scholarly journals Ulam stability of Caputo q-fractional delay difference equation: q-fractional Gronwall inequality approach

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Rabia Ilyas Butt ◽  
Thabet Abdeljawad ◽  
Manar A. Alqudah ◽  
Mujeeb ur Rehman
Keyword(s):  
Existence And Uniqueness ◽  
Gronwall Inequality ◽  
Ulam Stability ◽  
Delay Difference Equation ◽  
Fractional Difference ◽  
Grönwall Inequality ◽  
Fractional Delay ◽  
Delay Difference ◽  
Theoretical Results ◽  
Rassias Stability

AbstractIn this article, we discuss the existence and uniqueness of solution of a delay Caputo q-fractional difference system. Based on the q-fractional Gronwall inequality, we analyze the Ulam–Hyers stability and the Ulam–Hyers–Rassias stability. An example is provided to support the theoretical results.

scholarly journals Existence and Ulam stability for implicit fractional q-difference equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Saïd Abbas ◽  
Mouffak Benchohra ◽  
Nadjet Laledj ◽  
Yong Zhou
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Difference Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Ulam Stability ◽  
Uniqueness Of Solutions ◽  
Stability Results ◽  
Rassias Stability

AbstractThis paper deals with some existence, uniqueness and Ulam–Hyers–Rassias stability results for a class of implicit fractional q-difference equations. Some applications are made of some fixed point theorems in Banach spaces for the existence and uniqueness of solutions, next we prove that our problem is generalized Ulam–Hyers–Rassias stable. Two illustrative examples are given in the last section.

Existence, Uniqueness and Stability of Implicit Switched Coupled Fractional Differential Equations of ψ$\boldsymbol{\psi}$-Hilfer Type

2020 ◽  
Vol 21 (3-4) ◽  
pp. 327-337 ◽  
Author(s):  
Manzoor Ahmad ◽  
Akbar Zada ◽  
Xiaoming Wang
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Ulam Stability ◽  
Uniqueness Of Solutions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential System ◽  
Fractional Differential ◽  
Fixed Point Techniques ◽  
Rassias Stability

AbstractIn this article, we study the existence and uniqueness of solutions of a switched coupled implicit ψ-Hilfer fractional differential system. The existence and uniqueness results are obtained by using fixed point techniques. Further, we investigate different kinds of stability such as Hyers–Ulam stability and Hyers–Ulam–Rassias stability. Finally, an example is provided to illustrate the obtained results.

scholarly journals On the Hyers-Ulam Stability of Differential Equations of Second Order

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Qusuay H. Alqifiary ◽  
Soon-Mo Jung
Keyword(s):  
Differential Equations ◽  
Initial Conditions ◽  
Second Order ◽  
Gronwall Inequality ◽  
Ulam Stability ◽  
Grönwall Inequality

By using of the Gronwall inequality, we prove the Hyers-Ulam stability of differential equations of second order with initial conditions.

scholarly journals A generalized discrete fractional Gronwall inequality and its application on the uniqueness of solutions for nonlinear delay fractional difference system

2018 ◽  
Vol 12 (1) ◽  
pp. 36-48 ◽  
Author(s):  
Jehad Alzabut ◽  
Thabet Abdeljawad
Keyword(s):  
Gronwall Inequality ◽  
Difference System ◽  
Uniqueness Of Solutions ◽  
Fractional Difference ◽  
Numerical Example ◽  
Generalized Gronwall Inequality ◽  
Grönwall Inequality ◽  
Nonlinear Delay

In this paper, we state and prove a new discrete fractional version of the generalized Gronwall inequality. Based on this, a particular version expressed by means of discrete Mittag-Leer functions is provided. As an application, we prove the uniqueness and obtain an estimate for the solutions of nonlinear delay Caputo fractional difference system. Numerical example is presented to demonstrate the applicability of the main results.

On the stability for the fuzzy initial value problem

2020 ◽  
Vol 39 (5) ◽  
pp. 7747-7755
Author(s):  
Ngo Van Hoa ◽  
Tofigh Allahviranloo ◽  
Ho Vu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Initial Value Problem ◽  
Ulam Stability ◽  
Initial Value ◽  
The Stability ◽  
Theoretical Results ◽  
Rassias Stability

In this paper, we present the Hyers–Ulam stability and Hyers–Ulam-Rassias stability (HU-stability and HUR-stability for short) for fuzzy initial value problem (FIVP) by using fixed point theorem. We improve and extend some known results on the stability for FDEs by dropping some assumptions. Some examples illustrate the theoretical results.

scholarly journals Finite-Time Stability of Solutions for Nonlinear q -Fractional Difference Coupled Delay Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Jingfeng Wang ◽  
Chuanzhi Bai
Keyword(s):  
Finite Time ◽  
Gronwall Inequality ◽  
Delay Systems ◽  
Stability Of Solutions ◽  
Stability Criteria ◽  
Time Stability ◽  
Fractional Difference ◽  
Finite Time Stability ◽  
Grönwall Inequality

In this paper, we investigate and prove a new discrete q -fractional version of the coupled Gronwall inequality. By applying this result, the finite-time stability criteria of solutions for a class of nonlinear q -fractional difference coupled delay systems are obtained. As an application, an example is provided to demonstrate the effectiveness of our result.

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