Functional differential equations of second order (Q1420692)

The existence of a unique solution of boundary-value problems for functional-differential equations of the form \[ x' (t)=f(t,x_t), \quad t\in[0,1],\qquad x(0)=\phi_0,\quad x(1)=k_1, \tag{*} \] is considered in this paper where \(\phi_0\in C_0=C([-\tau ,0],\mathbb R),\) \(\tau>0\) and \(f\in C([0,1]\times C_0,\mathbb R)\), and for \(t\in [0,1]\), \(x_t\in C_0\) is defined by \(x_t(s)=x(t+s)\), \(s\in [-\tau ,0]\). Two monotone sequences are constructed which, under certain conditions, converge to the unique solution of (\(*\)). This convergence is superlinear. Examples are given to illustrate the results obtained.