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Existence of solutions for double perturbed neutral functional evolution equation. (English) Zbl 1197.34154

Summary: We discuss double perturbed neutral functional evolution equation with infinite delay

d dt(x(t)-h(t,x t ))=A(t)x(t)+f(t,x t )+g(t,x t ),tJ=[0,b](1·1)
x 0 =φ(1·2)

where {A(t):t>0} is a family of linear closed operators in a real Banach space X that generates an evolution system {U(t,s):0<st<} and D(A(t))X is dense in X. The history x t :(-,0]X, x t (θ)=x(t+θ), belongs to some abstract phase space defined axiomatically; g,f,h are appropriate functions.

The existence of mild solutions to such equations is obtained by using the theory of the Hausdorff measure of noncompactness and a fixed point theorem, without the compactness assumption on the associated evolution system. Our results improve and generalize some previous results.

34K30Functional-differential equations in abstract spaces
34K40Neutral functional-differential equations
47N20Applications of operator theory to differential and integral equations