Stepanov-like pseudo-almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay

In this paper, we prove the invariance of Stepanov-like pseudo-almost periodic functions under bounded linear operators. Furthermore, we obtain existence and uniqueness theorems of pseudo-almost periodic mild solutions to evolution equations u′(t)=A(t)u(t)+h(t)u′(t)=A(t)u(t)+h(t) and u′(t)=A(t)u(t)+f(t,Bu(t))+∫−∞tC(t,s)u(s)ds+F(t) on RR, assuming that A(t)A(t) satisfy “Acquistapace–Terreni” conditions, that the evolution family generated by A(t)A(t) has exponential dichotomy, that R(λ0,A(⋅))R(λ0,A(⋅)) is almost periodic, that B,C(t,s)t≥sB,C(t,s)t≥s are bounded linear operators, that ff is Lipschitz with respect to the second argument uniformly in the first argument and that hh, ff, FF are Stepanov-like pseudo-almost periodic for p>1p>1 and continuous. To illustrate our abstract result, a concrete example is given.