On a boundary value problem for functional differential equations (Q1820298)

This paper deals with the following boundary value problem for second order functional differential equations: \[ (p(t)x'(t))'+f(t,x_ t,x'(t))=0 \] \[ x(t)=h(t),\quad -r\leq t\leq 0,\quad h(0)=0,\quad x(T)=0 \] and \(| x(t)| \leq \phi (t)\) where \(f\in C[[0,T]\times C([- r,0],{\mathbb{R}}^ n)\times {\mathbb{R}}^ n,{\mathbb{R}}^ n]\) \(p\in C([0,T],(0,1])\) and \(\phi\) is a prespecified function. The result of this paper generalizes a previous one due to \textit{Ch. Fabry} and \textit{P. Habets} [J. Differ. Equations 42, 186-198 (1981; Zbl 0439.34018)] and it is very closely related to previous results of the author [Hiroshima Math. J. 12, 453-468 (1982; Zbl 0507.34055)].