Fuzzy Hyers-Ulam-Rassias stability for generalized additive functional equations

  • Zahra Zamani Payam Nour University
  • Bahman Yousefi Payam Nour University
  • Hassan Azadi Kenary Yasouj University

Abstract

In this paper we establish Hyers-Ulam-Rassias stability of a generalized functional equation in fuzzy Banach spaces. 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), 297-30.

Downloads

Download data is not yet available.

References

T. Bag and S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets Syst. 151 (2005) 513-547. https://doi.org/10.1016/j.fss.2004.05.004

S. C. Cheng and J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bulletin of Calcutta Mathematical Society 86 (1994), 429-436.

P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Mathematicae 27 (1984), 76-86. https://doi.org/10.1007/BF02192660

S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hambourg 62 (1992), 239-248. https://doi.org/10.1007/BF02941618

M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of mixed type cubic and quartic functional equations in random normed spaces, J. Ineq. Appl., Vol. 2009(2009), Article ID 527462, 9 pages. https://doi.org/10.1155/2009/527462

M. Eshaghi Gordji and M. Bavand Savadkouhi and Choonkil Park, Quadratic-quartic functional equations in RN-spaces, J. Ineq. Appl., Vol. 2009(2009), Article ID 868423, 14 pages. https://doi.org/10.1155/2009/868423

M. Eshaghi Gordji and H. Khodaei, Stability of functional equations, Lap Lambert Academic Publishing, 2010.

M. Eshaghi Gordji, S. Zolfaghari, J. M. Rassias and M. B. Savadkouhi, Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces, Abst. Appl. Anal., Vol. 2009(2009), Article ID 417473, 14 pages. https://doi.org/10.1155/2009/417473

C. Felbin, Finite-dimensional fuzzy normed linear space, Fuzzy Sets and Systems 48 (1992), 239-248. https://doi.org/10.1016/0165-0114(92)90338-5

P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211

D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222

S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.

A. K. Katsaras, Fuzzy topological vector spaces, Fuzzy Sets and Systems 12 (1984), 143-154. https://doi.org/10.1016/0165-0114(84)90034-4

I. Karmosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326-334.

S. V. Krishna and K. K. M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207-217. https://doi.org/10.1016/0165-0114(94)90351-4

D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567-572. https://doi.org/10.1016/j.jmaa.2008.01.100

A. Najati and A. Ranjbari, Stability of homomorphisms for a 3D Cauchy-Jensen functional equation on C∗-ternary algebras, J. Math. Anal. Appli., 341, (2008) 62-79.

C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), 711-720. https://doi.org/10.1016/S0022-247X(02)00386-4

C. Park, Modefied Trif's functional equations in Banach modules over a C∗-algebra and approximate algebra homomorphism, J. Math. Anal. Appl. 278 (2003), 93-108. https://doi.org/10.1016/S0022-247X(02)00573-5

C. Park, Fuzzy stability of a functional equation associated with inner product spaces, Fuzzy Sets and Systems 160 (2009), 1632-1642. https://doi.org/10.1016/j.fss.2008.11.027

C. Park, Generalized Hyers-Ulam-Rassias stability of n-sesquilinear-quadratic mappings on Banach modules over C∗-algebras, J. Comput. Appl. Math. 180 (2005), 279-291. https://doi.org/10.1016/j.cam.2004.11.001

C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). https://doi.org/10.1155/2007/50175

C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). https://doi.org/10.1155/2008/493751

J. M. Rassias and H. M. Kim, Generalized Hyers-Ulam stability for grnrral additive functional equations in quasi-βnormed spaces, J. Math. Anal. Appl., 356(2009)302-309. https://doi.org/10.1016/j.jmaa.2009.03.005

Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1

Th. M. Rassias, On the stability of the quadratic functional equation and it's application, Studia Univ. BabesBolyai XLIII (1998), 89-124.

Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264-284. https://doi.org/10.1006/jmaa.2000.7046

Th. M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325-338. https://doi.org/10.1006/jmaa.1993.1070

R. Saadati and C. Park, Non-Archimedean L-fuzzy normed spaces and stability of functional equations (in press).

R. Saadati, M. Vaezpour and Y. J. Cho, A note to paper "On the stability of cubic mappings and quartic mappings in random normed spaces", J. Ineq. Appl., Volume 2009, Article ID 214530, doi: 10.1155/2009/214530. https://doi.org/10.1155/2009/214530

R. Saadati and M. Vaezpour, Some results on Banach spaces, J. Appl. Math. Computing Vol. 17(2005), No. 1 - 2, pp. 475-484. https://doi.org/10.1007/BF02936069

Reza Saadati, M. M. Zohdi, and S. M. Vaezpour, Nonlinear L-Random Stability of an ACQ Functional Equation, J. Ineq. Appl., Volume 2011, Article ID 194394, 23 pages, https://doi.org/10.1155/2011/194394

F. Skof, Local properties and approximation of operators, Rendiconti del Seminario Matematico e Fisico di Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890

S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, NY, USA, 1964.

Published
2022-01-30
Section
Research Articles