h -stability for nonlinear abstract dynamic equations on time scales and applications
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h-stability for nonlinear abstract dynamic equations on time scales and applications Bilel Neggal2 · Khaled Boukerrioua1,2 · Brahim Kilani2 · Imen Meziri1,2 Received: 6 June 2019 / Accepted: 30 August 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract This paper focuses on the problem of h-stability of certain classes of dynamic perturbed systems on time scales using time scale versions of some Grönwall type inequalities. We prove under certain conditions on the nonlinear perturbations that the resulting perturbed nonlinear initial value problem still acquire h-stable, if the associated abstract dynamic equation has already owned this property. The paper ends up with two illustrative examples to highlight the utility of our results. Keywords Dynamic equations · Time scale · Integral inequality · Semi-group · h-stability Mathematics Subject Classification 34N05 · 34D20
1 Introduction The theory of dynamic equations on an arbitrary time scale was introduced in [1]. This theory was found promising because it unifies the theories of continuous-time and discrete-time systems. The calculus on time scales and dynamic equations on time scales have applications in any field that requires simultaneous modeling of continuous and discrete processes, because they bridge the divide between continuous and discrete aspects of processes. The applications include insect population models, epidemic models, neural networks, and heat transfer. Foundational definitions and results from the time scale calculus appear in an excellent introductory text by Bohner and Peterson [6,7]. During the last few years, several studies were
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Khaled Boukerrioua [email protected] Bilel Neggal [email protected] Brahim Kilani [email protected] Imen Meziri [email protected]
1
Lanos Laboratory, University of Badji Mokhtar, Annaba, Algeria
2
Department of Mathematics, University of Badji Mokhtar, P.O. Box 12 Annaba, Algeria
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B. Neggal et al.
achieved on stability, h-stability of certain classes of dynamical equations and oscillation of dynamic equations on time scales, see [2–5,8–16,19–21,24] and the references cited therein. The notion of h-stability was introduced by Pinto [23] which is an extension of the notions of exponential stability and uniform stability. This paper examines the h-stability for the following abstract dynamic equation on time scales: x (t) = Ax(t) + f (t, x), x(t0 ) = x0 ∈ D(A), t ∈ T+ (1.1) t0 , in terms of the h-stability of the homogeneous equation: x (t) = Ax(t), x(t0 ) = x0 ∈ D(A), t ∈ T+ t0 ,
(1.2)
where A is the generator of a C0 -semigroup T , f : T × X → X is r d-continuous with f (t, 0) = 0 and x is the delta derivative of x: T →X , X is a Banach space and assume T = +∞. We derive sufficient conditions for the h-stability notion of certain classes of abstract dynamic perturbed equations on time scales using time scale versions of some Grönwall type inequalities.
2 Preliminaries First, we will briefly mention some basic definitions and results of time scale c
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