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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 $0<\alpha\le 1$, $u$ is an $X$-valued function on $\Bbb R^+=[0,\infty)$, and $f$ is a given abstract function on $\Bbb R^+$ with values in $X$. We assume that $-A$ is a linear closed operator defined on a dense set $S$ in $X$ into $X$, $\{B(t): t\in\Bbb R^+\}$ 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.

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