# Nonlinear stability of a quadratic functional equation with complex involution

Archivum Mathematicum (2011)

• Volume: 047, Issue: 2, page 111-117
• ISSN: 0044-8753

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## Abstract

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Let $X,Y$ be complex vector spaces. Recently, Park and Th.M. Rassias showed that if a mapping $f:X\to Y$ satisfies $\begin{array}{c}\hfill f\left(x+iy\right)+f\left(x-iy\right)=2f\left(x\right)-2f\left(y\right)\end{array}$ for all $x$, $y\in X$, then the mapping $f:X\to Y$ satisfies $f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)+2f\left(y\right)$ for all $x$, $y\in X$. Furthermore, they proved the generalized Hyers-Ulam stability of the functional equation () in complex Banach spaces. In this paper, we will adopt the idea of Park and Th. M. Rassias to prove the stability of a quadratic functional equation with complex involution via fixed point method.

## How to cite

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abstract = {Let $X, Y$ be complex vector spaces. Recently, Park and Th.M. Rassias showed that if a mapping $f : X \rightarrow Y$ satisfies \begin\{eqnarray\} f(x+i y)+ f(x-iy) = 2 f(x) - 2f(y) \end\{eqnarray\} for all $x$, $y\in X$, then the mapping $f \colon X \rightarrow Y$ satisfies $f(x+y) + f(x-y) = 2 f(x) + 2 f(y)$ for all $x$, $y \in X$. Furthermore, they proved the generalized Hyers-Ulam stability of the functional equation () in complex Banach spaces. In this paper, we will adopt the idea of Park and Th. M. Rassias to prove the stability of a quadratic functional equation with complex involution via fixed point method.},
journal = {Archivum Mathematicum},
keywords = {quadratic mapping; fixed point; quadratic functional equation; generalized Hyers-Ulam stability; quadratic functional equation; generalized Hyers-Ulam stability; fixed point method},
language = {eng},
number = {2},
pages = {111-117},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Nonlinear stability of a quadratic functional equation with complex involution},
url = {http://eudml.org/doc/116539},
volume = {047},
year = {2011},
}

TY - JOUR
TI - Nonlinear stability of a quadratic functional equation with complex involution
JO - Archivum Mathematicum
PY - 2011
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 047
IS - 2
SP - 111
EP - 117
AB - Let $X, Y$ be complex vector spaces. Recently, Park and Th.M. Rassias showed that if a mapping $f : X \rightarrow Y$ satisfies \begin{eqnarray} f(x+i y)+ f(x-iy) = 2 f(x) - 2f(y) \end{eqnarray} for all $x$, $y\in X$, then the mapping $f \colon X \rightarrow Y$ satisfies $f(x+y) + f(x-y) = 2 f(x) + 2 f(y)$ for all $x$, $y \in X$. Furthermore, they proved the generalized Hyers-Ulam stability of the functional equation () in complex Banach spaces. In this paper, we will adopt the idea of Park and Th. M. Rassias to prove the stability of a quadratic functional equation with complex involution via fixed point method.
LA - eng
KW - quadratic mapping; fixed point; quadratic functional equation; generalized Hyers-Ulam stability; quadratic functional equation; generalized Hyers-Ulam stability; fixed point method
UR - http://eudml.org/doc/116539
ER -

## References

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1. Cădariu, L., Radu, V., Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (1) (2003), 7 pp, Art. ID 4. (2003) Zbl1043.39010MR1965984
2. Cholewa, P. W., 10.1007/BF02192660, Aequationes Math. 27 (1984), 76–86. (1984) Zbl0549.39006MR0758860DOI10.1007/BF02192660
3. Czerwik, S., 10.1007/BF02941618, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. (1992) Zbl0779.39003MR1182841DOI10.1007/BF02941618
4. Diaz, J., Margolis, B., 10.1090/S0002-9904-1968-11933-0, Bull. Amer. Math. Soc. 74 (1968), 305–309. (1968) Zbl0157.29904MR0220267DOI10.1090/S0002-9904-1968-11933-0
5. Fauiziev, V., Sahoo, K. P., 10.1007/s12044-007-0003-3, Proc. Indian Acad. Sci. Math. Sci. 117 (2007), 31–48. (2007) MR2300676DOI10.1007/s12044-007-0003-3
6. Găvruta, P., Găvruta, L., A new method for the generalized Hyers–Ulam–Rassias stability, Int. J. Nonlinear Anal. Appl. 1 (2) (2010), 11–18. (2010)
7. Hyers, D. H., 10.1073/pnas.27.4.222, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. (1941) Zbl0061.26403MR0004076DOI10.1073/pnas.27.4.222
8. Hyers, D. H., Isac, G., Rassias, Th. M., Stability of Functional Equations in Several Variables, Birkhäser, Basel, 1998. (1998) Zbl0907.39025MR1639801
9. Isac, G., Rassias, Th. M., 10.1155/S0161171296000324, Internat. J. Math. Math. Sci. 19 (1996), 219–228. (1996) Zbl0843.47036MR1375983DOI10.1155/S0161171296000324
10. Jun, K., Kim, H., On the stability of an $n$–dimensional quadratic and additive functional equation, Math. Inequal. Appl. 9 (2006), 153–165. (2006) Zbl1093.39026MR2198559
11. Jung, S., Lee, Z., A fixed point approach to the stability of quadratic functional equation with involution, Fixed Point Theory and Applications (2008), Article ID 732086 (2008). (2008) Zbl1149.39022MR2415405
12. Khodaei, H., Rassias, Th. M., Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl. 1 (1) (2010), 22–41. (2010)
13. Mirzavaziri, M., Moslehian, M. S., 10.1007/s00574-006-0016-z, Bull. Brazil. Math. Soc. 37 (2006), 361–376. (2006) MR2267783DOI10.1007/s00574-006-0016-z
14. Park, C., Rassias, Th. M., Fixed points and generalized Hyers–Ulam stability of quadratic functional equations, J. Math. Inequal. 37 (2006), 515–528. (2006) MR2408405
15. Radu, V., The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. (2003) Zbl1051.39031MR2031824
16. Rassias, Th. M., On the stability of the quadratic functional equation and its applications, Studia Univ. Babeş–Bolyai Math. XLIII (1998), 89–124. (1998) Zbl1009.39025MR1854544
17. Rassias, Th. M., 10.1023/A:1006499223572, Acta Appl. Math. 62 (1) (2000), 23–130. (2000) Zbl0981.39014MR1778016DOI10.1023/A:1006499223572
18. Skof, F., 10.1007/BF02924890, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. (1983) DOI10.1007/BF02924890
19. Ulam, S. M., Problems in Modern Mathematics, Wiley, New York, 1960. (1960) MR0280310

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