#### Vol. 14, No. 1, 2021

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Solutions of the variational equation for an $n$-th order boundary value problem with an integral boundary condition

### Benjamin L. Jeffers and Jeffrey W. Lyons

Vol. 14 (2021), No. 1, 155–166
##### Abstract

We discuss differentiation of solutions to the boundary value problem

$\begin{array}{cc}{y}^{\left(n\right)}=f\left(x,y,{y}^{\prime },{y}^{\prime \prime },\dots ,{y}^{\left(n-1\right)}\right),\phantom{\rule{1em}{0ex}}a

with respect to the boundary data. We show that under certain conditions, partial derivatives of the solution $y\left(x\right)$ of the boundary value problem with respect to the various boundary data exist and solve the associated variational equation along $y\left(x\right)$.

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##### Keywords
variational equation, integral condition, continuous dependence, smoothness, Peano theorem
Primary: 34B10
Secondary: 34B15