Halanay type inequalities on time scales with applications

Abstract

This paper aims to introduce Halanay type inequalities on time scales. By means of these inequalities we derive new global stability conditions for nonlinear dynamic equations on time scales. Giving several examples we show that beside generalization and extension to q-difference case, our results also provide improvements for the existing theory regarding differential and difference inequalites, which are the most important particular cases of dynamic inequalities on time scales.

Floquet theory based on new periodicity concept for hybrid systems involving q -difference equations

“…In this section, we aim to introduce basic definitions and properties of shift operators. The following definitions, lemmas and examples can be found in [1], [2], [3] and [5].…”

“…Thus, it is of importance to study the existence of periodic solutions of q-difference equations. In recent years, the shift operators, denoted δ ± (s, t), are introduced to construct delay dynamic equations and a new periodicity concept on time scales (see [1], [4], and [5]). We give a detailed information about the shift operators in further sections.…”

“…We give a detailed information about the shift operators in further sections. We may also refer to the studies [2], [3], and [5] for the basic definitions, properties and some applications of shift operators on time scales. In particular, we direct the readers to [1] for the construction of new periodicity concept on time scales.…”

A nonlinear version of Halanay's inequality for the uniform convergence to the origin

“…Most of the Halanay type inequalities available in the literature are linear in the righthand side. As far as the nonlinear case is concerned, the topic has been addressed in [1], [3], [37]. In [1] Halanay type inequalities in time scales are studied and related new global asymptotic stability results are given.…”

“…As far as the nonlinear case is concerned, the topic has been addressed in [1], [3], [37]. In [1] Halanay type inequalities in time scales are studied and related new global asymptotic stability results are given. A notable improvement towards nonlinear Halanay inequality is given in [3]: the right-hand side is given by a very general nonlinear function of the current value of the function at study, and of its supremum in the latest delay interval; the natural assumption of the increasing property of this general function in the second argument is introduced; the condition of negativity of this general function when the arguments are equal each other, inside an interval ranging from zero to an upper-bound, is proved to be a sufficient condition for the convergence to zero of the function at study.…”