Fuzzy Stability Results of Finite Variable Additive Functional Equation: Direct and Fixed Point Methods
Abstract
:1. Introduction and Preliminaries
- for ;
- iff for all ;
- if ;
- ;
- is a non-decreasing function of and ;
- for , is continuous on .
- the sequence converges to a fixed point of Λ;
- b is the unique fixed point of Λ in the set ;
- for all
2. General Solution
- (1)
- for all , k integers.
- (2)
- for all if Ψ is continuous.
3. Result and Discussion: Direct Method
4. Result and Discussion: Fixed Point Method
- and
- is the unique fixed point of in the set ;
5. Counter Example
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef] [Green Version]
- Ulam, S.M. Problems in Modern Mathematics; Science Editions; John Wiley & Sons, Inc.: New York, NY, USA, 1940. [Google Scholar]
- Hyers, D.H. On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 1941, 27, 222–224. [Google Scholar] [CrossRef] [Green Version]
- Rassias, T.M. On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72, 297–300. [Google Scholar] [CrossRef]
- Hyers, D.H.; Isac, G.; Rassias, T.M. Stability of Functional Equations in Several Variables; BirkhauserBoston, Inc.: Boston, MA, USA, 1998. [Google Scholar]
- Jun, K.W.; Kim, H.M. The generalized Hyers-Ulam- Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 2002, 274, 267–278. [Google Scholar] [CrossRef] [Green Version]
- Jun, K.W.; Kim, H.M.; Chang, I.S. On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation. J. Comput. Anal. Appl. 2005, 7, 21–33. [Google Scholar]
- Mirmostafaee, A.K.; Moslehian, M.S. Fuzzy approximately cubic mappings. Inform. Sci. 2008, 178, 3791–3798. [Google Scholar] [CrossRef]
- Mohiuddine, S.A. Stability of Jensen functional equation in intuitionistic fuzzy normed space. Chaos Solitons Fractals 2009, 42, 2989–2996. [Google Scholar] [CrossRef]
- Mohiuddine, S.A.; Alghamdi, M.A. Stability of functional equation obtained through a fixed-point alternative in intuitionistic fuzzy normed spaces. Adv. Differ. Equ. 2012, 2012, 141. [Google Scholar] [CrossRef] [Green Version]
- Mohiuddine, S.A.; Alotaibi, A. Fuzzy stability of a cubic functional equation via fixed point technique. Adv. Differ. Equ. 2012, 2012, 48. [Google Scholar] [CrossRef] [Green Version]
- Mohiuddine, S.A.; Alotaibi, A.; Obaid, M. Stability of various functional equations in non-Archimedean intuitionistic fuzzy normed spaces. Discrete Dyn. Nat. Soc. 2012, 2012, 234726. [Google Scholar] [CrossRef]
- Mohiuddine, S.A.; Cancan, M.; Sevli, H. Intuitionistic fuzzy stability of a Jensen functional equation via fixed point technique. Math. Comput. Model. 2011, 54, 2403–2409. [Google Scholar] [CrossRef]
- Mohiuddine, S.A.; Sevli, H. Stability of pexiderized quadratic functional equation in intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2011, 235, 2137–2146. [Google Scholar] [CrossRef] [Green Version]
- Mursaleen, M.; Mohiuddine, S.A. On stability of a cubic functional equation in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 42, 2997–3005. [Google Scholar] [CrossRef]
- Park, C.; Shin, D.Y. Functional equations in paranormed spaces. Adv. Differ. Equ. 2012, 123. [Google Scholar] [CrossRef] [Green Version]
- Park, K.H.; Jung, Y.S. Stability for a cubic functional equation. Bull. Korean Math. Soc. 2004, 41, 347–357. [Google Scholar] [CrossRef] [Green Version]
- Radu, V. The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4, 91–96. [Google Scholar]
- Rassias, J.M. On approximation of approximately linear mappings by linear mapping. J. Funct. Anal. 1982, 46, 126–130. [Google Scholar] [CrossRef] [Green Version]
- Rassias, J.M. On approximation of approximately linear mappings by linear mappings. Bull. Sci. Math. 1984, 108, 445–446. [Google Scholar] [CrossRef] [Green Version]
- Rassias, T.M. On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 2000, 62, 23–130. [Google Scholar] [CrossRef]
- Ravi, K.; Rassias, J.M.; Narasimman, P. Stability of cubic functional equation in fuzzy normed space. J. Appl. Anal. Comput. 2011, 1, 411–425. [Google Scholar]
- Katsaras, A.K. Fuzzy topological vector spaces. II. Fuzzy Sets Syst. 1984, 12, 143–154. [Google Scholar] [CrossRef]
- Felbin, C. Finite-dimensional fuzzy normed linear space. Fuzzy Sets Syst. 1992, 48, 239–248. [Google Scholar] [CrossRef]
- Krishna, S.V.; Sarma, K.K.M. Separation of fuzzy normed linear spaces. Fuzzy Sets Syst. 1994, 63, 207–217. [Google Scholar] [CrossRef]
- Xiao, J.-Z.; Zhu, X.-H. Fuzzy normed space of operators and its completeness. Fuzzy Sets Syst. 2003, 133, 389–399. [Google Scholar] [CrossRef]
- Bag, T.; Samanta, S.K. Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 2003, 11, 687–705. [Google Scholar]
- Cheng, S.C.; Mordeson, J.N. Fuzzy linear operators and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 1994, 86, 429–436. [Google Scholar]
- Kramosil, I.; Michalek, J. Fuzzy metrics and statistical metric spaces. Kybernetika 1975, 11, 336–344. [Google Scholar]
- Bag, T.; Samanta, S.K. Fuzzy bounded linear operators. Fuzzy Sets Syst. 2005, 151, 513–547. [Google Scholar] [CrossRef]
- Simic, S.; Radenovic, S. A functional inequality. J. Math. Anal. Appl. 1996, 197, 489–494. [Google Scholar] [CrossRef] [Green Version]
- Simic, S.; Radenovic, S. On locally subadditive functions. Mat. Vesn. 1994, 46, 89–92. [Google Scholar]
- Radenovic, S.; Simic, S. A note on connection between p-convex and subadditive functions. Univ. Beogr. Publ. Elektrotehn. Fak. Ser. Mat. 1999, 10, 59–62. [Google Scholar]
- Vujaković, J.; Mitrović, S.; Pavlović, M.; Radenović, S.N. On recent results concerning F-contraction in generalized metric spaces. Mathematics 2020, 8, 767. [Google Scholar] [CrossRef]
- Todorcevic, V. Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics; Springer Nature Switzerland AG: Cham, Switzerland, 2019. [Google Scholar]
- Latif, A.; Nazir, T.; Abbas, M. Stability of fixed points in generalized metric spaces. J. Nonlinear Var. Anal. 2018, 2, 287–294. [Google Scholar]
- Shukla, S. Fixed points of Prešić-Ćirić type fuzzy operators. J. Nonlinear Funct. Anal. 2019. [Google Scholar] [CrossRef] [Green Version]
- Takahahsi, W.; Yao, J.C. The split common fixed point problem for two finite families of nonlinear mappings in Hilbert spaces. J. Nonlinear Convex Anal. 2019, 20, 173–195. [Google Scholar]
- Rashid, M.H.M. A common fixed point theorem in strictly convex FM-spaces. Commun. Optim. Theory 2017. [Google Scholar] [CrossRef] [Green Version]
- Pinelas, S.P.; Govindan, V.; Tamilvanan, K. Stability of a quartic functional equation. J. Fixed Point Theory Appl. 2018, 20. [Google Scholar] [CrossRef]
- Pinelas, S.; Govindan, V.; Tamilvanan, K. Solution and stability of an n-dimensional functional equation. Analysis 2019. [Google Scholar] [CrossRef]
- Jung, R.L.; Choonkil, P.; Sandra, P.; Viya, G.; Tamilvanan, K.; Kokila, G. Stability of a sexvigintic functional equation. Nonlinear Funct. Anal. Appl. 2019, 24, 293–325. [Google Scholar]
- Mirmostafaee, A.K.; Moslehian, M.S. Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets Syst. 2008, 159, 720–729. [Google Scholar] [CrossRef]
- Mirmostafaee, A.K.; Mirzavaziri, M.; Moslehian, M.S. Fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst. 2008, 159, 730–738. [Google Scholar] [CrossRef]
- Diaz, J.; Margolis, B. A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 1968, 74, 305–309. [Google Scholar] [CrossRef] [Green Version]
- Miheţ, D.; Radu, V. On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 2008, 343, 567–572. [Google Scholar] [CrossRef] [Green Version]
- Gajda, Z. On stability of additive mappings. Intern. J. Math. Math. Sci. 1991, 14, 431–434. [Google Scholar] [CrossRef]
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Alanazi, A.M.; Muhiuddin, G.; Tamilvanan, K.; Alenze, E.N.; Ebaid, A.; Loganathan, K. Fuzzy Stability Results of Finite Variable Additive Functional Equation: Direct and Fixed Point Methods. Mathematics 2020, 8, 1050. https://doi.org/10.3390/math8071050
Alanazi AM, Muhiuddin G, Tamilvanan K, Alenze EN, Ebaid A, Loganathan K. Fuzzy Stability Results of Finite Variable Additive Functional Equation: Direct and Fixed Point Methods. Mathematics. 2020; 8(7):1050. https://doi.org/10.3390/math8071050
Chicago/Turabian StyleAlanazi, Abdulaziz M., G. Muhiuddin, K. Tamilvanan, Ebtehaj N. Alenze, Abdelhalim Ebaid, and K. Loganathan. 2020. "Fuzzy Stability Results of Finite Variable Additive Functional Equation: Direct and Fixed Point Methods" Mathematics 8, no. 7: 1050. https://doi.org/10.3390/math8071050
APA StyleAlanazi, A. M., Muhiuddin, G., Tamilvanan, K., Alenze, E. N., Ebaid, A., & Loganathan, K. (2020). Fuzzy Stability Results of Finite Variable Additive Functional Equation: Direct and Fixed Point Methods. Mathematics, 8(7), 1050. https://doi.org/10.3390/math8071050