scholarly journals RETRACTED ARTICLE: Existence of solutions of nonlocal initial value problems for differential equations with Hilfer–Katugampola fractional derivative

2019 ◽  
Vol 113 (4) ◽  
pp. 3903-3903 ◽  
Author(s):  
S. Harikrishnan ◽  
K. Kanagarajan ◽  
E. M. Elsayed
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Initial Value ◽  
Retracted Article

Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function

2021 ◽  
Vol 24 (4) ◽  
pp. 1220-1230
Author(s):  
Mohammed Al-Refai
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Maximum Principles ◽  
Initial Value ◽  
Derivative Operator ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
Two Parameters

Abstract In this paper, we formulate and prove two maximum principles to nonlinear fractional differential equations. We consider a fractional derivative operator with Mittag-Leffler function of two parameters in the kernel. These maximum principles are used to establish a pre-norm estimate of solutions, and to derive certain uniqueness and positivity results to related linear and nonlinear fractional initial value problems.

scholarly journals Existence of solutions to quasilinear differential equations in a Banach space

1976 ◽  
Vol 15 (3) ◽  
pp. 421-430
Author(s):  
James R. Ward
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Linear Systems ◽  
Function Space ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Reflexive Banach Space ◽  
Initial Value ◽  
Suitable Function

Initial value problems of the form x′ + A(t, x)x = f(t, x), x(0) = a, t ≥ 0, are considered in a real, separable, reflexive Banach space. Results concerning the existence of solutions on (0, ∞) are given by considering linear systems of the form x′ + A(t, u(t))x = f(t, u(t)). Here u(t) belongs to a suitable function space.

scholarly journals Analysis of Multiterm Initial Value Problems with Caputo–Fabrizio Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Mohammed Al-Refai ◽  
Muhammed Syam
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Initial Value ◽  
The Laplace Transform ◽  
Necessary And Sufficient

In this paper, we discuss the solvability of a class of multiterm initial value problems involving the Caputo–Fabrizio fractional derivative via the Laplace transform. We derive necessary and sufficient conditions to guarantee the existence of solutions to the problem. We also obtain the solutions in closed forms. We present two examples to illustrate the validity of the obtained results.

scholarly journals On the Nonlinear Fractional Differential Equations with Caputo Sequential Fractional Derivative

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Hailong Ye ◽  
Rui Huang
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Initial Value ◽  
Caputo Fractional Derivatives ◽  
Nonlinear Fractional Differential Equation ◽  
Global Existence Of Solutions ◽  
Sequential Fractional Derivative ◽  
Fractional Differential

The purpose of this paper is to investigate the existence of solutions to the following initial value problem for nonlinear fractional differential equation involving Caputo sequential fractional derivativeDc0α2Dc0α1yxp-2Dc0α1yx=fx,yx,x>0,y(0)=b0,Dc0α1y(0)=b1, whereDc0α1,Dc0α2are Caputo fractional derivatives,0<α1,α2≤1,p>1, andb0,b1∈R. Local existence of solutions is established by employing Schauder fixed point theorem. Then a growth condition imposed tofguarantees not only the global existence of solutions on the interval[0,+∞), but also the fact that the intervals of existence of solutions with any fixed initial value can be extended to[0,+∞). Three illustrative examples are also presented. Existence results for initial value problems of ordinary differential equations withp-Laplacian on the half-axis follow as a special case of our results.

L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Benoumran Telli ◽  
Mohammed Said Souid
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Point Theorem ◽  
Initial Value Problems ◽  
Initial Value ◽  
Hadamard Fractional Derivative ◽  
Implicit Differential Equations ◽  
Banach Contraction

Abstract In this paper, we study the existence of integrable solutions for initial value problems for fractional order implicit differential equations with Hadamard fractional derivative. Our results are based on Schauder’s fixed point theorem and the Banach contraction principle fixed point theorem.

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