Generalized Ulam-Hyers Stability of Jensen Functional Equation in Šerstnev PN Spaces
© M. Eshaghi Gordji et al. 2010
Received: 17 November 2009
Accepted: 1 March 2010
Published: 11 April 2010
We establish a generalized Ulam-Hyers stability theorem in a Šerstnev probabilistic normed space (briefly, Šerstnev PN-space) endowed with . In particular, we introduce the notion of approximate Jensen mapping in PN-spaces and prove that if an approximate Jensen mapping in a Šerstnev PN-space is continuous at a point then we can approximate it by an everywhere continuous Jensen mapping. As a version of a theorem of Schwaiger, we also show that if every approximate Jensen type mapping from the natural numbers into a Šerstnev PN-space can be approximated by an additive mapping, then the norm of Šerstnev PN-space is complete.
1. Introduction and Preliminaries
Mengerproposed transferring the probabilistic notions of quantum mechanic from physics to the underlying geometry. The theory of probabilistic normed spaces (briefly, PN-spaces) is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The notion of a probabilistic normed space was introduced by Šerstnev . Alsina, Schweizer, and Skalar gave a general definition of probabilistic normed space based on the definition of Meneger for probabilistic metric spaces in [2, 3].
Theorem 1.1 (see ).
Theorem 1.2 (see ).
Theorem 1.2was later extended for all . The stability phenomenon that was presented by Rassias is called the generalized Ulam-Hyers stability. In 1982, Rassias  gave a further generalization of the result of Hyers and proved the following theorem using weaker conditions controlled by a product of powers of norms.
Theorem 1.3 (25).
The above mentioned stability involving a product of powers of norms is called Ulam-Gavruta-Rassias stability by various authors (see [12–21]). In the last two decades, several forms of mixed type functional equations and their Ulam-Hyers stability are dealt with in various spaces like fuzzy normed spaces, random normed spaces, quasi-Banach spaces, quasi-normed linear spaces, and Banach algebras by various authors like in [6, 9, 14, 22–38].
It is easy to see that f with satisfies the Jensen equation if and only if it is additive; compare for [39, Theorem ]. Stability of Jensen equation has been studied at first by Kominek  and then by several other mathematicians example, (see [10, 33, 40–42] and references therein).
PN spaces were first defined by Šerstnev in1963(see ). Their definition was generalized in . We recall and apply the definition of probabilistic space briefly as given in , together with the notation that will be needed (see ). A distance distribution function (briefly, a d.d.f.) is a nondecreasing function from into that satisfies and , and is left-continuous on ; here as usual, . The space of d.d.f.'s will be denoted by and the set of all in for which by . The space is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all in . For any , is the d.d.f. given by
The space can be metrized in several ways , but we shall here adopt the Sibley metric . If are d.f.'s and is in , let denote the condition
In particular, under the usual pointwise ordering of functions, is the maximal element of . A triangle function is a binary operation on , namely, a function that is associative, commutative, nondecreasing in each place, and has as identity, that is, for all and in :
Typical continuous triangle functions are and . Here is a continuous t-norm, that is, a continuous binary operation on that is commutative, associative, nondecreasing in each variable, and has 1 as identity; is a continuous -conorm, namely, a continuous binary operation on which is related to the continuous -norm through For example, and or and .
A Probabilistic Normed space (briefly, PN space) is a quadruple , where is a real vector space, and are continuous triangle functions with and is a mapping (the probabilistic norm) from into such that for every choice of and in the following hold:
A PN space is called a Šerstnev space if it satisfies (N1), (N3) and the following condition:
2. Stability of Jensen Mapping in Šerstnev MPN Spaces
In this section, we provide a generalized Ulam-Hyers stability theorem in a Šerstnev MPN space.
which contradicts (2.29).
3. Completeness of Šerstnev MPN Spaces
This section contains two results concerning the completeness of a Šerstnev MPN space. Those are versions of a theorem of Schwaiger  stating that a normed space is complete if, for each whose Cauchy difference is bounded for all , there exists an additive mapping such that is bounded for all .
In order to prove our next results, we need to put the following conditions on an MPN space.
Clearly a definite MPN space is pseudodefinite.
In this work, we have analyzed a generalized Ulam-Hyers theorem in Šerstnev PN spaces endowed with . We have proved that if an approximate Jensen mapping in a Šerstnev PN space is continuous at a point then we can approximate it by an anywhere continuous Jensen mapping. Also, as a version of Schwaiger, we have showed that if every approximate Jensen-type mapping from natural numbers into a Šerstnev PN-space can be approximate by an additive mapping then the norm of Šerstnev PN-space is complete.
The authors are grateful to the referees for their helpful comments. The fourth author was supported by National Research Foundation of Korea (NRF-2009-0070788).
- Šerstnev AN: On the motion of a random normed space. Doklady Akademii Nauk SSSR 1963, 149: 280–283. English translation in Soviet Mathematics Doklady, vol. 4, pp. 388–390, 1963MathSciNetGoogle Scholar
- Alsina C, Schweizer B, Sklar A: On the definition of a probabilistic normed space. Aequationes Mathematicae 1993, 46(1–2):91–98. 10.1007/BF01834000MathSciNetView ArticleMATHGoogle Scholar
- Alsina C, Schweizer B, Sklar A: Continuity properties of probabilistic norms. Journal of Mathematical Analysis and Applications 1997, 208(2):446–452. 10.1006/jmaa.1997.5333MathSciNetView ArticleMATHGoogle Scholar
- Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1964:xvii+150.MATHGoogle Scholar
- Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar
- Cao H-X, Lv J-R, Rassias JM: Superstability for generalized module left derivations and generalized module derivations on a Banach module. I. Journal of Inequalities and Applications 2009, 2009:-10.MathSciNetMATHGoogle Scholar
- Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64–66. 10.2969/jmsj/00210064MathSciNetView ArticleMATHGoogle Scholar
- Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978, 72(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar
- Maligranda L: A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions—a question of priority. Aequationes Mathematicae 2008, 75(3):289–296. 10.1007/s00010-007-2892-8MathSciNetView ArticleMATHGoogle Scholar
- Ciepliński K: Stability of the multi-Jensen equation. Journal of Mathematical Analysis and Applications 2010, 363(1):249–254. 10.1016/j.jmaa.2009.08.021MathSciNetView ArticleMATHGoogle Scholar
- Rassias JM: On approximation of approximately linear mappings by linear mappings. Journal of Functional Analysis 1982, 46(1):126–130. 10.1016/0022-1236(82)90048-9MathSciNetView ArticleMATHGoogle Scholar
- Kim H-M, Rassias JM, Cho Y-S: Stability problem of Ulam for Euler-Lagrange quadratic mappings. Journal of Inequalities and Applications 2007, 2007:-15.MathSciNetMATHGoogle Scholar
- Lee Y-S, Chung S-Y: Stability of an Euler-Lagrange-Rassias equation in the spaces of generalized functions. Applied Mathematics Letters 2008, 21(7):694–700. 10.1016/j.aml.2007.07.022MathSciNetView ArticleMATHGoogle Scholar
- Moszner Z: On the stability of functional equations. Aequationes Mathematicae 2009, 77(1–2):33–88. 10.1007/s00010-008-2945-7MathSciNetView ArticleMATHGoogle Scholar
- Nakmahachalasint P: On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations. International Journal of Mathematics and Mathematical Sciences 2007, 2007:-10.MATHMathSciNetGoogle Scholar
- Pietrzyk A: Stability of the Euler-Lagrange-Rassias functional equation. Demonstratio Mathematica 2006, 39(3):523–530.MathSciNetMATHGoogle Scholar
- Rassias JM: On the stability of a multi-dimensional Cauchy type functional equation. In Geometry, Analysis and Mechanics. World Scientific, River Edge, NJ, USA; 1994:365–376.Google Scholar
- Rassias JM, Kim H-M: Approximate homomorphisms and derivations between -ternary algebras. Journal of Mathematical Physics 2008, 49(6):-10.Google Scholar
- Rassias JM, Lee J, Kim HM: Refined Hyers-Ulam stability for Jensen type mappings. Journal of the Chungcheong Mathematical Society 2009, 22(1):101–116.Google Scholar
- Rassias JM, Rassias MJ: On the Ulam stability of Jensen and Jensen type mappings on restricted domains. Journal of Mathematical Analysis and Applications 2003, 281(2):516–524. 10.1016/S0022-247X(03)00136-7MathSciNetView ArticleMATHGoogle Scholar
- Rassias JM, Rassias MJ: Asymptotic behavior of Jensen and Jensen type functional equations. Panamerican Mathematical Journal 2005, 15(4):21–35.MathSciNetMATHGoogle Scholar
- Bouikhalene B, Elqorachi E, Rassias JM: The superstability of d'Alembert's functional equation on the Heisenberg group. Applied Mathematics Letters 2010, 23(1):105–109. 10.1016/j.aml.2009.08.013MathSciNetView ArticleMATHGoogle Scholar
- Cao H-X, Lv J-R, Rassias JM: Superstability for generalized module left derivations and generalized module derivations on a Banach module. II. Journal of Inequalities in Pure and Applied Mathematics 2009, 10(3, article 85):1–8.MathSciNetMATHGoogle Scholar
- Eshaghi Gordji M: Stability of an additive-quadratic functional equation of two variables in -spaces. Journal of Nonlinear Science and Its Applications 2009, 2(4):251–259.MathSciNetMATHGoogle Scholar
- Faĭziev V, Rassias JM: Stability of generalized additive equations on Banach spaces and groups. Journal of Nonlinear Functional Analysis and Differential Equations 2007, 1(2):153–173.MathSciNetGoogle Scholar
- Farokhzad Rostami R, Hosseinioun SAR: Perturbations of Jordan higher derivations in Banach ternary algebras: an alternative fixed point approach. International Journal of Nonlinear Analysis and Applications 2010, 1(1):42–53.MATHGoogle Scholar
- Gavruta P: An answer to a question of John M. Rassias concerning the stability of Cauchy equation. In Advances in Equations and Inequalities, Hadronic Mathematics Series. Hadronic Press, Palm Harbor, Fla, USA; 1999:67–71.Google Scholar
- Ghobadipour N, Park C: Cubic-quartic functional equations in fuzzy normed spaces. International Journal of Nonlinear Analysis and Applications 2010, 1(1):12–21.MATHGoogle Scholar
- Gordji ME, Gharetapeh SK, Rassias JM, Zolfaghari S: Solution and stability of a mixed type additive, quadratic, and cubic functional equation. Advances in Difference Equations 2009, 2009:-17.MATHMathSciNetGoogle Scholar
- Eshaghi Gordji M, Zolfaghari S, Rassias JM, Savadkouhi MB: Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces. Abstract and Applied Analysis 2009, 2009:-14.MathSciNetMATHGoogle Scholar
- Gordji ME, Rassias JM, Savadkouhi MB: Approximation of the quadratic and cubic functional equations in RN-spaces. European Journal of Pure and Applied Mathematics 2009, 2(4):494–507.MathSciNetMATHGoogle Scholar
- Jun K-W, Kim H-M, Rassias JM: Extended Hyers-Ulam stability for Cauchy-Jensen mappings. Journal of Difference Equations and Applications 2007, 13(12):1139–1153. 10.1080/10236190701464590MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M: Hyers-Ulam-Rassias stability of Jensen's equation and its application. Proceedings of the American Mathematical Society 1998, 126(11):3137–3143. 10.1090/S0002-9939-98-04680-2MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M, Rassias JM: A fixed point approach to the stability of a functional equation of the spiral of Theodorus. Fixed Point Theory and Applications 2008, 2008:-7.MathSciNetMATHGoogle Scholar
- Khodaei H, Rassias ThM: Approximately generalized additive functions in several variables. International Journal of Nonlinear Analysis and Applications 2010, 1(1):22–41.MATHGoogle Scholar
- Kominek Z: On a local stability of the Jensen functional equation. Demonstratio Mathematica 1989, 22(2):499–507.MathSciNetMATHGoogle Scholar
- Park C, Rassias JM: Stability of the Jensen-type functional equation in -algebras: a fixed point approach. Abstract and Applied Analysis 2009, 2009:-17.MathSciNetMATHGoogle Scholar
- Shakeri S: Intuitionistic fuzzy stability of Jensen type mapping. Journal of Nonlinear Science and Its Applications 2009, 2(2):105–112.MathSciNetMATHGoogle Scholar
- Parnami JC, Vasudeva HL: On Jensen's functional equation. Aequationes Mathematicae 1992, 43(2–3):211–218. 10.1007/BF01835703MathSciNetView ArticleMATHGoogle Scholar
- Miheţ D: The fixed point method for fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems 2009, 160(11):1663–1667. 10.1016/j.fss.2008.06.014MathSciNetView ArticleMATHGoogle Scholar
- Mirmostafaee AK, Mirzavaziri M, Moslehian MS: Fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems 2008, 159(6):730–738. 10.1016/j.fss.2007.07.011MathSciNetView ArticleMATHGoogle Scholar
- Tabor J, Tabor J: Stability of the Cauchy functional equation in metric groupoids. Aequationes Mathematicae 2008, 76(1–2):92–104. 10.1007/s00010-007-2912-8MathSciNetView ArticleMATHGoogle Scholar
- Schweizer B, Sklar A: Probabilistic Metric Spaces. Dover, Mineola, NY, USA; 2005.MATHGoogle Scholar
- Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae 2000, 62(1):23–130. 10.1023/A:1006499223572MathSciNetView ArticleMATHGoogle Scholar
- Schwaiger J: Remark 12, in: Report the 25th Internat. Symp. on Functional Equations. Aequationes Mathematicae 1988, 35: 120–121.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.