Almost periodic and asymptotically almost periodic mild solutions for nonlinear abstract functional differential equations with infinite delay

2011 ◽  
Author(s):  
Peng Xiao ◽  
Yanfei Du
Keyword(s):  
Differential Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Almost Periodic ◽  
Functional Differential ◽  
Asymptotically Almost Periodic

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 The Kneser property for abstract retarded functional differential equations with infinite delay

2003 ◽  
Vol 2003 (26) ◽  
pp. 1645-1661 ◽  
Author(s):  
Hernán R. Henríquez
Keyword(s):  
Differential Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
First Order ◽  
Retarded Functional Differential Equations ◽  
Functional Differential

We establish existence of mild solutions for a class of semilinear first-order abstract retarded functional differential equations (ARFDEs) with infinite delay and we prove that the set consisting of mild solutions for this problem is connected in the space of continuous functions.

scholarly journals Almost periodic mild solutions of a class of partial functional differential equations

1998 ◽  
Vol 3 (3-4) ◽  
pp. 425-436 ◽  
Author(s):  
Bernd Aulbach ◽  
Nguyen Van Minh
Keyword(s):  
Differential Equations ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Partial Functional Differential Equations ◽  
Semilinear Equation ◽  
Functional Differential ◽  
Lipschitz Conditions

We study the existence of almost periodic mild solutions of a class of partial functional differential equations via semilinear almost periodic abstract functional differential equations of the form(*)                                                                       x′=f(t,x,xt).To this end, we first associate with every almost periodic semilinear equation(**)                                                                       x′=F(t,x).a nonlinear semigroup in the space of almost periodic functions. We then give sufficient conditions (in terms of the accretiveness of the generator of this semigroup) for the existence of almost periodic mild solutions of (**) as fixed points of the semigroup. Those results are then carried over to equation (*). The main results are stated under accretiveness conditions of the functionfin terms ofxand Lipschitz conditions with respect toxt.

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