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Weak almost periodic and optimal mild solutions of fractional evolution equations. (English) Zbl 1171.34331

From the text: The object of this paper is to study the fractional evolution equation

d α u(t) dt α +(A-B(t))u(t)=f(t),t>t 0 ,

in a Banach space X, where 0<α1, u is an X-valued function on + =[0,), and f is a given abstract function on + with values in X. We assume that -A is a linear closed operator defined on a dense set S in X into X, {B(t):t + } is a family of linear bounded operators defined on X into X.

We prove the existence of optimal mild solutions for linear fractional evolution equations with an analytic semigroup in a Banach space. We use the Gelfand-Shilov principle to prove existence, and then the Bochner almost periodicity condition to show that solutions are weakly almost periodic. As an application, we study a fractional partial differential equation of parabolic type.


MSC:
34G10Linear ODE in abstract spaces
26A33Fractional derivatives and integrals (real functions)
34C27Almost and pseudo-almost periodic solutions of ODE
35K90Abstract parabolic equations