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FORMULAS OF BENDIXSON AND ALEKSEEV FOR DIFFERENCE EQUATIONS

Published online by Cambridge University Press:  23 December 2003

M. BOHNER
Affiliation:
Department of Mathematics, Florida Institute of Technology, Melbourne, Florida 32901, USA bohner@umr.edu
V. LAKSHMIKANTHAM
Affiliation:
Department of Mathematics, Florida Institute of Technology, Melbourne, Florida 32901, USA lakshmik@fit.edu
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Abstract

A well-known formula of Bendixson states that solutions of first-order differential equations, as functions of their initial conditions, satisfy a certain partial differential equation. A consequence is Alekseev's nonlinear variation of parameters formula. In this paper, corresponding results are proved for difference equations. To achieve this, use is made of the recently introduced concept of alpha derivatives, rather than of differences or of the usual derivatives. This technique allows the results to be generalized to alpha dynamic equations, which include among others ordinary differential and difference equations.

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Papers
Copyright
© The London Mathematical Society 2004

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