Approximate Cauchy functional inequality in quasi-Banach spaces

Abstract

In this article, we prove the generalized Hyers-Ulam stability of the following Cauchy functional inequality:

$$\left|\right|f\left(x\right)+f\left(y\right)+nf\left(z\right)\left|\right|\le nf∥\left(\frac{x+y}{n}+x\right)∥$$

in the class of mappings from n-divisible abelian groups to p-Banach spaces for any fixed positive integer n ≥ 2.

1 Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms.

We are given a group G₁ and a metric group G₂ with metric ρ (·,·). Given ϵ > 0, does there exist a δ > 0 such that if f : G₁ → G₂ satisfies ρ(f(xy),f(x)f(y)) < δ for all x,y ∈ G₁, then a homomorphism h : G₁→G₂ exists with ρ(f(x), h(x)) < ϵ for all x ϵ G₁?

In other words, we are looking for situations when the homomorphisms are stable, i.e., if a mapping is almost a homomorphism, then there exists a true homomorphism near it.

In 1941, Hyers [2] considered the case of approximately additive mappings between Banach spaces and proved the following result. Suppose that E₁ and E₂ are Banach spaces and f : E₁ → E₂ satisfies the following condition: there is a constant ϵ ≥ 0 such that

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

for all x,y ∈ E₁. Then, the limit $h\left(x\right)=\underset{n\to \infty }{lim}\frac{f\left(2^{n}x\right)}{2^n}$ exists for all x ∈ E₁, and it is a unique additive mapping h:E₁→E₂ such that ||f(x) - h(x)|| ≤ ϵ.

The method which was provided by Hyers, and which produces the additive mapping h, was called a direct method. This method is the most important and most powerful tool for studying the stability of various functional equations. Hyers' theorem was generalized by Aoki [3] and Bourgin [4] for additive mappings by considering an unbounded Cauchy difference. In 1978, Rassias [5] also provided a generalization of Hyers' theorem for linear mappings which allows the Cauchy difference to be unbounded like this ||x||^p+ ||y||^p. Let E₁ and E₂ be two Banach spaces and f : E₁ → E₂ be a mapping such that f(tx) is continuous in t ∈ R for each fixed x. Assume that there exists ϵ > 0 and 0 ≤ p < 1 such that

$$\left|\right|f\left(x+y\right)-f\left(x\right)-f\left(y\right)\left|\right|\le \epsilon \left(\left|\right|x|{|}^p+|\left|y\right|{|}^p\right),\forall x,y\in E_1.$$

Then, there exists a unique R-linear mapping T : E₁ → E₂ such that

$$\left|\right|f\left(x\right)-T\left(x\right)\left|\right|\le \frac{2}{2-2^p}\left|\right|x|{|}^p$$

for all x ∈ E₁. A generalized result of Rassias' theorem was obtained by Găvruta in [6] and Jung in [7]. In 1990, Rassias [8] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. In 1991, Gajda [9], following the same approach as in [5], gave an affirmative solution to this question for p > 1. It was shown by Gajda [9], as well as by Rassias and [001]emrl [10], that one cannot prove a Rassias' type theorem when p = 1. The counterexamples of Gajda [9], as well as of Rassias and [001]emrl [10], have stimulated several mathematicians to invent new approximately additive or approximately linear mappings. In particular, Rassias [11, 12] proved a similar stability theorem in which he replaced the unbounded Cauchy difference by this factor ||x||^p||y||^qfor p,q ∈ R with p + q ≠ 1.

Let G be an n-divisible abelian group n ∈ N (i.e., a ↦ na : G → G is a surjection ) and X be a normed space with norm || · ||. Now, for a mapping f : G → X, we consider the following generalized Cauchy-Jensen equation

$$f\left(x\right)+f\left(y\right)+nf\left(z\right)=nf\left(\frac{x+y}{n}+z\right),n\ge 2$$

for all x,y, z ∈ G, which has been introduced in [13].

Proposition 1.1. For a mapping f : G → X, the following statements are equivalent.

• (a) f is additive,

• (b) $f\left(x\right)+f\left(y\right)+nf\left(z\right)=nf\left(\frac{x+y}{n}+z\right)$,

• (c) $\left|\right|f\left(x\right)+f\left(y\right)+nf\left(z\right)\left|\right|\le ∥nf\left(\frac{x+y}{n}+z\right)∥$

for all x, y, z ∈ G.

As a special case for n = 2, the generalized Hyers-Ulam stability of functional equation (b) and functional inequality (c) has been presented in [13]. We remark that there are some interesting papers concerning the stability of functional inequalities and the stability of functional equations in quasi-Banach spaces [14–18]. In this article, we are going to improve the theorems given in [13] without using the oddness of approximate additive functions concerning the functional inequality (c) for a more general case.

2 Generalized Hyers-Ulam stability of (c)

We recall some basic facts concerning quasi-Banach spaces and some preliminary results. Let X be a real linear space. A quasi-norm is a real-valued function on X satisfying the following:

• (1) ||x|| ≥ 0 for all x ∈ X and ||x|| = 0 if and only if x = 0.

• (2) ||λx|| = |λ|||x|| for all λ ∈ R and all x ∈ X.

• (3) There is a constant M ≥ 1 such that ||x + y|| ≤ M(||x|| + ||y||) for all x,y ∈ X.

The pair (X, || · ||) is called a quasi-normed space if || · || is a quasi-norm on X [19, 20]. The smallest possible M is called the modulus of concavity of || · ||. A quasi-Banach space is a complete quasi-normed space.

A quasi-norm || · || is called a p- norm (0 < p ≤ 1) if

$$\left|\right|x+y|{|}^p\le |\left|x\right|{|}^p+\left|\right|y|{|}^p$$

for all x,y ∈ X. In this case, a quasi-Banach space is called a p-Banach space.

Given a p-norm, the formula d(x,y) := ||x - y||^pgives us a translation invariant metric on X. By the Aoki-Rolewicz theorem [20], each quasi-norm is equivalent to some p-norm (see also [19]). Since it is much easier to work with p-norms, henceforth, we restrict our attention mainly to p-norms. We observe that if x₁, x₂,..., x _n are non-negative real numbers, then

$${\left(\sum _{i=1}^n{x}_i\right)}^p\le \sum _{i=1}^n{x_i}^p,$$

where 0 < p ≤ 1 [21].

From now on, let G be an n-divisible abelian group for some positive integer n ≥ 2, and let Y be a p-Banach space with the modulus of concavity M.

Theorem 2.1. Suppose that a mapping f : G → Y with f(0) = 0 satisfies the functional inequality

$$\left|\right|f\left(x\right)+f\left(y\right)+nf\left(z\right)\left|\right|\le ∥nf\left(\frac{x+y}{n}+z\right)∥+\phi \left(x,y,z\right)$$

(1)

for all x, y, z ∈ G, and the perturbing function φ : G³ →R⁺ satisfies

$$\Phi \left(x,y,z\right):=\sum _{i=0}^{\infty }\frac{\phi {\left(n^{i}x,n^{i}y,n^{i}z\right)}^p}{n^{ip}}<\infty$$

for all x,y,z ∈ G. Then, there exists a unique additive mapping h : G → Y, defined as $h\left(x\right)=\underset{k\to \infty }{lim}\frac{f\left(n^{k}x\right)-f\left(-n^{k}x\right)}{2n^k}$, such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M^2}{2n}{\left[\Phi \left(nx,0,-x\right)+\Phi \left(-nx,0,x\right)\right]}^{\frac{1}{p}}+\frac{M}{2}\phi \left(x,-x,0\right)$$

(2)

for all x ∈ G.

Proof. Let y = -x, z = 0 in (1) and dividing both sides by 2, we have

$$∥\frac{f\left(x\right)+f\left(-x\right)}{2}∥\le \frac{\phi \left(x,-x,0\right)}{2}$$

(3)

for all x ∈ G. Replacing x by nx and letting y = 0 and z = -x in (1), we get

$$\left|\right|f\left(nx\right)+nf\left(-x\right)\left|\right|\le \phi \left(nx,0,-x\right)$$

(4)

for all x ∈ G. Replacing x by -x in (4), one has

$$\left|\right|f\left(-nx\right)+nf\left(x\right)\left|\right|\le \phi \left(-nx,0,x\right)$$

(5)

for all x ∈ G. Put $g\left(x\right)=\frac{f\left(x\right)-f\left(-x\right)}{2}$. Combining (4) and (5) yields

$$\left|\right|ng\left(x\right)-g\left(nx\right)\left|\right|\le \frac{M}{2}\left(\phi \left(nx,0,-x\right)+\phi \left(-nx,0,x\right)\right)$$

that is,

$$∥g\left(x\right)-\frac{1}{n}g\left(nx\right)∥\le \frac{M}{2n}\left(\phi \left(nx,0,-x\right)+\phi \left(-nx,0,x\right)\right)$$

(6)

for all x ∈ G. It follows from (6) that

$$\begin{array}{l}{\Vert \frac{g(n^{l}x}{n^l}-\frac{g(n^{m}x)}{n^m}\Vert }^p\\ \le {{\displaystyle \sum _{k=l}^{m-1}\Vert \frac{1}{n^k}g(n^{k}x)-\frac{1}{n^{k+1}}g(n^{k+1}x)\Vert }}^p\\ ={\displaystyle \sum _{k=1}^{m-1}\frac{1}{n^{kp}}}{\Vert g(n^{k}x)-\frac{1}{n}g(n^{k+1}x)\Vert }^p\\ \le {\displaystyle \sum _{k=1}^{m-1}\frac{M^p}{2^p{n}^{{(k+1)}_p}}}[\phi (n^{k+1}x,0,-n^{k}x{)}^p+\phi (-n^{k+1}x,0,n^{k}x{)}^p]\end{array}$$

(7)

for all nonnegative integers m and l with m > l ≥ 0 and x ∈ G. Since the right-hand side of (7) tends to zero as l → ∞, we obtain the sequence $\left\{\frac{g(n^{m}x}{n^m}\right\}$ is Cauchy for all x ∈ G. Because of the fact that Y is complete, it follows that the sequence $\left\{\frac{g(n^{m}x}{n^m}\right\}$ converges in Y. Therefore, we can define a function h : G → Y by

$$h\left(x\right)=\underset{m\to \infty }{lim}\frac{g\left(n^{m}x\right)}{n^m}=\underset{m\to \infty }{lim}\frac{f\left(n^{m}x\right)-f\left(-n^{m}x\right)}{2n^m},x\in G.$$

Moreover, letting l = 0 and taking m → ∞ in (7), we get

$$∥\frac{f\left(x\right)-f\left(-x\right)}{2}-h\left(x\right)∥\le \left|\right|g\left(x\right)-h\left(x\right)\left|\right|\le \frac{M}{2n}{\left[\Phi \left(nx,0-x\right)+\Phi \left(-nx,0,x\right)\right]}^{\frac{1}{p}}$$

(8)

for all x ∈ G. It follows from (3) and (8) that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M^2}{2n}{\left[\Phi \left(nx,0,-x\right)+\Phi \left(-nx,0,x\right)\right]}^{\frac{1}{p}}+\frac{M}{2}\phi \left(x,-x,0\right)$$

for all x ∈ G.

It follows from (1) and (4) that

$$\begin{array}{l}\left|\right|h(x)+h(y)-h(x+y)|{|}^p=||h(x)+h(y)+h(-x-y)|{|}^p\\ =\underset{k\to \infty }{\lim }\frac{1}{n^{kp}}\left|\right|g(n^{k}x)+g(n^{k}y)+g(-n^k(x+y))|{|}^p\\ \le \underset{k\to \infty }{\lim }\frac{1}{2^p{n}^{kp}}(||f(n^{k}x)+f(n^{k}y)+nf(-n^{k-1}(x+y))|{|}^p\\ +\left|\right|-f(-n^{k}x)-f(-n^{k}y)-nf(n^{k-1}(x+y))|{|}^p\\ +\left|\right|nf(n^{k-1}(x+y))+f(-n^k(x+y))|{|}^p\\ +\left|\right|-nf(-n^{k-1}(x+y))+f(n^k(x+y))|{|}^p\\ \le \underset{k\to \infty }{\lim }\frac{1}{2^p{n}^{kp}}(\phi (n^{k}x,n^{k}y,-n^{k-1}(x+y){)}^p+\phi (-n^{k}x,-n^{k}y,n^{k-1}(x+y){)}^p\\ +\phi (-n^k(x+y),0,n^{k-1}(x+y){)}^p+\phi (n^k(x+y),0,-n^{k-1}(x+y){)}^p)\\ =0\end{array}$$

for all x,y ∈G. This implies that the mapping h is additive.

Next, let h' : G → Y be another additive mapping satisfying

$$\left|\right|f\left(x\right)-h^{\prime }\left(x\right)\left|\right|\le \frac{M^2}{2n}{\left[\Phi \left(nx,0,-x\right)+\Phi \left(-nx,0,x\right)\right]}^{\frac{1}{p}}+\frac{M}{2}\phi \left(x,-x,0\right)$$

for all x ∈ G. Then, we have

$$\begin{array}{ll}\hfill \left|\right|h\left(x\right)-h^{\prime }\left(x\right)|{|}^p& ={∥\frac{1}{n^k}h\left(n^{k}x\right)-\frac{1}{n^k}{h}^{\prime }\left(n^{k}x\right)∥}^p\\ \le \frac{1}{n^{kp}}\left(\left|\right|h\left(n^{k}x\right)-f\left(n^{k}x\right)|{|}^p+||f\left(n^{k}x\right)-h^{\prime }\left(n^{k}x\right)|{|}^p\right)\\ \le \frac{2M^{2p}}{2^p{n}^{{\left(k+1\right)}_p}}\left[\Phi \left(n^{k+1}x,0,-n^{k}x\right)+\Phi \left(-n^{k+1}x,0,n^{k}x\right)\right]+\frac{2M^p}{2^p{n}^{kp}}\phi {\left(n^{k}x,-n^{k}x,0\right)}^p\\ =\sum _{i=k}^{\infty }\frac{2M^{2p}}{2^p{n}^{\left(i+1\right)p}}\left[\phi {\left(n^{i+1}x,0,-n^{i}x\right)}^p+\phi {\left(-n^{i+1}x,0,n^{i}x\right)}^p\right]+\frac{2M^p\phi {\left(n^{k}x,-n^{k}x,0\right)}^p}{2^p{n}^{kp}}\end{array}$$

for all k ∈ N and all x ∈ G. Taking the limit as k → ∞, we conclude that

$$h\left(x\right)=h^{\prime }\left(x\right)$$

for all x ∈ G. This completes the proof.

Suppose that X is a normed space in the following corollaries. If we put φ(x,y,z) := θ(||x||^q||y||^r||z||^s) and φ (x,y,z) := θ(||x||^q+ ||y||^r+ ||z||^s) in Theorem 2.1, respectively, then we get the following Corollaries 2.2 and 2.3.

Corollary 2.2. Let q + r + s < 1, q, r, s > 0, θ > 0. If a mapping f : X → Y with f(0) = 0 satisfies the following functional inequality:

$$\left|\right|f(x)+f(y)+nf(z)\left|\right|\le \Vert nf\left(\frac{x+y}{n}+x\right)\Vert +\theta (|\left|x\right|{|}^q\left|\right|y\left|{|}^r\right|\left|z\right|{|}^s$$

for all x, y, z ∈ X, then f is additive.

Corollary 2.3. Let 0 < q,r,s <1, θ₁,θ₂ > 0. If a mapping f : X → Y with f(0) = 0 satisfies the following functional inequality:

$$\left|\right|f\left(x\right)+f\left(y\right)+nf\left(z\right)\left|\right|\le ∥nf\left(\frac{x+y}{n}+z\right)∥+{\theta }_1\left(\left|\right|x|{|}^q+|\left|y\right|{|}^r+\left|\right|z|{|}^s\right)+{\theta }_2$$

for all x,y,z ∈ X, then there exists a unique additive mapping h : X ∈ Y, defined as $h\left(x\right)=\underset{k\to \infty }{lim}\frac{f\left(n^{k}x\right)-f\left(-n^{k}x\right)}{2n^k}$, such that

$$\begin{array}{c}\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M^2\sqrt[p]{2}}{2}{\left(\frac{n^{pq}{\theta }_1^p\left|\right|x|{|}^{pq}}{n^p-n^{pq}}+\frac{{\theta }_1^p\left|\right|x|{|}^{ps}}{n^p-n^{ps}}+\frac{{\theta }_2^p}{n^p-1}\right)}^{\frac{1}{p}}\\ +\frac{M}{2}\left({\theta }_1\left|\right|x|{|}^q+{\theta }_1|\left|x\right|{|}^r+{\theta }_2\right)\end{array}$$

for all x ∈ X.

Noting the inequality

$$\left|\right|f\left(nx\right)-nf\left(x\right)\left|\right|\le M\left[\phi \left(nx,0,-x\right)+n\phi \left(x,-x,0\right)\right]$$

according to the inequalities (3) and (4), then we can similarly prove another stability theorem under the same condition as in Theorem 2.1:

Remark 2.4. Let φ : G³ → R+ and f : G → Y satisfy the assumptions of Theorem 2.1. Then, there exists a unique additive mapping h : G → Y, defined by $h\left(x\right)=\underset{k\to \infty }{lim}\frac{f\left(n^{k}x\right)}{n^k}$, such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M}{n}{\left[\Phi \left(nx,0,-x\right)+n^p\Phi \left(x,-x,0\right)\right]}^{\frac{1}{p}}$$

for all x ∈ G using the similar argument to Theorem 2.1.

In particular, if a mapping f : X → Y with f(0) = 0 satisfies the following functional inequality:

$$\left|\right|f\left(x\right)+f\left(y\right)+nf\left(z\right)\left|\right|\le ∥nf\left(\frac{x+y}{n}+z\right)∥+{\theta }_1\left(\left|\right|x|{|}^q+|\left|y\right|{|}^r+\left|\right|z|{|}^s\right)+{\theta }_2$$

for all x,y,z in a normed space X, where 0 < q,r,s < 1, θ₁,θ₂ > 0, then there exists a unique additive mapping h : X → Y such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le M{\left(\frac{\left(n^{pq}+n^p\right){\theta }_1^p\left|\right|x|{|}^{pq}}{n^p-n^{pq}}+\frac{n^p{\theta }_1^p\left|\right|x|{|}^{pr}}{n^p-n^{pr}}+\frac{{\theta }_1^p\left|\right|x|{|}^{ps}}{n^p-n^{ps}}+\frac{\left(1+n^p\right){\theta }_2^2}{n^p-1}\right)}^{\frac{1}{p}}$$

for all x ∈ X.

We may obtain more simple and sharp approximation than that of Theorem 2.1 for the stability result under the oddness condition.

Remark 2.5. Let φ : G³ → R⁺ and f : G → Y satisfy the assumptions of Theorem 2.1. Moreover, if the mapping f is odd, then there exists a unique additive mapping h : G → Y, defined by $h\left(x\right)=\underset{k\to \infty }{lim}\frac{f\left(n^{k}x\right)}{n^k}$, such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{1}{n}\Phi {\left(nx,0,-x\right)}^{\frac{1}{p}}$$

for all x ∈ G.

Now, we consider another stability result of functional inequality (c) in the followings.

Theorem 2.6. Suppose that a mapping f : G → Y satisfies

$$\left|\right|f\left(x\right)+f\left(y\right)+nf\left(z\right)\left|\right|\le ∥nf\left(\frac{x+y}{n}+z\right)∥+\phi \left(x,y,z\right)$$

(9)

and the perturbing function φ : G³ → R⁺ is such that

$$\Psi \left(x,y,z\right):=\sum _{i=1}^{\infty }{n}^{ip}\phi {\left(\frac{x}{n^i},\frac{y}{n^i},\frac{z}{n^i}\right)}^p<\infty$$

for all x,y,z ∈ G. Then, there exists a unique additive mapping h : G → Y, defined $h\left(x\right)\underset{k\to \infty }{lim}\frac{n^k}{2}\left(f\left(\frac{x}{n^k}\right)-f\left(-\frac{x}{n^k}\right)\right)$, such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M^2}{2n}{\left[\Psi \left(nx,0,-x\right)+\Psi \left(-nx,0,x\right)\right]}^{\frac{1}{p}}+\frac{M}{2}\phi \left(x,-x,0\right)$$

(10)

for all x ∈ G.

Proof. We observe that f(0) = 0 because of φ(0,0,0) = 0 by the convergence of Ψ(0,0,0) < ∞. Now, combining (4) and (5) yields the functional inequality

$$\left|\right|g\left(x\right)-ng\left(\frac{x}{n}\right)\left|\right|\le \frac{M}{2}\left(\phi \left(x,0,-\frac{x}{n}\right)+\phi \left(-x,0,\frac{x}{n}\right)\right),$$

where $g\left(x\right)=\frac{f\left(x\right)-f\left(-x\right)}{2}$, x ∈ G. It follows from the last inequality that

$${∥g\left(x\right)-n^{m}g\left(\frac{x}{n^m}\right)∥}^p\le \frac{M^p}{2^p}\sum _{i=0}^{m-1}{n}^{ip}\left[\phi {\left(\frac{x}{n^i},0,-\frac{x}{n^{i+1}}\right)}^p+\phi {\left(-\frac{x}{n^i},0,\frac{x}{n^{i+1}}\right)}^p\right]$$

(11)

for all x ∈ G.

The remaining proof is similar to the corresponding proof of Theorem 2.1. This completes the proof.

Suppose that X is a normed space in the following corollaries. If we put φ(x,y,z) := θ(||x||^q||y||^r||z||^s) and φ(x,y,z) := θ(||x||^q+ ||y||^r+ ||z||^s) in Theorem 2.6, respectively, then we get the following Corollaries 2.7 and 2.8.

Corollary 2.7. Let q + r + s > 1, q,r, s > 0, θ > 0. If a mapping f : X → Y satisfies the following functional inequality:

$$\left|\right|f(x)+f(y)+nf(z)\left|\right|\le \Vert nf\left(\frac{x+y}{n}+x\right)\Vert +\theta (|\left|x\right|{|}^q\left|\right|y\left|{|}^r\right|\left|z\right|{|}^s$$

for all x, y, z ∈ X, then f is additive.

Corollary 2.8. Let q,r,s > 1, θ₁ > 0. If a mapping f : X → Y satisfies the following functional inequality:

$$\left|\right|f\left(x\right)+f\left(y\right)+nf\left(z\right)\left|\right|\le ∥nf\left(\frac{x+y}{n}+z\right)∥+{\theta }_1\left(\left|\right|x|{|}^q+|\left|y\right|{|}^r+\left|\right|z|{|}^s\right)$$

for all x,y,z ∈ X, then there exists a unique additive mapping h : X → Y, defined as $h\left(x\right)\underset{k\to \infty }{lim}\frac{n^k}{2}\left(f\left(\frac{x}{n^k}\right)-f\left(-\frac{x}{n^k}\right)\right)$, such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M^2\sqrt[p]{2{\theta }_1}}{2}{\left(\frac{n^{pq}\left|\right|x|{|}^{pq}}{n^{pq}-n^p}+\frac{\left|\right|x|{|}^{ps}}{n^{ps}-n^p}\right)}^{\frac{1}{p}}+\frac{M{\theta }_1}{2}\left(\left|\right|x|{|}^q+|\left|x\right|{|}^r\right)$$

for all x ∈ X.

We can similarly prove another stability theorem under somewhat different conditions as follows:

Remark 2.9. Let φ : G³ → R⁺ and f : G → Y satisfy the assumptions of Theorem 2.6. Then, there exists a unique additive mapping h : G → Y, defined by h(x) = $h\left(x\right)=\underset{k\to \infty }{lim}{n}^{k}f\left(\frac{x}{n^k}\right)$, such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M}{n}{\left[\Psi \left(nx,0,-x\right)+n^p\Psi \left(x,-x,0\right)\right]}^{\frac{1}{p}}$$

for all x ∈ G.

In particular, if a mapping f : X → Y satisfies the following functional inequality:

$$\left|\right|f\left(x\right)+f\left(y\right)+nf\left(z\right)\left|\right|\le ∥nf\left(\frac{x+y}{n}+z\right)∥+{\theta }_1\left(\left|\right|x|{|}^q+|\left|y\right|{|}^r+\left|\right|z|{|}^s\right)$$

for all x,y, z in a normed space X, where q,r,s > 1, θ ₁ > 0, then there exists a unique additive mapping h : X → Y such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le M{\theta }_1{\left(\frac{\left(n^{pq}+n^p\right)\left|\right|x|{|}^{pq}}{n^{pq}-n^p}+\frac{\left|\right|x|{|}^{ps}}{n^{ps}-n^p}+\frac{n^p\left|\right|x\left|\right|pr}{n^{pr}-n^p}\right)}^{\frac{1}{p}}$$

for all x ∈ X.

We may obtain more simple and sharp approximation than that of Theorem 2.6 for the stability result under the oddness condition.

Remark 2.10. Let φ : G³ → R⁺ and f : G → Y satisfy the assumptions of Theorem 2.6. If the mapping f is odd, then there exists a unique additive mapping h : G → Y, defined by $h\left(x\right)=\underset{k\to \infty }{lim}{n}^{k}f\left(\frac{x}{n^k}\right)$, such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{1}{n}\Psi {\left(nx,0,-x\right)}^{\frac{1}{p}}$$

for all x ∈ G.

3 Alternative generalized Hyers-Ulam stability of (c)

From now on, we investigate the generalized Hyers-Ulam stability of the functional inequality (c).

Theorem 3.1. Suppose that a mapping f : G → Y with f(0) = 0 satisfies the functional inequality

$$\left|\right|f\left(x\right)+f\left(y\right)+nf\left(z\right)\left|\right|\le ∥nf\left(\frac{x+y}{n}+z\right)∥+\phi \left(x,y,z\right)$$

for all x,y,z ∈ G and there exists a constant L with 0 < L < 1 for which the perturbing function φ : G³→ R⁺ satisfies

$$\phi \left(nx,ny,nz\right)\le nL\phi \left(x,y,z\right)$$

(12)

for all x,y,z ∈ G. Then, there exists a unique additive mapping h : G → Y, defined as $h\left(x\right)=\underset{k\to \infty }{lim}\frac{f\left(n^{k}x\right)-f\left(-n^{k}x\right)}{2n^k}$, such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M^2}{2n\sqrt[p]{1-L^p}}\left[\phi \left(nx,0,-x\right)+\phi \left(-nx,0,x\right)\right]+\frac{M}{2}\phi \left(x,-x,0\right)$$

for all x ∈ G.

Proof. It follows from (7) and (12) that

$$\begin{array}{c}{∥\frac{g\left(n^{1}x\right)}{n^1}-\frac{g\left(n^{m}x\right)}{n^m}∥}^p\\ \le \sum _{k=1}^{m-1}\frac{M^p}{2^p{n}^{\left(k+1\right)p}}{\left[\phi \left(n^{k+1}x,0,-n^{k}x\right)+\phi \left(-n^{k+1}x,0,n^{k}x\right)\right]}^p\\ \le \sum _{k=1}^{m-1}\frac{M^p{L}^{kp}}{2^p{n}^p}{\left[\phi \left(nx,0,-x\right)+\phi \left(-nx,0,x\right)\right]}^p\end{array}$$

for all nonnegative integers m and l with m > l ≥ 0 and x ∈ G,where $g\left(x\right)=\frac{f\left(x\right)-f\left(-x\right)}{2}$. Since the sequence $\left\{\frac{g(n^{m}x}{n^m}\right\}$ is Cauchy for all x ∈ G, we can define a function h : G → Y by

$$h\left(x\right)=\underset{m\to \infty }{lim}\frac{g\left(n^{m}x\right)}{n^m}=\underset{m\to \infty }{lim}\frac{f\left(n^{m}x\right)-f\left(-n^{m}x\right)}{2n^m},x\in G.$$

Moreover, letting l = 0 and m → ∞ in the last inequality yields

$$∥\frac{f\left(x\right)-f\left(-x\right)}{2}-h\left(x\right)∥\le \frac{M}{2n\sqrt[p]{1-L^p}}\left[\phi \left(nx,0,-x\right)+\phi \left(-nx,0,x\right)\right]$$

(13)

for all x ∈ G. It follows from (3) and (13) that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M^2}{2n\sqrt[p]{1-L^p}}\left[\phi \left(nx,0,-x\right)+\phi \left(-nx,0,x\right)\right]+\frac{M}{2}\phi \left(x,-x,0\right)$$

for all x ∈ G.

The remaining proof is similar to the corresponding proof of Theorem 2.1. This completes the proof.

Remark 3.2. Let φ : G³→ R⁺ and f : G → Y satisfy the assumptions of Theorem 3.1. Then, there exists a unique additive mapping h : G → Y, defined by $h\left(x\right)=\underset{k\to \infty }{lim}\frac{f\left(n^{k}x\right)}{n^k}$, such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M}{n\sqrt[p]{1-L^p}}\left[\phi \left(nx,0,-x\right)+n\phi \left(x,-x,0\right)\right]$$

for all x ∈ G using the similar argument to Theorem 3.1.

In particular, if a mapping f : X → Y with f(0) = 0 satisfies the following functional inequality:

$$\left|\right|f\left(x\right)+f\left(y\right)+nf\left(z\right)\left|\right|\le ∥nf\left(\frac{x+y}{n}+z\right)∥+{\theta }_1\left(\left|\right|x|{|}^r+|\left|y\right|{|}^r+\left|\right|z|{|}^r\right)+{\theta }_2$$

for all x, y, z in a normed space X, where 0 < r < 1, θ₁, θ₂ > 0, then there exists a unique additive mapping h : X → Y such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M}{\sqrt[p]{n^p-n^{pr}}}\left(\left(n^r+2n+1\right){\theta }_1\left|\right|x|{|}^r+\left(n+1\right){\theta }_2\right)$$

for all x ∈ X, by considering L := n^{r- 1}.

Theorem 3.3. Suppose that a mapping f : G →Y satisfies the functional inequality

$$\left|\right|f\left(x\right)+f\left(y\right)+nf\left(z\right)\left|\right|\le ∥nf\left(\frac{x+y}{n}+z\right)∥+\phi \left(x,y,z\right)$$

for all x,y,z ∈ G and there exists a constant L with 0 < L < 1 for which the perturbing function φ : G³ → R⁺ satisfies

$$\phi \left(\frac{x}{n},\frac{y}{n},\frac{z}{n}\right)\le \frac{L}{n}\phi \left(x,y,z\right)$$

(14)

for all x,y,z ∈ G. Then, there exists a unique additive mapping h : G → Y, defined as $h\left(x\right)\underset{k\to \infty }{lim}\frac{n^k}{2}\left(f\left(\frac{x}{n^k}\right)-f\left(-\frac{x}{n^k}\right)\right)$, such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M^{2}L}{2n\sqrt[p]{1-L^p}}\left[\phi \left(nx,0,-x\right)+\phi \left(-nx,0,x\right)\right]+\frac{M}{2}\phi \left(x,-x,0\right)$$

for all x ∈ G.

Proof. We observe that f(0) = 0 because φ(0,0,0) = 0, which follows from the condition $\phi \left(0,0,0\right)\le \frac{L}{n}\phi \left(0,0,0\right)$. It follows from the inequality (11) and (14) that

$$\begin{array}{ll}\hfill {∥g\left(x\right)-n^{m}g\left(\frac{x}{n^m}\right)∥}^p& \le \frac{M^p}{2^p}\sum _{i=0}^{m-1}{n}^{ip}{\left[\phi \left(\frac{x}{n^i},0,-\frac{x}{n^{i+1}}\right)+\phi \left(-\frac{x}{n^i},0,\frac{x}{n^{i+1}}\right)\right]}^p\\ \le \frac{M^p}{2^p{n}^p}\sum _{i=0}^{m-1}{L}^{\left(i+1\right)p}{\left[\phi \left(nx,0,-x\right)+\phi \left(-nx,0,x\right)\right]}^p\end{array}$$

for all x ∈ G, where $g\left(x\right)=\frac{f\left(x\right)-f\left(-x\right)}{2}$, x ∈ G.

The remaining proof is similar to the corresponding proof of Theorem 2.1. This completes the proof.

Remark 3.4. Let φ : G³ → R⁺ and f : G → Y satisfy the assumptions of Theorem 3.3. Then, there exists a unique additive mapping h : G → Y, defined by $h\left(x\right)=\underset{k\to \infty }{lim}{n}^{k}f\left(\frac{x}{n^k}\right)$, such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{ML}{n\sqrt[p]{1-L^p}}\left[\phi \left(nx,0,-x\right)+n\phi \left(x,-x,0\right)\right]$$

for all x ∈ G using the similar argument to Theorem 3.3.

In particular, if a mapping f : X → Y satisfies the following functional inequality:

$$\left|\right|f\left(x\right)+f\left(y\right)+nf\left(z\right)\left|\right|\le ∥nf\left(\frac{x+y}{n}+z\right)∥+{\theta }_1\left(\left|\right|x|{|}^r+|\left|y\right|{|}^r+\left|\right|z|{|}^r\right)$$

for all x, y, z in a normed space X, where r > 1, θ ₁ > 0, then there exists a unique additive mapping h : X→Y such that

$$\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \frac{M}{\sqrt[p]{n^{pr}-n^p}}\left(n^r+2n+1\right){\theta }_1\left|\right|x|{|}^r$$

for all x ∈ X, by considering L : = n^1-r.