scholarly journals A Fixed-Point Approach to the Hyers–Ulam Stability of Caputo–Fabrizio Fractional Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 647 ◽  
Author(s):  
Kui Liu ◽  
Michal Fečkan ◽  
JinRong Wang
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Compact Interval ◽  
Ulam Stability ◽  
Fixed Point Approach ◽  
Fractional Differential ◽  
Stability Results ◽  
Rassias Stability

In this paper, we study Hyers–Ulam and Hyers–Ulam–Rassias stability of nonlinear Caputo–Fabrizio fractional differential equations on a noncompact interval. We extend the corresponding uniqueness and stability results on a compact interval. Two examples are given to illustrate our main results.

scholarly journals Boundary Value Problem for Caputo–Fabrizio Random Fractional Differential Equations

2020 ◽  
Vol 6 (2) ◽  
pp. 218-230
Author(s):  
Fouzia Bekada ◽  
Saïd Abbas ◽  
Mouffak Benchohra
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Random Operators ◽  
Ulam Stability ◽  
Fractional Differential ◽  
Stability Results

AbstractThis article deals with some existence of random solutions and Ulam stability results for a class of Caputo-Fabrizio random fractional differential equations with boundary conditions in Banach spaces. Our results are based on the fixed point theory and random operators. Two illustrative examples are presented in the last section.

scholarly journals Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Abdelkrim Salim ◽  
Mouffak Benchohra ◽  
Erdal Karapınar ◽  
Jamal Eddine Lazreg
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Measure Of Noncompactness ◽  
Ulam Stability ◽  
Stability Of Solutions ◽  
Fractional Differential ◽  
Implicit Fractional Differential Equations

Abstract In this manuscript, we examine the existence and the Ulam stability of solutions for a class of boundary value problems for nonlinear implicit fractional differential equations with instantaneous impulses in Banach spaces. The results are based on fixed point theorems of Darbo and Mönch associated with the technique of measure of noncompactness. We provide some examples to indicate the applicability of our results.

scholarly journals Hyers–Ulam stability of a coupled system of fractional differential equations of Hilfer–Hadamard type

2019 ◽  
Vol 52 (1) ◽  
pp. 283-295 ◽  
Author(s):  
Manzoor Ahmad ◽  
Akbar Zada ◽  
Jehad Alzabut
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Coupled System ◽  
Ulam Stability ◽  
Fractional Differential System ◽  
Different Types ◽  
Fractional Differential

AbstractIn this paper, existence and uniqueness of solution for a coupled impulsive Hilfer–Hadamard type fractional differential system are obtained by using Kransnoselskii’s fixed point theorem. Different types of Hyers–Ulam stability are also discussed.We provide an example demonstrating consistency to the theoretical findings.

scholarly journals Multiple solutions for a class of boundary value problems of fractional differential equations with generalized Caputo derivatives

2021 ◽  
Vol 6 (12) ◽  
pp. 13119-13142
Author(s):  
Yating Li ◽  
◽  
Yansheng Liu
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Multiple Solutions ◽  
Boundary Value ◽  
Ulam Stability ◽  
Caputo Derivatives ◽  
Fractional Differential ◽  
The Fixed Point Theorem

<abstract><p>This paper is mainly concerned with the existence of multiple solutions for the following boundary value problems of fractional differential equations with generalized Caputo derivatives:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \hskip 3mm \left\{ \begin{array}{lll} ^{C}_{0}D^{\alpha}_{g}x(t)+f(t, x) = 0, \ 0&lt;t&lt;1;\\ x(0) = 0, \ ^{C}_{0}D^{1}_{g}x(0) = 0, \ ^{C}_{0}D^{\nu}_{g}x(1) = \int_{0}^{1}h(t)^{C}_{0}D^{\nu}_{g}x(t)g'(t)dt, \end{array}\right. $\end{document} </tex-math></disp-formula></p> <p>where $ 2 &lt; \alpha &lt; 3 $, $ 1 &lt; \nu &lt; 2 $, $ \alpha-\nu-1 &gt; 0 $, $ f\in C([0, 1]\times \mathbb{R}^{+}, \mathbb{R}^{+}) $, $ g' &gt; 0 $, $ h\in C([0, 1], \mathbb{R}^{+}) $, $ \mathbb{R}^{+} = [0, +\infty) $. Applying the fixed point theorem on cone, the existence of multiple solutions for considered system is obtained. The results generalize and improve existing conclusions. Meanwhile, the Ulam stability for considered system is also considered. Finally, three examples are worked out to illustrate the main results.</p></abstract>

scholarly journals Analysis of Nonlinear Coupled Systems of Impulsive Fractional Differential Equations with Hadamard Derivatives

2019 ◽  
Vol 2019 ◽  
pp. 1-20 ◽  
Author(s):  
Usman Riaz ◽  
Akbar Zada ◽  
Zeeshan Ali ◽  
Manzoor Ahmad ◽  
Jiafa Xu ◽  
...  
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Coupled System ◽  
Coupled Systems ◽  
Ulam Stability ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Impulsive Fractional Differential Equations ◽  
Uniqueness Results ◽  
Fractional Differential

This work is committed to establishing the assumptions essential for at least one and unique solution of a switched coupled system of impulsive fractional differential equations having derivative of Hadamard type. Using Krasnoselskii’s fixed point theorem, the existence, as well as uniqueness results, is obtained. Along with this, different kinds of Hyers–Ulam stability are discussed. For supporting the theory, example is provided.

scholarly journals Existence and Stability of Implicit Fractional Differential Equations with Stieltjes Boundary Conditions Involving Hadamard Derivatives

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-36
Author(s):  
Danfeng Luo ◽  
Mehboob Alam ◽  
Akbar Zada ◽  
Usman Riaz ◽  
Zhiguo Luo
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Ulam Stability ◽  
Integral Boundary ◽  
Uniqueness Results ◽  
Cone Type ◽  
Fractional Differential ◽  
Rassias Stability ◽  
Implicit Fractional Differential Equations

In this article, we make analysis of the implicit fractional differential equations involving integral boundary conditions associated with Stieltjes integral and its corresponding coupled system. We use some sufficient conditions to achieve the existence and uniqueness results for the given problems by applying the Banach contraction principle, Schaefer’s fixed point theorem, and Leray–Schauder result of the cone type. Moreover, we present different kinds of stability such as Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability, and generalized Hyers–Ulam–Rassias stability by using the classical technique of functional analysis. At the end, the results are verified with the help of examples.

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