tailieunhanh - Báo cáo hóa học: "OSCILLATION AND NONOSCILLATION FOR IMPULSIVE DYNAMIC EQUATIONS ON CERTAIN TIME SCALES"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: OSCILLATION AND NONOSCILLATION FOR IMPULSIVE DYNAMIC EQUATIONS ON CERTAIN TIME SCALES | OSCILLATION AND NONOSCILLATION FOR IMPULSIVE DYNAMIC EQUATIONS ON CERTAIN TIME SCALES MOUFFAK BENCHOHRA SAMIRA HAMANI AND JOHNNY HENDERSON Received 1 December 2005 Revised 6 March 2006 Accepted 9 March 2006 We discuss the existence of oscillatory and nonoscillatory solutions for first-order impulsive dynamic equations on time scales with certain restrictions on the points of impulse. We will rely on the nonlinear alternative of Leray-Schauder type combined with a lower and upper solutions method. Copyright 2006 Mouffak Benchohra et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction This paper is concerned with the existence of oscillatory and nonoscillatory solutions of first-order impulsive dynamic equations on certain time scales. We consider the problem yA t f t y t te J 0 to oT t tk k 1 . y t- Ik y tk k 1 . where T is an unbounded-above time scale with 0 e T f JT X R . R is a given function Ik e C R R tk e T 0 t0 t1 tm tm 1 TO y t linih . - y tk h and y t- limh .0- y tk - h represent the right and left limits of y t at t tk in the sense of the time scale that is in terms of h 0 for which tk h tk - h e t0 to o T whereas if tk is left-scattered resp. right-scattered we interpret y t- y tk resp. y t y tk . Impulsive differential equations have become important in recent years in mathematical models of real processes and they rise in phenomena studied in physics chemical technology population dynamics biotechnology and economics. There have been significant developments in impulse theory also in recent years especially in the area of impulsive differential equations with fixed moments see the monographs of Bainov and Sime-onov 5 Lakshmikantham et al. 22 Samoilenko and Perestyuk 25 and the references therein. In recent years dynamic equations on times scales have received much .