Bounded solutions of delay dynamic equations on time scales
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Bounded solutions of delay dynamic equations on time scales Josef Diblík1,2* and Jiˇrí Vítovec1 *
Correspondence: [email protected]; [email protected] 1 Department of Mathematics, Faculty of Electrical Engineering and Communications, Brno University of Technology, Brno, Czech Republic 2 Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Brno, Czech Republic
Abstract In this paper we discuss the asymptotic behavior of solutions of the delay dynamic equation y (t) = f (t, y(τ (t))), where f : T × R → R, τ : T → T is a delay function and T is a time scale. We formulate the principle which gives the guarantee that the graph of at least one solution of the above mentioned equation stays in the prescribed domain. This principle uses the idea of the retraction method and is a suitable tool for investigating the asymptotic behavior of solutions of dynamic equations. This is illustrated by an example.
1 Introduction 1.1 Time scale calculus We assume that the reader is familiar with the notion of time scales. Thus, note just that T, [a, b]T := [a, b] ∩ T (resp. (a, b)T := (a, b) ∩ T or similarly, we define any combination of right and left open or closed interval), [a, ∞) T := [a, ∞) ∩ T, σ , ρ, μ and f stand for the time scale, a finite time scale interval, an infinite time scale interval, a forward jump operator, a backward jump operator, graininess and a -derivative of f . Further, the (T) stand for the class of continuous, rd-continuous and symbols C(T), Crd (T) and Crd rd-continuous -derivative functions. See [], which is the initiating paper of the time scale theory, the thesis [] and [] containing a lot of information on time scale calculus. Now, we remind further aspects of time scale calculus, which will be needed later; see, e.g., []. Definition Let T be a time scale. A function f : T × R → R is called (i) rd-continuous, if g defined by g(t) := f (t, y(t)) is rd-continuous for any rd-continuous function y : T → R; (ii) bounded on a set S ⊂ T × R, if there exists a constant M > such that f (t, y) ≤ M
for all (t, y) ∈ S;
(iii) Lipschitz continuous on a set S ⊂ T × R, if there exists a constant L > such that f (t, y ) – f (t, y ) ≤ L|y – y |
for all (t, y ), (t, y ) ∈ S.
© 2012 Diblík and Vítovec; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Diblík and Vítovec Advances in Difference Equations 2012, 2012:183 http://www.advancesindifferenceequations.com/content/2012/1/183
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1.2 Delay dynamic equations on time scales Let τ : T → T be an increasing rd-continuous function satisfying τ (t) ≤ t for all t ∈ T. Let the function f : T × R → R be rd-continuous. We consider the delay dynamic equation y (t) = f t, y τ (t)
()
on time scales T.
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