scholarly journals Stability Results for Implicit Fractional Pantograph Differential Equations via ϕ-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 94 ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Kamal Shah ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet ◽  
...  
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Fixed Point Theorems ◽  
Integral Condition ◽  
Hilfer Fractional Derivative ◽  
Fractional Differential ◽  
Generalized Fractional Derivative ◽  
Stability Results

This paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the problem. The existence and uniqueness of the problem is obtained using Schaefer’s and Banach fixed point theorems. In addition, the Ulam-Hyers and generalized Ulam-Hyers stability of the problem are established. Finally, some examples are given to illustrative the results.

scholarly journals Qualitative Analysis of Langevin Integro-Fractional Differential Equation under Mittag–Leffler Functions Power Law

2021 ◽  
Vol 5 (4) ◽  
pp. 266
Author(s):  
Mohammed A. Almalahi ◽  
F. Ghanim ◽  
Thongchai Botmart ◽  
Omar Bazighifan ◽  
Sameh Askar
Keyword(s):  
Differential Equation ◽  
Qualitative Analysis ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Different Types ◽  
Fractional Differential ◽  
Stability Results

This research paper intends to investigate some qualitative analysis for a nonlinear Langevin integro-fractional differential equation. We investigate the sufficient conditions for the existence and uniqueness of solutions for the proposed problem using Banach’s and Krasnoselskii’s fixed point theorems. Furthermore, we discuss different types of stability results in the frame of Ulam–Hyers by using a mathematical analysis approach. The obtained results are illustrated by presenting a numerical example.

scholarly journals Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Jamilu Abubakar ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Fixed Point Theorems ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

Abstract This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.

scholarly journals A Study of Fractional Differential Equation with a Positive Constant Coefficient via Hilfer Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Hoa Ngo Van ◽  
Vu Ho
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Positive Constant ◽  
Constant Coefficient ◽  
Initial Conditions ◽  
Hilfer Fractional Derivative ◽  
Fractional Differential ◽  
The Fixed Point Theorem ◽  
Rassias Stability

The aim of the paper is to consider the existence and uniqueness of solution of the fractional differential equation with a positive constant coefficient under Hilfer fractional derivative by using the fixed-point theorem. We also prove the bounded and continuous dependence on the initial conditions of solution. Besides, Hyers–Ulam stability and Hyers–Ulam–Rassias stability are discussed. Finally, we provide an example to demonstrate our main results.

scholarly journals Uniqueness of Positive Solutions for a Perturbed Fractional Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Chen Yang ◽  
Jieming Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Point Boundary ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

We are concerned with the existence and uniqueness of positive solutions for the following nonlinear perturbed fractional two-point boundary value problem:D0+αu(t)+f(t,u,u',…,u(n-2))+g(t)=0, 0<t<1, n-1<α≤n, n≥2,u(0)=u'(0)=⋯=u(n-2)(0)=u(n-2)(1)=0, whereD0+αis the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed-point theorem of generalized concave operators. An example is given to illustrate the main result.

scholarly journals On the Theory of ψ-Hilfer Nonlocal Cauchy Problem

2021 ◽  
pp. 161-177
Author(s):  
Mohammed A. Almalahi ◽  
Satish K. Panchal
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Constant Coefficient ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Representation Formula ◽  
Qualitative Properties ◽  
Fractional Differential ◽  
Nonlocal Cauchy Problem

In this paper, we derive the representation formula of the solution for ψ-Hilfer fractional differential equation with constant coefficient in the form of Mittag-Leffler function by using Picard’s successive approximation. Moreover, by using some properties of Mittag-Leffler function and fixed point theorems such as Banach and Schaefer, we introduce new results of some qualitative properties of solution such as existence and uniqueness. The generalized Gronwall inequality lemma is used in analyze Eα -Ulam-Hyers stability. Finally, one example to illustrate the obtained results

scholarly journals Existence and Uniqueness with Ulam Stability study of the solution for a class of Caputo fractional integro-differential equation with mixed conditions

2021 ◽  
Author(s):  
ABDELLOUAHAB Naimi
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Stability Study ◽  
Ulam Stability ◽  
Differential Problem ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Stability Results

In this article we show the existence, uniqueness and Ulam stability results of the solution for a class of a nonlinear Caputo fractional integro-differential problem with mixed conditions. we use three fixed point theorems to proof the existence and uniqueness results. By the results obtained, the reasons for the Ulam stability are verified. An example proposed to illustrate our main results.

scholarly journals k-Generalized Ψ-Hilfer differential equations in Banach spaces

2021 ◽  
Author(s):  
Salim Abdelkrim ◽  
Mouffak Benchohra ◽  
Jamal Lazreg ◽  
Gaston NGuerekata
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Measure Of Noncompactness ◽  
Initial Value ◽  
Hilfer Fractional Derivative ◽  
The Subject ◽  
Fractional Differential ◽  
Stability Results ◽  
Rassias Stability

In this paper, we prove some existence and Ulam-Hyers-Rassias stability results for a class of initial value problem for implicit nonlinear fractional differential equations and generalized Ψ-Hilfer fractional derivative in Banach spaces. The results are based on fixed point theorems of Darbo and Monch associated with the technique of measure of noncompactness. Illustrative examples are the subject of the last section.

scholarly journals Analysis of Coupled System of Implicit Fractional Differential Equations Involving Katugampola–Caputo Fractional Derivative

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Manzoor Ahmad ◽  
Jiqiang Jiang ◽  
Akbar Zada ◽  
Syed Omar Shah ◽  
Jiafa Xu
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Sufficient Conditions ◽  
Coupled System ◽  
Uniqueness Of Solutions ◽  
Fractional Differential System ◽  
Fractional Differential ◽  
Implicit Fractional Differential Equations

In this paper, we study the existence and uniqueness of solutions to implicit the coupled fractional differential system with the Katugampola–Caputo fractional derivative. Different fixed-point theorems are used to acquire the required results. Moreover, we derive some sufficient conditions to guarantee that the solutions to our considered system are Hyers–Ulam stable. We also provided an example that explains our results.

scholarly journals Positive Solution for the Nonlinear Hadamard Type Fractional Differential Equation withp-Laplacian

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Ya-ling Li ◽  
Shi-you Lin
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Continuous Function ◽  
Fractional Derivative ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Fixed Point Theorems ◽  
Laplacian Operator ◽  
Nonlinear Fractional Differential Equation ◽  
Fractional Differential

We study the following nonlinear fractional differential equation involving thep-Laplacian operatorDβφpDαut=ft,ut,1<t<e,u1=u′1=u′e=0,Dαu1=Dαue=0, where the continuous functionf:1,e×0,+∞→[0,+∞),2<α≤3,1<β≤2.Dαdenotes the standard Hadamard fractional derivative of the orderα, the constantp>1, and thep-Laplacian operatorφps=sp-2s. We show some results about the existence and the uniqueness of the positive solution by using fixed point theorems and the properties of Green's function and thep-Laplacian operator.

scholarly journals A study of symmetric contractions with an application to generalized fractional differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Aftab Hussain ◽  
Fahd Jarad ◽  
Erdal Karapinar
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Fractional Differential ◽  
Generalized Fractional Derivative ◽  
Fractional Boundary Value Problems

AbstractThis article proposes four distinct kinds of symmetric contraction in the framework of complete F-metric spaces. We examine the condition to guarantee the existence and uniqueness of a fixed point for these contractions. As an application, we look for the solutions to fractional boundary value problems involving a generalized fractional derivative known as the fractional derivative with respect to another function.

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