Comment on "Functional inequalities associated with Jordan-von Neumann type additive functional equations"

Abstract

Park et al. proved the Hyers-Ulam stability of some additive functional inequalities. There is a fatal error in the proof of Theorem 3.1. We revise the statements of the main theorems and prove the revised theorems.

2010 Mathematics Subject Classification: Primary 39B62; 39B72; 39B52.

1 Introduction and preliminaries

Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.

We are given a group G and a metric group G' with metric ρ(·,·). Given ϵ > 0, does there exist a δ > 0 such that if f : G → G' satisfies ρ(f (xy), f(x) f(y)) < δ for all x,y ∈ G, then a homomorphism h : G → G' exists with ρ(f(x), h(x)) < ϵ for all x ∈ G?

Hyers [2] considered the case of approximately additive mappings f: E → E', where E and E' are Banach spaces and f satisfies Hyers' inequality

$$∥f\left(x+y\right)-f\left(x\right)-f\left(y\right)∥\le \epsilon$$

for all x,y ∈ E. It was shown that the limit

$$L\left(x\right)=\underset{n\to \infty }{\text{lim}}\frac{f\left(2^{n}x\right)}{2^n}$$

exists for all x ∈ E and that L : E → E' is the unique additive mapping satisfying

$$∥f\left(x\right)-L\left(x\right)∥\le \epsilon .$$

Rassias [3] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded.

Theorem 1.1. (Rassias). Let f : E → E' be a mapping from a normed vector space E into a Banach space E' subject to the inequality

$$∥f\left(x+y\right)-f\left(x\right)-f\left(y\right)∥\le \epsilon \left({∥x∥}^p+{∥y∥}^p\right)$$

(1.1)

for all x,y ∈ E, where ϵ and p are constants with ϵ > 0 and p < 1. Then the limit

$$L\left(x\right)=\underset{n\to \infty }{\text{lim}}\frac{f\left(2^{n}x\right)}{2^n}$$

exists for all x ∈ E and L : E → E' is the unique additive mapping which satisfies

$$∥f\left(x\right)-L\left(x\right)∥\le \frac{2\epsilon }{2-2^p}{∥x∥}^p$$

(1.2)

for all x ∈ E. If p < 0 then inequality (1.1) holds for x,y ≠ 0 and (1.2) for x ≠ 0.

Rassias [4] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. Gajda [5] following the same approach as in Rassias [3] gave an affirmative solution to this question for p > 1. It was shown by Gajda [5], as well as by Rassias and Šemrl [6] that one cannot prove a Rassias' type theorem when p = 1 (cf. the books of Czerwik [7] and Hyers et al. [8]).

Rassias [9] followed the innovative approach of Rassias' theorem [3] in which he replaced the factor ∥x∥^p+ ∥y∥^pby ∥x∥^p· ∥y∥^qfor p,q ∈ ℝ with p + q ≠ 1. Găvruta [10] provided a further generalization of Rassias' theorem. During the last two decades a number of papers and research monographs have been published on various generalizations and applications of the Hyers-Ulam stability to a number of functional equations and mappings (see [11–13]).

Throughout this article, let G be a 2-divisible abelian group. Assume that X is a normed space with norm || · || _X and that Y is a Banach space with norm || · || _Y .

Gilányi [14] showed that if f satisfies the functional inequality

$$∥2f\left(x\right)+2f\left(y\right)-f\left(x{y}^{-1}\right)∥\le ∥f\left(xy\right)∥$$

(1.3)

then f satisfies the Jordan-von Neumann functional equation

$$2f\left(x\right)+2f\left(y\right)=f\left(xy\right)+f\left(x{y}^{-1}\right).$$

See also [15]. Gilányi [16] and Fechner [17] proved the Hyers-Ulam stability of the functional inequality (1.3).

Park et al. [18] proved the Hyers-Ulam stability of the following functional inequalities

$$∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥\le ∥2f\left(\frac{x+y+z}{2}\right)∥,$$

(1.4)

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

(1.5)

$$∥f\left(x\right)+f\left(y\right)+2f\left(z\right)∥\le ∥2f\left(\frac{x+y}{2}+z\right)∥.$$

(1.6)

But there is an error in the 8th line on the 6th page in the proof of [18, Theorem 3.1]. We revise the statements of the main theorems and prove the revised theorems.

In Section 2, we prove the Hyers-Ulam stability of the functional inequality (1.4).

In Section 3, we prove the Hyers-Ulam stability of the functional inequality (1.5).

In Section 4, we prove the Hyers-Ulam stability of the functional inequality (1.6).

2 Stability of a functional inequality associated with a 3-variable Jensen additive functional equation

Proposition 2.1. [18, Proposition 2.1] Let f : G → Y be a mapping such that

$${∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥}_Y\le {∥2f\left(\frac{x+y+z}{2}\right)∥}_Y$$

for all x, y, z ∈ G. Then f is Cauchy additive.

We prove the Hyers-Ulam stability of a functional inequality associated with a Jordan-von Neumann type 3-variable Jensen additive functional equation.

Theorem 2.2. Let r > 1 and θ be nonnegative real numbers, and let f : X →Y be an odd mapping such that

$${∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥}_Y\le {∥2f\left(\frac{x+y+z}{2}\right)∥}_Y+\theta \left({∥x∥}_X^r+{∥y∥}_X^r+{∥z∥}_X^r\right)$$

(2.1)

for all x,y, z ∈ X. Then there exists a unique Cauchy additive mapping h : X → Y such that

$${∥f\left(x\right)-h\left(x\right)∥}_Y\le \frac{2^r+2}{2^r-2}\theta {∥x∥}_X^r$$

(2.2)

for all x ∈ X.

Proof. Letting y = x and z = -2x in (2.1), we get

$${∥2f\left(x\right)-f\left(2x\right)∥}_Y={∥2f\left(x\right)+f\left(-2x\right)∥}_Y\le \left(2+2^r\right)\theta {∥x∥}_X^r$$

(2.3)

for all x ∈ X. So

$${∥f\left(x\right)-2f\left(\frac{x}{2}\right)∥}_Y\le \frac{2+2^r}{2^r}\theta {∥x∥}_X^r$$

for all x ∈ X. Hence

$$\begin{array}{ll}\hfill {∥2^{l}f\left(\frac{x}{2^l}\right)-2^{m}f\left(\frac{x}{2^m}\right)∥}_Y& \le \sum _{j=l}^{m-1}{∥2^{j}f\left(\frac{x}{2^j}\right)-2^{j+1}f\left(\frac{x}{2^{j+1}}\right)∥}_Y\\ \le \frac{2+2^r}{2^r}\sum _{j=l}^{m-1}\frac{2^j}{2^{rj}}\theta {∥x∥}_X^r\end{array}$$

(2.4)

for all nonnegative integers m and l with m > l and all x ∈ X.

It follows from (2.4) that the sequence $\left\{2^{n}f\left(\frac{x}{2^n}\right)\right\}$ is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence $\left\{2^{n}f\left(\frac{x}{2^n}\right)\right\}$ converges. So one can define the mapping h : X → Y by

$$h\left(x\right):=\underset{n\to \infty }{\text{lim}}{2}^{n}f\left(\frac{x}{2^n}\right)$$

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.4), we get (2.2).

It follows from (2.1) that

$$\begin{array}{c}{∥h\left(x\right)+h\left(y\right)+h\left(z\right)∥}_Y=\underset{n\to \infty }{\text{lim}}{2}^n{∥f\left(\frac{x}{2^n}\right)+f\left(\frac{y}{2^n}\right)+f\left(\frac{z}{2^n}\right)∥}_Y\\ \le \underset{n\to \infty }{\text{lim}}{2}^n{∥2f\left(\frac{x+y+z}{2^{n+1}}\right)∥}_Y+\underset{n\to \infty }{\text{lim}}\frac{2^n\theta }{2^{nr}}\left({∥x∥}_X^r+{∥y∥}_X^r+{∥z∥}_X^r\right)\\ ={∥2h\left(\frac{x+y+z}{2}\right)∥}_Y\end{array}$$

for all x, y, z ∈ X. So

$${∥h\left(x\right)+h\left(y\right)+h\left(z\right)∥}_Y\le {∥2h\left(\frac{x+y+z}{2}\right)∥}_Y$$

for all x, y, z ∈ X. By Proposition 2.1, the mapping h : X → Y is Cauchy additive.

Now, let T : X → Y be another Cauchy additive mapping satisfying (2.2). Then we have

$$\begin{array}{ll}\hfill {∥h\left(x\right)-T\left(x\right)∥}_Y& =2^n{∥h\left(\frac{x}{2^n}\right)-T\left(\frac{x}{2^n}\right)∥}_Y\\ \le 2^n\left({∥h\left(\frac{x}{2^n}\right)-f\left(\frac{x}{2^n}\right)∥}_Y+{∥T\left(\frac{x}{2^n}\right)-f\left(\frac{x}{2^n}\right)∥}_Y\right)\\ \le \frac{2\left(2^r+2\right)2^n}{\left(2^r-2\right)2^{nr}}\theta {∥x∥}_X^r,\end{array}$$

which tends to zero as n → ∞ for all x ∈ X. So we can conclude that h(x) = T(x) for all x ∈ X. This proves the uniqueness of h. Thus the mapping h : X → Y is a unique Cauchy additive mapping satisfying (2.2).

Theorem 2.3. Let r < 1 and θ be positive real numbers, and let f : X → Y be an odd mapping satisfying (2.1). Then there exists a unique Cauchy additive mapping h : X → Y such that

$${∥f\left(x\right)-h\left(x\right)∥}_Y\le \frac{2+2^r}{2-2^r}\theta {∥x∥}_X^r$$

(2.5)

for all x ∈ X.

Proof. It follows from (2.3) that

$${∥f\left(x\right)-\frac{1}{2}f\left(2x\right)∥}_Y\le \frac{2+2^r}{2}\theta {∥x∥}_X^r$$

for all x ∈ X. Hence

$$\begin{array}{ll}\hfill {∥\frac{1}{2^l}f\left(2^{l}x\right)-\frac{1}{2^m}f\left(2^{m}x\right)∥}_Y& \le \sum _{j=l}^{m-1}{∥\frac{1}{2^j}f\left(2^{j}x\right)-\frac{1}{2^{j+1}}f\left(2^{j+1}x\right)∥}_Y\\ \le \frac{2+2^r}{2}\sum _{j=l}^{m-1}\frac{2^{rj}}{2^j}\theta {∥x∥}_X^r\end{array}$$

(2.6)

for all nonnegative integers m and l with m > l and all x ∈ X.

It follows from (2.6) that the sequence $\left\{\frac{1}{2^n}f\left(2^{n}x\right)\right\}$ is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence $\left\{\frac{1}{2^n}f\left(2^{n}x\right)\right\}$ converges. So one can define the mapping h : X → Y by

$$h\left(x\right):=\underset{n\to \infty }{\text{lim}}\frac{1}{2^n}f\left(2^{n}x\right)$$

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), we get (2.5).

The rest of the proof is similar to the proof of Theorem 2.2.

Theorem 2.4. Let $r>\frac{1}{3}$ and θ be nonnegative real numbers, and let f : X → Y be an odd mapping such that

$${∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥}_Y\le {∥2f\left(\frac{x+y+z}{2}\right)∥}_Y+\theta \cdot {∥x∥}_X^r\cdot {∥y∥}_X^r\cdot {∥z∥}_X^r$$

(2.7)

for all x, y, z ∈ X. Then there exists a unique Cauchy additive mapping h : X → Y such that

$${∥f\left(x\right)-h\left(x\right)∥}_Y\le \frac{2^r\theta }{8^r-2}{∥x∥}_X^{3r}$$

(2.8)

for all x ∈ X.

Proof. Letting y = x and z = - 2x in (2.7), we get

$${∥2f\left(x\right)-f\left(2x\right)∥}_Y={∥2f\left(x\right)+f\left(-2x\right)∥}_Y\le 2^r\theta {∥x∥}_X^{3r}$$

(2.9)

for all x ∈ X. So

$${∥f\left(x\right)-2f\left(\frac{x}{2}\right)∥}_Y\le \frac{2^r}{8^r}\theta {∥x∥}_X^{3r}$$

for all x ∈ X. Hence

$$\begin{array}{ll}\hfill {∥2^{l}f\left(\frac{x}{2^l}\right)-2^{m}f\left(\frac{x}{2^m}\right)∥}_Y& \le \sum _{j=l}^{m-1}{∥2^{j}f\left(\frac{x}{2^j}\right)-2^{j+1}f\left(\frac{x}{2^{j+1}}\right)∥}_Y\\ \le \frac{2^r}{8^r}\sum _{j=l}^{m-1}\frac{2^j}{8^{rj}}\theta {∥x∥}_X^{3r}\end{array}$$

(2.10)

for all nonnegative integers m and l with m > l and all x ∈ X.

It follows from (2.10) that the sequence $\left\{2^{n}f\left(\frac{x}{2^n}\right)\right\}$ is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence $\left\{2^{n}f\left(\frac{x}{2^n}\right)\right\}$ converges. So one can define the mapping h : X → Y by

$$h\left(x\right):=\underset{n\to \infty }{\text{lim}}{2}^{n}f\left(\frac{x}{2^n}\right)$$

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.10), we get (2.8).

The rest of the proof is similar to the proof of Theorem 2.2.

Theorem 2.5. Let $r<\frac{1}{3}$ and θ be positive real numbers, and let f : X → Y be an odd mapping satisfying (2.7). Then there exists a unique Cauchy additive mapping h:X → Y such that

$${∥f\left(x\right)-h\left(x\right)∥}_Y\le \frac{2^r\theta }{2-8^r}{∥x∥}_X^{3r}$$

(2.11)

for all x ∈ X.

Proof. It follows from (2.9) that

$${∥f\left(x\right)-\frac{1}{2}f\left(2x\right)∥}_Y\le \frac{2^r}{2}\theta {∥x∥}_X^{3r}$$

for all x ∈ X. Hence

$$\begin{array}{ll}\hfill {∥\frac{1}{2^l}f\left(2^{l}x\right)-\frac{1}{2^m}f\left(2^{m}x\right)∥}_Y& \le \sum _{j=l}^{m-1}{∥\frac{1}{2^j}f\left(2^{j}x\right)-\frac{1}{2^{j+1}}f\left(2^{j+1}x\right)∥}_Y\\ \le \frac{2^r}{2}\sum _{j=l}^{m-1}\frac{8^{rj}}{2^j}\theta {∥x∥}_X^r\end{array}$$

(2.12)

for all nonnegative integers m and l with m > l and all x ∈ X.

It follows from (2.12) that the sequence $\left\{\frac{1}{2^n}f\left(2^{n}x\right)\right\}$ is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence $\left\{\frac{1}{2^n}f\left(2^{n}x\right)\right\}$ converges. So one can define the mapping h : X → Y by

$$h\left(x\right):=\underset{n\to \infty }{\text{lim}}\frac{1}{2^n}f\left(2^{n}x\right)$$

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.12), we get (2.11).

The rest of the proof is similar to the proof of Theorem 2.2.

3 Stability of a functional inequality associated with a 3-variable Cauchy additive functional equation

Proposition 3.1. [18, Proposition 2.2] Let f : G → Y be a mapping such that

$${∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥}_Y\le {∥f\left(x+y+z\right)∥}_Y$$

for all x, y, z ∈ G. Then f is Cauchy additive.

We prove the Hyers-Ulam stability of a functional inequality associated with a Jordan-von Neumann type 3-variable Cauchy additive functional equation.

Theorem 3.2. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping such that

$${∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥}_Y\le {∥f\left(x+y+z\right)∥}_Y+\theta \left({∥x∥}_X^r+{∥y∥}_X^r+{∥z∥}_X^r\right)$$

(3.1)

for all x, y, z ∈ X. Then there exists a unique Cauchy additive mapping h : X → Y such that

$${∥f\left(x\right)-h\left(x\right)∥}_Y\le \frac{2^r+2}{2^r-2}\theta {∥x∥}_X^r$$

for all x ∈ X.

Proof. Letting y = x and z = - 2x in (3.1), we get

$${∥2f\left(x\right)-f\left(2x\right)∥}_Y={∥2f\left(x\right)+f\left(-2x\right)∥}_Y\le \left(2+2^r\right)\theta {∥x∥}_X^r$$

(3.2)

for all x ∈ X.

The rest of the proof is the same as in the proof of Theorem 2.2.

Theorem 3.3. Let r < 1 and θ be positive real numbers, and let f : X → Y be an odd mapping satisfying (3.1). Then there exists a unique Cauchy additive mapping h:X → Y such that

$${∥f\left(x\right)-h\left(x\right)∥}_Y\le \frac{2+2^r}{2-2^r}\theta {∥x∥}_X^r$$

for all x ∈ X.

Proof. It follows from (3.2) that

$${∥f\left(x\right)-\frac{1}{2}f\left(2x\right)∥}_Y\le \frac{2+2^r}{2}\theta {∥x∥}_X^r$$

for all x ∈ X.

The rest of the proof is the same as in the proofs of Theorems 2.2 and 2.3.

Theorem 3.4. Let $r>\frac{1}{3}$ and θ be nonnegative real numbers, and let f : X → Y be an odd mapping such that

$${∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥}_Y\le {∥f\left(x+y+z\right)∥}_Y+\theta \cdot {∥x∥}_X^r\cdot {∥y∥}_X^r\cdot {∥z∥}_X^r$$

(3.3)

for all x, y, z ∈ X. Then there exists a unique Cauchy additive mapping h : X → Y such that

$${∥f\left(x\right)-h\left(x\right)∥}_Y\le \frac{2^r\theta }{8^r-2}{∥x∥}_X^{3r}$$

for all x ∈ X.

Proof Letting y = x and z = - 2x in (3.3), we get

$${∥2f\left(x\right)-f\left(2x\right)∥}_Y={∥2f\left(x\right)+f\left(-2x\right)∥}_Y\le 2^r\theta {∥x∥}_X^{3r}$$

(3.4)

for all x ∈ X.

The rest of the proof is the same as in the proofs of Theorems 2.2 and 2.4.

Theorem 3.5. Let $r<\frac{1}{3}$ and θ be positive real numbers, and let f : X → Y be an odd mapping satisfying (3.3). Then there exists a unique Cauchy additive mapping h : X → Y such that

$${∥f\left(x\right)-h\left(x\right)∥}_Y\le \frac{2^r\theta }{2-8^r}{∥x∥}_X^{3r}$$

for all x ∈ X.

Proof. It follows from (3.4) that

$${∥f\left(x\right)-\frac{1}{2}f\left(2x\right)∥}_Y\le \frac{2^r}{2}\theta {∥x∥}_X^{3r}$$

for all x ∈ X.

The rest of the proof is the same as in the proofs of Theorems 2.2 and 2.5.

4 Stability of a functional inequality associated with the Cauchy-Jensen functional equation

Proposition 4.1. [18, Proposition 2.3] Let f :G → Y be a mapping such that

$${∥f\left(x\right)+f\left(y\right)+2f\left(z\right)∥}_Y\le {∥2f\left(\frac{x+y}{2}+z\right)∥}_Y$$

for all x, y, z ∈ G. Then f is Cauchy additive.

We prove the Hyers-Ulam stability of a functional inequality associated with a Jordan-von Neumann type Cauchy-Jensen functional equation.

Theorem 4.2. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping such that

$${∥f\left(x\right)+f\left(y\right)+2f\left(z\right)∥}_Y\le {∥2f\left(\frac{x+y}{2}+z\right)∥}_Y+\theta \left({∥x∥}_X^r+{∥y∥}_X^r+{∥z∥}_X^r\right)$$

(4.1)

for all x, y, z ∈ X. Then there exists a unique Cauchy additive mapping h : X →Y such that

$${∥f\left(x\right)-h\left(x\right)∥}_Y\le \frac{2^r+1}{2^r-2}\theta {∥x∥}_X^r$$

for all x ∈ X.

Proof. Replacing x by 2x and letting y = 0 and z = -x in (4.1), we get

$${∥f\left(2x\right)-2f\left(x\right)∥}_Y={∥f\left(2x\right)+2f\left(-x\right)∥}_Y\le \left(1+2^r\right)\theta {∥x∥}_X^r$$

(4.2)

for all x ∈ X. So

$${∥f\left(x\right)-2f\left(\frac{x}{2}\right)∥}_Y\le \frac{1+2^r}{2^r}\theta {∥x∥}_X^r$$

for all x ∈ X.

The rest of the proof is similar to the proof of Theorem 2.2.

Theorem 4.3. Let r < 1 and θ be positive real numbers, and let f : X → Y be an odd mapping satisfying (4.1). Then there exists a unique Cauchy additive mapping h : X → Y such that

$${∥f\left(x\right)-h\left(x\right)∥}_Y\le \frac{1+2^r}{2-2^r}\theta {∥x∥}_X^r$$

for all x ∈ X.

Proof. It follows from (4.2) that

$${∥f\left(x\right)-\frac{1}{2}f\left(2x\right)∥}_Y\le \frac{1+2^r}{2}\theta {∥x∥}_X^r$$

for all x ∈ X.

The rest of the proof is similar to the proofs of Theorems 2.2 and 2.3.