scholarly journals Existence, Uniqueness, and Asymptotic Behavior of Mild Solutions to Stochastic Functional Differential Equations in Hilbert Spaces

2002 ◽  
Vol 181 (1) ◽  
pp. 72-91 ◽  
Author(s):  
Takeshi Taniguchi ◽  
Kai Liu ◽  
Aubrey Truman
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Hilbert Spaces ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Stochastic Functional Differential Equations ◽  
Functional Differential ◽  
Stochastic Functional

scholarly journals Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Aimin Liu ◽  
Yongjian Liu ◽  
Qun Liu
Keyword(s):  
Differential Equations ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Stochastic Functional Differential Equations ◽  
Almost Periodic ◽  
Quadratic Mean ◽  
Exponential Trichotomy ◽  
Functional Differential ◽  
Asymptotically Almost Periodic ◽  
Stochastic Functional

This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equationsdxt=Atxt+Ft,xt,xtdt+H(t,xt,xt)∘dW(t). A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup{Tt}t≥0is essentially removed, which is generated by the linear densely defined operatorA∶D(A)⊂L2(ℙ,ℍ)→L2(ℙ,ℍ), only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain. An example is also given to illustrate our results.

scholarly journals On nondensely defined semilinear stochastic functional differential equations with nonlocal conditions

2006 ◽  
Vol 2006 ◽  
pp. 1-13 ◽  
Author(s):  
M. Benchohra ◽  
S. K. Ntouyas ◽  
A. Ouahab
Keyword(s):  
Differential Equations ◽  
Hilbert Spaces ◽  
Existence Of Solutions ◽  
Functional Differential Equations ◽  
Nonlocal Conditions ◽  
Stochastic Functional Differential Equations ◽  
First Order ◽  
Nonlinear Alternative ◽  
Functional Differential ◽  
Stochastic Functional

The nonlinear alternative of Leray-Schauder type is used to investigate the existence of solutions for first-order semilinear stochastic functional differential equations in Hilbert spaces.

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