Existence of solutions of boundary value problems for second order functional differential equations (Q1826760)

This paper is devoted to the boundary value problem \[ x''= f(t,x,x_t, x',x_t'),\tag{1} \] \[ (x_0,x_0')\in \{(\varphi+ c_1, \psi+ c_2)\mid c_1,c_2\in\mathbb{R}\},\;\alpha(x|_J)= 0,\quad \beta(x'(1)- \delta x'|_J)= 0,\tag{2} \] where \(f\in \text{Car}(J\times\mathbb{R}\times C_r\times \mathbb{R}\times C_r)\), \(\varphi,\psi\in C_r\), \(J= [0,1]\), \(\delta\neq 1\), \(\alpha\in{\mathfrak A}_J\), \(\beta\in{\mathfrak A}_{[0, 1)}\), \({\mathfrak A}_J\) and \({\mathfrak A}_{[0,1)}\) are two special sets of functionals, and \(x|_J\) is the restriction of \(x\) to \(J\). For \(r> 0\), \(C_r\) denotes the Banach space of \(C^0\)-functions \(x: [-r,0]\to \mathbb{R}\). The goal of the paper is to find sufficient conditions for the existence of solutions to (1)--(2). For this purpose the authors use the Leray-Schauder degree theory.