Globally attractive mild solutions for non-local in time subdiffusion equations of neutral type

We prove the existence of at least one globally attractive mild solution to the equation $$ \partial_t (b*[x-h(\cdot,x(\cdot))])(t) + A(x(t) - h(t,x(t))) = f(t,x(t)), \quad t\geq 0, $$ under the assumption, among other hypothesis, that $A$ is an almost sectorial operator on a Banach space $X$ and the kernel $b$ belongs to a large class, which covers many relevant cases from physics applications, in particular the important case of time-fractional evolution equations of neutral type.