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Approximate homomorphisms and derivations on random Banach algebras
Journal of Inequalities and Applications volume 2012, Article number: 157 (2012)
The motivation of this paper is to investigate the stability of homomorphisms and derivations on random Banach algebras.
1 Introduction and preliminaries
The study of stability problems originated from a famous talk Under what condition does there exist a homomorphism near an approximate homomorphism? given by S. M. Ulam  in 1940. Next year, in 1941, D. H. Hyers  answered affirmatively the question of Ulam for additive mappings between Banach spaces.
Aoki  and Rassias  provided a generalization of the Hyers theorem for additive and linear functions respectively, by allowing the Cauchy difference to be unbounded.
Theorem 1.1 (Th. M. Rassias)
Let X be a normed space, Y be a Banach space andbe a function such that
for all, where ε and p are constants withand. Then there exists a unique additive functionsatisfying
for all. If, then the inequality (1.1) holds forand (1.2) for. Also, if for each fixedthe functionis continuous in, then A is linear.
The above theorem had a lot of influence on the development of the generalization of the Hyers-Ulam stability concept during the last three decades. This new concept is known as generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations (see [7, 16]). Furthermore, Gǎvruta  provided a generalization of Rassias’ theorem which allows the Cauchy difference to be controlled by a general unbounded function.
During the last three decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [4, 6, 8, 9, 12, 17–19, 24] and [27–34]). We also refer the readers to the books [1, 7, 10, 16, 20, 21, 28].
Recently, Khodaei and Rassias  introduced the generalized additive functional equation
where with , and they established a general solution and the generalized Hyers-Ulam stability for the functional equation (1.3) in various spaces. They proved that a function f between real vector spaces X and Y is a solution of (1.3) if and only if f is additive.
In the sequel we adopt the usual terminology, notations and conventions of the theory of random normed spaces, as in [36, 37]. Throughout this paper, let be the space of distribution functions, that is,
and the subset is the set
where, denotes the left limit of the function f at the point x. The space is partially ordered by the usual point-wise ordering of functions, i.e., if and only if for all . The maximal element for in this order is the distribution function given by
Definition 1.2 ()
A function is a continuous triangular norm (briefly, a t-norm) if T satisfies the following conditions:
T is commutative and associative;
T is continuous;
for all ;
whenever and for all .
Typical examples of continuous t-norms are , and (the Łukasiewicz t-norm).
Recall (see ) that if T is a t-norm and is a given sequence of numbers in is defined recurrently by
is defined as .
It is known  that for the Łukasiewicz t-norm the following implication holds:
Definition 1.3 ()
A random normed space (briefly, RN-space) is a triple , where X is a vector space, T is a continuous t-norm, and μ is a function from X into such that, the following conditions hold:
(RN1) for all if and only if ;
(RN2) for all , ;
(RN3) for all and .
Definition 1.4 Let be a RN-space.
A sequence in X is said to be convergent to x in X if, for every and , there exists a positive integer N such that whenever .
A sequence in X is called Cauchy if, for every and , there exists a positive integer N such that whenever .
A RN-space is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X. A complete RN-space is said to be a random Banach space.
Theorem 1.5 ()
Ifis a RN-space andis a sequence such that, thenalmost everywhere.
Definition 1.6 A random normed algebra is a random normed space with algebraic structure such that (RN4) for all and all .
Definition 1.7 Let and be random normed algebras:
An additive mapping is called a random homomorphism if for all .
An additive mapping is called a random derivation if for all .
The theory of random normed spaces is important as a generalization of the deterministic result of linear normed spaces and also in the study of random operator equations. The random normed spaces may also provide us with the appropriate tools to study the geometry of nuclear physics and have an important application in quantum particle physics. The generalized Hyers-Ulam stability of different functional equations in random and fuzzy normed spaces and random and fuzzy normed algebras has been recently studied in Alsina , Miheţ et al. , Baktash et al. , Saadati et al. , Gordji et al. , and Park et al. .
In this paper, we prove the generalized Hyers-Ulam stability of random homomorphisms and random derivations associated with the generalized additive functional equation (1.3) in random Banach algebras.
2 Main results
We use the following abbreviation for a given function f:
Theorem 2.1 Let X be a real algebra, be a random Banach algebra and (, anddenoted by) be a function such that
for all, and
for alland all. Suppose thatis a function satisfying
for alland. Then there exists a unique homomorphismsuch that
Proof Putting and () in (2.3), we obtain that
for all and , that is,
for all and . It follows from the last inequality that
for all and ; hence by using the relation , we have
for all and . So we have
for all and . We can show that the sequence is convergent. Therefore, one can define the function by
for all . Now, if we put , and replace with in (2.3) respectively, it follows that
for all and all . By letting in (2.5), we have ; thus H satisfies (1.3). Hence the function is additive (see also ). For the uniqueness property of H, see paper .
Finally, we show that H is multiplicative. Since for all and , from (2.1) it follows that
for all and all . Therefore, there exists a unique random homomorphism satisfying (2.4). □
In the following theorem, we establish the stability of derivations on random Banach algebras. We use the following abbreviation for a given function f:
Theorem 2.2 Letbe a random Banach algebra andbe a function such that (2.1) and (2.2) hold for alland all. Suppose thatis a function satisfying
for alland. Then there exists a unique derivationsuch that
Proof By the same reasoning as in the proof of Theorem 2.1, the sequence is convergent for all , and the function defined by
for all , is a unique additive function which satisfies (2.7). We have to show that is a derivation.
Since for all and , from (2.1) it follows that
for all and all . This means that D is a derivation on X. □
All authors read and approved the final manuscript.
Aczél J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, Cambridge; 1989.
Alsina C: On the stability of a functional equation arising in probabilistic normed spaces. In General Inequalities, vol. 5, Oberwolfach, 1986. Birkhäuser, Basel; 1987:263–271.
Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64–66. 10.2969/jmsj/00210064
Baker JA: The stability of certain functional equations. Proc. Am. Math. Soc. 1991, 112: 729–732. 10.1090/S0002-9939-1991-1052568-7
Baktash E, Cho YJ, Jalili M, Saadati R, Vaezpour SM: On the stability of cubic mappings and quadratic mappings in random normed spaces. J. Inequal. Appl. 2008., 2008:
Cholewa PW: Remarks on the stability of functional equations. Aequ. Math. 1984, 27: 76–86. 10.1007/BF02192660
Czerwik P: Functional Equations and Inequalities in Several Variables. World Scientific, Singapore; 2002.
Eshaghi Gordji M, Khodaei H: Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces. Nonlinear Anal. 2009, 71: 5629–5643. 10.1016/j.na.2009.04.052
Eshaghi Gordji M, Khodaei H: On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations. Abstr. Appl. Anal. 2009., 2009:
Eshaghi Gordji M, Khodaei H: Stability of Functional Equations. Lap Lambert Academic Publishing, Saarbrücken; 2010.
Eshaghi Gordji, M, Ghobadipour, N, Najati, A, Ebadian, A: Almost Jordan homomorphisms and Jordan derivations on fuzzy Banach algebras. Proyecciones (in press)
Gajda Z: On stability of additive mappings. Int. J. Math. Math. Sci. 1991, 14: 431–434. 10.1155/S016117129100056X
Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431–436. 10.1006/jmaa.1994.1211
Hadžić O, Pap E, Budincević M: Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces. Kybernetika 2002, 38(3):363–381.
Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222
Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.
Hyers DH, Rassias TM: Approximate homomorphisms. Aequ. Math. 1992, 44: 125–153. 10.1007/BF01830975
Hyers DH, Isac G, Rassias TM: Topics in Nonlinear Analysis and Applications. World Scientific, Singapore; 1997.
Isac G, Rassias TM: Stability of ψ -additive mappings: applications to nonlinear analysis. Int. J. Math. Math. Sci. 1996, 19: 219–228. 10.1155/S0161171296000324
Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor; 2001.
Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, Heidelberg; 2011.
Khodaei H, Rassias TM: Approximately generalized additive functions in several variables. Int. J. Nonlinear Anal. Appl. 2010, 1: 22–41.
Miheţ D, Saadati R, Vaezpour SM: The stability of an additive functional equation in Menger probabilistic φ -normed spaces. Math. Slovaca 2011, 61: 817–826. 10.2478/s12175-011-0049-7
Park C, Eshaghi Gordji M, Khodaei H: A fixed point approach to the Cauchy-Rassias stability of general Jensen type quadratic-quadratic mappings. Bull. Korean Math. Soc. 2010, 47: 987–996. 10.4134/BKMS.2010.47.5.987
Park C, Lee JR, Shin DY: Generalized Ulam-Hyers stability of random homomorphisms in random normed algebras associated with the Cauchy functional equation. Appl. Math. Lett. 2012, 25: 200–205. 10.1016/j.aml.2011.08.018
Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1
Rassias TM: New characterization of inner product spaces. Bull. Sci. Math. 1984, 108: 95–99.
Rassias TM: Functional Equations, Inequalities and Applications. Kluwer Academic, Dordrecht; 2003.
Rassias TM: On the stability of the quadratic functional equation and its applications. Stud. Univ. Babeş-Bolyai, Math. 1998, XLIII: 89–124.
Rassias TM: The problem of S. M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl. 2000, 246: 352–378. 10.1006/jmaa.2000.6788
Rassias TM: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 2000, 251: 264–284. 10.1006/jmaa.2000.7046
Rassias TM: On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 2000, 62: 23–130. 10.1023/A:1006499223572
Rassias TM, Šemrl P: On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc. Am. Math. Soc. 1992, 114: 989–993. 10.1090/S0002-9939-1992-1059634-1
Rassias TM, Šemrl P: On the Hyers-Ulam stability of linear mappings. J. Math. Anal. Appl. 1993, 173: 325–338. 10.1006/jmaa.1993.1070
Saadati R, Vaezpour SM, Cho YJ: A note on the “On the stability of cubic mappings and quadratic mappings in random normed spaces”. J. Inequal. Appl. 2009., 2009:
Schweizer B, Sklar A: Probabilistic Metric Spaces. Elsevier, Amsterdam; 1983.
Šerstnev AN: On the notion of a random normed space. Dokl. Akad. Nauk SSSR 1963, 149: 280–283. in Russian
Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1964. Chapter VI
The authors declare that they have no competing interests.
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Madadian, M., Ebadian, A., Eshaghi Gordji, M. et al. Approximate homomorphisms and derivations on random Banach algebras. J Inequal Appl 2012, 157 (2012). https://doi.org/10.1186/1029-242X-2012-157
- random normed algebra
- generalized Hyers-Ulam stability
- random homomorphism
- random derivation
- generalized additive functional equation