The connection between variational equations and solutions of second order nonlocal integral boundary value problems. (English) Zbl 1312.34051
Summary: We make certain continuity and disconjugacy assumptions upon the second-order boundary value problem with nonlocal integral boundary conditions,
\[
\begin{gathered} y''= f(x,y,y'),\quad y(x_1)= y_1,\;\text{and}\\ y(x_2)+ \int^d_c ry(x)\,dx,\;a< x_1< c< d< x_2< b,\;y_1,y_2,\;r\in\mathbb{R}.\end{gathered}
\]
Then, supposing we have a solution, \(y(x)\), of the boundary value problem, we differentiate the solution with respect to various boundary parameters. We show that the resulting function solves the associated variational equation of \(y(x)\).
MSC:
34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |