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

x∈E 󵄩󵄩󵄩󵄩g (x) − f (x) 󵄩󵄩󵄩󵄩 ≤ ε. (5) 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 ofHyers [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).

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][28][29][30][31].
Let us recall that the linear functional equation of the order  ∈ N has the form where  is a nonempty set,  is a linear space over a field F ∈ {R, C}, and the functions  :  → ,   :  → , and   :  → F for  = 1, . . .,  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  = 1, when the equation takes the form  () =  1 ()  ( 1 ()) +  () .
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].
In the case where C consists of all constant functions from R +     and T(C) contains only constant functions, the T-stability is usually called the Hyers-Ulam (or the Ulam-Hyers) stability.

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 Mihet ¸in [33] and Mihet ¸and Zaharia in [34,35]).
Theorem 3 (see [36,Theorem 2]).Let  be a nonempty set, (, ) a complete metric space,  :  → ,  :  ×  → ,  ∈ [0, 1), and If  :  → ,  > 0 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  : Theorem 4 (see [37,Theorem 2.2]).Let  be a nonempty set, (, ) a complete metric space, and  :  → ,  : × → .Assume also that where  : R + → R + is a comparison function, and let  :  → ,  > 0 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.
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].
Now, we present a result from [45].
The next two stability outcomes were obtained in [46].
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,  :  → ,  :  ×  → ,  ∈ R + , and and the series ∑ ∞ =0   (  ()) 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][51][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.
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 Then there exists a unique sequence {  } ∈N 0 ⊂  such that with an  ∈ R + .
Theorem 19 (see [55,Theorem 2]).Let (, ) be a metric space, Suppose that there exists {  } ∈N 0 ⊂ R + with Then there exist a sequence {  } ∈N 0 ⊂  and an  > 0 such that We refer the reader to [57] (and the references therein) for further stability results for linear difference equations of higher orders.

Iterative Stability
Let  = (0, ] for a  > 0 and  :  → , , ℎ :  → R given functions.Consider the linear nonhomogenous equation and its homogenous version where  :  → R is unknown.Brydak [58] (cf. 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.
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 0 does not belong to the domain of the solutions.
Let us yet introduce the following two assumptions.
(B) There exists an interval  ⊂  such that the sequence {  } ∈N converges uniformly to the zero function on .
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  > 0 such that for any  > 0 and solution  ∈ () of the inequality
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 .
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].
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.
The subsequent two theorems concern iterative stability (the first one has actually been proved in [63]).
Theorem 25 (see [62,  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.
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, C ⊂ R +  nonempty, T a function mapping C into R +  , and F a function mapping a nonempty set U ⊂   into   and such that where for simplicity we write F  := F(  ) for  = 1, 2 and ( 1 +  2 )() :=  1 () +  2 () for  ∈ .Assume also that U 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  0 ∈ U.Then, the equation is T-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 The proof of the necessary condition is analogous.But, again for the convenience of the readers, we present it below.So, assume that ( 123 (131) Remark 31.It is easily seen that the assumption that (122) admits a solution  0 ∈ U 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.

A Description of Solutions
Let, as before,  be a nonempty set,  :  → ,  a Banach space, and ℎ :  → .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 () ≡ 1).
First, let us rewrite Theorem 16 in a simplified form with () ≡ 1.
and  fulfils (134).Thus, it is enough to use Corollary 32.

Stability of Intervals and Regions
In this section, we assume that (H1) and the following hypothesis (instead of (H2)) are valid: (H3)  :  → (0, ∞) is a continuous function.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  > 0. The interval [, ∞) is called a stable interval of (85) if for every  > 0 and every  * ∈  there exists a  > 0 such that if a continuous function  0 :  * → R satisfies the condition then for its extension  * :  → R fulfilling equation (85) the condition holds (see [65,Definition 3]).

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][74][75][76]. Probably, it should refer to Definition 2 and therefore also to the operator T. Thus, we should speak of T-nonstability.Below, we present an example of such a nonstability result for a linear difference equation (as before  stands for a Banach space over F ∈ {R, C}).
There arises a natural question if we can replace condition (155) by one of the following two conditions:  For further examples of similar nonstability results (also for other equations), we refer the reader to [54,57,[73][74][75][76].

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][78][79][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 .
is said to be a selection of the multifunction .
The following result has been obtained in [77].For a survey on further similar results, we refer the reader to [81].
follows from [74, Examples 1-4] that this is not possible.