# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The fixed point method for fuzzy stability of the Jensen functional equation. (English) Zbl 1179.39039

The Jensen functional equation is

$2f\left(\frac{x+y}{2}\right)=f\left(x\right)+f\left(y\right)$

where the unknown $f$ is a mapping between linear spaces. In this paper, however, the unknown is considered as a mapping between fuzzy-normed linear spaces. The author comes up with an alternative proof and a slight improvement of a recently obtained generalized Hyers-Ulam-Rassias stability of such Jensen equation [A. K. Mirmostafaee, M. Mirzavaziri and M. S. Moslehian, “Fuzzy stability of the Jensen functional equation”, Fuzzy Sets Syst. 159, No. 6, 730–738 (2008; Zbl 1179.46060)]. The proof is based, besides several ideas of the original approach, on the fixed-point theory for the probabilistic metric spaces.

##### MSC:
 39B82 Stability, separation, extension, and related topics 46S40 Fuzzy functional analysis 39B52 Functional equations for functions with more general domains and/or ranges 46S50 Functional analysis in probabilistic metric linear spaces
##### References:
 [1] Aoki, T.: On the stability of the linear transformation in Banach spaces, J. math. Soc. Japan 2, 64-66 (1950) · Zbl 0040.35501 · doi:10.2969/jmsj/00210064 [2] Bag, T.; Samanta, S. K.: Finite dimensional fuzzy normed linear spaces, J. fuzzy math. 11, No. 3, 687-705 (2003) · Zbl 1045.46048 [3] L. Cădariu, V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Ineq. Pure Appl. Math. 4 (1) (2003) 7 (Article 4). · Zbl 1043.39010 · doi:emis:journals/JIPAM/v4n1/index.html [4] , Stability of functional equations of Ulam – Hyers – rassias type (2003) [5] Hadžić, O.; Pap, E.: Fixed point theory in probabilistic metric spaces, (2001) [6] Hadžić, O.; Pap, E.; Radu, V.: Generalized contraction mapping principles in probabilistic metric spaces, Acta math. Hungar. 101, No. 1 – 2, 131-148 (2003) · Zbl 1050.47052 · doi:10.1023/B:AMHU.0000003897.39440.d8 [7] Hyers, D. H.: On the stability of the linear functional equation, Proc. nat. Acad. sci. USA 27, 222-224 (1941) · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222 [8] Hyers, D. H.; Isac, G.; Rassias, Th.M.: Stability of functional equations in several variables, (1998) [9] Jung, S. M.: Hyers – Ulam – rassias stability of Jensen’s equation and its application, Proc. amer. Math. soc. 126, 3137-3143 (1998) · Zbl 0909.39014 · doi:10.1090/S0002-9939-98-04680-2 [10] Kominek, Z.: On a local stability of the Jensen functional equation, Demonstratio math. 22, 499-507 (1989) · Zbl 0702.39007 [11] Margolis, B.; Diaz, J. B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. amer. Math. soc. 126, No. 74, 305-309 (1968) · Zbl 0157.29904 · doi:10.1090/S0002-9904-1968-11933-0 [12] Miheţ, D.; Radu, V.: Generalized pseudo-metrics and fixed points in probabilistic metric spaces, Carpathian J. Math. 23, No. 1 – 2, 126-132 (2007) · Zbl 1199.54229 [13] Miheţ, D.; Radu, V.: On the stability of the additive Cauchy functional equation in random normed spaces, J. math. Anal. appl. 343, No. 1, 567-572 (2008) · Zbl 1139.39040 · doi:10.1016/j.jmaa.2008.01.100 [14] Mirmostafaee, A. K.; Mirzavaziri, M.; Moslehian, M. S.: Fuzzy stability of the Jensen functional equation, Fuzzy sets and systems 159, 730-738 (2008) · Zbl 1179.46060 · doi:10.1016/j.fss.2007.07.011 [15] Moslehian, M. S.; Székelyhidi, L.: Stability of ternary homomorphisms via generalized Jensen equation, Resultate math. 49, No. 3 – 4, 289-300 (2006) · Zbl 1114.39010 · doi:10.1007/s00025-006-0225-1 [16] Radu, V.: The fixed point alternative and the stability of functional equations, Sem. fixed point theory 4, No. 1, 91-96 (2003) · Zbl 1051.39031 [17] Rassias, Th.M.: On the stability of the linear mapping in Banach spaces, Proc. amer. Math. soc. 72, 297-300 (1978) · Zbl 0398.47040 · doi:10.2307/2042795 [18] Sherstnev, A. N.: On the notion of a random normed space, Dokl. akad. Nauk SSSR 149, 280-283 (1963) · Zbl 0127.34902