Smoothness of solutions with respect to multi-strip integral boundary conditions for second order ordinary differential equations

Abstract

Under certain conditions, solutions of the boundary value problem,$y'' = f(x,y,y'),$ $a < x < b,$ $y(x_1) = y_1, y(x_2) - \sum_{i=1}^m \gamma_i \int_{\xi_i}^{\eta_i} y(x) dx = y_2,$ $ a< x_1 <\xi_1 <\eta_1 < \xi_2 <\eta_2 <\cdots< \xi_m < \eta_m <x_2 < b,$ are differentiated with respect to the boundary conditions.