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On Some Recent Developments in Ulam's Type Stability
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.
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  and reads as follows.
For every real sequence withthere is a real number such thatMoreover, But the main motivation for study of that subject is due to Ulam (cf. ), 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 satisfiesthen a homomorphism exists with In 1941 Hyers  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 .
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  (see also [2, 6–8, 23]), without a proof, and next rediscovered and improved many years later by Găvruţa . 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 , 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.
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
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.
For some investigations of stability of the functional equation with , we refer to  (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
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
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  and answers a question raised by Rassias . 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 ; 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 , 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  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 .
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  (see also, e.g., ) 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 ). 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. ).
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 .
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 ), 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  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 , 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  (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 .
The stability of multiadditive, multi-Jensen, and multiquadratic mappings was recently investigated in [68, 84–93]. In particular, in  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 , 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, .
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 , 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 .
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).
Next, following , 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).
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 .
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 .
8. Stability in Non-Archimedean Spaces
Let us recall that a non-Archimedean valuation in a field is a function with