tailieunhanh - On the perron effect for exponential stability of differential systems on time scales

In 2007, N. H. Du and L. H. Tien [1] shown that the exponential stability of the linear equation on time scales implies the exponential stability of the suitable small enough Lipchitz perturbed equation. In this paper, we shall prove that if the perturbation is arbitrary small order 1 then the above argument is not true which is called Perron effect. | VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 4 (2017) 1-9 On the Perron Effect for Exponential Stability of Differential Systems on Time Scales Le Duc Nhien*, Le Huy Tien Department of Mathematics, Mechanics and Informatics, VNU University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam Received 02 September 2017 Revised 29 September 2017; Accepted 24 October 2017 Abstract: In 2007, N. H. Du and L. H. Tien [1] shown that the exponential stability of the linear equation on time scales implies the exponential stability of the suitable small enough Lipchitz perturbed equation. In this paper, we shall prove that if the perturbation is arbitrary small order 1 then the above argument is not true which is called Perron effect. Keywords: Exponential stability, Perron effect, time scales, linear dynamic equation. 1. Introduction and preliminaries Theory of dynamic equations on time scales was introduced by Stefan Hilger [2] in order to unify and extend results of differential equations, difference equations, q-difference equations, etc. There are many works concerned with the stability of dynamic equations on time scales such as exponential stability (see [3-5]); dichotomies of dynamic equations (see [6]). In this paper, we want to go further in the stability of dynamic equations. More precisely, we show that the exponential stability of the linear equation on time scales does not imply the exponential stability of the small enough Lipchitz perturbed equation if the perturbation is arbitrary small order 1 which is called Perron effect. Moreover, our results are different from examples of Perron type in both continuous and discrete cases (see [7-9]). We now introduce some basic concepts of time scales, which can be found in [10, 11]. A time scale is defined as a nonempty closed subset of the real numbers. Define the forward jump operator : is defined by ( t ) inf{ s : s t } and the graininess function ( t ) ( t ) t for any is .