Solutions of the Variational Equation for an nth Order Boundary Value Problem with an Integral Boundary Condition

In this paper, we discuss differentiation of solutions to the boundary value problem $y^{(n)}=f(x,y,y^{{}^{\mathrm{\prime }}},y^{{}^{\mathrm{\prime }\mathrm{\prime }}},\dots ,y^{(n-1)}),a<x<b,y^{(i)}(x_j)=y_{ij},0\le i\le m_j,1\le j\le k-1$, and $y^{(i)}(x_k)+{\int }_c^{d}py(x)dx=y_{ik},0\le i\le m_k,{\sum }_{i=1}^k{m}_i=n$ with respect to the boundary data. We show that under certain conditions, partial derivatives of the solution $y(x)$ of the boundary value problem with respect to the various boundary data exist and solve the associated variational equation along $y(x)$.