## Mathematics Faculty Articles

#### Title

Differentiation of Solutions of Nonlocal Boundary Value Problems with Respect to Boundary Data

Article

7-15-2011

#### Publication Title

Electronic Journal of Qualitative Theory of Differential Equations

#### Keywords

Nonlinear boundary value problem, Variational equation, Ordinary differential equation, Nonlocal boundary condition, Uniqueness, Existence

1417-3875

2011

51

1

11

#### Abstract

In this paper, we investigate boundary data smoothness for solutions of the nonlocal boundary value problem, $y^{(n)}=f(x,y,y',\ldots,y^{(n-1)}),y^{(i)}(x_j)=y_{ij}$ and $y^{(i)}(x_k)-\sum_{p=1}^m r_{ip}y(\eta_{ip})=y_{ik}.$ Essentially, we show under certain conditions that partial derivatives of the solution to the problem above exist with respect to boundary conditions and solve the associated variational equation. Lastly, we provide a corollary and nontrivial example.