Stability of Fréchet Functional Equation in Class of Differentiable Functions

Abstract

The goal of this work is to demonstrate the stability of the Fréchet functional equation in the class of differentiable functions with respect to a large class of seminorms defined by means of derivatives. To achieve this, I utilize previous results concerning the difference property of higher orders of the class of n-times continuously differentiable functions.

1. Introduction

The concept of the stability of functional equations has a very long and rich history. This theory originates from S.M. Ulam’s question (cf. [1]): we are given a group $(X,+)$ and a metric group $(Y,+,d)$. Given $\epsilon >0$, does there exist a $\delta >0$ such that if $f:X\to Y$ satisfies the inequality

$$d\left[f(x+y),f\left(x\right)+f\left(y\right)\right]<\delta \mathrm{for}\mathrm{all}x,y\in X,$$

then a homomorphism $A:X\to Y$ exists with

$$d\left[f\left(x\right),A\left(x\right)\right]<\epsilon \mathrm{for}\mathrm{all}x\in X?$$

The first partial solution to Ulam’s problem was provided by D.H. Hyers [2] under the assumption that X and Y are Banach spaces.

Theorem 1.

Let $\epsilon >0$. Suppose that a function $f:X\to Y$ satisfies the inequality

$$\parallel f(x+y)-f\left(x\right)-f\left(y\right)\parallel <\epsilon ,x,y\in X.$$

Then, there exists a unique additive function $A:X\to Y$ such that

$$\parallel f\left(x\right)-A\left(x\right)\parallel <\epsilon ,x\in X.$$

Moreover,

$$A\left(x\right)=\underset{n\to \infty }{lim}{\displaystyle \frac{f\left(2^{n}x\right)}{2^n}},x\in X.$$

In 1978, Th.M. Rassias [3] broadened Hyers’ theorem by examining an unbounded Cauchy difference, defined as $Cf(x,y):=f(x+y)-f\left(x\right)-f\left(y\right)$ (refer to [4] for further details). Over the past few decades, numerous authors have thoroughly explored the stability issues of various functional equations (see, for instance, [5,6,7,8,9,10], along with the references therein). Roughly speaking, the Hyers–Ulam stability addresses the following issue: how much an approximate solution to a given equation differs from the exact solutions. The concepts of approximate solutions and the proximity of two mappings can be interpreted in different ways. Consequently, the Hyers–Ulam stability of various functional equations can be approached in various directions. An alternative method to measure distance is the use of two-norms (see [11]). Extensive information on this topic is available in the following survey articles [12,13].

Assume that X and Y are normed spaces. For a function $f:X\to Y$, we establish

$${\parallel f\parallel }_{sup}:=\underset{x\in X}{sup}\parallel f\left(x\right)\parallel .$$

Note that ${\parallel f\parallel }_{sup}$ is allowed to take the infinite value. For the Cauchy difference $Cf$, the stability problem can be reformulated as follows. Let $\epsilon >0$ be given. Does there exist a $\delta >0$ such that if $f:X\to Y$ satisfies

$${\parallel Cf\parallel }_{sup}<\delta ,$$

then there exists an additive function $A:X\to Y$ with

$${\parallel f-A\parallel }_{sup}<\epsilon ?$$

A similar question can be asked for other important functional equations. We can consider Ulam’s problem for different norms. In this paper, I prove the stability of the Fréchet functional equation in the class of differentiable functions with respect to a large class of seminorms defined by means of derivatives. The same problems for the Cauchy-type functional equations and the quadratic functional equation were solved in [14,15], respectively. The Fréchet functional equation is closely related to generalized polynomial functions, which are an important part of the theory of functional equations, among others. The difference operator of higher orders plays a significant role in the theory of convex functions of higher orders—details can be found, e.g., in [16].

This article is organized as follows. In the next section, I introduce all necessary assumptions and provide auxiliary information on the difference operator. In Section 3, I prove several results that are crucial for the subsequent part of the paper, where I present the main result concerning the stability of the Fréchet functional equation. Finally, in Section 5, I offer some concluding remarks.

2. Preliminaries

Let X and Y be a real normed space and a real Banach space, respectively. By ${\mathbb{N}}_0$, $\mathbb{N}$, and $\mathbb{R}$, we denote the sets of all non-negative integers, positive integers, and real numbers, respectively. Let $f:X\to Y$ be an n-times Fréchet differentiable function. By $D^{n}f$, $n\in \mathbb{N}$, we denote the n-th derivative of f and $D^{0}f$ stands for f. By $C^n(X,Y)$, we denote the space of n-times continuously differentiable functions, and by $B{C}^n(X,Y)$, we denote the subspace of $C^n(X,Y)$ consisting of bounded functions. Moreover, $C^0(X,Y)$ and $C^{\infty }(X,Y)$ stand for the space of continuous functions and the space of infinitely many times continuously differentiable functions, respectively. By ${\partial }_k^{p}f$, $k=1,2$, we denote the p-th partial derivative of $f:X\times X\to Y$ with respect to the k-th variable.

In the following, we will use the assumptions presented by Jacek Tabor and Józef Tabor in [14]. Let us assume that we are given a norm in the space $X\times X$ such that $\parallel (x_1,x_2)\parallel$ is a function of $\parallel x_1\parallel$ and $\parallel x_2\parallel$, and the following condition is satisfied:

$$\parallel (x,0)\parallel =\parallel (0,x)\parallel =\parallel x\parallel ,x\in X.$$

The simplest example of such a norm is the mapping given by the formula

$$\parallel (x_1,x_2)\parallel =\parallel x_1\parallel +\parallel x_2\parallel .$$

Let $i_1:X\to X\times X$, $i_2:X\to X\times X$ be injections defined by

$$i_1\left(x\right):=(x,0),x\in X,$$

$$i_2\left(y\right):=(0,y),y\in X.$$

Let $L:X\times X\to X$ be a bounded linear mapping. Therefore, from the assumed conditions on the norm in $X\times X$, we obtain

$$\parallel L\circ i_1\parallel \le \parallel L\parallel \parallel i_1\parallel =\parallel L\parallel ,$$

$$\parallel L\circ i_2\parallel \le \parallel L\parallel \parallel i_2\parallel =\parallel L\parallel .$$

Hence, if $F:X\times X\to Y$ is n-times differentiable for $n\in \mathbb{N}$, then

$$\left.\begin{array}{c}\parallel {\partial }_{1}F(x,y)\parallel =\parallel DF(x,y)\circ i_1\parallel \le \parallel DF(x,y)\parallel ,\\ \parallel {\partial }_{2}F(x,y)\parallel =\parallel DF(x,y)\circ i_2\parallel \le \parallel DF(x,y)\parallel ,\end{array}\right\}$$

(1)

and, for fixed $p\in \mathbb{N}$ ($p\le n$), we have

$$\parallel {\partial }_1^{k-p}{\partial }_2^{p}F(x,y)\parallel \le \parallel D^{k}F(x,y)\parallel$$

(2)

for all $x,y\in X$ and $k=p,p+1,\dots ,n$.

For a given function $f:\mathbb{R}\to \mathbb{R}$ and a $y\in \mathbb{R}$, we define the well-known difference operator

$${\Delta }_{y}f\left(x\right):=f(x+y)-f\left(x\right),x\in \mathbb{R}.$$

Moreover, we define inductively the difference operator ${\Delta }_y^p$ as follows:

$${\Delta }_y^{p}f\left(x\right)={\Delta }_y\left({\Delta }_y^{p-1}f\left(x\right)\right),x\in \mathbb{R},p\in \mathbb{N}.$$

Additionally, we put ${\Delta }_y^{0}f\left(x\right)=f\left(x\right)$. In particular, we have ${\Delta }_y^{1}f\left(x\right)={\Delta }_{y}f\left(x\right)$.

Definition 1.

A function $f:\mathbb{R}\to \mathbb{R}$ is said to be a generalized polynomial of p-th order if and only if

$${\Delta }_y^{p+1}f\left(x\right)=0$$

for all $x,y\in \mathbb{R}$.

It is well known (cf. [17]) that $f:\mathbb{R}\to \mathbb{R}$ is a generalized polynomial of p-th order if and only if it has a unique representation

$$f=f_0+f_1+\dots +f_p,$$

where $f_0$ is a constant, and $f_k:\mathbb{R}\to \mathbb{R}$ for $k=1,2,\dots ,p$ are diagonalizations of k-additive symmetric functions $F_k:{\mathbb{R}}^k\to \mathbb{R}$—i.e.,

$$f_k\left(x\right):=F_k(\underset{k-\mathrm{times}}{\underbrace{x,\dots ,x}}),x\in \mathbb{R},k=1,2,\dots ,p.$$

The symmetric k-additive function $F_k$ is uniquely determined by $f_k$, and the polarization formula holds true:

$$F_k(\underset{k-\mathrm{times}}{\underbrace{y,\dots ,y}})={\displaystyle \frac{1}{p!}}{\Delta }_y^p{f}_k\left(x\right).$$

For additional fundamental properties of the difference operator, refer to the books [8,16].

In what follows, we will adopt the following notation:

$${\Delta }^{p}f(x,y):={\Delta }_y^{p}f\left(x\right),x,y\in \mathbb{R},p\in \mathbb{N}.$$

It is known (see [16]) that for any $p\in \mathbb{N}$, the following equality holds true:

$${\Delta }^{p}f(x,y)=\sum _{i=0}^p{(-1)}^{p-i}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{i}}\right)f(x+iy),x,y\in \mathbb{R}.$$

(3)

Let $p,n\in \mathbb{N}$ be fixed ($p\le n$) and let $f:X\to Y$ be n-times differentiable. Then, ${\Delta }^{p}f$ is also n-times differentiable and by (2) and (3) we have the following, for all $x,y\in X$:

(i)

if $p=1$, then $\Delta f(x,y)=f(x+y)-f\left(x\right)$ and

$$\parallel {\partial }_1^{k-1}{\partial }_2(\Delta f)(x,y)\parallel =\parallel D^{k}f(x+y)\parallel \le \parallel D^k\left(\Delta f\right)(x,y)\parallel ,1\le k\le n.$$

(ii)

if $p=2$, then ${\Delta }^{2}f(x,y)=f(x+2y)-2f(x+y)+f\left(x\right)$ and

$$\parallel {\partial }_1^{k-2}{\partial }_2^2\left({\Delta }^{2}f\right)(x,y)\parallel =\parallel 4D^{k}f(x+2y)-2D^{k}f(x+y)\parallel \le \parallel D^k\left({\Delta }^{2}f\right)(x,y)\parallel ,2\le k\le n.$$

(iii)

if $p=3$, then ${\Delta }^{3}f(x,y)=f(x+3y)-3f(x+2y)+3f(x+y)-f\left(x\right)$ and

$$\parallel {\partial }_1^{k-3}{\partial }_2^3\left({\Delta }^{3}f\right)(x,y)\parallel =\parallel 27D^{k}f(x+3y)-24D^{k}f(x+2y)+3D^{k}f(x+y)\parallel$$

$$\le \parallel D^k\left({\Delta }^{3}f\right)(x,y)\parallel ,3\le k\le n.$$

Generally, from (2), we have

$$\parallel {\partial }_1^{k-p}{\partial }_2^p\left({\Delta }^{p}f\right)(x,y)\parallel \le \parallel D^k\left({\Delta }^{p}f\right)(x,y)\parallel ,x,y\in X,k=p,p+1,\dots ,n.$$

(4)

Let $p\in \mathbb{N}$ be fixed, and let $p\le k$. Differentiating both sides of Equality (3) $(k-p)$-times with respect to x and p-times with respect to y we obtain

$${\partial }_1^{k-p}{\partial }_2^p\left({\Delta }^{p}f\right)(x,y)=\sum _{i=0}^p{(-1)}^{p-i}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{i}}\right)i^p{D}^{k}f(x+iy),x,y\in X.$$

In particular, taking $y=0$, we see that

$${\partial }_1^{k-p}{\partial }_2^p\left({\Delta }^{p}f\right)(x,0)=\sum _{i=0}^p{(-1)}^{p-i}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{i}}\right)i^p{D}^{k}f\left(x\right)=p!D^{k}f\left(x\right),x\in X,$$

(5)

due to the known equality

$$\sum _{i=0}^p{(-1)}^{p-i}\left({\displaystyle \genfrac{}{}{0pt}{}{p}{i}}\right)i^p=p!.$$

Therefore, from (4) and (5), we deduce that

$$\parallel D^{k}f\left(x\right)\parallel \le {\displaystyle \frac{1}{p!}}\parallel D^k\left({\mathrm{\Delta }}^{p}f\right)(x,0)\parallel ,x\in X,k=p,p+1,\dots ,n.$$

(6)

3. Difference Property

The concept of the difference property for various classes of real functions was introduced by N.G. de Bruijn in [18]. He posed the following question: let $f:\mathbb{R}\to \mathbb{R}$ be a function. Assume that for every $h\in \mathbb{R}$, the difference operator ${\Delta }_{h}f:\mathbb{R}\to \mathbb{R}$ given by the formula

$${\Delta }_{h}f\left(x\right)=f(x+h)-f\left(x\right),x\in \mathbb{R}$$

belongs to a given class $\mathcal{F}\subset {\mathbb{R}}^{\mathbb{R}}$. What can be said about the function f? In [18,19], it was demonstrated that for many important classes $\mathcal{F}$ (such as continuous, differentiable, analytic, absolute continuous, or Riemann-integrable functions), the function f can be expressed in the form

$$f=g+A,$$

(7)

where $g\in \mathcal{F}$ and $A:\mathbb{R}\to \mathbb{R}$ is an additive function—i.e., it satisfies the Cauchy functional equation

$$A(x+y)=A\left(x\right)+A\left(y\right),x,y\in \mathbb{R}.$$

We can formalize the concept as follows: a class $\mathcal{F}$ is said to have the difference property if any function $f:\mathbb{R}\to \mathbb{R}$, whose differences ${\Delta }_{h}f$ belong to $\mathcal{F}$, can be represented in the form (7). The foundational results of de Bruijn on this topic have been expanded and generalized in many directions. For a detailed overview of the difference property, refer to Laczkovich’s survey paper [20], as well as [21,22,23] for additional insights and references.

According to J.H.B. Kamperman [24], a class $\mathcal{F}\subset {\mathbb{R}}^{\mathbb{R}}$ is said to have the difference property of p-th order ($p\in \mathbb{N}$) if and only if any function $f:\mathbb{R}\to \mathbb{R}$ such that ${\Delta }_h^{p}f\in \mathcal{F}$ for each $h\in \mathbb{R}$ admits a decomposition of the form $f=g+\Gamma$, where $g\in \mathcal{F}$ and $\Gamma :\mathbb{R}\to \mathbb{R}$ is a polynomial function of p-th order.

The difference properties of higher orders have been thoroughly explored for various classes of functions. For example, Z. Gajda established these properties for the class of all continuous functions on locally compact Abelian groups and the class of all Riemann-integrable functions on compact, second-countable Abelian groups (see [25]).

In [26], we proved the following result, demonstrating that the class of p-times continuously differentiable functions has some kind of difference property of p-th order. Notably, we provide an explicit formula for the function ${\Gamma }_{p-1}$, whereas the original results by de Bruijn et al. only establish the existence of a corresponding function.

Theorem 2.

Let $p\in \mathbb{N}$ be fixed and let $f:\mathbb{R}\to \mathbb{R}$ be a function such that ${\Delta }^{p}f\in C^p(\mathbb{R}\times \mathbb{R},\mathbb{R})$. Then, there exists a generalized polynomial ${\Gamma }_{p-1}:\mathbb{R}\to \mathbb{R}$ of $(p-1)$-th order given by the formula

$${\Gamma }_{p-1}\left(x\right)=f\left(x\right)-{\displaystyle \frac{1}{p!}}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{t_1}{\int }}\underset{0}{\overset{t_2}{\int }}\dots \underset{0}{\overset{t_{p-1}}{\int }}{\partial }_2^p\left({\Delta }^{p}f\right)(ux,0)\left(x^p\right)dud{t}_{p-1}\dots d{t}_1,x\in \mathbb{R}$$

such that $f-{\Gamma }_{p-1}\in C^p(\mathbb{R},\mathbb{R})$. Moreover,

$$D^p(f-{\Gamma }_{p-1})\left(x\right)={\displaystyle \frac{1}{p!}}{\partial }_2^p\left({\Delta }^{p}f\right)(x,0),x\in \mathbb{R},$$

(8)

and

$$D^i(f-{\Gamma }_{p-1})\left(0\right)=0,0\le i\le p-1.$$

(9)

It is easy to prove that if a function f is p-times continuously differentiable, then the function ${\Gamma }_{p-1}$ is an ordinary polynomial of the form

$${\Gamma }_{p-1}\left(x\right)=a_0+a_{1}x+a_2{x}^2+\dots +a_{p-1}{x}^{p-1},$$

where the coefficients are given by

$$a_i={\displaystyle \frac{1}{i!}}{D}^{i}f\left(0\right),i=0,1,\dots ,p-1.$$

Now, we can formulate the following theorem.

Theorem 3.

Let $p,n\in \mathbb{N}$ ($p\le n$) be fixed and let $f:\mathbb{R}\to \mathbb{R}$ be a function such that ${\Delta }^{p}f\in C^n(\mathbb{R}\times \mathbb{R},\mathbb{R})$. Then, there exists a generalized polynomial ${\Gamma }_{p-1}:\mathbb{R}\to \mathbb{R}$ of $(p-1)$-th order such that $f-{\Gamma }_{p-1}\in C^n(\mathbb{R},\mathbb{R})$ and $D^i(f-{\Gamma }_{p-1})\left(0\right)=0$, $0\le i\le p-1$. Moreover,

$$\parallel D^k(f-{\Gamma }_{p-1}){\parallel }_{sup}\le {\displaystyle \frac{1}{p!}}\parallel D^k\left({\Delta }^{p}f\right){\parallel }_{sup},k=p,p+1,\dots ,n$$

(10)

and

$$\parallel D^k(f-{\Gamma }_{p-1})\left(0\right)\parallel \le {\displaystyle \frac{1}{p!}}\parallel D^k\left({\Delta }^{p}f\right)(0,0)\parallel ,k=0,1,\dots ,n.$$

(11)

Proof. 

From Equation (8), it follows that ${\partial }_2^p\left({\Delta }^{p}f\right)\in C^{n-p}(\mathbb{R}\times \mathbb{R},\mathbb{R})$; hence, ${\partial }_2^p\left({\Delta }^{p}f\right)(x,0)\in C^{n-p}(\mathbb{R},\mathbb{R})$ for all $x\in \mathbb{R}$. Therefore, $D^p(f-{\Gamma }_{p-1})\in C^{n-p}(\mathbb{R},\mathbb{R})$—that is, $f-{\Gamma }_{p-1}\in C^n(\mathbb{R},\mathbb{R})$. Applying Theorem 2 and Condition (6), one can obtain Inequality (10) due to the equality ${\Delta }^p(f-{\Gamma }_{p-1})={\Delta }^{p}f$. Meanwhile, Condition (11) follows from (6) and (9). □

In the following, we will require the following result.

Theorem 4

([8], Theorem 7.6). Let $p\in \mathbb{N}$ and $\epsilon >0$. Assume that G is a group that admits division by $p!$ and let X be a real Banach space. If $f:G\to X$ satisfies

$$\parallel {\Delta }^{p}f(x,y)\parallel \le \epsilon ,x,y\in G,$$

then there exists a generalized polynomial $P_{p-1}:G\to X$ of $(p-1)$-th order such that

$$\parallel f\left(x\right)-f\left(0\right)-P_{p-1}\left(x\right)\parallel \le {\displaystyle \frac{2\epsilon }{\underset{1\le k\le p}{sup}\left(\genfrac{}{}{0pt}{}{p}{k}\right)}},$$

where $P_{p-1}\left(x\right)=\sum _{k=1}^{p-1}{f}_k\left(x\right)$.

Now, we will consider the case where ${\Delta }^{p}f\in B{C}^n(\mathbb{R}\times \mathbb{R},\mathbb{R})$.

Theorem 5.

Let $p,n\in \mathbb{N}$ ($p\le n$) be fixed and let $f:\mathbb{R}\to \mathbb{R}$ be a function such that ${\Delta }^{p}f\in B{C}^n(\mathbb{R}\times \mathbb{R},\mathbb{R})$. Then, there exists a generalized polynomial ${{\rm Y}}_{p-1}:\mathbb{R}\to \mathbb{R}$ of $(p-1)$-th order such that $f-{{\rm Y}}_{p-1}\in B{C}^n(\mathbb{R},\mathbb{R})$. Moreover,

$$\parallel D^k(f-{{\rm Y}}_{p-1}){\parallel }_{sup}\le {\displaystyle \frac{1}{p!}}\parallel D^k\left({\Delta }^{p}f\right){\parallel }_{sup},k=p,p+1,\dots ,n$$

(12)

and

$$\parallel D^k(f-{{\rm Y}}_{p-1})\left(0\right)\parallel \le {\displaystyle \frac{1}{p!}}\parallel D^k\left({\Delta }^{p}f\right)(0,0)\parallel ,k=p,p+1,\dots ,n.$$

(13)

Proof. 

By Theorem 3, there exists a generalized polynomial ${\Gamma }_{p-1}:\mathbb{R}\to \mathbb{R}$ of $(p-1)$-th order such that $g:=f-{\Gamma }_{p-1}\in C^n(\mathbb{R},\mathbb{R})$ and $D^{i}g\left(0\right)=0$, where $0\le i\le p-1$. Then, ${\Delta }^{p}g={\Delta }^{p}f\in B{C}^n(\mathbb{R}\times \mathbb{R},\mathbb{R})$. It means that ${\Delta }^{p}g$ is bounded. In virtue of Theorem 4, there exists a generalized polynomial $P_{p-1}$ such that $P_{p-1}\left(0\right)=0$ and

$$\underset{x\in \mathbb{R}}{sup}\parallel g\left(x\right)-P_{p-1}\left(x\right)\parallel \le {\displaystyle \frac{2}{\underset{1\le k\le p}{sup}\left(\genfrac{}{}{0pt}{}{p}{k}\right)}}\underset{(x,y)\in \mathbb{R}\times \mathbb{R}}{sup}\parallel {\Delta }^{p}g(x,y)\parallel .$$

(14)

Since $g\in C^n(\mathbb{R},\mathbb{R})$, then also $P_{p-1}\in C^n(\mathbb{R},\mathbb{R})$ (see [8], Corollary 7.4). Define a function ${{\rm Y}}_{p-1}:\mathbb{R}\to \mathbb{R}$ by the formula

$${{\rm Y}}_{p-1}\left(x\right):={\Gamma }_{p-1}\left(x\right)+P_{p-1}\left(x\right),x\in \mathbb{R}.$$

Then, ${{\rm Y}}_{p-1}$ is a generalized polynomial of $(p-1)$-th order and $f-{{\rm Y}}_{p-1}=g-P_{p-1}\in C^n(\mathbb{R},\mathbb{R})$. Therefore, from (6), we have

$$\parallel D^k(f-{{\rm Y}}_{p-1}){\parallel }_{sup}\le {\displaystyle \frac{1}{p!}}\parallel D^k\left({\Delta }^{p}f\right){\parallel }_{sup},k=p,p+1,\dots ,n,$$

$$\parallel D^k(f-{{\rm Y}}_{p-1})\left(0\right)\parallel \le {\displaystyle \frac{1}{p!}}\parallel D^k\left({\Delta }^{p}f\right)(0,0)\parallel ,k=p,p+1,\dots ,n.$$

The proof is completed. □

Remark 1.

Conditions (12) and (13) also hold for $k=0$. This follows from (14) and from the equality $(f-{{\rm Y}}_{p-1})\left(0\right)={\Delta }^{p}f(0,0)=0$, respectively.

4. Stability

In this section, I show the main stability result concerning the Fréchet functional equation. We will consider a class of seminorms defined by means of derivatives. Let $f:X\to Y$ be a function such that $f\in C^n(X,Y)$, $n\in {\mathbb{N}}_0$. In subspaces of $C^n(X,Y)$, one can consider different norms defined in terms of $\parallel D^{k}f\left(0\right)\parallel$ and $\parallel D^k{f\parallel }_{sup}$ for $k\le n$. For example, the following norms are frequently encountered in the literature:

$$\parallel f\parallel :=\sum _{k=0}^n\parallel D^{k}f\left(0\right)\parallel +\parallel D^{n}f{\parallel }_{sup},$$

$$\parallel f\parallel :=\sum _{k=0}^n{\parallel D^{k}f\parallel }_{sup},$$

$$\parallel f\parallel :=\underset{k=0,\dots ,n}{max}{\parallel D^{k}f\parallel }_{sup}.$$

In what follows, we will use slightly modified assumptions introduced in [14]. Let $n\in {\mathbb{N}}_0\cup \{\infty \}$ be fixed. In the set ${[0,\infty ]}^{2n+2}$, we introduce the order

$$(x_1,x_2,\dots )\le (y_1,y_2,\dots )$$

if and only if $x_i\le y_i$ for $i\in \mathbb{N}$, $i\le 2n+2$.

Let $\phi :{[0,\infty ]}^{2n+2}\to [0,\infty ]$ be any function that satisfies the following conditions:

(i)

$\phi (x+y)\le \phi \left(x\right)+\phi \left(y\right)$, $x,y\in {[0,\infty ]}^{2n+2}$;

(ii)

$\phi \left(\alpha x\right)=\alpha \phi \left(x\right)$, $x,y\in {[0,\infty ]}^{2n+2}$, $\alpha \in [0,\infty )$;

(iii)

$x\le y⟹\phi \left(x\right)\le \phi \left(y\right)$, $x,y\in {[0,\infty ]}^{2n+2}$.

We additionally assume that $0·\infty =0$. From (ii), we obtain $\phi \left(0\right)=0$.

We define the mapping $\Phi :C^n(X,Y)\to {[0,\infty ]}^{2n+2}$ by the formula

$$\Phi \left(f\right):=\left({\parallel f\left(0\right)\parallel ,\parallel f\parallel }_{sup}{,\parallel Df\left(0\right)\parallel ,\parallel Df\parallel }_{sup},\dots ,\parallel D^{n}f\left(0\right)\parallel ,\parallel D^n{f\parallel }_{sup}\right),$$

and set

$$S_{\phi }(X,Y):=\left\{f\in C^n(X,Y):\phi \left(\Phi \left(f\right)\right)<\infty \right\}.$$

Since $\phi \left(0\right)=0$, the set $S_{\phi }$ necessarily includes the zero function. It is straightforward to verify that $S_{\phi }$ forms a linear space, and that the map $\phi \circ \Phi |S_{\phi }$ defines a seminorm on $S_{\phi }$. We will denote this seminorm by ${\parallel ·\parallel }_{\phi }$. For the sake of consistency, we will adopt the same notation for the space $C^n(X\times X,Y)$.

With these preliminaries established, we are now prepared to prove the main theorem of this section.

Theorem 6.

Let $p,n\in \mathbb{N}$ be fixed ($p\le n$) and let $f:\mathbb{R}\to \mathbb{R}$ be a function such that ${\Delta }^{p}f\in S_{\phi }(\mathbb{R}\times \mathbb{R},\mathbb{R})$. We additionally assume that the function φ does not depend on the k-th variable ($k=2,4,\dots ,2p$ or $k=3,4,\dots ,2p$). Then, there exists a generalized polynomial ${\Psi }_{p-1}:\mathbb{R}\to \mathbb{R}$ of $(p-1)$-th order such that $f-{\Psi }_{p-1}\in S_{\phi }(\mathbb{R},\mathbb{R})$ and

$$\parallel f-{\Psi }_{p-1}{\parallel }_{\phi }\le {\displaystyle \frac{1}{p!}}{\parallel {\Delta }^{p}f\parallel }_{\phi }.$$

Proof. 

Let $p\in \mathbb{N}$ be fixed and assume that ${\Delta }^{p}f\in C^n(\mathbb{R}\times \mathbb{R},\mathbb{R})$. Suppose that $\phi$ does not depend on the k-th variable ($k=2,4,\dots ,2p$). Then,

$$\phi (\underset{2p-\mathrm{times}}{\underbrace{0,\infty ,0,\infty ,\dots ,0,\infty }},0,0,\dots ,0)=\phi \left(\underset{(2n+2)-\mathrm{times}}{\underbrace{0,0,\dots ,0}}\right)=0.$$

By Theorem 3, there exists a generalized polynomial ${\Gamma }_{p-1}:\mathbb{R}\to \mathbb{R}$ of $(p-1)$-th order satisfying Conditions (10) and (11). Therefore,

$$\Phi (f-{\Gamma }_{p-1})\le {\displaystyle \frac{1}{p!}}\left[\Phi \left({\Delta }^{p}f\right)+(0,\infty ,0,\infty ,\dots ,0,\infty ,0,0,\dots ,0)\right],$$

and hence, from (i), (ii), and (iii), we have

$$\phi \left(\Phi (f-{\Gamma }_{p-1})\right)\le \phi \left({\displaystyle \frac{1}{p!}}\left[\Phi \left({\Delta }^{p}f\right)+(0,\infty ,0,\infty ,\dots ,0,\infty ,0,0,\dots ,0)\right]\right)$$

$$\le {\displaystyle \frac{1}{p!}}\phi \left(\Phi \left({\Delta }^{p}f\right)\right)+{\displaystyle \frac{1}{p!}}\phi (0,\infty ,0,\infty ,\dots ,0,\infty ,0,0,\dots ,0)={\displaystyle \frac{1}{p!}}\phi \left(\Phi \left({\Delta }^{p}f\right)\right)+0,$$

i.e.,

$$\parallel f-{\Gamma }_{p-1}{\parallel }_{\phi }\le {\displaystyle \frac{1}{p!}}{\parallel {\Delta }^{p}f\parallel }_{\phi }.$$

If ${\Delta }^{p}f\in B{C}^n(\mathbb{R}\times \mathbb{R},\mathbb{R})$, then by Theorem 5, there exists a generalized polynomial ${{\rm Y}}_{p-1}:\mathbb{R}\to \mathbb{R}$ of $(p-1)$-th order satisfying Conditions (12) and (13). Hence,

$$\Phi (f-{{\rm Y}}_{p-1})\le {\displaystyle \frac{1}{p!}}\left[\Phi \left({\Delta }^{p}f\right)+(0,0,\underset{(2p-2)-\mathrm{times}}{\underbrace{\infty ,\infty ,\dots ,\infty }},0,0,\dots ,0)\right],$$

and consequently, from (i), (ii), and (iii), we obtain

$$\phi \left(\Phi (f-{{\rm Y}}_{p-1})\right)\le {\displaystyle \frac{1}{p!}}\phi \left(\Phi \left({\Delta }^{p}f\right)+(0,0,\infty ,\infty ,\dots ,\infty ,0,0,\dots ,0)\right)\le {\displaystyle \frac{1}{p!}}\phi \left(\Phi \left({\Delta }^{p}f\right)\right),$$

i.e.,

$$\parallel f-{{\rm Y}}_{p-1}{\parallel }_{\phi }\le {\displaystyle \frac{1}{p!}}{\parallel {\Delta }^{p}f\parallel }_{\phi }.$$

If ${\Delta }^{p}f$ is unbounded, then $\parallel {\Delta }^p{f\parallel }_{sup}=\infty$. By Theorem 3, we can find a generalized polynomial such that Conditions (10) and (11) hold. Then, $\Phi (f-{\Gamma }_{p-1})\le {\displaystyle \frac{1}{p!}}\Phi \left({\Delta }^{p}f\right)$, and hence,

$$\parallel f-{\Gamma }_{p-1}{\parallel }_{\phi }\le {\displaystyle \frac{1}{p!}}{\parallel {\Delta }^{p}f\parallel }_{\phi }.$$

The proof is completed. □

One can readily observe that, for $n\in {\mathbb{N}}_0$, if we define

$$\phi (x_1,x_2,\dots ,x_{2n+2})=\sum _{k=0}^n{x}_{2k+1}+x_{2n+2},$$

(15)

$$\phi (x_1,x_2,\dots ,x_{2n+2})=\sum _{k=0}^n{x}_{2k+2},$$

or

$$\phi (x_1,x_2,\dots ,x_{2n+2})=\underset{0\le k\le n}{max}{x}_{2k+2},$$

then we would obtain the stability of the Fréchet functional equation in the norms defined at the beginning of this section, respectively.

It is natural to inquire whether the constant ${\displaystyle \frac{1}{p!}}$ obtained in Theorem 6 is sharp. In the following example, we obtain a worse estimate of $f-{\Psi }_{p-1}$ for $n=p=2$ and the norm given by (15). The question of the sharpness of the constant ${\displaystyle \frac{1}{p!}}$ remains open.

Example 1.

Let $n=p=2$ and let φ be defined as in (15)—i.e.,

$$\phi (x_1,x_2,\dots ,x_6)=x_1+x_3+x_5+x_6.$$

Let the norm in the space $\mathbb{R}\times \mathbb{R}$ be given by the formula

$$\parallel (x_1,x_2)\parallel =|x_1|+|x_2|,x_1,x_2\in \mathbb{R}.$$

We consider the function

$$f\left(x\right)=sinx,x\in \mathbb{R}.$$

Then, $f\left(0\right)=0$, $Df\left(0\right)=1$, and $D^{2}f\left(0\right)=0$. Moreover, for fixed $x,y\in \mathbb{R}$, we have

$$\parallel D\left({\Delta }^{2}f\right)(x,y)\parallel =\parallel {\partial }_1\left({\Delta }^{2}f\right)(x,y)h_1+{\partial }_2\left({\Delta }^{2}f\right)(x,y)h_2\parallel$$

$$=\underset{|h_1|+|h_2|=1}{sup}|(cos(x+2y)-2cos(x+y)+cosx)h_1+(2cos(x+2y)-2cos(x+y))h_2|$$

$$\ge max\left(|cos(x+2y)-2cos(x+y)+cosx|,|2cos(x+2y)-2cos(x+y\left)\right|\right).$$

On the other hand, we obtain

$$\parallel D\left({\Delta }^{2}f\right)(x,y)\parallel \le \underset{|h_1|+|h_2|=1}{sup}\left(\right|cos(x+2y)-2cos(x+y)+cosx||h_1|$$

$$+|2cos(x+2y)-2cos(x+y)||h_2\left|\right)$$

$$\le max\left(|cos(x+2y)-2cos(x+y)+cosx|,|2cos(x+2y)-2cos(x+y\left)\right|\right).$$

Therefore,

$$\parallel D\left({\Delta }^{2}f\right)(x,y)\parallel =max\left(|cos(x+2y)-2cos(x+y)+cosx|,|2cos(x+2y)-2cos(x+y\left)\right|\right)$$

for $x,y\in \mathbb{R}$. In a similar way, it can be shown that

$$\parallel D^2\left({\Delta }^{2}f\right)(x,y)\parallel =\parallel {\partial }_1^2\left({\Delta }^{2}f\right)(x,y)h_1^2+2{\partial }_{12}^2\left({\Delta }^{2}f\right)(x,y)h_1{h}_2+{\partial }_2^2\left({\Delta }^{2}f\right)(x,y)h_2^2\parallel$$

$$=max\left\{\right|sin(x+2y)-2sin(x+y)+sinx|,$$

$$|2sin(x+2y)-2sin(x+y\left)\right|,|4sin(x+2y)-2sin(x+y\left)\right|\}$$

for $x,y\in \mathbb{R}$. Hence, we obtain $\left({\Delta }^{2}f\right)(0,0)=D\left({\Delta }^{2}f\right)(0,0)=D^2\left({\Delta }^{2}f\right)(0,0)=0$, and it is not difficult to find that

$$\underset{(x,y)\in \mathbb{R}\times \mathbb{R}}{sup}\parallel D^2\left({\Delta }^{2}f\right)(x,y)\parallel =6.$$

Since the function f is bounded on $\mathbb{R}$, the unique generalized polynomial that approximates f is ${\Psi }_1=0$. Then,

$$\parallel f-{\Psi }_1{\parallel }_{\phi }=\parallel f\left(0\right)\parallel +\parallel Df\left(0\right)\parallel +\parallel D^{2}f\left(0\right)\parallel +\parallel D^2{f\parallel }_{sup}=0+1+0+1=2,$$

while

$$\parallel {\Delta }^2{f\parallel }_{\phi }=\parallel \left({\Delta }^{2}f\right)(0,0)\parallel +\parallel D\left({\Delta }^{2}f\right)(0,0\parallel +\parallel D^2\left({\Delta }^{2}f\right)(0,0\parallel +\parallel D^2\left({\Delta }^{2}f\right){\parallel }_{sup}=6.$$

Remark 2.

One might naturally ask whether the stability result concerning the seminorm can be extended to other types of norms. At present, the author has no answer to this question.

5. Conclusions

In this paper, I have presented results concerning the Hyers–Ulam stability of the Fréchet functional equation in the class of differentiable functions. These results are intimately connected to the concept of the difference property of higher orders, as introduced by J.H.B. Kamperman in [24]. To prove the main stability theorem, I utilized earlier results from [26], and employed the proof techniques outlined in [14].

A natural generalization of Theorem 2, and consequently of Theorems 3 and 5, involves considering the difference ${\Delta }^{p}f$ as an element of the class $C^n(X\times X,Y)$. Here, X is a real normed space, and Y is a real Banach space. In this context, we expect the generalized polynomial ${\Gamma }_{p-1}$ to take the form

$${\Gamma }_{p-1}\left(x\right)=f\left(x\right)-{\displaystyle \frac{1}{p!}}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{t_1}{\int }}\underset{0}{\overset{t_2}{\int }}\dots \underset{0}{\overset{t_{p-1}}{\int }}{\partial }_2^p\left({\Delta }^{p}f\right)(ux,0)\left(\underset{p-\mathrm{times}}{\underbrace{x,x,\dots ,x}}\right)dud{t}_{p-1}\dots d{t}_1,x\in X.$$

The principal difficulty lies in demonstrating the differentiability of this function. The solution to this problem remains open.