Cauchy problem on fractional integro-differential evolution equations with infinite delay in fractional power space (Q380617)

From the introduction: The main purpose of this paper is to investigate the fractional integro-differential evolution equations with infinite delay \[ \begin{gathered} {^CD^q_t} x(t)=- Ax(t)+ f\Biggl(t, x_t,\,\int^1_0 g(t,s,x_s)\,ds\Biggr),\quad t\in J= [0,T],\quad q\in (0,1),\\ x(t)= \varphi(t)\in{\mathcal B},\quad -\infty< t\leq 0,\end{gathered} \] where \({^CD^q_t}\) denotes the Caputo fractional derivative of order \(q\), \(-A:D(A)\to X\) is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators \(\{T(t), t\geq 0\}\) on a Banach space \(X\) and compact for \(t> 0\). The histories \(x_t: (-\infty,0]\to X\), \(x_t= x(t+ s)\) belongs to some abstract phase space \({\mathcal B}\), that will be introduced in Section 2, Finally, \(f: J\times{\mathcal B}\times X_\alpha\to X\) (or \(X_\alpha\), \(0< \alpha< 1\)) and \(g: \{(t,s)\in J\times J\mid 0\leq s\leq t\}\times\to X_\alpha\) are continuous functions specified latter.