scholarly journals On Local Generalized Ulam–Hyers Stability for Nonlinear Fractional Functional Differential Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Dongming Nie ◽  
Azmat Ullah Khan Niazi ◽  
Bilal Ahmed
Keyword(s):  
Differential Equations ◽  
Existence Of A Solution ◽  
Neutral Differential Equations ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Functional Differential ◽  
Schauder Theorem ◽  
Equations With Delay ◽  
Rassias Stability ◽  
Fractional Functional

We discuss the existence of positive solution for a class of nonlinear fractional differential equations with delay involving Caputo derivative. Well-known Leray–Schauder theorem, Arzela–Ascoli theorem, and Banach contraction principle are used for the fixed point property and existence of a solution. We establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for the same class of nonlinear fractional neutral differential equations. The simulation of an example is also given to show the applicability of our results.

scholarly journals Numerical Studies for Fractional Functional Differential Equations with Delay Based on BDF-Type Shifted Chebyshev Approximations

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
V. G. Pimenov ◽  
A. S. Hendy
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
True Solution ◽  
New Approach ◽  
Numerical Studies ◽  
Runge Kutta Method ◽  
Functional Differential ◽  
Fractional Functional Differential Equations ◽  
Equations With Delay ◽  
Fractional Functional

Fractional functional differential equations with delay (FDDEs) have recently played a significant role in modeling of many real areas of sciences such as physics, engineering, biology, medicine, and economics. FDDEs often cannot be solved analytically so the approximate and numerical methods should be adapted to solve these types of equations. In this paper we consider a new method of backward differentiation formula- (BDF-) type for solving FDDEs. This approach is based on the interval approximation of the true solution using the Clenshaw and Curtis formula that is based on the truncated shifted Chebyshev polynomials. It is shown that the new approach can be reformulated in an equivalent way as a Runge-Kutta method and the Butcher tableau of this method is given. Estimation of local and global truncating errors is deduced and this leads to the proof of the convergence for the proposed method. Illustrative examples of FDDEs are included to demonstrate the validity and applicability of the proposed approach.

scholarly journals On Some Qualitative Properties of Mild Solutions of Nonlocal Semilinear Functional Differential Equations

2014 ◽  
Vol 47 (4) ◽  
Author(s):  
Rupali S. Jain ◽  
M. B. Dhakne
Keyword(s):  
Differential Equations ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Functional Differential Equations ◽  
Qualitative Properties ◽  
Contraction Theorem ◽  
Banach Contraction ◽  
Functional Differential ◽  
Banach Contraction Theorem ◽  
Equations With Delay

AbstractIn the present paper, we investigate the qualitative properties such as existence, uniqueness and continuous dependence on initial data of mild solutions of first and second order nonlocal semilinear functional differential equations with delay in Banach spaces. Our analysis is based on semigroup theory and modified version of Banach contraction theorem.

A New Investigation on Fractional-Ordered Neutral Differential Systems with State-Dependent Delay

2019 ◽  
Vol 20 (7-8) ◽  
pp. 803-809 ◽  
Author(s):  
N. Valliammal ◽  
C. Ravichandran ◽  
Zakia Hammouch ◽  
Haci Mehmet Baskonus
Keyword(s):  
Differential Equations ◽  
Mild Solutions ◽  
Existence Results ◽  
Differential Systems ◽  
Neutral Differential Equations ◽  
State Dependent ◽  
State Dependent Delay ◽  
Fractional Differential ◽  
Noncompact Measure ◽  
Equations With Delay

AbstractFractional differential equations with delay behaviors occur in fields like physical and biological ones with state-dependent delay or nonconstant delay and has drawn the attention of researchers. The main goal of the present work is to study the existence of mild solutions of neutral differential system along state-dependent delay in Banach space. By employing the fractional theory, noncompact measure and Mönch’s theorem, we investigate the existence results for neutral differential equations of fractional order with state-dependent delay. An illustration of derived results is offered.

scholarly journals Boundary Value Problem for Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Impulses

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Abdelkrim Salim ◽  
Mouffak Benchohra ◽  
Jamal Eddine Lazreg ◽  
Gaston N’Guérékata
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Banach Contraction ◽  
Fractional Differential ◽  
The Banach Contraction Principle ◽  
Stability Results ◽  
Rassias Stability ◽  
Implicit Fractional Differential Equations

This article deals with some existence, uniqueness, and Ulam-Hyers-Rassias stability results for a class of boundary value problem for nonlinear implicit fractional differential equations with impulses and generalized Hilfer Fractional derivative. The results are obtained using the Banach contraction principle and Krasnoselskii’s and Schaefer’s fixed-point theorems.

scholarly journals Ulam stability for delay fractional differential equations with a generalized Caputo derivative

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5265-5274 ◽  
Author(s):  
Raad Ameen ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Caputo Derivative ◽  
Ulam Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Equations With Delay ◽  
Rassias Stability

The objective of this paper is to extend Ulam-Hyers stability and Ulam-Hyers-Rassias stability theory to differential equations with delay and in the frame of a certain class of a generalized Caputo fractional derivative with dependence on a kernel function. We discuss the conditions such delay generalized Caputo fractional differential equations should satisfy to be stable in the sense of Ulam-Hyers and Ulam-Hyers-Rassias. To demonstrate our results two examples are presented.

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