International Journal of Nonlinear Analysis and Applications
Hyers--Ulam--Rassias stability of Jensen's functional equation on fuzzy normed linear spaces
Document Type : Research Paper
Authors
1 Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran
2 Department of Mathematics, Payame Noor University, Tehran, Iran
Abstract
In this paper, the Hyers--Ulam--Rassias stability of Jensen's functional equation is discussed in detail on fuzzy normed linear spaces. It is shown through the analysis that every fuzzy approximate Jensen-type mapping can be approximated by an additive mapping. The obtained results are the fuzzy versions of some classical theorems.
Keywords
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[11] M. Eshaghi Gordji, H. Azadi Kenary, H. Rezaei, Y.W. Lee and G.H. Kim, Solution and Hyers-Ulam-Rassias= stability of generalized mixed type additive-quadratic functional equations in fuzzy Banach spaces, Abstr. Appl. Anal. 2012 (2012)
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[24] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), no. 1, 91-96.
[25] J.M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126–130.
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[27] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
[28] J.M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), 268–273.
[29] K. Ravi, M. Arunkumar and J.M. Rassias, Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Int. J. Math. Stat. 3 (2008), 36–46.
[30] K. Ravi and B.V.S. Kumar, Stability and geometrical interpretation of reciprocal functional equation, Asian J. Current Engg. Maths 1 (2012), no. 5, 300–304.
[31] K. Ravi and B.V.S. Kumar, Ulam-Gavruta-Rassias stability of Rassias reciprocal functional equation, Global J. Appl. Math. Math. Sci. 3 (2010), no. 1-2, 57–79.
[32] K. Ravi, J.M. Rassias and B.V.S. Kumar, A fixed point approach to the generalized Hyers-Ulam stability of reciprocal difference and adjoint functional equations, Thai J. Math. 8 (2010), no. 3, 469–481.
[33] K. Ravia, J.M. Rassias and B.V.S. Kumar, Stability of reciprocal difference and adjoint functional equations in paranormed spaces: Direct and fixed point methods, Funct. Anal. Approx. Comput. 5 (2013), no. 1, 57–72.
[34] BV. Senthil Kumar, A. Bodaghi, Approximation of the Jensen type rational functional equation by a fixed point technique, Bol. Soc. Paranaense Mat. 38 (2020), no. 3, 125–132.
[35] B.V. Senthil Kumar and H. Dutta, Fuzzy stability of a rational functional equation and its relevance to system design, Int. J. Gen. Syst. 48 (2019), no. 2, 157–169.
[36] B.V. Senthil Kumar, H. Dutta and S. Sabarinathan, Fuzzy approximations of a multiplicative inverse cubic functional equation, Soft Comput. 24 (2020), 13285–13292.
[37] S.M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, Inc., New York, 1964.
[38] J. Xiao and X. Zhu, On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets Syst. 125 (2002), 153–161.
[8] A. Bodaghi, P. Narasimman, J.M. Rassias and K. Ravi, Ulam stability of the reciprocal functional equation in non-Archimedean fields, Acta Math. Univ. Comenianae 85 (2016), no. 1, 113–124.
[9] L. Cadariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Ineq. Pure Appl. Math. 4 (2003), no. 1, Art. 4.
[10] L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2006), 43–52.
[11] M. Eshaghi Gordji, H. Azadi Kenary, H. Rezaei, Y.W. Lee and G.H. Kim, Solution and Hyers-Ulam-Rassias= stability of generalized mixed type additive-quadratic functional equations in fuzzy Banach spaces, Abstr. Appl. Anal. 2012 (2012)
[12] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets Syst. 48 (1992), 239–248.
[13] Z. Gajda, On stability of additive mappings, Int. 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. Natl. Acad. Sci. USA 27 (1941), 222–224.
[16] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: applications to nonlinear analysis, Int. J. Math. Math. Sci. 19 (1996), no. 2, 219–228.
[17] S.M. Jung, A fixed point approach to the stability of the equation f(x + y) = f(x)f(y)/(f(x) + f(y)), Aust. J. Math. Anal. Appl. 6 (1998), no. 1, 1–6, Art. 8.
[18] S.M. Jung, Hyers-Ulam-Rassias stability of Jensen’s equation and its application, Proc. Amer. Math. Soc. 126 (1998), 3137–3143.
[19] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets Syst. 12 (1984), 215–229.
[20] S.O. Kim, A. Bodaghi and C. Park, Stability of functional inequalities associated with the Cauchy-Jensen additive functional equalities in non-Archimedean Banach spaces, J. Nonlinear Sci. Appl. 8 (2015), 776–786.
[21] Z. Kominek, On a local stability of the Jensen functional equation, Demonst. Math. 22 (1989), 499–507.
[22] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst. 159 (2008), 730–738.
[23] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159 (2008), 720–729.
[24] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), no. 1, 91-96.
[25] J.M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126–130.
[26] J.M. Rassias, On approximately of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), no. 4, 445–446.
[27] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
[28] J.M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), 268–273.
[29] K. Ravi, M. Arunkumar and J.M. Rassias, Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Int. J. Math. Stat. 3 (2008), 36–46.
[30] K. Ravi and B.V.S. Kumar, Stability and geometrical interpretation of reciprocal functional equation, Asian J. Current Engg. Maths 1 (2012), no. 5, 300–304.
[31] K. Ravi and B.V.S. Kumar, Ulam-Gavruta-Rassias stability of Rassias reciprocal functional equation, Global J. Appl. Math. Math. Sci. 3 (2010), no. 1-2, 57–79.
[32] K. Ravi, J.M. Rassias and B.V.S. Kumar, A fixed point approach to the generalized Hyers-Ulam stability of reciprocal difference and adjoint functional equations, Thai J. Math. 8 (2010), no. 3, 469–481.
[33] K. Ravia, J.M. Rassias and B.V.S. Kumar, Stability of reciprocal difference and adjoint functional equations in paranormed spaces: Direct and fixed point methods, Funct. Anal. Approx. Comput. 5 (2013), no. 1, 57–72.
[34] BV. Senthil Kumar, A. Bodaghi, Approximation of the Jensen type rational functional equation by a fixed point technique, Bol. Soc. Paranaense Mat. 38 (2020), no. 3, 125–132.
[35] B.V. Senthil Kumar and H. Dutta, Fuzzy stability of a rational functional equation and its relevance to system design, Int. J. Gen. Syst. 48 (2019), no. 2, 157–169.
[36] B.V. Senthil Kumar, H. Dutta and S. Sabarinathan, Fuzzy approximations of a multiplicative inverse cubic functional equation, Soft Comput. 24 (2020), 13285–13292.
[37] S.M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, Inc., New York, 1964.
[38] J. Xiao and X. Zhu, On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets Syst. 125 (2002), 153–161.
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Volume 14, Issue 1
January 2023Pages 3011-3023
- Receive Date: 19 August 2020
- Revise Date: 25 June 2022
- Accept Date: 29 June 2022