Method of upper and lower solutions for nonlinear Caputo fractional difference equations and its applications

2019 ◽  
Vol 22 (5) ◽  
pp. 1307-1320 ◽  
Author(s):  
Churong Chen ◽  
Martin Bohner ◽  
Baoguo Jia
Keyword(s):  
Difference Equations ◽  
Caputo Derivative ◽  
Upper And Lower Solutions ◽  
Fractional Difference ◽  
Numerical Examples ◽  
Maximal Solutions ◽  
Fractional Difference Equations ◽  
Monotone Sequences ◽  
Minimal And Maximal Solutions

Abstract In this paper, we extend the applications of the method of upper and lower solutions for a class of nonlinear nabla fractional difference equations involving Caputo derivative. We obtain the existence of coupled minimal and maximal solutions which constructed by two monotone sequences. In order to illustrate our main results, we present two numerical examples in the end.

Some new Gompertz fractional difference equations

2020 ◽  
Vol 13 (4) ◽  
pp. 705-719
Author(s):  
Tom Cuchta ◽  
Brooke Fincham
Keyword(s):  
Difference Equations ◽  
Fractional Difference ◽  
Fractional Difference Equations

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.

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.

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.

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