On a functional equation by Baak, Boo and Rassias

On a functional equation by Baak, Boo and Rassias In Baak et al. (J Math Anal Appl 314(1):150–161, 2006) the authors considered the functional equation $$\begin{aligned} r f\left( \frac{1}{r}\,\sum _{j=1}^{d}x_j\right)+ & {} \sum _{i(j)\in \{0,1\} \atop \sum _{1\le j\le d} i(j)=\ell }r f\left( \frac{1}{r}\,\sum _{j=1}^d (-1)^{i(j)}x_j\right) \\= & {} \left( {d-1\atopwithdelims ()\ell }-{d-1\atopwithdelims ()\ell -1} +1\right) \sum _{j=1}^{d} f(x_j) \end{aligned}$$ r f 1 r ∑ j = 1 d x j + ∑ i ( j ) ∈ { 0 , 1 } ∑ 1 ≤ j ≤ d i ( j ) = ℓ r f 1 r ∑ j = 1 d ( - 1 ) i ( j ) x j = d - 1 ℓ - d - 1 ℓ - 1 + 1 ∑ j = 1 d f ( x j ) where $$d,\ell \in \mathbb {N}$$ d , ℓ ∈ N , $$1<\ell <d/2$$ 1 < ℓ < d / 2 and $$r\in \mathbb {Q}{\setminus }\{0\}$$ r ∈ Q \ { 0 } . The authors determined all odd solutions $$f:X\rightarrow Y$$ f : X → Y for vector spaces X, Y over $$\mathbb {Q}$$ Q . In Oubbi (Can Math Bull 60:173–183, 2017) the author considered the same equation but now for arbitrary real $$r\not =0$$ r ≠ 0 and real vector spaces X, Y. Generalizing similar results from (J Math Anal Appl 314(1):150–161, 2006) he additionally investigates certain stability questions for the equation above, but as for that equation itself for odd approximate solutions only. The present paper deals with the general solution of the equation and the corresponding stability inequality. In particular it is shown that under certain circumstances non-odd solutions of the equation exist. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png aequationes mathematicae Springer Journals

On a functional equation by Baak, Boo and Rassias

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Analysis; Combinatorics
ISSN
0001-9054
eISSN
1420-8903
D.O.I.
10.1007/s00010-017-0534-3
Publisher site
See Article on Publisher Site

Abstract

In Baak et al. (J Math Anal Appl 314(1):150–161, 2006) the authors considered the functional equation $$\begin{aligned} r f\left( \frac{1}{r}\,\sum _{j=1}^{d}x_j\right)+ & {} \sum _{i(j)\in \{0,1\} \atop \sum _{1\le j\le d} i(j)=\ell }r f\left( \frac{1}{r}\,\sum _{j=1}^d (-1)^{i(j)}x_j\right) \\= & {} \left( {d-1\atopwithdelims ()\ell }-{d-1\atopwithdelims ()\ell -1} +1\right) \sum _{j=1}^{d} f(x_j) \end{aligned}$$ r f 1 r ∑ j = 1 d x j + ∑ i ( j ) ∈ { 0 , 1 } ∑ 1 ≤ j ≤ d i ( j ) = ℓ r f 1 r ∑ j = 1 d ( - 1 ) i ( j ) x j = d - 1 ℓ - d - 1 ℓ - 1 + 1 ∑ j = 1 d f ( x j ) where $$d,\ell \in \mathbb {N}$$ d , ℓ ∈ N , $$1<\ell <d/2$$ 1 < ℓ < d / 2 and $$r\in \mathbb {Q}{\setminus }\{0\}$$ r ∈ Q \ { 0 } . The authors determined all odd solutions $$f:X\rightarrow Y$$ f : X → Y for vector spaces X, Y over $$\mathbb {Q}$$ Q . In Oubbi (Can Math Bull 60:173–183, 2017) the author considered the same equation but now for arbitrary real $$r\not =0$$ r ≠ 0 and real vector spaces X, Y. Generalizing similar results from (J Math Anal Appl 314(1):150–161, 2006) he additionally investigates certain stability questions for the equation above, but as for that equation itself for odd approximate solutions only. The present paper deals with the general solution of the equation and the corresponding stability inequality. In particular it is shown that under certain circumstances non-odd solutions of the equation exist.

Journal

aequationes mathematicaeSpringer Journals

Published: Jan 24, 2018

References

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