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Formulas of Bendixson and Alekseev for Difference Equations
Author(s) -
Bohner M.,
Lakshmikantham V.
Publication year - 2004
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609303002753
Subject(s) - mathematics , nonlinear system , differential equation , ordinary differential equation , alpha (finance) , order (exchange) , variation of parameters , partial differential equation , mathematical analysis , pure mathematics , statistics , physics , finance , quantum mechanics , economics , construct validity , psychometrics
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.