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APPROXIMATE QUADRATIC FUNCTIONAL EQUATION IN FELBIN’S TYPE NORMED LINEAR SPACES

Year 2013, Volume: 42 Issue: 5, 501 - 516, 01.05.2013

Fridoun Moradlou Saber Rezaee İldar Sadeqi

Abstract

In this paper, we prove the generalized Hyers–Ulam–Rassias stability ofquadratic functional equation in Felbin’s type normed linear spaces byusing the direct and fixed point methods. The concept of Hyers-UlamRassias stability originated from Th. M. Rassias’ stability theorem thatappeared in his paper: On the stability of the linear mapping in Banachspaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.

Keywords

Analysis fixed point fuzzy numbers fuzzy normed space Hyers–Ulam–Rassias stability

References

  • Acz´ el, J., Dhombres, J. Functional Equations in Several Variables, Cambridge University Press, 1989.
  • Aoki, T. On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2, 64–66, 1950.
  • Bag, T. and Samanta, S. K. Fixed point theorems in Felbin type fuzzy normed linear spaces, J. Fuzzy Math. 16(1), 243–260, 2008.
  • Bag, T., Samanta, S. K. A comparative study of fuzzy norms on a linear space, Fuzzy Sets Syst. 159(6), 670–684, 2008.
  • Baker, J. A. The stability of certain functional equations, Proc. Amer. Math. Soc. , 112, 729–732, 1991.
  • Borelli, C., Forti, G.L. On a general Hyers–Ulam stability result, Internat. J. Math. Math. Sci. 18, 229–236, 1995.
  • Bourgin, D.G. Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57, 223–237, 1951.
  • C˘ adariu, L., Radu, V. Fixed points and the stability of quadratic functional equations, An. Univ. Timi¸ soara, Ser. Mat.-Inform. 41, 25–48, 2003.
  • Czerwik, S. The stability of the quadratic functional equation, in: Th. M. Rassias, J. Tabor (Eds.), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 81–91, 1994. Czerwik, S. Functional Equations and Inequalities in Several Variables, World Scientific Publ.Co., New Jersey, London, Singapore, 2002.
  • Diaz, J., Margolis, B. A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74, 305–309, 1968.
  • Eskandani, G.Z. On the Hyers-ˆ uUlam-ˆ uRassias stability of an additive functional equation in quasi-Banach spaces, J. Math. Anal. Appl. 345, 405–409, 2008.
  • Faizev, V.A., Rassias, Th. M., Sahoo, P.K. The space of (Ψ, γ) - additive mappings on semigroups, Transactions of the Amer. Math. Soc. 354(11), 4455–4472, 2002.
  • Felbin, C. Finite dimensional fuzzy normed linear spaces, Fuzzy Sets Syst. 48, 239–248, 199 Gantner, T., Steinlage, R., Warren, R. Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62, 547–562, 1978.
  • Gˇ avruta, P. A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings J. Math. Anal. Appl. 184, 431–436, 1994.
  • Hoehle, U. Fuzzy real numbers as Dedekind cuts with respect to a multiple-valued logic, Fuzzy Sets Syst. 24, 263–278, 1987.
  • Hyers, D. H. On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27, 222–224, 1941.
  • Hyers, D. H., Isac, G., Rassias, Th. M. Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998.
  • Jun, K., Lee, Y. On the Hyers–Ulam–Rassias stability of a Pexiderized quadratic inequality, Math. Inequal. Appl. 4, 93–118, 2001.
  • Jun, K., Kim, H., Rassias, J. M. Extended Hyers–Ulam stability for Cauchy–Jensen mappings, J. Difference Equ. Appl. 13, 1139–1153, 2007.
  • Jung, S.- M. Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Inc., Florida, 2001.
  • Kim, G.H. On the stability of quadratic mapping in normed spaces, Internat. J. Math. Math. Sci. 25, 217–229, 2001.
  • Kaleva, O., Seikkala, S. On fuzzy metric spaces, Fuzzy Sets Syst. 12, 215–229, 1984.
  • Kaleva, O. The completion of fuzzy metric spaces, J. Math. Anal. Appl. 109, 194–198, 1985. Kaleva, O. A comment on the completion of fuzzy metric spaces, Fuzzy Sets Syst. 159(16), 2190–2192, 2008.
  • Kim, H.-M., Rassias, J.M. Generalization of Ulam stability problem for Euler–Lagrange quadratic mappings, J. Math. Anal. Appl. 336, 277–296, 2007.
  • Lowen, R. Fuzzy Set Theory, (Ch. 5 : Fuzzy real numbers), Kluwer, Dordrecht, 1996.
  • Mirmostafaee, A. K., Moslehian, M.S. Fuzzy version of Hyers-Ulam-Rassias theorem, Fuzzy sets Syst. 159(6), 720–729, 2008.
  • Moradlou, F. , Vaezi, H., Eskandani, G.Z. Hyers–Ulam–Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces. Mediterr. J. Math. 6(2), 233–248, 2009. Moradlou, F., Vaezi, H. and Park, C. Fixed Points and Stability of an Additive Functional Equation of n-Apollonius Type in C*-Algebras, Abstr. Appl. Anal. , Art. ID 672618, 2008. Moradlou, F., Vaezi, H., Eskandani, G. Z. Hyers–Ulam–Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces. Mediterr. J. Math. 6(2), 233–248, 2009. Moradlou, F. Additive functional inequalities and derivations on Hilbert C ∗ -modules, Glasg. Math. J. 55, 341–348, 2013.
  • Moradlou, F. Approximate Euler - Lagrange - Jensen type additive mapping in multi Banach spaces, A Fixed Point Approach, Commun. Korean Math. Soc. 28 (2), 319–333, 20 Moslehian, M. S., Rassias, Th. M. Generalized Hyers-Ulam stability of mappings on normed Lie triple systems, Math. Inequal. and Appl. 11(2), 371–380, 2008.
  • Moslehian, M. S. On the orthogonal stability of the Pexiderized quadratic equation, J. Difference Equ. Appl. 11, 999–1004, 2005.
  • Maligranda, L. A result of Tosio Aoki about a generalization of Hyers–Ulam stability of additive functions- a question of priority, Aequationes Math. 75, 289–296, 2008.
  • Nakmahachalasint, P. On the generalized Ulam–Gˇ avruta–Rassias stability of mixed–type linear and Euler–Lagrange–Rassias functional equations, Internat. J. Math. Math. Sci. 2007 (ID 63239), 1–10.
  • Park, C. On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275, 711–720, 2002.
  • Park, C. On an approximate automorphism on a C ∗ -algebra, Proc. Amer. Math. Soc. 132, 1739–1745, 2004.
  • Park, C. Fixed points and Hyers–Ulam–Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl., Art. ID 50175, 2007.
  • Park, C., Rassias, Th.M. Fixed points and stability of the Cauchy functional equation, Aust. J. Math. Anal. Appls. 6(1), Art.14, 9pp, 2009.
  • Pietrzyk, A. Stability of the Euler–Lagrange–Rassias functional equation, Demonstratio Mathematica 39, 523–530, 2006.
  • Popa, D. Functional inclusions on square-symmetric groupoids and Hyers-Ulam stability, Math. Inequal. Appl. 7, 419–428, 2004.
  • Prastaro, A., Rassias, Th.M. Ulam stability in geometry of PDE, Nonlinear Funct.Anal. Appl. 8, 259–278, 2003.
  • Radu, V. The fixed point alternative and stability of functional equations, Fixed Point Theory, Cluj-Napoca IV(1), 91–96, 2003.
  • Rassias, J.M. On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108, 445–446, 1984.
  • Rassias, J. M. Alternative contraction principle and alternative Jensen and Jensen type mappings, Internat. J. Appl. Math. Stat. 4, 1–10, 2006.
  • Rassias, Th. M. On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72, 297–300, 1978.
  • Rassias, Th. M. Problem 16; 2, Report of the 27 th International Symp. on Functional Equations, Aequationes Math. 39, 292–293, 1990.
  • Rassias, Th. M. The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246, 352–378, 2000.
  • Rassias, Th. M. On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251, 264–284, 2000.
  • Rassias, Th. M. On the stability of minimum points, Mathematica 45(68)(1), 93–104, 2003. [54] Rodabaugh, S. E. Fuzzy addition in the L-fuzzy real line, Fuzzy Sets Syst. 8, 39–51, 1982. [55] Sadeqi, I., Salehi, M. Fuzzy compacts operators and topological degree theory , Fuzzy Sets Syst. 160(9), 1277–1285, 2009.
  • Sadeqi, I., Moradlou, F., Salehi, M. On approximate Cauchy equation in Felbin’s type fuzzy normed linear spaces, to appear in Iran. J. Fuzzy Syst. 10 (3), 51–63.
  • 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, Interscience Publ. New York, 1960.
  • Xiao, J., Zhu, X. On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets Syst. 125, 153–161, 2002.
  • Xiao, J., Zhu, X. Topological degree theory and fixed point theorems in fuzzy normed space, Fuzzy Sets Syst. 147, 437–452, 2004.

Year 2013, Volume: 42 Issue: 5, 501 - 516, 01.05.2013

Fridoun Moradlou Saber Rezaee İldar Sadeqi

Abstract

References

  • Acz´ el, J., Dhombres, J. Functional Equations in Several Variables, Cambridge University Press, 1989.
  • Aoki, T. On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2, 64–66, 1950.
  • Bag, T. and Samanta, S. K. Fixed point theorems in Felbin type fuzzy normed linear spaces, J. Fuzzy Math. 16(1), 243–260, 2008.
  • Bag, T., Samanta, S. K. A comparative study of fuzzy norms on a linear space, Fuzzy Sets Syst. 159(6), 670–684, 2008.
  • Baker, J. A. The stability of certain functional equations, Proc. Amer. Math. Soc. , 112, 729–732, 1991.
  • Borelli, C., Forti, G.L. On a general Hyers–Ulam stability result, Internat. J. Math. Math. Sci. 18, 229–236, 1995.
  • Bourgin, D.G. Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57, 223–237, 1951.
  • C˘ adariu, L., Radu, V. Fixed points and the stability of quadratic functional equations, An. Univ. Timi¸ soara, Ser. Mat.-Inform. 41, 25–48, 2003.
  • Czerwik, S. The stability of the quadratic functional equation, in: Th. M. Rassias, J. Tabor (Eds.), Stability of Mappings of Hyers–Ulam Type, Hadronic Press, Florida, 81–91, 1994. Czerwik, S. Functional Equations and Inequalities in Several Variables, World Scientific Publ.Co., New Jersey, London, Singapore, 2002.
  • Diaz, J., Margolis, B. A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74, 305–309, 1968.
  • Eskandani, G.Z. On the Hyers-ˆ uUlam-ˆ uRassias stability of an additive functional equation in quasi-Banach spaces, J. Math. Anal. Appl. 345, 405–409, 2008.
  • Faizev, V.A., Rassias, Th. M., Sahoo, P.K. The space of (Ψ, γ) - additive mappings on semigroups, Transactions of the Amer. Math. Soc. 354(11), 4455–4472, 2002.
  • Felbin, C. Finite dimensional fuzzy normed linear spaces, Fuzzy Sets Syst. 48, 239–248, 199 Gantner, T., Steinlage, R., Warren, R. Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62, 547–562, 1978.
  • Gˇ avruta, P. A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings J. Math. Anal. Appl. 184, 431–436, 1994.
  • Hoehle, U. Fuzzy real numbers as Dedekind cuts with respect to a multiple-valued logic, Fuzzy Sets Syst. 24, 263–278, 1987.
  • Hyers, D. H. On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27, 222–224, 1941.
  • Hyers, D. H., Isac, G., Rassias, Th. M. Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998.
  • Jun, K., Lee, Y. On the Hyers–Ulam–Rassias stability of a Pexiderized quadratic inequality, Math. Inequal. Appl. 4, 93–118, 2001.
  • Jun, K., Kim, H., Rassias, J. M. Extended Hyers–Ulam stability for Cauchy–Jensen mappings, J. Difference Equ. Appl. 13, 1139–1153, 2007.
  • Jung, S.- M. Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Inc., Florida, 2001.
  • Kim, G.H. On the stability of quadratic mapping in normed spaces, Internat. J. Math. Math. Sci. 25, 217–229, 2001.
  • Kaleva, O., Seikkala, S. On fuzzy metric spaces, Fuzzy Sets Syst. 12, 215–229, 1984.
  • Kaleva, O. The completion of fuzzy metric spaces, J. Math. Anal. Appl. 109, 194–198, 1985. Kaleva, O. A comment on the completion of fuzzy metric spaces, Fuzzy Sets Syst. 159(16), 2190–2192, 2008.
  • Kim, H.-M., Rassias, J.M. Generalization of Ulam stability problem for Euler–Lagrange quadratic mappings, J. Math. Anal. Appl. 336, 277–296, 2007.
  • Lowen, R. Fuzzy Set Theory, (Ch. 5 : Fuzzy real numbers), Kluwer, Dordrecht, 1996.
  • Mirmostafaee, A. K., Moslehian, M.S. Fuzzy version of Hyers-Ulam-Rassias theorem, Fuzzy sets Syst. 159(6), 720–729, 2008.
  • Moradlou, F. , Vaezi, H., Eskandani, G.Z. Hyers–Ulam–Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces. Mediterr. J. Math. 6(2), 233–248, 2009. Moradlou, F., Vaezi, H. and Park, C. Fixed Points and Stability of an Additive Functional Equation of n-Apollonius Type in C*-Algebras, Abstr. Appl. Anal. , Art. ID 672618, 2008. Moradlou, F., Vaezi, H., Eskandani, G. Z. Hyers–Ulam–Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces. Mediterr. J. Math. 6(2), 233–248, 2009. Moradlou, F. Additive functional inequalities and derivations on Hilbert C ∗ -modules, Glasg. Math. J. 55, 341–348, 2013.
  • Moradlou, F. Approximate Euler - Lagrange - Jensen type additive mapping in multi Banach spaces, A Fixed Point Approach, Commun. Korean Math. Soc. 28 (2), 319–333, 20 Moslehian, M. S., Rassias, Th. M. Generalized Hyers-Ulam stability of mappings on normed Lie triple systems, Math. Inequal. and Appl. 11(2), 371–380, 2008.
  • Moslehian, M. S. On the orthogonal stability of the Pexiderized quadratic equation, J. Difference Equ. Appl. 11, 999–1004, 2005.
  • Maligranda, L. A result of Tosio Aoki about a generalization of Hyers–Ulam stability of additive functions- a question of priority, Aequationes Math. 75, 289–296, 2008.
  • Nakmahachalasint, P. On the generalized Ulam–Gˇ avruta–Rassias stability of mixed–type linear and Euler–Lagrange–Rassias functional equations, Internat. J. Math. Math. Sci. 2007 (ID 63239), 1–10.
  • Park, C. On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275, 711–720, 2002.
  • Park, C. On an approximate automorphism on a C ∗ -algebra, Proc. Amer. Math. Soc. 132, 1739–1745, 2004.
  • Park, C. Fixed points and Hyers–Ulam–Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl., Art. ID 50175, 2007.
  • Park, C., Rassias, Th.M. Fixed points and stability of the Cauchy functional equation, Aust. J. Math. Anal. Appls. 6(1), Art.14, 9pp, 2009.
  • Pietrzyk, A. Stability of the Euler–Lagrange–Rassias functional equation, Demonstratio Mathematica 39, 523–530, 2006.
  • Popa, D. Functional inclusions on square-symmetric groupoids and Hyers-Ulam stability, Math. Inequal. Appl. 7, 419–428, 2004.
  • Prastaro, A., Rassias, Th.M. Ulam stability in geometry of PDE, Nonlinear Funct.Anal. Appl. 8, 259–278, 2003.
  • Radu, V. The fixed point alternative and stability of functional equations, Fixed Point Theory, Cluj-Napoca IV(1), 91–96, 2003.
  • Rassias, J.M. On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108, 445–446, 1984.
  • Rassias, J. M. Alternative contraction principle and alternative Jensen and Jensen type mappings, Internat. J. Appl. Math. Stat. 4, 1–10, 2006.
  • Rassias, Th. M. On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72, 297–300, 1978.
  • Rassias, Th. M. Problem 16; 2, Report of the 27 th International Symp. on Functional Equations, Aequationes Math. 39, 292–293, 1990.
  • Rassias, Th. M. The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246, 352–378, 2000.
  • Rassias, Th. M. On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251, 264–284, 2000.
  • Rassias, Th. M. On the stability of minimum points, Mathematica 45(68)(1), 93–104, 2003. [54] Rodabaugh, S. E. Fuzzy addition in the L-fuzzy real line, Fuzzy Sets Syst. 8, 39–51, 1982. [55] Sadeqi, I., Salehi, M. Fuzzy compacts operators and topological degree theory , Fuzzy Sets Syst. 160(9), 1277–1285, 2009.
  • Sadeqi, I., Moradlou, F., Salehi, M. On approximate Cauchy equation in Felbin’s type fuzzy normed linear spaces, to appear in Iran. J. Fuzzy Syst. 10 (3), 51–63.
  • 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, Interscience Publ. New York, 1960.
  • Xiao, J., Zhu, X. On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets Syst. 125, 153–161, 2002.
  • Xiao, J., Zhu, X. Topological degree theory and fixed point theorems in fuzzy normed space, Fuzzy Sets Syst. 147, 437–452, 2004.
There are 51 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Fridoun Moradlou This is me

Saber Rezaee This is me

İldar Sadeqi This is me

Publication Date May 1, 2013
Published in Issue Year 2013 Volume: 42 Issue: 5

Cite

APA Moradlou, F., Rezaee, S., & Sadeqi, İ. (2013). APPROXIMATE QUADRATIC FUNCTIONAL EQUATION IN FELBIN’S TYPE NORMED LINEAR SPACES. Hacettepe Journal of Mathematics and Statistics, 42(5), 501-516.
AMA Moradlou F, Rezaee S, Sadeqi İ. APPROXIMATE QUADRATIC FUNCTIONAL EQUATION IN FELBIN’S TYPE NORMED LINEAR SPACES. Hacettepe Journal of Mathematics and Statistics. May 2013;42(5):501-516.
Chicago Moradlou, Fridoun, Saber Rezaee, and İldar Sadeqi. “APPROXIMATE QUADRATIC FUNCTIONAL EQUATION IN FELBIN’S TYPE NORMED LINEAR SPACES”. Hacettepe Journal of Mathematics and Statistics 42, no. 5 (May 2013): 501-16.
EndNote Moradlou F, Rezaee S, Sadeqi İ (May 1, 2013) APPROXIMATE QUADRATIC FUNCTIONAL EQUATION IN FELBIN’S TYPE NORMED LINEAR SPACES. Hacettepe Journal of Mathematics and Statistics 42 5 501–516.
IEEE F. Moradlou, S. Rezaee, and İ. Sadeqi, “APPROXIMATE QUADRATIC FUNCTIONAL EQUATION IN FELBIN’S TYPE NORMED LINEAR SPACES”, Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 5, pp. 501–516, 2013.
ISNAD Moradlou, Fridoun et al. “APPROXIMATE QUADRATIC FUNCTIONAL EQUATION IN FELBIN’S TYPE NORMED LINEAR SPACES”. Hacettepe Journal of Mathematics and Statistics 42/5 (May 2013), 501-516.
JAMA Moradlou F, Rezaee S, Sadeqi İ. APPROXIMATE QUADRATIC FUNCTIONAL EQUATION IN FELBIN’S TYPE NORMED LINEAR SPACES. Hacettepe Journal of Mathematics and Statistics. 2013;42:501–516.
MLA Moradlou, Fridoun et al. “APPROXIMATE QUADRATIC FUNCTIONAL EQUATION IN FELBIN’S TYPE NORMED LINEAR SPACES”. Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 5, 2013, pp. 501-16.
Vancouver Moradlou F, Rezaee S, Sadeqi İ. APPROXIMATE QUADRATIC FUNCTIONAL EQUATION IN FELBIN’S TYPE NORMED LINEAR SPACES. Hacettepe Journal of Mathematics and Statistics. 2013;42(5):501-16.

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