Solvability of a neutral differential equation with deviated argument (Q860575)

The paper deals with a functional-differential equation of the form \[ x'(t)=f(t,x(H(t)),x'(h(t)))\text{ for }t\in[0,\infty)\tag{1.} \] Using the measure of non-compactness together with Schauder's fixed point theorem, the author proves the existence of a solution \(y(t)\) of (1) such that \[ \lim_{t\to\infty}y(t)\exp(-h(t))=0, \] where the function \(h:[0,\infty)\to[0,\infty)\) is continuous, non-decreasing, \(h(t)\leq t\) and \(\lim_{t\to\infty}h(t)=\infty\), \(\lim_{t\to\infty}(t-h(t))=\infty\).