tailieunhanh - Báo cáo hóa học: " Local stability of the Pexiderized Cauchy and Jensen’s equations in fuzzy spaces"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Local stability of the Pexiderized Cauchy and Jensen’s equations in fuzzy spaces | Najati et al. Journal of Inequalities and Applications 2011 2011 78 http content 2011 1 78 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Local stability of the Pexiderized Cauchy and Jensen s equations in fuzzy spaces Abbas Najati 1 Jung Im Kang2 and Yeol Je Cho3 Correspondence jikang@. kr 2National Institute for Mathematical Sciences KT Daeduk 2 Research Center 463-1 Jeonmin-dong Yuseong-gu Daejeon 305-811 Korea Full list of author information is available at the end of the article Springer Abstract Lex X be a normed space and Y be a Banach fuzzy space. Let D x y e X X X x y d where d 0. We prove that the Pexiderized Jensen functional equation is stable in the fuzzy norm for functions defined on D and taking values in Y. We consider also the Pexiderized Cauchy functional equation. 2000 Mathematics Subject Classification 39B22 39B82 46S10. Keywords Pexiderized Cauchy functional equation generalized Hyers-Ulam stability Jensen functional equation non-Archimedean space 1. Introduction The functional equation f is stable if any function g satisfying the equation f approximately is near to the true solution of f . The stability problem of functional equations originated from a question of Ulam 1 concerning the stability of group homomorphisms Let G1 be a group and let G2 be a metric group with the metric d v . Given 0 does there exist Ỏ 0 such that if a function h G1 G2 satisfies the inequality d h xy h x h y ỗ for all x y e G1 then there exists a homomorphism H G1 G2 with d h x H x for all x e G1 In other words we are looking for situations when the homomorphisms are stable i. e. if a mapping is almost a homomorphism then there exists a true homomorphism near it. If we turn our attention to the case of functional equations then we can ask the question When the solutions of an equation differing slightly from a given one must be close to the true solution of the given equation. .