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

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