scholarly journals Fractional Difference Equations with Real Variable

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Jin-Fa Cheng ◽  
Yu-Ming Chu
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Real Variable ◽  
Fractional Difference ◽  
Basic Properties ◽  
Fractional Difference Equations ◽  
Fractional Difference Equation ◽  
Definition Of

We independently propose a new kind of the definition of fractional difference, fractional sum, and fractional difference equation, give some basic properties of fractional difference and fractional sum, and give some examples to demonstrate several methods of how to solve certain fractional difference equations.

Asymptotic stability of fractional difference equations with bounded time delays

2020 ◽  
Vol 23 (2) ◽  
pp. 571-590
Author(s):  
Mei Wang ◽  
Baoguo Jia ◽  
Feifei Du ◽  
Xiang Liu
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Integral Inequality ◽  
Time Delays ◽  
Asymptotical Stability ◽  
Stability Conditions ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
Fractional Difference Equation

AbstractIn this paper, an integral inequality and the fractional Halanay inequalities with bounded time delays in fractional difference are investigated. By these inequalities, the asymptotical stability conditions of Caputo and Riemann-Liouville fractional difference equation with bounded time delays are obtained. Several examples are presented to illustrate the results.

Stability analysis of impulsive fractional difference equations

2018 ◽  
Vol 21 (2) ◽  
pp. 354-375 ◽  
Author(s):  
Guo–Cheng Wu ◽  
Dumitru Baleanu
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Difference Equation ◽  
Fractional Calculus ◽  
Difference Equations ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
Numerical Result ◽  
Fractional Difference Equations ◽  
Fractional Difference Equation

AbstractWe revisit motivation of the fractional difference equations and some recent applications to image encryption. Then stability of impulsive fractional difference equations is investigated in this paper. The fractional sum equation is considered and impulsive effects are introduced into discrete fractional calculus. A class of impulsive fractional difference equations are proposed. A discrete comparison principle is given and asymptotic stability of nonlinear fractional difference equation are discussed. Finally, an impulsive Mittag–Leffler stability is defined. The numerical result is provided to support the analysis.

scholarly journals Existence of local solutions for fractional difference equations with left focal boundary conditions

2021 ◽  
Vol 24 (1) ◽  
pp. 324-331
Author(s):  
Johnny Henderson ◽  
Jeffrey T. Neugebauer
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Real Number ◽  
Natural Number ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
Fractional Difference Equation ◽  
Local Solutions

Abstract For 1 < ν ≤ 2 a real number and T ≥ 3 a natural number, conditions are given for the existence of solutions of the νth order Atıcı-Eloe fractional difference equation, Δ ν y(t) + f(t + ν − 1, y(t + ν − 1)) = 0, t ∈ {0, 1, …, T}, and satisfying the left focal boundary conditions Δy(ν − 2) = y(ν + T) = 0.

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 Asymptotic Stability for a Class of Nonlinear Difference Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Chang-you Wang ◽  
Shu Wang ◽  
Zhi-wei Wang ◽  
Fei Gong ◽  
Rui-fang Wang
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Real Numbers ◽  
Nonlinear Difference Equations ◽  
Positive Real ◽  
Fractional Difference ◽  
Fractional Difference Equation

We study the global asymptotic stability of the equilibrium point for the fractional difference equationxn+1=(axn-lxn-k)/(α+bxn-s+cxn-t),n=0,1,…, where the initial conditionsx-r,x-r+1,…,x1,x0are arbitrary positive real numbers of the interval(0,α/2a),l,k,s,tare nonnegative integers,r=max⁡⁡{l,k,s,t}andα,a,b,care positive constants. Moreover, some numerical simulations are given to illustrate our results.

scholarly journals On the Fractional Difference Equations of Order(2,q)

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Jin-Fa Cheng ◽  
Yu-Ming Chu
Keyword(s):  
Difference Equations ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
Definition Of

This paper presents a kind of new definition of fractional difference, fractional summation, and fractional difference equations and gives methods for explicitly solving fractional difference equations of order(2,q).

Construction of fixed point operators for nonlinear difference equations of non integer order with impulses

2020 ◽  
Vol 23 (3) ◽  
pp. 886-907
Author(s):  
Syed Sabyel Haider ◽  
Mujeeb Ur Rehman
Keyword(s):  
Fixed Point ◽  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Nonlinear Difference Equations ◽  
Fractional Difference ◽  
Integer Order ◽  
Fractional Difference Equation ◽  
Point Operator

AbstractIn this article, we establish a technique for transforming arbitrary real order delta difference equations with impulses to corresponding summation equations. The technique is applied to non-integer order delta difference equation with some boundary conditions. Furthermore, the summation formulation for impulsive fractional difference equation is utilized to construct fixed point operator which in turn are used to discuss existence of solutions.

scholarly journals Oscillatory behavior of nonlinear Hilfer fractional difference equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tuğba Yalçın Uzun
Keyword(s):  
Difference Equations ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Oscillatory Behavior ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
All Solutions ◽  
Alpha Beta

AbstractIn this paper, we study the oscillation behavior for higher order nonlinear Hilfer fractional difference equations of the type $$\begin{aligned}& \Delta _{a}^{\alpha ,\beta }y(x)+f_{1} \bigl(x,y(x+\alpha ) \bigr) =\omega (x)+f_{2} \bigl(x,y(x+ \alpha ) \bigr),\quad x\in \mathbb{N}_{a+n-\alpha }, \\& \Delta _{a}^{k-(n-\gamma )}y(x) \big|_{x=a+n-\gamma } = y_{k}, \quad k= 0,1,\ldots,n, \end{aligned}$$ Δ a α , β y ( x ) + f 1 ( x , y ( x + α ) ) = ω ( x ) + f 2 ( x , y ( x + α ) ) , x ∈ N a + n − α , Δ a k − ( n − γ ) y ( x ) | x = a + n − γ = y k , k = 0 , 1 , … , n , where $\lceil \alpha \rceil =n$ ⌈ α ⌉ = n , $n\in \mathbb{N}_{0}$ n ∈ N 0 and $0\leq \beta \leq 1$ 0 ≤ β ≤ 1 . We introduce some sufficient conditions for all solutions and give an illustrative example for our results.

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