Badora (2002) proved the following stability result. Let and be nonnegative real numbers, then for every mapping of a ring onto a Banach algebra satisfying and for all , there exists a unique ring homomorphism such that . Moreover, , for all and all from the algebra generated by . In this paper, we generalize Badora's stability result above on ring homomorphisms for Riesz algebras with extended norms.

1. Introduction

The approximation of solution of the Cauchy's equation lying near to some solution has received a lot of attention from mathematicians in the areas of modern analysis and applied mathematics. Any solution of this equation is called an additive function. Let and be Banach spaces, and let be a positive number. A function of into is called -additive if for all . In the 1940s, Ulam [1] proposed the following stability problem of this equation. 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. There are several definitions and critics of the notion of this stability in the literature (see, e.g., [3, 4]). In 1949, Bourgin [5] generalized Hyers' results to the ring homomorphisms and proved the following.

Theorem 1.1. Let and be nonnegative real numbers. Then every mapping of a Banach algebra with an identity element onto a Banach algebra with an identity element satisfying for all , is a ring homomorphism of onto , that is, and for all .

Finally, Badora [6] proved the following theorem on the Bourgin's result related to stability problem (Theorem 1.1) without additional assumptions.

Theorem 1.2. Let be a ring, let be a Banach algebra, and let and be nonnegative real numbers. Assume that satisfies (1.1) for all . Then there exists a unique ring homomorphism such that Moreover, for all and all from the algebra generated by .

The present paper is in essence a revised and extended compilation of Hyers' result and Theorem 1.2 to the Riesz algebras with extended norms. After outlining the basic information on Riesz space theory, we present the main definitions and facts concerning approximate Riesz algebra (with an extended norm)-valued ring homomorphisms.

2. Preliminaries

A real Banach space endowed with a (partial) order ≀ is called a Banach lattice whenever(1)the order ≀ agrees with the linear operations, that is, for all and ;(2)the order ≀ makes a lattice, that is, for all , the supremum and infimum exist in (hence, the modulus exists for each );(3)the norm is monotonous with respect to the order ≀, that is, for all , implies (hence, for all ).

Recall that a (partially) ordered vector space satisfying (1) and (2) above is called a Riesz space. the spaces of real valued continuous functions on a compact Hausdorff space , -spaces, the spaces of convergent sequences, and the spaces of 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 . Throughout the present paper, all the Riesz spaces are assumed to be Archimedean. 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

We assume that is a fixed positive element in the Riesz space . First of all, we present the following definition.

Definition 2.1. (1) It is said that the sequence in converges -uniformly to the element whenever, for every , there exists such that holds for each .
(2) It is said that the sequence in converges relatively uniformly 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 formula

Note that if and only if , the ideal generated by . Also if and only if .

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 in an associative multiplication with the usual algebra properties such that for all .

For more detailed information about Riesz spaces, the reader can consult the book β€œRiesz Spaces” by Luxemburg and Zaanen [7].

3. Main Results

We begin with the following theorem concerning stability of the functional equation . For a function , let us denote by for , the composition of by itself and in general let for .

The theorem can easily be obtained from [8] or [9]. We give the proof here for the benefit of the reader.

Theorem 3.1. Let be a complete metric space, a nonempty set and such that and are two given functions. Assume that is a function satisfying for each and for some function . If the function satisfies the inequality and the series is convergent for each , then for each integer , one has
(2) is a Cauchy sequence. exists for every , and is the unique function satisfying and the inequality

Proof. (1) Replacing by in (3.1), we get Then by (3.2), we obtain The proof follows by induction.
(2) Let , then thus is a Cauchy sequence for each and it is convergent as is complete. Let for each .
By using (3.4), we get taking the limit as goes to infinity, then we obtain (3.5). By continuity of , we have Suppose that another function satisfies and (3.5). By induction it is easy to show that and . Hence for , Since for every , with , this completes the proof.

Let be a linear space over either complex or real numbers. The operation of addition of elements will be denoted, as usual, by . The operation of multiplication of an element by a scalar will be denoted by . Suppose that in the linear space , we are given a metric . The space is called a metric linear space if the operations of addition and multiplication by numbers are continuous with respect to the metric . A metric linear space is called complete if every Cauchy sequence converges to an element , that is, .

We now give the following corollary in [9] which will be useful in the sequel.

Corollary 3.2. Let a complete metric linear space and be a linear space. Suppose that there exists such that , where . Let satisfy Then there is a unique solution of with

Proof. From (3.12) and (3.14), we get for . By using Theorem 3.1 with , , and , the limit function exists for each and
As and for every , we get Next, by (3.14), for every we have for , so letting we obtain .
Suppose is also a solution of and Then , whence, by Theorem 3.1, we have which implies the uniqueness of .

The following theorem is an extended application of Hyers' result to the Riesz spaces.

Theorem 3.3. Let a linear space, be 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 .

Now, if is a Banach space or extended -normed Banach lattice, then we can take or , , , and . We may obtain the classical Hyers' result [2] and Theorem 3.3 with such , and by using Corollary 3.2.

Finally, we give the following theorem which is an extended application of Badora's result (Theorem 1.2) to Riesz algebras with extended norms. For a proof, we use Theorem 3.3 and the similar techniques of Badora [6] with suitable modifications.

Theorem 3.4. Let be a linear algebra, and 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(1) and in , then ;(2) and in , then .Let and be nonnegative real numbers. Assume that a map satisfies for all . Then there exists a unique ring homomorphism such that , . Moreover, for all and all from the algebra generated by .

Proof. From Theorem 3.3, it follows that there exists a unique additive function such that Hence, it is enough to show that is a multiplicative function. Using the additivity of , it follows that which means that with respect to norm.
Let Then using inequality (3.22), we get with respect to norm.
Applying (3.26) and (3.28), we have for all , since is weaker than . Hence, we get the following functional equation: From this equation and the additivity of , we have Therefore, Sending to infinity, by (3.26), we see that Combining this equation with (3.30), we see that is a multiplicative function.
Moreover, from (3.22) we get with respect to norm.
Thus, by (3.26) and the fact that is weaker than , we get that for all . Hence, by (3.33), so that which completes the proof.