tailieunhanh - Báo cáo hóa học: " On the super-stability of exponential Hilbert-valued functional equations"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : On the super-stability of exponential Hilbert-valued functional equations | Rezaei and Sharifzadeh Journal of Inequalities and Applications 2011 2011 114 http content 2011 1 114 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access On the super-stability of exponential Hilbert-valued functional equations H Rezaei and M Sharifzadeh Correspondence rezaei@. Department of Mathematics College of Sciences Yasouj University Yasouj-75914-74831 Iran Springer Abstract We generalize the well-known Baker s super-stability result for exponential mappings with values in the field of complex numbers to the case of an arbitrary Hilbert space with the Hadamard product. Then we will prove an even more general result of this type. 2000 MSC primary 39B72 secondary 46E40. Keywords exponential functions stability Hilbert-valued function 1. Introduction The stability problem of functional equations goes back to a question of Ulam 1 concerning the stability of group homomorphisms. Hyers 2 gave a first affirmative partial answer to the question of Ulam for Banach spaces see also 3 . Hyers s theorem was generalized by Aoki 4 for additive mappings and by Rassias 5 6 for linear mappings by considering an unbounded Cauchy difference. Baker et al. 7 have proved the super-stability of the exponential functional equation If a function f R R is approximately exponential function . there exists a nonnegative number a such that f x y - f x f y a for x y e R then f is either bounded or exponential. This theorem was the first result concerning the super-stability phenomenon of functional equations. Baker 8 generalized this famous result to any function f G c where G is a semigroup. The same result is also true for approximately exponential mappings with values in a normed algebra with the property that the norm is multiplicative. Theorem . Let G be a semigroup and Y be a normed algebra in which the norm is multiplicative. Then for a function f G Y satisfying the inequality f x y -f

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