1 Introduction

Let \({G}\) be an Abelian group and \({E}\) a vector space. Let \(S({E})\) be a family of subsets of \({E}\). We say that \(S({E})\) is linearly invariant if it is closed under the addition and scalar multiplication defined as usual sense and translation invariant, i.e, \(x+A\in S({E})\), for every \(A\in S({E})\) and every \(x\in {E}\) (see [1]). It is easy to verify that \(S({E})\) contains all singleton subsets of \({E}\). In particular, \(CB({E})\) the family of all closed balls with center at zero is a linearly invariant family in a normed vector space \({E}\). By \(B(G\times G,S({E}))\) we denote the family of all functions \(f:G\times G\longrightarrow {E}\) such that \(\mathtt{Im }f\subset A\) for some \(A\in S({E})\), where \(G\times G\) is the Cartesian product of \(G\) with itself. Obviously, this family is a vector space and contains all constant functions.

We say that \(B(G\times G,S({E}))\) admits a left invariant mean (briefly LIM), if the family \(S({E})\) is linearly invariant and there exists a linear operator \(M:B(G\times G,S({E}))\longrightarrow {E}\) such that

  1. (i)

    if \(\mathtt{Im }f\subset A\) for some \(A\in S({E})\), then \(M[f]\in A\),

  2. (ii)

    if \(f\in B(G\times G,S({E}))\) and \((a,b)\in G\times G\), then \(M[f^{a,b}]=M[f]\),

where \(f^{a,b}(x,y)=f(x+a,y+b)\). Following [2, 3] and for 2-variable functions let \(\mathbf{d }:(G\times G)\times (G\times G)\longrightarrow S({E})\) be a set-valued function such that

$$\begin{aligned} \mathbf{d }\left( (x+a,y+b),(w+a,z+b)\right)&=\mathbf{d }\left( (a+x,b+y),(a+w,b+z)\right) \\&=\mathbf{d }((x,y),(w,z)) \end{aligned}$$

for all \((a,b),(x,y),(w,z)\in G\times G\). A function \(f:G\times G\longrightarrow {E}\) is said to be \(\mathbf{d }\)-Lipschitz if \(f(x,y)-f(w,z)\in \mathbf{d }((x,y),(w,z))\) for all \((x,y),(w,z)\in G\times G\).

Let \((G\times G,d)\) be a metric group and \({E}\) a normed space. A function \(m_{f}:{\mathbb {R}}^+\longrightarrow {\mathbb {R}}^+\) is a module of continuity of \(f:G\times G\longrightarrow {E}\) if \(d((x,y),(w,z))\le \delta \) implies \(||f(x,y)-f(w,z)||\le m_{f}(\delta )\) for every \(\delta >0\) and every \((x,y),(w,z)\in G\times G\). A function \(f:G\times G\longrightarrow {E}\) is called Lipschitz function if it satisfies the condition

$$\begin{aligned} ||f(x,y)-f(w,z)||\le L d((x,y),(w,z)) \end{aligned}$$
(1.1)

for every \((x,y),(w,z)\in G\times G\). The smallest constant \(L\) with this property is denoted by \(\mathtt{lip }(f)\).

We define \(Lip(G\times G,{E})\) to be the Lipschitz space consisting of all bounded Lipschitz functions with the norm

$$\begin{aligned} ||f||_{Lip}:=||f||_{\sup }+\mathtt{lip }(f). \end{aligned}$$

The study of stability problems for functional equations is related to a question of Ulam [4] concerning the stability of group homomorphisms, which was affirmatively answered for Banach spaces by Hyers [5] (see for example [6, 7] and references therein).

In Lipschitz spaces the stability type problems for some functional equations were studied by Czerwik and Dlutek [8] and Tabor [3, 9]. Czerwik and Dlutek [8] established the stability of the quadratic functional equation

$$\begin{aligned} f(x+y)+f(x-y)=2f(x)+2f(y) \end{aligned}$$

and the author of the present paper [10] proved the stability of the cubic functional equation in Lipschitz spaces. The stability problem for the following quartic functional equation

$$\begin{aligned} f(2x + y)+f(2x-y)= 4(f(x + y)+f(x-y))+24f(x)-6f(y) \end{aligned}$$

first was considered by Rassias [11] for mappings from a real normed space into a Banach space. Najati [12] proved the generalized Hyers–Ulam stability for the above quartic functional equation for functions from a linear space into a Banach space. Bae [13] obtained the general solution and the stability of the following 2-variable quadratic functional equation

$$\begin{aligned} f(x+z,y+w)+f(x-z,y-w)=2f(x,y)+2f(z,w) \end{aligned}$$

in complete normed spaces. In this paper, we verify the stability of the quartic functional equation in the Lipschitz norms.

2 Approximation with \(\mathbf{d }\)-Lipschitz approach

For a given function \(f:G\times G\longrightarrow {E}\) we define its quadratic difference as follows

$$\begin{aligned} Qf(x,y,z,w):=2f(x,y)+2f(z,w)-f(x+z,y+w)-f(x-z,y-w) \end{aligned}$$

for all \((x,y),(z,w)\in G\times G\). By \(\Delta (G)\) we denote the diagonal set on \(G\), i.e.,

$$\begin{aligned} \Delta (G):=\{(x,x)\in G\times G: x\in G\}. \end{aligned}$$

Theorem 2.1

Let \({G}\) be an Abelian group, and let \({E}\) be a vector space. Assume that the family \(B(G\times G,S({E}))\) admits LIM. If \(f:G\times G\longrightarrow {E}\) is a function and \(Qf(t,s,\cdot ,\cdot ):G\times G\longrightarrow {E}\) is \(\mathbf{d }\)-Lipschitz for every \((t,s)\in G\times G\), then there exists a quartic function \(Q\) such that \(f_{|_{\Delta (G)}}-Q\) is \(\frac{1}{2}\,\mathbf d \)-Lipschitz.

Proof

For every \((a,b)\in G\times G\) we define \(F_{a,b}:G\times G\longrightarrow {E}\) by

$$\begin{aligned} F_{a,b}(x,y):=\frac{1}{2}f(x+a,y+b)+\frac{1}{2}f(x-a,y-b)-f(x,y). \end{aligned}$$

We prove that \(F_{a,b}\in B(G\times G,S({E}))\). We have for \((x,y),(a,b)\in G\times G\),

$$\begin{aligned} F_{a,b}(x,y)&=\frac{1}{2}f(x+a,y+b)+\frac{1}{2}f(x-a,y-b)-f(x,y)-f(a,b)\\&\quad -\frac{1}{2}f(x,y)-\frac{1}{2}f(x,y)+f(x,y)+f(0,0)\\&\quad +f(a,b)-f(0,0)\\&=\frac{1}{2}Qf(x,y,0,0)-\frac{1}{2}Qf(x,y,a,b)+f(a,b)-f(0,0). \end{aligned}$$

Set \(A:=\frac{1}{2}\mathbf{d }((0,0),(a,b))+f(a,b)+f(0,0)\). It is clear that \(A\in S({E})\). In view of our assumptions it follows that \(\mathtt{Im }F_{a,b}\subset A\) and so we obtain the result. The fact that the family \(B(G\times G,S({E}))\) admits LIM ensures there exists a linear operator \(M:B(G\times G,S({E}))\longrightarrow {E}\) such that

  1. (i)

    \(M[F_{a,b}]\in A\) for some \(A\in S({E})\),

  2. (ii)

    if for \((z,w)\in G\times G\), \(F_{a,b}^{z,w}:G\times G\longrightarrow {E}\) is defined by \(F_{a,b}^{z,w}(t,s):=F_{a,b}(t+z,s+w)\) for every \((t,s)\in G\times G\), then \(F_{a,b}^{z,w}\in B(G\times G,S({E}))\) and \(M[F_{a,b}]=M[F_{a,b}^{z,w}]\).

Define the function \(K:G\times G\longrightarrow {E}\) by \(K(x,y):=M[F_{x,y}]\) for \((x,y)\in G\times G\). We know that \(B(G\times G,S({E}))\) contains constant functions. By using property (i) of \(M\) it is easy to verify that if \(f:G\times G\longrightarrow {E}\) is constant, i.e., \(f(x,y)=c\) for \((x,y)\in G\times G\), where \(c\in {E}\), then \(M[f]=c\). We now show that \(f-K\) is \(\frac{1}{2}\,\mathbf d \)-Lipschitz. Let for any \((x,y)\in G\times G\) the constant function \(R_{x,y}:G\times G\longrightarrow {E}\) be the function \(R_{x,y}(z,w):=f(x,y)\) for all \((z,w)\in G\times G\). We have

$$\begin{aligned}&\left( f(x,y)-K(x,y)\right) -\left( f(z,w)-K(z,w)\right) \\&\quad =\left( M[R_{x,y}]-M[F_{x,y}]\right) -\left( M[R_{z,w}]-M[F_{z,w}]\right) \\&\quad =M[R_{x,y}-F_{x,y}]-M[R_{z,w}-F_{z,w}]\\&\quad =M\left[ \frac{1}{2}Qf(\cdot ,\cdot ,x,y)-\frac{1}{2}Qf(\cdot ,\cdot ,z,w)\right] \end{aligned}$$

for all \((x,y),(z,w)\in G\times G\). On the other hand

$$\begin{aligned} \frac{1}{2}Qf(t,s,x,y)-\frac{1}{2}Qf(t,s,z,w)\in \frac{1}{2}\mathbf {d}\left( (x,y),(z,w)\right) \end{aligned}$$
(2.1)

for all \((x,y),(z,w)\in G\times G\). From this we deduce that

$$\begin{aligned} \mathtt{Im }\left( \frac{1}{2}Qf(\cdot ,\cdot ,x,y)-\frac{1}{2}Qf(\cdot ,\cdot ,z,w)\right) \subseteq \frac{1}{2}\mathbf {d}\left( (x,y),(z,w)\right) . \end{aligned}$$

In view of property (i) of \(M\) we conclude that

$$\begin{aligned} M\left[ \frac{1}{2}Qf(\cdot ,\cdot ,x,y)-\frac{1}{2}Qf(\cdot ,\cdot ,z,w)\right] \in \frac{1}{2}\mathbf {d}\left( (x,y),(z,w)\right) \end{aligned}$$

for all \((x,y),(z,w)\in G\times G\). This shows that

$$\begin{aligned} \left( f(x,y)-K(x,y)\right) -\left( f(z,w)-K(z,w)\right) \in \frac{1}{2}\mathbf {d}\left( (x,y),(z,w)\right) \end{aligned}$$

for all \((x,y),(z,w)\in G\times G\), i.e., \(f-K\) is a \(\frac{1}{2}\,\mathbf d \)-Lipschitz function. We now have

$$\begin{aligned} 2K(x,y)+2K(z,w)=2M[F_{x,y}(t,s)]+2M[F_{z,w}(t,s)]. \end{aligned}$$

Furthermore, applying property (ii) of \(M\), one gets

$$\begin{aligned} M[F_{x,y}]&=M\left[ F_{x,y}^{z,w}\right] \\ M[F_{x,y}]&=M\left[ F_{x,y}^{-z,-w}\right] \end{aligned}$$

for \((z,w)\in G\times G\). Consequently, we have

$$\begin{aligned} 2K(x,y)+2K(z,w)&=2M[F_{x,y}]+2M[F_{z,w}]\\&=M\left[ F_{x,y}^{z,w}\right] +M\left[ F_{x,y}^{-z,-w}\right] +2M[F_{z,w}]. \end{aligned}$$

On the other hand we have

$$\begin{aligned}&M[F_{x,y}^{z,w}]+M[F_{x,y}^{-z,-w}]+2M[F_{z,w}]\\&\quad =M\left[ \frac{1}{2}f(t+x+z,s+y+w)+\frac{1}{2}f(t-x+z,s-y+w)-f(t+z,s+w)\right] \\&\qquad +M\left[ \frac{1}{2}f(t+x-z,s+y-w)+\frac{1}{2}f(t-x-z,s-y-w)-f(t-z,s-w)\right] \\&\qquad +M[f(t+z,s+w)+f(t-z,s-w)-2f(t,s)]\\&\quad =K(x+z,y+w)+K(x-z,y-w). \end{aligned}$$

This shows that \(K\) is 2-variable quadratic. Define \(Q: G\longrightarrow E\) by \(Q(x):=K(x,x)\). We have

$$\begin{aligned} f_{|_{\Delta (G)}}-Q=f_{|_{\Delta (G)}}-K_{|_{\Delta (G)}}=(f-K)_{|_{\Delta (G)}}. \end{aligned}$$

The function \(f-K\) is \(\frac{1}{2}\,\mathbf d \)-Lipschitz and so is \(f_{|_{\Delta (G)}}-Q\). The following equality entails that \(Q\) is quartic.

$$\begin{aligned}&4(Q(x + y)+ Q(x -y)) + 24Q(x) -6Q(y)\\&\quad =4(K(x + y,x + y) + K(x -y,x -y)) + 24K(x,x) -6K(y,y)\\&\quad =32K(x,x) + 2K(y,y)\\&\quad =2K(2x,2x) + 2K(y,y)\\&\quad =K(2x+y,2x+y) + K(2x-y,2x-y)\\&\quad =Q(2x+y)+Q(2x-y). \end{aligned}$$

\(\square \)

Remark 2.2

Assuming the hypotheses of Theorem 2.1 and \(\mathtt{Im }Qf\subset A\) for some \(A\in S({E})\), we then obtain \(\mathtt{Im }(f_{|_{\Delta (G)}}-Q)\subset \frac{1}{2}A\). In fact,

$$\begin{aligned} \mathtt{Im }\left( \frac{1}{2}Qf(x,y,\cdot ,\cdot )\right) \subset \mathtt{Im }\left( \frac{1}{2}Qf\right) \subset \frac{1}{2}A \end{aligned}$$

and so \(\frac{1}{2}Qf(x,y,\cdot ,\cdot )\in B(G\times G,S({E}))\) for all \((x,y)\in G\times G\). Thus, property (i) of \(M\) implies

$$\begin{aligned} f(x,y)-K(x,y)=M\left[ \frac{1}{2}Qf(x,y,\cdot ,\cdot )\right] \in \frac{1}{2}A \end{aligned}$$

for all \((x,y)\in G\times G\). Therefore, \(\mathtt{Im }(f_{|_{\Delta (G)}}-Q)\subset \mathtt{Im }(f-K)\subset \frac{1}{2}A\).

3 Approximation with Lipschitz norm

Consider an Abelian group \((G\times G,+)\) with a metric \(d\) invariant under translation, i.e., satisfying the condition

$$\begin{aligned} d\left( (x+a,y+b),(w+a,z+b)\right)&=d\left( (a+x,b+y),(a+w,b+z)\right) \\&=d((x,y),(w,z)) \end{aligned}$$

for all \((a,b),(x,y),(w,z)\in G\times G\). We say that a metric \(D\) on \(G\times G\times G\times G\) is a product metric if it is an invariant metric and the following condition holds

$$\begin{aligned} D\left( (a,b,x,y),(a,b,w,z)\right)&=D\left( (x,y,a,b),(w,z,a,b)\right) \\&=d((x,y),(w,z))\,\,(a,b),(x,y),(w,z)\in G\times G. \end{aligned}$$

Theorem 3.1

Let \((G\times G,+,d,D)\) be a product metric, and let \({E}\) be a normed space such that \(B(G\times G,CB({E}))\) admits LIM. Assume that \(f:G\times G\longrightarrow E\) be a function. If \(Qf\in Lip(G\times G\times G\times G, E)\), then there exists a quartic function \(Q\) such that

$$\begin{aligned} ||f_{|_{\Delta (G)}}-Q||_{Lip}\le \frac{1}{2}||Qf||_{Lip}. \end{aligned}$$

Proof

Assume that \(m_{{Qf}}: {\mathbb {R}}^{+}\rightarrow {\mathbb {R}}^{+}\) is the module of continuity of \(Qf:G\times G\times G\times G\rightarrow E\) with the product metric \(D\) on \(G\times G\times G\times G\). It is immediate that

$$\begin{aligned} ||Qf(t,s,x,y)-Qf(t,s,w,z)||&\le \inf _{D((t,s,x,y),(t,s,w,z))\le \delta }m_{Qf}(\delta )\\&=\inf _{d((x,y),(w,z))\le \delta }m_{Qf}(\delta ) \end{aligned}$$

for all \((t,s),(x,y),(w,z)\in G\times G\). Define the set-valued function \(\mathbf{d }:{G}\times {G}\longrightarrow CB({E})\) by

$$\begin{aligned} \mathbf{d }((x,y),(w,z)):=\inf _{d((x,y),(w,z))\le \delta }m_{Qf}(\delta )B(0,1), \end{aligned}$$

where \(B(0,1)\) is the closed unit ball with center at zero. We now conclude that \(Qf(t,s,\cdot ,\cdot )\) is d-Lipschitz and so Theorem 2.1 implies there exists a quartic function \(Q\) such that \(f_{|_{\Delta (G)}}-Q\) is \(\frac{1}{2}\,\mathbf d \)-Lipschitz. Hence,

$$\begin{aligned} ||(f(x,x)-Q(x))-(f(z,z)-Q(z))||\le \inf _{d((x,x),(z,z))\le \delta }\frac{1}{2}m_{Qf}(\delta ), \end{aligned}$$

which shows that \(m_{{f_{|_{\Delta (G)}}-Q}}=\frac{1}{2}m_{{Qf}}\). Moreover, \(||Qf||_{\sup }<\infty \) and clearly \(\mathtt{Im }Qf\subset ||Qf||_{\sup }B(0,1)\). Using Remark 2.2 we get

$$\begin{aligned} ||f_{|_{\Delta (G)}}-Q||_{\sup }\le \frac{1}{2}||Qf||_{\sup }. \end{aligned}$$
(3.1)

We may also prove that \(m_{Qf}=\mathtt{lip }(Qf)\) and so \(\mathtt{lip }(f_{|_{\Delta (G)}}-Q)\le \frac{1}{2}\mathtt{lip }(Qf)\). Applying the inequality (3.1) we get

$$\begin{aligned} ||f_{|_{\Delta (G)}}-Q||_{Lip}&=||f_{|_{\Delta (G)}}-Q||_{\sup }+\mathtt{lip }(f_{|_{\Delta (G)}}-Q)\\&\le \frac{1}{2}||Qf||_{\sup }+\frac{1}{2}\mathtt{lip }(Qf)\\&=\frac{1}{2}||Qf||_{Lip}. \end{aligned}$$

\(\square \)