Existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations

2012 ◽  
Vol 75 (7) ◽  
pp. 3339-3347 ◽  
Author(s):  
R. Sakthivel ◽  
P. Revathi ◽  
S. Marshal Anthoni
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Mild Solutions ◽  
Almost Automorphic ◽  
Pseudo Almost Automorphic ◽  
Fractional Differential ◽  
Stochastic Fractional Differential Equations

scholarly journals Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-23
Author(s):  
Junfei Cao ◽  
Zaitang Huang ◽  
Gaston M. N’Guérékata
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Mild Solutions ◽  
Relaxation Oscillation ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Almost Automorphic ◽  
Densely Defined Operator ◽  
Fractional Differential ◽  
The Fixed Point Theorem

This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations Dtαxt=Axt+Dtα-1Ft,xt,Bxt,  t∈R, where 1<α<2, A is a linear densely defined operator of sectorial type on a complex Banach space X and B is a bounded linear operator defined on X, F is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity F is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

Pseudo Almost Automorphic Solution of Semilinear Fractional Differential Equations with the Caputo Derivatives

2015 ◽  
Vol 18 (4) ◽  
Author(s):  
Dingjiang Wang ◽  
Zhinan Xia
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Alternative Theorem ◽  
Caputo Derivatives ◽  
Almost Automorphic ◽  
Fractional Powers Of Operators ◽  
Pseudo Almost Automorphic ◽  
Fractional Differential

AbstractIn this paper, we deal with existence and uniqueness of (μ, ν)-pseudo almost automorphic mild (classical) solution to semilinear fractional differential equations with the Caputo derivatives. The main results are obtained by means of the fixed point theory, Leray-Schauder alternative theorem and fractional powers of operators. Moreover, an application to fractional predator-prey system with diffusion is given.

scholarly journals Approximate controllability of multivalued stochastic fractional differential equations with Atangana-Baleanu-Caputo derivatives

2021 ◽  
Author(s):  
Pallavi Bedi ◽  
Anoop Kumar ◽  
Aziz Khan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Approximate Controllability ◽  
Mild Solutions ◽  
Resolvent Operators ◽  
Analysis Theory ◽  
Approximately Controllable ◽  
Fractional Differential ◽  
Stochastic Fractional Differential Equations ◽  
Fixed Point Technique

This article aims to discuss the approximate controllability of multivalued impulsive stochastic fractional differential equations with ABC derivatives in Hilbert space. Firstly, with the help of stochastic analysis, theory of resolvent operators and the fixed point technique, we confirm the existence of mild solutions for the proposed control system. Secondarily, we show that the proposed system of equations is approximately controllable under a certain hypothesis. To confirm the applicability of the obtained results, an example is provided at the end of this paper.

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