Operator approach for time-fractional evolution equations in Banach spaces

Our first main purpose is to establish a framework for initial value problems for time-fractional evolution equation of order $\alpha \in (0,1)$ in Banach space $X$: $$\text{pppa}(u(t)-a)=Au(t)+F(t),0<t<T\mathrm{.}\text{eqno}(\ast )$$ Here $u:(0,T)\text{rrrr}X$ is an $X$-valued function defined in $(0,T)$, and $a\in X$ is an initial value. The operator $A$ satisfies a decay condition of resolvent which is the same as a generator of analytic semigroup. Based on $X$-valued Laplace transforms, we establish a solution formula yielding the well-posedness for (*). In particular, we can directly treat a case $X=L^p(\text{OOO})$ over a bounded domain $\text{OOO}$ and a uniform elliptic operator $A$. Our theory is feasibly applicable to other topics such as regularity of solutions, inverse problems and control problems.