solution and stability of a mixed type cubic and quartic functional equation in quasi-banach spaces混合类型的解决方案和稳定三次和四次函数方程quasi-banach空间.pdfVIP

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2009, Article ID 417473, 14 pages doi:10.1155/2009/417473 Research Article Solution and Stability of a Mixed Type Cubic and Quartic Functional Equation in Quasi-Banach Spaces M. Eshaghi Gordji,1 S. Zolfaghari,1 J. M. Rassias,2 and M. B. Savadkouhi1 1 Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran 2 Section of Mathematics and Informatics, Pedagogical Department, National and Capodistrian University of Athens, 4, Agamemnonos Street, Aghia Paraskevi, 15342 Athens, Greece Correspondence should be addressed to M. Eshaghi Gordji, madjid.eshaghi@ Received 24 January 2009; Accepted 6 August 2009 Recommended by Elena Litsyn We obtain the general solution and the generalized Ulam-Hyers stability of the mixed type cubic and quartic functional equation f x 2yf x −2y 4 f x yf x −y−24f y−6f x3f 2y in quasi-Banach spaces. Copyright q 2009 M. Eshaghi Gordji 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 We recall some basic facts concerning quasiBanach space. A quasinorm is a real-valued function on X satisfying the following. 1 x ≥ 0 for all x ∈ X and x 0 if and only if x 0. 2 λ · x |λ| · x for all λ ∈ R and all x ∈ X. 3 There is a constant K ≥ 1 such that x y ≤ K x y for all x, y ∈ X. The pair X, · is called a quasinormed space if · is a quasinorm on X . A quasiBanach space is a complete quasinormed space. A quasinorm · is called a p-norm 0 p ≤ 1 if p p