tailieunhanh - Limit behaviors of nonoscillatory solutions of three-dimensional time scale systems
In this article, we investigate the oscillatory behavior of a three-dimensional system of dynamic equations on an unbounded time scale. A time scale T is a nonempty closed subset of real numbers. An example is given to illustrate some of the results. | Turk J Math (2018) 42: 2576 – 2587 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Limit behaviors of nonoscillatory solutions of three-dimensional time scale systems 1 Özkan ÖZTÜRK1,∗,, Raegan HIGGINS2 , Department of Mathematics, Faculty of Arts and Sciences, Giresun University, Giresun, Turkey 2 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA Received: • Accepted/Published Online: • Final Version: Abstract: In this article, we investigate the oscillatory behavior of a three-dimensional system of dynamic equations on an unbounded time scale. A time scale T is a nonempty closed subset of real numbers. An example is given to illustrate some of the results. Key words: Three-dimensional dynamical system, time scales, fixed points, existence of nonoscillatory solutions, classification 1. Introduction In this paper, we study the nonlinear system ∆ x (t) = p(t)f (y(t)) ∆ y (t) = q(t)g(z(t)) ∆ z (t) = −r(t)h(x(t)) (1) on [t0 , ∞)T such that t0 ∈ T and t0 ≥ 0 , where p, q ∈ Crd ([t0 , ∞)T , [0, ∞)), r ∈ Crd ([t0 , ∞)T , (0, ∞)), and ∫ ∞ t0 ∫ p(s) ∆s = ∞ = ∞ q(s) ∆s. (2) t0 We also assume that f, g, h ∈ C(R, R) are nondecreasing functions such that uf (u) > 0, ug(u) > 0 and uh(u) > 0 for u ̸= 0 . Here we only consider unbounded time scales, and by t ≥ t0 , we mean t ∈ [t0 , ∞)T := [t0 , ∞) ∩ T. Classifications of nonoscillatory solutions for some other versions of system (1) are also considered in [8–11]. The theory of time scales was initiated by Stefan Hilger in his PhD thesis [6] in 1988. The main purpose was to unify and extend continuous and discrete cases in one comprehensive theory. Since 1988, there has been much research in many areas of time scales including the classification and existence of dynamical systems. For an introduction to the theory of time scales, we refer readers to the books .
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