Abstract and Applied Analysis

Volume 2012 (2012), Article ID 716936, 41 pages

doi:10.1155/2012/716936

Review Article

On Some Recent Developments in Ulam's Type Stability

Nicole Brillouët-Belluot,¹ Janusz Brzdęk,² and Krzysztof Ciepliński²

¹Département d'Informatique et de Mathématiques, Ecole Centrale de Nantes, 1 Rue de la Noë, B.P. 92101, 44321 Nantes Cedex 3, France

²Department of Mathematics, Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland

Received 9 August 2012; Accepted 11 October 2012

Academic Editor: Bing Xu

Copyright © 2012 Nicole Brillouët-Belluot et al. 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.

We present a survey of some selected recent developments (results and methods) in the theory of Ulam's type stability. In particular we provide some information on hyperstability and the fixed point methods.

1. Introduction

The theory of Ulam's type stability (also quite often connected, e.g., with the names of Bourgin, Gvruţa, Ger, Hyers, and Rassias) is a very popular subject of investigations at the moment. In this expository paper we do not give an introduction to it or an ample historical background; for this we refer to [1–11]. Here we only want to attract the readers attention to some selected topics by presenting some new results and methods in several areas of the theory, which have not been treated at all or only marginally in those publications and which are somehow connected to the research interests of the authors of this paper. Also the number of references is significantly limited (otherwise the list of references would be the major part of the paper) and is only somehow representative (but certainly not fully) to the subjects discussed in this survey.

First we present a brief historical background for the stability of the Cauchy equation. Next we discuss some aspects of stability and nonstability of functional equations in single variable, some methods of proofs applied in that theory (the Forti method and the methods of fixed points), stability in non-Archmedean spaces, selected results on functional congruences, stability of composite type functional equations (in particular of the Goąb-Schinzel equation and its generalizations), and finally the notion of hyperstability. We end the paper with remarks also on some other miscellaneous issues.

2. Some Classical Results Concerning the Cauchy Equation

Throughout this paper , , , and denote, as usual, the sets of positive integers, integers, reals, and complex numbers, respectively. Moreover, and .

For the beginning let us mention that the first known result on stability of functional equations is due to Pólya and Szegő [12] and reads as follows.

For every real sequence with

there is a real number such that

Moreover,

But the main motivation for study of that subject is due to Ulam (cf. [13]), who in 1940 in his talk at the University of Wisconsin presented some unsolved problems and among them was the following question.

Let be a group and a metric group. Given , does there exist such that if satisfies

then a homomorphism exists with

In 1941 Hyers [14] published the following answer to it.

Let and be Banach spaces and . Then for every with

there exists a unique function such that

We can describe that latter result saying that the Cauchy functional equation (2.8) is Hyers-Ulam stable (or has the Hyers-Ulam stability) in the class of functions . For examples of various possible definitions of stability for functional equations and some discussions on them we refer to [9].

The result of Hyers was extended by Aoki [15] (for ; see also [16–18]), Gajda [19] (for ), and Rassias [20] (for ; see also [21, p. 326] and [22]), in the following way.

Theorem 2.1. Let and be two normed spaces, let be complete, , and let be a real number. Let be an operator such that

Then there exists a unique additive operator with

A further generalization was suggested by Bourgin [22] (see also [2, 6–8, 23]), without a proof, and next rediscovered and improved many years later by Găvruţa [24]. Below, we present the Găvruţa type result in a bit generalized form (on the restricted domain), which can be easily derived from [25, Theorem 1].

Corollary 2.2. Let be a linear space over a field with and let be a Banach space. Let be nonempty, , and satisfy

Suppose that there is such that and

Then there exists a unique such that

where

Corollary 2.2 generalizes several already classical results on stability of (2.8). In fact, if we take and

with some , , and with , then has the form

On the other hand, if , and

with some , , and with , then

It is easily seen that, in this way, with and we get the result of Hyers [14], with , , and we obtain Theorem 2.1, with and we have the results of Rassias [26, 27].

Remark 2.3. Actually, as it is easily seen in the proof of [25, Theorem 1], it is enough to assume in Corollary 2.2 that is a commutative semigroup that is uniquely divisible by 2 (i.e., for each there exists a unique with .)

For recent results on stability of some conditional versions of the Cauchy functional equation (2.8) we refer to, for example, [28–31].

3. Stability of the Linear Functional Equation in Single Variable

In this section , stands for a Banach space over , is a nonempty set, , , , and , unless explicitly stated otherwise.

The functional equation

for , is known as the linear functional equation of order . For some information on it we refer to [32, 33] and the references therein.

A simply particular case of functional equation (3.1), with , is the difference equation:

for sequences in , where is a fixed sequence in , namely, (3.1) becomes difference equation (3.2) with

There are only few results on stability of (3.1), and actually only of some particular cases of it. For example, [34, Corollary 4] (cf. [34, Remark 5]) yields the following stability result.

Corollary 3.1. Assume that

and are such that

(e.g., and are constant). If a function satisfies the inequality

then there exists a unique solution to (3.1) with

The assumption (3.4) seems to be quite restrictive. So far we only know that it can be avoided for some special cases of (3.1). For instance, this is the case when each function is constant, is nonzero, and for (with some function ), where as usual, for each , denotes the th iterate of , that is,

Then (3.1) can be written in the following form

with some , and [35, Theorem 2] implies the following stability result.

Theorem 3.2. Let , , satisfy

and denote the roots of the characteristic equation

Assume that one of the following three conditions is valid.

  for .   for and is injective.   for and is bijective.

Then there is a solution of (3.9) with

Moreover, in the case where or holds, is the unique solution of (3.9) such that

The following example (see [35, Example 1]) shows that the statement of Theorem 3.2 need not to be valid in the general situation if for some .

Example 3.3. Fix . Let and let the functions and be given by

Then it is easily seen that

Suppose that is a solution of

Clearly,

is the characteristic equation of (3.16) with the roots . Let

Then it is easily seen that for , whence by a simple induction on we get

Consequently

which means that

Thus we have shown that the statement of Theorem 3.2 is not valid in this case.

Estimation (3.12) is not optimal at least in some cases; for details we refer to [36, Remark 1.5, and Theorem 3.1] (see also [37]).

For some investigations of stability of the functional equation

with , we refer to [38] (note that the equation is a special case of (3.1) and a generalization of (3.9)). Here we only present one simplified result from there.

To this end we need a hypothesis concerning the roots of the equations

with , which reads as follows.

Functions satisfy the condition

Hypothesis means that, for every , are the complex roots of (3.23). Clearly, the functions are not unique, but for every the sequence

is uniquely determined up to a permutation. Moreover, if and only if for each (see [38, Remark 1]).

We say that a function is -invariant provided

Now we are in a position to present a result that can de deduced from [38, Theorem 1].

Theorem 3.4. Let , let be valid, and let be -invariant for .

Assume that and fulfills the inequality

Further, suppose that

Then (3.22) has a solution with

As it folows from [38, Remark 8], the form of in Theorem 3.4 can be explicitly described in some recurrent way.

Some further results on stability of (3.9), particular cases of it and some other similar equations in single variable can be found in [1, 35, 39–51]. For instance, it has been shown in [34, 52, 53] that stability of numerous functional equations of this kind is a direct consequence of some fixed point results. We deal with that issue in the section on the fixed point methods.

At the end of this part we would like to suggest some terminology that might be useful in the investigation of stability also for some other equations (as before, denotes the class of functions mapping a nonempty set into a nonempty set ). Moreover, that terminology could be somehow helpful in clarification of the notion of nonstability, which is very briefly discussed in the next section.

Definition 3.5. Let be nonempty and let be an operator mapping into . We say that (3.1) is -stable (with uniqueness, resp.) provided for every and with

there exists a (unique, resp.) solution of (3.1) such that

In connection with the original statement of Ulam's problem we might think of yet another definition that seems to be quite natural and useful sometimes.

Definition 3.6. Let and . We say that functional equation (3.1) is -stable (with uniqueness, resp.,) provided for every function satisfying (3.30), there exists a (unique, resp.,) solution to (3.1) such that

Given , for each we write

Then is an operator mapping into and, according to Theorem 3.4, the functional equation

(i.e., (3.22) with ) is -stable with uniqueness (cf. [48, Theorem 2.1]).

Further, note that for every with

we have

and consequently (3.34) is -stable with

4. Nonstability

There are only few outcomes of which we could say that they concern nonstability of functional equation. The first well-known one is due to Gajda [19] and answers a question raised by Rassias [54]. Namely, he gave an example of a function showing that a result analogous to that described in Theorem 2.1 cannot be obtained for (for further such examples see [21]; cf. also, e.g., [55, 56]).

In general it is not easy to define the notion of nonstability precisely, mostly because at the moment there are several notions of stability in use (see [9, 57]). For instance, we could understand nonstability as in Example 3.3. The other possibility is to refer to Definitions 3.5 and 3.6 and define -nonstability and -nonstability, respectively. Finally, if there does not exist an such that the equation is -stable, then we could say that it is -nonstable.

For some further propositions of such definitions and preliminary results on nonstability we refer to [58–62]. As an example we present below the result from [60, Theorem 1] concerning nonstability of the difference equation

where and are sequences in and is a sequence in .

Theorem 4.1. Let be a sequence of positive real numbers, a sequence in , and a sequence in with the property

Then there exists a sequence in with

such that, for every sequence in satisfying recurrence (4.1),

The issue of nonstability seems to be a new promising area for research.

5. Stability and Completeness

It is well known that the completeness of the target space is of great importance in the theory of Hyers-Ulam stability of functional equations; we could observe this fact for the stability of the Cauchy equation in the second section.

In [63], Forti and Schwaiger proved that if is a commutative group containing an element of infinite order, is a normed space, and the Cauchy functional equation is Hyers-Ulam stable in the class , then the space has to be complete (let us also mention here that Moszner [64] showed that all four assumptions are essential to get the completeness of ).

The above-described effect, stability implies completeness, was recently proved for some other equations (see [65–68]). Here we present only one result of this kind. It concerns the quadratic equation

and comes from [67].

Theorem 5.1. Let be a finitely generated free commutative group and be a normed space. If (5.1) is Hyers-Ulam stable in the class , then the space is complete.

6. The Method of Forti

As Forti [43] (see also, e.g., [69]) has clearly demonstrated, the stability of functional equations in single variable, in particular of the form:

plays a basic role in many investigations of the stability of functional equations in several variables. Some examples presenting that method can be found in [25, 70, 71] (see also [72]). Here we give only one such example that corrects [70, Corollary 3.2], which unfortunately has been published with some details confused. The main tool is the following theorem (see [70, Theorem 2.1]; cf. [43]).

Theorem 6.1. Assume that is a complete metric space, is a nonempty set, , , , , ,

Then, for every , the limit

exists and is the unique function such that (6.1) holds and

The next corollary presents the corrected version of [70, Corollary 3.2] and its proof. Let us make some preparations for it.

First, let us recall that a groupoid (i.e., a nonempty set endowed with a binary operation ) is uniquely divisible by 2 provided, for each , there is a unique with ; such we denote by . Next, we use the notion:

and (only if the groupoid is uniquely divisible by 2)

for every , .

A groupoid is square symmetric provided the operation + is square symmetric, that is, for ; it is easy to show by induction that, for each (for all , if the groupoid is uniquely divisible by 2), we have

Clearly every commutative semigroup is a square symmetric groupoid. Next, let be a linear space over a field , , , and define a binary operation by

Then it is easy to check that provides a simple example of a square symmetric groupoid.

The square symmetric groupoids have been already considered in several papers investigating the stability of some functional equations (see, e.g., [73–79]). For a description of square symmetric operations we refer to [80].

Finally, we say that is a complete metric groupoid provided is a groupoid, is a complete metric space, and the operation is continuous, in both variables simultaneously, with respect to the metric .

Now we are in a position to present the mentioned above corrected version of [70, Corollary 3.2].

Corollary 6.2. Let and be square symmetric groupoids, be uniquely divisible by 2, be a complete metric groupoid, , (i.e., for ), and . Suppose that there exist such that ,

and satisfies

Then there is a unique function with

Proof. From (6.10), with , we obtain for , which yields

Hence, by Theorem 6.1 (with , , , , and ) the limit

exists for every and

Next, by (6.7) and (6.10), for every with , we have

for , whence letting we deduce that is a solution of (6.11).

It remains to show the uniqueness of . So suppose that ,

Then

and by induction it is easy to show that (6.11) and (6.18) yield and for every and . Hence, for each ,

Since , letting we get .

7. The Fixed Point Methods

Apart from the classical method applied by Hyers and its modification proposed by Forti (see also [72]), the fixed point methods seem to be the most popular at the moment in the investigations of the stability of functional equations, both in single and several variables. Although the fixed point method was used for the first time by Baker [39] who applied a variant of Banach's fixed point theorem to obtain the Hyers-Ulam stability of the functional equation

most authors follow Radu's approach (see [81], where a new proof of Theorem 2.1 for was given) and make use of a theorem of Diaz and Margolis. Here we only present one of the recent results obtained in this way.

Let us recall that a mapping , where is a commutative group, is a linear space, and is a positive integer, is called multiquadratic if it is quadratic in each variable. Similarly we define multiadditive and multi-Jensen mappings. Some basic facts on multiadditive functions can be found for instance in [82] (where their application to the representation of polynomial functions is also presented), whereas for the general form of multi-Jensen mappings and their connection with generalized polynomials we refer to [83].

The stability of multiadditive, multi-Jensen, and multiquadratic mappings was recently investigated in [68, 84–93]. In particular, in [88] Radu's approach was applied to the proof of the following theorem.

Theorem 7.1. Let be a Banach space and for every , let be a mapping such that

for an . If is a mapping satisfying, for any ,

then for every there exists a unique multiquadratic mapping such that

Baker's idea (to prove his result it is enough to define suitable (complete) metric space and (contractive) operator, which form follows from the considered equation (in this case ), and apply the (Banach) fixed point theorem) was used by several mathematicians, who applied other fixed point theorems to extend and generalize Baker's result. Now, we present some of these recent outcomes.

To formulate the first of them, let us recall that a mapping is called a comparison function if it is nondecreasing and

In [94], Matkowski's fixed point theorem was applied to the proof of the following generalization of Baker's result.

Theorem 7.2. Let be a nonempty set, let be a complete metric space, , and . Assume also that

where is a comparison function, and let , be such that

Then there is a unique function satisfying (7.1) and

Moreover, .

On the other hand, in [95], Baker's idea and a variant of Ćirić's fixed point theorem were used to obtain the following result concerning the stability of (7.1).

Theorem 7.3. Let be a nonempty set, let be a complete metric space, , and and

where satisfy

for all and a . If , , and (7.9) holds, then there is a unique function satisfying (7.1) and

A consequence of Theorem 7.3 is the following result on the stability of the linear functional equation of order 1.

Corollary 7.4. Let be a nonempty set, let be a real or complex Banach space, , , (here denotes the Banach algebra of all bounded linear operators on ), , and

If , , and

then there exists a unique function satisfying

and the condition

In [96], Miheţ gave one more generalization of Baker's result. In order to do this he proved a fixed point alternative and used it in the proof of this generalization. To formulate Miheţ's theorem, let us recall that a mapping is called a generalized strict comparison function if it is nondecreasing, ,

Theorem 7.5. Let be a nonempty set, let be a complete metric space, , and . Assume also that

where is a generalized strict comparison function and let , be such that (7.9) holds. Then there is a unique function satisfying (7.1) and

A somewhat different fixed point approach to the Hyers-Ulam stability of functional equations, in which the stability results are simple consequences of some new fixed point theorems, can be found in [34, 52, 53, 97].

Given a nonempty set and a metric space , we define by

Now, we are in a position to present the following fixed point theorem from [34].

Theorem 7.6. Let be a nonempty set, let be a complete metric space, , , , and let be given by

If is an operator satisfying the inequality

and functions and are such that

then for every the limit

exists and the function , defined in this way, is a unique fixed point of with

A consequence of Theorem 7.6 is the following result on the stability of a quite wide class of functional equations in a single variable.

Corollary 7.7. Let be a nonempty set, let be a complete metric space, , , , and let a function satisfy the inequality

for any and , and be an operator defined by

Assume also that is given by (7.22) and functions and are such that

and (7.25) holds. Then for every limit (7.26) exists and the function is a unique solution of the functional equation

satisfying inequality (7.27).

Let us also mention here that very recently Cădariu et al. [97] improved the above two outcomes considering, instead of that given by (7.22), a more general operator .

Next, following [53], we deal with the case of non-Archimedean metric spaces. In order to do this, we introduce some notations and definitions.

Let be a nonempty set. For any we write provided

and we say that an operator is nondecreasing if it satisfies the condition

Moreover, given a sequence in , we write provided

We will also use the following hypothesis concerning operators :

() for every sequence in with .

Finally, recall that a metric on a nonempty set is called non-Archimedean (or an ultrametric) provided

We can now formulate the following fixed point theorem.

Theorem 7.8. Let be a nonempty set, let be a complete non-Archimedean metric space, and let be a nondecreasing operator satisfying hypothesis . If is an operator satisfying inequality (7.23) and functions and are such that

and (7.24) holds, then for every limit (7.26) exists and the function , defined in this way, is a fixed point of with

If, moreover,

then is the unique fixed point of satisfying (7.37).

An immediate consequence of Theorem 7.8 is the following result on the stability of (7.31) in complete non-Archimedean metric spaces.

Corollary 7.9. Let be a nonempty set, be a complete non-Archimedean metric space, , , , and a function satisfy the inequality

for any and , and be an operator defined by (7.29). Assume also that is given by

and functions and are such that (7.30) and (7.36) hold. Then for every limit (7.26) exists and the function is a solution of functional equation (7.31) satisfying inequality (7.37).

Given nonempty sets and functions , , we define an operator by

and we say that is an operator of substitution provided with some and . Moreover, if is continuous for each (with respect to a topology in ), then we say that is continuous.

The following fixed point theorem was proved in [52].

Theorem 7.10. Let be a nonempty set, let be a complete metric space, , , , and

Assume also that for every , is nondecreasing, , ,

and (7.24) holds. Then for every limit (7.26) exists and inequality (7.27) is satisfied. Moreover, the following two statements are true.

(i)If is a continuous operator of substitution or is continuous at for each , then is a fixed point of . (ii)If is subadditive (that is,

for all ) for each , then has at most one fixed point such that

for a positive integer .

Theorem 7.10 with immediately gives the following generalization of Baker's result.

Corollary 7.11. Let be a nonempty set, let be a complete metric space, , , and

Assume also that , , (7.43) holds, , for every , is nondecreasing, is continuous, and

Then for every the limit

exists, (7.27) holds and is a solution of (7.1). Moreover, if for every , is subadditive and , then is the unique solution of (7.1) fulfilling (7.45).

Let us finally mention that the fixed point method is also a useful tool for proving the Hyers-Ulam stability of differential (see [98, 99]) and integral equations (see for instance [100–102]). Some further details and information on the connections between the fixed point theory and the Hyers-Ulam stability can be found in [103].

8. Stability in Non-Archimedean Spaces

Let us recall that a non-Archimedean valuation in a field is a function with

A field endowed with a non-Archimedean valuation is said to be non-Archimedean. Let be a linear space over a field with a non-Archimedean valuation that is nontrivial (i.e., we additionally assume that there is an such that ). A function is said to be a non-Archimedean norm if it satisfies the following conditions:

If is a non-Archimedean norm in , then the pair is called a non-Archimedean normed space.

If is a non-Archimedean normed space, then it is easily seen that the function , given by , is a non-Archimedean metric on . Therefore non-Archimedean normed spaces are special cases of metric spaces. The most important examples of non-Archimedean normed spaces are the -adic numbers (here is any prime number), which have gained the interest of physicists because of their connections with some problems coming from quantum physics, -adic strings, and superstrings (see, for instance, [104]).

In [105], correcting the mistakes in the proof given by the second author in 1968, Arriola and Beyer showed that the Cauchy functional equation is Hyers-Ulam stable in . Schwaiger [106] did the same in the class of functions from a commutative group which is uniquely divisible by to a Banach space over . In 2007, Moslehian and Rassias [107] proved the generalized Hyers-Ulam stability of the Cauchy equation in a more general setting, namely, in complete non-Archimedean normed spaces. After their results a lot of papers (see, for instance, [87–89, 93] and the references given there) on the stability of other equations in such spaces have been published. Here we present only one example of these outcomes which is a generalization of the result of Moslehian and Rassias and was obtained in [87] (cf. also Theorem 7 in [106]).

Theorem 8.1. Let be a commutative semigroup and be a complete non-Archimedean normed space over a non-Archimedean field of characteristic different from . Assume also that and for every , is a mapping such that for each ,

and the limit

denoted by , exists. If is a function satisfying

then for every there exists a multiadditive mapping for which

For every the function is given by

If, moreover,

then for every , is the unique multiadditive mapping satisfying condition (8.6).

It seems that [53] was the first paper where the Hyers-Ulam stability was considered in the most general setting, namely, in complete non-Archimedean metric spaces. One of its results (Corollary 7.9) was mentioned in Section 6; the others, which can be also derived from Theorem 7.8, read as follows (from now on denotes a nonempty set and stands for a complete non-Archimedean metric space).

Corollary 8.2. Suppose that is a groupoid and

Let , , , , and satisfy inequality (7.39) for any and . Assume also that functions , and are such that

and (7.36) holds with given by (7.40). Then the limit exists for every , where is defined by

and the functions , and fulfil

Corollary 8.3. Suppose that is a commutative group and is invariant (i.e., for ). Let , , , and satisfy

Assume also that there is a number such that

with a function . Then the limit exists for every , where is given by

and the function , defined in this way, is the unique solution of the functional equation

such that for .

Corollary 8.4. Let be a groupoid, , , , , , a function satisfy inequality (7.39) for any and , and be an operator defined by

Assume also that a function is such that

Then there exists a function such that

9. Functional Congruences

In this section denotes a real Banach space, stands for a subgroup of the group , and is a real linear space, unless explicitly stated otherwise. We write

for and .

Baron et al. [108] have started the study of conditions on a convex set and a function with

which guarantee that there exists an additive function (i.e., for ) such that

or, in other words, that can be represented in the form

with some , . That is a continuation and an extension of some earlier investigations in [109–111]. Here we present some examples of results from [112] (see also [113, 114]), which generalize those in [108].

They correspond simultaneously to the classical Ulam's problem of stability for the Cauchy equation (with ) and to the subjects considered, for example, in [115–128], where functions satisfying (9.2) with (mainly on restricted domains), have been investigated. The latter issue appears naturally in connection with descriptions of subgroups of some groups (see [129]) and some representations of characters (see, e.g., [109, 115, 116, 122–125]).

It is proved in [112, Example 1] that without any additional assumptions on , the mentioned above decomposition of is not possible in general.

In what follows we say that two nonempty sets are separated provided

In the rest of this section stands for a nonempty closed, symmetric (i.e., for each ), and convex subset of . The next theorem (see [112, Theorem 10]) involves the notion of Christensen measurability and we refer to [130] (cf. [131, 132]) for the details concerning it.

Theorem 9.1. Suppose that is a Polish real linear space, is Christensen measurable, (9.2) holds, and one of the following three conditions is valid.

(i)The sets and are separated and is separable. (ii)The sets and are separated, is countable, and is bounded. (iii)The sets and are separated for some and is countable.

Then there exists an additive function satisfying (9.3).

Moreover, in the case where is bounded, is unique and continuous.

Remark 9.2. There arises a natural question to what extent each of assumptions (i)–(iii) in Theorem 9.1, but also in Theorems 9.3 and 9.4, can be weakened (if at all)?

Certainly, the boundedness of in Theorem 9.1 is necessary for the uniqueness and continuity of as it follows from [112, Remark 4]. It is also the case for the uniqueness, linearity, and continuity of in Theorems 9.3 and 9.4.

For the next theorem we need the notion of Baire property. Let us recall that has the Baire property provided, for every open set , the set has the Baire property, that is, there are an open set and sets of the first category, with

Let us yet recall that a topology in a real linear space is called semilinear provided the mapping

is separately continuous with respect to each variable (see, e.g., [133]). A real linear space endowed with a semilinear topology is called a semilinear topological space.

Now we are in a position to present [112, Theorem 13].

Theorem 9.3. Suppose that is a real semilinear topological space of the second category of Baire (in itself), one of conditions (i)–(iii) of Theorem 9.1 is valid, and fulfills (9.2) and has the Baire property. Then there exists an additive function such that (9.3) holds.

Moreover, in the case where is bounded, is unique and linear; in the case where is bounded and is a linear topological space, is unique and continuous.

For our last theorem (see [112, Theorem 15]), let us recall that , mapping a topological space into , is universally measurable provided, for every open set , the set is universally measurable, that is, it is in the universal completion of the Borel field in (see e.g., [131, 132]); is Borel provided, for every Borel set , the set is Borel in .

Theorem 9.4. Let be endowed with a topology such that the mapping

is Borel for every , one of conditions (i)–(iii) of Theorem 9.1 be valid, and fulfill (9.2) and be universally measurable. Then there exists an additive function such that (9.3) holds.

Moreover, if is bounded, then is unique and linear; if is bounded and the topology in is linear and metrizable with a complete metric, then is unique and continuous.

10. Hyperstability

In this part, and are normed spaces, is nonempty, and . We say that the following conditional Cauchy functional equation

is -hyperstable in the class of functions provided each satisfying the inequality

must fulfil (10.1).

According to our best knowledge, the first hyperstability result was published in [134] (for the constant function ) and concerned the ring homomorphisms. However, the term hyperstability has been used for the first time probably in [135].

Now we present two very elementary hyperstability results for (10.1). The first one is a simple consequence of Corollary 2.2.

Corollary 10.1. Let and be fixed positive real numbers, , , and for . Assume that satisfies (10.2) with given by

Then is a solution to (10.1).

Proof. It is easily seen that condition (2.13) is valid with for and with for . Hence it is enough to use Corollary 2.2.

We have as well the following.

Proposition 10.2. Let and let . Suppose that there exist functions with and

Then is additive.

Proof. Inequality (10.4) yields

Hence, by [136, Theorem 3.1], is additive.

Below we provide two simple examples of applications of those hyperstability results; they correspond to the investigations in [137–149] concerning the inhomogeneous Cauchy equation and the cocycle equation.

Corollary 10.3. Let be such that for some with . Assume that one of the following two conditions is valid.

(a) and there exist and positive reals and with

(b), and there are functions with , and

Then the conditional functional equation

has no solutions in the class of functions .

Proof. Let be a solution to (10.8). Then

Hence, by Corollary 10.1 (if (a) holds) and Proposition 10.2 (if (b) holds), is a solution to (10.1). This means that , which is a contradiction.

Corollary 10.4. Let , and be a symmetric (i.e., for ) solution to the cocycle functional equation

Assume that one of conditions (a) and (b) holds. Then , for .

Proof. is coboundary (see [146] or [149]), that is, there is with for . Clearly is a solution to (10.8). Hence Corollary 10.3 implies the statement.

For some further (more involved) examples of hyperstability results, concerning also some other functional equations, we refer to [150–153]. The issue of hyperstability seems to be a very promising subject to study within the theory of Ulam's type stability.

11. Stability of Composite Functional Equations

The problem of studying the stability of the composite functional equations was raised by Ger in 2000 (at the 38th International Symposium on Functional Equations) and in particular it concerned the Hyers-Ulam stability of the Gołąb-Schinzel equation

for the information concerning that equation and generalizations of it we refer to the survey paper [154].

In 2005, Chudziak [155] answered this question and proved that in the class of continuous real functions equation (11.1) is superstable. More precisely, he showed that if is a continuous function satisfying

with a positive real number , then either is bounded or it is a solution of (11.1).

In [156], Chudziak and Tabor generalized this result. Namely, they proved that if is a subfield of , is a vector space over and , is a function satisfying the inequality

and such that the limit

exists (not necessarily finite) for every , then either is bounded or it is a solution of (11.1) on . Therefore, (11.1) is superstable also in this class of functions.

Later on, in [157, 158], the same results have been proved for the generalized Gołąb-Schinzel equation

where is a positive integer and is a nonzero complex number. If , then functional equation (11.5) is superstable in the class of continuous real functions. If , , and is a vector space over , then (11.5) is superstable in the class of functions such that the limit (11.4) (not necessarily finite) exists for every .

It is known (see [159]) that the phenomenon of superstability is caused by the fact that we mix two operations. Namely, on the right-hand side of (11.1) we have a product, but in (11.2) we measure the distance between the two sides of (11.1) using the difference. Therefore, it is more natural to measure the difference between 1 and the quotients of the sides of (11.1). In [159] it has been proved that for the exponential equation this approach leads to the traditional stability. The result is different in the case of the Gołąb-Schinzel equation.

In [160] it is proved that if is continuous at 0 and satisfies the following two inequalities

for a given , then either is close to 1 or it is a solution of (11.1). Therefore, with this definition of (quotient) stability, the Gołąb-Schinzel equation is also superstable in the class of real functions that are continuous at 0. This approach to stability, using quotients, is now called the stability in the sense of Ger.

Chudziak generalized this result in [161] where he proved that if is a vector space over , , is a positive integer and is such that admits an algebraically interior point (i.e., a point such that, for every , there exists such that for with ) and satisfies the following two inequalities

for a given , then either is bounded or it is a solution of (11.5). Thus, in the class of functions such that admits an algebraically interior point, (11.5) is superstable in the sense of Ger.

In [162] those results were extended to a class of functional equations which includes (11.1), (11.4), and the exponential equation. Consider, namely, the functional equation

where and is a continuous nonzero multiplicative function. It turns out (see [162]) that if is continuous and satisfies the inequality

then either is a solution of the functional equation

or the following three conditions are valid.

(i)If is odd, then either is bounded or it is a solution of (11.8) with .(ii)If is even and , then either is bounded or it is a solution of (11.8) with some .(iii)If , then there exists a unique such that

(For some results on (11.10) see [163]).

In [164], the stability in the sense of Ger of (11.8) was studied in the following more general setting.

Theorem 11.1. Let be a real linear space and let be multiplicative and continuous at a point . Assume also that with satisfies the inequalities

for some . If , then there exists a unique function with satisfying (11.8) and

If and the set has an algebraically interior point, then either is bounded or it is solution of (11.8) with replaced by sign().

In view of the above result, some questions arise. Can we obtain analogous results in the complex case? Are the assumptions on and the set necessary?

The results related to the stability of composite functional equations which have been obtained up to now and which have been described previously concern essentially the Gołąb-Schinzel type functional equations. A few other equations have been investigated in [165–167]. For instance, another very important example of composite functional equations is the translation equation

(see [168–171] for more information on it and its applications) and its stability has been studied in [172–177].

It would be interesting to study also the stability of other composite type functional equations such as the Baxter functional equation [178] and the Ebanks functional equation [179].

12. Miscellaneous

At the end of this survey we would like to attract the attention of the readers to the results and new techniques of proving the stability results in [77, 180–186]; those techniques involve the methods of multivalued function.

A new approach to the stability of functional equations has been proposed by Paneah (see, e.g., [187]) with some critique of the notions that have been commonly accepted so far. Another method, using the concept of shadowing, was presented in [188] and recently applied in [79, 189–192].

An approach to stability in the ring of formal power series is suggested in [173].

Stability of some conditional versions of the Cauchy equation has been studied in [193–197], for example, of the following Mikusiński functional equation

For some connections between Ulam's type stability and the number theory see [198–200].

For some recent results on stability of derivations in rings and algebras see, for example, [201, 202] and the references therein.

Stability for ODE and PDE has been studied, for example, in [98, 99, 203–225], for stability results for some integral equations see [100–102, 226].

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