tailieunhanh - Báo cáo hóa học: " Research Article On Stability of a Functional Equation Connected with the Reynolds Opera"

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 Stability of a Functional Equation Connected with the Reynolds Opera | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 79816 3 pages doi 2007 79816 Research Article On Stability of a Functional Equation Connected with the Reynolds Operator Adam Najdecki Received 18 July 2006 Revised 30 November 2006 Accepted 3 December 2006 Recommended by Saburou Saitoh Let X o be an Abelain semigroup g X X and let K be either R or C. We prove superstability of the functional equation f x o g y f x f y in the class of functions f X K. We also show some stability results of the equation in the class of functions f X K . Copyright 2007 Adam Najdecki. 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. Throughout this paper is a positive integer X o is a commutative semigroup K is either the field of reals R or the field of complex numbers C and g X X is an arbitrary function. We study stability of the functional equation f x o g y f x f y for x y e X 1 in the class of functions f X K where a1 a2 . an b1 b2 . b a1b2 a2b2 . anbn for a1 a2 . an b1 b2 . b e K . For details concerning the problem of stability of functional equations we refer to . 1 . Particular cases of 1 are the well-known multiplicative Cauchy equation f xy f x f y exponential equation f x y f x f y see . 2 and the equation f xf y f x f y . 2 The origin of 2 is in the averaging theory applied to turbulent fluid motion. This equation is connected with some linear operators that is the Reynolds operator see 3 and 4 the averaging operator the multiplicatively symmetric operator see 2 . Ger and Semrl in 5 cf. 6 7 considered the problem of stability for the exponential equation in the class of functions mapping X into a semisimple complex commutative 2 Journal of Inequalities and Applications Banach algebra . They have shown that if a mapping f X satisfies Ilf x 0 y - f x f y e 3