The Fixed Point Method for Fuzzy Approximation of a Functional Equation Associated with Inner Product Spaces

M. Eshaghi Gordji and H. Khodaei

Discrete Dynamics in Nature and Society, 2010, vol. 2010, 1-15

Abstract:

Th. M. Rassias (1984) proved that the norm defined over a real vector space 𠑋 is induced by an inner product if and only if for a fixed integer 𠑛 ≥ 2 , ∑ 𠑛 𠑖 = 1 ‖ 𠑥 𠑖 ∑ − ( 1 / 𠑛 ) 𠑛 𠑗 = 1 𠑥 𠑗 ‖ 2 = ∑ 𠑛 𠑖 = 1 ‖ 𠑥 𠑖 ‖ 2 ∑ − 𠑛 ‖ ( 1 / 𠑛 ) 𠑛 𠑖 = 1 𠑥 𠑖 ‖ 2 holds for all 𠑥 1 , … , 𠑥 𠑛 ∈ 𠑋 . The aim of this paper is to extend the applications of the fixed point alternative method to provide a fuzzy stability for the functional equation ∑ 𠑛 𠑖 = 1 𠑓 ( 𠑥 𠑖 ∑ − ( 1 / 𠑛 ) 𠑛 𠑗 = 1 𠑥 𠑗 ∑ ) = 𠑛 𠑖 = 1 𠑓 ( 𠑥 𠑖 ∑ ) − 𠑛 𠑓 ( ( 1 / 𠑛 ) 𠑛 𠑖 = 1 𠑥 𠑖 ) which is said to be a functional equation associated with inner product spaces.

Date: 2010
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnddns:140767

DOI: 10.1155/2010/140767

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