scholarly journals Existence and Attractivity for Fractional Evolution Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Yong Zhou ◽  
Jia Wei He ◽  
Bashir Ahmad ◽  
Ahmed Alsaedi
Keyword(s):  
Fractional Derivative ◽  
Evolution Equations ◽  
Global Attractivity ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Cauchy Problems ◽  
Fractional Evolution Equations ◽  
Liouville Fractional Derivative

We study the existence and attractivity of solutions for fractional evolution equations with Riemann-Liouville fractional derivative. We establish sufficient conditions for the global attractivity of mild solutions for the Cauchy problems in the case that semigroup is compact.

Attractivity for fractional evolution equations with almost sectorial operators

2018 ◽  
Vol 21 (3) ◽  
pp. 786-800 ◽  
Author(s):  
Yong Zhou
Keyword(s):  
Evolution Equations ◽  
Sufficient Conditions ◽  
Cauchy Problems ◽  
Fractional Evolution Equations ◽  
Integer Order ◽  
Sectorial Operators ◽  
Almost Sectorial Operators ◽  
Globally Attractive

Abstract In this paper, we initiate the question of the attractivity of solutions for fractional evolution equations with almost sectorial operators. We establish sufficient conditions for the existence of globally attractive solutions for the Cauchy problems in cases that semigroup is compact as well as noncompact. Our results essentially reveal certain characteristics of solutions for fractional evolution equations, which are not possessed by integer order evolution equations.

A Study on Impulsive Hilfer Fractional Evolution Equations with Nonlocal Conditions

2020 ◽  
Vol 21 (2) ◽  
pp. 205-218
Author(s):  
Haide Gou ◽  
Yongxiang Li
Keyword(s):  
Fractional Derivative ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Sobolev Type ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Characteristic Solution Operators ◽  
Impulsive Evolution Equations

AbstractIn this paper, we concern with the existence of mild solution to nonlocal initial value problem for nonlinear Sobolev-type impulsive evolution equations with Hilfer fractional derivative which generalized the Riemann–Liouville fractional derivative. At first, we establish an equivalent integral equation for our main problem. Second, by means of the properties of Hilfer fractional calculus, combining measure of noncompactness with the fixed-point methods, we obtain the existence results of mild solutions with two new characteristic solution operators. The results we obtained are new and more general to known results. At last, an example is provided to illustrate the results.

Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators

2020 ◽  
Vol 23 (1) ◽  
pp. 268-291 ◽  
Author(s):  
Pengyu Chen ◽  
Xuping Zhang ◽  
Yongxiang Li
Keyword(s):  
Resolvent Operator ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Control Function ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Resolvent Operators ◽  
Fractional Evolution Equations ◽  
Theoretical Results

AbstractIn this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of fractional evolution equations with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green’s function and constructing a control function involving Gramian controllability operator. The discussions are based on Schauder’s fixed point theorem as well as the theory of α-order solution operator and α-order resolvent operator. An example is given to illustrate the feasibility of our theoretical results.

scholarly journals Periodic Solutions andS-Asymptotically Periodic Solutions to Fractional Evolution Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jia Mu ◽  
Yong Zhou ◽  
Li Peng
Keyword(s):  
Periodic Solutions ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Fractional Evolution Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Asymptotically Periodic ◽  
Liouville Fractional Derivative ◽  
Asymptotically Periodic Solutions

This paper deals with the existence and uniqueness of periodic solutions,S-asymptotically periodic solutions, and other types of bounded solutions for some fractional evolution equations with the Weyl-Liouville fractional derivative defined for periodic functions. Applying Fourier transform we give reasonable definitions of mild solutions. Then we accurately estimate the spectral radius of resolvent operator and obtain some existence and uniqueness results.

scholarly journals On the Approximate Controllability of Fractional Evolution Equations with Generalized Riemann-Liouville Fractional Derivative

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
N. I. Mahmudov ◽  
M. A. McKibben
Keyword(s):  
Fixed Point ◽  
Linear System ◽  
Fractional Derivative ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Semigroup Theory ◽  
Abstract Theory ◽  
Fractional Evolution Equations ◽  
Liouville Fractional Derivative ◽  
Approximately Controllable

We discuss the approximate controllability of fractional evolution equations involving generalized Riemann-Liouville fractional derivative. The results are obtained with the help of the theory of fractional calculus, semigroup theory, and the Schauder fixed point theorem under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate the abstract theory.

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.

Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2

2019 ◽  
Vol 22 (4) ◽  
pp. 1086-1112 ◽  
Author(s):  
Linxin Shu ◽  
Xiao-Bao Shu ◽  
Jianzhong Mao
Keyword(s):  
Approximate Controllability ◽  
Stochastic Systems ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Mönch Fixed Point Theorem ◽  
Liouville Derivative ◽  
Liouville Fractional Derivative

Abstract In this paper, we consider the existence of mild solutions and approximate controllability for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2. As far as we know, there are few articles investigating on this issue. Firstly, the mild solutions to the equations are proved using Laplace transform of the Riemann-Liouville derivative. Moreover, the estimations of resolve operators involving the Riemann-Liouville fractional derivative of order 1 < α < 2 are given. Then, the existence results are obtained via the noncompact measurement strategy and the Mönch fixed point theorem. The approximate controllability of this nonlinear Riemann-Liouville fractional nonlocal stochastic systems of order 1 < α < 2 is concerned under the assumption that the associated linear system is approximately controllable. Finally, the approximate controllability results are obtained by using Lebesgue dominated convergence theorem.

scholarly journals ON UNIQUENESS OF MILD SOLUTIONS FOR DISSIPATIVE STOCHASTIC EVOLUTION EQUATIONS

2010 ◽  
Vol 13 (03) ◽  
pp. 363-376 ◽  
Author(s):  
CARLO MARINELLI ◽  
MICHAEL RÖCKNER
Keyword(s):  
Evolution Equations ◽  
Broad Class ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Strong Solutions ◽  
Fundamental Issue ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Monotonicity Assumption ◽  
Semigroup Approach

In the semigroup approach to stochastic evolution equations, the fundamental issue of uniqueness of mild solutions is often "reduced" to the much easier problem of proving uniqueness for strong solutions. This reduction is usually carried out in a formal way, without really justifying why and how one can do that. We provide sufficient conditions for uniqueness of mild solutions to a broad class of semilinear stochastic evolution equations with coefficients satisfying a monotonicity assumption.

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