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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
$\frac{d^\alpha u(t)}{dt^\alpha}+(A-B(t))u(t)= f(t), \quad t>t_0,$
in a Banach space $$X$$, where $$0<\alpha\leq 1$$, $$u$$ is an $$X$$-valued function on $$\mathbb R^+=[0,\infty)$$, and $$f$$ is a given abstract function on $$\mathbb R^+$$ with values in $$X$$. We assume that $$-A$$ is a linear closed operator defined on a dense set $$S$$ in $$X$$ into $$X$$, $$\{B(t): t\in\mathbb R^+\}$$ 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:
 34G10 Linear differential equations in abstract spaces 26A33 Fractional derivatives and integrals 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 35K90 Abstract parabolic equations
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