Existence and Uniqueness Results of Volterra–Fredholm Integro-Differential Equations via Caputo Fractional Derivative

In this paper, we study a Volterra–Fredholm integro-differential equation. The considered problem involves the fractional Caputo derivatives under some conditions on the order. We prove an existence and uniqueness analytic result by application of the Banach principle. Then, another result that deals with the existence of at least one solution is delivered, and some sufficient conditions for this result are established by means of the fixed point theorem of Schaefer. Ulam stability of the solution is discussed before including an example to illustrate the results of the proposal.

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

10.22541/au.161133857.79110223/v1 ◽

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.

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A Nonlinear Implicit Fractional Equation with Caputo Derivative

Journal of Mathematics ◽

10.1155/2021/5547003 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Ameth Ndiaye

Keyword(s):

Differential Equation ◽

Fixed Point ◽

Fixed Point Theorem ◽

Existence And Uniqueness ◽

Initial Conditions ◽

Sufficient Conditions ◽

Implicit Differential Equation ◽

The Fixed Point Theorem ◽

Banach Principle ◽

Fractional Equation

In this paper, we study a nonlinear implicit differential equation with initial conditions. The considered problem involves the fractional Caputo derivatives under some conditions on the order. We prove an existence and uniqueness analytic result by application of Banach principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions for this result are established by means of the fixed point theorem of Schaefer. At the end, we discuss two examples to illustrate the applicability of the main results.

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

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2018-0371 ◽

2020 ◽

Vol 21 (3-4) ◽

pp. 327-337 ◽

Cited By ~ 3

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.

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Existence and Uniqueness of Solutions for Coupled Impulsive Fractional Pantograph Differential Equations with Antiperiodic Boundary Conditions

Advances in Mathematical Physics ◽

10.1155/2021/6616899 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Karim Guida ◽

Lahcen Ibnelazyz ◽

Khalid Hilal ◽

Said Melliani

Keyword(s):

Differential Equation ◽

Fixed Point ◽

Differential Equations ◽

Existence And Uniqueness ◽

Uniqueness Of Solutions ◽

Krasnoselskii’S Fixed Point Theorem ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Fractional Differential ◽

Banach’S Contraction Principle

In this paper, we investigate the solutions of coupled fractional pantograph differential equations with instantaneous impulses. The work improves some existing results and contributes toward the development of the fractional differential equation theory. We first provide some definitions that will be used throughout the paper; after that, we give the existence and uniqueness results that are based on Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Two examples are given in the last part to support our study.

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Existence, uniqueness, approximation of solutions and Ealpha-Ulam stability results for a class of nonlinear fractional differential equations involving psi-Caputo derivative with initial conditions

Mathematica Moravica ◽

10.5937/matmor2101001d ◽

2021 ◽

Vol 25 (1) ◽

pp. 1-30

Author(s):

Choukri Derbazi ◽

Zidane Baitiche ◽

Mouffak Benchohra ◽

Gaston N'guérékata

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Existence And Uniqueness ◽

Initial Conditions ◽

Ulam Stability ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Fractional Differential ◽

Stability Results

The main purpose of this paper is to study the existence, uniqueness, Ea-Ulam stability results, and other properties of solutions for certain classes of nonlinear fractional differential equations involving the ps-Caputo derivative with initial conditions. Modern tools of functional analysis are applied to obtain the main results. More precisely using Weissinger's fixed point theorem and Schaefer's fixed point theorem the existence and uniqueness results of solutions are proven in the bounded domain. While the well known Banach fixed point theorem coupled with Bielecki type norm are used with the end goal to establish sufficient conditions for existence and uniqueness results on unbounded domains. Meanwhile, the monotone iterative technique combined with the method of upper and lower solutions is used to prove the existence and uniqueness of extremal solutions. Furthermore, by means of new generalizations of Gronwall's inequality, different kinds of Ea-Ulam stability of the proposed problem are studied. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility and correctness of the main results.

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Existence and Ulam–Hyers Stability of a Fractional-Order Coupled System in the Frame of Generalized Hilfer Derivatives

Mathematics ◽

10.3390/math9202543 ◽

2021 ◽

Vol 9 (20) ◽

pp. 2543

Author(s):

Abdulkafi M. Saeed ◽

Mohammed S. Abdo ◽

Mdi Begum Jeelani

Keyword(s):

Fixed Point ◽

Fractional Order ◽

Existence And Uniqueness ◽

Fractional Derivatives ◽

Fixed Point Theorems ◽

Coupled System ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Special Cases ◽

Stability Of The Solution

In this research paper, we consider a class of a coupled system of fractional integrodifferential equations in the frame of Hilfer fractional derivatives with respect to another function. The existence and uniqueness results are obtained in weighted spaces by applying Schauder’s and Banach’s fixed point theorems. The results reported here are more general than those found in the literature, and some special cases are presented. Furthermore, we discuss the Ulam–Hyers stability of the solution to the proposed system. Some examples are also constructed to illustrate and validate the main results.

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A generalized sequential problem of Lane-Emden type via fractional calculus

Moroccan Journal of Pure and Applied Analysis ◽

10.2478/mjpaa-2020-0013 ◽

2020 ◽

Vol 6 (2) ◽

pp. 168-183 ◽

Cited By ~ 1

Author(s):

Yazid Gouari ◽

Zoubir Dahmani ◽

Ameth Ndiaye

Keyword(s):

Differential Equation ◽

Fixed Point ◽

Fixed Point Theorem ◽

Existence And Uniqueness ◽

Sufficient Conditions ◽

Group Property ◽

Semi Group ◽

Singular Differential Equation ◽

Banach Contraction ◽

The Fixed Point Theorem

AbstractIn this paper, we combine the Riemann-Liouville integral operator and Caputo derivative to investigate a nonlinear time-singular differential equation of Lane Emden type. The considered problem involves n fractional Caputo derivatives under the conditions that neither commutativity nor semi group property is satisfied for these derivatives. We prove an existence and uniqueness analytic result by application of Banach contraction principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions related to this result are established by means of the fixed point theorem of Schaefer. We end the paper by presenting to the reader some illustrative examples.

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Analysis of Impulsive Boundary Value Pantograph Problems via Caputo Proportional Fractional Derivative under Mittag–Leffler Functions

Fractal and Fractional ◽

10.3390/fractalfract5040251 ◽

2021 ◽

Vol 5 (4) ◽

pp. 251

Author(s):

Bounmy Khaminsou ◽

Weerawat Sudsutad ◽

Chatthai Thaiprayoon ◽

Jehad Alzabut ◽

Songkran Pleumpreedaporn

Keyword(s):

Differential Equation ◽

Fixed Point ◽

Fractional Derivative ◽

Existence And Uniqueness ◽

Fixed Point Theorems ◽

Boundary Value ◽

Nonlinear Functional ◽

Numerical Examples ◽

Existence And Uniqueness Results ◽

Uniqueness Results

This manuscript investigates an extended boundary value problem for a fractional pantograph differential equation with instantaneous impulses under the Caputo proportional fractional derivative with respect to another function. The solution of the proposed problem is obtained using Mittag–Leffler functions. The existence and uniqueness results of the proposed problem are established by combining the well-known fixed point theorems of Banach and Krasnoselskii with nonlinear functional techniques. In addition, numerical examples are presented to demonstrate our theoretical analysis.

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Modeling the dynamics of hepatitis E via the Caputo–Fabrizio derivative

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2018074 ◽

2019 ◽

Vol 14 (3) ◽

pp. 311 ◽

Cited By ~ 39

Author(s):

Muhammad Altaf Khan ◽

Zakia Hammouch ◽

Dumitru Baleanu

Keyword(s):

Fixed Point ◽

Existence And Uniqueness ◽

Numerical Scheme ◽

Arbitrary Order ◽

Fixed Point Theory ◽

Hepatitis E ◽

Point Theory ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Essential Properties

A virus that causes hepatitis E is known as (HEV) and regarded on of the reason for lever inflammation. In mathematical aspects a very low attention has been paid to HEV dynamics. Therefore, the present work explores the HEV dynamics in fractional derivative. The Caputo–Fabriizo derivative is used to study the dynamics of HEV. First, the essential properties of the model will be presented and then describe the HEV model with CF derivative. Application of fixed point theory is used to obtain the existence and uniqueness results associated to the model. By using Adams–Bashfirth numerical scheme the solution is obtained. Some numerical results and tables for arbitrary order derivative are presented.

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On the Nonlocal Fractional Delta-Nabla Sum Boundary Value Problem for Sequential Fractional Delta-Nabla Sum-Difference Equations

Mathematics ◽

10.3390/math8040476 ◽

2020 ◽

Vol 8 (4) ◽

pp. 476

Author(s):

Jiraporn Reunsumrit ◽

Thanin Sitthiwirattham

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Boundary Value Problem ◽

Boundary Conditions ◽

Difference Equations ◽

Existence And Uniqueness ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Banach Contraction ◽

The Banach Contraction Principle

In this paper, we propose sequential fractional delta-nabla sum-difference equations with nonlocal fractional delta-nabla sum boundary conditions. The Banach contraction principle and the Schauder’s fixed point theorem are used to prove the existence and uniqueness results of the problem. The different orders in one fractional delta differences, one fractional nabla differences, two fractional delta sum, and two fractional nabla sum are considered. Finally, we present an illustrative example.

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