tailieunhanh - Báo cáo hóa học: " Common zeros of the solutions of two differential equations with transcendental coefficients"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Common zeros of the solutions of two differential equations with transcendental coefficients | Asiri Journal of Inequalities and Applications 2011 2011 134 http content 2011 1 134 RESEARCH Journal of Inequalities and Applications a SpringerOpen Journal Open Access Common zeros of the solutions of two differential equations with transcendental coefficients Asim Asiri1 2 Correspondence amkasiri@. sa 1School of Mathematical Sciences King Abdulaziz University University of Nottingham Nottingham NG72rD UK Full list of author information is available at the end of the article Abstract Purpose We consider a pair of homogeneous linear differential equations with transcendental entire coefficients of finite order and the question of when solutions of these equations can have the same zeros or nearly the same zeros. Method We apply the Nevanlinna theory and properties of entire solutions of linear differential equations. Conclusion The results determine all pairs of such equations having solutions with the same zeros or nearly the same zeros. Keywords Nevanlinna theory differential equations 1 Introduction This paper continues our study from 1 of the question of when two linear differential equations in the complex domain can have solutions with nearly the same zeros. Our starting point is the equation w Pw 0 1 and 1 considered the case where P is a polynomial. In this paper P will be a transcendental entire function but we will retain the same notation since we will sometimes refer the reader to 1 . The following theorem was proved in 1955 by Wittich 2 . Theorem Iff 0 is a non-trivial solution of 1 and P is a transcendental entire function then we have i fhas infinite order and T r P S r f . ii If a is a non-zero complex number then f takes the value a infinitely often and in fact N r f-i T r f S r ft- In this paper we use standard notation of Nevanlinna theory from 3 . The reader is referred to the books of Hille 4 and Laine 5 the keynote paper 6 and to 7-14 for comprehensive results on the zeros of solutions .