Continuous Dependence and Differentiating Solutions of a Second Order Boundary Value Problem with an Average Value Condition

Continuous Dependence and Differentiating Solutions of a Second Order Boundary Value Problem with an Average Value Condition

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’’

=f(x,y,y’),a

each of the boundary data 𝑥1, 𝑦1, 𝑦2, c, d solve the associated variational equation with interesting boundary conditions. Of note is the second boundary condition with an average value condition.