Fixed points and quadratic rho-functional equations
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Authors
Choonkil Park
- Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea.
Sang Og Kim
- Department of Mathematics, Hallym University, Chuncheon 24252, Korea.
Abstract
In this paper, we solve the quadratic \(\rho\)-functional equations
\[f(x + y) + f(x - y) - 2f(x) - 2f(y) = \rho
\left(
2f
(\frac{x + y}{2})
+ 2f
(\frac{x - y}{2})
- f(x) - f(y)\right), \qquad (1)\]
where \(\rho\) is a fixed non-Archimedean number or a fixed real or complex number with \(\rho\neq 1;2\), and
\[2f
(\frac{x + y}{2})
+ 2f
(\frac{x - y}{2})
- f(x) - f(y) = \rho
\left(f(x + y) + f(x - y) - 2f(x) - 2f(y)\right); \qquad (2)\]
where \(\rho\) is a fixed non-Archimedean number or a fixed real or complex number with \(\rho\neq 1; \frac{-1}{2}\).
Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic \(\rho\)-functional equations
(1) and (2) in non-Archimedean Banach spaces and in Banach spaces.
Share and Cite
ISRP Style
Choonkil Park, Sang Og Kim, Fixed points and quadratic rho-functional equations, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 4, 1858--1871
AMA Style
Park Choonkil, Kim Sang Og, Fixed points and quadratic rho-functional equations. J. Nonlinear Sci. Appl. (2016); 9(4):1858--1871
Chicago/Turabian Style
Park, Choonkil, Kim, Sang Og. "Fixed points and quadratic rho-functional equations." Journal of Nonlinear Sciences and Applications, 9, no. 4 (2016): 1858--1871
Keywords
- Hyers-Ulam stability
- non-Archimedean normed space
- fixed point
- quadratic \(\rho\)-functional equation.
MSC
- 46S10
- 39B62
- 39B52
- 47H10
- 47S10
- 12J25
References
-
[1]
T. Aoki , On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.
-
[2]
N. Brillouet-Belluot, J. Brzdek, K. Cieplinski, On some recent developments in Ulam's type stability, Abstr. Appl. Anal., 2012 (2012), 41 pages.
-
[3]
J. Brzdek , Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar., 141 (2013), 58-67.
-
[4]
L. Cădariu, V. Radu , Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math., 2003 (2003), 7 pages.
-
[5]
L. Cădariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber., 346 (2004), 43-52.
-
[6]
L. Cădariu, V. Radu , Fixed point methods for the generalized stability of functional equations in a single variable , Fixed Point Theory Appl., 2008 (2008), 15 pages.
-
[7]
P. W. Cholewa, Remarks on the stability of functional equations , Aequationes Math., 27 (1984), 76-86.
-
[8]
S. Czerwik , Functional Equations and Inequalities in Several Variables, World Scientific Publishing Co., Inc., River Edge, NJ (2002)
-
[9]
S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Palm Harbor, Florida (2003)
-
[10]
J. Diaz, B. Margolis , A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305-309.
-
[11]
Z. Gajda , On stability of additive mappings, Internat. J. Math. Math. Sci., 14 (1991), 431-434.
-
[12]
P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.
-
[13]
A. Gilányi , Eine zur Parallelogrammgleichung äquivalente Ungleichung, Aequationes Math., 62 (2001), 303-309.
-
[14]
A. Gilányi, On a problem by K. Nikodem, Math. Inequal. Appl., 5 (2002), 707-710.
-
[15]
D. H. Hyers , On the stability of the linear functional equation , Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224.
-
[16]
D. H. Hyers, G. Isac, Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel (1998)
-
[17]
G. Isac, Th. M. Rassias, Stability of \(\psi\)-additive mappings: Appications to nonlinear analysis , Internat. J. Math. Math. Sci., 19 (1996), 219-228.
-
[18]
D. Miheţ, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl., 343 (2008), 567-572.
-
[19]
M. S. Moslehian, G. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal., 69 (2008), 3405-3408.
-
[20]
C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl., 2007 (2007), 15 pages.
-
[21]
C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory Appl., 2008 (2008), 9 pages.
-
[22]
V. Radu , The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91-96.
-
[23]
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.
-
[24]
Th. M. Rassias, P. Šemrl , On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl., 173 (1993), 325-338.
-
[25]
F. Skof , Propriet locali e approssimazione di operatori , Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129.
-
[26]
N. E. Tovmasian, V. H. Iritsyan, Some functional equations with shifts in a class of analytic functions, J. Contemp. Math. Anal., 35 (2000), 15-25.
-
[27]
N. E. Tovmasian, V. S. Zakarian, Some singular functional equations in a class of analytic functions, J. Contemp. Math. Anal., 35 (2000), 40-56.
-
[28]
S. M. Ulam, A Collection of the Mathematical Problems , Interscience Publ., New York (1960)