Abstract and Applied Analysis

Volume 2012 (2012), Article ID 240258, 5 pages

doi:10.1155/2012/240258

Research Article

Approximate Riesz Algebra-Valued Derivations

Faruk Polat

Department of Mathematics, Faculty of Science, Çankırı Karatekin University, 18000 Çankırı, Turkey

Received 7 May 2012; Accepted 26 August 2012

Academic Editor: Janusz Brzdek

Copyright © 2012 Faruk Polat. 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.

Abstract.

Let be a Riesz algebra with an extended norm such that is complete. Also, let be another extended norm in weaker than such that whenever (a) and in , then ; (b) and in , then . Let and be two nonnegative real numbers. Assume that a map satisfies and for all . In this paper, we prove that there exists a unique derivation such that , (). Moreover, for all .

1. Introduction

Let and be Banach spaces and let . A function is called -additive if for all . The well-known problem of stability of functional equation started with the following question of Ulam [1]. Does there exist for each , a such that, to each -additive function of into there corresponds an additive function of into satisfying the inequality for each ? In 1941, Hyers [2] answered this question in the affirmative way and showed that may be taken equal to . The answer of Hyers is presented in a great number of articles and books. For the theory of the stability of functional equations see Hyers et al [3].

Let be an algebra. A mapping is called a derivation if and only if it satisfies the following functional equations:

for all .

The stability of derivations was first studied by Jun and Park [4]. Further, approximate derivations were investigated by a number of mathematicians (see, e.g., [5–7]).

The aim of the present paper is to examine the stability problem of derivations for Riesz algebras with extended norms.

2. Preliminaries

A vector space with a partial order satisfying the following two conditions:

(1) for all and ,(2)for all , the supremum and infimum exist in (hence, the modulus exists for each ),

is called a Riesz space or vector lattice. Typical examples of Riesz spaces are provided by the function spaces. the spaces of real valued continuous functions on a topological space , real valued absolutely summable sequences, the spaces of real valued convergent sequences, and the spaces of real valued sequences converging to zero are natural examples of Riesz spaces under the pointwise ordering. A Riesz space is called Archimedean if and for each imply . A subset in a Riesz space is said to be solid if it follows from in and that . A solid linear subspace of a Riesz space is called an ideal. Every subset of a Riesz space is included in a smallest ideal , called ideal generated by . A principal ideal of a Riesz space is any ideal generated by a singleton . This ideal will be denoted by . It is easy to see that

Let be a Riesz space and . Firstly, we give the following definition.

Definition 2.1. The sequence in is said to be -uniformly convergent to the element whenever, for every , there exists such that holds for each .

The sequence in is said to be relatively uniformly convergent to whenever converges -uniformly to for some .

When dealing with relative uniform convergence in an Archimedean Riesz space , it is natural to associate with every positive element an extended norm in by the following formula:

Note that if and only if . Also if and only if .

A Banach lattice is a vector lattice that is simultaneously a Banach space whose norm is monotone in the following sense.

For all , implies . Hence, for all .

The sequence in is called an extended -normed Cauchy sequence, if for every there exists such that for all . If every extended -normed Cauchy sequence is convergent in , then is called an extended -normed Banach lattice.

A Riesz space is called a Riesz algebra or a lattice ordered algebra if there exists an associative multiplication in with the usual algebra properties such that for all .

For more detailed information about Riesz spaces, the reader can consult the book Riesz Spaces by Luxemburg and Zaanen [8]. In the sequel, all the Riesz spaces are assumed to be Archimedean.

3. Main Result

Recently, Polat [9] generalized the Hyers' result [2] to Riesz spaces with extended norms and proved the following.

Theorem 3.1. Let be a linear space and a Riesz space equipped with an extended norm such that the space is complete. If, for some , a map is -additive, then limit exists for each . is the unique additive function satisfying the inequality for all .

By using Theorem 3.1, we give the main result of the paper as follows.

Theorem 3.2. Let be a Riesz algebra with an extended norm such that is complete. Also, let be another extended norm in weaker than such that whenever

(a) and in , then ;(b) and in , then .

Let and be two nonnegative real numbers. Assume that a map satisfies

for all . Then, there exists a unique derivation such that , (). Moreover, for all .

Proof. By Condition (3.1), Theorem 3.1 shows that there exists a unique additive function such that

for each . It is enough to show that satisfies Condition (1.2). The inequality (3.3) implies that

By the additivity of , we then have

which means that

with respect to norm and so is with respect to norm. Condition (3.2) implies that the function defined by is bounded. Hence

with respect to norm. Applying (3.6) and (3.7), we have

Indeed, we have the following with respect to norm,

Let and be fixed. Then using (3.8) and additivity of , we have

Therefore,

Sending to infinity, by (3.6), we see that

Combining this formula with (3.8), we have that satisfies (1.2) which is the desired result. Moreover, the last formula yields for all .

References

S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, London, UK, 1960.

D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941.

D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser Boston, Boston, Mass, USA, 1998.

K.-W. Jun and D.-W. Park, “Almost derivations on the Banach algebra ,” Bulletin of the Korean Mathematical Society, vol. 33, no. 3, pp. 359–366, 1996.

M. S. Moslehian, “Ternary derivations, stability and physical aspects,” Acta Applicandae Mathematicae, vol. 100, no. 2, pp. 187–199, 2008.

M. E. Gordji and M. S. Moslehian, “A trick for investigation of approximate derivations,” Mathematical Communications, vol. 15, no. 1, pp. 99–105, 2010.

A. Fošner, “On the generalized Hyers-Ulam stability of module left derivations,” Aequationes Mathematicae, vol. 84, no. 1-2, pp. 91–98, 2012.

W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces, vol. 1, North-Holland Publishing Company, Amsterdam, The Netherlands, 1971.

F. Polat, “Some generalizations of Ulam-Hyers stability functional equations to Riesz algebras,” Abstract and Applied Analysis, vol. 2012, Article ID 653508, 9 pages, 2012.