# Approximate Riesz Algebra-Valued Derivations

Authors: Faruk Polat
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

LetF be a Riesz algebra with an extended norm ||·||u such that (F,||·||u) is complete. Also, let ||·||v be another extended norm in F weaker than ||·||u such that whenever (a) xn→x and xn·y→z in ||·||v, then z=x·y; (b) yn→y and x·yn→z in ||·||v, then z=x·y. Let ε and δ> be two nonnegative real numbers. Assume that a map f:F→F satisfies ||f(x+y)-f(x)-f(y)||u≤ε and ||f(x·y)-x·f(y)-f(x)·y||v≤δ for all x,y∈F. In this paper, we prove that there exists a unique derivation d:F→F such that ||f(x)-d(x)||u≤ε, (x∈F). Moreover, x·(f(y)-d(y))=0 for all x,y∈F.


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## Body

## 1. Introduction

LetE and E′ be Banach spaces and let δ>0. A function f:E→E′ is called δ-additive if ||f(x+y)-f(x)-f(y)||<δ for all x,y∈E. The well-known problem of stability of functional equation f(x+y)=f(x)+f(y) started with the following question of Ulam [1]. Does there exist for each ɛ>0, a δ>0 such that, to each δ-additive function f of E into E′ there corresponds an additive function l of E into E′ satisfying the inequality ||f(x)-l(x)||≤ɛ for each x∈E? In 1941, Hyers [2] answered this question in the affirmative way and showed that δ may be taken equal to ɛ. 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].LetF be an algebra. A mapping d:F→F is called a derivation if and only if it satisfies the following functional equations:
(1.1)d(a+b)=d(a)+d(b),(1.2)d(ab)=ad(b)+d(a)b,
for all a,b∈F.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 spaceF with a partial order ≤ satisfying the following two conditions:(1)
x≤y⇒αx+z≤αy+z for all z∈F and 0≤α∈ℝ,(2)
for allx,y∈F, the supremum x∨y and infimum x∧y exist in F (hence, the modulus |x|:=x∨(-x) exists for each x∈F),is called a Riesz space or vector lattice. Typical examples of Riesz spaces are provided by the function spaces. C(K) the spaces of real valued continuous functions on a topological space K, lp real valued absolutely summable sequences, c the spaces of real valued convergent sequences, and c0 the spaces of real valued sequences converging to zero are natural examples of Riesz spaces under the pointwise ordering. A Riesz space F is called Archimedean if 0≤u,v∈F and nu≤v for each n∈ℕ imply u=0. A subset S in a Riesz space F is said to be solid if it follows from |u|≤|v| in F and v∈S that u∈S. A solid linear subspace of a Riesz space F is called an ideal. Every subset D of a Riesz space F is included in a smallest ideal FD, called ideal generated byD. A principal ideal of a Riesz space F is any ideal generated by a singleton {u}. This ideal will be denoted by Iu. It is easy to see that
(2.1)Iu={v∈F:λ≥0such  that|v|≤λ|u|}.
Let F be a Riesz space and 0≤u∈F. Firstly, we give the following definition.Definition 2.1.
(
1
) The sequence (xn) in F is said to be u-uniformly convergent to the element x∈F whenever, for every ɛ>0, there exists n0 such that |xn0+k-x|≤ɛu holds for each k.
(
2
) The sequence (xn) in F is said to be relatively uniformly convergent to x whenever xn converges u-uniformly to x∈F for some 0≤u∈F.When dealing with relative uniform convergence in an Archimedean Riesz spaceF, it is natural to associate with every positive element u∈F an extended norm ||·||u in F by the following formula:
(2.2)∥x∥u=inf{λ≥0:|x|≤λu}(x∈F).
Note that ||x||u<∞ if and only if x∈Iu. Also |x|≤δu if and only if ||x||u≤δ.A Banach lattice is a vector latticeF that is simultaneously a Banach space whose norm is monotone in the following sense.For allx,y∈F, |x|≤|y| implies ||x||≤||y||. Hence, ||x||=∥|x|∥ for all x∈F.The sequence(xn) in (F,||·||u) is called an extended u-normed Cauchy sequence, if for every ɛ>0 there exists k such that ||xn+k-xm+k||u<ɛ for all m,n. If every extended u-normed Cauchy sequence is convergent in F, then F is called an extended u-normed Banach lattice.A Riesz spaceF is called a Riesz algebra or a lattice ordered algebra if there exists an associative multiplication in F with the usual algebra properties such that 0≤u·v for all 0≤u,v∈F.For more detailed information about Riesz spaces, the reader can consult the bookRiesz 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.
LetE be a linear space and F a Riesz space equipped with an extended norm ||·||u such that the space (F,||·||u) is complete. If, for some δ>0, a map f:E→(F,||·||u) is δ-additive, then limit l(x)=limn→∞f(2nx)/2n exists for each x∈E. l(x) is the unique additive function satisfying the inequality ||f(x)-l(x)||u≤δ for all x∈E.By using Theorem3.1, we give the main result of the paper as follows.Theorem 3.2.
LetF be a Riesz algebra with an extended norm ||·||u such that (F,||·||u) is complete. Also, let ||·||v be another extended norm in F weaker than ||·||u such that whenever(a)
xn→x and xn·y→z in ||·||v, then z=x·y;(b)
yn→y and x·yn→z in ||·||v, then z=x·y.Let ɛ and δ be two nonnegative real numbers. Assume that a map f:F→F satisfies
(3.1)∥f(x+y)-f(x)-f(y)∥u≤ɛ,(3.2)∥f(x·y)-x·f(y)-f(x)·y∥v≤δ,
for all x,y∈F. Then, there exists a unique derivation d:F→F such that ||f(x)-d(x)||u≤ɛ, (x∈F). Moreover, x·(f(y)-d(y))=0 for all x,y∈F.Proof.
By Condition (3.1), Theorem 3.1 shows that there exists a unique additive function d:F→F such that
(3.3)∥f(x)-d(x)∥u≤ɛ,
for each x∈F. It is enough to show that d satisfies Condition (1.2). The inequality (3.3) implies that
(3.4)∥f(nx)-d(nx)∥u≤ɛ(x∈F,n∈ℕ).
By the additivity ofd, we then have
(3.5)∥1nf(nx)-d(x)∥u≤1nɛ(x∈F,n∈ℕ),
which means that
(3.6)d(x)=limn→∞1nf(nx),(x∈F),
with respect to ||·||u norm and so is with respect to ||·||v norm. Condition (3.2) implies that the function r:F×F→F defined by r(x,y)=f(x·y)-x·f(y)-f(x)·y is bounded. Hence
(3.7)limn→∞1nr(nx,y)=0,(x,y∈F),
with respect to ||·||v norm. Applying (3.6) and (3.7), we have
(3.8)d(x·y)=x·f(y)+d(x)·y,(x,y∈F).
Indeed, we have the following  with respect to ||·||v norm,
(3.9)d(x·y)=limn→∞1nf(n(x·y))=limn→∞1nf((nx)·y)=limn→∞1n(nx·f(y)+f(nx)·y+r(nx,y))=limn→∞(x·f(y)+f(nx)n·y+r(nx,y)n)=x·f(y)+d(x)·y,(x,y∈F).
Let x,y∈F and n∈ℕ be fixed. Then using (3.8) and additivity of d, we have
(3.10)x·f(ny)+nd(x)·y=x·f(ny)+d(x)·ny=d(x·ny)=d(nx·y)=nx·f(y)+d(nx)·y=nx·f(y)+nd(x)·y.
Therefore,
(3.11)x·f(y)=x·f(ny)n,(x,y∈F,n∈ℕ).
Sending n to infinity, by (3.6), we see that
(3.12)x·f(y)=x·d(y),(x,y∈F).
Combining this formula with (3.8), we have that d satisfies (1.2) which is the desired result. Moreover, the last formula yields x·(f(y)-d(y))=0 for all x,y∈F.



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*Source: 240258-2012-09-30.xml*