Existence of solutions of boundary value problems for differential equations with delayed arguments. (Q1398434)

The author deals with the following problem \[ \begin{gathered} x'(t)= f(t, x(t), x(\alpha(t)),\quad t\in J= [0,T],\quad T> 0,\\ x(0)= \lambda x(T)+ k,\end{gathered}\tag{1} \] with \(f\in C(J\times \mathbb{R}\times \mathbb{R},\mathbb{R})\), \(\alpha\in C(J,J)\), \(\lambda, k\in\mathbb{R}\). He establishes some results for problem (1) for \(\lambda\geq 0\), and discusses also the case \(\lambda< 0\). Moreover, the author formulates some existence results when several delayed arguments \(\alpha_i\) on the right-hand side in (1) are presented. Some examples satisfying the assumptions are given.