Differential equations on variable time scales

Date

2009-02-01

Author

Akhmet, Marat

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We introduce a class of differential equations on variable time scales with a transition condition between two consecutive parts of the scale. Conditions for existence and uniqueness of solutions are obtained. Periodicity, boundedness and stability of solutions are considered. The method of investigation is by means of two successive reductions: B-equivalence of the system [E. Akalfn, M.U. Akhmet, The principles of B-smooth discontinuous flows, Computers and Mathematics with Applications 49 (2005) 981-995; M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163-178; M.U. Akhmet, N.A. Perestyuk, The comparison method for differential equations with impulse action, Differential Equations 26 (9) (1990) 1079-1096] on a variable time scale to a system on a time scale, a reduction to an impulsive differential equation [M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163-178; M.U. Akhmet, M. Turan, The differential equations on time scales through impulsive differential equations, Nonlinear Analysis 65 (2006) 2043-2060]. Appropriate examples are constructed to illustrate the theory.

Subject Keywords

Variable time scales, Differential equations on time scales, Impulsive differential equations, Existence and uniqueness, Bounded solutions, Periodic solutions, Stability

URI

https://hdl.handle.net/11511/43903

Journal

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS

DOI

https://doi.org/10.1016/j.na.2008.02.020

Collections

Department of Mathematics, Article

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Citation Formats

M. Akhmet, “Differential equations on variable time scales,” NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, pp. 1175–1192, 2009, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/43903.