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Báo cáo hóa học: " Research Article On the Kneser-Type Solutions for Two-Dimensional Linear Differential Systems with Deviating Arguments"

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: Research Article On the Kneser-Type Solutions for Two-Dimensional Linear Differential Systems with Deviating Arguments | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 52304 22 pages doi 10.1155 2007 52304 Research Article On the Kneser-Type Solutions for Two-Dimensional Linear Differential Systems with Deviating Arguments Alexander Domoshnitsky and Roman Koplatadze Received 7 January 2007 Revised 26 March 2007 Accepted 25 April 2007 Recommended by Alberto Cabada For the differential system u1 t p t u2 T t u2 t q t u1 ơ t t e 0 to where p q e Lloc R R T ơ e C R R limt T t limt-.to.x t to we get necessary and sufficient conditions that this system does not have solutions satisfying the condition u1 t u2 t 0 for t e t0 TO . Note one of our results obtained for this system with constant coefficients and delays p t p q t q T t t - A Ơ t t s where s A e R and A s 0 . The inequality S A - pq 2 e is necessary and sufficient for nonexistence of solutions satisfying this condition. Copyright 2007 A. Domoshnitsky and R. Koplatadze. 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 The equation u t pu t t e 0 to with positive constant coefficient p has two linearly independent solutions u1 e t and u2 e tĩt. The second solution satisfies the property u t u t 0 for t e 0 to and it is the Kneser-type solution. The ordinary differential equation with variable coefficient u t p t u t p t 0 t e 0 to preserves the solutions of the Kneser-type. The differential equation with deviating argument u t p t u T t p t 0 t e 0 to 1.1 where u t y t for Ỉ 0 generally speaking does not inherit this property. The problems of existence nonexistence of the Kneser-type solutions were studied in 1-4 . Assertions on existence of bounded solutions their uniqueness and oscillation were obtained in the monograph by Ladde et al. see 5 pages 130-139 . Several possible types 2 Journal of Inequalities and