scholarly journals Stability, Neimark–Sacker Bifurcation, and Approximation of the Invariant Curve of Certain Homogeneous Second-Order Fractional Difference Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Mirela Garić-Demirović ◽  
Samra Moranjkić ◽  
Mehmed Nurkanović ◽  
Zehra Nurkanović
Keyword(s):  
Difference Equation ◽  
Equilibrium Point ◽  
Asymptotic Approximation ◽  
Second Order ◽  
Invariant Curve ◽  
Unique Equilibrium ◽  
Fractional Difference ◽  
Global Character ◽  
Fractional Difference Equation

We investigate the local and global character of the unique equilibrium point and boundedness of the solutions of certain homogeneous fractional difference equation with quadratic terms. Also, we consider Neimark–Sacker bifurcations and give the asymptotic approximation of the invariant curve.

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.

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.

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.

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