Stochastic fractional evolution equations with fractional brownian motion and infinite delay

Abstract In this paper, we consider a class of stochastic fractional evolution equations with infinite delay and a fractional Brownian motion in a Hilbert space. By the stochastic analysis technique, we establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz condition with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory.

[1]  Kexue Li,et al.  Stochastic delay fractional evolution equations driven by fractional Brownian motion , 2014, 1406.3336.

[2]  F. Comte,et al.  Long memory continuous time models , 1996 .

[3]  R. Sakthivel,et al.  Existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations , 2012 .

[4]  Jace W. Nunziato,et al.  On heat conduction in materials with memory , 1971 .

[5]  刘 凯湘 Stability of infinite dimensional stochastic differential equations with applications , 2006 .

[6]  Yong Zhou,et al.  Existence of mild solutions for fractional neutral evolution equations , 2010, Comput. Math. Appl..

[7]  M. Rypdal,et al.  Testing hypotheses about sun-climate complexity linking. , 2010, Physical review letters.

[8]  Litan Yan,et al.  Existence result for fractional neutral stochastic integro-differential equations with infinite delay , 2011 .

[9]  Rathinasamy Sakthivel,et al.  Approximate controllability of fractional stochastic evolution equations , 2012, Comput. Math. Appl..

[10]  Litan Yan,et al.  On a jump-type stochastic fractional partial differential equation with fractional noises , 2012 .

[11]  R. Sakthivel,et al.  On time‐dependent stochastic evolution equations driven by fractional Brownian motion in a Hilbert space with finite delay , 2014 .

[12]  Yong Zhou,et al.  Nonlocal Cauchy problem for fractional evolution equations , 2010 .

[13]  Ke Wang,et al.  The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay , 2007 .

[14]  I. Simonsen Measuring anti-correlations in the nordic electricity spot market by wavelets , 2001, cond-mat/0108033.

[15]  Litan Yan,et al.  Asymptotic behavior for neutral stochastic partial differential equations with infinite delays , 2013 .

[16]  Yong Ren,et al.  Retarded stochastic differential equations with infinite delay driven by Rosenblatt process , 2018 .

[17]  T. Caraballo,et al.  The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional brownian motion , 2011 .

[18]  Salah H. Abid,et al.  Approximate Controllability of Fractional Stochastic Integro-Differential Equations Driven by Mixed Fractional Brownian Motion , 2015 .

[19]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[20]  F. Mainardi Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .

[21]  B. Boufoussi,et al.  Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space , 2012 .

[22]  Jinhu Lü,et al.  Stability analysis of linear fractional differential system with multiple time delays , 2007 .

[23]  Kai Liu,et al.  Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps , 2008 .

[24]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[25]  Rongchan Zhu,et al.  Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise , 2013, 1307.4392.

[26]  I. M. Fuente,et al.  Long-Range Correlations in Rabbit Brain Neural Activity , 2005, Annals of Biomedical Engineering.

[27]  O Naghshineh,et al.  EXISTENCE AND MEASUREABILITY OF THE SOLUTION OF THE STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION , 2009 .

[28]  D. Nualart The Malliavin Calculus and Related Topics , 1995 .

[29]  Patrice R. Rougier,et al.  Relation between postural control assessment with eyes open and centre of pressure visual feedback effects in healthy individuals , 2009, Experimental Brain Research.

[30]  Rathinasamy Sakthivel,et al.  Sobolev-type fractional stochastic differential equations with non-Lipschitz coefficients , 2017, J. Comput. Appl. Math..

[31]  Rathinasamy Sakthivel,et al.  Existence of solutions for nonlinear fractional stochastic differential equations , 2013 .

[32]  Ruhollah Jahanipur,et al.  Nonlinear functional differential equations of monotone-type in Hilbert spaces☆ , 2010 .

[33]  Jinghuai Gao,et al.  Existence results for semilinear fractional differential equations via Kuratowski measure of noncompactness , 2012 .

[34]  Yong Zhou,et al.  Existence of solutions for a class of fractional boundary value problems via critical point theory , 2011, Comput. Math. Appl..