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Bulletin of the London Mathematical Society 2004 36(1):65-71; doi:10.1112/S0024609303002753
© 2004 by London Mathematical Society
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© London Mathematical Society

Formulas of Bendixson and Alekseev for Difference Equations

M. Bohner and V. Lakshmikantham

Department of Mathematics, Florida Institute of Technology Melbourne, Florida 32901, USA; bohner{at}umr.edu, lakshmik{at}fit.edu

Received 18 April 2002. Revision received 11 February 2003.

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. 2000 Mathematics Subject Classification 39A12, 39A13.


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