scholarly journals Semilinear Fractional Evolution Inclusion Problem in the Frame of a Generalized Caputo Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Adel Lachouri ◽  
Abdelouaheb Ardjouni ◽  
Fahd Jarad ◽  
Mohammed S. Abdo
Keyword(s):  
Initial Value Problems ◽  
Existence Result ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Inclusion Problem ◽  
Topological Properties ◽  
Initial Value ◽  
Derivative Operator ◽  
Fractional Differential ◽  
Theoretical Results

In this paper, we study the existence of solutions to initial value problems for a nonlinear generalized Caputo fractional differential inclusion with Lipschitz set-valued functions. The applied fractional operator is given by the kernel k ρ , s = ξ ρ − ξ s and the derivative operator 1 / ξ ′ ρ d / d ρ . The existence result is obtained via fixed point theorems due to Covitz and Nadler. Moreover, we also characterize the topological properties of the set of solutions for such inclusions. The obtained results generalize previous works in the literature, where the classical Caputo fractional derivative is considered. In the end, an example demonstrating the effectiveness of the theoretical results is presented.

Some misinterpretations and lack of understanding in differential operators with no singular kernels

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 594-612 ◽  
Author(s):  
Abdon Atangana ◽  
Emile Franc Doungmo Goufo
Keyword(s):  
Fractional Calculus ◽  
Power Law ◽  
Initial Value Problems ◽  
Differential Operators ◽  
Integral Operators ◽  
Initial Value ◽  
Complex Processes ◽  
Singular Kernels ◽  
Fractional Differential ◽  
Theoretical Results

AbstractHumans are part of nature, and as nature existed before mankind, mathematics was created by humans with the main aim to analyze, understand and predict behaviors observed in nature. However, besides this aspect, mathematicians have introduced some laws helping them to obtain some theoretical results that may not have physical meaning or even a representation in nature. This is also the case in the field of fractional calculus in which the main aim was to capture more complex processes observed in nature. Some laws were imposed and some operators were misused, such as, for example, the Riemann–Liouville and Caputo derivatives that are power-law-based derivatives and have been used to model problems with no power law process. To solve this problem, new differential operators depicting different processes were introduced. This article aims to clarify some misunderstandings about the use of fractional differential and integral operators with non-singular kernels. Additionally, we suggest some numerical discretizations for the new differential operators to be used when dealing with initial value problems. Applications of some nature processes are provided.

Solvability of initial value problems with fractional order differential equations in banach spaces by α-dense curves

2017 ◽  
Vol 20 (3) ◽  
Author(s):  
Gonzalo García
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Initial Value ◽  
Measures Of Noncompactness ◽  
Fractional Order Differential Equations ◽  
Fractional Differential

AbstractIn this paper we study the existence of solutions for an initial value problem, posed in a given Banach space, with a fractional differential equation via densifiability techniques. For our goal, we will prove a new fixed point result (not based on measures of noncompactness) which is, in forms, a generalization of the well-known Darbo’s fixed point theorem but essentially different. Some illustrative examples are given.

scholarly journals Existence of solutions for fractional differential inclusions with initial value condition and non-instantaneous impulses

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5499-5510 ◽  
Author(s):  
Danfeng Luo ◽  
Zhiguo Luo
Keyword(s):  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Fractional Differential Inclusions ◽  
Initial Value ◽  
Generalized Gronwall Inequality ◽  
Nonlinear Alternative ◽  
Fractional Differential ◽  
Theoretical Results

In this paper, we mainly consider the existence of solutions for a kind of ?-Hilfer fractional differential inclusions involving non-instantaneous impulses. Utilizing another nonlinear alternative of Leray-Schauder type, we present a new constructive result for the addressed system with the help of generalized Gronwall inequality and Lagrange mean-value theorem, and some achievements in the literature can be generalized and improved. As an application, a typical example is delineated to demonstrate the effectiveness of our theoretical results.

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 Hilfer fractional differential inclusions with Erdélyi–Kober fractional integral boundary condition

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Adel Lachouri ◽  
Mohammed S. Abdo ◽  
Abdelouaheb Ardjouni ◽  
Bahaaeldin Abdalla ◽  
Thabet Abdeljawad
Keyword(s):  
Differential Inclusions ◽  
Fractional Integral ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Integral Boundary Conditions ◽  
Fractional Differential Inclusions ◽  
Hilfer Fractional Derivative ◽  
Integral Boundary ◽  
Fractional Differential ◽  
Theoretical Results

AbstractIn this article, we debate the existence of solutions for a nonlinear Hilfer fractional differential inclusion with nonlocal Erdélyi–Kober fractional integral boundary conditions (FIBC). Both cases of convex- and nonconvex-valued right-hand side are considered. Our obtained results are new in the framework of Hilfer fractional derivative and Erdélyi–Kober fractional integral with FIBC via the fixed point theorems (FPTs) for a set-valued analysis. Some pertinent examples demonstrating the effectiveness of the theoretical results are presented.

Solvability of Anti-periodic BVPs for Impulsive Fractional Differential Systems Involving Caputo and Riemann–Liouville Fractional Derivatives

2018 ◽  
Vol 19 (2) ◽  
pp. 125-152 ◽  
Author(s):  
Yuji Liu
Keyword(s):  
Fixed Point ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential ◽  
Conclusion Section

AbstractSufficient conditions are given for the existence of solutions of anti-periodic value problems for impulsive fractional differential systems involving both Caputo and Riemann–Liouville fractional derivatives. We allow the nonlinearities$p(t)f(t,x,y,z,w)$and$q(t)g(t,x,y,z,w)$in fractional differential equations to be singular at$t=0$and$t=1$. Both$f$and$g$may be super-linear and sub-linear. The analysis relies on some well known fixed point theorems. The initial value problem discussed may be seen as a generalization of some ecological models. An example is given to illustrate the efficiency of the main theorems. Many unsuitable lemmas in recent published papers are pointed out in order not to mislead readers. A conclusion section is given at the end of the paper.

scholarly journals Nonlinear Impulsive Multi-Order Caputo-Type Generalized Fractional Differential Equations with Infinite Delay

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1108 ◽  
Author(s):  
Bashir Ahmad ◽  
Madeaha Alghanmi ◽  
Ahmed Alsaedi ◽  
Ravi P. Agarwal
Keyword(s):  
Contraction Mapping ◽  
Existence Result ◽  
Existence Of Solutions ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Contraction Mapping Principle ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Fractional Differential

We establish sufficient conditions for the existence of solutions for a nonlinear impulsive multi-order Caputo-type generalized fractional differential equation with infinite delay and nonlocal generalized integro-initial value conditions. The existence result is proved by means of Krasnoselskii’s fixed point theorem, while the contraction mapping principle is employed to obtain the uniqueness of solutions for the problem at hand. The paper concludes with illustrative examples.

Existence of solutions to initial value problems for nonlinear fractional differential equations on the semi-axis

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Tiberiu Trif
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Growth Conditions ◽  
Initial Value ◽  
Nonlinear Fractional Differential Equations ◽  
Global Existence Of Solutions ◽  
Fractional Differential

AbstractThe purpose of the paper is to investigate the global existence of solutions to initial value problems for nonlinear fractional differential equations on the semi-axis. More precisely, it deals with the initial value problem (*)$\left\{ \begin{gathered} D_{0 + }^\alpha x(t) = f(t,x(t)),t \in [0,\infty ], \hfill \\ \lim _{t \to 0 + } t^{1 - \alpha } x(t) = x_0 , \hfill \\ \end{gathered} \right. $ where 0 < α < 1, D 0+α denotes the Riemann-Liouville fractional derivative of order α, and f: (0,∞) × ℝ → ℝ is a continuous function. Unlike all the previous papers dealing with the problem of existence of solutions to (*), this problem is solved here by constructing a special locally convex space which is metrizable and complete. Then Schauder’s fixed point theorem enables to provide sufficient conditions on f, ensuring that (*) possesses at least one solution. The growth conditions imposed to f are weaker than other similar conditions already used in the literature.

On coupled system of nonlinear Ψ-Hilfer hybrid fractional differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ashwini D. Mali ◽  
Kishor D. Kucche ◽  
José Vanterler da Costa Sousa
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Coupled System ◽  
Initial Value ◽  
Fractional Differential

Abstract This paper is dedicated to investigating the existence of solutions to the initial value problem (IVP) for a coupled system of Ψ-Hilfer hybrid fractional differential equations (FDEs) and boundary value problem (BVP) for a coupled system of Ψ-Hilfer hybrid FDEs. Analysis of the current paper depends on the two fixed point theorems involving three operators characterized on Banach algebra. In the view of an application, we provided useful examples to exhibit the effectiveness of our achieved results.

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