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

y(n) = f(x,y,y,y,,y(n1)),a < x < b, y(i)(x j) = yij,0 i mj,1 j k 1, y(i)(x k) +cdpy(x)dx = yik,0 i mk, i=1kmi = 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).

Keywords
variational equation, integral condition, continuous dependence, smoothness, Peano theorem
Mathematical Subject Classification
Primary: 34B10
Secondary: 34B15
Milestones
Received: 20 August 2020
Revised: 14 September 2020
Accepted: 28 September 2020
Published: 4 March 2021

Communicated by Johnny Henderson
Authors
Benjamin L. Jeffers
Trinity University
San Antonio, TX
United States
Jeffrey W. Lyons
The Citadel
Charleston, SC
United States