Analyticity and nonanalyticity of solutions of delay-differential equations (Q2927859)

The equation NEWLINE\[NEWLINEx'(t)=f(t,x(t),x(\eta(t))NEWLINE\]NEWLINE is considered, where the nonlinear functions \(f, \eta\) are both analytic functions of their arguments (typically the function \(\eta\) represents a delay). The purpose of the paper is to establish sufficient conditions ensuring at which values of \(t\) a given solution \(x=x(t)\) of the considered equation is analytic and, also, at which values \(t\) the solution \(x\) is not analytic. These solutions are generally \(C^{\infty}\) and, very often, for such a solution \(x=x(t)\) there coexist points at which \(x\) is analytic and also points at which \(x\) is non analytic. The properties of the function \(\eta=\eta(t)\) play a crucial role in the study of this problem.