Keywords

1 Introduction

In what follows \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{R}\), and \(\mathbb{C}\) denote the sets of positive integers, integers, reals, and complex numbers, respectively; moreover, \(\mathbb{R}_+:=[0,\infty)\), and \(\mathbb{N}_0:=\mathbb{N}\cup \{0\}\).

The issue of stability of functional equations has been a very popular subject of investigations for more than 50 years. The first known result on it is due to Gy. Pólya and G. Szegö [54] and reads as follows.

For every real sequence \((a_n)_{n\in \mathbb{N}}\) with

$$\sup_{n,m\in \mathbb{N}}|a_{n+m}- a_n - a_{m} |\leq 1,$$

there is a real number ω such that

$$\sup_{n\in \mathbb{N}}|a_n - \omega n |\leq 1.$$

Moreover,

$$\omega = \lim_{n\to\infty} \frac{a_n}n.$$

But the main motivation for investigation of that subject was given by S. M. Ulam, who in 1940 in his talk at the University of Wisconsin discussed a number of unsolved problems. The following question concerning the stability of homomorphism was among them. Let G 1 be a group and \((G_2,d)\) a metric group. Given \(\varepsilon>0\) , does there exist \(\delta>0\) such that if \(f:G_1 \rightarrow G_2\) satisfies

$$d(f(xy),f(x)f(y))<\delta$$

for all \(x,y \in G_1\) , then a homomorphism \(T:G_1 \rightarrow G_2\) exists with

$$d(f(x),T(x))< \varepsilon$$

for all \(x,y \in G_1\) ?

The first answer to it was published in 1941 by D. H. Hyers [40]. The subsequent theorem contains an extension of it.

Theorem 1

Let E 1 and E 2 be two normed spaces, \(c\ge 0\) and \(p\in \mathbb{R}\setminus \{1\}\) . Assume that \(f:E_1\to E_2\) satisfies the inequality

$$\|f(x+y)-f(x)-f(y)\|\le c(\|x\|^p+\|y\|^p), \qquad x,y\in E_1\setminus \{0\}.$$

If E 2 is complete and \(p\ge 0\) , then there is a unique \(T:E_1\to E_2\) that is additive (i.e., \(T(x+y)=T(x)+T(y)\) for \(x,y\in E_1\) ) and fulfills

$$\|f(x)-T(x)\|\le \frac{c}{\big|2^{p-1}-1\big|}\;\|x\|^p, \qquad x\in E_1\setminus \{0\}.$$
(1)

If \(p<0\) , then f is additive.

It contains the results of Hyers [40] (p = 0), Aoki [2] and Rassias [59] (\(p\in(0,1)\)), Gajda [38] (\(1<p\)), and Brzd\c{e}k [11] (\(p<0\)).

From [38] it follows that an analogous result is not true for p = 1 (see [41–43] for more details). Moreover, it has been proved in [10] that estimation (1) is optimum.

Results similar to Theorem 1 have been proved for numerous other functional equations. Also, the theorem has been generalized and extended in various directions. For more detailed information we refer to [3, 7 39 41–43, 48, 60, 62].

We can introduce the following general definition of the notion of stability that corresponds to the outcomes collected in Theorem 1 (for some comments on various possible definitions of stability we refer to [51–53]).

Definition 1

Let \(n\in \mathbb{N}\), A be a nonempty set, \((X,d)\) be a metric space, \(\mathcal{C}\subset{\mathbb{R}_+}^{A^n}\) be nonempty, \(\mathcal{T}\) be a function mapping \(\mathcal{C}\) into \({\mathbb{R}_+}^A\), and \(\mathcal{F}_1,\mathcal{F}_2\) be functions mapping nonempty \(\mathcal{D}\subset X^A\) into \(X^{A^n}\). We say that the equation

$$\mathcal{F}_1\varphi (x_1,\ldots,x_n)=\mathcal{F}_2\varphi (x_1,\ldots,x_n)$$
(2)

is \(\mathcal{T}\) – stable provided for every \(\varepsilon\in \mathcal{C}\) and \(\varphi_0\in \mathcal{D}\) with

$$\begin{aligned} d\big(\mathcal{F}_1\varphi_0 (x_1,\ldots,x_n),\mathcal{F}_2\varphi_0 (x_1,\ldots,x_n)\big)\le \varepsilon (x_1,\ldots,x_n)\,\\ x_1,\ldots,x_n\in A,\end{aligned}$$

there is a solution \(\varphi\in \mathcal{D}\) of (2) such that

$$d\big(\varphi (x),\varphi_0(x)\big)\le \mathcal{T}\varepsilon (x),\qquad x\in A. $$

Let us mention that given two nonempty sets, by A B we denote, as usual, the family of all functions mapping B into A.

2 Stability of Zeros of Polynomials

That notion of stability of functional equations, described above, inspired numerous authors to investigate stability of other mathematical objects, in a similar manner (see, e.g., [7, 35, 41–43]).

For instance Li and Hua [49] started to study stability of the solutions of the following polynomial equation

$$x^{n}+\alpha x+ \beta=0,$$
(3)

with \(x\in [-1,1]\), where α and β are fixed real numbers and n is a positive integer. They have proved the following theorem.

Theorem 2

Assume that \(\vert \alpha\vert>n\) and

$$\vert \beta \vert<\vert \alpha\vert -1.$$

Then there exists a real constant \(K>0\) , such that for each \(\varepsilon>0\) and \(y\in[-1,1]\) with

$$\vert y^{n} +\alpha y +\beta \vert\leq\varepsilon, $$

there is a solution \(v\in[-1,1]\) of Eq. ( 3 ) such that

$$\vert y-v \vert\leq K\varepsilon. $$

They have asked if an analogous property is true for more general polynomials of the form

$$a_{n} z^{n}+ a_{n-1} z^{n-1}+ {\ldots}+a_{1}z+a_{0}=0. $$

In this way they have inspired authors of the papers [6, 44]. For example, the following result has been proved in [6].

Theorem 3

Let \(\varepsilon>0\) and \(a_0,\ldots,a_n\in \mathbb{R}\) be such that

$$\vert a_{0} \vert < \vert a_{1} \vert - (\vert a_{2} \vert +\vert a_{3} \vert +{\ldots}+ \vert a_{n} \vert), $$
$$\vert a_{1} \vert> 2 \vert a_{2} \vert + 3 \vert a_{3} \vert +{\ldots}+ (n-1)\vert a_{n-1} \vert + n \vert a_{n} \vert. $$

If \(y \in [-1,1]\) fulfills the inequality

$$ \vert a_{n}y^{n} + a_{n-1}y^{n-1}+{\ldots}+a_{1}y+a_{0} \vert \leq \varepsilon,$$

then there is \(z\in [-1,1]\) with

$$a_{n} z^{n}+ a_{n-1} z^{n-1}+ {\ldots}+a_{1}z+a_{0}=0 $$

and

$$|y-z|\le \lambda \varepsilon,$$

where

$$\lambda:= \frac{2 \vert a_{2} \vert + 3 \vert a_{3} \vert + {\ldots} + (n-1) \vert a_{n-1} \vert + n \vert a_{n} \vert}{|a_{1}|} < 1.$$

S.-M. Jung [44] has proved the subsequent theorem.

Theorem 4

Let \(\mathbb{K} \in \{\mathbb{R,C}\}\), \(n \in \mathbb{N}\), \(a_{0}, a_{1}, {\ldots},a_{n} \in \mathbb{K}\), \(r>0\) and

$$\begin{aligned} B_{r}=\{\omega \in \mathbb{K}: \vert \omega \vert \leq r\}.\end{aligned}$$

Assume that

$$\vert a_{1} \vert> \sum_{i=2}^{n} i r^{i-1} \vert a_{i} \vert, $$
$$\vert a_{0} \vert \leq \sum_{i=2}^{n} (i-1) r^{i} \vert a_{i} \vert. $$

If \(\varepsilon>0\) and \(z \in B_{r}\) fulfill the inequality

$$\vert a_{n} z^{n}+ a_{n-1} z^{n-1}+ {\ldots}+a_{1}z+a_{0} \vert \leq \varepsilon, $$

then there is \(z_{0} \in B_{r}\) such that

$$a_{n} z_0^{n}+ a_{n-1} z_0^{n-1}+ {\ldots}+a_{1}z_0+a_{0}=0$$

and

$$\begin{aligned} \vert z-z_{0} \vert \leq \frac{\varepsilon}{(1-\lambda) \vert a_{1} \vert},\end{aligned}$$

where

$$\lambda:= \frac{1}{\vert a_{1} \vert} \sum_{i=2}^{n} i r^{i-1} \vert a_{i} \vert <1.$$

Further generalization of those two theorems have been obtained in [16], where stability of the following functional equation

$$f(x)+\sum_{j=1}^{m} a_j(x)f(\xi_j(x))^{p(j)}=G(x),$$
(4)

has been studied in the class of functions f mapping a nonempty set S into a commutative Banach algebra X over a field \(\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}\), with the unit element denoted by e, where \(m\in \mathbb{N}\), \(G\in X^S\) and \(a_1,\ldots,a_{m}\in X^S\), \(p:\{1,\ldots,m\}\to \mathbb{N}\), \(\xi_1,\ldots,\xi_m\in S^S\) \(\xi_1,\ldots,\xi_m\in S^S\). We write \(f(y)^0=e\) and

$$f(y)^k:=(f(y))^k,\qquad k\in \mathbb{N}.$$

Note that the linear functional equation (in single variable)

$$f(x)+\sum_{j=1}^{m} a_j(x)f(\xi_j(x))=G(x)$$
(5)

is a particular case of Eq. (4) (when \(p(i)=1\) for \(i=1,\ldots,m\)). It is very well known and its stability has already been studied in several papers, under various additional assumptions. For more information on its solutions we refer to [46,47].

In this chapter we present a survey of those stability results concerning Eqs. (4) and (5), published by various authors.

For examples of other stability results for functional equations in single variable see for instance to [1, 4, 5, 7–9, 12, 13, 18, 22, 27–30, 36, 37, 45, 65–69]. For information on polynomials and their solutions we refer to [50,61].

3 Stability of the Linear Equation: The General Case

In what follows we assume that S is a nonempty set, \(\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}\), \(m\in \mathbb{N}\), and \(\xi_1,\ldots,\xi_m\in S^S\), unless explicitly stated otherwise.

We start our survey with the following general result that can be easily deduced from [22, Corollary 4].

Theorem 5

Let X be a commutative Banach algebra over a field \(\mathbb{K}\), \(a_1,\ldots,a_{m}\in X^S\), \(\varepsilon:S\to\mathbb{R}_+\), \(\phi:S\to X\),

$$q(x):=\sum_{i=1}^{m}\|a_i(x)\|<1,$$
$$\varepsilon(\xi_i(x)) \le\varepsilon(x)$$

and

$$q(\xi_i(x)) \le q(x)$$
(6)

for \(x\in S\), \(i=1,\ldots,m\) . Assume that

$$\Big\|\phi(x)+\sum_{i=1}^{m}a_i(x)\phi(\xi_i(x))-G(x)\Big\|\leq \varepsilon(x), \qquad x\in S. $$

Then, for each \(x\in S\) , the limit

$$f(x):=\lim_{n\to\infty}\mathcal{T}^n\phi(x)$$

exists and the function \(f:S\to X\) , defined in this way, is the unique solution to Eq. ( 5 ) such that

$$\|\phi(x)-f(x)\|\leq \frac{\varepsilon(x)}{1-q(x)},\qquad x\in S,$$

where \(\mathcal{T}:X^S\to X^S\) is given by:

$$\mathcal{T}g(x):=G(x)-\sum_{i=1}^{m}a_i(x)g(\xi_i(x)), \qquad g\in X^S,\,x\in S. $$

Clearly, assumption (6) is fulfilled when

$$\|a_i(\xi_i(x))\| \le \|a_i(x)\|,\qquad x\in S,i=1,\ldots,m; $$

this is the case, e.g., when the functions \(a_1,\ldots,a_{m}\) are constant.

In the case m = 1 Eq. (5) takes the form

$$\begin{aligned}\varphi(x)+a_1(x) \varphi(\xi_1(x))=G(x).\end{aligned}$$
(7)

If ξ1 is bijective, then it can be rewritten in the form

$$\begin{aligned}\varphi(\xi(x))=a(x) \varphi(x)+F(x)\end{aligned}$$
(8)

with \(\xi:=\xi_1^{-1}\),

$$a(x):=-a_1(\xi(x)),\qquad x\in S,$$

and

$$F(x):= G(\xi(x)),\qquad x\in S.$$

Also, if a 1 takes only the scalar values and \(0\not\in a_1(S)\), then (7) can be written as (8) with

$$a(x):=-\frac 1{a_1(x)},\qquad x\in S,$$

and

$$F(x):=\frac{G(x)}{a_1(x)},\qquad x\in S.$$

Stability of (8) has been investigated in [5, 8, 24, 56–58, 63] (for some related results see, e.g., [8, 9, 29–34, 64–69]); it seems that the most general result has been provided in [24, Lemma 1] and it is presented below. As usual, for each \(p\in \mathbb{N}_0\), we write ξp for the p-th iterate of ξ, i.e.,

$$\xi^0(x)=x,\qquad x\in S,$$

and

$$ \xi^{p+1}(x)=\xi(\xi^p(x)),\qquad p\in \mathbb{N}_0, x\in S,$$

and, only if ξ is bijective,

$$ \xi^{-p}=\left(\xi^{-1}\right)^p,$$

where \(\xi^{-1}\) denotes the function inverse to ξ.

From now on we assume that X is a Banach space over \(\mathbb{K}\), \(F\in X^S\)and \(\xi\in S^S\), unless explicitly stated otherwise.

Theorem 6

Let \(\varepsilon_0: S\to \mathbb{R}_+\), \(a: S\to \mathbb{K}\),

$$S':=\{x\in S: a(\xi^p(x))\neq 0 \;\mbox{for}p\in \mathbb{N}_0\},$$
$$\varepsilon'(x):= \sum_{k=0}^{\infty}\frac{\varepsilon_0(\xi^k(x))}{\prod_{p=0}^k|a(\xi^p(x))|}<\infty, \qquad x\in S',$$

and \(\varphi_s: S\to X\) be a function satisfying the inequality

$$\begin{aligned}\|\varphi_s(\xi(x))-a(x)\varphi_s(x)-F(x)\|\le \varepsilon_0(x),\qquad x\in S.\end{aligned}$$
(9)

Suppose that the function

$$\begin{aligned}\xi_0:=\xi|_{S\setminus S'}\end{aligned}$$
(10)

(i.e., the restriction of ξ to the set \(S\!\setminus\! S'\) ) is injective and

$$\begin{aligned}\xi(S\!\setminus\! S')\subset S\!\setminus\! S',\qquad a(S\!\setminus\! S')\subset \{0\}.\end{aligned}$$

Then the limit

$$\varphi'(x):=\lim_{n\to \infty}\left[\frac{\varphi_s(\xi^n(x))}{\prod_{j=0}^{n-1}a(\xi^j(x))}- \sum_{k=0}^{n-1}\frac{F(\xi^k(x))}{\prod_{j=0}^{k}a(\xi^j(x))}\right] $$

exists for every \(x\in S'\) and the function \(\varphi:S \to X\) , given by:

$$\varphi(x): =\begin{cases} \varphi'(x), & \text{if $x\in S'$;}\\ F(\xi_0^{-1}(x)), & \text{if $x\in \xi(S)\setminus S'$;}\\ \varphi_s(x)+u(x), & \text{if $x\in S\setminus [S'\cup \xi(S)]$,}\end{cases} $$

with any \(u:S\to X\) such that

$$\|u(x)\|\leq \varepsilon_0(x)\;,\qquad x\in S,$$

is a solution of functional Eq. ( 8 ) with

$$\begin{aligned} \|\varphi_s(x)-\varphi(x)\|\le \varepsilon'(x)\;, \qquad x\in S,\end{aligned}$$
(11)

where

$$\begin{aligned} \varepsilon'(x):= \left\{\!\!\!\begin{array}{ll} \varepsilon_0(\xi_0^{-1}(x)), & \text{if $x\in \xi(S)\setminus S'$;}\\ \varepsilon_0(x), & \text{if $x\in S\setminus [S'\cup \xi(S)]$}.\end{array}\right.\end{aligned}$$

Moreover, ϕ is the unique solution of ( 8 ) that satisfies (11) if and only if

$$S= S'\cup \xi(S).$$

To simplify the statements, in Theorem 6 it is assumed that assumption (10) is fulfilled by every function \(\xi:S\to S\) when the set \(S\setminus S'\) is empty. Note that in the case \(S\setminus S'=\emptyset\), Theorem 6 takes the following much simpler form, which is actually [63, Theorem 2.1].

Theorem 7

Let \(\varepsilon_0: S\to \mathbb{R}_+\), \(a: S\to \mathbb{K}\setminus\{0\}\),

$$\varepsilon'(x):= \sum_{k=0}^{\infty}\frac{\varepsilon_0(\xi^k(x))}{\prod_{p=0}^k|a(\xi^p(x))|}<\infty, \qquad x\in S,$$

and \(\varphi_s: S\to X\) be a function satisfying inequality ( 9 ). Then the limit

$$\begin{aligned} \varphi(x):=\lim_{n\to \infty}\left[\frac{\varphi_s(\xi^n(x))}{\prod_{j=0}^{n-1}a(\xi^j(x))}- \sum_{k=0}^{n-1}\frac{F(\xi^k(x))}{\prod_{j=0}^{k}a(\xi^j(x))}\right]\end{aligned}$$

exists for every \(x\in S\) and the function \(\varphi:S \to X\) , defined in this way, is the unique solution of functional Eq. ( 8 ) that satisfies inequality (11).

The next result has been stated in [24, Corollary 1].

Theorem 8

Let \(a: S\to \mathbb{K}\), \(\varepsilon_0: S\to \mathbb{R}_+\), \(\varphi_s:S \to X\) satisfy ( 9 ), ξ be bijective,

$$S'':=\{x\in S: a(\xi^{-p}(x))\neq 0 \;\mbox{for}p\in \mathbb{N}\},$$

\(\xi(S'')\subset S''\), \(a(S\setminus S'')\subset \{0\}\) , and

$$\varepsilon''(x):=\sum_{k=1}^{\infty} \varepsilon_0(\xi^{-k}(x))\prod_{p=1}^{k-1} |a(\xi^{-p}(x))| < \infty, \qquad x\in S''.$$

Then, for every \(x\in S''\) , the limit

$$\varphi''(x):=\lim_{n\to \infty}\left[\varphi_s(\xi^{-n}(x)) \prod_{j=1}^{n}a(\xi^{-j}(x))+ \sum_{k=1}^{n}F(\xi^{-k}(x)) \prod_{j=1}^{k-1}a(\xi^{-j}(x))\right] $$

exists and the function \(\varphi:S \to X\) , given by

$$\varphi(x): =\begin{cases} \varphi''(x), & \text{if $x\in S''$;}\\ F(\xi^{-1}(x)), & \text{if $x\in S\setminus S''$,}\end{cases} $$

is the unique solution of Eq. ( 8 ) such that

$$\|\varphi_s(x)-\varphi(x)\|\le \varepsilon''(x), \qquad x\in S, $$

where

$$\varepsilon''(x)=\varepsilon_0(\xi^{-1}(x))\;,\qquad x\in S\setminus S''.$$

For some remarks and examples complementing the above results see [26, pp. 96, 97].

Let us yet present one more simple result from [24, Lemma 2] (a function h mapping S into a nonempty set P is ξ-invariant provided \(h(\xi(x))=h(x)\) for \(x\in S\)).

Theorem 9

Assume that ξ is bijective, \(\varepsilon_0:S\to \mathbb{R}_+\) and \(a:S\to \mathbb{K}\) are ξ-invariant,

$$\overline{S}:=\{x\in S: |a(x)|\neq 1\},$$

and \(\varphi_s:S\to X\) satisfies ( 9 ). Then there exists a unique solution \(\varphi:\overline{S} \to X\) of Eq. ( 8 ) such that

$$\|\varphi_s(x)-\varphi(x)\| \leq \frac{\varepsilon_0(x)}{|1-|a(x)|\,|}, \qquad x\in \overline{S}. $$

It follows from [24, Remark 7.7] that, in the statement of Theorem 9, in some situations ϕ cannot be extended to a solution of (8) that maps S into X.

In several cases it can be proved that the assumptions, that appear in the theorems containing the stability results, are necessary. So, one could guess that in the case when some of them are not fulfilled, we should be able to obtain a kind of nonstability outcomes. It is true, but the point is that in general it is very difficult to give a (reasonably simple) general definition of nonstability; for examples of such definitions we refer to [17, 20, 23–25]. If we base on Definition 1, then such nonstability notion should refer to the operator \(\mathcal{T}\) and it seems that we should speak of \(\mathcal{T}\)-nonstability. Below we give an example of such nonstability result for m = 1, given in [21, Theorem 1], and the reader will easily identify the suitable operator \(\mathcal{T}\).

Theorem 10

Assume that \((\overline{a}_n)_{n\in \mathbb{N}_0}\) is a sequence in \(\mathbb{K}\setminus \{0\}\), \((b_n)_{n\in \mathbb{N}_0}\) is a sequence in X and \((\varepsilon_n)_{n\in \mathbb{N}_0}\) is a sequence of positive real numbers such that

$$ \lim_{n\to \infty}\frac{\varepsilon_n|\overline{a}_{n+1}|}{\varepsilon_{n+1}}=1.$$

Then there exists a sequence \((x_n)_{n\in \mathbb{N}_0}\) in X satisfying

$$\|x_{n+1}-\overline{a}_nx_n-b_n\|\le \varepsilon_n,\qquad n\in \mathbb{N}_0,$$

and such that, for every sequence \((y_n)_{n\in \mathbb{N}_0}\) in X, given by

$$y_{n+1}=\overline{a}_ny_n+b_n,\qquad n\in \mathbb{N}_0,$$

we have

$$\lim_{n\to \infty}\frac{\|x_n-y_n\|}{\varepsilon_{n-1}}=\infty.$$

For further examples of nonstability results we refer to [17, 20, 23–25]. At the end of the next section we give examples of nonstability results for \(m>1\).

4 Stability of the Linear Equation: Iterative Case

In this section we focus on a special iterative case of (5), when there is a function \(\xi:S\to S\) such that

$$\xi_j:=\xi^{j},\qquad j=1,\ldots,m.$$

Then (5) takes the form

$$f(x)+\sum_{j=1}^{m} a_j(x)f(\xi^{j}(x))=G(x).$$
(12)

If ξ is bijective, then it can be rewritten in the form

$$f(\eta^m(x))=\sum_{j=1}^m b_j(x)f(\eta^{m-j}(x))+F(x)$$
(13)

(analogously as in the previous section by replacing x by \(\xi^{-m}(x)\)) with \(\eta:=\xi^{-1}\) and

$$b_i(x):=-a_i(\eta^m(x)),\qquad F(x):= G(\eta^m(x)),\qquad x\in S, i=1,\ldots,m.$$

Also, if a m takes only the scalar values and \(0\not\in a_m(S)\), then (12) can be written in the form of (13) with \(\eta:=\xi\) and

$$b_i(x):=-\frac {a_{m-i}}{a_m(x)},\qquad F(x):=\frac{G(x)}{a_m(x)},\qquad x\in S,i=1,\ldots,m-1.$$

In what follows we use the following hypothesis concerning the roots of the equation

$$z^m-\sum_{j=1}^m b_j(x)z^{m-j}=0,$$
(14)

which (for \(x\in S\)) is the characteristic equation of functional Eq. (13). The hypothesis reads as follows.

\((\mathcal{H})\) \(\eta:S\to S\), \(b_1,\ldots, b_m:S\to \mathbb{K}\) \(F:S\to X\) and functions \(r_1,\ldots,r_m:S\to \mathbb{C}\) satisfy the following condition

$$\prod_{i=1}^m (z-r_i(x))=z^m-\sum_{j=1}^m b_j(x)z^{m-j}, \qquad x\in S, z\in \mathbb{C}. $$

It is easily seen that \((\mathcal{H})\) means that \(r_1(x),\ldots,r_m(x)\in \mathbb{C}\) are the complex roots of Eq. (14) for every \(x\in S\). Moreover, the functions \(r_1,\ldots,r_m\) are not unique, but for every \(x\in S\) the sequence

$$(r_1(x),\ldots,r_m(x))$$

is uniquely determined up to a permutation. Clearly,

$$0\not\in b_m(S) \mbox{if and only if} 0\not\in r_j(S) \mbox{for} j=1,\ldots,m.$$

As before, we say that that a function \(\varphi:S\to X\) is f-invariant provided

$$\varphi(f(x))=\varphi(x)\;, \qquad x\in S.$$

Note, that under the assumption that \((\mathcal{H})\) holds, \(b_1,\ldots,b_m\) are f-invariant if and only if \(r_1,\ldots,r_m\) can be chosen f-invariant (see [24, Remark 3]).

To simplify some statements we write

$$\prod_{p=1}^0 \lambda(h^p(x)):=1$$

for every \(h: S\to S\), \(\lambda:S\to \mathbb{K}\), \(x\in S\). Moreover, we assume that the restriction to the empty set of any function is injective.

Now we are in a position to present [24, Theorem 1] (see also [24, Remark 7]), which reads as follows.

Theorem 11

Let \(\varepsilon_0: S\to \mathbb{R}_+\), \((\mathcal{H})\) be valid, \(\varphi_s: S\to X\),

$$\Big\|\varphi_s(\eta^m(x))-\sum_{i=1}^m b_i(x)\varphi_s(\eta^{m-i}(x))-F(x)\Big\|\leq \varepsilon_0(x), \qquad x\in S,$$
(15)

r j be η-invariant for \(j>1\), \((i_1,\ldots,i_m)\in \{-1,1\}^m\) . Write

$$s_j:=\frac{1}{2}(1-i_j),\qquad j=1,\ldots,m,$$
$$S_1:=\{x\in S: r_1(\eta^{i_1p}(x))\neq 0 \;\mbox{for}p\in \mathbb{N}_0\}.$$

Assume that, for each \(j\in\{1,\ldots,m\}\) , one of the following three conditions holds:

  • \(1^{\circ}\) \(i_j=1\) for \(j=1,\ldots,m\) and \(0\not\in b_m(S)\) ;

  • [ \(2^{\circ}\) ] \(i_j=1\) for \(j=1,\ldots,m\) , η is injective, \(\eta(S\setminus S_1)\subset S\setminus S_1\), \(r_1(S\setminus S_1)\subset \{0\}\) ;

  • [ \(3^{\circ}\) ] η is bijective, \(\eta(S_1)\subset S_1\) , and \(r_1(S\setminus S_1)\subset \{0\}\) .

Further, suppose that

$$\varepsilon_1(x):=\sum_{k=s_1}^{\infty} \varepsilon_{0}(\eta^{i_1k}(x)) \prod_{p=s_1}^{k-s_1}|r_1(\eta^{i_1p}(x))|^{-i_1}< \infty, \qquad x\in S_1, $$
$$\varepsilon_j(x):=\sum_{k=s_j}^{\infty} \varepsilon_{j-1}(\eta^{i_jk}(x)) |r_j(x)|^{-i_j(k+i_j)}< \infty, \qquad x\in S_j, j\in\{2,\ldots,m\},$$

where

$$S_j:=\{x\in S: r_j(x)\neq 0\}, \qquad j>1,$$

and, in the case \(S\setminus S_j\neq \emptyset\),

$$\varepsilon_j(x): =\begin{cases} \varepsilon_{j-1}(\eta^{-1}(x)), & \text{if $x\in \eta(S)\setminus S_j$;}\\ \varepsilon_{j-1}(x), & \text{if $x\in S\setminus [S_j\cup \eta(S)]$,}\end{cases}$$

for \(x\in S\setminus S_j\), \(j\in\{1,\ldots,m\}\). Then Eq. ( 13 ) has a solution \(\varphi: S\to X\) with

$$\|\varphi_s(x)- \varphi(x)\|\le \varepsilon_m(x), \qquad x\in S. $$

Moreover, if r 1 is η-invariant and

$$S\setminus S_j\subset \eta(S\setminus S_j),\qquad j=1,\ldots,m,$$

then for each η-invariant function \(h:S\to \mathbb{R}\) Eq. ( 13 ) has at most one solution \(\varphi:S\to X\) such that

$$\|\varphi_s(x)-\varphi(x)\|\le h(x)\varepsilon_m(x),\qquad x\in S.$$

A simplified version of Theorem 11, with constant coefficient functions b j , can be found in [19].

If we assume that the functions \(\varepsilon_0,b_1,\ldots,b_m\) are η-invariant and η is bijective, then we obtain the following result, which is much simpler than Theorem 11 (see [24, Theorem 2]).

Theorem 12

Suppose that hypothesis \((\mathcal{H})\) holds, η is bijective, \(\varepsilon_0:S\to \mathbb{R}_+\) and \(b_1,\ldots,b_m\) are η-invariant,

$$\widetilde{S}:=\{x\in S:|r_j(x)| \neq 1\;\mbox{for} j=1,\ldots,m\},$$

and a function \(\varphi_s:S\to X\) is an ϵ 0 -solution of Eq. ( 13 ) that is ( 15 ) holds. Then there is a unique solution \(\varphi:\widetilde{S} \to X\) of ( 13 ) such that

$$\|\varphi_s(x)-\varphi(x)\| \leq \frac{\varepsilon_{0}(x)}{\left|(1-|r_1(x)|)\cdot \ldots \cdot (1-|r_m(x)|)\right|}, \qquad x\in \widetilde{S}\;.$$
(16)

Moreover, for each η-invariant function \(\varepsilon:\widetilde{S}\to \mathbb{R}\) , ϕ is the unique solution of ( 13 ) such that

$$\|\varphi_s(x)-\varphi(x)\| \leq \varepsilon(x),\qquad x\in \widetilde{S}\;.$$

It follows from [26, Remark 7.13] that, in the case \(\mathbb{K}=\mathbb{R}\) and

$$r_j(S)\subset [0,\infty),\qquad j=1,\ldots,m,$$

estimation (16) in Theorem 12 is the best possible in the general situation. But in some other situations we can get sometimes much better estimations than (16), as for instance in [14, Theorem 3.1] (cf. [14, p. 3]), which is stated for m = 2, \(F(x)\equiv 0\) and \(\varepsilon_0\) .b 1 and b 2 being constant functions; it reads as follows.

Theorem 13

Let \(\eta:S\to S\), \(b_1,b_2\in \mathbb{K}\), \(b_2\ne 0\), \(\bar{\varepsilon}> 0\) and \(g: S \to X\) satisfy the inequality

$$ \sup_{x \in S} \| g(\eta^2(x)) -b_1g(\eta(x))- b_2g(x) \| \leq \bar{\varepsilon} \qquad x\in S.$$

Suppose that one of the following three conditions is valid:

\((i)\) :

\(| s_i | < 1\) for \(i = 1, 2\) and \(s_1 \neq s_2\) ;

\((ii)\) :

\(| s_i | \neq 1\) for \(i = 1, 2\) and η is bijective;

\((iii)\) :

\((ii)\) holds and \(s_1 \neq s_2\),

where s 1 and s 2 denote the complex roots of the equation

$$b_2z^2+b_1z-1=0.$$

Then there exists a solution \(f: S \to X\) of the equation

$$f(\eta^2(x))= b_1f(\eta(x)) + b_2f(x), \qquad x\in S$$
(17)

such that

$$\sup_{x \in S} \| g(x) - f(x) \| \leq M \varepsilon, $$

where

$$M =\begin{cases} \min \{M_1, M_2\}, & \mbox{if}\;(i)\; \mbox{or}\;(iii)\; \mbox{holds}; \\ M_2, & \mbox{if}\;(ii) \mbox{ holds}\end{cases}$$

and

$$M_1:= \frac{1}{| s_1 - s_2 |} \left(\frac{| s_1 |}{| | s_1 | - 1 |} + \frac{| s_2 |}{| | s_2 | - 1 |} \right),$$
$$M_2:= \frac{1}{| (| s_1 | - 1)(| s_2 | - 1) |}.$$

Moreover, if \(| s_i | < 1\) for \(i = 1, 2\) , then there exists exactly one solution \(f: S \to X\) of Eq. ( 17 ) such that

$$\sup_{x \in S} \| g(x) - f(x) \| < \infty.$$

Related and even more general (to some extent) results for Eq. (17) can be derived from [15, Theorem 2.1].

Now, let us recall [24, Definition 3] (cf. [26, Definition 7.3]).

Definition 2

Eq. (13) is said to be strongly Hyers–Ulam stable (in the class of functions \(\psi:S\to X\)) provided there exists \(\alpha\in \mathbb{R}\) such that, for every \(\delta>0\) and for every \(\psi:S\to X\) satisfying

$$\sup_{x\in S}\Big\|\psi(\eta^m(x))-\sum_{i=1}^m b_i(x)\psi(\eta^{m-i}(x))-F(x)\Big\|\leq \delta,$$

there exists a solution \(\varphi:S\to X\) of (13) with

$$\sup_{x\in S}\|\varphi(x)-\psi(x)\|\leq \alpha\delta.$$

In [26, Corollary 7.4] the following result is stated.

Theorem 14

Suppose that hypothesis \((\mathcal{H})\) is valid, η is bijective and

$$\inf_{x\in S} |1-|r_j(x)||>0,\qquad j=1,\ldots,m.$$
(18)

Then, in the case where \(b_1,\ldots,b_m\) are η-invariant, Eq. ( 13 ) is strongly Hyers–Ulam stable.

From [26, Example 7.5] it follows (see [26, Remark 7.14]) that assumption (18) is necessary in the theorem above.

In the special case when the functions \(b_1,\ldots,b_m\) are constant, Eq. (13) becomes the following functional equation:

$$\varphi(\eta^m(x))=\sum_{i=1}^m b_i\varphi(\eta^{m-i}(x))+F(x)$$
(19)

with given fixed \(b_1,\ldots,b_m\in \mathbb{K}\). Then Theorems 11 and 12 obtain much simpler forms described in [24, Corollaries 3 and 4]. They show in particular that Eq. (19) is strongly Hyers–Ulam stable under the assumption that its characteristic equation

$$r^m-\sum_{i=1}^m b_i r^{m-i}=0$$
(20)

has no roots of module one. The assumption is necessary (see [26, Examples 7.6 and 7.7]).

Clearly, a simple particular case of functional Eq. (13), with S being either the set of nonnegative integers \(\mathbb{N}_0\) or the set of integers \(\mathbb{Z}\), is the difference equation

$$y_{n+m}=\sum_{i=1}^{m} b_{i}(n)y_{n+m-i}+d_n, \qquad n\in S,$$
(21)

for sequences \((y_n)_{n\in S}\) in X, where \((d_n)_{n\in S}\) is a fixed sequence in X; namely Eq. (13) becomes difference Eq. (21) with

$$f(n)=n+1, y_n:=f(n)=f(\eta^n(0)), d_n:=F(n), \qquad n\in S.$$

Stability and nonstability results for such difference equations can be found in [20], with constant functions b i . Let us recall here a nonstability outcome from [20, Theorem 4].

Theorem 15

Let \(T\in \{\mathbb{N}_0,\mathbb{Z}\}\), \(b_1,\ldots, b_m\in \mathbb{K}\) and \(r_1, \dots,r_m\) denote all the complex roots of Eq. ( 20 ). Assume that \(|r_j|=1\) for some \(j\in \{1, \dots,m\}\) . Then, for any \(\delta>0\) , there exists a sequence \((y_n)_{n\in T}\) in X, satisfying the inequality

$$\Big\|y_{n+m}-\sum_{i=1}^m b_i y_{n+m-i}-d_n\Big\|\leq \delta, \qquad n\in T, $$

such that

$$\sup_{n\in T}\;\|y_n-x_n\|=\infty $$

for every sequence \((x_n)_{n\in T}\) in X, fulfilling the recurrence

$$x_{n+m}=\sum_{i=1}^m b_i x_{n+m-i}+d_n, \qquad n\in T.$$
(22)

Moreover, if \(r_1, \dots,r_m \in \mathbb{K}\) or there is a bounded sequence \((x_n)_{n\in T}\) in X fulfilling ( 22 ), then \((y_n)_{n\in T}\) can be chosen unbounded.

The next theorem provides one more nonstability result from [26, Theorem 7.4].

Theorem 16

Suppose that \(\eta \in S^S$, $F\in X^S\), \(b_1,\ldots, b_m\in \mathbb{K}\) , Eq. ( 19 ) has a solution in the class of functions mapping S into X, characteristic Eq. ( 20 ) has a complex root of module 1, and there exists \(x_0\in S\) such that

$$\eta^k(x_0)\neq \eta^n(x_0)\;,\qquad k,n\in \mathbb{N}_0, k\neq n,$$

and

$$\eta(S\setminus S_0)\subset S\setminus S_0,$$

where

$$S_0:=\{\eta^n(x_0): n\in \mathbb{N}_0\}.$$

Then, for each \(\delta>0\) , there is a function \(\psi:S\to X\) , satisfying the inequality

$$\sup_{x\in S}\Big\|\psi(\eta^{m}(x))-\sum_{i=1}^{m} b_i\psi(\eta^{m-i}(x))-F(x)\Big\|\leq \delta, $$

such that

$$\sup_{x\in S}\;\|\psi(x)-\varphi(x)\|=\infty$$

for arbitrary solution \(\varphi:S\to X\) of Eq. ( 19 ).

Moreover, if all the roots of characteristic Eq. ( 20 ) are in \(\mathbb{K}\) , then ψ can be chosen unbounded.

A similar, but more general result has been obtained in [23, Theorem 1]. Below we present next two nonstability outcomes from [23, Theorems 2 and 3]. As before, \(\eta\in S^S$, $F\in X^S\) and, in the second theorem (see [23, Remark 1]), \(d_1,\ldots,d_{m-1}\) are the unique complex numbers such that

$$b_1=r_1+d_1,\qquad b_m= - r_1 d_{m-1}$$

and, in the case \(m>2\),

$$r_1 d_{j-1}+d_j,\qquad j=2,\ldots,m-1.$$

Theorem 17

Let \(b_1,\ldots, b_m\in \mathbb{K}\), \(m>1\), \(S_0\subset S\) be nonempty, \(\eta(S_0)\subset S_0\),

$$\sup_{x\in S_0}\|F(x)\|< \infty,$$
$$\sum_{j=1}^{m}b_j=1,$$

and

$$\lim_{n\to\infty} \Big\|\sum_{k=0}^n F(\eta^k(x_0))\Big\|=\infty$$

for some \(x_0\in S_0\) . Then Eq. ( 19 ) is nonstable on S 0 , that is there is a function \(\psi:S\to X\) such that

$$\sup_{x\in S_0}\Big\|\psi(\eta^{m}(x))-\sum_{i=1}^{m} b_i\psi(\eta^{m-i}(x))-F(x)\Big\|<\infty $$

and

$$ \sup_{x\in S_0}\;\|\psi(x)-\varphi(x)\|=\infty$$

for arbitrary solution \(\varphi:S\to X\) of Eq. ( 19 ).

Theorem 18

Let \(b_1,\ldots, b_m\in \mathbb{K}\), \(m>1\), \(S_0\subset S\) be nonempty. Suppose that Eq. ( 20 ) have a root \(r_1\in \mathbb{K}\) , there is a function \(\psi_0:S\to X\) such that

$$\sup_{x\in S_0}\|\psi_0(\eta(x))-r_1\psi_0(x)-F(x)\|< \infty$$

and the equation

$$\psi_1(\eta(x))=r_1\psi(x)+F(x)$$

has no solutions \(\psi_1:S\to X\) with

$$\sup_{x\in S_0}\|\psi_0(x)-\psi_1(x)\|<\infty.$$

Further, assume that the equation

$$\gamma(\eta^{m-1}(x))=\sum_{i=1}^{m-1} d_i\gamma(\eta^{m-i}(x))+\psi_0(x)$$

is nonstable on S 0 (in the sense described in Theorem 17) or has a solution \(\gamma:S\to X\) . Then Eq. ( 19 ) is nonstable on S 0 .

5 Set-Valued Case

In this part we present two theorems that contain results on selections of set-valued maps satisfying linear inclusions, which can be derived from Theorems 1 and 2 in [55]. They are closely related to the issue of stability of the corresponding functional equations.

Let K be a nonempty set and \((Y, d)\) be a metric space. We will denote by n(Y) the family of all nonempty subsets of Y. The nonnegative real number

$$\delta(A):=\sup\;\{d(x,y):\ x,y\in A\}$$

is said to be the diameter of a nonempty set \(A\subset Y\). For \(F: K\to n(Y)\) we denote by \({\rm cl} F\) the multifunction defined by

$$({\rm cl} F)(x):={\rm cl} F(x),\qquad x\in K.$$

Each function \(f: K\to Y\) such that

$$f(x)\in F(x),\qquad x\in K,$$

is said to be a selection of the multifunction F.

The theorems read as follows.

Theorem 19

Let \(F: K\to n(Y)\), \(m\in\mathbb{N}\), \(a_1, \ldots, a_m: K\to \mathbb{R}\), \(\xi_1, \ldots, \xi_m: K\to K\) and

$$\begin{aligned} \liminf_{n\to \infty}\sum_{i_1=1}^{k}|a_{i_1}(x)|\sum_{i_2=1}^{k}|a_{i_2}\circ\xi_{i_1}(x)|\ldots \sum_{i_n=1}^{k}|a_{i_n}\circ\xi_{i_{n-1}}\circ \ldots \circ\xi_{i_1}(x)|\\ \times \delta(F\circ\xi_{i_n}\circ\ldots \circ \xi_{i_1}(x))=0, \qquad x\in K.\end{aligned}$$
  1. (a)

    If Y is complete and

    $$\sum_{i=1}^{m}a_i(x)F(\xi_i(x))\subset F(x),\qquad x\in K, $$

    then there exists a unique selection \(f: K\to Y\) of the multifunction \({\rm cl} F\) such that

    $$\sum_{i=1}^{m}a_i(x)f(\xi_i(x))=f(x),\qquad x\in K.$$
  2. (b)

    If

    $$F(x)\subset \sum_{i=1}^{m}a_i(x)F(\xi_i(x)), \qquad x\in K, $$

    then F is a single-valued function and

    $$\sum_{i=1}^{m}a_i(x)F(\xi_i(x))= F(x),\qquad x\in K.$$

Theorem 20

Let \(m\in\mathbb{N}\), \(a_1, \ldots, a_m: K\to \mathbb{R}\), \(\xi_1, \ldots, \xi_m: K\to K\) , \(F, G: K\to n(Y)\),

$$\begin{aligned} k(x)&:=\delta(F(x)+G(x))\\ &\mbox{}+\sum_{l=1}^{\infty}\sum_{i_1=1}^{m}|a_{i_1}(x)|\sum_{i_2=1}^{m}|a_{i_2}\circ\xi_{i_1}(x)|\ldots \sum_{i_l=1}^{m}|a_{i_l}\circ\xi_{i_{l-1}}\circ\ldots \circ\xi_{i_1}(x)|\\ &\quad\mbox{}\times \delta(F\circ\xi_{i_l}\circ\ldots \circ\xi_{i_1}(x)+G\circ\xi_{i_l}\circ\ldots\circ\xi_{i_1}(x))<\infty,\end{aligned}$$
$$\sum_{i=1}^{m}a_i(x)F(\xi_i(x))\subset F(x)+G(x),$$

and \(0\in G(x)\) for all \(x\in K\) . Then there exists a unique function \(f: K\to Y\) such that, for each \(x\in K\),

$$\sum_{i=1}^{m}a_i(x)f(\xi_i(x))=f(x),$$
$$\sup_{y\in F(x)}d(f(x), y)\leq k(x).$$

6 Stability of the Polynomial Equation

We end this chapter with a result proved in [16] and concerning stability functional Eq. (4), i.e., the equation

$$ f(x)+\sum_{j=1}^{m} a_j(x)f(\xi_j(x))^{p(j)}=G(x).$$

In this section, as at the end of Sect. 2, X denotes a Banach commutative algebra over \(\mathbb{K}\), \(m\in \mathbb{N}\), \(a_1,\ldots,a_{m}:S\to X\), \(p:\{1,\ldots,m\}\to \mathbb{N}\), \(G\in X^S\)and \(\xi_1,\ldots,\xi_m\in S^S\).

In what follows, \(r>0\) is a fixed real number and

$$\mathcal{B}_r:=\big\{f\in X^S: \|f(x)\|\leq r \;\mbox{for}x\in S\big\}.$$

To simplify statements of the main results we define operators \(\mathcal{L}:X^S\to X^S\) and \(\Psi:\mathbb{R}_+^S\to \mathbb{R}_+^S\) by the formulas:

$$\begin{aligned} \mathcal{L}h(x)=G(x)-\sum_{i=1}^{m} a_i (x) h(\xi_i(x))^{p(i)},\qquad h\in X^S, x\in S,\\ \Psi\gamma(x)=\sum_{i=1}^{m} p(i)r^{p(i)-1}\|a_i (x)\|\gamma(\xi_i(x)),\qquad \gamma\in \mathbb{R}_+^S, x\in S.\end{aligned}$$

Now we are in a position to present [16, Theorem 2].

Theorem 21

Suppose that \(\delta \in \mathbb{R}_+^S\), \(\gamma\in \mathcal{B}_r\),

$$\begin{aligned} \Big\|\gamma(x)+\sum_{j=1}^{m} a_j(x)\gamma(\xi_j(x))^{p(j)}-G(x)\Big\|\leq \delta(x),\qquad x\in S,\\ \|G(x)\|\leq r - \sum_{i=1}^m\|a_i(x)\| r^{p(i)}, \qquad x\in S,\\ \chi(x):=\sum_{n=0}^{\infty}\Psi^n\delta(x)<\infty, \qquad x\in S.\end{aligned}$$

Then there is a unique solution \(f\in \mathcal{B}_r\) of Eq. ( 4 ) with

$$ \| f(x)-\gamma(x)\| \leq \chi(x), \qquad x\in S;$$

in particular

$$f(x)=\lim_{n\to\infty}\mathcal{L}^n\gamma(x),\qquad x\in S.$$

If in Theorem 21 we take \(S=\{t_0\}\), then it is easily seen that we obtain the following.

Corollary 1

Suppose that \(\xi_0,\xi_1,\ldots,\xi_m\in X\), \(z_0\in X\), \(\|z_0\|\leq r\),

$$ \|\xi_0\|\leq r - \sum_{i=1}^m\|\xi_i\| r^i,\qquad \lambda_0:=\sum_{j=1}^{m} jr^{j-1} \|\xi_j\| < 1.$$

Then there is a unique \(z\in X\) such that \(\|z\|\leq r\),

$$ z=\sum_{j=0}^{m} \xi_j z^{j}, \qquad \|z-z_0\|\leq \frac{1}{1-\lambda_0}\Big\| z_0-\sum_{j=0}^{m} \xi_j z_0^{j}\Big\|.$$

In particular

$$z=\lim_{n\to\infty}L^n (z_0),$$

with

$$L(w)=\sum_{i=0}^{m} \xi_i w^i,\qquad w\in X.$$