Stability of the Exponential Functional Equation in Riesz Algebras

Abstract

We deal with the stability of the exponential Cauchy functional equation in the class of functions mapping a group (, +) into a Riesz algebra . The main aim of this paper is to prove that the exponential Cauchy functional equation is stable in the sense of Hyers-Ulam and is not superstable in the sense of Baker. To prove the stability we use the Yosida Spectral Representation Theorem.

1. Introduction

In 1979 Baker et al. (cf. [1]) proved that the exponential functional equation in the class of functions mapping a vector space to the real numbers is superstable; that is, any function satisfying, with given , the inequality is either bounded or exponential (satisfies (1)). Then Baker generalized this famous result in [2]. We quote this theorem here since it will be used in the sequel.

Theorem 1 (cf. [2, Theorem  1]). Let be a semigroup and let be given. If a function satisfies the inequality for all , then either for all or for all .

After that the stability of the exponential functional equation has been widely investigated (cf., e.g., [3–6]).

This paper will primarily be concerned with the question if similar result holds true in the class of functions taking values in Riesz algebra with the common notion of the absolute value of an element stemming from the order structure of .

The main aim of the present paper is to show that the superstability phenomenon does not hold in such an order setting. However, we prove that the exponential functional equation (1) is stable in the Ulam-Hyers sense; that is, for any given satisfying inequality (3) there exists an exponential function which approximates uniformly on in the sense that the set is bounded in .

As a method of investigation we apply spectral representation theory for Riesz spaces; to be more precise, we use the Yosida Spectral Representation Theorem for Riesz spaces with a strong order unit.

For some recent results concerning stability of functional equations in vector lattices we refer the interested reader to [7–12].

2. Preliminaries

Throughout the paper , , , and are used to denote the sets of all positive integers, integers, real numbers and nonnegative real numbers, respectively.

For the readers convenience we quote basic definitions and properties concerning Riesz spaces (cf. [13]).

Definition 2 (cf. [13, Definitions  11.1 and  22.1]). We say that a real linear space , endowed with a partial order , is a Riesz space if exists for all and We define the absolute value of by the formula . A Riesz space is called Archimedean if, for each , the inequality holds whenever the set is bounded above. We say that is a Riesz algebra if is a Riesz space endowed with the common algebra multiplication satisfying for . A Riesz algebra is termed an -algebra, whenever implies for every .

There are several types of convergence that may be defined according to the order structure. One of them is the relatively uniform convergence defined as follows.

Definition 3 (cf. [13, Definition  39.1]). Let be a Riesz space and let . A sequence in is said to converge -uniformly to an element whenever, for every , there exists a positive integer such that holds for all . A sequence in is called -uniform Cauchy sequence whenever, for every , there exists a positive integer such that holds for all .

Definition 4 (cf. [13, Definition  39.3]). A Riesz space is called -uniformly complete (with a given ) whenever every -uniform Cauchy sequence has a -uniform limit in . Furthermore is called uniformly complete if it is -uniformly complete with any .

There is a large class of spaces satisfying the above conditions. In particular every Dedekind -complete space (see Definition 5 below) is an Archimedean and uniformly complete Riesz space.

Definition 5 (cf. [13, Definition  1.1]). We say that a Riesz space is Dedekind -complete if any non-empty at most countable subset which is bounded above has a supremum.

Definition 6 (cf. [13, Definition  21.4]). The element is called a strong unit if for every there exists such that .

For more detailed information and, in particular, examples of Riesz spaces posessing the above properties we refer the interested reader to [13].

In further considerations the Yosida Spectral Representation Theorem, which is quoted below, will be used.

Theorem 7. Yosida Spectral Representation Theorem (cf. [13, Theorem  45.3]). Let be an Archimedean Riesz space with a strong unit . Then there exists a topological space and a Riesz subspace of the space of all real continuous functions on (with the pointwise order and pointwise operations of addition and scalar multiplication) and a Riesz isomorphism of onto .

We will not distinguish and its Yosida representant, if no confusion can occur.

Directly from the construction of the Yosida repesentatives one can deduce that the Yosida representative of a strong unit is a constant function . We omit the formal details here as they exceed the scope of the paper.

In general, the space of Yosida representatives is a Riesz subspace of . The following theorem gives us conditions under which is the whole .

Theorem 8 (cf. [13, Theorem  45.4]). Let be an Archimedean Riesz space with a strong unit and let and be as in the previous Theorem. The following conditions are now mutually equivalent.(i) is uniformly complete.(ii).(iii)Every -uniform Cauchy sequence in has an -uniform limit in .Hence, if is a Dedekind -complete space with a strong unit, then .

In the case where , the Yosida representation of is not only a Riesz space but also a Riesz algebra with respect to the pointwise multiplication of functions in . But then, since and are isomorphic as Riesz spaces, we may introduce ring multiplication for the elements induced by the multiplication of representatives, that is, where is the Riesz isomorphism. Notice that given by (5) is uniquely determined. Such a multiplication makes into a commutative Riesz algebra with a unit element (a strong unit is an algebra unit element, that is, for ).

From now on a multiplication in a Riesz space will be construed in the above sense.

3. The Main Result

We start with some, easy to prove, properties of exponential real functions on a -divisible group.

Remark 9. Let be a -divisible group and let satisfy exponential functional equation (1). Then the following conditions hold.(i) for .(ii)If there exists such that , then .(iii)If , then .(iv)If is bounded, then or .(v)If then for .
Our main result reads us the following.

Theorem 10. Let be an Abelian 2-divisible group and let be an Archimedean Riesz space with a strong unit . We assume that is -uniformly complete. If a function satisfies with given , then there exists an exponential function such that

Proof. The idea of the proof is based on the use of the Yosida Spectral Representation Theorem which enables us to apply Theorem 1 of Baker.
The proof runs in four steps.
Step  1. Consider satisfying (6). According to the Yosida Spectral Representation Theorem, for every , we have . Therefore, by (6), one has It means that, for any , satisfies all the assumptions of Theorem 1. By Theorem 1  either is bounded on with for or is exponential on the whole . Let Of course we have and .
We will prove that is an open subset of . For the indirect proof consider and suppose that each neighbourhood of has a nonempty intersection with . Let be given by Since , there exist and such that . On the other hand, according to the supposition, in each neighbourhood of there exists with , which brings a contradiction with the continuity of .
Step  2. For given we define by We shall prove the continuity of . First consider the case . Take an arbitrary neighbourhood of . Since is open, there exists a neighbourhood of with . By the choice of we have and by the continuity of at there exists a neighbourhood of such that . Then forms a neighbourhood of with .
Thus, it remains to consider . For arbitrary let be a neighbourhood of . We will prove that there exists a neighbourhood of such that . Contrary, suppose that in each neighbourhood of there exists with or . Consider the case . Then, taking into account the positivity of , which follows from the fact that is exponential and unbounded on , we have Let be fixed. Then there exists such that or depending on the case where or , respectively. On the other hand, by the continuity of and the fact that there exists a neighbourhood of such that for . By the same reasons, there exists a neighbourhood of such that for . Then is a neighbourhood of such that and for all , which brings a contradiction.
Consequently as and . This completes the proof that .
Therefore, by Theorem 8 one may treat as an element of . Since has been chosen arbitrarily, in fact formula (11) defines a function .
Step  3. We will prove that function given by (11) is exponential.
Let . If then we have . Else and then . As a consequence we have .
Step  4. We will prove (7).
Let . Then for and for . It means that which yields

4. Final Remarks

Let us recall the following theorem, which provides us with the condition under which a Riesz homomorphism (as a homomorphism between Riesz spaces) is multiplicative.

Theorem 11 (cf. [14, Proposition  353P]). Let be an Archimedean -algebra with multiplicative identity . If is another Archimedean -algebra with multiplicative identity , and is a positive linear operator such that , then is a Riesz homomorphism if and only if for all .

By the above theorem and the Yosida Spectral Representation Theorem one can obtain the following corollary.

Corollary 12. Let be an Abelian -divisible group and let be an Archimedean -algebra with a multiplicative identity which is a strong order unit. We assume that is -uniformly complete. If a function satisfies with given , then there exist an exponential function such that

Theorem 10 (Corollary 12) states that the exponential functional equation (1) in Riesz algebras is stable in the Ulam-Hyers sense. Taking into account Theorem 1 it is natural to ask if (1) is superstable in the sense of Baker. It appears that the superstability phenomenon in Riesz algebras fails to hold. In the next example we show that there exists a group , an -algebra satisfying all the assertions of Theorem 10 and a function which fulfills (6) but is neither exponential nor bounded.

Example 13. Let be an Archimedean -algebra of all bounded real functions on the interval with a strong unit with the pointwise order, pointwise addition and multiplication. Then is -uniformly complete. Let be given by We define by Then such an is, clearly, neither bounded nor exponential. To observe that satisfies (6) fix . If then we have . If then .

Remark 14. In general the exponential function satisfying assertions of Theorem 10 is not unique. Indeed, consider , , and as defined in Example 13. Then the exponential functions given by approximate uniformly on , that is, satisfy (7).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

I wish to express my gratitude to the anonymous referees for several valuable suggestions.