scholarly journals On Discrete Fractional Integral Inequalities for a Class of Functions

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
Saima Rashid ◽  
Hijaz Ahmad ◽  
Aasma Khalid ◽  
Yu-Ming Chu
Keyword(s):  
Fractional Calculus ◽  
Difference Equations ◽  
Logistic Map ◽  
Neural Systems ◽  
The Novel ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
Systems With Memory ◽  
Positive Functions

Discrete fractional calculus ℱ C is proposed to depict neural systems with memory impacts. This research article aims to investigate the consequences in the frame of the discrete proportional fractional operator. ℏ -discrete exponential functions are assumed in the kernel of the novel generalized fractional sum defined on the time scale ℏ ℤ . The nabla ℏ -fractional sums are accounted in particular. The governing high discretization of problems is an advanced version of the existing forms that can be transformed into linear and nonlinear difference equations using appropriately adjusted transformations invoking property of observing the new chaotic behaviors of the logistic map. Based on the theory of discrete fractional calculus, explicit bounds for a class of positive functions n n ∈ ℕ concerned are established. These variants can be utilized as a convenient apparatus in the qualitative analysis of solutions of discrete fractional difference equations. With respect to applications, we can apply the introduced outcomes to explore boundedness, uniqueness, and continuous reliance on the initial value problem for the solutions of certain underlying worth problems of fractional difference equations.

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 On the nonoscillatory behavior of solutions of three classes of fractional difference equations

2020 ◽  
Vol 40 (5) ◽  
pp. 549-568
Author(s):  
Said Rezk Grace ◽  
Jehad Alzabut ◽  
Sakthivel Punitha ◽  
Velu Muthulakshmi ◽  
Hakan Adıgüzel
Keyword(s):  
Fractional Calculus ◽  
Difference Equations ◽  
Riccati Transformation ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
Transformation Technique ◽  
Fractional Difference Equations ◽  
Difference Formula

In this paper, we study the nonoscillatory behavior of three classes of fractional difference equations. The investigations are presented in three different folds. Unlike most existing nonoscillation results which have been established by employing Riccati transformation technique, we employ herein an easily verifiable approach based on the fractional Taylor's difference formula, some features of discrete fractional calculus and mathematical inequalities. The theoretical findings are demonstrated by examples. We end the paper by a concluding remark.

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.

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 Oscillation properties of solutions of fractional difference equations

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 185-192 ◽  
Author(s):  
Mustafa Bayram ◽  
Aydin Secer
Keyword(s):  
Difference Equations ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
Further Development ◽  
Oscillation Properties

In this article, studied the properties of the oscillation of fractional difference equations, and we obtain some results. The results we obtained are an expansion and further development of highly known results. Then we showed them with examples.

Asymptotic behavior of mild solutions for nonlinear fractional difference equations

2018 ◽  
Vol 21 (2) ◽  
pp. 527-551 ◽  
Author(s):  
Zhinan Xia ◽  
Dingjiang Wang
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Difference Equations ◽  
Mild Solutions ◽  
Contraction Mapping Principle ◽  
Fractional Difference ◽  
Alternative Theorem ◽  
Fractional Difference Equations ◽  
Banach Contraction ◽  
Asymptotically Periodic

AbstractIn this paper, we establish some sufficient criteria for the existence, uniqueness of discrete weighted pseudo asymptotically periodic mild solutions and asymptotic behavior for nonlinear fractional difference equations in Banach space, where the nonlinear perturbation is Lipschitz type, or non-Lipschitz type. The results are a consequence of application of different fixed point theorems, namely, the Banach contraction mapping principle, Leray-Schauder alternative theorem and Matkowski’s fixed point technique.

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