Continuous dependence and differentiating solutions of a second order boundary value problem with average value condition

Abstract

Using a few conditions, continuous dependence, and a result regarding smoothness of initial conditions, we show that derivatives of solutions to the second order boundary value problem $y^{\prime \prime }=f\left(x,y,y^{\prime }\right)$, $a<x<b$, satisfying $y\left(x_1\right)=y_1$, $1∕\left(d-c\right){\int }_c^{d}y\left(x\right)\underset{̣}{x}=y_2$, where $a<x_1<c<d<b$ and $y_1,y_2\in ℝ$ with respect to each of the boundary data $x_1$, $y_1$, $y_2$, $c$, $d$ solve the associated variational equation with interesting boundary conditions. Of note is the second boundary condition, which is an average value condition.