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DIFFERENTIATING SOLUTIONS OF A BOUNDARY VALUE PROBLEM ON A TIME SCALE

Part of: Dynamic equations on time scales or measure chains Boundary value problems

Published online by Cambridge University Press:  01 April 2016

LEE H. BAXTER ,
JEFFREY W. LYONS  and
JEFFREY T. NEUGEBAUER
LEE H. BAXTER
Affiliation:
Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475, USA email lee_baxter6@mymail.eku.edu
JEFFREY W. LYONS
Affiliation:
Division of Math, Science and Technology, Nova Southeastern University, Fort Lauderdale, FL 33314, USA email jlyons@nova.edu
JEFFREY T. NEUGEBAUER*
Affiliation:
Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475, USA email jeffrey.neugebauer@eku.edu
*
✉jeffrey.neugebauer@eku.edu
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Abstract

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We show that the solution of the dynamic boundary value problem $y^{{\rm\Delta}{\rm\Delta}}=f(t,y,y^{{\rm\Delta}})$, $y(t_{1})=y_{1}$, $y(t_{2})=y_{2}$, on a general time scale, may be delta-differentiated with respect to $y_{1},~y_{2},~t_{1}$ and $t_{2}$. By utilising an analogue of a theorem of Peano, we show that the delta derivative of the solution solves the boundary value problem consisting of either the variational equation or its dynamic analogue along with interesting boundary conditions.

Keywords

time scalesvariational equationuniqueness

MSC classification

Primary: 34N05: Dynamic equations on time scales or measure chains
Secondary: 34B99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 1 , August 2016 , pp. 101 - 109
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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