Fixed Points, Inner Product Spaces, and Functional Equationsopen access
- Authors
- Park, Choonkil
- Issue Date
- Jul-2010
- Publisher
- SPRINGER INTERNATIONAL PUBLISHING AG
- Citation
- FIXED POINT THEORY AND APPLICATIONS, v.2010, pp.1 - 14
- Indexed
- SCIE
SCOPUS
- Journal Title
- FIXED POINT THEORY AND APPLICATIONS
- Volume
- 2010
- Start Page
- 1
- End Page
- 14
- URI
- https://scholarworks.bwise.kr/hanyang/handle/2021.sw.hanyang/174462
- DOI
- 10.1155/2010/713675
- ISSN
- 1687-1820
- Abstract
- Rassias introduced the following equality Sigma(n)(i,j)=1 parallel to x(i) - x(j)parallel to(2) = 2n Sigma(n)(i=1) parallel to x(i)parallel to(2), Sigma(n)(i=1) x(i) = 0, for a fixed integer n >= 3. Let V, W be real vector spaces. It is shown that, if a mapping f : V -> W satisfies the following functional equation Sigma(n)(i,j-1) f(x(i) -x(j)) = 2n Sigma(n)(i-1) f(x(i)) for all x(1), ... , x(n) is an element of V with Sigma(n)(i-1) x(i) = 0, which is defined by the above equality, then the mapping f : V. W is realized as the sum of an additive mapping and a quadratic mapping. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.
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