Existence of solutions to second-order neutral functional-differential equations (Q2719900)

By using Schaefer's fixed-point theorem, the authors prove an existence result for a class of second-order neutral functional-differential equations of the form NEWLINE\[NEWLINE \frac{d}{dt}\left[x'(t)-g(t,x_t)\right]=Ax(t)+f(t,x_t,x'(t)),\;\;t\in J=(0,T],\;x_0=\phi,\;\;x'(0)=y_0,NEWLINE\]NEWLINE where \(A\) is the infinitesimal generator of a strongly continuous cosine family acting in a Banach space \(X\), \(f:J\times C([-r,0];X)\times X\to X\) and \(g:J\times C([-r,0];X)\to X\) are given functions with \(\phi\in C([-r,0];X)\) and \(y_0\in X\).