Existence results for nonlinear functional evolution equations with delay conditions

Let X be a real Banach space, A(t):D(A(t))⊂X→2X be an m-accretive operator, G:[0,T]×Lp(-r,0;X)×X→X be a mapping, Lt:Lp(-r,t;X)→X be a mapping, ut:[-r,0]→X satisfy ut(s)=u(t+s) for every s∈[-r,0], and φ0∈Lp(-r,0;X) for 1⩽p<∞ with φ0(0)∈D¯, where D¯=D(A(t))¯ (independent of t). The local existence of integral solutions of nonlinear functional evolution equation with delay conditiondu(t)dt+A(t)u(t)∋G(t,ut,Ltu),0⩽t⩽T,u(0)=φ0(t),-r⩽t⩽0is established in the case when the evolution operator {U(t,s)} generated by {A(t)} is equicontinuous.