Abstract and Applied Analysis
Volume 2012 (2012), Article ID 392179, 17 pages
http://dx.doi.org/10.1155/2012/392179
Research Article

Approximation of Mixed-Type Functional Equations in Menger PN-Spaces

M. Eshaghi Gordji,1 H. Khodaei,1 Y. W. Lee,2 and G. H. Kim3

1Department of Mathematics, Semnan University, Semnan 35195-363, Iran
2Department of Computer Hacking, Information Security, Daejeon University, Youngwoondong Donggu Daejeon 300-716, Republic of Korea
3Department of Mathematics, Kangnam University, Yongin, Gyeonggi 446-702, Republic of Korea

Received 28 September 2011; Accepted 8 February 2012

Academic Editor: Agacik Zafer

Copyright © 2012 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.

Abstract

Let 𝑋 and 𝑌 be vector spaces. We show that a function 𝑓 ∶ 𝑋 → 𝑌 with 𝑓 ( 0 ) = 0 satisfies Δ 𝑓 ( 𝑥 1 , … , 𝑥 𝑛 ) = 0 for all 𝑥 1 , … , 𝑥 𝑛 ∈ 𝑋 , if and only if there exist functions 𝐶 ∶ 𝑋 × 𝑋 × 𝑋 → 𝑌 , 𝐵 ∶ 𝑋 × 𝑋 → 𝑌 and 𝐴 ∶ 𝑋 → 𝑌 such that 𝑓 ( 𝑥 ) = 𝐶 ( 𝑥 , 𝑥 , 𝑥 ) + 𝐵 ( 𝑥 , 𝑥 ) + 𝐴 ( 𝑥 ) for all 𝑥 ∈ 𝑋 , where the function 𝐶 is symmetric for each fixed one variable and is additive for fixed two variables, 𝐵 is symmetric bi-additive, 𝐴 is additive and Δ 𝑓 ( 𝑥 1 , … , 𝑥 𝑛 ) = ∑ 𝑛 𝑘 = 2 ( ∑ 𝑘 𝑖 1 = 2 ∑ 𝑖 𝑘 + 1 2 = 𝑖 1 + 1 ⋯ ∑ 𝑛 𝑖 𝑛 − 𝑘 + 1 = 𝑖 𝑛 − 𝑘 + 1 ∑ ) 𝑓 ( 𝑛 𝑖 = 1 , 𝑖 ≠ 𝑖 1 , … , 𝑖 𝑛 − 𝑘 + 1 𝑥 𝑖 − ∑ 𝑛 − 𝑘 + 1 𝑟 = 1 𝑥 𝑖 𝑟 ) + ∑ 𝑓 ( 𝑛 𝑖 = 1 𝑥 𝑖 ) − 2 𝑛 − 2 ∑ 𝑛 𝑖 = 2 ( 𝑓 ( 𝑥 1 + 𝑥 𝑖 ) + 𝑓 ( 𝑥 1 − 𝑥 𝑖 ) ) + 2 𝑛 − 1 ( 𝑛 − 2 ) 𝑓 ( 𝑥 1 ) ( 𝑛 ∈ ℕ , 𝑛 ≥ 3 ) for all 𝑥 1 , … , 𝑥 𝑛 ∈ 𝑋 . Furthermore, we solve the stability problem for a given function 𝑓 satisfying Δ 𝑓 ( 𝑥 1 , … , 𝑥 𝑛 ) = 0 , in the Menger probabilistic normed spaces.