Existence results for neutral functional integrodifferential equations with infinite delay via fractional operators (Q545580)

The following initial value problem for a class of semilinear neutral functional differential equations with infinite delay of the form \[ \frac{d}{dt}[x(t)-g(t,x_t)]=Ax(t)+\int_0^tB(t-s)x(s)\,ds+f\left(t,x_t,\int_0^th(s,x_s)\,ds\right)\!, \;t\in J=[0,b], \;x_0=\phi, \] is considered, where \(A: D(A)\subset X\to X\) is the infinitesimal generator of a compact, analytic resolvent, \(g, h: J\times {\mathcal B}\to X,\) \(f: J\times {\mathcal B}\times X\to X,\) \((X,\|\cdot\|)\) is a Banach space, \(B(t)\) a bounded linear operator and \({\mathcal B}\) is the phase space. Existence results are proved via a Krasnosel'skii-Schaefer type fixed point theorem. An example illustrating the results is also presented.