Abstract
In this note, we study the Ulam stability of a functional equation both in Banach and mBanach spaces. Particular cases of this equation are, among others, equations which characterize multiadditive and multiJensen functions. Moreover, it is satisfied by the socalled multilinear mappings.
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1 Background and Motivation
Assume that X is a linear space over the field \({\mathbb {F}}\), and Y is a linear space over the field \({\mathbb {K}}\). Let us recall (see for instance [20]) that a mapping \(f:X\rightarrow Y\) satisfies a linear functional equation provided
for some \(a_1,a_2\in {\mathbb {F}}\) and \(A_1,A_2\in {\mathbb {K}}\).
It is obvious that the functional equation
and, under the additional assumption that the characteristics of \({\mathbb {F}}\) and \({\mathbb {K}}\) are different from 2, the equation
(their solutions are said to be additive and Jensen mappings, respectively) are particular cases of (1). For more information about equations (2) and (3) and some applications of them ((3) is called the Jensen equation and it is connected with the notion of convexity) we refer the reader for example to [14, 15].
Given an \(n\in \mathbb {N}\) (throughout this note \({\mathbb {N}}\) stands for the set of all positive integers and \({\mathbb {N}}_0 :=\mathbb {N}\cup \{0\}\)) such that \(n\ge 2\), we will say that a function \(f:X^n\rightarrow Y\) is nlinear (roughly, multilinear) if it satisfies the linear functional equation in each of its arguments, i.e.
with some \(a_{i1},a_{i2}\in {\mathbb {F}}\) and \(A_{i1},A_{i2}\in {\mathbb {K}}\).
It is clear that multiadditive functions, introduced by S. Mazur and W. Orlicz (see for example [15], where one can also find their application to the representation of polynomial mappings), and multiJensen functions, defined in 2005 by W. Prager and J. Schwaiger (see for instance [21]) with the connection with generalized polynomials, are multilinear. Moreover, with \(k\in \mathbb {N}\) such that \(1\le k<n\), \(a_{11}=a_{12}=\ldots =a_{k1}=a_{k2}=1,\; a_{k+11}=a_{k+12}=\ldots =a_{n1}=a_{n2}=\frac{1}{2}\) and \(A_{11}=A_{12}=\ldots =A_{k1}=A_{k2}=1,\; A_{k+11}=A_{k+12}=\ldots =A_{n1}=A_{n2}=\frac{1}{2}\) in the above definition we obtain the notion of a kCauchy and \(nk\)Jensen (briefly, multiCauchyJensen) function (see [1, 2]).
Let \(a_{11},a_{12},\ldots ,a_{n1},a_{n2}\in {\mathbb {F}}\) and \(A_{i_1,\ldots ,i_n}\in {\mathbb {K}}\) for \(i_1,\ldots ,i_n\in \{1,2\}\) be given scalars. In this paper, we deal with the following quite general functional equation
Now, we present three functional equations which are special cases of the considered equation.
Example 1
Equation (4) with \(a_{11}=a_{12}=\ldots =a_{n1}=a_{n2}=1\) and \(A_{i_1,\ldots ,i_n}=1\) for \(i_1,\ldots ,i_n\in \{1,2\}\) leads to the following functional equation
which (see [9]) characterizes multiadditive mappings.
Example 2
Another particular case of equation (4), i.e. the functional equation
was introduced and investigated in [21].
Example 3
The functional equation
which characterizes multiCauchyJensen mappings (see [1]), is also a special case of equation (4).
Next, note that we have the following.
Proposition 1
If \(f:X^n\rightarrow Y\) is a multilinear mapping, then there exist \(A_{i_1,\ldots ,i_n}\in {\mathbb {K}}\) for \(i_1,\ldots ,i_n\in \{1,2\}\) such that equation (4) holds for any \(x_{11},x_{12},\ldots ,x_{n1},x_{n2}\in X\).
The question about an error we commit replacing an object fulfilling some properties only approximately by an actual object possessing these properties is natural and interesting in many scientific investigations. To deal with it one can use the notion of the Ulam stability.
In 1940, S.M. Ulam posed the problem of the stability of homomorphisms of metric groups (a year later, D.H. Hyers gave its solution in the case of Banach spaces). Another very important example is a question concerning the stability of isometries. This problem was investigated for instance in [3, 11, 13, 19, 23] (see also [18] for more information and references on this topic).
Let us recall that an equation is said to be Ulam stable in a class of functions provided each function from this class fulfilling our equation “approximately” is “near” to its actual solution.
In recent years, the stability of various objects has been studied by many researchers (for more information on the notion of the Ulam stability as well as its applications we refer the reader to [4, 6, 7, 12, 18]). Furthermore, some stability results on equations (5), (6) and (7) can be found, among others, in [1, 2, 9, 21].
In this note, the Ulam stability of equation (4) is shown. Moreover, we apply our main results (Theorems 2 and 7) to get some stability outcomes on functional equations (5) and (7).
Let us finally mention that as the concepts of an approximate solution and the nearness of two mappings can be obviously understood in various ways, we deal with the stability of the considered functional equations not only in classical Banach spaces, but also in mBanach spaces (i.e. spaces with nonstandard measures of the distance, namely the ones given by mnorms).
2 Stability in Banach Spaces
In this section, we prove the Ulam stability of functional equations (4), (5) and (7) in Banach spaces.
2.1 Main Result
We start with equation (4).
Theorem 2
Assume that Y is a Banach space, \(\varepsilon >0\) and
If \(f:X^n\rightarrow Y\) is a function satisfying
for \(x_{11},x_{12},\ldots , x_{n1},x_{n2}\in X\), then there is a unique mapping \(F:X^n\rightarrow Y\) fulfilling equation (4) and
for \(x_{1},\ldots , x_{n}\in X\).
Proof
Put
Let us first note that (9) with \(x_{i2}=x_{i1}\) for \(i\in \{1,\ldots ,n\}\) gives
and consequently
Fix \(l,p\in {\mathbb {N}}_0\) such that \(l<p\). Then
and thus for each \((x_{11},\ldots ,x_{n1})\in X^n\), \(\Big ( \frac{f(a_1^kx_{11},\ldots ,a_n^kx_{n1})}{A^{k}}\Big )_{k\in {\mathbb {N}}_0}\) is a Cauchy sequence. Using the fact that Y is a Banach space we conclude that this sequence is convergent, which allows us to define
Putting now \(l=0\) and letting \(p\rightarrow \infty \) in (12) we see that
i.e. condition (10) is satisfied.
Let us next observe that from (9) we get
for \(x_{11},x_{12},\ldots , x_{n1},x_{n2}\in X\) and \(k\in {\mathbb {N}}_0\). Letting now \(k\rightarrow \infty \) and applying definition (13) we deduce that
for \(x_{11},x_{12},\ldots , x_{n1},x_{n2}\in X\), and thus we see that the mapping \(F:X^n\rightarrow Y\) is a solution of functional equation (4).
Let us finally assume that \(G:X^n\rightarrow Y\) is another mapping fulfilling equation (4) and inequality (10) for \(x_{1},\ldots , x_{n}\in X\). Then for any \(l\in \mathbb {N}\) and \((x_{11},\ldots ,x_{n1})\in X^n\) we have
Letting now \(l\rightarrow \infty \) and using (8) we conclude that \(G=F\). \(\square \)
2.2 Corollaries
Now, we present some consequences of the main result.
First, consider the case \(a_{11}=a_{12}=\ldots =a_{n1}=a_{n2}=1\) and \(A_{i_1,\ldots ,i_n}=1\) for \(i_1,\ldots ,i_n\in \{1,2\}\). Then from Theorem 2 we get the following outcome on the Ulam stability of functional equation (5).
Corollary 3
Assume that Y is a Banach space and \(\varepsilon >0\). If \(f:X^n\rightarrow Y\) is a function satisfying
for \(x_{11},x_{12},\ldots , x_{n1},x_{n2}\in X\), then there is a unique solution \(F:X^n\rightarrow Y\) of equation (5) such that
Another consequence of Theorem 2 is a result on the stability of equation (7). Namely, we have the following.
Corollary 4
Assume that Y is a Banach space and \(\varepsilon >0\). If \(f:X^n\rightarrow Y\) is a function satisfying
for \(x_{11},x_{12},\ldots , x_{n1},x_{n2}\in X\), then there is a unique solution \(F:X^n\rightarrow Y\) of equation (7) such that
Let us finally observe that a particular case of Theorem 2 is Theorem 2.1 in [10].
3 Stability in mBanach Spaces
In this section, we deal with the Ulam stability of the considered equations in mBanach spaces.
3.1 Preliminaries
In 1989, A. Misiak (see [17]) generalized the notion of 2normed space introduced by S. Gähler a quarter of a century earlier and defined mnormed spaces. Now, we recall (see for instance [5, 17, 22]) some basic definitions and facts concerning such spaces.
Let \(m\in \mathbb {N}\) be such that \(m\ge 2\) and Y be an at least mdimensional real linear space. If a mapping \(\Vert \cdot \,, \ldots , \cdot \Vert :Y^m\rightarrow \mathbb R\) fulfils the following four conditions:
(i) \(\Vert x_1, \ldots , x_m\Vert = 0\) if and only if \(x_1, \ldots , x_m\) are linearly dependent,
(ii) \(\Vert x_1, \ldots , x_m\Vert \) is invariant under permutation,
(iii) \(\Vert \alpha x_1, \ldots , x_m\Vert = \alpha  \Vert x_1, \ldots , x_m\Vert \),
(iv) \(\Vert x+y, x_2, \ldots , x_m\Vert \le \Vert x, x_2, \ldots , x_m\Vert + \Vert y, x_2, \ldots , x_m\Vert \)
for any \(\alpha \in {\mathbb {R}}\) and \(x,y, x_1, \ldots , x_m \in Y\), then it is said to be an mnorm on Y, whereas the pair \((Y, \Vert \cdot , \ldots , \cdot \Vert )\) is called an mnormed space.
Let us mention the following two known properties of mnorms.
Remark 5
Assume that \((Y, \Vert \cdot , \ldots , \cdot \Vert )\) is an mnormed space. Then:

(i)
the mapping \(\Vert \cdot , \ldots , \cdot \Vert \) is nonnegative;

(ii)
for any \(k\in {\mathbb {N}}, x_2 ,\ldots ,x_m \in Y\) and \(y_i\in Y\) for \(i\in \{1, \ldots , k\}\) we have
$$\begin{aligned} \Big \Vert \sum _{i=1}^{k}y_i , x_2, \ldots , x_m \Big \Vert \le \sum _{i=1}^{k}\Vert y_i , x_2, \ldots , x_m\Vert . \end{aligned}$$
Let \((y_k )_{k\in {\mathbb {N}}}\) be a sequence of elements of an mnormed space \((Y, \Vert \cdot , \ldots , \cdot \Vert )\). We say that it is Cauchy sequence provided
On the other hand, the sequence \((y_k)_{k\in {\mathbb {N}}}\) is called convergent if there is a \(y\in Y\) such that
Then the element y is said to be the limit of \((y_k)_{k\in {\mathbb {N}}}\) and it is denoted by \(\lim _{k\rightarrow \infty }y_k\). Obviously each convergent sequence has exactly one limit and the standard properties of the limit of a sum and a scalar product hold true.
By an mBanach space we mean an mnormed space such that each its Cauchy sequence is convergent.
We will also use the following known facts.
Remark 6
Assume that \((Y, \Vert \cdot , \ldots , \cdot \Vert )\) is an mnormed space. Then:

(i)
if \(x_1, \ldots , x_m\in Y, \alpha \in {\mathbb {R}}\), \(i,j\in \{1,\ldots ,m\}\) and \(i< j\), then
$$\begin{aligned} \Vert x_1,\ldots , x_i,\ldots ,x_j,\ldots , x_m\Vert = \Vert x_1,\ldots , x_i,\ldots , x_j + \alpha x_i, \ldots , x_m \Vert ; \end{aligned}$$ 
(ii)
if \(x, y, y_2,\ldots , y_m\in Y\), then
$$\begin{aligned} \big \Vert x, y_2,\ldots ,y_m\Vert  \Vert y, y_2,\ldots ,y_m\Vert \big  \le \Vert xy, y_2,\ldots ,y_m\Vert ; \end{aligned}$$ 
(iii)
if \(x\in Y\) and
$$\begin{aligned} \Vert x, y_2,\ldots ,y_m\Vert =0,\qquad y_2,\ldots , y_m\in Y, \end{aligned}$$then \(x=0\);

(iv)
if \((x_k)_{k\in {\mathbb {N}}}\) is a convergent sequence of elements of Y, then
$$\begin{aligned} \lim _{k\rightarrow \infty } \Vert x_k, y_2,\ldots ,y_m\Vert = \big \Vert \lim _{k\rightarrow \infty } x_k, y_2,\ldots ,y_m\big \Vert ,\qquad y_2,\ldots ,y_m \in Y. \end{aligned}$$
Let us finally mention that more information on mnormed spaces as well as on some problems investigated in them can be found for example in [5, 8, 16, 17, 22].
3.2 Main Result
Now, we prove the Ulam stability of equation (4).
Theorem 7
Assume that \(m\in \mathbb {N}\), \(\varepsilon >0\), Y is an \((m+1)\)Banach space, and (8) holds true. If \(f:X^n\rightarrow Y\) is a function satisfying
for \(x_{11},x_{12},\ldots , x_{n1},x_{n2}\in X\) and \(z\in Y^m\), then there is a unique mapping \(F:X^n\rightarrow Y\) fulfilling equation (4) and
for \(x_{1},\ldots , x_{n}\in X\) and \(z\in Y^m\).
Proof
Let A and \(a_i\) for \(i\in \{1,\ldots ,n\}\) be as in the proof of Theorem 2, and fix \(l,p\in {\mathbb {N}}_0\) such that \(l<p\). It is easy to see that
which shows that for each \((x_{11},\ldots ,x_{n1})\in X^n\), \(\Big ( \frac{f(a_1^kx_{11},\ldots ,a_n^kx_{n1})}{A^{k}}\Big )_{k\in {\mathbb {N}}_0}\) is a Cauchy sequence. Using now the fact that Y is an \((m+1)\)Banach space we conclude that this sequence is convergent, which allows us to define the mapping \(F:X^n\rightarrow Y\) by (13).
Next, putting \(l=0\) and letting \(p\rightarrow \infty \) in (16), and using Remark 6 we see that
which means that condition (15) is satisfied.
Let us also note that from (14) we get
for \(x_{11},x_{12},\ldots , x_{n1},x_{n2}\in X\), \(z\in Y^m\) and \(k\in {\mathbb {N}}_0\). Thus, letting \(k\rightarrow \infty \) and applying definition (13) and Remark 6 we finally conclude that the mapping \(F:X^n\rightarrow Y\) is a solution of equation (4).
Let us finally assume that \(G:X^n\rightarrow Y\) is another mapping fulfilling equation (4) and inequality (15) for \(x_{1},\ldots , x_{n}\in X\) and \(z\in Y^m\). Then for any \(l\in \mathbb {N},\, (x_{11},\ldots ,x_{n1})\in X^n\) and \(z\in Y^m\) we have
Letting now \(l\rightarrow \infty \) and using (8) together with Remarks 5 and 6 we conclude that \(G=F\). \(\square \)
3.3 Corollaries
Now, we present three consequences of the main result of this section.
First, consider the case \(a_{11}=a_{12}=\ldots =a_{n1}=a_{n2}=1\) and \(A_{i_1,\ldots ,i_n}=1\) for \(i_1,\ldots ,i_n\in \{1,2\}\). Then from Theorem 7 we get the following outcome on the stability of equation (5).
Corollary 8
Assume that \(m\in \mathbb {N}\), \(\varepsilon >0\) and Y is an \((m+1)\)Banach space. If \(f:X^n\rightarrow Y\) is a function satisfying
for \(x_{11},x_{12},\ldots , x_{n1},x_{n2}\in X\) and \(z\in Y^m\), then there is a unique solution \(F:X^n\rightarrow Y\) of equation (5) such that
for \((x_{1},\ldots , x_{n})\in X^n\) and \(z\in Y^m\).
Another consequence of Theorem 7 is the following result on the Ulam stability of equation (7).
Corollary 9
Assume that \(m\in \mathbb {N}\), \(\varepsilon >0\) and Y is an \((m+1)\)Banach space. If \(f:X^n\rightarrow Y\) is a function satisfying
for \(x_{11},x_{12},\ldots , x_{n1},x_{n2}\in X\) and \(z\in Y^m\), then there is a unique solution \(F:X^n\rightarrow Y\) of equation (7) such that
for \((x_{1},\ldots , x_{n})\in X^n\) and \(z\in Y^m\).
Let us finally mention that Theorem 7 generalizes Theorem 3.3 in [10].
3.4 Final Remark
In the paper we consider, among others due to historical reasons, mnormed and mBanach spaces for \(m\ge 2\). However, one can also admit the case \(m=1\) and then get the results of the previous section as corollaries of the ones proved in Sect. 3.
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The author is grateful to the referee for a number of helpful suggestions for improvement in the article.
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Ciepliński, K. On Ulam Stability of a Functional Equation. Results Math 75, 151 (2020). https://doi.org/10.1007/s00025020012754
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DOI: https://doi.org/10.1007/s00025020012754