GENERALIZED HYERS ULAM STABILITY OF AN AQCQ-FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN BANACH SPACES
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Authors
CHOONKIL PARK
- Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea.
MADJID ESHAGHI GORDJI
- Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran.
ABBAS NAJATI
- Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, 56199-11367, Iran.
Abstract
In this paper, we prove the generalized Hyers-Ulam stability of the following
additive-quadratic-cubic-quartic functional equation
\[f(x + 2y) + f(x - 2y) = 4f(x + y) + 4f(x - y) - 6f(x) + f(2y) + f(-2y) - 4f(y) - 4f(-y)\]
in non-Archimedean Banach spaces.
Share and Cite
ISRP Style
CHOONKIL PARK, MADJID ESHAGHI GORDJI, ABBAS NAJATI, GENERALIZED HYERS ULAM STABILITY OF AN AQCQ-FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN BANACH SPACES, Journal of Nonlinear Sciences and Applications, 3 (2010), no. 4, 272-281
AMA Style
PARK CHOONKIL, GORDJI MADJID ESHAGHI, NAJATI ABBAS, GENERALIZED HYERS ULAM STABILITY OF AN AQCQ-FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN BANACH SPACES. J. Nonlinear Sci. Appl. (2010); 3(4):272-281
Chicago/Turabian Style
PARK, CHOONKIL, GORDJI , MADJID ESHAGHI, NAJATI, ABBAS. "GENERALIZED HYERS ULAM STABILITY OF AN AQCQ-FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN BANACH SPACES." Journal of Nonlinear Sciences and Applications, 3, no. 4 (2010): 272-281
Keywords
- non-Archimedean Banach space
- additive-quadratic-cubic-quartic functional equation
- generalized Hyers-Ulam stability.
MSC
- 46S10
- 39B52
- 54E40
- 47S10
- 26E30
- 12J25
References
-
[1]
T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.
-
[2]
P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76-86.
-
[3]
S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg , 62 (1992), 59-64.
-
[4]
P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London (2002)
-
[5]
M. Eshaghi-Gordji, S. Abbaszadeh, C. Park, On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces, Journal of Inequalities and Applications, Art. ID 153084 (2009)
-
[6]
M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of cubic and quartic functional equations in non-Archimedean spaces, Acta Appl. Math., DOI 10.1007/s10440-009-9512-7 (2009)
-
[7]
M. Eshaghi-Gordji, S. Kaboli-Gharetapeh, C. Park, S. Zolfaghri, Stability of an additive-cubic- quartic functional equation, Advances in Difference Equations , Art. ID 395693 (2009)
-
[8]
M. Eshaghi Gordji, H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Analysis.-TMA , 71 (2009), 5629-5643.
-
[9]
P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. , 184 (1994), 431-436.
-
[10]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224.
-
[11]
D. H. Hyers, G. Isac, Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel (1998)
-
[12]
K. Jun, H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. , 274 (2002), 867-878.
-
[13]
S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida (2001)
-
[14]
H. Khodaei, Th. M. Rassias , Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl. , 1 (2010), 22-41.
-
[15]
S. Lee, S. Im, I. Hwang, Quartic functional equations, J. Math. Anal. Appl. , 307 (2005), 387-394.
-
[16]
F. Moradlou, H. Vaezi, G. M. Eskandani, Hyers-Ulam-Rassias stability of a quadartic and additive functional equation in quasi-Banach spaces, Mediterr. J. Math. , 6 (2009), 233-248.
-
[17]
M. S. Moslehian, Th. M. Rassias, Stability of functional equations in non-Archimedean spaces, Applicable Analysis and Discrete Mathematics. , 1 (2007), 325-334.
-
[18]
M. S. Moslehian, Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Non- linear Anal.-TMA , 69 (2008), 3405-3408.
-
[19]
C. Park, On an approximate automorphism on a \(C^*\)-algebra, Proc. Amer. Math. Soc. , 132 (2003), 1739-1745.
-
[20]
C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras, Bull. Sci. Math. , 132 (2008), 87-96.
-
[21]
C. Park, J. Cui, Generalized stability of \(C^*\)-ternary quadratic mappings, Abstract Appl. Anal., Art. ID 23282 (2007)
-
[22]
C. Park, A. Najati, Homomorphisms and derivations in \(C^*\)-algebras, Abstract Appl. Anal. , Art. ID 80630 (2007)
-
[23]
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. , 46 (1982), 126-130.
-
[24]
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. , 108 (1984), 445-446.
-
[25]
J. M. Rassias, Refined Hyers-Ulam approximation of approximately Jensen type mappings , Bull. Sci. Math. , 131 (2007), 89-98.
-
[26]
J. M. Rassias, M. J. Rassias, Asymptotic behavior of alternative Jensen and Jensen type functional equations, Bull. Sci. Math. , 129 (2005), 545-558.
-
[27]
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. , 72 (1978), 297-300.
-
[28]
Th. M. Rassias, Problem 16, Report of the 27th International Symp. on Functional Equations, Aequationes Math. , 39 (1990), 292-293
-
[29]
Th. M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babes-Bolyai XLIII , (1998), 89-124.
-
[30]
Th. M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl., 246 (2000), 352-378.
-
[31]
Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. , 251 (2000), 264-284.
-
[32]
Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. , 62 (2000), 23-130.
-
[33]
Th. M. Rassias, P. Šemrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. , 114 (1992), 989-993.
-
[34]
Th. M. Rassias, P. Šemrl , On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. , 173 (1993), 325-338.
-
[35]
Th. M. Rassias, K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. , 228 (1998), 234-253.
-
[36]
F. Skof, Proprietà locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano , 53 (1983), 113-129.
-
[37]
S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. , New York (1960)