1 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, 14–16, 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,

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

2 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, β, γ. 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, r, s. 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]. □

3 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.

4 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. □

5 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.