Some Remarks on the Best Ulam Constant

Abstract

The problem of Ulam stability for equations can be stated in terms of how much the mappings satisfying the equations approximately (in a sense) differ from the exact solutions of these equations. One of the best known results in this area is the following: Let g be a mapping from a normed space V into a Banach space B. Let $\xi \ge 0$ and $t\ne 1$ be fixed real numbers and $g:V\to B$ satisfy the inequality $\parallel g(u+v)-g\left(u\right)-g\left(v\right)\parallel \le \xi (\parallel u{\parallel }^t+\parallel v{\parallel }^t)$ for $u,v\in V\setminus \left\{0\right\}$. Then, there exists a unique additive $f:V\to B$ fulfilling the inequality $\parallel g\left(u\right)-f\left(u\right)\parallel \le \xi |1-2^{t-1}{|}^{-1}{\parallel u\parallel }^t$ for $u\in V\setminus \left\{0\right\}$. There arises a natural problem if the constant, on the right hand side of the latter inequality, is the best possible. It is known as the problem of the best Ulam constant. We discuss this problem, as well as several related issues, show possible generalizations of the existing results, and indicate open problems. To make this publication more accessible to a wider audience, we limit the related information, avoid advanced generalizations, and mainly focus only on the additive Cauchy equation $f(x+y)=f\left(x\right)+f\left(y\right)$ and on the general linear difference equation $x_{n+p}=a_1{x}_{n+p-1}+\dots +a_p{x}_n+b_n$ (considered for sequences in a Banach space). In particular, we show that there is a significant symmetry between Ulam constants of several functional equations and of their inhomogeneous or radical forms. We hope that in this way we will stimulate further research in this area.

1. Introduction

This is a short review paper presenting and discussing several selected results concerning the best Ulam constant of some functional (also including difference) equations. The notion of the best Ulam constant appeared in the theory of Ulam stability in connection with a problem raised by Th.M. Rassias (cf., e.g., [1]) and concerning the given below result, which can be regarded as quite representative for such types of stability (see, e.g., [2,3,4]).

Theorem 1. 

Assume that B is a Banach space, V is a normed space, and $V_0:=V\setminus \left\{0\right\}$. Let $\xi \ge 0$ and $t\ne 1$ be real numbers and $g:V\to B$ be such that

$$\parallel g(u+v)-g\left(u\right)-g\left(v\right)\parallel \le {\xi (\parallel u\parallel }^t+{\parallel v\parallel }^t),\forall u,v\in V_0.$$

(1)

Then, there exists a unique additive $f:V\to B$ fulfilling the inequality

$$\parallel g\left(u\right)-f\left(u\right)\parallel \le {\displaystyle \frac{\xi }{|1-2^{t-1}|}}{\parallel u\parallel }^t,\forall u\in V_0.$$

(2)

Let us recall that a mapping $f:V\to B$ is additive if it satisfies the equation

$$\begin{array}{c}\hfill f(u+v)=f\left(u\right)+f\left(v\right),\forall u,v\in V.\end{array}$$

(3)

Before we move further, let us remind the reader that the main issue in Ulam stability theory can be briefly expressed in the following way: how much does an approximate solution to an equation differ from the exact solutions to it? It is clear that such a question naturally arises in many areas of scientific investigation and can be asked with regard to any difference, differential, integral, functional, etc., equation. Similar problems are studied in the theories of optimization, perturbation, approximation, and shadowing. For further information on Ulam stability (also called, e.g., Hyers–Ulam or Ulam–Hyers or Ulam–Hyers–Rassias stability), we refer to monographs [2,3,5,6] and papers [7,8,9,10].

To express some ideas more clearly, let us introduce the following simplified definition: (${\mathbb{R}}_{+}$ means the set of nonnegative real numbers and $A^B$ stands for the family of all functions mapping a set $B\ne \varnothing$ into a set $A\ne \varnothing$).

Definition 1. 

Let k be a positive integer, $(D,\rho )$ be a metric space, U be a nonempty set, ${\mathcal{T}}_1,{\mathcal{T}}_2:D^U\to D^{U^k}$, $\mathcal{E}\subset {\mathbb{R}}_{+}^{U^k}$ be nonempty, and $\mathcal{S}:\mathcal{E}\to {\mathbb{R}}_{+}^U$. The functional equation

$$\begin{array}{c}\hfill \left({\mathcal{T}}_{1}g\right)(t_1,\dots ,t_k)=\left({\mathcal{T}}_{2}g\right)(t_1,\dots ,t_k)\end{array}$$

(4)

is said to be $\mathcal{S}$-stable if, for any $f:U\to D$ and $\varphi \in \mathcal{E}$ with

$$\begin{array}{c}\hfill \rho (\left({\mathcal{T}}_{1}f\right)(t_1,\dots ,t_k),\left({\mathcal{T}}_{2}f\right)(t_1,\dots ,t_k))\le \varphi (t_1,\dots ,t_k),\forall t_1,\dots ,t_k\in U,\end{array}$$

(5)

there is a mapping $g:U\to D$ satisfying Equation (4) for all $t_1,\dots ,t_k\in U$ such that

$$\rho \left(g\right(u),f(u\left)\right)\le \left(\mathcal{S}\varphi \right)\left(u\right),\forall u\in U.$$

The function ϕ is called the control function (of the stability inequality (5)).

If $\left(\mathcal{S}\varphi \right)\left(u\right)=0$ for some $\varphi \in \mathcal{E}$ and every $u\in U$, then we say that the equation is ϕ-hyperstable.

Note that Equation (4) takes the form of the Cauchy functional Equation (3) with $k=2$, $U=V$, $\left({\mathcal{T}}_1\beta \right)(s,t)=\beta (s+t)$ and $\left({\mathcal{T}}_2\beta \right)(u,v)=\beta \left(u\right)+\beta \left(v\right)$ for $\beta \in B^V$ and $u,v\in V$.

According to Theorem 1, the Cauchy Equation (3) is $\mathcal{S}$-stable with

$$\begin{array}{c}\hfill \left(\mathcal{S}\varphi \right)\left(x\right)={\gamma }_{\varphi }\varphi (x,x),\forall \varphi \in \mathcal{E},x\in U,\end{array}$$

(6)

where

$$\begin{array}{c}\hfill \mathcal{E}:=\{{\psi }_{\xi }\in {\mathbb{R}}_{+}^{U^2}:\xi \in {\mathbb{R}}_{+}\}\end{array}$$

(7)

with ${\psi }_{\xi }(u,v)={\xi (\parallel u\parallel }^t+{\parallel v\parallel }^t)$ for $u,v\in V$, $\xi \in {\mathbb{R}}_{+}$, and

$$\begin{array}{c}\hfill {\gamma }_{\varphi }={\displaystyle \frac{1}{|2-2^t|}},\forall \varphi \in \mathcal{E}.\end{array}$$

(8)

In the cases in which such stability has been proved for functional equations, operator $\mathcal{S}$ mainly has form (6) with some real constants ${\gamma }_{\varphi }\ge 0$. If $\varphi$ is fixed and $\mathcal{S}$ has form (6), then we say that the constant ${\gamma }_{\varphi }$ is an Ulam $\varphi$-constant of Equation (3). In such a case, it is very natural to ask if such a constant is the best possible (i.e., the smallest).

As we have already mentioned, this problem was raised by Th.M. Rassias [1] (cf., e.g., [3]) in 1990 with regard to the result depicted by Theorem 1. Later, in [11], this problem was extended to some other equations and now it is quite common to consider it with regard to any equation for which the Ulam stability is investigated. If (for a given equation and function $\varphi$) the infimum of all such Ulam $\varphi$-constants is an Ulam $\varphi$-constant, then we call it the best Ulam $\varphi$-constant (cf., e.g., [6]) and denote by $K_{\varphi }$.

Sometimes it is good to add information on the class of functions in which we investigate this constant because its value may also depend on the type of domain of the functions considered, as is easily seen in the next section, in which we discuss some results concerning the best Ulam constants for the additive Cauchy Equation (3).

In further sections, similar issues are discussed for several functional equations, including some difference equations. We also point out some open problems.

In this paper, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ always mean the sets of positive integers, integers, rational numbers, reals, and complex numbers, respectively. Moreover, ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$.

2. Best Ulam Constant for the Additive Cauchy Equation

A simple example of a constant function $g:V\to B=\mathbb{R}$, $g\left(u\right)=2\xi$ for $u\in V$, shows that for $t=0$, the constant in (2) is optimal (it is equal 1), with $f\left(u\right)=0$ for $u\in V$. This optimality is a consequence of the uniqueness of the additive function f in Theorem 1.

Next, it has been proved in [12] (see also [3]) that this is also the case for $t>0$, $t\ne 1$, when the general form (in the situation considered in Theorem 1) of the best Ulam constant $K_{\varphi }$ is given by the formula

$$\begin{array}{c}\hfill K_{\varphi }={\displaystyle \frac{1}{|2-2^t|}}\end{array}$$

(9)

for each $\varphi \in \mathcal{E}$, with $\mathcal{E}$ defined by (7).

In the case of $t<0$, the best Ulam constant $K_{\varphi }$ in (2) is equal to 0 for each $\varphi \in \mathcal{E}$, because we have the following result (see, e.g., [4] Theorem 3.1) (in the case $U=V\setminus \left\{0\right\}$, this theorem can be deduced from an earlier result in ([13] Theorem 5)).

Theorem 2. 

Let V and W be normed spaces, $U\subset V\setminus \left\{0\right\}$ be nonempty, and $\xi \ge 0$ and $t<0$ be real numbers. Let

$$-U:=\{-u:u\in U\}=U$$

(10)

and assume that there exists a positive integer $n_0$ such that

$$nu\in U,\forall u\in U,n\in \mathbb{N},n\ge n_0.$$

(11)

Then, each $f:U\to W$, fulfilling the inequality

$$\parallel f(u+v)-f\left(u\right)-f\left(v\right)\parallel \le \xi (\parallel u{\parallel }^t+\parallel v{\parallel }^t),\forall z,v\in U,z+v\in U,$$

is additive on U, i.e.,

$$f(u+v)=f\left(u\right)+f\left(v\right),\forall u,v\in U,u+v\in U.$$

(12)

Note that Theorem 2 provides information on the best Ulam constant for Equation (3) even on the restricted domains U satisfying conditions (10) and (11). The result contained in it is a $\varphi$-hyperstability property of (12) (see Definition 1), with $\varphi (u,v)\equiv \xi (\parallel u{\parallel }^t+\parallel v{\parallel }^t)$ and $t<0$. Broadly speaking, hyperstability of an equation means that approximate (in the sense considered in a particular situation) solutions of it must be the exact solutions to the equation.

The next theorem (see [4] Theorem 3.4) shows that also for $t>0$ the best Ulam constant can be determined on some restricted domains. Namely, we have the following result that complements Theorem 2 and generalizes Theorem 1.

Theorem 3. 

Let W be a Banach space, V be a normed space, $U\subset V\setminus \left\{0\right\}$ be nonempty, $\xi \ge 0$ and $t\ge 0$ be real numbers and $t\ne 1$. Assume that one of the following two conditions holds.

(i) 

$t<1$ and $2U:=\{2u:u\in U\}\subset U$.

(ii) 

$t>1$ and $U\subset 2U$.

• Let $f:U\to W$ satisfy

  $$\parallel f(u+v)-f\left(u\right)-f\left(v\right)\parallel \le \xi (\parallel u{\parallel }^t+\parallel v{\parallel }^t),\forall u,v\in U,u+v\in U.$$

  (13)

• Then, there is exactly one $A:U\to W$, which is additive on U and such that

  $$\parallel f\left(u\right)-A\left(u\right)\parallel \le {\displaystyle \frac{\xi }{|1-2^{t-1}|}}{\parallel u\parallel }^t,\forall u\in U.$$

  (14)

Remark 1. 

Also, in Theorem 3, estimation (14) is the best possible in the general situation, which means that the best Ulam constant in this situation is the same as in Theorem 1 and given by (8). Namely, it is easily seen that we have equality in (14), with $A\left(u\right)=0$ for $u\in U$ if we take $U=V=\mathbb{R}$ and $f:U\to W$ given by $f\left(u\right)=sgn\left(u\right){\left|u\right|}^t{w}_0$ for $u\in U$ (sgn is the signum function, which returns the sign of a real number) with any fixed $w_0\in W$ such that

$$\parallel w_0\parallel ={\displaystyle \frac{\xi }{|1-2^{t-1}|}}.$$

Moreover, the next lemma (see [12] or [3] Theorem 2.10) guarantees that (13) is fulfilled.

Lemma 1. 

Let $\xi ,t\in \mathbb{R}$, $t>0$, $t\ne 1$, and $h_0:\mathbb{R}\to \mathbb{R}$ be given by: $h_0\left(u\right)=csgn\left(u\right){\left|u\right|}^t$ for $u\in \mathbb{R}$, where

$$c={\displaystyle \frac{\xi }{|1-2^{t-1}|}}.$$

Then,

$$\parallel h_0(u+v)-h_0\left(u\right)-h_0{\left(v\right)\parallel \le \xi \left(\right|u|}^t+{\left|v\right|}^t),\forall u,v\in \mathbb{R}.$$

(15)

Remark 2. 

Actually, Remark 1 only provides the best Ulam constant in the case when $V=\mathbb{R}$, and there arises an open question if this constant can be diminished for some other V. The next proposition shows that the best Ulam constant is the same when V is a real inner product space (it is enough to take the additive function A such that $A\left(u\right)=0$ for all arguments u).

Proposition 1. 

Let $(X,⟨·,·⟩)$ be a real inner product space and Y be a normed space. Fix, $t,\xi \in {\mathbb{R}}_{+}$, $t\ne 1$, $x_0\in X$ with $\parallel x_0\parallel =1$ and $y_0\in Y$ with $\parallel y_0\parallel =1$. Let $h:X\to Y$ be given by:

$$h\left(v\right)=h_0\left(⟨v,x_0⟩\right)y_0,\forall v\in X,$$

where $h_0:\mathbb{R}\to \mathbb{R}$ is defined as in Lemma 1. Then,

$$\parallel h(u+v)-h\left(u\right)-h\left(v\right)\parallel \le \xi (\parallel u{\parallel }^t+\parallel v{\parallel }^t),\forall u,v\in X,$$

(16)

and

$$\parallel h\left(u\right)\parallel \le {\displaystyle \frac{\xi }{|1-2^{t-1}|}}{\parallel u\parallel }^t,\forall u\in X,$$

(17)

$$\parallel h\left(s{x}_0\right)\parallel ={\displaystyle \frac{\xi }{|1-2^{t-1}|}}{\parallel s{x}_0\parallel }^t,\forall s\in \mathbb{R}.$$

(18)

Proof. 

Fix $u,v\in X$. Then, by the Cauchy–Schwartz inequality and Lemma 1,

$$\begin{array}{cc}\hfill \parallel h(u+v)& - h\left(u\right)-h\left(v\right)\parallel = |h_0\left(⟨u+v,x_0⟩\right)-h_0\left(⟨u,x_0⟩\right)-h_0\left(⟨v,x_0⟩\right)|\parallel y_0\parallel \hfill \\ \hfill & = |h_0(⟨u,x_0⟩+⟨v,x_0⟩)-h_0\left(⟨u,x_0⟩\right)-h_0\left(⟨v,x_0⟩\right)|\parallel y_0\parallel \hfill \\ \hfill & \le \xi \left(\right|⟨u,x_0⟩{|}^t+|⟨v,x_0⟩{|}^t{) \le \xi (\parallel u\parallel }^t+{\parallel v\parallel }^t)\parallel x_0{\parallel }^t\hfill \\ \hfill & =\xi (\parallel u{\parallel }^t+\parallel v{\parallel }^t).\hfill \end{array}$$

Next,

$$\begin{array}{cc}\hfill \parallel h\left(u\right)\parallel & = \parallel h_0\left(⟨u,x_0⟩\right)y_0\parallel = |h_0\left(⟨u,x_0⟩\right)|\parallel y_0\parallel ={\displaystyle \frac{\xi }{|1-2^{t-1}|}}{\left|⟨u,x_0⟩\right|}^t\hfill \\ \hfill & \le {\displaystyle \frac{\xi }{|1-2^{t-1}|}}{\parallel u\parallel }^t\parallel x_0{\parallel }^t ={\displaystyle \frac{\xi }{|1-2^{t-1}|}}{\parallel u\parallel }^t,\forall u\in X,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel h\left(s{x}_0\right)\parallel & = \parallel h_0\left(⟨s{x}_0,x_0⟩\right)y_0\parallel = |h_0\left(⟨s{x}_0,x_0⟩\right)|\parallel y_0\parallel ={\displaystyle \frac{\xi }{|1-2^{t-1}|}}{\left|⟨s{x}_0,x_0⟩\right|}^t\hfill \\ \hfill & ={\displaystyle \frac{\xi }{|1-2^{t-1}|}}{\left|s\right|}^t={\displaystyle \frac{\xi }{|1-2^{t-1}|}}{\parallel s{x}_0\parallel }^t,\forall s\in \mathbb{R}.\hfill \end{array}$$

□

The assumption (10) is very important in Theorem 2, which follows from the next result, which has been proved in [14].

Theorem 4. 

Let W be a Banach space, V be a normed space, $U\subset V\setminus \left\{0\right\}$ be nonempty, and $t,\xi \in \mathbb{R}$, $\xi \ge 0$, $t<0$. Assume that there exists $n_0\in \mathbb{N}$ such that condition (11) holds. If $g:U\to W$ satisfies the inequality

$$\parallel g(u+v)-g\left(u\right)-g\left(v\right)\parallel \le \xi (\parallel u{\parallel }^t+\parallel v{\parallel }^t),\forall u,v\in U,u+v\in U,$$

then there is exactly one mapping $A:U\to W$, which is additive on U and fulfils the inequality

$$\parallel g\left(u\right)-A\left(u\right)\parallel \le {\xi \parallel u\parallel }^t,\forall u\in U.$$

It can be easily seen that the Ulam constant in this theorem is equal to 1 and is the best Ulam constant for (12) in the general situation, under the assumptions of Theorem 4 (see [14] Remark 3.2).

The next Table 1 summarizes the information given above on the values of the best Ulam $\varphi$-constant $K_{\varphi }$ for Equation (12), for each $\varphi \in \mathcal{E}$ (where $\mathcal{E}$ is defined by (7)), in the class of functions that map a nonempty subset U of a real inner product space V into a Banach space W. As we can see, $K_{\varphi }$ depends on the assumptions on U and the value of t.

Table 1. The known information on the dependence of the best Ulam $\varphi$-constant $K_{\varphi }$ for Equation (12) (and the control function $\varphi (z,v)\equiv \xi (\parallel z{\parallel }^t+\parallel v{\parallel }^t)$) on the assumptions on U and the value of $t\ne 1$.

Value of t	Assumptions on U	Value of ${\mathit{K}}_{\mathit{\varphi }}$
$t<0$	(10)	$K_{\varphi }=1$
$t<0$	(10) and (11)	$K_{\varphi }=0$
$0\le t<1$	$2U\subset U$	$K_{\varphi }={|2^t-2|}^{-1}$
$1<t$	$U\subset 2U$	$K_{\varphi }={|2^t-2|}^{-1}$

It was shown in ([15] Theorem 13) that Theorems 2–4 yield the following result concerning the difference equation:

$$u_{n+m}=u_n+u_m,\forall n,m\in S,$$

(19)

with $S\in \{\mathbb{N},\mathbb{Z}\}$, considered for sequences ${\left(u_n\right)}_{n\in S}$ in a normed space W.

Theorem 5. 

Let W be a normed space, $S\in \{\mathbb{N},\mathbb{Z}\}$, $S_0:=S\setminus \left\{0\right\}$, and $\xi \ge 0$ and $t<1$ be real numbers. If ${\left(u_n\right)}_{n\in S}$ is a sequence in W such that

$$\parallel u_{n+m}-u_n-u_m{\parallel \le \xi \left(\right|n|}^t+{\left|m\right|}^t),\forall n,m\in S_0,$$

(20)

then the following three statements hold true.

(i) 

If $t<0$ and $S=\mathbb{Z}$, then

$$u_n=n{u}_1,\forall n\in S.$$

(21)

(ii) 

If W is complete, $t<0$ and $S=\mathbb{N}$, then there exists a unique $z_0\in W$ with

$$\parallel u_n-n{z}_0\parallel \le \xi n^t,\forall n\in S.$$

(22)

Moreover,

$$z_0=\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}{u}_n.$$

(23)

(iii) 

If W is complete and $t\ge 0$, then there is a unique $z_0\in W$ with

$$\parallel u_n-n{z}_0\parallel \le {\displaystyle \frac{{\xi \left|n\right|}^t}{1-2^{t-1}}},\forall n\in S_0.$$

(24)

Moreover, (23) holds.

All Ulam $\varphi$-constants in this theorem are the best possible (see [15] Remark 4). So, the best Ulam $\varphi$-constant $K_{\varphi }$ (for $\varphi (n,m)\equiv \xi \left(\right|n{|}^t+\left|m{|}^t\right)$) for the difference Equation (19), considered for sequences ${\left(u_n\right)}_{n\in S}$ in the Banach space W, with $S\in \{\mathbb{N},\mathbb{Z}\}$, is given in the following Table 2.

Table 2. The dependence of the best Ulam $\varphi$-constant $K_{\varphi }$, for Equation (19) and the control function $\varphi (n,m)\equiv \xi \left(\right|n{|}^t+\left|m{|}^t\right)$, on the value of $t<1$ and the form of S (with $S\in \{\mathbb{N},\mathbb{Z}\}$).

	$\mathit{t}<0$	$0\le \mathit{t}<1$
$S=\mathbb{N}$	$K_{\varphi }=1$	$K_{\varphi }={|2-2^t|}^{-1}$
$S=\mathbb{Z}$	$K_{\varphi }=0$	$K_{\varphi }={|2-2^t|}^{-1}$

Remark 3. 

It is an open question if an analogoue of Theorem 5 also can be obtained for $t>1$.

3. Inhomogeneous Version of Equation (12)

In this section, we show that the results presented in the previous sections also allow us to obtain information on the best Ulam constant for the inhomogeneous version of (12), i.e., for the equation

$$g(u+v)=g\left(u\right)+g\left(v\right)+d(u,v),\forall u,v\in U,u+v\in U,$$

(25)

where d is a fixed function with domain U.

That is, we have the following proposition.

Proposition 2. 

Let V and W be normed spaces, $U\subset V$ be nonempty, and $\varphi :U^2\to {\mathbb{R}}_{+}$. Let $d:U^2\to W$ be such that Equation (25) has at least one solution $g_0:U\to W$.

Then, $K\in {\mathbb{R}}_{+}$ is an Ulam ϕ-constant of Equation (12) if and only if it is an Ulam ϕ-constant of (25).

Moreover, $K\in {\mathbb{R}}_{+}$ is the best Ulam ϕ-constant of Equation (12) if and only if it is the best Ulam ϕ-constant of Equation (25).

Proof. 

First, assume that $K\in {\mathbb{R}}_{+}$ is an Ulam $\varphi$-constant of Equation (12). Fix $g:U\to W$ with

$$\parallel g(u+v)-g\left(u\right)-g\left(v\right)-d(u,v)\parallel \le \varphi (u,v),\forall u,v\in U,u+v\in U.$$

Write $g_1\left(u\right)=g\left(u\right)-g_0\left(u\right)$ for $u\in U$. Then, $\parallel g_1(u+v)-g_1\left(u\right)-g_1\left(v\right)\parallel = \parallel g(u+v)-g\left(u\right)-g\left(v\right)-d(u,v)\parallel \le \varphi (u,v)$ for $u,v\in U$ with $u+v\in U$. Consequently, there exists a solution $h_1:U\to W$ of Equation (12) such that $\parallel g_1\left(u\right)-h_1\left(u\right)\parallel \le K\varphi (u,u)$ for $u\in U$. Let $h\left(u\right)=h_1\left(u\right)+g_0\left(u\right)$ for $u\in U$. Then, it is easily seen that h is a solution to (25) and $\parallel g\left(u\right)-h\left(u\right)\parallel = \parallel g_1\left(u\right)-h_1\left(u\right)\parallel \le K\varphi (u,u)$ for $u\in U$. This means that K is an Ulam $\varphi$-constant of Equation (25).

Now, assume that $K\in {\mathbb{R}}_{+}$ is the best Ulam $\varphi$-constant of Equation (12). From the previous reasoning, it follows that it is an Ulam $\varphi$-constant of Equation (25). Suppose that there exists $K^{\prime }\in {\mathbb{R}}_{+}$, which is an Ulam $\varphi$-constant of Equation (25) and $K^{\prime }<K$. Fix $g_1:U\to W$ such that $\parallel g_1(u+v)-g_1\left(u\right)-g_1\left(v\right)\parallel \le \varphi (u,v)$ for $u,v\in U$ with $u+v\in U$. Write $g\left(u\right)=g_1\left(u\right)+g_0\left(u\right)$ for $u\in U$. Then,

$$\parallel g(u+v)-g\left(u\right)-g\left(v\right)-d(u,v)\parallel = \parallel g_1(u+v)-g_1\left(u\right)-g_1\left(v\right)\parallel \le \varphi (u,v)$$

for $u,v\in U$ with $u+v\in U$. Consequently, there exists a solution $h:U\to W$ of Equation (25) such that $\parallel g\left(u\right)-h\left(u\right)\parallel \le K^{\prime }\varphi (u,u)$ for $u\in U$. Let $h_1\left(u\right)=h\left(u\right)-g_0\left(u\right)$ for $u\in U$. Then, it is easily seen that $h_1$ is a solution to (12) and $\parallel g_1\left(u\right)-h_1\left(u\right)\parallel \le K^{\prime }\varphi (u,u)$ for $u\in U$. Thus, we have shown that $K^{\prime }$ is an Ulam $\varphi$-constant of Equation (12), which is a contradiction, because $K^{\prime }<K$ and K is assumed to be the best Ulam $\varphi$-constant of (12). This completes the proof that the best Ulam $\varphi$-constant of (12) is the best Ulam $\varphi$-constant of (25).

The proofs of the converse implications are analogous. □

It follows from Proposition 2 that all the information on the best Ulam constants given in the previous section for Equation (12) is valid also for (25), with any function $d:U^2\to W$ such that Equation (25) has at least one solution $g_0:U\to W$. Reasonings similar to that in the proof of Proposition 2 work also for numerous other functional equations, which allows us to extend the information about the best Ulam constant from the homogeneous versions of the equations to the inhomogeneous versions of them. This is the case, for instance, for the results in [16] concerning the best Ulam constant for the Fréchet equation.

Recently, also the subsequent finer stability results for the inhomogeneous Cauchy Equation (25) have been obtained in [17].

Theorem 6. 

Let V be a normed space and $d:V^2\to \mathbb{R}$ be such that:

$$\begin{array}{cc}\hfill d(u+v,u+v)& - d(u,u)-d(v,v)\hfill \\ \hfill & =d(2u,2v)-2d(u,v),u,v\in V.\hfill \end{array}$$

• Let $U\ne \varnothing$ be a subset of $V\setminus \left\{0\right\}$, $2U\subset U$, $\chi ,\nu ,t\in \mathbb{R}$, $t<1$, $\chi \le \nu$ and $f:U\to \mathbb{R}$ satisfy

  $$\begin{array}{cc}\hfill \chi (\parallel u{\parallel }^t+\parallel v{\parallel }^t)& \le f(u+v)-f\left(u\right)-f\left(v\right)-d(u,v)\hfill \\ \hfill & \le \nu (\parallel u{\parallel }^t+\parallel v{\parallel }^t),\forall u,v\in U,u+v\in U.\hfill \end{array}$$

  (26)

Then, there exists exactly one mapping $g:U\to \mathbb{R}$ satisfying Equation (25) such that

$$\begin{array}{c}\hfill {\displaystyle \frac{\chi }{1-2^{t-1}}}{\parallel u\parallel }^t\le g\left(u\right)-f\left(u\right)\le {\displaystyle \frac{\nu }{1-2^{t-1}}}{\parallel u\parallel }^t,\forall u\in U.\end{array}$$

(27)

Theorem 7. 

Let V, d and $\chi ,\nu$ be as in Theorem 6 and $t\in \mathbb{R}$, $t>1$. Let $U\subset V\setminus \left\{0\right\}$ be nonempty, $U\subset 2U$ and $f:U\to \mathbb{R}$ satisfy (26). Then, there exists exactly one mapping $g:U\to \mathbb{R}$ such that (25) is fulfilled and

$$\begin{array}{c}\hfill {\displaystyle \frac{\chi }{2^{t-1}-1}}{\parallel u\parallel }^t\le f\left(u\right)-g\left(u\right)\le {\displaystyle \frac{\nu }{2^{t-1}-1}}{\parallel u\parallel }^t,\forall u\in U.\end{array}$$

(28)

In those two theorems, two constants occur on both sides of inequalities (27) and (28). If $\chi =-\nu$, then from the previous results it follows that they are the best possible in the general situation and also in the special case where V is a real inner product space (see Remark 2). It would be good to find some significant information on their optimality when $\chi \ne -\nu$.

4. Radical Versions

In this section, we show that the results on the best Ulam constant for Equation (12) imply analogous outcomes for the radical versions of (12), i.e., for the equation

$$g\left(\sqrt[n]{u^n+v^n}\right)=g\left(u\right)+g\left(v\right),\forall u,v\in U,\sqrt[n]{u^n+v^n}\in U,$$

(29)

where $U\subset \mathbb{R}$ is nonempty and $n\in \mathbb{N}$, $n>1$ is fixed. Various analogous equations of similar (radical) type have been recently studied, e.g., in [18,19,20,21,22,23,24,25,26,27,28,29].

For the sake of simplicity, here we only consider the case of (29) with odd n and $U\in \{\mathbb{R},{\mathbb{R}}_{+}\}$. We have the following proposition for functions g mapping U to a normed space W.

Proposition 3. 

Let $\varphi :U^2\to {\mathbb{R}}_{+}$ and $n\in \mathbb{N}$, $n>1$ be odd.

Then, $K\in {\mathbb{R}}_{+}$ is an Ulam ϕ-constant of Equation (12) if and only if it is an Ulam ${\varphi }_n$-constant of Equation (29), where ${\varphi }_n(u,v)\equiv \varphi (u^n,v^n)$.

Moreover, $K\in {\mathbb{R}}_{+}$ is the best Ulam ϕ-constant of Equation (12) if and only if it is the best Ulam ${\varphi }_n$-constant of Equation (29).

Proof. 

First, assume that $K\in {\mathbb{R}}_{+}$ is an Ulam $\varphi$-constant of Equation (12). Let $g:U\to W$ be such that

$$\parallel g\left(\sqrt[n]{u^n+v^n}\right)-g\left(u\right)-g\left(v\right)\parallel \le {\varphi }_n(u,v),\forall u,v\in U.$$

Write

$$g_1\left(u\right)=g\left(\sqrt[n]{u}\right),\forall u\in U.$$

• Then,

  $$\begin{array}{cc}\hfill \parallel g_1(u^n+v^n)-g_1\left(u^n\right)-g_1\left(v^n\right)\parallel & = \parallel g\left(\sqrt[n]{u^n+v^n}\right)-g\left(u\right)-g\left(v\right)\parallel \hfill \\ \hfill & \le {\varphi }_n(u,v)=\varphi (u^n,v^n),\forall u,v\in U,\hfill \end{array}$$

  whence

  $$\parallel g_1(u+v)-g_1\left(u\right)-g_1\left(v\right)\parallel \le \varphi (u,v),\forall u,v\in U.$$

Consequently, there exists a solution $h_1:U\to W$ of Equation (12) such that $\parallel g_1\left(u\right)-h_1\left(u\right)\parallel \le K\varphi (u,u)$ for $u\in U$. Let $h\left(u\right)=h_1\left(u^n\right)$ for $u\in U$. Then, it is easy to check that h is a solution to (29) and

$$\parallel g\left(u\right)-h\left(u\right)\parallel = \parallel g_1\left(u^n\right)-h_1\left(u^n\right)\parallel \le K\varphi (u^n,u^n)=K{\varphi }_n(u,u),\forall u\in U.$$

This means that K is an Ulam ${\varphi }_n$-constant of Equation (29).

Now, assume that $K\in {\mathbb{R}}_{+}$ is the best Ulam $\varphi$-constant of (12). We show that K is the best Ulam ${\varphi }_n$-constant of (29).

As we have just shown, K is an Ulam ${\varphi }_n$-constant for (29). So, suppose that there is $K^{\prime }\in {\mathbb{R}}_{+}$, which is an Ulam ${\varphi }_n$-constant of Equation (29) and $K^{\prime }<K$. Fix $g_1:U\to W$ with

$$\parallel g_1(u+v)-g_1\left(u\right)-g_1\left(v\right)\parallel \le \varphi (u,v),\forall u,v\in U.$$

Write $g\left(u\right)=g_1\left(u^n\right)$ for $u\in U$. Then,

$$\begin{array}{cc}\hfill \parallel g\left(\sqrt[n]{u^n+v^n}\right)-g\left(u\right)-g\left(v\right)\parallel & = \parallel g_1(u^n+v^n)-g_1\left(u^n\right)-g_1\left(v^n\right)\parallel \hfill \\ \hfill & \le \varphi (u^n,v^n)={\varphi }_n(u,v),\forall u,v\in U,\hfill \end{array}$$

whence there exists a solution $h:U\to W$ of Equation (29) such that $\parallel g\left(u\right)-h\left(u\right)\parallel \le K^{\prime }{\varphi }_n(u,u)$ for $u\in U$.

Let $h_1\left(u\right)=h\left(\sqrt[n]{u}\right)$ for $u\in U$. Then, it is easy to check that $h_1$ is a solution to (12) and

$$\parallel g_1\left(u\right)-h_1\left(u\right)\parallel = \parallel g\left(\sqrt[n]{u}\right)-h\left(\sqrt[n]{u}\right)\parallel \le K^{\prime }{\varphi }_n(\sqrt[n]{u},\sqrt[n]{u})=K^{\prime }\varphi (u,u),\forall u\in U.$$

Thus, we have shown that $K^{\prime }$ is an Ulam $\varphi$-constant of Equation (12), which is a contradiction, because $K^{\prime }<K$ and K is assumed to be the best Ulam $\varphi$-constant of (12). This completes the proof that the best Ulam $\varphi$-constant of (12) is the best Ulam ${\varphi }_n$-constant of (29).

The proofs of the converse implications are analogous. □

Proposition 3 means that we can use all the information on the best Ulam constant of Equation (12) to obtain analogous results for Equation (29). In particular, from Table 1, we derive the following Table 3 with values of the best Ulam $\varphi$-constant $K_{\varphi }$ of Equation (29), with $\varphi (u,v)\equiv \xi (\parallel u{\parallel }^t+\parallel v{\parallel }^t)$ and $U\in \{\mathbb{R},{\mathbb{R}}_{+}\}$.

Table 3. The dependence of the best Ulam $\varphi$-constant $K_{\varphi }$ (for Equation (29) with the control function $\varphi (u,v)\equiv \xi (\parallel u{\parallel }^t+\parallel v{\parallel }^t)$) on the form of $U\in \{\mathbb{R},{\mathbb{R}}_{+}\}$ and the value of $t\ne n$.

Value of t	Form of U	Value of ${\mathit{K}}_{\mathit{\varphi }}$
$t<0$	$U={\mathbb{R}}_{+}$	$K_{\varphi }=1$
$t<0$	$U=\mathbb{R}$	$K_{\varphi }=0$
$0\le t$, $t\ne n$	$U\in \{\mathbb{R},{\mathbb{R}}_{+}\}$	$K_{\varphi }={|2^{t/n}-2|}^{-1}$

It is easily seen that a similar approach can be applied to the problem of the best Ulam constant of numerous other functional equations of the radical type, e.g.,those considered in [18,19,20,21,22,23,24,25,26,27,28,29].

5. Other Equations

Clearly, in a similar way we can study the best Ulam $\varphi$-constants of many other equations and with other control functions $\varphi$. We refer to [2,3,5,6] for examples of such equations and control functions and further references on this subject.

Let us only mention here that several authors (see, e.g., [30,31,32,33,34,35,36]) studied stability of the subsequent quite general functional equation

$$\sum _{i=1}^p{C}_{i}g\left(\sum _{j=1}^m{c}_{ij}{u}_j\right)=D(u_1,\dots ,u_m),\forall u_1,\dots ,u_m\in U,\sum _{j=1}^m{c}_{ij}{u}_j\in U,$$

(30)

which can be considered, for instance, for maps g from a suitable nonempty subset U of a module V, over a ring P, into a linear space W over a field $\mathbb{K}$, where $p,m\in \mathbb{N}$, $C_1,\dots ,C_p\in \mathbb{K}\setminus \left\{0\right\}$, $c_{ij}\in \mathbb{P}$ for $i=1,2,\cdots ,p$, $j=1,2,\cdots ,m$, and $D:U^m\to W$ is fixed. Clearly, the functional Equations (3) and (25) are special cases of Equation (30).

Several examples of various other particular cases of (30) can be found, e.g., in [30,31,32,33,34,35,36,37,38]. Some hyperstability results have been obtained for (30), e.g., in [30,32,33,35,37]), with some control functions $\varphi$, which gives us information that in those situations the best Ulam constant of (30) is equal to 0. Using reasoning analogous to the previous two sections, it is possible to extend those results at least for the radical and some inhomogeneous versions of (30).

Now, let us focus on the best Ulam constant of a particular case of (30) with $m=1$, i.e., on the linear difference equation of higher order

$$x_{n+p}=a_1{x}_{n+p-1}+\dots +a_p{x}_n+b_n,\forall n\in \mathbb{N},$$

(31)

considered for sequences ${\left(x_n\right)}_{n\in \mathbb{N}}$ in a Banach space X over a field $\mathbb{K}$ of real or complex numbers, with fixed $a_1,\dots ,a_p\in \mathbb{K}$, $a_p\ne 0$, an integer $p\ge 1$, and a fixed sequence ${\left(b_n\right)}_{n\in \mathbb{N}}$ in X.

Let $r_1,\cdots ,r_p$ be all the complex roots of the characteristic equation of (31), i.e., of the equation

$$r^p-\sum _{i=1}^p{a}_i{r}^{p-i}=0.$$

(32)

We have, for instance, the following Ulam stability result (see, e.g., [6] Theorem 95).

Theorem 8. 

Let $\delta \in {\mathbb{R}}_{+}$ and $|r_i| \ne 1$ for $i=1,\dots ,p$. Let ${\left(y_n\right)}_{n\in \mathbb{N}}$ be a sequence in X with

$$\parallel y_{n+p}-a_1{y}_{n+p-1}-\dots -a_p{y}_n-b_n\parallel \le \delta ,\forall n\in \mathbb{N}.$$

(33)

Then, there is a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ in X satisfying Equation (31) and such that

$$\parallel y_n-x_n\parallel \le {\displaystyle \frac{\delta }{\left|\right(1-|r_1\left|\right)·\dots ·(1-|r_p\left|\right)|}},\forall n\in \mathbb{N}.$$

(34)

Let us mention yet that if $|r_i| =1$ for some $i\in \{1,\dots ,p\}$, then analogous results are not possible and we can say that non-stability occurs in the following sense (see [6] Theorem 110).

Theorem 9. 

Let $|r_i| =1$ for some $i\in \{1,\dots ,p\}$. Then, there exists a sequence ${\left(y_n\right)}_{n\in \mathbb{N}}$ in X such that (33) holds with some $\delta \in {\mathbb{R}}_{+}$ and

$$\underset{n\in \mathbb{N}}{sup}\parallel y_n-x_n\parallel =\infty$$

for each sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ in X satisfying Equation (31).

For some other related results and information, we refer to [6,39,40,41,42,43,44,45,46].

Clearly, Theorem 8 provides information on the existence of the Ulam $\varphi$-constant for (31) with constant functions $\varphi$. Now, as before, we can ask if this constant is the best Ulam $\varphi$-constant for constant functions $\varphi$. It follows from [47] and ([48] Corollary 2.5) that the best Ulam $\varphi$-constant (with constant functions $\varphi$) for (31) is the same as in (34) when $p=1$ or when all the roots of Equation (32) are real numbers greater than 1 (a similar result can be deduced from ([6] Remark 22), but only when (31) is considered for every $n\in \mathbb{Z}$). Moreover, it follows from [47,48,49,50] that the constant in (34) is not optimal in some cases.

However, some other partial information on the best Ulam constant for Equation (31) (considered for sequences in a Banach space X with constant control functions $\varphi$) have also been proved in [48,49,50]. We present them below.

Actually, they have been proved only when $b_n=0$ for each $n\in \mathbb{N}$, but we can easily use a reasoning similar to the proof of Proposition 2 to extend them to the case of an arbitrary sequence ${\left(b_n\right)}_{n\in \mathbb{N}}$ in X.

Let us start with the results for the cases where $p=2,3$, obtained in [49,50].

Theorem 10. 

Let $\varphi :\mathbb{N}\to {\mathbb{R}}_{+}$ be a constant function and

$$|r_i| >1,\forall i\in \{1,\dots ,p\}.$$

(35)

Then, the following two statements are valid.

(i) 

If $p=2$, then the best Ulam ϕ-constant of (31) is given by the formula:

$$\begin{array}{c}\hfill K_{\varphi }=\left\{\begin{array}{cc}|r_1-r_2{|}^{-1}{\displaystyle \sum _{s=1}^{\infty }}|r_1^{-s}-r_2^{-s}|,\hfill & if{r}_1\ne r_2;\hfill \\ \left(\right|r_1{|-1)}^{-2},\hfill & if{r}_1=r_2.\hfill \end{array}\right.\end{array}$$

(ii) 

If $p=3$, then the best Ulam ϕ-constant of (31) is given by the formula:

$$\begin{array}{c}\hfill K_{\varphi }=\left\{\begin{array}{cc}{\displaystyle \frac{1}{|(r_2-r_1)(r_3-r_1)(r_3-r_2)|}}{\displaystyle \sum _{s=1}^{\infty }}|{\displaystyle \frac{r_3-r_2}{r_1^s}}+{\displaystyle \frac{r_1-r_3}{r_2^s}}+{\displaystyle \frac{r_2-r_1}{r_3^s}}|,\hfill & if r_1,r_2,r_3 are distinct;\hfill \\ |r_2-r_1{|}^{-2}{\displaystyle \sum _{s=1}^{\infty }}|{\displaystyle \frac{1}{r_1^s}}+{\displaystyle \frac{s(r_1-r_2)-r_2}{r_2^{s+1}}}|,\hfill & if r_1\ne r_2=r_3;\hfill \\ \left(\right|r_1{|-1)}^{-3},\hfill & if r_1=r_2=r_3.\hfill \end{array}\right.\end{array}$$

For general $p>1$, we have the following result obtained in ([48] Theorem 2.3) but only under the assumption that each characteristic root of the equation is of multiplicity 1.

Theorem 11. 

Let $\varphi :\mathbb{N}\to {\mathbb{R}}_{+}$ be a constant function. Assume that all the roots $r_1,\cdots ,r_p\in \mathbb{C}$ of Equation (32) are distinct and (35) holds. Then, the best Ulam ϕ-constant of (31) is given by

$$K_{\varphi }={\displaystyle \frac{1}{\left|v\right|}}\sum _{s=1}^{+\infty }|{\displaystyle \frac{v_1}{r_1^s}}-{\displaystyle \frac{v_2}{r_2^s}}+\cdots +{\displaystyle \frac{{(-1)}^{p+1}{v}_p}{r_p^s}}|,$$

(36)

with $v=detV$ and $v_k=det{V}_k$, where $V=V(r_1,r_2,\dots ,r_p)$ is a Vandermonde matrix of order p and $V_k=V(r_1,r_2,\dots ,r_{k-1},r_{k+1},\dots ,r_p)$ is a Vandermonde matrix of order $p-1$ for $k=1,\dots ,p$.

Remark 4. 

It remains an open problem to determine the best Ulam constant for (31) without these additional assumptions, especially without condition (35) and possibly for non-constant control functions ϕ. In some cases, the best Ulam constant for (31) may not exist, but anyway it would be desirable to find the infimum of all Ulam constants in such situations.

Finally, let us mention that there are several results on the best Ulam constant of differential equations and some operators, and we refer to [6,51,52,53,54] for examples of such outcomes, more details and further references. Some other results on the best Ulam constants can be found in [55,56].

6. Conclusions

The approximate solutions of various equations are quite important in several areas of scientific investigation, and it is desirable to know how big the difference is between such approximate solutions and the mappings that satisfy the equations exactly. This is the main subject of the theory of Ulam stability, which is related to the theories of shadowing, optimization, perturbation, and approximation.

One of the main problems in Ulam stability is the optimality of the constants that appear in some stability outcomes. In this short survey, we have discussed several aspects of this problem, mainly using two simple examples of the Cauchy additive functional equation $f(x+y)=f\left(x\right)+f\left(y\right)$ and the general linear difference equation (with constant coefficients) $x_{n+p}=a_1{x}_{n+p-1}+\dots +a_p{x}_n+b_n$. We have shown possible generalizations of existing results and indicated open problems. We believe that this limited approach makes this publication more accessible to a wider audience.

In particular, our results show that there is a significant symmetry between the Ulam constants of several functional equations and their inhomogeneous or radical forms.

Plainly, the closeness of two mappings and the notion of an approximate solution can also be understood in many nonstandard ways with respect to other ways of measuring distance (see, e.g., [57]), and we refer for suitable examples of such results to [28,58,59,60,61,62,63,64,65,66,67,68]). It would also be interesting to investigate the problem of the best Ulam constant in such situations.