Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 536791, 14 pages

http://dx.doi.org/10.1155/2014/536791

Review Article

On Ulam’s Type Stability of the Linear Equation and Related Issues

Janusz Brzdęk,¹ Krzysztof Ciepliński,1,2 and Zbigniew Leśniak¹

¹Department of Mathematics, Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland

²Faculty of Applied Mathematics, AGH University of Science and Technology, Mickiewicza 30, 30-059 Kraków, Poland

Received 12 June 2014; Accepted 5 July 2014; Published 25 August 2014

Academic Editor: Ajda Fošner

Copyright © 2014 Janusz Brzdęk 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.

This is a survey paper concerning stability results for the linear functional equation in single variable. We discuss issues that have not been considered or have been treated only briefly in other surveys concerning stability of the equation. In this way, we complement those surveys.

1. Introduction

It is a commonly accepted conviction that the issue of stability of functional equations has been motivated by a problem raised by Ulam (cf. [1]) in 1940 in his talk at the University of Wisconsin. The problem can be stated as follows.

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

then a homomorphism exists with

The first (partial) answer to it was published in 1941 by Hyers [2]. It reads as follows.

Let and be Banach spaces and . Then, for every with

there is a unique solution of the Cauchy equation

such that

Nowadays, we describe that result of Hyers simply saying that Cauchy functional equation (4) is Hyers-Ulam stable (or has the Hyers-Ulam stability). Next, Hyers and Ulam published some further stability results for polynomial functions, isometries, and convex functions in [3–6].

For the last 50 years, that issue has been a very popular subject of investigations and we refer the reader to monographs and surveys [7–17] for further information, references, some discussions, and examples of recent results. Below, we present only one such example, which is an extension of the result of Hyers [2] and is composed of the outcomes from [18–21] (cf. [22, 23]; see also [24]).

Before we do this, let us yet recall that a function is called additive provided it is a solution of (4).

Theorem 1. Let and be normed spaces, , fixed real numbers. Assume also that is a mapping such that

If and is complete, then there is a unique additive function with

If , then is additive.

In this paper, we focus on stability of a linear functional equation of the first order, in single variable and some related results; in this way, we complement to some extent the information provided in surveys [7–9, 25, 26]. Let us yet mention that the equation plays a significant role in the investigations of stability of the functional equations in several variables; for suitable examples, we refer the reader, for example, to [8, 27–31].

2. Preliminaries

In what follows, , , , , and denote, as usually, the sets of positive integers, integers, rationals, reals, and complex numbers, respectively; moreover, and .

Let us recall that the linear functional equation of the order has the form

where is a nonempty set, is a linear space over a field , and the functions , , and for are given. The unknown function is . We refer the reader to [7–9, 25, 26] for surveys on stability results for that equation (with arbitrary ) and its generalizations. In this paper, we focus only on the case , when the equation takes the form

It is easily seen that the following functional equation

with suitable functions and , is its natural generalization. Next, if is bijective, then we can rewrite (9) in the form

and a natural generalization of it is the functional equation

with suitable functions and .

We discuss stability results for those three functional equations and some related issues that have not been treated at all or only briefly in [7–9, 25, 26].

The following general definition (cf. [25]) describes the main idea of the notion of stability that we use in this paper; for comments on various possible definitions of stability, we refer the reader to [16, 17, 32] (given two nonempty sets, and , by we denote, as usual, the family of all functions mapping into ).

Definition 2. Let , be a nonempty set, a metric space, nonempty, a function mapping into , and functions mapping nonempty set into . We say that the equation

is -stable provided, for any and with

there is a solution of (13) such that

In the case where consists of all constant functions from and contains only constant functions, the -stability is usually called the Hyers-Ulam (or the Ulam-Hyers) stability.

3. Stability Results

In this section, we present various examples of stability results. We do not compare them, in general. The readers can easily do it themselves.

The first theorem is a well known example of the Hyers-Ulam stability result for a particular case of functional equation (10) (its probabilistic versions have been given by Miheţ in [33] and Miheţ and Zaharia in [34, 35]).

Theorem 3 (see [36, Theorem 2]). Let be a nonempty set, a complete metric space, , , , and

If , and

then there is a unique solution of the functional equation

such that

To formulate the next result (which is a generalization of Theorem 3), we recall that a mapping is called a comparison function if it is nondecreasing and

Theorem 4 (see [37, Theorem 2.2]). Let be a nonempty set, a complete metric space, and , . Assume also that

where is a comparison function, and let , be such that (17) holds. Then, there is a unique solution of (18) such that

Moreover,

Below, we present several other (less known) similar stability results for particular cases of (10), obtained in an analogous way as Theorems 3 and 4, that is, by the fixed point methods.

Theorem 5 (see [38, Theorem 2.1]). Let be a nonempty set, a complete metric space, and functions , , and fulfil

for any , , and a fixed . If satisfies the inequality

then there exists a solution of (18) such that

The subsequent theorem also concerns (18).

Theorem 6 (see [39, Theorem 2.2]). Let S be a nonempty set, a complete metric space, , , and

where    fulfil the inequality

for a fixed . If , and (17) holds, then there is a unique function satisfying (18) and such that

Recall that a mapping is called a generalized strict comparison function if it is nondecreasing, , and

The following is one more generalization of Theorem 3.

Theorem 7 (see [40, Theorem 3.1]). Let be a nonempty set, a complete metric space, and , . Assume also that

where is a generalized strict comparison function, and let , be such that (17) holds. Then, there is a unique function satisfying (18) and

The next result involves a generalization of condition (17) (with a constant replaced by a suitable function on the right hand side of the inequality).

Theorem 8 (see [41, Theorem 4.1]). Let be a nonempty set, a complete metric space, , , , where , and

Assume that satisfies

with a mapping for which there exists an such that

Then, there is a unique solution of (18) such that

Let us mention here that an analogous result for the complete probabilistic metric spaces has been obtained in [42].

Another result on the stability of (18) comes from [43] (for some related results cf. [44]). To formulate it, we define, for given nonempty sets , and functions , , an operator by

Theorem 9 (see [43, Corollary 2.1]). Let be a nonempty set, a complete metric space, , , and

Assume also that and are such that

fulfils (34) and, for every , is nondecreasing and is continuous. Then, the limit

exists for every ,

and is a solution of (18). Moreover, if, for every , is subadditive (i.e., for ) and , then is the unique solution of (18) with

Now, we present a result from [45].

Theorem 10 (see [45, Theorem 2.2]). Let S be a nonempty set, a complete metric space, , , , and

Assume also that , are such that

with an . Then, there is a unique solution of the equation

such that

The next two stability outcomes were obtained in [46].

Theorem 11 (see [46, Theorem 2]). Let be a nonempty set, a complete metric space, , , , and . Assume also that functions satisfy the inequality

and is such that

If fulfils

then there exists a unique solution of the functional equation

such that

Theorem 12 (see [46, Theorem 5]). Let be a nonempty set, a Banach space over , , , , , and . Assume also that functions satisfy the inequalities

If fulfils

then there exists a unique solution of the functional equation

such that

Moreover,

The next theorem has been applied in [47] to prove stability of the Pexiderized linear functional equation

Theorem 13 (see [47, Theorem 2.1]). Let be a nonempty set, a Banach space over , , , , , , and

If satisfies

then there is a unique function such that

Theorem 14 (see [47, Theorem 2.5]). Let S be a nonempty set, a Banach space over , a bijection, , , , , and

If satisfies (59), then there is a unique function such that (60) holds and

The authors have also proved in [47] a stability result for the system of homogeneous linear equations

which gives a partial affirmative answer to a problem posed by G.L. Forti during the 13th International Conference on Functional Equations and Inequalities (Małe Ciche, Poland, September 13–19, 2009).

The below theorem has been used in [48] to prove a stability result for the following functional equation

with suitable functions and .

Theorem 15 (see [48, Theorem  1]). Let be a nonempty set, a complete metric space, , , , and

If , are such that

and the series converges for every , then there is a unique solution of the functional equation

with

Let us also mention that the probabilistic stability of the following particular cases of (10) and (18)

was investigated in [49]. Further results on stability of this equation can be found, for instance, in [50–52].

The next result deals with linear equation (54) and is due to Trif [53]. We will show its application in the sequel, in the section concerning solutions of a simplified version of the linear equation.

Theorem 16 (see [53, Theorem 2.1]). Let be a nonempty set, , a Banach space over , , , and such that

If satisfies the inequality

then there exists a unique solution of (54) with

Moreover,

Actually, condition (74) has not been included in the statement of [53, Theorem 2.1], but it can be easily derived from the proof of the theorem. For some investigations of condition (71), we refer the reader to [54].

We end this section with quite general stability results for difference equations that have been obtained in [55].

Theorem 17 (see [55, Theorem 1]). Let be an abelian group, a complete, and invariant metric in , a continuous isomorphism for every , , , and , . Suppose that

Then there exists a unique sequence such that

with an .

Remark 18 (see [55, Remark 3]). It follows from [56, Remark 2.3] that, in the case

the conclusion of Theorem 17 is not generally true.

Theorem 19 (see [55, Theorem 2]). Let be a metric space, , , , and

Suppose that there exists with

Then there exist a sequence and an such that

Remark 20 (see [55, Remark 3]). There is no uniqueness of the sequence in Theorem 19, which follows from [56, Remark 2.2].

If

then the conclusion of Theorem 19 is not generally true (cf. [56, Remark 2.3]).

We refer the reader to [57] (and the references therein) for further stability results for linear difference equations of higher orders.

4. Iterative Stability

Let for a and , given functions. Consider the linear nonhomogenous equation

and its homogenous version

where is unknown.

Brydak [58] (cf. [59, Definition 2]) introduced the notion of stability (later called iterative stability), which for (84) means that for every there exists a such that if a continuous function satisfies the condition

then there exists a continuous solution of (84) such that

where

In general, the following two hypotheses have been used in investigations of that stability.

(H1) is a strictly increasing continuous function and for .(H2) is a continuous function such that for .

It is known that if (H1) and (H2) hold, then continuous solutions of (84) and (85) defined on depend on an arbitrary function (cf. [60, Theorem 2.1]). The crucial assumption here is that does not belong to the domain of the solutions.

Let us yet introduce the following two assumptions.

(A)The limit exists, is continuous in and for .(B)There exists an interval such that the sequence converges uniformly to the zero function on .

Brydak [58] proved that if either (A) holds and

or (B) holds, then (84) is iteratively stable (cf. also [7]). Turdza [61] considered the same problem in the case where , , , and is a Banach space over . He proved that if (H1), (H2), and (A) hold and , then (84) is iteratively stable (cf. [7] for suitable comments).

Choczewski et al. [59, Theorem 1] have also introduced the following definition of stability, which according to the comment following Definition 2 can be called the Hyers-Ulam stability.

Definition 21 (see [59, Definition 21]). Equation (84) is called stable in the class consisting of the all functions continuous in the interval , if there exists a such that for any and solution of the inequality

there exists a solution of (84) with

They showed (under hypotheses (H1) and (H2)) that if (85) is stable (iteratively stable, resp.) and has a continuous solution , then so is (84); a very recent and more general result of this type will be presented at the end of this section.

For an ample and much more detailed discussion of the results concerning iterative stability, we refer the reader to survey paper [7]. Below, we present some outcomes obtained by Turdza in [62], which have not been included in [7].

The notion of iterative stability has been introduced in [62] for functional equation (12), that is, for the equation

with suitable given functions and and the unknown function .

The author has used in his considerations the following hypotheses.

()The function is continuous and strictly increasing in the interval , for , and .() The function is defined in a set and takes values in ( is a nonempty set), and for every fixed the function is invertible in the set (provided ).() For any and function , which is continuous in the interval and such that , there exists exactly one function that is continuous in and satisfies (92) and the condition for .() For every , there exists an such that for any continuous solution of the inequality

where and

and continuous solution of (92), fulfilling the condition

the subsequent inequality is valid

Let be a nontrivial interval and denote the class of all functions defined and continuous in . The next two definitions have been introduced in [62].

Definition 22 (see [62, Definition 1]). Equation (92) is iteratively stable in the interval in the class , if there exists an such that, for any and solution of the system of inequalities (93), there exists a solution of (92) satisfying (96).

Definition 23 (see [62, Definition 2]). Equation (92) is stable in the interval in the class , if there exists an such that, for any and solution of the inequality

there exists a solution of (92) satisfying (96).

Actually the term “iterative stable” has been used in [62] instead of “iteratively stable,” but it seems that the latter one is more correct and consistent with [59, Definition 2].

The notions of stability described in Definitions 22 and 23 are closely related. Namely, we have the following.

Theorem 24 (see [62, Theorem 2]). Let hypotheses and be valid, (97) hold, and

for a function such that

with a . Then, Definitions 22 and 23 are equivalent.

The subsequent two theorems concern iterative stability (the first one has actually been proved in [63]).

Theorem 25 (see [62, Theorem 1]). Let hypotheses and be valid. If there exists an with

then for every (92) is iteratively stable in .

Theorem 26 (see [62, Theorem 4]). Let hypotheses be valid with . Assume also that there exist and continuous function such that

in a neighbourhood of zero, (98) holds, and

with a . Then, for every (92) is iteratively stable in the interval .

The next theorem corresponds to Theorem 26.

Theorem 27 (see [62, Theorem 3]). Let hypotheses and be valid. If condition (98) is fulfilled with a function such that (99) holds, then for every (92) is stable in the interval in the class .

A connection between the stability and the continuous dependance of (92) on a given function has been investigated in [62, Theorems 5 and 6]. Below, we present those results.

Theorem 28 (see [62, Theorem 5]). Let be valid and functions satisfy hypothesis with for . Let be such that

Assume also that there is an such that (99) holds and

If , , and

then there exists an such that

Theorem 29 (see [62, Theorem 6]). Assume that and are valid and functions satisfy hypothesis with for . Let the sequence converge to uniformly on . If the equations

are stable in with constants and

then for every solution of (92) there exists a sequence of solutions of (107), which converges to uniformly on .

Now, we show how some considerations concerning the iterative stability can be expressed in terms of difference equations; we will only deal with (85). Let us assume that . Then, hypothesis (H1) implies that is an attractive fixed point of . Indeed, for every , the sequence , where

tends to , since for all . Moreover, for every .

Let be a solution of (85). For a fixed , put

Then, by (85), we have

On the other hand, by (85) and (88),

Let be a function such that

Fix an and put

Then, we get

Next, condition (86) with yields

Hence, by (112), we obtain

and consequently, by (116),

So, in the particular case where , that is, , we have

Thus, we have shown that, in particular, if there is a with for , then (87) holds with .

We end this section with a very simple, but useful (we hope) observation, which is a simplified version of [64, Theorem 1]; it corresponds to the already mentioned [59, Theorem 1] and, in view of Theorem 24, it concerns relation between iterative stabilities of some special cases of (84) and (85). Using it, we can also deduce easily from Theorem 17 some stability results for (76) in the special case when all are additive.

Let be a nonempty set, a normed space, nonempty, a function mapping into , and a function mapping a nonempty set into and such that

where for simplicity we write for and for . Assume also that is a subgroup of ; that is,

Now, we are in a position to present the following theorem (cf. Definition 2).

Theorem 30. Let . Suppose that the equation

admits a solution . Then, the equation

is -stable if and only if so is (122).

Proof. Since the proof is very elementary and short, we present it here for the convenience of the readers.

Assume first that (122) is -stable. Let and satisfy the condition

Write . Then, and

Hence, there exists a solution of (122) such that

Clearly, is a solution of (123) and

The proof of the necessary condition is analogous. But, again for the convenience of the readers, we present it below. So, assume that (123) is -stable. Let and satisfy

Write . Then,

Hence, there exists a solution of (123) such that

Clearly, is a solution of (122) and

Remark 31. It is easily seen that the assumption that (122) admits a solution is very important in the proof of Theorem 30; an analogous hypothesis is also applied in [59, Theorem 1].

In the next section, we present some remarks on the issue of the existence of solutions of (122), resulting from some stability outcomes obtained for the equation.

5. A Description of Solutions

Let, as before, be a nonempty set, , a Banach space, and . In this section, we show how to derive from Theorem 16, in a very easy way, a description of solutions of the equation

under assumption (139). Note that (132) is a particular case of (84) (with ).

First, let us rewrite Theorem 16 in a simplified form with .

Corollary 32. Let be such that

If satisfies

then there exists a unique solution of (132) with

Moreover,

Let us next introduce some notions.

We say that a function is -invariant provided . Define an equivalence relation by

and write

It is easily seen that a function is -invariant if and only if is constant on for every .

Now, we are ready to present the following description of solutions of functional equation (132).

Corollary 33. Let be -invariant. Suppose that

Then, there exists a unique solution of (132) such that (135) holds. Moreover,

Proof. Observe that (133) holds with

and fulfils (134). Thus, it is enough to use Corollary 32.

6. Stability of Intervals and Regions

In this section, we assume that (H1) and the following hypothesis (instead of (H2)) are valid:

(H3) is a continuous function.

Then, for , where is given by (88). For each , put . Let be a continuous function such that

Then, there exists a unique continuous function satisfying (85) such that for (see [60, Theorem 2.1]).

Czerni [65, 66] has considered stability and uniform stability of real intervals for (85). First, we present the results concerning the case where the studied intervals do not depend on . Next, we proceed to the stability of regions, that is, to the case where the interval changes continuously with .

For simplicity, let us restrict our attention to the case where the studied intervals have the form for some . The interval is called a stable interval of (85) if for every and every there exists a such that if a continuous function satisfies the condition

then for its extension fulfilling equation (85) the condition

holds (see [65, Definition 3]).

Theorem 34 (see [65, Theorem 4]). Let , satisfy (H1) and (H3), respectively. Then, , where , is a stable interval of (85) if and only if

To explain the above theorem, we suppose that for some . By the continuity of , we can assume without loss of generality that . Then, there exists an such that . Let . Then, for each , if for a continuous function , then by (85)

Therefore, the interval cannot be stable. In other words, if for some , then we can take such and that .

The condition that for implies that for each solution of (85) and each we have

where is given by (109). Hence, if for an and a , then for all . Consequently, for any , we can take any in (143) to obtain (144). Moreover, such a does not depend on the choice of . Furthermore, by (147), we obtain that is an invariant set.

The condition for in Theorem 34 can be slightly weakened. In the case where for all from a vicinity of , we can replace, in Theorem 34, with the interval , where is arbitrarily taken from this vicinity.

A different situation is if we consider the problem of interval stability for some particular . It may happen that, in the case where condition (145) does not hold, we can still find for all and for some a such that (143) implies (144) for every satisfying (142). More precisely, for a fixed , to obtain the stability of , we need to assume that for . Then, by (147),

if . We will say that the interval is stable with respect to the set if for all and there exists such a that (143) implies (144) for every satisfying (142).

Theorem 34 can be now restated in the following form.

Theorem 35. Let , satisfy (H1) and (H3), respectively. Then, , where , is a stable interval of (85) with respect to an if and only if

Czerni [65] has also considered the stability of regions of the form

where is a continuous function which satisfies the inequality

The constant interval is now replaced by interval varying continuously with . Let us note that if for , then for any the constant function given by satisfies inequality (151).

Using the assumption that the function fulfills inequality (151), Czerni proved the following theorem.

Theorem 36 (see [66, Theorem 2.2]). Let , satisfy (H1) and (H3), respectively, and . Assume that is a continuous solution of inequality (151). Then, if there exists a such that the region is stable with respect to , then is stable with respect to each .

The above theorem has been used in the proof of the next result about stability of the region .

Theorem 37 (see [66, Theorem 3.2]). Let , satisfy (H1) and (H3), respectively, and . Assume that for a continuous solution of inequality (151) there exists a such that

Then, the region is stable with respect to each .

Let us note that, the assumption on function is the counterpart of condition (149) in the case of stability of . In the proof of the above theorem, it is showed that inequality (152) implies that the region is stable with respect to . Indeed, the compactness of gives that there exists a positive minimum of over . Taking any smaller than this minimum (and, of course, smaller than a given ), we obtain the stability of .

We say that solutions of (85) depend continuously on initial conditions if, for each solution of (85), each and, for an arbitrary sequence converging uniformly to , where each element of the sequence is a continuous function satisfying (142), the sequence of the extensions of the functions fulfilling equation (85) tends uniformly to on the interval (see [66, Definition 1.2]).

In the case where solutions of (85) depend continuously on initial conditions, we have the following result.

Theorem 38 (see [66, Theorem 3.3]). Let , satisfy (H1) and (H3), respectively. Assume that solutions of (85) depend continuously on initial conditions. If is a continuous solution of inequality (151), then the region is stable with respect to each .

However, without assuming continuous dependency on initial conditions the following theorem holds.

Theorem 39 (see [67, Theorem 2.1]). Let , satisfy (H1) and (H3), respectively, and . Assume that is a continuous solution of inequality (151). Then, the following conditions are equivalent.

(i)The region is not stable with respect to .(ii)There exists a sequence of elements of and a strictly increasing sequence of positive integers such that

where is given by

The results concerning the interval stability of (85) presented above and similar results for (12) (see [67, 68]) have been motivated by Shanholt’s paper [69] concerning the stability of sets for difference equations. To compare these results with stability results in the theory of difference equation, see, for example, [70–72].

7. Nonstability

It seems to be difficult to give a suitable (but simple) definition of nonstability of functional equations; some examples of such definitions can be found in [54, 57, 73–76]. Probably, it should refer to Definition 2 and therefore also to the operator . Thus, we should speak of -nonstability. Below, we present an example of such a nonstability result for a linear difference equation (as before stands for a Banach space over ).

Theorem 40 (see [74, Theorem 1]). Assume that is a sequence in , is a sequence in , and is a sequence of nonnegative real numbers such that

Then, there exists a sequence in satisfying

and such that, for every sequence in , given by

we have

Clearly, Theorem 40 shows that (under assumption (155)) difference equation (157) is not -stable, for instance, for every operator such that

with a bounded sequence , where for and can be completely arbitrary.

There arises a natural question if we can replace condition (155) by one of the following two conditions:

It follows from [74, Examples 1–4] that this is not possible.

For further examples of similar nonstability results (also for other equations), we refer the reader to [54, 57, 73–76].

8. Multivalued Solutions

The issue of stability of functional equations in one variable has been investigated also for multivalued functions, and for suitable results we refer the reader to [77–80].

In this part of the paper, we present only one example of such results (on selections of set-valued maps satisfying linear inclusions), which is closely connected to the issue of stability of the corresponding functional equations.

Let be a nonempty set and be a metric space. We will denote by the family of all nonempty subsets of . The real number

is said to be the diameter of a nonempty set . Given , we write cl for the multifunction defined by

Each with

is said to be a selection of the multifunction .

The following result has been obtained in [77].

Theorem 41 (see [77, Theorem 2]). Let , , , , and

(1)If is complete and

then, for each , the limit

exists and is a unique selection of the multifunction cl such that

(2)If

then is a single-valued function and

For a survey on further similar results, we refer the reader to [81].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, London, UK, 1960.

D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941.

D. H. Hyers, “Transformations with bounded m th differences,” Pacific Journal of Mathematics, vol. 11, no. 2, pp. 591–602, 1961.

D. H. Hyers and S. M. Ulam, “On approximate isometries,” Bulletin of the American Mathematical Society, vol. 51, pp. 288–292, 1945.

D. H. Hyers and S. M. Ulam, “Approximate isometries of the space of continuous functions,” Annals of Mathematics, vol. 48, pp. 285–289, 1947.

D. H. Hyers and S. M. Ulam, “Approximately convex functions,” Proceedings of the American Mathematical Society, vol. 3, pp. 821–828, 1952.

R. P. Agarwal, B. Xu, and W. Zhang, “Stability of functional equations in single variable,” Journal of Mathematical Analysis and Applications, vol. 288, no. 2, pp. 852–869, 2003.

N. Brillouët-Belluot, J. Brzdęk, and K. Ciepliński, “On some recent developments in Ulam's type stability,” Abstract and Applied Analysis, vol. 2012, Article ID 716936, 41 pages, 2012.

J. Brzdęk and K. Ciepliński, “Hyperstability and superstability,” Abstract and Applied Analysis, vol. 2013, Article ID 401756, 2013.

S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing, River Edge, NJ, USA, 2002.

G. L. Forti, “Hyers-Ulam stability of functional equations in several variables,” Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143–190, 1995.

D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, Mass, USA, 1998.

D. H. Hyers and T. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125–153, 1992.

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001.

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, NY, USA, 2011.

Z. Moszner, “Sur les définitions différentes de la stabilité des équations fonctionnelles,” Aequationes Mathematicae, vol. 68, no. 3, pp. 260–274, 2004.

Z. Moszner, “On the stability of functional equations,” Aequationes Mathematicae, vol. 77, no. 1-2, pp. 33–88, 2009.

T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.

J. Brzdęk, “Hyperstability of the Cauchy equation on restricted domains,” Acta Mathematica Hungarica, vol. 141, no. 1-2, pp. 58–67, 2013.

Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431–434, 1991.

T. M. Rassias, “On a modified Hyers-Ulam sequence,” Journal of Mathematical Analysis and Applications, vol. 158, no. 1, pp. 106–113, 1991.

T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.

T. M. Rassias, “Problem,” Aequationes Mathematicae, vol. 39, no. 2-3, p. 309, 1990.

T. M. Rassias and P. Šemrl, “On the behavior of mappings which do not satisfy Hyers-Ulam stability,” Proceedings of the American Mathematical Society, vol. 114, no. 4, pp. 989–993, 1992.

J. Brzdęk and M. Piszczek, “On stability of the linear and polynomial functional equations in single variable,” in Handbook of Functional Equations: Stability Theory, T. M. Rassias, Ed., Springer, New York, NY, USA, 2014.

J. Brzdęk, D. Popa, B. Xu, P. Pardalos, H. M. Srivastava, and P. Georgiev, “Remarks on stability of the linear functional equation in single variable,” in Nonlinear Analysis: Stability, Approximation, and Inequalities, pp. 91–119, Springer, New York, NY, USA, 2012.

J. Brzdęk, “On a method of proving the Hyers-Ulam stability of functional equations on restricted domains,” The Australian Journal of Mathematical Analysis and Applications, vol. 6, no. 1, article 4, 10 pages, 2009.

J. Brzdęk and A. Pietrzyk, “A note on stability of the general linear equation,” Aequationes Mathematicae, vol. 75, no. 3, pp. 267–270, 2008.

G. L. Forti, “Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 127–133, 2004.

G. L. Forti, “Elementary remarks on Ulam-Hyers stability of linear functional equations,” Journal of Mathematical Analysis and Applications, vol. 328, no. 1, pp. 109–118, 2007.

A. Pietrzyk, “Stability of the Euler-Lagrange-Rassias functional equation,” Demonstratio Mathematica, vol. 39, no. 3, pp. 523–530, 2006.

Z. Moszner, “On stability of some functional equations and topology of their target spaces,” Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, vol. 11, pp. 69–94, 2012.

D. Miheţ, “The probabilistic stability for a functional nonlinear equation in a single variable,” Journal of Mathematical Inequalities, vol. 3, no. 3, pp. 475–483, 2009.

D. Miheţ and C. Zaharia, “Probabilistic (quasi)metric versions for a stability result of Baker,” Abstract and Applied Analysis, vol. 2012, Article ID 269701, 10 pages, 2012.

D. Miheţ and C. Zaharia, “The probabilistic stability for the gamma functional equation,” Journal of Science and Arts, vol. 20, no. 3, pp. 297–302, 2012.

J. A. Baker, “The stability of certain functional equations,” Proceedings of the American Mathematical Society, vol. 112, no. 3, pp. 729–732, 1991.

L. Găvruţa, “Matkowski contractions and Hyers-Ulam stability,” Buletinul Ştiinţific al Universităţii Politehnica din Timişoara. Seria Matematică-Fizică, vol. 53, no. 2, pp. 32–35, 2008.

P. Gǎvruta and L. Gǎvruta, “A new method for the generalized Hyers-Ulam-Rassias stability,” International Journal of Nonlinear Analysis and Applications, vol. 1, no. 2, pp. 11–18, 2010.

M. Akkouchi, “Stability of certain functional equations via a fixed point of Ciric,” Filomat, vol. 25, no. 2, pp. 121–127, 2011.

D. Mihet, “The Hyers-Ulam stability for two functional equations in a single variable,” Banach Journal of Mathematical Analysis, vol. 2, no. 1, pp. 48–52, 2008.

L. Cădariu and V. Radu, “Fixed point methods for the generalized stability of functional equations in a single variable,” Fixed Point Theory and Applications, vol. 2008, Article ID 749392, 15 pages, 2008.

L. Cădariu and V. Radu, “Fixed points and stability for functional equations in probabilistic metric and random normed spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 589143, 18 pages, 2009.

R. Badora and J. Brzdęk, “A note on a fixed point theorem and the Hyers-Ulam stability,” Journal of Difference Equations and Applications, vol. 18, no. 6, pp. 1115–1119, 2012.

R. Badora and J. Brzdęk, “Fixed points of a mapping and Hyers-Ulam stability,” Journal of Mathematical Analysis and Applications, vol. 413, no. 1, pp. 450–457, 2014.

L. Cădariu, M. S. Moslehian, and V. Radu, “An application of Banach's fixed point theorem to the stability of a general functional equation,” Analele Universităţii din Timişoara: Seria Matematică-Informatică, vol. 47, no. 3, pp. 21–26, 2009.

L. Cădariu, L. Găvruţa, and P. Găvruţa, “Weighted space method for the stability of some nonlinear equations,” Applicable Analysis and Discrete Mathematics, vol. 6, no. 1, pp. 126–139, 2012.

M. Alimohammady and A. Sadeghi, “Stability and common stability for the systems of linear equations and its applications,” Mathematical Sciences, vol. 6, article 43, 2012.

C. Borelli, “On Hyers-Ulam stability for a class of functional equations,” Aequationes Mathematicae, vol. 54, no. 1-2, pp. 74–86, 1997.

D. Miheţ, “The probabilistic stability for a functional equation in a single variable,” Acta Mathematica Hungarica, vol. 123, no. 3, pp. 249–256, 2009.

L. Cădariu and V. Radu, “The fixed points method for the stability of some functional equations,” Carpathian Journal of Mathematics, vol. 23, no. 1-2, pp. 63–72, 2007.

L. Cădariu and V. Radu, “Remarks on the stability of a functional equation of quadratic type,” Österreich Akademie der Wissenschaften Mathematisch-Naturwissenschaftliche Klasse. Sitzungsberichte, vol. 216, pp. 15–32 (2008), 2007.

L. Cadariu and V. Radu, “Fixed points and generalized stability for functional equations in abstract spaces,” Journal of Mathematical Inequalities, vol. 3, no. 3, pp. 463–473, 2009.

T. Trif, “On the stability of a general gamma-type functional equation,” Publicationes Mathematicae, vol. 60, no. 1-2, pp. 47–61, 2002.

J. Brzdęk, D. Popa, and B. Xu, “Remarks on stability and non-stability of the linear functional equation of the first order,” Applied Mathematics and Computation, vol. 238, pp. 141–148, 2014.

J. Brzdęk, D. Popa, and B. Xu, “The Hyers-Ulam stability of nonlinear recurrences,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 443–449, 2007.

D. Popa, “Hyers-Ulam-Rassias stability of a linear recurrence,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 591–597, 2005.

J. Brzdęk, D. Popa, and B. Xu, “Remarks on stability of linear recurrence of higher order,” Applied Mathematics Letters, vol. 23, no. 12, pp. 1459–1463, 2010.

D. Brydak, “On the stability of the functional equation ,” Proceedings of the American Mathematical Society, vol. 26, pp. 455–460, 1970.

B. Choczewski, E. Turdza, and R. Węgrzyk, “On the stability of a linear functional equation,” Rocznik Naukowo-Dydaktyczny: Wyższa Szkoła Pedagogiczna w Krakowie, Prace Matematyczne, vol. 9, pp. 15–21, 1979.

M. Kuczma, Functional Equations in a Single Variable, Państwowe Wydawnictwo Naukowe, Warszawa, Poland, 1968.

E. Turdza, “The stability of an iterative linear equation,” in General Inequalities 4 (Oberwolfach, 1983), pp. 277–285, Birkhäuser, Basel, Switzerland, 1984.

E. Turdza, “Some remarks on the stability of the non-linear functional equation of the first order,” Demonstratio Mathematica, vol. 6, pp. 883–891, 1973.

E. Turdza, “On the stability of the functional equation of the first order,” Annales Polonici Mathematici, vol. 24, pp. 35–38, 1970-1971.

J. Brzdęk, “Remarks on stability of some inhomogeneous functional equations,” Aequationes Mathematicae. In press

M. Czerni, “Asymptotic interval stability for a linear homogeneous functional equation,” Wyższa Szkoła Pedagogiczna w Krakowie. Rocznik Naukowo-Dydaktyczny. Prace Matematyczne, vol. 12, pp. 19–36, 1987.

M. Czerni, “Stability of normal regions for linear homogeneous functional equations,” Aequationes Mathematicae, vol. 36, no. 2-3, pp. 176–187, 1988.

M. Czerni, “Further results on stability of normal regions for linear homogeneous functional equations,” Aequationes Mathematicae, vol. 49, no. 1-2, pp. 1–11, 1995.

M. Czerni, “Stability of normal regions for nonlinear functional equation of iterative type,” in Stability of Mappings of Hyers-Ulam Type, Th. M. Rassias and J. Tabor, Eds., Hadronic Press Collection of Original Articles, pp. 67–79, Hadronic Press, Palm Harbor, Fla, USA, 1994.

G. A. Shanholt, “Set stability for difference equations,” International Journal of Control, vol. 19, no. 2, pp. 309–314, 1974.

P. Cull, M. Flahive, and R. Robson, Difference Equations: From Rabbits to Chaos, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 2005.

S. Elaydi, An Introduction to Difference Equations, Springer, New York, NY, USA, 2005.

W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Harcourt/Academic Press, San Diego, Calif, USA, 2001.

J. Brzdęk, D. Popa, and B. Xu, “Note on nonstability of the linear recurrence,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 76, no. 1, pp. 183–189, 2006.

J. Brzdęk, D. Popa, and B. Xu, “On nonstability of the linear recurrence of order one,” Journal of Mathematical Analysis and Applications, vol. 367, no. 1, pp. 146–153, 2010.

J. Brzdęk, D. Popa, and B. Xu, “Note on nonstability of the linear functional equation of higher order,” Computers & Mathematics with Applications, vol. 62, no. 6, pp. 2648–2657, 2011.

J. Brzdęk, D. Popa, and B. Xu, “A note on stability of the linear functional equations of higher order and fixed points of an operator,” Fixed Point Theory, vol. 13, no. 2, pp. 347–355, 2012.

M. Piszczek, “On selections of set-valued inclusions in a single variable with applications to several variables,” Results in Mathematics, vol. 64, no. 1-2, pp. 1–12, 2013.

M. Piszczek, “The properties of functional inclusions and Hyers-Ulam stability,” Aequationes Mathematicae, vol. 85, no. 1-2, pp. 111–118, 2013.

M. Piszczek, “On selections of set-valued maps satisfying some inclusions in a single variable,” Mathematica Slovaca. In Press

M. Piszczek, “Inclusions in a single variable in ultrametric spaces and Hyers-Ulam stability,” The Scientific World Journal, vol. 2013, Article ID 129637, 5 pages, 2013.

J. Brzdęk and M. Piszczek, “Selections of set-valued maps satisfying some inclusions and the Hyers-Ulam stability,” in Handbook of Functional Equations: Stability Theory, T. M. Rassias, Ed., Springer.