Abstract

We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra). The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equation in Riesz spaces, the Cauchy equation with squares in -algebras, and the quadratic functional equation in Riesz spaces.

1. Introduction

In this paper we deal with a method of treating approximate solutions of functional equations in a class of functions taking values in Riesz spaces (algebras). Some recent results concerning stability of functional equations in ordered spaces can be found in [17].

In view of the fact that the idea of applying the Spectral Representation Theory (SRT for short) for Riesz spaces to investigate approximate solutions of functional equations in vector lattices has appeared fruitful for various functional equations (cf. [2, 3, 5]) it seems to be valuable to formulate a general theorem that could play a role of a tool hopefully applicable to a wide class of functional equations. The main purpose of this paper is to provide such a result (see Section 3).

As it has already been mentioned, in the following we are going to make use of the SRT for Riesz spaces that provides a representation of vectors of a given Riesz space by extended (admitting infinite values) real continuous functions on a certain topological space which are finite on a dense subset of (). The above means that a given Riesz space (under some additional assumptions) is Riesz isomorphic with a Riesz subspace of . Unfortunately, it appears that, in general, the whole of is not necessarily a Riesz space and that causes some difficulties. The second inconvenience we have to defeat stems from the fact that functions from may attain infinite values.

Once the main results of the paper are achieved, we show their benefits. We apply them to investigate approximate solutions of three selected functional equations. The first two of them have the common origin, but they exhibit different stability behaviours (at least in the class of real-valued functions). We show that an alternative Cauchy functional equation is stable in Riesz spaces (see Section 4). In Section 5 we prove that the Cauchy equation with squares is stable in -algebras; however unlike in the case of real-valued functions it is not superstable. The third one is the quadratic functional equation We prove its stability in Riesz spaces in Section 6.

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 reader’s convenience we quote basic definitions and properties concerning Riesz spaces following [8].

We say that a partially ordered real linear space (we denote the order in by ≤) is a Riesz space (vector lattice) if exists for all (cf. [8, Definition 11.1]). 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 from above (cf. [8, Definition 22.1]). We say that is a Riesz algebra if is a Riesz space equipped with the common algebra multiplication satisfying whenever . A Riesz algebra is termed an -algebra, whenever implies for every .

A Riesz space is said to be Dedekind complete (-complete) if any nonempty (at most countable) subset of which is bounded from above has a supremum (cf. [8, Definition 1.1]).

In the following the notion of the relatively uniform convergence will be used (cf. [8, 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 . We say that is relatively uniformly convergent if is -uniformly convergent with some . A sequence in is called -uniform Cauchy sequence whenever, for every , there exists a positive integer such that holds for all .

In general the -uniform limit of a sequence may depend on the choice of and does not have to be unique. However, if is Archimedean, the -uniform limit, if it exists, is unique. In this case the fact that converges -uniformly to will be denoted by .

A Riesz space is called -uniformly complete (with a given ) whenever every -uniform Cauchy sequence has a -uniform limit. We say that is uniformly complete if it is -uniformly complete with every (cf. [8, Definition 39.3]).

There is a large class of spaces satisfying the above conditions. In particular every Dedekind -complete space is Archimedean and uniformly complete.

The element is called a strong unit if for every there exists such that .

The element is called a weak unit if the band generated by is the whole of (cf. [8, Definition 21.4]). Recall that a Riesz subspace of is an ideal if it is solid, that is, whenever it follows from , , and that . An ideal is termed a band in , whenever a subset of has a supremum in , that supremum is an element of (cf. [8, Definition 17.1]). If is Archimedean then is a weak unit if and only if , where stands for the disjoint complement of (cf. [9, 353L]).

A linear mapping between Riesz spaces and is called a Riesz homomorphism if

Now we define the family of extended (admitting infinite values) real continuous functions on a given topological space that are finite-valued on a dense subset of and discuss their elementary properties.

Given a topological space , any continuous mapping of into with the usual topology, such that the set is dense in , is called an extended (real-valued) continuous function on . The set of all extended (real-valued) continuous functions on will be denoted by . We consider the pointwise order in and the pointwise multiplication by scalars, where it is understood that . For any , the set is open and dense. If and holds for all , then (by definition) is called the sum of and (notation ). Since the set is dense, the function is uniquely defined if it exists. It occurs that is not necessarily closed with respect to the operation of addition (cf. [8, p. 295]). Any subset of closed under the operation of addition, multiplication by scalars and the taking of finite infima and suprema, is obviously a Riesz space with respect to pointwise ordering. Accordingly, any subset of this kind is called a Riesz space of extended real continuous functions on .

The difficulties with the addition disappear if is extremally disconnected; that is, the closure of every open set is open. Then is a Riesz space (cf. [8, Theorem 47.2]) and even an -algebra with the multiplicative identity under the appropriate definition of the multiplication. Given , the set is open and dense. Thus the function which is equal to for every is finite-valued and continuous on . Hence, this function can uniquely be extended to an extended continuous function on (cf. [8, Theorem 47.1]), and let, by definition, this extended function be .

Proposition 1 (cf. [3, Proposition 1]). If is an extremally disconnected topological space then is an -algebra with a multiplicative identity.

At the end of this section we briefly remind the notion of the Hyers-Ulam stability originated by the well-known problem posed by Ulam (cf. [10]) during his talk at the University of Wisconsin in 1940 and the answer given by Hyers (cf. [11]), which we quote below.

Let and be Banach spaces and let . Then for every with there is a unique such that and To describe this result we used to say that the Cauchy functional equation (6) is Hyers-Ulam stable in the class of functions . It is worth to mention here that, probably, the first known result in this direction is due to Pólya and Szegö (cf. [12]).

Next, the stability of functional equations has been widely investigated and generalized in various directions by many authors. For the extensive discussion concerning possible definitions of the stability of functional equations and differences between them we refer the interested reader to [13]. Examples of various recent results concerning the subject as well as a list of numerous references connected with it can be found in the survey paper [14].

3. Main Results

From now on let be a groupoid and a Riesz space (-algebra) and let ; . We will say that a function is a solution of equation if for . Given , any with will be called a -solution of (7). will be termed an approximate solution of (7) if it is a -solution of (7) with some . Finally, we will say that (7) is stable (or Hyers-Ulam stable) if for any there is such that for each -solution of (7) there exists a solution of (7) with

We will focus on a class of functional equations that possess the following property.

Definition 2. We will say that (7) has the uniform -approximation property (URAP for short) if there exist , (), , and real sequences , , , and such that if we take , with the ordinary order, as a realisation of then for any and any -solution of (7) the following conditions hold:(P1) ,(P2) ,(P3) ,for ,  , and .

The URAP is closely related to the Hyers-Ulam stability of (7) in the class of real-valued functions, where the role of the operators , (), is played by the so called Hyers operators (cf. [10, 11]). The term uniform in the name of the property refers to the fact that the right-hand sides of (P1)–(P3) do not depend on . It is evident that URAP implies the Hyers-Ulam stability. As it will be shown below, in many cases the converse is also true.

Lemma 3. Let be a groupoid. Assume that (7) is Hyers-Ulam stable in the class of real-valued functions defined on and that there exist and , , such that for any solution of (7) and for any -solution of (7) Then (7) possesses the URAP.

Proof. Let be an -solution of (7). We define for , . By the Hyers-Ulam stability of (7) there exist a solution of (7) and with Applying (12) for in place of and taking into account (10) we obtain Dividing the above inequality by , side by side, we have on account of the definition of . By (14) and (12) we have (P2) with .
If we rewrite (14) for given and and then add the resulting inequalities, side by side, we obtain (P1) with and .
Since is an -solution of (7) then using (8) for in place of we have Dividing the above inequality by and taking into account (11), in view of the definition of , we arrive at which means that (P3) holds with provided that . The case means that satisfies (7) and, therefore, (P3) holds with any nonnegative .

Remark 4. Let us observe that all the assumptions of Lemma 3 are fulfilled if we assume that (7) is Hyers-Ulam stable in the class of real-valued functions and that there exists such that any solution of (7) is -homogeneous with some ; that is, and is -homogeneous with some ; that is,

In particular, any functional equation which is Hyers-Ulam stable in the class of real-valued functions, whose solution form additive functions, has the URAP.

Corollary 5. Let map a groupoid into . Assume that there exist an additive function and a real number such that Let for and . Then (P1)–(P3) hold with , , , for and .

Proof. Routine.

Let stand for the substructure of generated by and let . We will consider the following hypotheses:(H1)there exist a topological space and a Riesz (-algebra) isomorphism ,(H2) and commute; that is, for any and ,(H3) and commute; that is, for any , , ,(H4)for each there exists an open and dense subset of such that (H5)for each , if exists, then

Let us note that the SRT for Riesz spaces provides results which guarantee (H1). Various classical spectral representation theorems offer effective constructions of a topological space as well as a space of representatives and a Riesz isomorphism , depending on the properties of a given Riesz space (cf., e.g., [8, Ch.7]).

It is easy to see that if and are defined with the use of the ordinary Riesz space (algebra) operations, that is, linear operations or lattice operations, then (H2), (H3), and (H5) are automatically satisfied.

Now we are going to prove a lemma that provides some properties of a function that yield (H4) (for ). Assume that (H1) holds and that we are given mappings , , and open and dense subsets , of such that For fixed we consider the following hypotheses. (L1) is an Abelian group and (L2) is an Abelian semigroup and (L3) is an Abelian group and

Lemma 6. Let map a groupoid into a Riesz space and let (H1) hold. If for any at least one of the hypotheses (L1), (L2), and (L3) holds then (H4) (for ) is fulfilled.

Proof. Assume, at first, that (L1) holds. For fixed we define By (23) and (H1), for every , we have which means that Therefore, by (22), we see that Replacing by and by in (29) we observe that . Now, suppose that for given and apply (29) with and replaced by and , respectively, in order to obtain . By induction we arrive at On the other hand, using (29) with in place of and in place of , we receive . This along with (30) yields For arbitrary we define , where and are given by (26). By (31) for and for . Let us consider and . Using (29) with and replaced by and , respectively, we observe that , which completes the proof of (20).
Now we assume that (L2) holds. Let and let Similarly as in case (L1) one can observe that (H1), (24), and (22) yield By the definition . Suppose that for given and apply (33) with and replaced by and , respectively, in order to obtain . By induction we receive thus (31) holds.
Fix arbitrary and define , where and are given by (32). By (31) for and for . Let us consider , . Using (33) for and , we observe that . This completes the proof of (20).
Finally, assume that (L3) holds. Let and let Observe that due to (H1), (25), and (22) we have By the definition . Let , suppose that for , and apply (36) with and replaced by and , respectively, in order to obtain . By induction we obtain Exchanging with and repeating the lines of the proof of (37) we conclude that for , which together with (37) results with (31).
Fix arbitrary and define , where and are given by (35). By (31) for and for . We will prove that for . It is evident that . Let and suppose that for . Applying (36) with and replaced by and , respectively, we observe that . Thus, by induction To observe that (38) holds also for negative integers it is enough to apply (36) with and in place of and , respectively, and take into account (38). Thus, Now, let be fixed and assume that, for given , it is for . Then by (36) with and replaced by and , respectively, we obtain . By induction, If is a negative integer, we use (36) with and replaced by and , respectively, together with (40) to complete the proof of (20).

Now we are in a position to formulate and prove the main theorem of the paper.

Theorem 7. Let be a groupoid, let be an Archimedean Riesz space, and let (7) possess the URAP. Assume that, for given , is a -solution of (7) and that hypotheses (H1)–(H5) hold. Let be such that for and assume that is -uniformly complete. Then there exists a solution of (7) such that Moreover, the solution of (7) satisfying (41) is unique provided that it is unique in the case where we consider as a realisation of .

Proof. The proof runs in three steps.
Step 1. Assume that is a -solution of (7). We will prove that, for every , the sequence is -uniformly convergent.
Fix . By (H4) there exists an open and dense subset of with Let . By (H1) and (H2) for arbitrarily fixed and we have where stands for the function mapping into given by for . This means that is a real-valued -solution of (7) on . Since (7) has the URAP, by (P1), with some and . Similarly, by (P2) we obtain with some . According to (H3) and the definition of the above inequalities imply respectively. Since is arbitrary, is open and dense in ; moreover all the functions in the above inequalities (as functions of variable ) are continuous; we obtain Due to the fact that is a Riesz homomorphism, the above inequalities result in respectively. Inequality (48) means that is a -uniform Cauchy sequence in a -uniformly complete Riesz space and, therefore, relatively uniformly convergent. This, due to the fact that was arbitrarily fixed, proves that given by is well defined.
Letting in (49) we obtain (41) as is Archimedean.
Step 2. We will prove that is a solution of (7). Let . By (H4) there exists an open and dense subset of with (20). Let . By (H1) and (H2) for arbitrarily fixed and we have which means that is a real-valued -solution of (7) on . Since (7) has the URAP, by (P3), with some and consequently, taking into account the definition of , (H2), and (H3), we have Since the last inequality is valid for any from the open and dense subset of and all the functions in the above inequality are continuous, we obtain Finally, taking into account the fact that is a Riesz homomorphism, we arrive at Letting and taking into account the definition of , (H5) and the fact that is Archimedean, we have which proves that satisfies (7) as was chosen arbitrarily.
Step 3. We will prove the uniqueness of satisfying (41) under the assumption that in the class of real-valued functions a solution of (7) which approximates is uniquely determined. Contrary, suppose that two solutions of (7) satisfy Fix . By (H4) there exists an open and dense subset of such that Let . According to (H1) and (H2), the above means that, for arbitrarily fixed , function is a real-valued -solution of (7) on . On the other hand, by (56), as is a Riesz homomorphism. This along with (H2) means that , () are real-valued solutions of (7) that approximate on . Hence . But then as was chosen arbitrarily, is open and dense in , and and are continuous. Since is injective we infer that . This completes the proof, as was arbitrarily fixed.

Remark 8. Theorem 7 remains valid for more involved functional equations, for instance, alternative (conditional) functional equations for . By (59) we mean that satisfies (59) if Given , any is a -near solution of (59) if We assume that both operators and satisfy (H2) and (H5). Since, in fact, is now two-place function, we assume that satisfies for . Moreover, defining open and dense subsets of we replace any occurrence of by .

Remark 9. Similarly, instead of an alternative in (59) one may consider a conjunction, which is useful to investigate systems of functional equations.

Remark 10. Observe that Theorem 7 remains valid if one considers a slightly more general definition of the URAP. Namely, one can allow sequences , , , to be dependent on . Now we assume the convergence of , , to and to at each point . Moreover, one can consider different deltas on the right-hand sides of (P1)–(P3).

Remark 11. Let us note that one can replace condition (P3) in Definition 2 of the URAP with the following one:(P3′) the sequence is convergent to 0.
Then, accordingly, Step 2 of the proof of Theorem 7, that is, the proof that defined by (50) satisfies (7), should be replaced by the following reasoning.
Let . By (H4) there exists an open and dense subset of with (20). Let . By the definition of (50) and (H5) we have Let be fixed. Then, On the other hand, by (H1) and (H2), for any , we have which means that is a real-valued -solution of (7) on . Since (7) has the URAP, then (P3′), (H2), and (H3) yield Taking into account (64), we have Since the last equality is valid for any from the open and dense subset of and all the functions in the above inequality are continuous, we obtain Consequently, we infer that as is a Riesz isomorphism. This, due to the fact that was chosen arbitrarily, completes the proof that satisfies (7).

4. Approximate Solutions of an Alternative Cauchy Equation

In this section we deal with approximate solutions of an alternative Cauchy functional equation This equation belongs to the class of conditional Cauchy equations with the condition dependent on the unknown function. The general solution of (70) is described in [15, Theorem 8]. Stability of this equation, in the class of functions mapping an Abelian semigroup into a Banach space, has been investigated in [16] and in a more general setting in [17]. For the readers convenience we quote the main result of [16] as it will be used in the sequel.

Theorem 12 (cf. [16, Theorem 1]). Let be an Abelian semigroup and let be a Banach space. If, for some and all , a function satisfies then there exists a unique additive function such that for all .

The natural question arises if a similar result holds true in ordered spaces. One can rewrite all the sentences of Theorem 12 for functions mapping an Abelian semigroup into a Riesz space , replacing the norm by the absolute value in . The main goal of this section is to apply Theorem 7 with the purpose to give an affirmative answer to this question.

We will use one of the most general spectral representation theorems, namely, the Johnson-Kist Spectral Representation Theorem which we quote here.

Theorem 13 (Johnson-Kist representation theorem) (cf. [8, Theorem 44.4]). Let be an Archimedean Riesz space. There exists a Riesz space of extended real continuous functions and a Riesz isomorphism of onto .

The main theorem of this section reads as follows.

Theorem 14. Let be an Abelian semigroup and let be an Archimedean Riesz space. Assume that, for given , is -uniformly complete. If, for every , a function satisfies then there exists a unique additive function such that

Proof. By the Johnson-Kist Spectral Representation Theorem there exist a topological space and a Riesz isomorphism ; hence (H1) is satisfied. (H2′) and (H5) hold because (70) has the form (59) with and , for . If satisfies (73) then and, therefore, (L2) holds with , for and . Thus, by Lemma 6, we have (H4). It is clear that given by , for , , satisfies (H3). Moreover, by Theorem 12 applied for and Lemma 3, one can easily verify that (59) possesses URAP with , , for and . Putting we have which means that for .
Now all the assertions of Theorem 14 follow directly from Theorem 7.

It is easy to observe that the constant of approximation in Theorem 14 is the best possible one.

In view of Theorem 12 and the meaning of the approximate solution of an alternative functional equation (70), that is condition (71) one may expect the following implication: for , in Theorem 14 instead of condition (73), with the common meaning of as and . Of course, if the order in a Riesz space is linear then conditions (73) and (76) coincide. However, as it will be shown in the example below, in general, assumption (73) cannot be replaced by (76).

Example 15. Let be the Archimedean Riesz space of all real functions of real variable with the pointwise order and let be given by Then is -uniformly complete. We define by Then is not additive and satisfies (76) with defined above and . On the other hand, cannot be approximated by any additive function. Suppose, for contradiction, that there exists an additive mapping satisfying (74). Let us fix . For inequality (74) results with according to the definition of . Directly from the definition of we have . Then by (74) and the additivity of we obtain . Eventually, we infer that and coincide and, therefore, is additive. We have obtained a contradiction.

Let us point out that the assumption that the Riesz space is Archimedean is necessary in order to have the uniqueness of an existing additive function in Theorem 14, which can be observed in the following simple example.

Example 16. Let us consider the lexicographically ordered plane . is then -uniformly complete Riesz space. Moreover function given by satisfies inequality (73) with and . On the other hand inequality (74) holds true with any additive mapping , , of the form .

5. Approximate Solutions of the Cauchy Equation with Squares

Equation (70) has stemmed from with a real function , and next has been investigated in the form which admits further generalisations from the real case to more general structures. Affirmative results concerning stability of (80) are contained in [18] for real-valued functions and, for the class of functions taking values in Riesz spaces, in [5]. There are also known results concerning the stability of the generalized equation (80) for functions acting into a normed space, called Fischer-Muszély functional equation: It occurs that, despite the fact that on the assumption that the norm is strictly convex (81) is equivalent to the Cauchy functional equation (cf. [19]), even if we consider with the Euclidean norm as a target space of , (81) fails to be stable in the Hyers-Ulam sense (cf. [18]). However, if we consider the stability of (81) in the class of surjective functions, then the answer is positive (cf. [20]).

Finally, concerning (79) in the class of complex functions we have the following stability result.

Theorem 17 (cf. [16, Theorem 2]). Let be an Abelian semigroup. If for a given a function satisfies then there exists a unique additive function such that

Remark 18. In fact, for complex functions, (79) occurs to be superstable in the sense of Baker which was proved in [21] and, with the use of Theorem 5, in [22].

The main aim of this section is to apply Theorem 7 in order to prove that in the class of functions taking values in an -algebra equation (79) is stable; however it is not superstable. Our main stability result reads as follows.

Theorem 19. Let be an Abelian semigroup and let be an Archimedean -algebra with a multiplicative identity . Assume that is -uniformly complete for given . If a function satisfies then there exists a unique additive function such that

In the proof of Theorem 19 we are going to use Proposition from [3], which is quoted below. This proposition is due to the Ogasawara-Maeda Spectral Representation Theorem for Archimedean Riesz spaces with a weak unit (cf. [8, Theorem 50.1]). The multiplicative identity is a weak order unit (cf. [9, Proposition ]) and the topological space—the domain of the Ogasawara-Maeda representatives—appears to be extremally disconnected; hence, according to Proposition 1, is an -algebra.

Proposition 20 (cf. [3, Proposition 2]). Let be an Archimedean -algebra with a multiplicative identity . Then there exist a topological space and an -subalgebra of the -algebra and an -algebra isomorphism of onto .

Proof of Theorem 19. By Proposition 20 there exist a topological space and an -algebra isomorphism ; hence (H1) is satisfied. Equation (79) has the form (7) with , for ; thus (H2) and (H5) hold. By (84) we have and, therefore, (L2) holds with , for and . Thus, by Lemma 6 we have (H4). It is evident that given by for , satisfies (H3). Moreover, by Theorem 17 and Lemma 3 (79) possesses the URAP with , , and for and . For we have and consequently for .
Having applied Theorem 7 we finish the proof.

Theorem 19 states that in -algebras the Cauchy equation with squares (79) is stable in the Hyers-Ulam sense. According to Remark 18 one can ask if (79) is superstable in the sense of Baker. But it appears that this is not the case.

Example 21. Let be the Archimedean -algebra of all bounded real functions on the interval with a multiplicative identity , the pointwise order, pointwise addition, and multiplication. Let be given by Then is -uniformly complete and given by is, clearly, neither bounded nor additive. Moreover, one can easily check that and satisfy (84).

6. Approximate Solutions of the Quadratic Functional Equation

In this section we deal with approximate solutions of the quadratic functional equation Stability of this equation, in the class of functions mapping an Abelian group into a Banach space, has been investigated in [23] (cf., e.g., [2426]).

Theorem 22 (cf. [23]). Let be an Abelian group and let be a Banach space. If a function satisfies the inequality for some and all then there exists a unique quadratic function such that for all .

The main aim of this section is to show that a similar result holds true in the class of Riesz space-valued mappings.

Theorem 23. Let be an Abelian group and let be an Archimedean -uniformly complete Riesz space, with some