Random stability of a functional equation associated with inner product: A fixed point approach
- Authors
- Park, Choonkil; Lee, Jung Rye; Shin, Dong Yun
- Issue Date
- Jan-2012
- Publisher
- Watam Press
- Keywords
- Additive mapping; Fixed point; Functional equation related to inner product space; Hyers-Ulam stability; Quadratic mapping; Random Banach space
- Citation
- Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, v.19, no.1, pp.65 - 79
- Indexed
- SCOPUS
- Journal Title
- Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis
- Volume
- 19
- Number
- 1
- Start Page
- 65
- End Page
- 79
- URI
- https://scholarworks.bwise.kr/hanyang/handle/2021.sw.hanyang/166471
- ISSN
- 1201-3390
- Abstract
- Th.M. Rassias [Bull. Sci. Math. 108 (1984), 95{99] proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed positive integer l2l12l∑ 2li= 1xi2+∑2li=1xi ? 12l∑2lj=1xj2=∑ 2li=1?xi? 2holds for all x1; ; x2l ? V. For the above equality, we can define the following functionalequation 2lf(12l∑2li=1xi)+∑2li=1fxi ? 12l∑2lj=1xj =∑2li=1f(xi); (1) whose solution is realized as the sum of an additive mapping and a quadratic mapping. Using fixed point method, we prove the Hyers-Ulam stability of the functional equation (1) in random Banach spaces.
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