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Tính ổn định Hyers-Ulam tổng quát cho một phương trình chức năng hỗn hợp trong không gian gần β-normed
Tóm tắt
Trong bài báo này, chúng tôi thiết lập nghiệm tổng quát và điều tra tính ổn định Hyers-Ulam tổng quát của phương trình chức năng hỗn hợp sau đây
$$f(\lambda x + y) + f(\lambda x - y) = f(x + y) + f(x - y) + (\lambda - 1)[(\lambda +2)f(x) + \lambda f(-x)],$$
$${(\lambda \in {\mathbb N}, \lambda \ne 1)}$$
trong không gian gần β-normed.
Từ khóa
#tính ổn định Hyers-Ulam #phương trình chức năng hỗn hợp #không gian gần β-normedTài liệu tham khảo
Aoki T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2, 64–66 (1950)
J. Aczél and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, 1989.
Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, RI, 2000.
H. X. Cao, J. R. Lv and J. M. Rassias, Superstability for generalized module left derivations and generalized module derivations on a banach module (I), Journal of Inequalities and Applications, Volume 2009, Art. ID 718020, 1-10
H. X. Cao, J. R. Lv and J. M. Rassias, Superstability for generalized module left derivations and generalized module derivations on a banach module(II), J. Pure. Appl. Math. 10 (2009), Issue 2, 1-8.
P.W. Cholewa, Remarks on the stability of functional equations. Aequationes Math. 27 76-86. (1984)
S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, New Jersey, London, Singapore, Hong Kong, 2002.
S. Czerwik, On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg 62 59-64 (1992)
G.Z. Eskandani, On the Hyers-Ulam-Rassias stability of an additive functional equation in quasi-Banach spaces. J. Math. Anal. Appl. 345 405-409 (2008)
G.Z. Eskandani and P. Găvruta, On the stability problem in quasi-Banach spaces, Nonlinear Funct. Anal. Appl. (to appear)
G.Z. Eskandani, H. Vaezi and Y.N. Dehghan, Stability of mixed additive and quadratic functional equation in non-Archimedean Banach modules, Taiwanese J. Math. (to appear)
P. Găvruta, On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings. J. Math. Anal. Appl. 261, 543-553 (2001)
P. Găvruta, An answer to question of John M. Rassias concerning the stability of Cauchy equation, Advanced in Equation and Inequality (1999) 67-71.
P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184 431-436 (1994)
A. Grabiec, The generalized Hyers-Ulam stability of a class of functional equations. Publ. Math. Debrecen 48, 217-235 (1996)
D. H. Hyers, On the stability of the linear functional equation. Proc. Nat. Acad. Sci. 27, 222-224 (1941)
D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998.
K. Jun and Y. Lee, On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality. Math. Inequal. Appl. 4, 93-118 (2001)
S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathimatical Analysis, Hadronic Press, Palm Harbor, 2001.
S.-M. Jung, Asymptotic properties of isometries, J. Math. Anal. Appl. 276 642-653 (2002)
S.-M. Jung, Stability of the quadratic equation of Pexider type. Abh. Math. Sem. Univ. Hamburg 70, 175-190 (2000)
S.-M. Jung, Quadratic functional equations of Pexider type, International Journal of Mathematics and Mathematical Sciences 24 (2000), no. 5, 351-359.
S.-M. Jung and P. K. Sahoo, Hyers-Ulam stability of the quadratic equation of Pexider type, Journal of the Korean Mathematical Society 38 (2001), no. 3, 645-656.
Kannappan Pl.: Quadratic functional equation and inner product spaces. Results Math. 27, 368–372 (1995)
F. Moradlou, H. Vaezi and G.Z. Eskandani, Hyers-Ulam-Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces. Mediterr. J. Math. 6 (2009), no. 2, 233-248.
C. Park, Homomorphisms between Poisson JC*-algebras. Bull. Braz. Math. Soc. 36, 79-97 (2005)
J.M. Rassias, On approximation of approximately linear mappings by linear mappings, Journal of Functional Analysis, vol. 46, no. 1, pp. 126-130, 1982.
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bulletin des Sciences Mathematiques, vol. 108, no. 4, pp. 445-446, 1984.
J. M. Rassias, Solution of a problem of Ulam, Journal of Approximation Theory, vol. 57, no. 3, pp. 268-273, 1989.
J.M. Rassias, Solution of a stability problem of Ulam, Discussiones Mathematicae, vol. 12, pp. 95-103, 1992.
J.M. Rassias and H.M. Kim, Generalized Hyers-Ulam stability for general additive functional equations in quasi-β-normed spaces, J. Math. Anal. Appl. 356, 302-309 (2009)
Th.M. Rassias, On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297-300 (1978)
Th.M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158, 106-113 (1991)
Th.M. Rassias and P. ̆Semrl, On the behaviour of mappings which do not satisfy HyersUlam stability. Proc. Amer. Math. Soc. 114, 989-993 (1992)
Th.M. Rassias and J. Tabor, What is left of Hyers-Ulam stability?. J. Natural Geometry, 1, 65-69 (1992)
K. Ravi, M. Arunkumar and J. M. Rassias, Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Intern. J. Math. Stat. 3 (A08)(2008), 36-46.
K. Ravi, J. M. Rassias, M. Arunkumar, and R. Kodandan, Stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation, J. Pure. Appl. Math. 10(2009), Issue 4, Article 114, 1-29.
S. Rolewicz, Metric Linear Spaces, PWN-Polish Sci. Publ., Warszawa, Reidel, Dordrecht, 1984.
M. B. Savadkouhi, M. E. Gordji, J. M. Rassias and N. Ghobadipour, Approximate ternary Jordan derivations on Banach ternary algebras, J. Math. Phys. 50 (2009), 042303: 1-9.
Skof F.: Local properties and approximations of operators. Rend. Sem. Mat. Fis. Milano 53, 113–129 (1983)
Ulam S.M.: A Collection of the Mathematical Problems, pp. 431–436. Interscience Publ., New York (1960)
