]>
2011
4
1
ISSN 2008-1898
90
TERNARY JORDAN HOMOMORPHISMS IN \(C^*\)-TERNARY ALGEBRAS
TERNARY JORDAN HOMOMORPHISMS IN \(C^*\)-TERNARY ALGEBRAS
en
en
In this note, we prove the Hyers-Ulam-Rassias stability of Jordan
homomorphisms in \(C^*\)-ternary algebras for the following generalized Cauchy-
Jensen additive mapping:
\[rf( \frac{s \Sigma^p_{ j=1} x_j + t \Sigma^d_{ j=1} x_j}{ r} ) = s \Sigma^p_{ j=1} f(x_j) + t \Sigma^d_{ j=1} f(x_j)\]
and generalize some results concerning this functional equation.
1
10
S.
KABOLI GHARETAPEH
Department of Mathematics
Payame Noor University, Mashhad Branch
Iran
simin.kaboli@gmail.com
MADJID
ESHAGHI GORDJI
Department of Mathematics
Semnan University
Iran
madjid.eshaghi@gmail.com
M. B.
GHAEMI
Department of Mathematics
Iran University of Science and Technology
Iran
mghaemi@iust.ac.ir
E.
RASHIDI
Department of Mathematics
Semnan University
Iran
ehsanerashidi@gmail.com
Hyers-Ulam-Rassias stability
\(C^*\)-ternary algebra.
Article.1.pdf
[
[1]
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##[3]
T. Aoki, On the stability of the linear transformationin Banach spaces, J. Math. Soc. Japan , 2 (1950), 64-66
##[4]
P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. , 27 (1984), 76-86
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S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59-64
##[6]
S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, London (2002)
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A. Ebadian, A. Najati, M. Eshaghi Gordji, On approximate additive-quartic and quadratic-cubic functional equations in two variables on abelian groups, Results. Math, DOI 10.1007/s00025-010-0018-4 (2010)
##[8]
M. Eshaghi Gordji, M. B. Ghaemi, S. Kaboli Gharetapeh, S. Shams, A. Ebadian , On the stability of \(J^*\)-derivations, Journal of Geometry and Physics, 60 (2010), 454-459
##[9]
M. Eshaghi Gordji, S. Kaboli Gharetapeh, T. Karimi , E. Rashidi, M. Aghaei, Ternary Jordan derivations on \(C^*\)-ternary algebras, Journal of Computational Analysis and Applications, 12 (2010), 463-470
##[10]
M. Eshaghi Gordji, S. Kaboli-Gharetapeh, C. Park, S. Zolfaghri, Stability of an additive-cubic-quartic functional equation, Advances in Difference Equations, Article ID 395693, 2009 (2009), 1-20
##[11]
M. Eshaghi Gordji, S. Kaboli Gharetapeh, J. M. Rassias, S. Zolfaghari , Solution and stability of a mixed type additive, quadratic and cubic functional equation, Advances in difference equations, Article ID 826130, 2009 (2009), 1-17
##[12]
M. Eshaghi Gordji, T. Karimi, S. Kaboli Gharetapeh, Approximately n-Jordan homomorphisms on Banach algebras, J. Ineq. Appl. Article ID 870843, 2009 (2009), 1-8
##[13]
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
##[14]
M. Eshaghi Gordji, H. Khodaei, On the Generalized Hyers-Ulam-Rassias Stability of Quadratic Functional Equations, Abstract and Applied Analysis, Article ID 923476, 2009 (2009), 1-11
##[15]
M. Eshaghi Gordji, A. Najati, Approximately \(J^*\)-homomorphisms: A fixed point approach, Journal of Geometry and Physics, 60 (2010), 809-814
##[16]
Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci., 14 (1991), 431-434
##[17]
P. Gavruta , A generalization of the Hyers Ulam Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436
##[18]
D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27 (1941), 222-224
##[19]
D. H. Hyers, G. Isac, T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel (1998)
##[20]
H. Khodaei, Th. M. Rassias, Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl., 1 (2010), 22-41
##[21]
K. W. Jun, H. M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl., 274 (2002), 867-878
##[22]
K. W. Jun, H. M. Kim, I. S. Chang, On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation, J. Comput. Anal. Appl., 7 (2005), 21-33
##[23]
S.-M. Jung, Hyers-Ulam-Rassias stability of Jensen;s equation and its application, Proc. Amer. Math. Soc., 126 (1998), 3137-3143
##[24]
S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press Inc., Palm Harbor, Florida (2001)
##[25]
S.-M. Jung, Stability of the quadratic equation of Pexider type, Abh. Math. Sem. Univ. Hamburg, 70 (2000), 175-190
##[26]
A. Najati, C. Park, Homomorphisms and derivations \(C^*\)-ternary algebras, , (preprint), -
##[27]
C. Park, Lie *-homomorphisms between Lie \(C^*\)-algebras and Lie *-derivations on Lie \(C^*\)-algebras, J. Math. Anal. Appl., 293 (2004), 419-434
##[28]
C. Park, Homomorphisms between Lie \(JC^*\)-algebras and Cauchy-Rassias stability of Lie \(JC^*\)- algebra derivations, J. Lie Theory, 15 (2005), 393-414
##[29]
C. Park, Homomorphisms between Poisson \(JC^*\)-algebras, Bull. Braz. Math. Soc., 36 (2005), 79-97
##[30]
C. Park, Isomorphisms between \(C^*\)-ternary algebras, J. Math. Phys., Article ID 103512 (2006)
##[31]
C. Park, Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between \(C^*\)-algebras, Bull. Belgian Math. Soc.-Simon Stevin, 13 (2006), 619-631
##[32]
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal., 46 (1982), 126-130
##[33]
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sc. Math., 108 (1984), 445-446
##[34]
J. M. Rassias, On a new approximation of approximately linear mappings by linear mappings, Discuss. Math., 7 (1985), 193-196
##[35]
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. , 72 (1978), 297-300
##[36]
Th. M. Rassias, New characterization of inner product spaces, Bull. Sci. Math., 108 (1984), 95-99
##[37]
Th. M. Rassias, P. ·Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc., 114 (1992), 989-993
##[38]
Th. M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Mayh. Anal. Appl., 246 (2000), 352-378
##[39]
Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251 (2000), 264-284
##[40]
Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23-130
##[41]
Th. M. Rassias , Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordecht, Boston and London (2003)
##[42]
F. Skof, Local properties and approximations of operators, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129
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L. Takhtajan, On foundation of the generalized Nambu mechanics, Comm. Math. Phys., 160 (1994), 295-315
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S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Editions. Wiley, New York (1964)
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]
STABILITY OF THE LOBACEVSKI EQUATION
STABILITY OF THE LOBACEVSKI EQUATION
en
en
The aim of this paper is to investigate the superstability of the
Lobacevski equation
\[f (\frac{x + y}{ 2})^2 = f(x)f(y),\]
which is bounded by the unknown functions \(\varphi(x)\) or \(\varphi(y)\). The obtained result
is a generalization of P. G·avruta's result in 1994.
11
18
GWANG HUI
KIM
Department of Mathematics
Kangnam University
Republic of Korea
ghkim@kangnam.ac.kr
Hyers-Ulam-Rassias stability
superstability
Lobacevski equation
d'Alembert functional equation
sine functional equation.
Article.2.pdf
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[1]
T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japen, 2 (1950), 64-66
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R. Badora, On the stability of cosine functional equation, Rocznik Naukowo-Dydak., Prace Mat., 15 (1998), 1-14
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R. Badora, R. Ger, On some trigonometric functional inequalities, Functional Equations- Results and Advances, (2002), 3-15
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P. W. Cholewa, The stability of the sine equation, Proc. Amer. Math. Soc., 88 (1983), 631-634
##[8]
G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Mathematicae. , (), -
##[9]
P. Gavruta, A Generalization of the Hyers-Ulam-Rassias Stability of Approximately Additive Mapping, Journal of Mathematical Analysis and Applications , 184 (1994), 431-436
##[10]
P. Gavruta, On the stability of some functional equation, Stability of Mappings of Hyers- Ulam Type, Hardronic Press, (1994), 93-98
##[11]
R. Ger, Superstability is not natural, Roczik Naukowo-Dydaktyczny WSP w Krakowie, Prace Mat., 159 (1993), 109-123
##[12]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA., 27 (1941), 222-224
##[13]
Pl. Kannappan, G. H. Kim, On the stability of the generalized cosine functional equations, Ann. Acad. Paedagog. Crac. Stud. Math., 1 (2001), 49-58
##[14]
G. H. Kim, A generalization of Hyers-Ulam-Rassias Stability of the G-Functional Equation, Math. Ineq. & Appl., 10 (2007), 351-58
##[15]
, The Stability of the d'Alembert and Jensen type functional equations, Jour. Math. Anal & Appl., 325 (2007), 237-248
##[16]
, A Stability of the generalized sine functional equations, Jour. Math. Anal & Appl., 331 (2007), 886-894
##[17]
K. W. Jun, Y. H. Lee, On The Hyers-Ulam-Rassias Stability of a Pexiderizes Mixes Type Quadratic functional equation, J. Chungcheong Math. Soc., 20 (2007), 117-137
##[18]
, On the Hyers-Ulam-Rassias stability of functional equations in n-variables, Jour. Math. Anal & Appl., 299 (2004), 375-391
##[19]
, A generlization of the Hyers- Ulam-Rassia Stability of the beta functional equation, Publ. Mathematics, 59 (2001), 111-119
##[20]
Y. W. Lee, K. S. Ji, Modified Hyers-Ulam-Rassias Stability of functional equations with square-symmetric Operation, Comm. Korean Math. Soc., 16 (2001), 211-223
##[21]
, On the Modified Hyers-Ulam-Rassias stability of the equation \(f(x^2 -y^2 +rxy) = f(x^2) - f(y^2) + rf(xy)\), J. Chungcheong Math. Soc., 10 (1997), 109-116
##[22]
Th. M. Rassias , On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. , 72 (1978), 297-300
##[23]
, On a Modified Hyers-Ulam Sequence, Jour. Math. Anal & Appl. , 158 (1991), 106-113
##[24]
P. Semrl, On the Behavior of Mappings Which Do Not Satisfy Hyers-Ulam Stability, Proc. Amer. Math. Soc., 114 (1992), 989-993
##[25]
, On the Hyers-Ulam Stability of Linear Mappings, Jour. Math. Anal & Appl., 173 (1993), 325-338
##[26]
, Th. M. Rassias, On the stability of functional equations in Banach spaces, Jour. Math. Anal & Appl., 251 (2000), 264-284
##[27]
, The Problem of S. M. Ulam for Approximately Multiplicative Mappings, Jour. Math. Anal & Appl., 246 (2000), 352-378
##[28]
S. M. Ulam, Problems in Modern Mathematics, Chap. VI, Science editions, Wiley, New York (1964)
]
ISOMORPHISMS AND GENERALIZED DERIVATIONS IN PROPER \(CQ^*\)-ALGEBRAS
ISOMORPHISMS AND GENERALIZED DERIVATIONS IN PROPER \(CQ^*\)-ALGEBRAS
en
en
In this paper, we prove the Hyers-Ulam-Rassias stability of homomorphisms in proper \(CQ^*\)-algebras and of generalized derivations on proper
\(CQ^*\)-algebras for the following Cauchy-Jensen additive mappings:
\[f (\frac{ x + y + z}{ 2 }) + f (\frac{ x - y + z}{ 2}) = f(x) + f(z),\]
\[f (\frac{ x + y + z}{ 2 }) - f (\frac{ x - y + z}{ 2}) = f(y),\]
\[2f (\frac{ x + y + z}{ 2 }) = f(x)+f(y)+f(z),\]
which were introduced and investigated in [3, 30].
This is applied to investigate isomorphisms in proper \(CQ^*\)-algebras.
19
36
CHOONKIL
PARK
Department of Mathematics
Research Institute for Natural Sciences, Hanyang University
South Korea
baak@hanyang.ac.kr
DEOK-HOON
BOO
Department of Mathematics
Chungnam National University
South Korea
dhboo@cnu.ac.kr
Hyers-Ulam-Rassias stability
Cauchy-Jensen functional equation
proper \(CQ^*\)-algebra isomorphism
generalized derivation.
Article.3.pdf
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[1]
J. P. Antoine, A. Inoue, C. Trapani, \(O^*\)-dynamical systems and *-derivations of unbounded operator algebras, Math. Nachr. , 204 (1999), 5-28
##[2]
J. P. Antoine, A. Inoue, C. Trapani, Partial *-Algebras and Their Operator Realizations, Kluwer, Dordrecht (2002)
##[3]
C. Baak, Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces , Acta Math. Sinica., 22 (2006), 1789-1796
##[4]
F. Bagarello, Applications of topological *-algebras of unbounded operators, J. Math. Phys., 39 (1998), 6091-6105
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F. Bagarello, A. Inoue, C. Trapani, Some classes of topological quasi *-algebras, Proc. Amer. Math. Soc., 129 (2001), 2973-2980
##[6]
F. Bagarello, A. Inoue, C. Trapani, *-Derivations of quasi-*-algebras, Internat. J. Math. Math. Sci., 21 (2004), 1077-1096
##[7]
F. Bagarello, A. Inoue, C. Trapani, Exponentiating derivations of quasi-*-algebras: possible approaches and applications, Internat. J. Math. Math. Sci., 2005 (2005), 2805-2820
##[8]
F. Bagarello, C. Trapani, States and representations of \(CQ^*\)-algebras, Ann. Inst. H. Poincare, 61 (1994), 103-133
##[9]
F. Bagarello, C. Trapani, \(CQ^*\)-algebras: structure properties, Publ. RIMS Kyoto Univ., 32 (1996), 85-116
##[10]
S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, London, Singapore and Hong Kong (2002)
##[11]
S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Palm Harbor, Florida (2003)
##[12]
R. Farokhzad Rostami, S. A. R. Hosseinioun, Perturbations of Jordan higher derivations in Banach ternary algebras: An alternative fixed point approach, Internat. J. Nonlinear Anal. Appl., 1 (2010), 42-53
##[13]
R. J. Fleming, J. E. Jamison, Isometries on Banach Spaces: Function Spaces, Monographs and Surveys in Pure and Applied Mathematics Vol. 129, Chapman & Hall/CRC, Boca Raton, London, New York and Washington D.C. (2003)
##[14]
Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. , 14 (1991), 431-434
##[15]
P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436
##[16]
N. Ghobadipour, C. Park, Cubic-quartic functional equations in fuzzy normed spaces, Internat. J. Nonlinear Anal. Appl., 1 (2010), 12-21
##[17]
M. E. Gordji, S. K. Gharetapeh, J. M. Rassias, S. Zolfaghari, Solution and stability of a mixed type additive, quadratic and cubic functional equation, Advances in Difference Equations , Art. ID 826130 (2009)
##[18]
M. E. Gordji, J. M. Rassias, N. Ghobadipour, Generalized Hyers-Ulam stability of generalized (N;K)-derivations, Abstract and Applied Analysis , Art. ID 437931 (2009)
##[19]
M. E. Gordji, S. Zolfaghari, J. M. Rassias, M. B. Savadkouhi, Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces, Abstract and Applied Analysis, Art. ID 417473 (2009)
##[20]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224
##[21]
D. H. Hyers, G. Isac, Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel (1998)
##[22]
D. H. Hyers, Th. M. Rassias, Approximate homomorphisms , Aequationes Math., 44 (1992), 125-153
##[23]
S. Jung, Hyers-Ulam-Rassias stability of Jensen's equation and its application, Proc. Amer. Math. Soc., 126 (1998), 3137-3143
##[24]
S. Jung, J. M. Rassias, A fixed point approach to the stability of a functional equation of the spiral of Theodorus, Fixed Point Theory and Applications, Art. ID 945010 (2008)
##[25]
H. Khodaei, Th. Rassias, Approximately generalized additive functions in several variables, Internat. J. Nonlinear Anal. Appl., 1 (2010), 22-41
##[26]
C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl., 275 (2002), 711-720
##[27]
C. Park, Homomorphisms between Poisson \(JC^*\)-algebras, Bull. Braz. Math. Soc., 36 (2005), 79-97
##[28]
C. Park, Homomorphisms between Lie \(JC^*\)-algebras and Cauchy-Rassias stability of Lie \(JC^*\)-algebra derivations, J. Lie Theory, 15 (2005), 393-414
##[29]
C. Park, Isomorphisms between unital \(C^*\)-algebras, J. Math. Anal. Appl., 307 (2005), 753-762
##[30]
C. Park, Isomorphisms between \(C^*\)-ternary algebras, J. Math. Phys., Art. ID 103512 (2006)
##[31]
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal., 46 (1982), 126-130
##[32]
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math., 108 (1984), 445-446
##[33]
J. M. Rassias, Solution of a problem of Ulam , J. Approx. Theory, 57 (1989), 268-273
##[34]
J. M. Rassias, Solution of a stability problem of Ulam, Discuss. Math., 12 (1992), 95-103
##[35]
J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnik Matem- aticki, 34 (1999), 243-252
##[36]
J. M. Rassias , Solution of the Ulam stability problem for cubic mappings, Glasnik Matematicki, 36 (2001), 63-72
##[37]
J. M. Rassias, M. J. Rassias , Asymptotic behavior of alternative Jensen and Jensen type functional equations, Bull. Sci. Math. , 129 (2005), 545-558
##[38]
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300
##[39]
Th. M. Rassias, Problem 16, Report of the 27th International Symp. on Functional Equations, Aequationes Math., 39 (1990), 292-293
##[40]
Th. M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl., 246 (2000), 352-378
##[41]
Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251 (2000), 264-284
##[42]
Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23-130
##[43]
Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London (2003)
##[44]
Th. M. Rassias, P. ·Semrl , On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl., 173 (1993), 325-338
##[45]
F. Skof, Proprietµa locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129
##[46]
C. Trapani, Quasi-*-algebras of operators and their applications, Rev. Math. Phys., 7 (1995), 1303-1332
##[47]
C. Trapani, Some seminorms on quasi-*-algebras, Studia Math., 158 (2003), 99-115
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C. Trapani, Bounded elements and spectrum in Banach quasi *-algebras, Studia Math., 172 (2006), 249-273
##[49]
S. M. Ulam, Problems in Modern Mathematics, Wiley, New York (1960)
]
RANDOM STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS A FIXED POINT APPROACH
RANDOM STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS A FIXED POINT APPROACH
en
en
Using the fixed point method, we prove the generalized Hyers-
Ulam stability of the following quadratic functional equations
\[cf (\sum^n_{ i=1} x_i) + \sum^n_{ j=2} f (\sum^n_{ i=1} x_i - (n + c - 1)x_j)\\
= (n + c - 1)(f(x_1) + c \sum^n _{i=2} f(x_i) + \sum^n_{ i<j,j=3} (\sum^{n-1}_{ i=2} f(x_i - x_j) )),\\
Q(\sum^n _{i=1} d_ix_i ) + \sum_{1\leq i<j\leq n} d_id_jQ(x_i - x_j) =(\sum^n_{ i=1} d_i)(\sum^n_{ i=1} d_iQ(x_i))\]
in random Banach spaces.
37
49
SEUNG WON
SCHIN
Seoul Science High School
Republic of Korea
maplemenia@naver.com
DOHYEONG
KI
Seoul Science High School
Republic of Korea
wooki7098@naver.com
JAEWON
CHANG
Seoul Science High School
Republic of Korea
jjwjjw9595@naver.com
MIN JUNE
KIM
Seoul Science High School
Republic of Korea
frigen@naver.com
random Banach space
fixed point
quadratic functional equation
generalized Hyers-Ulam stability.
Article.4.pdf
[
[1]
J. Aczel, J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge (1989)
##[2]
T. Aoki , On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66
##[3]
L. Cădariu, V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math., (2003)
##[4]
L. Cădariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber., 346 (2004), 43-52
##[5]
L. Cădariu, V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications, Art. ID 749392 (2008)
##[6]
P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76-86
##[7]
S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59-64
##[8]
P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong , Singapore and London (2002)
##[9]
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
##[10]
P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436
##[11]
M. E. Gordji, M. B. Ghaemi, H. Hajani , Generalized Hyers-Ulam-Rassias theorem in Menger probabilistic normed spaces, Discrete Dynamics in Nature and Society, Art. ID 162371 (2010)
##[12]
O. Hadžić, E. Pap, Fixed Point Theory in PM Spaces, Kluwer Academic Publishers, Dordrecht (2001)
##[13]
O. Hadžić, E. Pap, M. Budincević, Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces, Kybernetica, 38 (2002), 363-381
##[14]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224
##[15]
D. H. Hyers, G. Isac, Th. M. Rassias, Stability of Functional Equations in Several Variables, BirkhÄauser, Basel (1998)
##[16]
G. Isac, Th. M. Rassias, Stability of \(\psi\)-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci., 19 (1996), 219-228
##[17]
S. Jung , Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida (2001)
##[18]
D. Miheţ, The fixed point method for fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, 160 (2009), 1663-1667
##[19]
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
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]
ON THE STABILITY OF SOME QUADRATIC FUNCTIONAL EQUATION
ON THE STABILITY OF SOME QUADRATIC FUNCTIONAL EQUATION
en
en
In this paper we establish the general solution of the functional
equation which is closely associated with the quadratic functional equation
and we investigate the Hyers-Ulam-Rassias stability of this equation in Banach
spaces.
50
59
M.
ADAM
Department of Mathematics and informatics
School of Occupational Safety of Katowice
Poland
madam@wszop.edu.pl
Quadratic functional equation
Hyers-Ulam-Rassias stability.
Article.5.pdf
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]
APPROXIMATELY PARTIAL TERNARY QUADRATIC DERIVATIONS ON BANACH TERNARY ALGEBRAS
APPROXIMATELY PARTIAL TERNARY QUADRATIC DERIVATIONS ON BANACH TERNARY ALGEBRAS
en
en
Let \(A_1,A_2,...,A_n\) be normed ternary algebras over the complex
field \(\mathbb{C}\) and let \(B\) be a Banach ternary algebra over \(\mathbb{C}\). A mapping \(\delta_k\) from
\(A_1 \times ...\times A_n\) into \(B\) is called a k-th partial ternary quadratic derivation if
there exists a mapping \(g_k : A_k \rightarrow B\) such that
\[\delta_k(x_1,..., [a_kb_kc_k],..., x_n) =[g_k(a_k)g_k(b_k)\delta_k(x_1 ,..., c_k,..., xn)]
+ [g_k(a_k)\delta_k(x_1,..., b_k,..., x_n)g_k(c_k)]
+ [\delta_k(x_1,...,a_k,..., x_n)g_k(b_k)g_k(c_k)]\]
and
\[\delta_k(x_1,..., a_k + b_k,..., x_n) + \delta_k(x_1,... a_k - b_k,..., x_n)
= 2\delta_k(x_1,..., a_k,..., x_n) + 2\delta_k(x_1,...,b_k,..., x_n)\]
for all \(a_k, b_k, c_k \in A_k\) and all \(x_i \in A_i (i \neq k)\). We prove the Hyers-Ulam-
Rassias stability of the partial ternary quadratic derivations in Banach ternary
algebras.
60
69
A.
JAVADIAN
Department of Physics
Semnan University
Iran
M.
ESHAGHI GORDJI
Department of Mathematics
Semnan University
Iran
madjid.eshaghi@gmail.com
M.
BAVAND SAVADKOUHI
Department of Mathematics
Semnan University
Iran
bavand.m@gmail.com
Hyers-Ulam-Rassias stability
Banach ternary algebra
Partial ternary quadratic derivation.
Article.6.pdf
[
[1]
S. Abbaszadeh, Intuitionistic fuzzy stability of a quadratic and quartic functional equation, Int. J. Nonlinear Anal. Appl. , 1 (2010), 100-124
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R. Badora, On approximate derivations, Math. Inequal. Appl. , 9 (2006), 167-173
##[3]
M. Bavand Savadkouhi, M. Eshaghi Gordji, J. M. Rassias, N. Ghobadipour, Approximate ternary Jordan derivations on Banach ternary algebras, J. Math. Phys, 50 (2009), 1-9
##[4]
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##[5]
H. Chu, S. Koo, J. Park , Partial stabilities and partial derivations of n-variable functions, Nonlinear Anal.-TMA , (to appear), -
##[6]
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##[7]
A. Ebadian, A. Najati, M. E. Gordji, On approximate additive-quartic and quadratic- cubic functional equations in two variables on abelian groups, Results Math. , DOI 10.1007/s00025-010-0018-4 (2010)
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##[10]
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##[12]
M. Eshaghi Gordji, M. Bavand Savadkouhi, M. Bidkham, Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces, Journal of Computational Analysis and Applications, 12 (2010), 454-462
##[13]
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##[14]
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##[15]
M. Eshaghi Gordji, M. B. Ghaemi, S. Kaboli Gharetapeh, S. Shams, A. Ebadian, On the stability of \(J^*\)-derivations, Journal of Geometry and Physics. , 60(3) (2010), 454-459
##[16]
M. Eshaghi Gordji, M. Ghanifard, H. Khodaei, C. Park, A fixed point approach to the random stability of a functional equation driving from quartic and quadratic mappings, Discrete Dynamics in Nature and Society, Article ID: 670542. (2010)
##[17]
M. Eshaghi Gordji, N. Ghobadipour, Stability of (\(\alpha,\beta,\psi\))-derivations on Lie \(C^*\)-algebras, To appear in International Journal of Geometric Methods in Modern Physics , (IJGMMP), -
##[18]
M. Eshaghi Gordji, S. Kaboli Gharetapeh, T. Karimi , E. Rashidi, M. Aghaei , Ternary Jordan derivations on \(C^*\)-ternary algebras, Journal of Computational Analysis and Applications, 12 (2010), 463-470
##[19]
M. Eshaghi Gordji, S. Kaboli-Gharetapeh, C. Park, S. Zolfaghri , Stability of an additive-cubic-quartic functional equation, Advances in Difference Equations. Article ID 395693, (2009), 1-20
##[20]
M. Eshaghi Gordji, S. Kaboli Gharetapeh, J. M. Rassias, S. Zolfaghari, Solution and stability of a mixed type additive, quadratic and cubic functional equation, Advances in difference equations, Article ID 826130, doi:10.1155/2009/826130. , 2009 (2009), 1-17
##[21]
M. Eshaghi Gordji, H. Khodaei , On the Generalized Hyers-Ulam-Rassias Stability of Quadratic Functional Equations, Abs. Appl. Anal. Article ID 923476, (2009), 1-11
##[22]
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
##[23]
M. Eshaghi Gordji, M. S. Moslehian, A trick for investigation of approximate derivations, Math. Commun. , 15 (2010), 99-105
##[24]
M. Eshaghi Gordji, M. Ramezani, A. Ebadian, C. Park, Quadratic double centralizers and quadratic multipliers, Advances in Difference Equations , (in press), -
##[25]
M. Eshaghi Gordji, J. M. Rassias, N. Ghobadipour, Generalized Hyers-Ulam stability of the generalized (n; k)-derivations , Abs. Appl. Anal., Article ID 437931, 2009 (2009), 1-8
##[26]
R. Farokhzad, S. A. R. Hosseinioun, Perturbations of Jordan higher derivations in Banach ternary algebras: An alternative fixed point approach, Internat. J. Nonlinear Anal. Appl. , 1 (2010), 42-53
##[27]
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##[31]
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##[32]
G. Isac, Th. M. Rassias, On the Hyers-Ulam stability of \(\psi\)-additive mappings, J. Approx. Theory, 72 (1993), 131-137
##[33]
K. Jun, D. Park, Almost derivations on the Banach algebra \(C^n[0, 1]\), Bull. Korean Math. Soc. , 33 (1996), 359-366
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H. Khodaei, M. Kamyar , Fuzzy approximately additive mappings, Int. J. Nonlinear Anal. Appl. , 1 (2010), 44-53
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##[37]
C. Park, M. Eshaghi Gordji , Comment on Approximate ternary Jordan derivations on Banach ternary algebras [Bavand Savadkouhi et al. , J. Math. Phys. 50, 042303 (2009)], J. Math. Phys. , 51 (2010), 1-7
##[38]
C. Park, A. Najati , Generalized additive functional inequalities in Banach algebras, Int. J. Nonlinear Anal. Appl. , 1 (2010), 54-62
##[39]
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##[40]
F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129
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S. M. Ulam , Problems in modern mathematics, Chapter VI, science ed., Wiley, New York (1940)
]
JORDAN HOMOMORPHISMS IN PROPER \(JCQ^*\)-TRIPLES
JORDAN HOMOMORPHISMS IN PROPER \(JCQ^*\)-TRIPLES
en
en
In this paper, we investigate Jordan homomorphisms in proper
\(JCQ^*\)-triples associated with the generalized 3-variable Jesnsen functional
equation
\[rf(\frac{x + y + z}{ r} ) = f(x) + f(y) + f(z),\]
with \(r \in (0; 3) /\{1\}\). We moreover prove the Hyers-Ulam-Rassias stability of
Jordan homomorphisms in proper \(JCQ^*\)-triples.
70
81
S.
KABOLI GHARETAPEH
Department of Mathematics
Payame Noor University, Mashhad Branch
Iran
simin.kaboli@gmail.com
S.
TALEBI
Department of Mathematics
Payame Noor University, Mashhad Branch
Iran
talebis@pnu.ac.ir
CHOONKIL
PARK
Department of Mathematics
Research Institute for Natural Sciences, Hanyang University
Republic of Korea
baak@hanyang.ac.kr
MADJID
ESHAGHI GORDJI
Department of Mathematics
Semnan University
Iran
madjid.eshaghi@gmail.com
Hyers-Ulam-Rassias stability
proper \(JCQ^*\)-triple Jordan homomorphism.
Article.7.pdf
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HYERS-ULAM-RASSIAS STABILITY OF THE APOLLONIUS TYPE QUADRATIC MAPPING IN RN-SPACES
HYERS-ULAM-RASSIAS STABILITY OF THE APOLLONIUS TYPE QUADRATIC MAPPING IN RN-SPACES
en
en
Recently, in [5], Najati and Moradlou proved Hyers-Ulam-Rassias
stability of the following quadratic mapping of Apollonius type
\[Q(z - x) + Q(z - y) =\frac{ 1}{ 2}Q(x - y) + 2Q ( z -\frac{ x + y}{ 2})\]
in non-Archimedean space. In this paper we establish Hyers-Ulam-Rassias stability of this functional equation in random normed spaces by direct method
and fixed point method. The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias stability theorem that appeared in his paper: On
the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc.
72 (1978), 297-300.
82
91
H. AZADI
KENARY
Department of Mathematics, College of Science
Yasouj University
Iran
azadi@mail.yu.ac.ir
K.
SHAFAAT
Department of Mathematics, College of Science
Yasouj University
Iran
M.
SHAFEI
Department of Mathematics, College of Science
Yasouj University
Iran
G.
TAKBIRI
Department of Mathematics, College of Science
Yasouj University
Iran
Fixed point theory
Stability
Random normed space.
Article.8.pdf
[
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