix classes related to the [Formula: see text] space. Finally, we characterize certain classes of compact operators on [Formula: see text] using Hausdorff measure of noncompactness.

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