Development and application of Halanay-type theory: Evolutionary differential and difference equations with time lag

Authors:
Christopher T. H. Baker
Affiliations:
Department of Mathematics, University of Chester, Chester CH1 4BJ, United Kingdom and University of Manchester, Manchester M13 9PL, United Kingdom
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

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Abstract

We consider extensions, developments and modifications of a result due to Halanay, and the application of ''Halanay-type inequalities'' in the analysis and numerics of retarded functional-differential equations, difference equations, and retarded functional-difference equations. Our emphasis is on the variety, structure and development, and future development, of Halanay-type results and their applications. We classify and present novel results of Halanay type (linear and non-linear, discrete, semi-discrete, and continuous) and establish their relevance to delay-differential equations, discretized analogues (we consider @q-methods), and difference equations. A role for such results in stability and contractivity analysis is made apparent.