tailieunhanh - Stability of perturbed dynamic system on time scales with initial time difference

The behavior of solutions of a perturbed dynamic system with respect to an original unperturbed dynamic system, which have initial time difference, are investigated on arbitrary time scales. Notions of stability, asymptotic stability, and instability with initial time difference are introduced. | Turkish Journal of Mathematics Research Article Turk J Math (2015) 39: 1 – 15 ¨ ITAK ˙ c TUB ⃝ doi: Stability of perturbed dynamic system on time scales with initial time difference ˘ Co¸skun YAKAR∗, B¨ ulent OGUR Department of Mathematics, Faculty of Sciences, Gebze Institute of Technology, C ¸ ayırova, Gebze, Kocaeli, Turkey Received: • Accepted: • Published Online: • Printed: Abstract: The behavior of solutions of a perturbed dynamic system with respect to an original unperturbed dynamic system, which have initial time difference, are investigated on arbitrary time scales. Notions of stability, asymptotic stability, and instability with initial time difference are introduced. Sufficient conditions of stability properties are given with the help of Lyapunov-like functions. Key words: Time scales, stability, Lyapunov-like functions, comparison results, initial time difference 1. Introduction In [1, 7], Hilger introduced the theory of time scales, closed subsets of R, to unify the theory of differential and difference equations into a single set-up and to extend these theories to other kinds of so-called dynamic equations. This extension gives us a chance to consider the continuous and discrete cases simultaneously. Stability theory is one of the important branches of the theory of differential equations. Numerous studies have been done about this theory [4, 8]. Some of these results were extended to dynamic equations on time scales [9]. An important problem in stability theory is to determine which stability properties of a particular differential system are preserved under sufficiently small perturbations. This problem was investigated in several ways in [4, 5, 6, 8, 9]. However, the possibility of making errors in initial time as well as in initial position needs to be considered. When such a change of initial time for each solution is considered, then