scholarly journals Mild Solutions for Impulsive Integro-Differential Equations Involving Hilfer Fractional Derivative with almost Sectorial Operators

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 313
Author(s):  
Kulandhaivel Karthikeyan ◽  
Panjaiyan Karthikeyan ◽  
Nichaphat Patanarapeelert ◽  
Thanin Sitthiwirattham
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Jump Conditions ◽  
Hilfer Fractional Derivative ◽  
Sectorial Operators ◽  
Almost Sectorial Operators ◽  
Suitable Definition ◽  
Definition Of

In this manuscript, we establish the mild solutions for Hilfer fractional derivative integro-differential equations involving jump conditions and almost sectorial operator. For this purpose, we identify the suitable definition of a mild solution for this evolution equations and obtain the existence results. In addition, an application is also considered.

Existence of Mild Solutions for Sobolev-Type Hilfer Fractional Nonautonomous Evolution Equations with Delay

2018 ◽  
Vol 19 (5) ◽  
pp. 481-492
Author(s):  
Haide Gou ◽  
Baolin Li
Keyword(s):  
Existence Result ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Sobolev Type ◽  
Operator Family ◽  
Hilfer Fractional Derivative ◽  
Definition Of ◽  
Sadovskii Fixed Point Theorem ◽  
Equations With Delay

AbstractThis paper treats the existence of mild solutions for Sobolev-type Hilfer fractional nonautonomous evolution equations with delay in Banach spaces. We first characterize the definition of mild solutions for the studied problem which was given based on an operator family generated by the operator pair (A,B) and probability density function. And then via Hilfer fractional derivative and combining the techniques of fractional calculus, measure of noncompactness and Sadovskii fixed-point theorem, we obtain new existence result of mild solutions for Sobolev-type Hilfer fractional nonautonomous evolution equations. Particularly, the existence or compactness of an operator $B^{-1} $ is not necessarily needed in our results. Furthermore, our results obtained improve and extend some related conclusions on this topic. At last, an example is given to illustrate our main results.

scholarly journals Existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Pallavi Bedi ◽  
Anoop Kumar ◽  
Thabet Abdeljawad ◽  
Zareen A. Khan ◽  
Aziz Khan
Keyword(s):  
Banach Space ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Control Function ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Existence Results ◽  
Fractional Evolution Equations ◽  
Sectorial Operators ◽  
Almost Sectorial Operators

Abstract In this article, we are concerned with the existence of mild solutions and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators and nonlocal conditions. The existence results are obtained by first defining Green’s function and approximate controllability by specifying a suitable control function. These results are established with the help of Schauder’s fixed point theorem and theory of almost sectorial operators in a Banach space. An example is also presented for the demonstration of obtained results.

scholarly journals Fuzzy Fractional Evolution Equations and Fuzzy Solution Operators

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Atimad Harir ◽  
Said Melliani ◽  
Lalla Saadia Chadli
Keyword(s):  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Fractional Evolution Equations ◽  
Fuzzy Solution ◽  
Definition Of ◽  
Fuzzy Laplace Transform ◽  
Derivative Order

In this paper, the fuzzy fractional evolution equations of order q (FFEE) with fuzzy Caputo fractional derivative are introduced. We study the existence and uniqueness of mild solutions for FFEE under some conditions. Also, we generalize the definition of the fuzzy fractional integral and derivative order q. The fuzzy Laplace transform is presented and proved. The solvability of the problem (FFEE) and the properties of the fuzzy solution operator and its generator are investigated and developed.

Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Shengli Xie
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Existence Results ◽  
Order Case

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

scholarly journals Fractional Stochastic Differential Equations with Hilfer Fractional Derivative: Poisson Jumps and Optimal Control

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Fathalla A. Rihan ◽  
Chinnathambi Rajivganthi ◽  
Palanisamy Muthukumar
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Poisson Jumps ◽  
Hilfer Fractional Derivative ◽  
Analysis Techniques ◽  
Mild Conditions ◽  
Fractional Stochastic Differential Equations ◽  
Existence Of Optimal Control

In this work, we consider a class of fractional stochastic differential system with Hilfer fractional derivative and Poisson jumps in Hilbert space. We study the existence and uniqueness of mild solutions of such a class of fractional stochastic system, using successive approximation theory, stochastic analysis techniques, and fractional calculus. Further, we study the existence of optimal control pairs for the system, using general mild conditions of cost functional. Finally, we provide an example to illustrate the obtained results.

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