On Ulam Stability of Functional Equations in Non-Archimedean Spaces

Abstract

We present a survey of outcomes on Ulam stability of functional equations in non-Archimedean normed spaces. We focus mainly on functional equations in several variables (including the Cauchy equation, the Jordan–von Neumann equation, the Jensen equation, and their generalizations), but we also report a result on a general equation in a single variable, which can be applied to the very important linear functional equation. Let us note that one can observe the symmetry between the presented results and the analogous ones obtained for both classical and two-normed spaces.

1. Introduction

It is well-known that the theory of Ulam stability was initiated in 1940 by S.M. Ulam’s question concerning the stability of homomorphisms of metric groups, and that the first partial affirmative answer to this question was given a year later in [1] by D.H. Hyers in the case of Banach spaces. Since then this topic has drawn the attention of many researchers (see, for example, [2,3,4,5,6,7]).

Since the ideas of an approximate solution and the proximity of two objects appearing in Ulam’s problem can be interpreted in different ways, one can deal with the stability problem not only in classical Banach spaces, but also in other types of spaces with unconventional distance measures.

In [8,9], the authors presented a survey of the published results on Ulam stability for functional equations in two-normed spaces. The aim of this paper is to perform this in the case of non-Archimedean normed spaces. In this way, we demonstrate the symmetry between the presented results and the analogous ones obtained for two-normed spaces.

1.1. Functional Equations

The basic division of functional equations are equations of one and several variables.

The most famous functional equation is the Cauchy equation,

$$A(x+y)=A\left(x\right)+A\left(y\right),$$

(1)

and its solutions are called additive mappings. Let us recall (see, for example, [10,11]) that, for $A:\mathbb{R}\to \mathbb{R}$, Equation (1) was first treated by A.M. Legendre and C.F. Gauss, but its general continuous solution was found by A.L. Cauchy, and the equation has been named after him.

Among other functional equations in several variables, the Jensen equation,

$$J\left({\displaystyle \frac{x+y}{2}}\right)={\displaystyle \frac{J\left(x\right)+J\left(y\right)}{2}}$$

(2)

and the Jordan–von Neumann equation,

$$V(x+y)+V(x-y)=2V\left(x\right)+2V\left(y\right)$$

(3)

deserve special attention. Let us recall that Equation (2) is closely connected with the notion of convex function, whereas Equation (3) (which is also called quadratic) is helpful in some characterizations of inner product spaces. One can find a lot of information about these two functional equations and their applications, for example, in [10,11,12,13]. Let us yet mention that solutions for (2) are said to be Jensen mappings, while solutions for (3) are called quadratic mappings.

As for functional equations in a single variable, we focus our attention on the linear equation,

$$L\left(f\right(x\left)\right)=g\left(x\right)L\left(x\right)+h\left(x\right),$$

(4)

where f, g, and h are given functions, and the mapping L is unknown.

The investigations into solutions for Equation (4) in different classes of functions were surveyed in [14,15], where some of its applications in ergodic theory, probability theory and differential equations were also presented. As for the dynamical context of (4), one can consult [16,17].

Let us next mention a few important particular cases of the linear equation.

Putting $f\left(x\right)=x+1$, $g\left(x\right)=x$ and $h\left(x\right)=0$ in (4) gives the well-known gamma functional equation,

$$L(x+1)=xL\left(x\right),$$

(5)

which is helpful in some characterizations (see, for example, the appreciated Bohr–Mollerup theorem) of Euler’s gamma function.

Another very important (notably in the theory of dynamical systems) particular case of (4) is the cohomological equation,

$$L\left(f\right(x\left)\right)=L\left(x\right)+h\left(x\right).$$

(6)

Several results on its monotonic, smooth, and analytic solutions were described in [14,15,17]. On the other hand, in [18], the cohomological equation is discussed from the point of view of functional analysis and dynamical systems, and some applications of Equation (6) in dynamical systems can be found in [19].

Let us yet pay attention to two more special cases of (4), i.e., the Schröder functional equation,

$$L\left(f\right(x\left)\right)=sL\left(x\right)$$

(7)

and the Abel functional equation,

$$L\left(f\right(x\left)\right)=L\left(x\right)+1.$$

(8)

It is known that these two equations have been present in complex dynamics from its beginnings (see [20] for more information), and have been investigated in various settings since then. One can find a lot of information about the solutions for Equations (7) and (8), as well as their applications (for instance, in dynamical systems and ergodic theory, differential equations, and probability theory), for example, in [14,15,16,17,21].

Let us finally mention that other functional equations in a single variable were studied very recently, for example, in [22,23,24].

1.2. Ulam Stability

One of the ways to determine an error we make replacing an object having some properties by an object satisfying them only approximately is the notion of the Ulam (or Hyers–Ulam) stability.

Let us recall that, in 1940, S.M. Ulam asked the following question, currently known as Ulam’s stability problem: given two groups, $G_1$ and $G_2$ (we assume that $G_2$ is equipped with a metric), and an approximate homomorphism, $f:G_1\to G_2$, is f close to a homomorphism? The answer to this general question depends on the groups $G_1$ and $G_1$, as well as the chosen notions of an approximate homomorphism and proximity between functions.

In 1974, K. Grove, H. Karcher and E.A. Ruh (see [25]) showed that unitary representations of compact groups are stable with respect to the operator norm. Their result was generalized to amenable groups by D. Kazhdan in 1982 (see [26]) as follows:

Let $f:G\to U\left(\mathcal{H}\right)$ be a function from an amenable group G into the group $U\left(\mathcal{H}\right)$ of unitary operators on the Hilbert space $\mathcal{H}$. If $\delta <{\displaystyle \frac{1}{200}}$ and

$${\parallel f\left(xy\right)-f\left(x\right)f\left(y\right)\parallel }_{\mathrm{op}}\le \delta ,x,y\in G,$$

then there is a group homomorphism $h:G\to U\left(\mathcal{H}\right)$, such that

$${\parallel h\left(x\right)-f\left(x\right)\parallel }_{\mathrm{op}}\le 2\delta ,x\in G.$$

In 2017, W. T. Gowers and O. Hatami (see [27]) obtained a stability theorem of this kind for the Hilbert–Schmidt norm, in the case where $G_1$ is a finite group and $G_2$ is the unitary group $U\left(n\right)$.

Let us next recall that an equation is said to be Ulam stable or Hyers–Ulam stable in a class of functions, provided that each function from this class satisfying the equation “approximately” is “near” to its solution.

It is also known that Ulam’s stability problem can be considered as a problem in terms of the stability of Cauchy Equation (1).

In 1941, D.H. Hyers in [1] gave its partial solution (instead of groups he considered Banach spaces). Then, his result was extended in various ways. The following theorem is one of such generalizations.

Theorem 1.

Let $E_1$ and $E_2$ be normed spaces, $E_2$ be complete, $c>0$, and $p\ne 1$. If $f:E_1\to E_2$ satisfies

$$\parallel f(x+y)-f\left(x\right)-f\left(y\right)\parallel \le {c(\parallel x\parallel }^p+{\parallel y\parallel }^p),x,y\in E_1,$$

then there exists a unique additive mapping $a:E_1\to E_2$, such that

$$\parallel f\left(x\right)-a\left(x\right)\parallel \le {\displaystyle \frac{{c\parallel x\parallel }^p}{|1-2^{p-1}|}},x\in E_1.$$

Let us recall that, for $0\le p<1$, Theorem 1 was proved by T. Aoki in [28] and Th.M. Rassias in [29], for $p>1$ by Z. Gajda in [30], and for $p<0$ by Th.M. Rassias in [31]. It is also well-known (see [30]) that for $p=1$, an analogous outcome is not valid (a phenomenon of instability occurs). Let us finally mention that a further generalization of Hyers’ result was suggested by D.G. Bourgin in [32].

Since then, the notion of the Ulam stability has been a popular topic of research, and it has been applied for various functional equations (see books [2,3,4,12] and survey papers [5,7,8,9] for a lot of examples and references).

Let us mention that Ulam type stability of some other objects (for example differential, difference and integral equations, isometries, operators, flows, random dynamics, vector measures, and $C^{*}$-algebras) has been also intensively studied (several references on these topics can be found, for instance, in [2,33]).

As for differential equations, D. Popa, I. Rasa, and A. Viorel, in [34], introduced the so-called conditional stability and showed that the logistic equation is conditionally Ulam stable. Their research was continued by M. Onitsuka in [35]. These two papers show an application of the Ulam stability in biology, as the logistic growth model is one of the classical models for population dynamics. Let us finally mention that the conditional Ulam stability of some other biological models has been also studied (see [36,37] for the von Bertalanffy growth model, and [38] for the nonautonomous Richards model).

1.3. Non-Archimedean Fields and Spaces

Let us recall (see, for instance, [39,40]) that by a non-Archimedean field, we mean a field $\mathbb{K}$ equipped with an absolute value (valuation) $|·|:\mathbb{K}\to [0,\infty )$, which is non-Archimedean, i.e., $|·|$ satisfies the inequality

$$|r+s|\le max\left\{\right|r|,|s\left|\right\},r,s\in \mathbb{K}.$$

Let p be a prime number. Put

$${\left|0\right|}_p:=0$$

and

$$|p^{\alpha }{\displaystyle \frac{a}{b}}{|}_p:=p^{-\alpha },$$

where $\alpha$ is an integer and $a,b$ are non-zero integers not divisible by p. The completion ${\mathbb{Q}}_p$ of $\mathbb{Q}$ with respect to ${|·|}_p$ is the field of p-adic numbers, the most interesting and important example of non-Archimedean fields.

Assume that $\mathbb{K}$ is a field with a non-trivial non-Archimedean absolute value, and let X be a non-Archimedean normed space, i.e., a linear space over $\mathbb{K}$ equipped with a non-Archimedean norm$\parallel ·\parallel :X\to [0,\infty )$, which means that the triangle inequality is strengthened to

$$\parallel x+y\parallel \le max\{\parallel x\parallel ,\parallel y\parallel \},x,y\in X.$$

Let us mention that in any non-Archimedean normed space X, one can define a metric in the standard way, and that a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ of elements of X is Cauchy if and only if ${(x_{n+1}-x_n)}_{n\in \mathbb{N}}$ converges to zero.

Let us also recall (see [41]) that the completion of a normed space over $(\mathbb{Q},|·{|}_p)$ is a Banach space over $({\mathbb{Q}}_p,|·{|}_p)$. Furthermore, the completion of a non-Archimedean normed space is a complete non-Archimedean space.

The p-adic numbers, although they might have seemed exotic when were introduced by K. Hensel at the end of the 19th century, are currently well-established in the mathematical world, and used by physicists as well.

There are, for instance, non-Archimedean branches of number theory, algebra (field and group theory), algebraic geometry, real and complex analysis, dynamical systems, functional analysis and operator theory.

Moreover, the p-adics are often more useful than the real numbers for attacking problems in number theory. It is known, for example, that A. Wiles (see [42]) applied them in the proof of Fermat’s Last Theorem.

Physicists have used p-adic numbers for instance in quantum mechanics (p-adic strings), condensed matter physics (disordered systems and spin glasses) and theoretical physics (AdS/CFT correspondence). For more details, we refer the reader to [43,44,45,46], as well as to survey paper [47], where some applications in biology, data mining, geology, cryptography and information security are also presented.

2. Stability of Functional Equations in Non-Archimedean Normed Spaces

2.1. Cauchy, Jordan–von Neumann, and Jensen Equations

It seems that the first stability results in non-Archimedean context was obtained by J. Schwaiger in [48]. It has been proven there that some functional equations are Ulam stable in the class of mappings from a commutative group that is uniquely divisible by p to a Banach space over ${\mathbb{Q}}_p$.

Next, L.M. Arriola and W.A. Beyer, in [49], showed that if $f:{\mathbb{Q}}_p\to \mathbb{R}$ is a continuous function for which there exists a fixed $ϵ$, such that

$$|f(x+y)-f\left(x\right)-f\left(y\right)|\le ϵ,x,y\in {\mathbb{Q}}_p,$$

then there exists a unique additive function $T:{\mathbb{Q}}_p\to \mathbb{R}$, such that

$$|f\left(x\right)-T\left(x\right)|\le ϵ,x\in {\mathbb{Q}}_p.$$

In 2007, M.S. Moslehian and Th.M. Rassias (see [50]) studied the generalized Ulam stability of the Cauchy and Jordan–von Neumann equations in a more general setting, namely in complete non-Archimedean normed spaces.

Let us first recall their result on the stability of the Cauchy functional equations.

Theorem 2.

Let V be a commutative semigroup and W be a complete non-Archimedean normed space. Assume also that $\phi :V^2\to [0,\infty )$ is a mapping such that, for any $x,y\in V$,

$$\underset{j\to \infty }{lim}{\displaystyle \frac{1}{{\left|2\right|}^j}}\phi (2^{j}x,2^{j}y)=0$$

(9)

and the limit

$$\underset{k\to \infty }{lim}max\{{\displaystyle \frac{1}{{\left|2\right|}^j}}\phi (2^{j}x,2^{j}x):0\le j<k\},$$

(10)

denoted by $\tilde{\phi }\left(x\right)$, exists. If $f:V\to W$ is a function satisfying

$$\parallel f(x+y)-f\left(x\right)-f\left(y\right)\parallel \le \phi (x,y),x,y\in V,$$

(11)

then there exists an additive mapping $F:V\to W$ for which

$$\parallel f\left(x\right)-F\left(x\right)\parallel \le {\displaystyle \frac{1}{\left|2\right|}}\tilde{\phi }\left(x\right),x\in V.$$

(12)

The function F is given by

$$F\left(x\right):=\underset{j\to \infty }{lim}{\displaystyle \frac{1}{2^j}}f\left(2^{j}x\right),x\in V.$$

(13)

If, moreover,

$$\underset{l\to \infty }{lim}\underset{k\to \infty }{lim}max\{{\displaystyle \frac{1}{{\left|2\right|}^j}}\phi (2^{j}x,2^{j}x):l\le j<k+l\}=0,x\in V,$$

(14)

then F is the unique additive mapping satisfying Condition (12).

Let us mention here that if V is a normed space and $\left|2\right|<1$, then, for any $\delta >0$, $p>1$ and

$$\phi (x,y):=\delta (\parallel x{\parallel }^p+\parallel y{\parallel }^p)$$

Conditions (9) and (14) hold, and Limit (10) exists.

In 2009, A.K. Mirmostafaee (see [51]) obtained another result concerning the stability of the Cauchy functional equation.

Theorem 3.

Let X be a linear space over a non-Archimedean field $\mathbb{K}$, and Y be a complete non-Archimedean normed space over $\mathbb{K}$. Assume also that $k\in \mathbb{N}$ and $\phi :X^2\to [0,\infty )$ are such that, for any $x,y\in X$,

$$\underset{j\to \infty }{lim}{\left|k\right|}^j\phi (k^{-j}x,k^{-j}y)=0.$$

(15)

If $f:X\to Y$ is a function satisfying (11), then there exists a unique additive mapping $F:X\to Y$ for which

$$\parallel f\left(x\right)-F\left(x\right)\parallel \le max\left\{\right|k{|}^{i-1}\tilde{\phi }\left(k^{-i}x\right):i\ge 1\},x\in X,$$

(16)

where

$$\tilde{\phi }\left(x\right)=max\{\phi (x,x),\phi (x,2x),\dots ,\phi (x,(k-1)x)\},x\in X.$$

(17)

Assume that k is the smallest positive integer with $\left|k\right|<1$, $c>0$ and $p\in [0,1)$. In [51], the author also noticed that, in the case where X and Y are non-Archimedean normed spaces over $\mathbb{K}$ and Y is complete, from Theorem 3, it follows that if $f:X\to Y$ satisfies

$$\parallel f(x+y)-f\left(x\right)-f\left(y\right)\parallel \le c(\parallel x{\parallel }^p+\parallel y{\parallel }^p),x,y\in X,$$

then there exists a unique additive mapping $a:X\to Y$, such that

$$\parallel f\left(x\right)-a\left(x\right)\parallel \le {2c\left|k\right|}^{-p}{\parallel x\parallel }^p,x\in X.$$

The stability of the Cauchy functional equation in non-Archimedean normed spaces was also studied by J. Schwaiger in [41].

Theorem 4.

Let p be a prime, S be a commutative semigroup uniquely divisible by p, and X be a non-Archimedean normed space over $(\mathbb{Q},|·{|}_p)$ with completion $X^c$. Assume also that $\phi :S^2\to [0,\infty )$ is a mapping such that

$$\underset{j\to \infty }{lim}{\displaystyle \frac{1}{p^j}}\phi \left({\displaystyle \frac{x}{p^j}},{\displaystyle \frac{y}{p^j}}\right)=0,x,y\in S$$

(18)

and

$$\underset{k\to \infty }{lim}{\displaystyle \frac{1}{p^k}}{\tilde{\phi }}_p\left({\displaystyle \frac{x}{p^k}}\right)=0,x\in S,$$

(19)

where

$${\tilde{\phi }}_p\left(x\right):=max\{\phi (jx,x):1\le j<p\},x\in S.$$

If $f:S\to X$ is a function satisfying (11), then there exists an additive mapping $F:S\to X^c$ for which

$$\parallel f\left(x\right)-F\left(x\right)\parallel \le {\tilde{\Phi }}_p\left(x\right):=sup\left\{{\displaystyle \frac{1}{p^k}}{\tilde{\phi }}_p\left({\displaystyle \frac{x}{p^k}}\right),k\in {\mathbb{N}}_0\right\},x\in S.$$

(20)

If, moreover, an additive function $G:S\to X^c$ fulfills

$$\parallel f\left(x\right)-G\left(x\right)\parallel \le r{\tilde{\Phi }}_p\left(x\right),x\in S$$

with an $r>0$, then $G=F$.

Theorem 4 with

$$\phi (x,y):=ϵ,x,y\in S$$

gives the following.

Corollary 1.

Let p be a prime, S be a commutative semigroup uniquely divisible by p, X be a non-Archimedean normed space over $(\mathbb{Q},|·{|}_p)$ with completion $X^c$, and $ϵ>0$. If $f:S\to X$ is a function satisfying

$$\parallel f(x+y)-f\left(x\right)-f\left(y\right)\parallel \le ϵ,x,y\in S,$$

then there exists an additive mapping $F:S\to X^c$ for which

$$\parallel f\left(x\right)-F\left(x\right)\parallel \le ϵ,x\in S.$$

If, moreover, an additive function $G:S\to X^c$ fulfills

$$\parallel f\left(x\right)-G\left(x\right)\parallel \le rϵ,x\in S$$

with an $r>0$, then $G=F$.

The generalized Hyers–Ulam stability of the following Pexiderized Cauchy functional equation:

$$f(x+y)=g\left(x\right)+h\left(y\right)$$

was proved in [51,52].

As for the Jordan–von Neumann equation, M.S. Moslehian and Th.M. Rassias proved the following.

Theorem 5.

Let V be a commutative group, and W be a complete non-Archimedean normed space. Assume also that $\phi :V^2\to [0,\infty )$ is a mapping such that, for any $x,y\in V$,

$$\underset{j\to \infty }{lim}{\displaystyle \frac{1}{{\left|4\right|}^j}}\phi (2^{j}x,2^{j}y)=0$$

(21)

and the limit

$$\underset{k\to \infty }{lim}max\{{\displaystyle \frac{1}{{\left|4\right|}^j}}\phi (2^{j}x,2^{j}x):0\le j<k\},$$

(22)

denoted by $\tilde{\phi }\left(x\right)$, exists. If $f:V\to W$ is a function satisfying $f\left(0\right)=0$ and

$$\parallel f(x+y)+f(x-y)-2f\left(x\right)-2f\left(y\right)\parallel \le \phi (x,y),x,y\in V,$$

(23)

then there exists a quadratic mapping $F:V\to W$ for which

$$\parallel f\left(x\right)-F\left(x\right)\parallel \le {\displaystyle \frac{1}{\left|4\right|}}\tilde{\phi }\left(x\right),x\in V.$$

(24)

The function F is given by

$$F\left(x\right):=\underset{j\to \infty }{lim}{\displaystyle \frac{1}{4^j}}f\left(2^{j}x\right),x\in V.$$

(25)

If, moreover,

$$\underset{l\to \infty }{lim}\underset{k\to \infty }{lim}max\{{\displaystyle \frac{1}{{\left|4\right|}^j}}\phi (2^{j}x,2^{j}x):l\le j<k+l\}=0,x\in V,$$

(26)

then F is the unique quadratic mapping satisfying condition (24).

If V is a normed space, and $\left|2\right|<1$, then for any $\delta >0$, $p>1$ and

$$\phi (x,y):={\delta \parallel x\parallel }^p{\parallel y\parallel }^p$$

Conditions (21) and (26) hold, and Limit (22) exists.

In 2010, another stability outcome on the Jordan–von Neumann equation was obtained by A.K. Mirmostafaee in [53].

Let us also note that in [54], the generalized Hyers–Ulam stability of the following Pexider version of the Jordan–von Neumann equation was studied:

$$f(x+y)+f(x-y)=2g\left(x\right)+2g\left(y\right)$$

In 2009, M.S. Moslehian (see [55]) investigated the stability of the Jensen functional equation in non-Archimedean normed spaces, as well as its asymptotic behavior. He proved, among other things, the following.

Theorem 6.

Let V be a non-Archimedean normed space, and W be a complete non-Archimedean normed space over a non-Archimedean field with $\left|3\right|<1$. Assume also that $\alpha ,\beta \ge 0$ and $p\in [0,1)$. If $f:V\to W$ is a function satisfying $\parallel f\left(0\right)\parallel \le \beta$ and

$$\parallel 2f\left({\displaystyle \frac{x+y}{2}}\right)-f\left(x\right)-f\left(y\right)\parallel \le {\alpha max\{\parallel x\parallel }^p{,\parallel y\parallel }^p\},x,y\in V\setminus \left\{0\right\},$$

(27)

then there exists a unique Jensen mapping $F:V\to W$ for which

$$\parallel f\left(x\right)-F\left(x\right)\parallel \le max\left\{{\displaystyle \frac{\alpha }{{\left|3\right|}^p}}{\parallel x\parallel }^p,\left|2\right|\beta \right\},x\in V\setminus \left\{0\right\}.$$

(28)

Other results on the generalized Hyers–Ulam stability of the Jensen equation can be found in [51,52]. In [51], the stability of the Pexider–Jensen functional equation,

$$2f\left({\displaystyle \frac{x+y}{2}}\right)=g\left(x\right)+h\left(y\right)$$

was also shown.

2.2. Multi-Additive, Multi-Quadratic and Multi-Jensen Mappings

Let us recall that a function $f:V^n\to W$, where V is a commutative group, W is a linear space and $n\in \mathbb{N}$, is called multi-additive or n-additive if it is additive (satisfies Cauchy’s functional equation) in each variable. Some basic facts on such mappings, which were introduced by S. Mazur and W. Orlicz, can be found, for instance, in [10], where their application to the representation of polynomial functions is also presented.

In [56], it was proved that a mapping $f:V^n\to W$ is multi-additive if and only if, for any $(x_{11},\dots ,x_{n1})$ and $(x_{12},\dots ,x_{n2})\in V^n$, we have

$$f(x_{11}+x_{12},\dots ,x_{n1}+x_{n2})=\sum _{1\le i_1,\dots ,i_n\le 2}f(x_{1i_1},\dots ,x_{n{i}_n}).$$

(29)

It is obvious that functional Equation (29) is a generalization of the Cauchy equation. The following result from [57] concerns its stability.

Theorem 7.

Let V be a commutative group, and W be a complete non-Archimedean space. Assume also that $n\in \mathbb{N}$ and $\phi :V^{2n}\to [0,\infty )$ is a mapping such that, for each $(x_{11},x_{12},\dots ,x_{n1},x_{n2})\in V^{2n}$,

$$\underset{j\to \infty }{lim}{\displaystyle \frac{1}{{\left|2\right|}^{jn}}}\phi (2^j{x}_{11},2^j{x}_{12},\dots ,2^j{x}_{n1},2^j{x}_{n2})=0$$

(30)

and the limit

$$\underset{k\to \infty }{lim}max\{{\displaystyle \frac{1}{{\left|2\right|}^{jn}}}\phi (2^j{x}_{11},2^j{x}_{11},\dots ,2^j{x}_{n1},2^j{x}_{n1}):0\le j<k\},$$

(31)

denoted by $\tilde{\phi }(x_{11},\dots ,x_{n1})$, exists. If $f:V^n\to W$ is a function satisfying

$$\parallel f(x_{11}+x_{12},\dots ,x_{n1}+x_{n2})-\sum _{1\le i_1,\dots ,i_n\le 2}f(x_{1i_1},\dots ,x_{n{i}_n})\parallel$$

(32)

$$\le \phi (x_{11},x_{12},\dots ,x_{n1},x_{n2}),(x_{11},x_{12},\dots ,x_{n1},x_{n2})\in V^{2n},$$

then there exists a multi-additive mapping $F:V^n\to W$ for which

$$\parallel f(x_{11},\dots ,x_{n1})-F(x_{11},\dots ,x_{n1})\parallel \le {\displaystyle \frac{1}{{\left|2\right|}^n}}\tilde{\phi }(x_{11},\dots ,x_{n1}),(x_{11},\dots ,x_{n1})\in V^n.$$

(33)

The function F is given by

$$F(x_{11},\dots ,x_{n1}):=\underset{j\to \infty }{lim}{\displaystyle \frac{1}{2^{nj}}}f(2^j{x}_{11},\dots ,2^j{x}_{n1}),(x_{11},\dots ,x_{n1})\in V^n.$$

(34)

If, moreover,

$$\underset{l\to \infty }{lim}\underset{k\to \infty }{lim}max\left\{{\displaystyle \frac{1}{{\left|2\right|}^{jn}}}\phi (2^j{x}_{11},2^j{x}_{11},\dots ,2^j{x}_{n1},2^j{x}_{n1}):l\le j<k+l\right\}=0$$

(35)

for $(x_{11},\dots ,x_{n1})\in V^n$, then F is the unique multi-additive mapping satisfying Condition (33).

Now, assume that G is a commutative group, and X is a linear space over a field of the characteristic different from 2, and recall (see [58]) that a mapping $Q:G^n\to X$ is said to be multi-quadratic or n-quadratic if it is quadratic in each variable.

In [59,60], the authors showed that a mapping $Q:G^n\to X$ is n-quadratic if and only if, for any $(x_{11},\dots ,x_{n1}),(x_{12},\dots ,x_{n2})\in G^n$, we have

$$\sum _{i_1,\dots ,i_n\in \{-1,1\}}Q(x_{11}+i_1{x}_{12},\dots ,x_{n1}+i_n{x}_{n2})=2^n\sum _{j_1,\dots ,j_n\in \{1,2\}}Q(x_{1j_1},\dots ,x_{n{j}_n}).$$

(36)

In [61], the following generalization of Theorem 5, was proved.

Theorem 8.

Let G be a commutative group, X be a complete non-Archimedean normed space over a non-Archimedean field of the characteristic different from 2, and $\phi :G^{2n}\to [0,\infty )$ be a function such that, for any $(x_{11},x_{12},\dots ,x_{n1},x_{n2})\in G^{2n}$,

$$\underset{k\to \infty }{lim}{\displaystyle \frac{\phi (2^k{x}_{11},2^k{x}_{12},\dots ,2^k{x}_{n1},2^k{x}_{n2})}{{\left|4\right|}^{nk}}}=0$$

(37)

and there exists the limit

$$\underset{k\to \infty }{lim}max\left\{{\displaystyle \frac{\phi (2^j{x}_{11},2^j{x}_{11},\dots ,2^j{x}_{n1},2^j{x}_{n1})}{{\left|4\right|}^{nj}}}:0\le j<k\right\},$$

(38)

denoted by $\psi (x_{11},\dots ,x_{n1})$. Assume also that $f:G^n\to X$ is a mapping such that $f(x_{11},\dots ,x_{n1})=0$ for any $(x_{11},\dots ,x_{n1})\in G^n$, with at least one component which is equal to zero, and

$$\parallel \sum _{i_1,\dots ,i_n\in \{-1,1\}}f(x_{11}+i_1{x}_{12},\dots ,x_{n1}+i_n{x}_{n2})-2^n\sum _{j_1,\dots ,j_n\in \{1,2\}}f(x_{1j_1},\dots ,x_{n{j}_n})\parallel$$

(39)

$$\le \phi (x_{11},x_{12},\dots ,x_{n1},x_{n2}),(x_{11},x_{12},\dots ,x_{n1},x_{n2})\in G^{2n}.$$

Then, there is a mapping $Q:G^n\to X$ satisfying Equation (36), and the inequality

$$\parallel f(x_{11},\dots ,x_{n1})-Q(x_{11},\dots ,x_{n1})\parallel \le {\displaystyle \frac{1}{{\left|4\right|}^n}}\psi (x_{11},\dots ,x_{n1}),(x_{11},\dots ,x_{n1})\in G^n.$$

(40)

If, moreover, for any $(x_{11},\dots ,x_{n1})\in G^n$ we have

$$\underset{l\to \infty }{lim}\underset{k\to \infty }{lim}max\left\{{\displaystyle \frac{\phi (2^j{x}_{11},2^j{x}_{11},\dots ,2^j{x}_{n1},2^j{x}_{n1})}{{\left|4\right|}^{nj}}}:l\le j<k+l\right\}=0,$$

(41)

then Q is a unique solution of Equation (36) for which (40) holds.

Assume that $V,W$ are commutative groups and V is uniquely divisible by 2. Let us recall (see, for instance, [62]) that a function $f:V^n\to W$ is called multi-Jensen (such mappings were introduced by W. Prager and J. Schwaiger in 2005 (see [63]) with the connection with generalized polynomials) if it is a Jensen mapping in each variable.

Denote by $\left|S\right|$ the cardinality of a set S, and put

$$\mathbf{n}:=\{1,\dots ,n\},n\in \mathbb{N}.$$

For a subset $S=\{j_1,\dots ,j_i\}$ of $\mathbf{n}$ with $1\le j_1<\dots <j_i\le n$ and $x=(x_1,\dots ,x_n)\in V^n$,

$$x_S:=(0,\dots ,0,x_{j_1},0,\dots ,0,x_{j_i},0,\dots ,0)\in V^n$$

denotes the element which coincides with x in exactly those components, which are indexed by the elements of S, and whose other components are set equal zero.

It is known (see [64]) that, under some additional assumptions on V and W, a function $f:V^n\to W$ is multi-Jensen if and only if

$$f\left({\displaystyle \frac{1}{2}}(x+y)\right)={\displaystyle \frac{1}{2^n}}\sum _{S\subseteq \mathbf{n}}f(x_S+y_{\mathbf{n}\setminus S}),x,y\in V^n.$$

(42)

In 2012, T.Z. Xu (see [65]) obtained the following result on the generalized stability of Equation (42).

Theorem 9.

Let X be a linear space over a non-Archimedean field $\mathbb{K}$, Y be a complete non-Archimedean normed space over $\mathbb{K}$, and ${\phi }_k:X^2\to [0,\infty )$ for $k\in \mathbf{n}$ be functions such that ${\phi }_k(0,0)=0$, and for any $i\in \{0,\dots ,n-1\}$ and $(x_1,\dots ,x_n),(y_1,\dots ,y_n)\in X^n$,

$$\sum _{l=0}^{\infty }{\left({\displaystyle \frac{1}{{\left|2\right|}^{n-i}}}\right)}^l\sum _{k=1}^n{\phi }_k(2^l{x}_k,2^l{y}_k)<\infty .$$

(43)

Assume also that, for any subset $T=\{j_1,\dots ,j_i\}$ of $\mathbf{n}$ with $1\le j_1<\dots <j_i\le n$ (we adopt the convention $\{j_1,\dots ,j_0\}:=\varnothing$) and $(x_1,\dots ,x_n)\in X^n$, the limit

$$\underset{m\to \infty }{lim}max\left\{{\displaystyle \frac{1}{{\left|2\right|}^i{\left|2\right|}^{(n-i)(j+1)}}}\sum _{k\in \mathbf{n}\setminus \{j_1,\dots ,j_i\}}{\phi }_k(2^{j+1}{x}_k,0):0\le j<m\right\},$$

(44)

denoted by ${\psi }_T(x_1,\dots ,x_n)$, exists. If $f:X^n\to Y$ is a mapping such that

$$\parallel 2^{n}f\left({\displaystyle \frac{1}{2}}(x+y)\right)-\sum _{S\subseteq \mathbf{n}}f(x_S+y_{\mathbf{n}\setminus S})\parallel \le \sum _{k=1}^n{\phi }_k(x_k,y_k)$$

(45)

for $x=(x_1,\dots ,x_n),y=(y_1,\dots ,y_n)\in X^n$, then there is a multi-Jensen mapping $F:X^n\to Y$ satisfying the inequality

$$\parallel f\left(x\right)-F\left(x\right)\parallel \le max\left\{{\psi }_T\left(x\right):T\mathbf{n}\right\},x\in X^n.$$

(46)

The function F is given by

$$F\left(x\right)=\sum _{T\subseteq \mathbf{n}}{F}_T\left(x\right),x\in X^n,$$

(47)

where, for any $x\in X^n$, we have $F_{\mathbf{n}}\left(x\right):=f\left(0\right)$ and

$$F_T\left(x\right):=\underset{l\to \infty }{lim}{\displaystyle \frac{1}{{\left|2\right|}^{(n-|T\left|\right)l}}}\sum _{\varnothing \ne S\subseteq \mathbf{n}\setminus T}{(-1)}^{n-\left|T\right|-\left|S\right|}f\left(2^l{x}_S\right),T\mathbf{n}.$$

(48)

2.3. Some Other Equations

In [66,67], the generalized Hyers–Ulam stability of the equation

$$f(x+y+z)+f\left(x\right)+f\left(y\right)+f\left(z\right)=f(x+y)+f(y+z)+f(z+x),$$

which is sometimes called Deeba’s or Fréchet’s functional equation, was studied.

In [67], the authors investigated the stability of the Drygas functional equation

$$f(x+y)+f(x-y)=2f\left(x\right)+f\left(y\right)+f(-y).$$

Now, let X and Y be linear spaces, and $f:X\to Y$. Put

$${\Delta }_{x}f\left(y\right)=f(x+y)-f\left(y\right),x,y\in X,$$

and define inductively

$${\Delta }_{x_1,\dots ,x_n}^{n}f\left(y\right)={\Delta }_{x_1,\dots ,x_{n-1}}^{n-1}\left({\Delta }_{x_n}f\left(y\right)\right),x_1,\dots ,x_n,y\in X.$$

Moreover, for $x_1=\dots =x_n=x$, we write

$${\Delta }_x^{n}f\left(y\right)={\Delta }_{x_1,\dots ,x_n}^{n}f\left(y\right),x,y\in X.$$

The equation

$${\Delta }_x^{n}f\left(y\right)=n!f\left(x\right)$$

is said to be a monomial functional equation of degree n, and every one of its solutions is called a monomial mapping of degree n (see [68]).

A function $f:X\to Y$ is called a polynomial of degree n if it is a solution of the following Fréchet functional equation of degree $n+1$:

$${\Delta }_{x_1,\dots ,x_{n+1}}^{n+1}f\left(0\right)=0$$

(see [69]).

In 2010, A.K. Mirmostafaee (see [68]) obtained the generalized Hyers–Ulam stability of the monomial functional equation, i.e., he proved that if X is a linear space over the rationals and Y is a complete non-Archimedean normed space, then any mapping $f:X\to Y$, such that the inequality

$$\parallel {\Delta }_x^{n}f\left(y\right)-n!f\left(x\right)\parallel \le \phi (x,y),x,y\in X$$

holds with a suitable control function $\phi$, can be approximated by a unique monomial mapping of degree n.

In 2012, A.K. Mirmostafaee (see [69]) showed that if X and Y are non-Archimedean normed spaces over the same non-Archimedean field $\mathbb{K}$ and Y is complete, then for any function $f:X\to Y$ satisfying

$$\parallel {\Delta }_{x_1,\dots ,x_n}^{n}f\left(0\right)\parallel \le {\phi }_n(\parallel x_1\parallel ,\dots ,\parallel x_n\parallel ),x_1,\dots ,x_n\in X$$

with a suitable control function ${\phi }_n:{\mathbb{R}}_{+}^n\to {\mathbb{R}}_{+}$, there exists a unique polynomial $p_{n-1}:X\to Y$ of a degree of, at most, $n-1$, such that

$$\parallel f\left(x\right)-p_{n-1}{\left(x\right)\parallel \le \left|k\right|}^{-r(n-1)}{\phi }_n(\parallel x\parallel ,\dots ,\parallel x\parallel ),x\in X,$$

where k is the smallest positive integer with $\left|k\right|<1$ and $r\in [0,1)$.

Stability results for other functional equations can be found, for example, in [70,71,72,73,74].

2.4. Functional Equations in a Single Variable

Now, let S denote a nonempty set, and X stand for a complete non-Archimedean normed space.

Given a set $Z\ne \varnothing$ and functions $f:S\to S$ and $F:S\times Z\to Z$, we define an operator ${\mathcal{L}}_f^F:Z^S\to Z^S$ by

$${\mathcal{L}}_f^F\left(\alpha \right)\left(t\right):=F(t,\alpha \left(f\left(t\right)\right)),\alpha \in Z^S,t\in S.$$

Moreover, if $\Lambda :S\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, then we write

$${\Lambda }_t:=\Lambda (t,·),t\in S.$$

In [75], the authors proved a fixed point theorem in complete non-Archimedean normed spaces and applied it to obtain some stability results. One of them concerns a quite general functional equation in a single variable.

Theorem 10.

Let $F:S\times X\to X$, $f:S\to S$, $\Lambda :S\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, $\phi :S\to X$, $\epsilon :S\to {\mathbb{R}}_{+}$ and

$$\parallel F(t,x)-F(t,y)\parallel \le \Lambda (t,\parallel x-y\parallel ),t\in S,x,y\in X.$$

(49)

Assume also that, for every $t\in S$, ${\Lambda }_t$ is nondecreasing,

$$\underset{n\to \infty }{lim}{\left({\mathcal{L}}_f^{\Lambda }\right)}^n\left(\epsilon \right)\left(s\right)=0,s\in S$$

(50)

and

$$\parallel \phi \left(t\right)-F(t,\phi (f\left(t\right)\left)\right)\parallel \le \epsilon \left(t\right),t\in S.$$

(51)

Then, for each $t\in S$ the limit

$$\underset{n\to \infty }{lim}{\left({\mathcal{L}}_f^F\right)}^n\left(\phi \right)\left(t\right)=:\Phi \left(t\right)$$

(52)

exists, and the function $\Phi \in X^S$ is the unique solution of the functional equation

$$\Phi \left(t\right)=F(t,\Phi (f\left(t\right)\left)\right)$$

(53)

such that

$$\parallel \phi \left(t\right)-\Phi \left(t\right)\parallel \le \underset{n\in {\mathbb{N}}_0}{sup}{\left({\mathcal{L}}_f^{\Lambda }\right)}^n\left(\epsilon \right)\left(t\right)=:h\left(t\right),t\in S.$$

(54)

Let us remark (see [75]) that Theorem 10 can be, for example, applied to the equations

$$\Phi \left(t\right)=b\left(t\right)\Phi \left(f\right(t\left)\right)+H\left(t\right),$$

$$\Phi \left(f\right(t\left)\right)=a\left(t\right)\Phi \left(t\right)+G\left(t\right)$$

and

$$\Phi \left(t\right)=A(\Phi (f\left(t\right)\left)\right)+H\left(t\right).$$

2.5. Hyperstability

Let X be a normed space over a field $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, Y be a complete non-Archimedean normed space over a field $\mathbb{K}$, $a,b\in \mathbb{F}\setminus \left\{0\right\}$ and $A,B\in \mathbb{K}\setminus \left\{0\right\}$.

In [76], the authors studied the hyperstability (for a survey on this notion, see [77]) of the linear functional equation

$$f(ax+by)=Af\left(x\right)+Bf\left(y\right).$$

(55)

They showed, among other things, the following.

Theorem 11.

Let $l,r,s\in \mathbb{R}$ be such that $l\ge 0$ and $r,s>0$. Assume also that ${\left|a\right|}^{r+s}\ne \left|A\right|$ or ${\left|b\right|}^{r+s}\ne \left|B\right|$. If a function $f:X\to Y$ satisfies

$$\parallel f(ax+by)-Af\left(x\right)-Bf\left(y\right)\parallel \le {l\parallel x\parallel }^r{\parallel y\parallel }^s,x,y\in X,$$

(56)

then f is a solution of (55).

A consequence of Theorem 11 is clearly a hyperstability result on the Jensen functional equation (see also [78]).

Let us finally mention that several hyperstability outcomes on other functional equations can be found, for instance, in [79,80,81].

2.6. Instability

In [82], the following functional equations:

$$I_q\left({\displaystyle \frac{x+y}{2}}\right)-I_q(x+y)={\displaystyle \frac{3I_q\left(x\right)I_q\left(y\right)}{{\left(\sqrt{I_q\left(x\right)}+\sqrt{I_q\left(y\right)}\right)}^2}}$$

(57)

and

$$I_q\left({\displaystyle \frac{x+y}{2}}\right)+I_q(x+y)={\displaystyle \frac{5I_q\left(x\right)I_q\left(y\right)}{{\left(\sqrt{I_q\left(x\right)}+\sqrt{I_q\left(y\right)}\right)}^2}}$$

(58)

were investigated. The authors showed that, under some assumptions, they are stable in non-Archimedean context. However, for the control function of the form $k\left({\left|x\right|}^{-2}+{\left|y\right|}^{-2}\right)$, the instability occurs.

3. Conclusions

In this survey, we presented some classical and recent results on Ulam stability of functional equations in non-Archimedean normed spaces. This should be useful for investigating the stability of other equations in such spaces.

We finish the article with two more remarks.

Namely, it is worth noting that in many of the results presented the conditions on the bounding/control function are not satisfied by a constant map and, therefore, these outcomes do not imply the classical Ulam stability. This concerns, in particular, Theorem 2, as has been pointed out in [41].

Among the tools used in the Ulam stability theory, the direct method introduced by Hyers (which, as was shown in [83], sometimes does not work) and the fixed point method were applied in the case of non-Archimedean spaces.