zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 probabilistic stability for a functional equation in a single variable. (English) Zbl 1212.39036

By using the fixed point method, the author deals with the probabilistic Hyers-Ulam stability and the generalized Hyers-Ulam-Rassias stability of the functional equation


where η:XX,μ:YY are given functions and f is the unknown mapping from X to a probabilistic metric space (Y,F,T M ) with T M (a,b)=min(a,b) and probabilistic distance F.

39B52Functional equations for functions with more general domains and/or ranges
39B82Stability, separation, extension, and related topics
47H10Fixed point theorems for nonlinear operators on topological linear spaces
54E70Probabilistic metric spaces
[1]T. Aoki, On the stability of the linear transformation in Banach spaces, J. M. Soc. Japan, 2 (1950), 64–66. · Zbl 0040.35501 · doi:10.2969/jmsj/00210064
[2]J. A. Baker, The stability of certain functional equations, Proc. Amer. Math. Soc., 112 (1991), 729–732. · doi:10.1090/S0002-9939-1991-1052568-7
[3]L. Cădariu and V. Radu, Fixed point method for the generalized stability of functional equations in single variable, Fixed Point Theory and Applications, vol. 2008, Article ID 749392 (2008), 15 pages.
[4]S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific (River Edge, NJ, 2002).
[5]G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math., 50 (1995), 143–190. · Zbl 0836.39007 · doi:10.1007/BF01831117
[6]P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431–436. · Zbl 0818.46043 · doi:10.1006/jmaa.1994.1211
[7]A. Gianyi, Z. Kaiser and Z. Páles, Estimates to the stability of functional equations, Aequationes Math., 73 (2007), 125–143. · Zbl 1121.39029 · doi:10.1007/s00010-006-2854-6
[8]O. Hadžić, A generalization of the contraction principle in PM-spaces, Zb. Rad. Prirod. Mat. Fak. Univ. u Novom Sadu, 10 (1980), 13–21.
[9]O. Hadžić and E. Pap, Fixed Point Theory in PM Spaces, Kluwer Academic Publ. (2001).
[10]O. Hadzic, E. Pap and V. Radu, Generalized contraction mapping principles in probabilistic metric spaces, Acta Math. Hungar., 101 (2003), 131–148. · Zbl 1050.47052 · doi:10.1023/B:AMHU.0000003897.39440.d8
[11]D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222–224. · doi:10.1073/pnas.27.4.222
[12]D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables (Basel, 1998).
[13]I. Istrăţescu, On generalized Menger spaces, Boll. UMI (5) 13-A (1976), 95–100.
[14]C. F. K. Jung, On generalized complete metric spaces, Bull. Amer. Math. Soc., 75 (1969), 113–116. · Zbl 0194.23801 · doi:10.1090/S0002-9904-1969-12165-8
[15]Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math., 22 (1989), 499–507.
[16]W. A. J. Luxemburg, On the convergence of successive approximations in the theory of ordinary differential equations, II, Nederl. Akad. Wetensch. Proc. Ser. A 61 = Indag. Math., 20 (1958), 540–546.
[17]B. Margolis and J. B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305–309. · Zbl 0157.29903 · doi:10.1090/S0002-9904-1968-11920-2
[18]D. Miheţ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl., 343 (2008), 567–572.
[19]M. Mirmostafaee, M. Mirzavaziri and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, 159 (2008), 730–738. · Zbl 1179.46060 · doi:10.1016/j.fss.2007.07.011
[20]A. K. Mirmostafee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems, 159 (2008), 720–729. · Zbl 1178.46075 · doi:10.1016/j.fss.2007.09.016
[21]V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, Cluj-Napoca, IV(1) (2003), 91–96.
[22]Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. · doi:10.1090/S0002-9939-1978-0507327-1
[23]B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland (1983).
[24]V. M. Sehgal and A. T. Bharucha-Reid, Fixed points of contraction mappings on PM-Spaces, Math. Syst. Theory, 6 (1972), 97–100. · Zbl 0244.60004 · doi:10.1007/BF01706080
[25]N. X. Tan, Generalized probabilistic metric spaces and fixed point theorems, Math. Nachr., 129 (1986), 205–218. · Zbl 0603.54049 · doi:10.1002/mana.19861290119