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Báo cáo hóa học: " Erratum A Note to Paper “On the Stability of Cubic Mappings and Quartic Mappings in Random Normed Spaces”"

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: Erratum A Note to Paper “On the Stability of Cubic Mappings and Quartic Mappings in Random Normed Spaces” | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 214530 6 pages doi 10.1155 2009 214530 Erratum A Note to Paper On the Stability of Cubic Mappings and Quartic Mappings in Random Normed Spaces R. Saadati 1 S. M. Vaezpour 1 and Y. J. Cho2 1 Department of Mathematics and Computer Science Amirkabir University ofTechnology 424 Hafez Avenue Tehran 15914 Iran 2 Department of Mathematics Education and the RINS Gyeongsang National University Chinju 660-701 South Korea Correspondence should be addressed to Y. J. Cho yjcho@gnu.ac.kr Received 16 February 2009 Accepted 21 April 2009 Recently Baktashet al. 2008 proved the stability of the cubic functional equation f 2x y f 2x-y 2f x y 2f x - y 12f x and the quartic functional equation f 2x y f 2x - y 4f x y 4f x - y 24f x - 6f y in random normed spaces. In this note we correct the proofs. Copyright 2009 R. Saadati et al. 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 and Preliminaries If inf t 0 F f a inf t 0 G f a in general we cannot conclude that F f G f . For example let F Jf 3 4 G f t i 1 and a 1 2. We know that inf t 0 3 4 1 2 0 inf t 0 t i 1 1 2 1 but F 4 3 4 G 4 4 5. This example shows that in 1 inequalities 2.13 and 3.13 do not follow from inequalities 2.12 and 3.12 . The functional equation f 2x y f 2x - y 2f x y 2f x - y 12f x h1 is said to be the cubic functional equation since the function f x ex3 is its solution. Every solution of the cubic functional equation is said to be a cubic mapping. The stability problem for the cubic functional equation was solved by Jun and Kim 2 and Lee 3 for mappings f X Y where X is a real normed space and Y is a Banach space. Later a number of mathematicians have worked on the stability of some types of the cubic equation 4 . The functional equation f 2x y f 2x -