International Journal of Nonlinear Analysis and Applications

Fuzzy HUR stability of partitioned functional equations

Document Type : Research Paper

Authors

1 Department of Mathematics, Payame Noor University, P.O. Box 19395-4697, Tehran Iran

2 Department of Mathematics, Yasouj University, Yasouj, Iran

Abstract

 In this paper, we establish the Hyers-Ulam-Rassias stability of the following functional equation
$$
(4p)^nf\left(\frac{x_1+\cdots +x_{(4p)^n}}{(4p)^n}\right)  + 4p\sum_{i=1}^{(4p)^{n-1}} f\left(\frac{x_{4pi-4p+1} + \cdots +x_{4pi}}{4p}\right)  = 2  \sum_{i=1}^{(4p)^n} f\left(\frac{x_i+x_{i+1}}{2}\right)
$$
in fuzzy Banach spaces.

Keywords

[1] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687–705.
[2] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets Syst. 151 (2005), 513–547.
[3] R. Biswas, Fuzzy inner product spaces and fuzzy norm functions, Inf. Sci. 53 (1991), 185–190.
[4] S.C. Cheng and J.N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436.
[5] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86.
[6] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hambourg 62 (1992), 239–248.
[7] H. Dutta, B.V.S. Kumar and S. Sabarinathan, Fuzzy stability of a new Hexic functional equation in various spaces, An. St. Univ. Ovidius Constanta 30 (2022), no. 3, 143–171.
[8] M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of mixed type cubic and quartic functional equations in random normed spaces, J. Inequal. Appl. 2009 (2009), Article ID 527462.
[9] M. Eshaghi Gordji, M. Bavand Savadkouhi and C. Park, Quadratic-quartic functional equations in RN-spaces, J. Inequal. Appl. 2009 (2009), Article ID 868423.
[10] M. Eshaghi Gordji and H. Khodaei, Stability of Functional Equations, Lap Lambert Academic Publishing, London, United Kingdom, 2010.
[11] 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. 2009 (2009), Article ID 417473.
[12] C. Felbin, Finite-dimensional fuzzy normed linear space, Fuzzy Sets Syst. 48 (1992), 239–248.
[13] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434.
[14] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436.
[15] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.
[16] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
[17] A.K. Katsaras, Fuzzy topological vector spaces, Fuzzy Sets Syst. 12 (1984), 143–154.
[18] I. Karmosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334.
[19] S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets Syst. 63 (1994), 207–217.
[20] R. Liu and D. O’Regan, Ulam type stability of first-order linear impulsive fuzzy differential equations, Fuzzy Sets Syst. 400 (2020), 34–89.
[21] D. Mihet, The fixed point method for fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst. 160 (2009), no. 11, 1663–1667.
[22] 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.
[23] A.K. Mirmostafaee, A fixed point approach to almost quartic mappings in quasi fuzzy normed spaces, Fuzzy Sets Syst. 160 (2009), no. 11, 1653–1662.
[24] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inf. Sci. 178 (2008), no. 19, 3791–3798.
[25] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy almost quadratic functions, Results Math. 52 (2008), no. 1-2, 161–177.
[26] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst. 159 (2008), no. 6, 730–738.
[27] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159 (2008), no. 6, 720–729.
[28] E. Movahednia and M. Mursaleen, Stability of a generalized quadratic functional equation in intuitionistic fuzzy 2-normed space, Filomat 13 (2016), 449–457.
[29] M. Mursaleen and K.J. Ansari, Stability results in intuition fuzzy normed spaces for a cubic functional equation, Appl. Math. Inf. Sci. 5 (2013), 1677–1684.
[30] C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), 711–720.
[31] 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.
[32] C. Park, Fuzzy stability of a functional equation associated with inner product spaces, Fuzzy Sets Syst. 160 (2009), 1632–1642.
[33] C. Park, Linear functional equations in Banach modules over a C∗-algebra, Acta Appl. Math. 77 (2003), 125–161.
[34] C. Park, D.Y. Shin, R. Saadati and J.R. Lee, Fixed point approach to the fuzzy stability of an AQCQ-functional equation, Filomat 30 (2016), no. 7, 1833–1851.
[35] M. Ramdoss, D. Pachaiyappan, C. Park and J.R. Lee, Stability of a generalized n-variable mixed-type functional equation in fuzzy modular spaces, J. Inequal. Appl. 2021 (2021), no. 61, doi:10.1186/s13660-021-02594-y.
[36] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
[37] Th.M. Rassias, On the stability of the quadratic functional equation and it’s application, Studia Univ. Babes Bolyai 43 (1998), 89–124.
[38] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284.
[39] Th.M. Rassias and P. ˇSemrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325–338.
[40] W. Ren, Z. Yang, X. Sun and M. Qi, Hyers-Ulam stability of Hermite fuzzy differential equations and fuzzy Mellin transform, J. Intell. Fuzzy Syst. 35 (2018), no. 3, 3721–3731.
[41] R. Saadati and C. Park, Non-Archimedean L-fuzzy normed spaces and stability of functional equations, Compu. Math. Appl.   (2010), 2488–2496.
[42] 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. Inequal. Appl. 2009 (2009), Article ID 214530.
[43] R. Saadati, S. Sedghi and H. Zhou, A common fixed point theorem for ψ-weakly commuting maps in L-fuzzy metric spaces, Iran. J. Fuzzy Syst. 5 (2008), no. 1, 47–53.
[44] R. Saadati, M. M. Zohdi and S. M. Vaezpour, Nonlinear L-random stability of an ACQ functional equation, J. Inequal. Appl. 2011 (2011), Article ID 194394.
[45] P. Seo, S. Lee and R. Saadati, Fuzzy stability of an additive-quadratic functional equation with the fixed point alternative, Pure and Appl. Math. 22 (2015), no. 3, 285–298.
[46] A. Sharifi, H. Azadi, B. Yousefi and R. Soltani, HUR-approximation of an ELTA functional equation, Filomat 34 (2020), no. 13, 4311–4328.
[47] F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129.
[48] T. Trif, On the stability of a functional equation deriving from an inequality of T. Popoviciu for convex functions, J. Math. Anal. Appl. 272 (2002), 604–616.
[49] S.M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, NY, USA, 1964.
[50] J. Wu and L. Lu, Hyers-Ulam-Rassias stability of additive mappings in fuzzy normed spaces, J. Math. 2021 (2021), Article ID 5930414.
[51] Z. Zamani, B. Yousefi, and H. Azadi, Fuzzy Hyers-Ulam-Rassias stability for generalized additive functional equations, Bol. Soc. Paran. Mat. 40 (2022), no. 3s, 1–14.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 7
July 2023
Pages 57-72
  • Receive Date: 01 July 2022
  • Revise Date: 18 February 2023
  • Accept Date: 16 April 2023