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Hyers–Ulam stability and hyperstability of a Jensen-type functional equation on 2-Banach spaces

Abstract

The main aim of this paper is to establish the Hyers–Ulam stability and hyperstability of a Jensen-type quadratic mapping in 2-Banach spaces. That is, we prove the various types of Hyers–Ulam stability and hyperstability of the Jensen-type quadratic functional equation of the form

$$ g \biggl( \frac{x+y}{2} + z \biggr) + g \biggl( \frac{x+y}{2} - z \biggr) + g \biggl( \frac{x-y}{2} + z \biggr) + g \biggl( \frac{x-y}{2} - z \biggr) = g(x) + g(y) + 4 g(z), $$

in 2-Banach spaces by using the Hyers direct method.

Introduction and preliminaries

The concept of stability in a functional equation is introduced by substituting the corresponding functional inequality, which acts as a perturbation of the functional equation. In 1940, S.M. Ulam [23] was the first to propose the stability of homomorphisms between groups. The first affirmative solution to this question was given by D.H. Hyers [10] in 1941. Hyers’ theorem for additive and linear mappings was generalized independently by T. Aoki [1] and Th.M. Rassias [19]. In 1994, Găvruta [9] provided a more general theorem than the previous results.

It is well known that the parallelogram low

$$ \Vert a+b \Vert ^{2} + \Vert a-b \Vert ^{2} = 2 \Vert a \Vert ^{2} + 2 \Vert b \Vert ^{2}$$

is valid for the square norm on an inner product space. The functional equation \(f(x+y) + f(x-y) = 2 f(x) + 2 f(y)\), which is derived from the parallelogram low, is said to be the quadratic functional equation. A function g between two groups (or linear spaces) is called a quadratic function if g is a solution of the quadratic functional equation.

Skof [22] proved that the quadratic functional equation has the Hyers–Ulam stability for functions \(f: X \to Y\), where X is a normed space and Y is a Banach space. It should be noted that the stability theorem of Skof is still valid if an Abelian group replaces the normed space X (see [3]). A generalized Hyers–Ulam stability of the quadratic functional equation was obtained later by Czerwik [4] in Banach spaces. Some quadratic type functional equations have been introduced by several authors, and they obtained many interesting results concerning the Hyers–Ulam stability (see for example [2, 6, 12, 1416, 20, 21] and the references therein).

Park et al. [17] introduced the following Jensen-type quadratic functional equation:

$$ \begin{aligned} &g \biggl(\frac{x+y}{2}+z \biggr)+g \biggl(\frac{x+y}{2}-z \biggr)+g \biggl(\frac{x-y}{2}+z \biggr)+g \biggl(\frac{x-y}{2}-z \biggr) \\ &\quad =g(x)+g(y)+4g(z), \end{aligned} $$
(1.1)

where \(g:X\to Y\) is a function between linear spaces X and Y. They showed that if an even function \(g:X\to Y\) satisfies \(g(0)=0\) and (1.1) for all \(x,y,z\in X\), then g is quadratic. We prove, without any additional assumptions, that if a function g satisfies (1.1), then g is quadratic.

In this paper, we investigate the Hyers–Ulam stability and hyperstability of the Jensen-type quadratic functional equation (1.1) in 2-Banach spaces.

The concept of linear 2-normed spaces was introduced by S. Gähler [7, 8] in the middle of 1960s. In the following, we recall some basic facts about 2-normed spaces and some preliminary results.

Definition 1.1

([7])

Let A be a linear space over \(\mathbb{R}\) with \(\dim A>1\). A function \(\Vert \cdot ,\cdot \Vert : A\times A\rightarrow \mathbb{R}\) is called a 2-norm on A if it satisfies the following properties:

  1. (1)

    \(\Vert a,b \Vert =0\) if and only if a and b are linearly dependent,

  2. (2)

    \(\Vert \lambda a,b \Vert = \vert \lambda \vert \Vert a,b \Vert \),

  3. (3)

    \(\Vert a,b \Vert = \Vert b,a \Vert \),

  4. (4)

    \(\Vert a,b+c \Vert \leqslant \Vert a,b \Vert + \Vert a,c \Vert \),

for each \(a,b,c\in A\) and \(\lambda \in \mathbb{R}\). In this case, \((A, \Vert \cdot ,\cdot \Vert )\) is a called a linear 2-normed space. By \((1)\), \((3)\), and \((4)\), we infer that \(\Vert a,b \Vert \geqslant 0\) for all \(a,b\in A\).

Definition 1.2

Let A be a linear 2-normed space and \(\{a_{n}\}_{n}\) be a sequence in A.

  • The sequence \(\{a_{n}\}_{n}\) is called a Cauchy sequence if

    $$ \lim_{l,m \rightarrow \infty } \Vert a_{l}-a_{m},y \Vert =0,\quad y\in A.$$
  • The sequence \(\{a_{n}\}_{n}\) is called a convergent sequence if there is \(a\in A\) such that

    $$ \lim_{n \rightarrow \infty } \Vert a_{n}-a,b \Vert =0,\quad b \in A.$$

    In this case, a is called the limit of \(\{a_{n}\}\), and we write \(\lim_{n \rightarrow \infty }a_{n}=a\).

  • A is called a 2-Banach space if each Cauchy sequence in A is convergent.

We need the following lemma, and we will use it in our results.

Lemma 1.3

[18] Let \((A, \Vert \cdot ,\cdot \Vert )\) be a 2-normed space.

\((i)\):

If \(\Vert a,x \Vert =0\) for all \(x\in A\), then \(a=0\).

\((\mathit{ii})\):

For a convergent sequence \(\{a_{n}\}\) in A,

$$ \lim_{n\rightarrow \infty } \Vert a_{n},x \Vert = \Bigl\Vert \lim_{n\rightarrow \infty } a_{n},x \Bigr\Vert ,\quad x \in A.$$

General solution of (1.1)

In this section, we provide the general solution of functional equation (1.1) without any additional assumptions. We also provide a Jordan and von Neumann type characterization theorem for inner product spaces. Throughout this part, \((G,+)\) is a 2-divisible group (not necessarily Abelian) and \((H,+)\) is an Abelian group.

Theorem 2.1

Suppose that \(g: G \to H\) satisfies (1.1) for all \(x,y,z\in G\). Then g is quadratic.

Proof

Letting \(x=y=z=0\) in (1.1), we get \(2g(0)=0\). Putting \(z=0\) in (1.1) and using \(2g(0)=0\), we obtain

$$ 2g \biggl(\frac{x+y}{2} \biggr)+2g \biggl( \frac{x-y}{2} \biggr)=g(x)+g(y), \quad x,y\in G. $$
(2.1)

Letting \(y=0\) in (2.1), we get

$$ 4g \biggl(\frac{x}{2} \biggr)=g(x)+g(0),\quad x\in G. $$
(2.2)

It follows from (2.1) and (2.2) that

$$ 2g(x)+2g(y)= 4g \biggl(\frac{x+y}{2} \biggr)+4g \biggl( \frac{x-y}{2} \biggr)=g(x+y)+g(x-y)+2g(0),\quad x,y\in G.$$

Since \(2g(0)=0\), we infer that g is quadratic. □

By [13, Theorem 4.1] and Theorem 2.1, we get the general solution of (1.1).

Corollary 2.2

Let \((H,+)\) be a 2-divisible Abelian group. Suppose that \(g: G \to H\) satisfies (1.1) and \(g(x + y + z) = g(x + z + y)\) for all \(x,y,z\in G\). Then g has the form \(g(x)=B(x,x)\), where \(B:G\times G\to H\) is a symmetric and biadditive function. This function B is unique and is given by

$$ B(x, y)=\frac{g(x+y)-g(x-y)}{4}=\frac{g(x+y)-g(x)-g(y)}{2},\quad x,y \in G.$$

Now we give a Jordan and von Neumann type characterization theorem for inner product spaces which is related to (1.1).

Theorem 2.3

Let X be a normed space and

$$ \begin{aligned} & \biggl\Vert \frac{x+y}{2}+z \biggr\Vert ^{p}+ \biggl\Vert \frac{x+y}{2}-z \biggr\Vert ^{q}+ \biggl\Vert \frac{x-y}{2}+z \biggr\Vert ^{r}+ \biggl\Vert \frac{x-y}{2}-z \biggr\Vert ^{s} \\ &\quad = \Vert x \Vert ^{\alpha }+ \Vert y \Vert ^{\beta }+4 \Vert z \Vert ^{\gamma },\quad x,y,z\in X, \end{aligned} $$
(2.3)

for some nonnegative real numbers p, q, r, s, α, βγ. Then X is an inner product space.

Proof

Letting \(z=0\) in (2.3), we get

$$ \biggl\Vert \frac{x+y}{2} \biggr\Vert ^{p}+ \biggl\Vert \frac{x+y}{2} \biggr\Vert ^{q}+ \biggl\Vert \frac{x-y}{2} \biggr\Vert ^{r}+ \biggl\Vert \frac{x-y}{2} \biggr\Vert ^{s} = \Vert x \Vert ^{\alpha }+ \Vert y \Vert ^{\beta },\quad x,y,z\in X. $$
(2.4)

Letting \(y=0\) and \(\Vert x \Vert =2\) (\(x=0\) and \(\Vert y \Vert =2\)) in (2.4), we obtain \(\alpha =\beta =2\). Replacing \(y=x\) in (2.4) and then setting \(\Vert x \Vert =2,4\), we get \(2^{p}+2^{q}=4\) and \(4^{p}+4^{q}=32\). Then it is easily obtained \(p=q=2\). Similarly, replacing \(y=-x\) in (2.4) and then setting \(\Vert x \Vert =2,4\), we obtain \(r=s=2\). Therefore (2.4) means

$$ \Vert x+y \Vert ^{2}+ \Vert x-y \Vert ^{2}=2 \Vert x \Vert ^{2}+2 \Vert y \Vert ^{2},\quad x,y \in X.$$

Hence X is an inner product space by the Jordan-von Neumann result [11]. □

Theorem 2.4

Let X be a normed space and

$$ \begin{gathered} \biggl\Vert \frac{x+y}{2}+z \biggr\Vert ^{p}+ \biggl\Vert \frac{x+y}{2}-z \biggr\Vert ^{q}+ \biggl\Vert \frac{x-y}{2}+z \biggr\Vert ^{r}+ \biggl\Vert \frac{x-y}{2}-z \biggr\Vert ^{s}=6,\\ \Vert x \Vert = \Vert y \Vert = \Vert z \Vert =1,\end{gathered} $$
(2.5)

for some nonnegative real numbers p, q, rs. Then X is an inner product space.

Proof

Letting \(y=x\) in (2.5), we get

$$ \Vert x+z \Vert ^{p}+ \Vert x-z \Vert ^{q}=4,\quad\quad \Vert x \Vert = \Vert z \Vert =1. $$
(2.6)

Replacing \(z=x\) (\(z=-x\)) in (2.6), we get \(p=q=2\). Then (2.6) means

$$ \Vert x+z \Vert ^{2}+ \Vert x-z \Vert ^{2}=4, \quad\quad \Vert x \Vert = \Vert z \Vert =1.$$

Hence X is an inner product space by [5, Theorem 2.1]. □

Hyperstability and stability of (1.1)

Throughout this section, \(\mathcal{A}\) denotes a linear 2-normed space, and \((G,+)\) is a 2-divisible Abelian group. For convenience, we set

$$\begin{aligned} D_{g}(x,y, z):={}& g \biggl( \frac{x+y}{2} + z \biggr) + g \biggl( \frac{x+y}{2} - z \biggr) + g \biggl( \frac{x-y}{2} + z \biggr) \\ & {}+ g \biggl( \frac{x-y}{2} - z \biggr) - g(x) - g(y) - 4 g(z) \end{aligned}$$

for a given function g.

We introduce some hyperstability results for the Jensen-type quadratic functional equation (1.1).

Theorem 3.1

Let \(\varphi :G\to \mathcal{A}\) be a surjective function. Assume that \(\eta : G^{4}\to [0,+\infty )\) and \(g : G \to \mathcal{A}\) are functions satisfying

$$\begin{aligned} & \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \eta (x,y,z,w), \end{aligned}$$
(3.1)
$$\begin{aligned} & \lim_{n\to \infty }\eta \bigl((n+2)x,ny,0,w \bigr)=0, \quad\quad \lim_{n\to \infty } \eta (nx,ny,0,w)=0 \end{aligned}$$
(3.2)

for all \(x,y , z, w \in G\). Then g is quadratic and satisfies functional equation (1.1); that is, \(D_{g} (x, y, z)=0\) for all \(x,y,z\in G\).

Proof

Letting \(y=nx\), \(z=0\) and replacing x by \((n+2)x\) in (3.1), we obtain

$$ \bigl\Vert 2g \bigl((n+1)x \bigr)+2g(x)-g \bigl((n+2)x \bigr)-g(nx)-4g(0), \varphi (w) \bigr\Vert \leqslant \eta \bigl((n+2)x,nx,0,w \bigr) $$

for all \(x, y, w \in G\) and \(n\in \mathbb{N}\). Therefore

$$ 2g(x)=\lim_{n\to \infty } \bigl[g \bigl((n+2)x \bigr)+g(nx)-2g \bigl((n+1)x \bigr)+4g(0) \bigr],\quad x \in G.$$

By (3.1) and (3.2), we get

$$\begin{aligned} & \biggl\Vert 2g \biggl(\frac{x+y}{2} \biggr)+2g \biggl( \frac{x-y}{2} \biggr)-g(x)-g(y)-4g(0),\varphi (w) \biggr\Vert \\ &\quad \leqslant \frac{1}{2}\limsup_{n\to \infty } \bigl\Vert D_{g} \bigl((n+2)x,(n+2)y,0 \bigr), \varphi (w) \bigr\Vert \\ &\quad\quad {} + \frac{1}{2}\limsup_{n\to \infty } \bigl\Vert D_{g}(nx,ny,0),\varphi (w) \bigr\Vert \\ &\quad\quad {} + \limsup_{n\to \infty } \bigl\Vert D_{g} \bigl((n+1)x,(n+1)y,0 \bigr),\varphi (w) \bigr\Vert \\ &\quad \leqslant \frac{1}{2}\limsup_{n\to \infty }\eta \bigl((n+2)x,(n+2)y,0,w \bigr) \\ &\quad\quad {} + \frac{1}{2}\limsup_{n\to \infty }\eta (nx,ny,0,w)+\limsup_{n \to \infty }\eta \bigl((n+1)x,(n+1)y,0,w \bigr)=0. \end{aligned}$$

Then

$$ 2g \biggl(\frac{x+y}{2} \biggr)+2g \biggl( \frac{x-y}{2} \biggr)=g(x)+g(y)+4g(0), \quad x,y\in G. $$
(3.3)

Letting \(x=y=z=0\) in (3.3), we get \(2g(0)=0\). The rest of the proof follows from the proof of Theorem 2.1. □

Remark 3.2

By a similar argument, it can be shown that if the conditions

$$ \lim_{n\to \infty }\eta \bigl(2(n+1)x,0,nx,w \bigr)=0, \quad\quad \lim_{n\to \infty } \eta (nx,0,ny,w)=0,\quad x,y,w\in G$$

are substituted for (3.2), Theorem 3.1 is still valid.

Corollary 3.3

Let \(p,q<0\) and \(r,\alpha ,\beta ,\gamma \geqslant 0\) be real numbers. Suppose that X is a linear normed space, \(\psi :X\to [0,+\infty )\) and \(g,\varphi : X \to \mathcal{A}\) are functions such that φ is surjective and

$$ \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \bigl(\alpha \Vert x \Vert ^{p}+\beta \Vert y \Vert ^{q}+\gamma \Vert z \Vert ^{r} \bigr)\psi (w) $$

for all \(x,y \in X\setminus \{0\}\) and \(z, w\in X\). Then \(D_{g} (x, y, z)=0\) for all \(x,y\in X\setminus \{0\}\) and \(z\in X\). In particular,

$$ 2g \biggl(\frac{x+y}{2} \biggr)+2g \biggl(\frac{x-y}{2} \biggr)=g(x)+g(y), \quad x,y\in X\setminus \{0\}.$$

Corollary 3.4

Let \(p,r<0\) and \(q,\alpha ,\beta ,\gamma \geqslant 0\) be real numbers. Suppose that X is a linear normed space, \(\psi :X\to [0,+\infty )\) and \(g,\varphi : X \to \mathcal{A}\) are functions such that φ is surjective and

$$ \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \bigl(\alpha \Vert x \Vert ^{p}+\beta \Vert y \Vert ^{q}+\gamma \Vert z \Vert ^{r} \bigr)\psi (w) $$

for all \(x,z \in X\setminus \{0\}\) and \(y, w\in X\). Then \(D_{g} (x, y, z)=0\) for all \(x,z\in X\setminus \{0\}\) and \(y\in X\). In particular,

$$ 2g \biggl(\frac{x+z}{2} \biggr)+2g \biggl(\frac{x-z}{2} \biggr)=g(x)+g(z)+2g(0), \quad x,z\in X\setminus \{0\}.$$

Theorem 3.5

Let \(\varphi :G\to \mathcal{A}\) be a surjective function. Assume that \(\eta : G^{4}\to [0,+\infty )\) and \(g : G \to \mathcal{A}\) are functions satisfying

$$\begin{aligned} & \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \eta (x,y,z,w), \end{aligned}$$
(3.4)
$$\begin{aligned} & \lim_{n\to \infty }\eta \bigl((n+1)x,(n+1)x,nx,w \bigr)=0, \quad \quad \lim_{n\to \infty }\eta (nx,ny,nz,w)=0 \end{aligned}$$
(3.5)

for all \(x,y , z, w \in G\). Then g is quadratic and satisfies functional equation (1.1); that is, \(D_{g} (x, y, z)=0\) for all \(x,y,z\in G\).

Proof

Letting \(y=(n+1)x\), \(z=nx\) and replacing x by \((n+1)x\) in (3.4), we obtain

$$ \bigl\Vert g \bigl((2n+1)x \bigr)+g(x)+g(-nx)-2g \bigl((n+1)x \bigr)-3g(nx), \varphi (w) \bigr\Vert \leqslant \eta \bigl((n+1)x,(n+1)x,nx,w \bigr) $$

for all \(x, y, w \in G\) and \(n\in \mathbb{N}\). Therefore

$$ g(x)=\lim_{n\to \infty } \bigl[2g \bigl((n+1)x \bigr)+3g(nx)-g \bigl((2n+1)x \bigr)-g(-nx) \bigr],\quad x \in G.$$

Hence (3.4) and (3.5) yield

$$\begin{aligned} \bigl\Vert D_{g} (x, y, z),\varphi (w) \bigr\Vert &\leqslant 2 \limsup_{n\to \infty } \bigl\Vert D_{g} \bigl((n+1)x,(n+1)y,(n+1)z \bigr),\varphi (w) \bigr\Vert \\ &\quad {} + 3\limsup_{n\to \infty } \bigl\Vert D_{g}(nx,ny,nz), \varphi (w) \bigr\Vert \\ &\quad {} + \limsup_{n\to \infty } \bigl\Vert D_{g} \bigl((2n+1)x,(2n+1)y,(2n+1)z \bigr), \varphi (w) \bigr\Vert \\ &\quad {} + \limsup_{n\to \infty } \bigl\Vert D_{g}(-nx,-ny,-nz), \varphi (w) \bigr\Vert \\ &\leqslant 2\limsup_{n\to \infty }\eta \bigl((n+1)x,(n+1)y,(n+1)z,w \bigr) \\ &\quad {} + 3\limsup_{n\to \infty }\eta (nx,ny,nz,w) \\ &\quad {} +\limsup_{n\to \infty }\eta \bigl((2n+1)x,(2n+1)y,(2n+1)z,w \bigr) \\ &\quad {} +\limsup_{n\to \infty }\eta (-nx,-ny,-nz,w)=0. \end{aligned}$$

Then \(D_{g} (x, y, z)=0\) for all \(x,y,z\in G\), which yields that g is quadratic by Theorem 2.1. □

Corollary 3.6

Let \(p,q,r<0\) and \(\alpha ,\beta ,\gamma \geqslant 0\) be real numbers. Suppose that X is a linear normed space, \(\psi :X\to [0,+\infty )\) and \(g,\varphi : X \to \mathcal{A}\) are functions such that φ is surjective and

$$ \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \bigl(\alpha \Vert x \Vert ^{p}+\beta \Vert y \Vert ^{q}+\gamma \Vert z \Vert ^{r} \bigr)\psi (w) $$

for all \(x,y , z\in X\setminus \{0\}\) and \(w\in X\). Then \(D_{g} (x, y, z)=0\) for all \(x,y,z\in X\setminus \{0\}\).

Let us consider A be a real normed linear space and also consider that there is a 2-norm on A which makes \((A, \Vert \cdot, \cdot \Vert )\) a 2-Banach space.

Theorem 3.7

Take \(\varepsilon , \theta , s \geqslant 0\) and \(r \neq 1\). Let \(g : A \to A \) be a function satisfying the inequality

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \textstyle\begin{cases} \varepsilon ( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} ) \Vert w \Vert ^{r}, & r>1; \\ \theta +\varepsilon ( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} ) \Vert w \Vert ^{r}, & r< 1, \end{cases} $$
(3.6)

for all \(x,y , z, w \in A\). Then g satisfies functional equation (1.1); that is, \(D_{g} (x, y, z)=0\) for all \(x,y,z\in A\).

Proof

Two cases arise according to whether \(r>1\) or \(r<1\). First, take the case \(r>1\). Replacing w by \(\frac{w}{n}\) in (3.6), we infer that

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \frac{\varepsilon }{n^{r-1}} \bigl( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} \bigr) \Vert w \Vert ^{r},\quad x,y , z, w \in A, n\geqslant 1.$$

So, by taking the limit as \(n\to \infty \), we obtain \(\Vert D_{g} (x, y, z), w \Vert =0\) for all \(x,y , z, w \in A\). Then \(D_{g} (x, y, z)=0\) for all \(x,y , z \in A\) by Lemma 1.3.

Now consider the case \(r<1\). Replacing w by nw in (3.6), we infer that

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \frac{\theta }{n}+\varepsilon n^{r-1} \bigl( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} \bigr) \Vert w \Vert ^{r},\quad x,y , z, w \in A, n\geqslant 1.$$

It should be noted that if \(r<0\), then the recent inequality satisfies for \(w\neq 0\). Taking the limit as \(n\to \infty \), we obtain the result. □

Corollary 3.8

Let \(\varepsilon \geqslant 0\) and \(g : A \to A \) be a function satisfying the inequality

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \varepsilon $$

for all \(x,y , z, w \in A\). Then g is quadratic and satisfies functional equation (1.1).

Remark 3.9

It should be noted that the hyperstability Theorem 3.7 does not hold in normed spaces (see [17, Corollaries 2.3, 2.5]).

In the case of \(r=1\), we have the following stability result.

Theorem 3.10

Let \(\theta , \varepsilon , s \geqslant 0\) with \(s\neq 2\). If \(g : A \to A \) is a function satisfying the inequality

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \theta +\varepsilon \bigl( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} \bigr) \Vert w \Vert $$
(3.7)

for all \(x,y , z, w \in A\), then there exists a unique quadratic function \(Q : A \to A \) satisfying functional equation (1.1) such that

$$ \bigl\Vert Q(x) - g(x), w \bigr\Vert \leqslant \frac{2^{s} \varepsilon }{ \vert 4 - 2^{s} \vert } \Vert x \Vert ^{s} \Vert w \Vert . $$
(3.8)

Proof

Replacing w by nw in (3.7), we infer that

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \frac{\theta }{n}+\varepsilon \bigl( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} \bigr) \Vert w \Vert ,\quad x,y , z, w \in A, n \geqslant 1. $$

Hence, by taking the limit as \(n\to \infty \), we obtain

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \varepsilon \bigl( \Vert x \Vert ^{s} + \Vert y \Vert ^{s} + \Vert z \Vert ^{s} \bigr) \Vert w \Vert . $$
(3.9)

Letting \(x=y=z=0\) in (3.9), we obtain \(\Vert 2g(0),w \Vert =0\) for all \(w\in A\). Then \(g(0)=0\) by Lemma 1.3. We assume that \(s<2\). Setting \(y = z=0 \) and replacing x by 2x in (3.9), we have

$$ \bigl\Vert g(2x)-4g(x),w \bigr\Vert \leqslant 2^{s}\varepsilon \Vert x \Vert ^{s} \Vert w \Vert , \quad x,w\in A. $$

Therefore

$$ \biggl\Vert \frac{1}{4^{n}}g \bigl(2^{n}x \bigr) - \frac{1}{4^{m}}g \bigl(2^{m} x \bigr), w \biggr\Vert \leqslant \varepsilon \Vert x \Vert ^{s} \Vert w \Vert \sum _{k=m}^{n-1} \biggl(\frac{2^{s}}{4} \biggr)^{k+1} $$
(3.10)

for all \(x, w \in A\) and all integers \(n\geqslant m\geqslant 0\). Hence, \(\{ \frac{1}{4^{n}}g(2^{n} x) \}\) is a Cauchy sequence in A for all \(x \in A\). Now, we will define a function \(Q : A \to A\) by

$$ Q(x) = \lim_{n \to \infty } \frac{1}{4^{n}}f \bigl(2^{n} x \bigr),\quad x\in A.$$

Letting \(m=0\) and taking the limit in (3.10) as \(n\to \infty \), we obtain (3.8).

For the case \(s>2\), the argument is similar. The uniqueness of Q is clearly obtained from (3.8). □

Remark 3.11

In the case \(s>2\) and \(\theta >0\), the stability Theorem 3.10 is not valid in Banach spaces.

Stability and hyperstability of (1.1) for functions \(g:(A, \Vert \cdot, \cdot \Vert )\rightarrow (A, \Vert \cdot, \cdot \Vert )\)

In this section, we study similar problems which we have discussed in the last section for functions \(g: (A, \Vert \cdot ,\cdot \Vert ) \to (A, \Vert \cdot, \cdot \Vert )\), where \((A, \Vert \cdot, \cdot \Vert )\) is a 2-Banach space.

First, we will establish the Hyers–Ulam stability and generalized Hyers–Ulam stability of a Jensen’s quadratic functional equation (1.1), which is controlled by the sums of powers of norms on 2-Banach spaces.

Theorem 4.1

Let \(g:\mathcal{A}\to \mathcal{A}\). Assume that \(\varepsilon , \theta \geqslant 0\) and \(\varphi :\mathcal{A}\to \mathcal{A}\) is a surjective function.

\((i)\):

If \(s\in (0,2)\) and

$$ \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \theta +\varepsilon \bigl[ \Vert x,w \Vert ^{s}+ \Vert y,w \Vert ^{s}+ \Vert z,w \Vert ^{s} \bigr], \quad x,y,z, w \in \mathcal{A}, $$
(4.1)

then there exists a unique quadratic function \(Q : A \to A \) satisfying functional equation (1.1) such that

$$ \bigl\Vert g(x)-Q(x), \varphi (w) \bigr\Vert \leqslant \frac{\theta }{3}+ \frac{2^{s}\varepsilon }{4-2^{s}} \Vert x,w \Vert ^{s}, \quad x, w \in \mathcal{A}. $$
(4.2)
\((\mathit{ii})\):

If \(s\in (2,+\infty )\) and

$$ \bigl\Vert D_{g} (x, y, z), \varphi (w) \bigr\Vert \leqslant \varepsilon \bigl[ \Vert x,w \Vert ^{s}+ \Vert y,w \Vert ^{s}+ \Vert z,w \Vert ^{s} \bigr],\quad x,y,z, w \in \mathcal{A}, $$

then there exists a unique quadratic function \(Q : A \to A \) satisfying functional equation (1.1) such that

$$ \bigl\Vert g(x)-Q(x), \varphi (w) \bigr\Vert \leqslant \frac{2^{s}\varepsilon }{2^{s}-4} \Vert x,w \Vert ^{s},\quad x, w \in \mathcal{A}. $$

Proof

\((i)\) Letting \(x=y=z=0\) in (4.1), we obtain \(\Vert 2g(0),\varphi (w) \Vert \leqslant \varepsilon \) for all \(w\in A\). Since φ is surjective, for each \(a\in \mathcal{A}\) we have \(na\in \varphi (\mathcal{A})\) for all \(n\in \mathbb{N}\). Then \(\Vert 2g(0),na \Vert \leqslant \varepsilon \) for all \(n\in \mathbb{N}\). Letting now \(n\to \infty \) and applying Lemma 1.3, we infer that \(g(0)=0\). Setting \(y = z=0 \) and replacing x by 2x in (4.1), we have

$$ \bigl\Vert g(2x)-4g(x),\varphi (w) \bigr\Vert \leqslant \varepsilon + 2^{s}\theta \Vert x, w \Vert ^{s},\quad x,w\in A. $$

Therefore

$$ \biggl\Vert \frac{1}{4^{n}}g \bigl(2^{n}x \bigr) - \frac{1}{4^{m}}g \bigl(2^{m} x \bigr), \varphi (w) \biggr\Vert \leqslant \sum_{k=m}^{n-1} \frac{\varepsilon }{4^{k+1}}+ \theta \Vert x, w \Vert ^{s} \sum _{k=m}^{n-1} \biggl(\frac{2^{s}}{4} \biggr)^{k+1} $$
(4.3)

for all \(x, w \in A\) and all integers \(n\geqslant m\geqslant 0\). Thus, \(\{ \frac{1}{4^{n}}g(2^{n} x) \}\) is a Cauchy sequence in A for all \(x \in A\). Now, we will define a function \(Q : A \to A\) by

$$ Q(x) = \lim_{n \to \infty } \frac{1}{4^{n}}f \bigl(2^{n} x \bigr),\quad x\in A.$$

Letting \(m=0\) and taking the limit in (4.3) as \(n\to \infty \), we obtain (4.2). The uniqueness of Q is clearly obtained from (4.2).

For the case \(s>2\), the argument is similar. □

Corollary 4.2

Let \(g:\mathcal{A}\to \mathcal{A}\) satisfy

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \theta + \varepsilon \bigl[ \Vert x,w \Vert + \Vert y,w \Vert + \Vert z,w \Vert \bigr],\quad x,y,z, w \in \mathcal{A}.$$

Then there exists a unique quadratic function \(Q : A \to A \) satisfying functional equation (1.1) such that

$$ \bigl\Vert g(x)-Q(x), w \bigr\Vert \leqslant \varepsilon \Vert x,w \Vert ,\quad x, w \in \mathcal{A}. $$

For the case \(s\neq 1\), we have the following result.

Theorem 4.3

Take \(\varepsilon , \theta \geqslant 0\) and \(s \neq 1\). Let \(g : A \to A \) be a function satisfying the inequality

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \textstyle\begin{cases} \varepsilon ( \Vert x,w \Vert ^{s} + \Vert y, w \Vert ^{s} + \Vert z, w \Vert ^{s} ), & s>1; \\ \theta +\varepsilon ( \Vert x, w \Vert ^{s} + \Vert y, w \Vert ^{s} + \Vert z, w \Vert ^{s} ), & s< 1, \end{cases} $$
(4.4)

for all \(x,y , z, w \in A\) (with \(\Vert x,w \Vert \Vert y,w \Vert \Vert z,w \Vert \neq 0\) when \(s<0\)). Then \(D_{g} (x, y, z)=0\) for all \(x,y,z\in A\).

Proof

First we consider the case \(s>1\). Replacing w by \(\frac{w}{n}\) in (4.4), we infer that

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \frac{\varepsilon }{n^{s-1}} \bigl( \Vert x, w \Vert ^{s} + \Vert y, w \Vert ^{s} + \Vert z, w \Vert ^{s} \bigr),\quad x,y , z, w \in A, n \geqslant 1.$$

Taking the limit as \(n\to \infty \), we obtain \(\Vert D_{g} (x, y, z), w \Vert =0\) for all \(x,y , z, w \in A\). Then \(D_{g} (x, y, z)=0\) for all \(x,y , z \in A\) by Lemma 1.3.

Now we consider the case \(s<1\). Replacing w by nw in (4.4), we infer that

$$ \bigl\Vert D_{g} (x, y, z), w \bigr\Vert \leqslant \frac{\theta }{n}+\varepsilon n^{s-1} \bigl( \Vert x, w \Vert ^{s} + \Vert y, w \Vert ^{s} + \Vert z, w \Vert ^{s} \bigr),\quad x,y , z, w \in A, n\geqslant 1.$$

It should be noted that if \(s<0\), then the recent inequality is satisfied for all \(x,y , z, w \in A\) with \(\Vert x,w \Vert \Vert y,w \Vert \Vert z,w \Vert \neq 0\). Taking the limit as \(n\to \infty \), we obtain the result. □

Conclusion

We have proved the various types of Hyers–Ulam stability and hyperstability of the Jensen-type quadratic functional equation of the form

$$ g \biggl( \frac{x+y}{2} + z \biggr) + g \biggl( \frac{x+y}{2} - z \biggr) + g \biggl( \frac{x-y}{2} + z \biggr) + g \biggl( \frac{x-y}{2} - z \biggr) = g(x) + g(y) + 4 g(z) $$

in 2-Banach spaces by using the Hyers direct method.

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Acknowledgements

The authors would like to thank the editor and the reviewers for the detailed and valuable suggestions that helped to improve the original manuscript to its present form.

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APS and AN contributed to the study conception and design. The first draft of the manuscript was written by APS and all authors commented on the previous versions of the manuscript. All authors read and approved the final manuscript. AN supervised the project.

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Correspondence to Abbas Najati.

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Selvan, A.P., Najati, A. Hyers–Ulam stability and hyperstability of a Jensen-type functional equation on 2-Banach spaces. J Inequal Appl 2022, 32 (2022). https://doi.org/10.1186/s13660-022-02769-1

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MSC

  • 39B82
  • 39B52
  • 39B62

Keywords

  • Hyers–Ulam stability
  • Hyperstability
  • Jensen-type function
  • Quadratic function
  • 2-Banach space
  • Direct method