Superstability of the functional equation with a cocycle related to distance measures

Abstract

In this paper, we obtain the superstability of the functional equation $f(pr,qs)+f(ps,qr)=\theta (pq,rs)f(p,q)f(r,s)$ for all $p,q,r,s\in G$, where G is an Abelian group, f a functional on $G^2$, and θ a cocycle on $G^2$. This functional equation is a generalized form of the functional equation $f(pr,qs)+f(ps,qr)=f(p,q)f(r,s)$, which arises in the characterization of symmetrically compositive sum-form distance measures, and as products of some multiplicative functions. In reduction, they can be represented as exponential functional equations. Also we investigate the superstability with following functional equations: $f(pr,qs)+f(ps,qr)=\theta (pq,rs)f(p,q)g(r,s)$, $f(pr,qs)+f(ps,qr)=\theta (pq,rs)g(p,q)f(r,s)$, $f(pr,qs)+f(ps,qr)=\theta (pq,rs)g(p,q)g(r,s)$, $f(pr,qs)+f(ps,qr)=\theta (pq,rs)g(p,q)h(r,s)$.

MSC:39B82, 39B52.

1 Introduction

Let $(G,\cdot )$ be an Abelian group. Let I denote the open unit interval $(0,1)$. Let ℝ and ℂ denote the set of real and complex numbers, respectively. Let ${\mathbb{R}}_{+}=\{x\in \mathbb{R}\mid x>0\}$ be a set of positive real numbers and ${\mathbb{R}}_k=\{x\in \mathbb{R}\mid x>k>0\}$ for some $k\in \mathbb{R}$.

Further, let

$${\mathrm{\Gamma }}_n^o=\{P=(p_1,p_2,\dots ,p_n)|0<p_k<1,\sum _{k=1}^n{p}_k=1\}$$

denote the set of all n-ary discrete complete probability distributions (without zero probabilities), that is, ${\mathrm{\Gamma }}_n^o$ is the class of discrete distributions on a finite set Ω of cardinality n with $n\ge 2$. Over the years, many distance measures between discrete probability distributions have been proposed. The Hellinger coefficient, the Jeffreys distance, the Chernoff coefficient, the directed divergence, and its symmetrization J-divergence are examples of such measures (see [1] and [2]).

Almost all similarity, affinity or distance measures ${\mu }_n:{\mathrm{\Gamma }}_n^o\times {\mathrm{\Gamma }}_n^o\to {\mathbb{R}}_{+}$ that have been proposed between two discrete probability distributions can be represented in the sum form

$${\mu }_n(P,Q)=\sum _{k=1}^n\varphi (p_k,q_k),$$

(1.1)

where $\varphi :I\times I\to \mathbb{R}$ is a real-valued function on unit square, or a monotonic transformation of the right side of (1.1), that is,

$${\mu }_n(P,Q)=\psi (\sum _{k=1}^n\varphi (p_k,q_k)),$$

(1.2)

where $\psi :\mathbb{R}\to {\mathbb{R}}_{+}$ is an increasing function on ℝ. The function ϕ is called a generating function. It is also referred to as the kernel of ${\mu }_n(P,Q)$.

In information theory, for P and Q in ${\mathrm{\Gamma }}_n^o$, the symmetric divergence of degree α is defined as

$$J_{n,\alpha }(P,Q)=\frac{1}{2^{\alpha -1}-1}[\sum _{k=1}^n(p_k^{\alpha }{q}_k^{1-\alpha }+p_k^{1-\alpha }{q}_k^{\alpha })-2].$$

It is easy to see that $J_{n,\alpha }(P,Q)$ is symmetric. That is, $J_{n,\alpha }(P,Q)=J_{n,\alpha }(Q,P)$ for all $P,Q\in {\mathrm{\Gamma }}_n^o$. Moreover, it satisfies the composition law

$$\begin{array}{r}{J}_{nm,\alpha }(P\ast R,Q\ast S)+J_{nm,\alpha }(P\ast S,Q\ast R)\\ =2J_{n,\alpha }(P,Q)+2J_{m,\alpha }(R,S)+\lambda J_{n,\alpha }(P,Q)J_{m,\alpha }(R,S)\end{array}$$

for all $P,Q\in {\mathrm{\Gamma }}_n^o$ and $R,S\in {\mathrm{\Gamma }}_m^o$ where $\lambda =2^{\alpha -1}-1$ and

$$P\ast R=(p_1{r}_1,p_1{r}_2,\dots ,p_1{r}_m,p_2{r}_1,\dots ,p_2{r}_m,\dots ,p_n{r}_m).$$

In view of this, symmetrically compositive statistical distance measures are defined as follows. A sequence of symmetric measures $\{{\mu }_n\}$ is said to be symmetrically compositive if for some $\lambda \in \mathbb{R}$,

$$\begin{array}{r}{\mu }_{nm}(P\star R,Q\star S)+{\mu }_{nm}(P\star S,Q\star R)\\ =2{\mu }_n(P,Q)+2{\mu }_m(R,S)+\lambda {\mu }_n(P,Q){\mu }_m(R,S)\end{array}$$

for all $P,Q\in {\mathrm{\Gamma }}_n^o$, $S,R\in {\mathrm{\Gamma }}_m^o$, where

$$P\ast R=(p_1{r}_1,p_1{r}_2,\dots ,p_1{r}_m,p_2{r}_1,\dots ,p_2{r}_m,\dots ,p_n{r}_m).$$

Chung, Kannappan, Ng and Sahoo [1] characterized symmetrically compositive sum-form distance measures with a measurable generating function. The following functional equation:

(FE) $f(pr,qs)+f(ps,qr)=f(p,q)f(r,s)$

holding for all $p,q,r,s\in I$ was instrumental in the characterization of symmetrically compositive sum-form distance measures. They proved the following theorem giving the general solution of this functional equation (FE).

Suppose $f:I^2\to \mathbb{R}$ satisfies the functional equation (FE), that is,

$$f(pr,qs)+f(ps,qr)=f(p,q)f(r,s)$$

for all $p,q,r,s\in I$. Then

$$f(p,q)=M_1(p)M_2(q)+M_1(q)M_2(p),$$

(1.3)

where $M_1,M_2:\mathbb{R}\to \mathbb{C}$ are multiplicative functions. Further, either $M_1$ and $M_2$ are both real or $M_2$ is the complex conjugate of $M_1$. The converse is also true.

The stability of the functional equation (FE), as well as the four generalizations of (FE), namely,

($F{E}_{fg}$) $f(pr,qs)+f(ps,qr)=f(p,q)g(r,s)$,

($F{E}_{gf}$) $f(pr,qs)+f(ps,qr)=g(p,q)f(r,s)$,

($F{E}_{gg}$) $f(pr,qs)+f(ps,qr)=g(p,q)g(r,s)$,

($F{E}_{gh}$) $f(pr,qs)+f(ps,qr)=g(p,q)h(r,s)$

for all $p,q,r,s\in G$, were studied by Kim and Sahoo in [3, 4]. For other functional equations similar to (FE), the interested reader should refer to [5–8], and [9].

The present work continues the study for the stability of the Pexider type functional equation of (FE) added a cocycle property to the conditions in the results [3, 4]. These functional equations arise in the characterization of symmetrically compositive sum-form distance measures, products of some multiplicative functions. In reduction, they can be represented as a (hyperbolic) cosine (sine, trigonometric) functional equation, exponential, and Jensen functional equation, respectively.

Tabor [10] investigated the cocycle property. The definition of cocycle as follows:

Definition 1 A function $\theta :G^2\to \mathbb{R}$ is a cocycle if it satisfies the equation

$$\theta (a,bc)\theta (b,c)=\theta (ab,c)\theta (a,b),\mathrm{\forall }a,b,c\in G.$$

For example, if $F(x,y)=\frac{f(x)f(y)}{f(xy)}$ for a function $f:\mathbb{R}\to {\mathbb{R}}_{+}$, then F is a cocycle. Also if $\theta (x,y)=ln(x)ln(y)$ for a function $\theta :{{\mathbb{R}}_{+}}^2\to (\mathbb{R},+)$, then θ is a cocycle, that is, $\theta (a,bc)+\theta (b,c)=\theta (ab,c)+\theta (a,b)$, and in this case, it is well known that $\theta (x,y)$ is represented by $B(x,y)+M(xy)-M(x)-M(y)$ where B is an arbitrary skew-symmetric biadditive function and M is some function [11]. If $\theta (x,y)=a^{ln(x)ln(y)}$, then $\theta :{\mathbb{R}}_{+}^2\to (\mathbb{R},\cdot )$ is a cocycle and in this case, $\theta (x,y)$ is represented by $e^{B(x,y)}{e}^{M(xy)-M(x)-M(y)}$.

Let us consider the generalized characterization of a symmetrically compositive sum form related to distance measures with a cocycle:

($CDM$) $f(pr,qs)+f(ps,qr)=\theta (pq,rs)f(p,q)f(r,s)$

for all $p,q,r,s\in G$ and where f, θ are functionals on $G^2$, which can be represented as exponential functional equation in reduction.

In fact, if $f(x,y)=\frac{1}{x}+\frac{1}{y}$, then $f(pr,qs)+f(ps,qr)=f(p,q)f(r,s)$, and also if $f(x,y)=a^{lnxy}$, and $\theta (x,y)=2$ then f, θ satisfy the equation $f(pr,qs)+f(ps,qr)=\theta (pq,rs)f(p,q)f(r,s)$.

This paper aims to investigate the superstability of four generalized functional equations of ($CDM$), namely, as well as that of the following type functional equations:

($G{M}_{fffg}$) $f(pr,qs)+f(ps,qr)=\theta (pq,rs)f(p,q)g(r,s)$,

($G{M}_{ffgf}$) $f(pr,qs)+f(ps,qr)=\theta (pq,rs)g(p,q)f(r,s)$,

($G{M}_{ffgg}$) $f(pr,qs)+f(ps,qr)=\theta (pq,rs)g(p,q)g(r,s)$,

($G{M}_{ffgh}$) $f(pr,qs)+f(ps,qr)=\theta (pq,rs)g(p,q)h(r,s)$.

2 Superstability of the equations

In this section, we investigate the superstability of ($CDM$) and four generalized functional equations ($G{M}_{fffg}$), ($G{M}_{ffgf}$), ($G{M}_{ffgg}$), and ($G{M}_{ffgh}$).

Theorem 1 Let $f,g:G^2\to \mathbb{R}$, $\varphi :G^2\to {\mathbb{R}}_{+}$ be functions and a function $\theta :G^2\to {\mathbb{R}}_k$ be a cocycle satisfying

$$|f(pr,qs)+f(ps,qr)-\theta (pq,rs)g(p,q)h(r,s)|\le \varphi (r,s)\mathrm{\forall }p,q,r,s\in G.$$

(2.1)

and $|f(p,q)-g(p,q)|\le M$ for all $p,q\in G$ and some constant M.

Then either g is bounded or h satisfies ($CDM$).

Proof Let g be an unbounded solution of inequality (2.1). Then there exists a sequence $\{(x_n,y_n)|n\in N\}$ in $G^2$ such that $0\ne |g(x_n,y_n)|\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$.

Letting $p=x_n$, $q=y_n$ in (2.1) and dividing by $|\theta (x_n{y}_n,rs)g(x_n,y_n)|$, we have

$$|\frac{f(x_{n}r,y_{n}s)+f(x_{n}s,y_{n}r)}{\theta (x_n{y}_n,rs)g(x_n,y_n)}-h(r,s)|\le \frac{\varphi (r,s)}{k|g(x_n,y_n)|}.$$

Passing to the limit as $n\to \mathrm{\infty }$, we obtain

$$h(r,s)=\underset{n\to \mathrm{\infty }}{lim}\frac{f(x_{n}r,y_{n}s)+f(x_{n}s,y_{n}r)}{\theta (x_n{y}_n,rs)g(x_n,y_n)}.$$

(2.2)

Letting $p=x_{n}p$, $q=y_{n}q$ in (2.1) and dividing by $|g(x_n,y_n)|$, we have

$$\begin{array}{r}|\frac{f(x_{n}pr,y_{n}qs)+f(x_{n}ps,y_{n}qr)}{g(x_n,y_n)}-\frac{\theta (x_{n}p{y}_{n}q,rs)g(x_{n}p,y_{n}q)}{g(x_n,y_n)}h(r,s)|\\ \le \frac{\varphi (r,s)}{|g(x_n,y_n)|}\to 0\end{array}$$

(2.3)

as $n\to \mathrm{\infty }$.

Letting $p=x_{n}q$, $q=y_{n}p$ in (2.1) and dividing by $|g(x_n,y_n)|$, we have

$$\begin{array}{r}|\frac{f(x_{n}qr,y_{n}ps)+f(x_{n}qs,y_{n}pr)}{g(x_n,y_n)}-\frac{\theta (x_{n}q{y}_{n}p,rs)g(x_{n}q,y_{n}p)}{g(x_n,y_n)}h(r,s)|\\ \le \frac{\varphi (r,s)}{|g(x_n,y_n)|}\to 0\end{array}$$

(2.4)

as $n\to \mathrm{\infty }$.

Note that for any a, b, c in G, $\theta (ba,c)\theta (b,a)=\theta (b,ac)\theta (a,c)$ by the definition of the cocycle. Letting $pq=a$, $x_n{y}_n=b$, and $rs=c$ we have

$$\frac{\theta (x_n{y}_{n}pq,rs)\theta (x_n{y}_n,pq)}{\theta (x_n{y}_n,pqrs)}=\theta (pq,rs)$$

for any p, q, r, s, $x_n$, $y_n$ in G. Thus, from (2.2), (2.3), and (2.4), we obtain

$$\begin{array}{r}|h(pr,qs)+h(ps,qr)-\theta (pq,rs)h(p,q)h(r,s)|\\ =\underset{n\to \mathrm{\infty }}{lim}|\frac{f(x_{n}pr,y_{n}qs)+f(x_{n}qs,y_{n}pr)+f(x_{n}ps,y_{n}qr)+f(x_{n}qr,y_{n}ps)}{\theta (x_n{y}_n,prqs)g(x_n,y_n)}\\ -\theta (pq,rs)h(p,q)h(r,s)|\\ \le \underset{n\to \mathrm{\infty }}{lim}\left|\frac{1}{\theta (x_n{y}_n,prqs)}\right|\cdot |\frac{f(x_{n}pr,y_{n}qs)+f(x_{n}ps,y_{n}qr)}{g(x_n,y_n)}\\ -\frac{\theta (x_{n}p{y}_{n}q,rs)g(x_{n}p,y_{n}q)h(r,s)}{g(x_n,y_n)}|\\ +\underset{n\to \mathrm{\infty }}{lim}\left|\frac{1}{\theta (x_n{y}_n,prqs)}\right|\cdot |\frac{f(x_{n}qr,y_{n}ps)+f(x_{n}qs,y_{n}pr)}{g(x_n,y_n)}\\ -\frac{\theta (x_{n}q{y}_{n}p,rs)g(x_{n}q,y_{n}p)h(r,s)}{g(x_n,y_n)}|\\ +|h(r,s)|\underset{n\to \mathrm{\infty }}{lim}|\frac{\theta (x_n{y}_{n}pq,rs)\theta (x_n{y}_n,pq)}{\theta (x_n{y}_n,pqrs)}\cdot \frac{g(x_{n}p,y_{n}q)+g(x_{n}q,y_{n}p)}{\theta (x_n{y}_n,pq)g(x_n,y_n)}\\ -\theta (pq,rs)h(p,q)|\\ \le h(r,s)\theta (pq,rs)\underset{n\to \mathrm{\infty }}{lim}|\frac{f(x_{n}p,y_{n}q)+f(x_{n}q,y_{n}p)}{\theta (x_n{y}_n,pq)g(x_n,y_n)}\\ +\frac{(g-f)(x_{n}p,y_{n}q)+(g-f)(x_{n}q,y_{n}p)}{\theta (x_n{y}_n,pq)g(x_n,y_n)}-h(p,q)|\\ \le h(r,s)\theta (pq,rs)\underset{n\to \mathrm{\infty }}{lim}\left|\frac{2M}{kg(x_n,y_n)}\right|\\ +h(r,s)\theta (pq,rs)\underset{n\to \mathrm{\infty }}{lim}|\frac{f(x_{n}p,y_{n}q)+f(x_{n}q,y_{n}p)}{\theta (x_n{y}_n,pq)g(x_n,y_n)}-h(p,q)|\\ =0.\end{array}$$

 □

Theorem 2 Let $f,g:G^2\to \mathbb{R}$, $\varphi :G^2\to {\mathbb{R}}_{+}$ be functions and a function $\theta :G^2\to {\mathbb{R}}_k$ be a cocycle satisfying

$$|f(pr,qs)+f(ps,qr)-\theta (pq,rs)g(p,q)h(r,s)|\le \varphi (p,q)\mathrm{\forall }p,q,r,s\in G,$$

(2.5)

and $|f(p,q)-h(p,q)|\le M$ for all $p,q\in G$ and some constant M.

Then either h is bounded or g satisfies ($CDM$).

Proof For h to be an unbounded solution of inequality (2.5), we can choose a sequence $\{(x_n,y_n)|n\in N\}$ in $G^2$ such that $0\ne |h(x_n,y_n)|\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$.

Letting $r=x_n$, $s=y_n$ in (2.5) and dividing by $|\theta (pq,x_n{y}_n)h(x_n,y_n)|$, we have

$$|\frac{f(p{x}_n,q{y}_n)+f(p{y}_n,q{x}_n)}{\theta (pq,x_n{y}_n)h(x_n,y_n)}-g(p,q)|\le \frac{\varphi (p,q)}{k|h(x_n,y_n)|}.$$

Passing to the limit as $n\to \mathrm{\infty }$, we obtain

$$g(p,q)=\underset{n\to \mathrm{\infty }}{lim}\frac{f(p{x}_n,q{y}_n)+f(p{y}_n,q{x}_n)}{\theta (pq,x_n{y}_n)h(x_n,y_n)}.$$

(2.6)

Replacing $r=r{x}_n$, $s=s{y}_n$ in (2.5) and dividing by $|h(x_n,y_n)|$, we have

$$\begin{array}{r}|\frac{f(pr{x}_n,qs{y}_n)+f(ps{y}_n,qr{x}_n)}{h(x_n,y_n)}-\theta (pq,r{x}_{n}s{y}_n)g(p,q)\frac{h(r{x}_n,s{y}_n)}{h(x_n,y_n)}|\\ \le \frac{\varphi (p,q)}{|h(x_n,y_n)|}\to 0\end{array}$$

(2.7)

as $n\to \mathrm{\infty }$.

Replacing $r=r{y}_n$, $s=s{x}_n$ in (2.5) and dividing by $|h(x_n,y_n)|$, we have

$$\begin{array}{r}|\frac{f(pr{y}_n,qs{x}_n)+f(ps{x}_n,qr{y}_n)}{h(x_n,y_n)}-g(p,q)\theta (pq,r{y}_{n}s{x}_n)\frac{h(r{y}_n,s{x}_n)}{h(x_n,y_n)}|\\ \le \frac{\varphi (p,q)}{|h(x_n,y_n)|}\to 0\end{array}$$

(2.8)

as $n\to \mathrm{\infty }$.

Thus from (2.6), (2.7), and (2.8), we obtain

$$\begin{array}{r}|g(pr,qs)+g(ps,qr)-\theta (pq,rs)g(p,q)g(r,s)|\\ =\underset{n\to \mathrm{\infty }}{lim}|\frac{f(pr{x}_n,qs{y}_n)+f(pr{y}_n,qs{x}_n)+f(ps{x}_n,qr{y}_n)+f(ps{y}_n,qr{x}_n)}{\theta (prqs,x_n{y}_n)h(x_n,y_n)}\\ -\theta (pq,rs)g(p,q)g(r,s)|\\ \le \underset{n\to \mathrm{\infty }}{lim}\left|\frac{1}{\theta (pqrs,x_n{y}_n)}\right|\cdot |\frac{f(pr{x}_n,qs{y}_n)+f(ps{y}_n,qr{x}_n)}{h(x_n,y_n)}\\ -g(p,q)\theta (pq,r{x}_{n}s{y}_n)\frac{h(r{x}_n,s{y}_n)}{h(x_n,y_n)}|\\ +\underset{n\to \mathrm{\infty }}{lim}\left|\frac{1}{\theta (pqrs,x_n{y}_n)}\right|\cdot |\frac{f(pr{y}_n,qs{x}_n)+f(ps{x}_n,qr{y}_n)}{h(x_n,y_n)}\\ -g(p,q)\theta (pq,r{y}_{n}s{x}_n)\frac{h(r{y}_n,s{x}_n)}{h(x_n,y_n)}|\\ +|g(p,q)|\underset{n\to \mathrm{\infty }}{lim}|\frac{\theta (pq,r{x}_{n}s{y}_n)\theta (rs,x_n{y}_n)}{\theta (pqrs,x_n{y}_n)}\cdot \frac{h(r{x}_n,s{y}_n)+h(r{y}_n,s{x}_n)}{\theta (rs,x_n{y}_n)h(x_n{y}_n)}\\ -\theta (pq,rs)g(r,s)|\\ =|g(p,q)|\theta (pq,rs)\underset{n\to \mathrm{\infty }}{lim}|\frac{(h-f)(r{x}_n,s{y}_n)+(h-f)(r{y}_n,s{x}_n)}{\theta (rs,x_n{y}_n)h(x_n,y_n)}\\ +\frac{f(r{x}_n,s{y}_n)+f(r{y}_n,s{x}_n)}{\theta (rs,x_n{y}_n)h(x_n,y_n)}-g(r,s)|\\ \le |g(p,q)|\theta (pq,rs)\frac{2M}{k|h(x_n,y_n)|}\\ +|g(p,q)|\theta (pq,rs)\underset{n\to \mathrm{\infty }}{lim}|\frac{f(r{x}_n,s{y}_n)+f(r{y}_n,s{x}_n)}{\theta (rs,x_n{y}_n)h(x_n,y_n)}-g(r,s)|\\ =0.\end{array}$$

 □

Corollary 1 Let $f,g:G^2\to \mathbb{R}$, $\varphi :G^2\to {\mathbb{R}}_{+}$ be functions and a function $\theta :G^2\to {\mathbb{R}}_k$ be a cocycle satisfying

$$|f(pr,qs)+f(ps,qr)-\theta (pq,rs)g(p,q)g(r,s)|\le \varphi (p,q)\mathit{\text{ or }}\varphi (r,s)$$

for any $p,q,r,s\in G$ and $|f(p,q)-g(p,q)|\le M$ for all $p,q\in G$ and some constant M. Then either g is bounded or g satisfies ($CDM$).

Corollary 2 Let $f,g:G^2\to \mathbb{R}$, $\varphi :G^2\to {\mathbb{R}}_{+}$ be functions and a function $\theta :G^2\to {\mathbb{R}}_k$ be a cocycle satisfying

$$|f(pr,qs)+f(ps,qr)-\theta (pq,rs)f(p,q)g(r,s)|\le \varphi (p,q)$$

for any $p,q,r,s\in G$. Then either g is bounded, or f satisfies ($CDM$) and also f and g satisfy ($G{M}_{fffg}$).

Corollary 3 Let $f,g:G^2\to \mathbb{R}$, $\varphi :G^2\to {\mathbb{R}}_{+}$ be functions and a function $\theta :G^2\to {\mathbb{R}}_k$ be a cocycle satisfying

$$|f(pr,qs)+f(ps,qr)-\theta (pq,rs)f(p,q)g(r,s)|\le \varphi (r,s)$$

for any $p,q,r,s\in G$. Then either f is bounded, or g satisfies ($CDM$) and also g and f satisfy

($G{M}_{gggf}$) $g(pr,qs)+g(ps,qr)-\theta (pq,rs)g(p,q)f(r,s)$.

Corollary 4 Let $f,g:G^2\to \mathbb{R}$, $\varphi :G^2\to {\mathbb{R}}_{+}$ be functions and a function $\theta :G^2\to {\mathbb{R}}_k$ be a cocycle satisfying

$$|f(pr,qs)+f(ps,qr)-\theta (pq,rs)g(p,q)f(r,s)|\le \varphi (p,q)\mathrm{\forall }p,q,r,s\in G$$

for any $p,q,r,s\in G$. Then either f is bounded, or g satisfies ($CDM$) and also f and g satisfy ($G{M}_{gggf}$).

Corollary 5 Let $f,g:G^2\to \mathbb{R}$, $\varphi :G^2\to {\mathbb{R}}_{+}$ be functions and a function $\theta :G^2\to {\mathbb{R}}_k$ be a cocycle satisfying

$$|f(pr,qs)+f(ps,qr)-\theta (pq,rs)g(p,q)f(r,s)|\le \varphi (r,s)\mathrm{\forall }p,q,r,s\in G$$

for any $p,q,r,s\in G$. Then either g is bounded, or f satisfies ($CDM$) and also f and g satisfy ($G{M}_{fffg}$).

Corollary 6 Let $f,g:G^2\to \mathbb{R}$, $\varphi :G^2\to {\mathbb{R}}_{+}$ be functions and a function $\theta :G^2\to {\mathbb{R}}_k$ be a cocycle satisfying

$$|f(pr,qs)+f(ps,qr)-\theta (pq,rs)g(p,q)f(r,s)|\le \varphi (p,q)\mathrm{\forall }p,q,r,s\in G$$

for any $p,q,r,s\in G$. Then either f is bounded, or g satisfies ($CDM$) and also f and g satisfy ($G{M}_{gggf}$).

Corollary 7 Let $k>0$ and $f,g:G^2\to \mathbb{R}$, $\varphi :G^2\to {\mathbb{R}}_{+}$ be functions satisfying

$$|f(pr,qs)+f(ps,qr)-k^{ln(pq)ln(rs)}f(p,q)f(r,s)|\le \varphi (p,q)\mathit{\text{ or }}\varphi (r,s)$$

for any $p,q,r,s\in G$. Then either f is bounded or f satisfies the following equation:

$$f(pr,qs)+f(ps,qr)=k^{ln(pq)ln(rs)}f(p,q)f(r,s).$$

Corollary 8 Let $f,g:G^2\to \mathbb{R}$, $\varphi :G^2\to {\mathbb{R}}_{+}$ be functions satisfying

$$|f(pr,qs)+f(ps,qr)-f(p,q)f(r,s)|\le \varphi (p,q)\mathit{\text{ or }}\varphi (r,s)$$

for any $p,q,r,s\in G$. Then either f is bounded or f satisfies (FE).

Theorem 3 Let $f,g:G^2\to \mathbb{R}$, $\varphi :G^2\to {\mathbb{R}}_{+}$ be functions and a function $\theta :G^2\to {\mathbb{R}}_k$ be a cocycle satisfying

$$|f(pr,qs)+f(ps,qr)-\theta (pq,rs)f(p,q)g(r,s)|\le \epsilon$$

for any $p,q,r,s\in G$. Then f (or g) is bounded, or f and g satisfy ($CDM$) and also f, g, θ satisfy ($G{M}_{fffg}$).

Proof Replacing $g(p,q)$ by $f(p,q)$ and $h(r,s)$ by $g(r,s)$ for all $p,q,r,s\in G$ in Theorem 1, we find that f is bounded or g satisfies ($CDM$). Note that f is bounded iff g is bounded. Namely, for all $p,q,r,s\in G$

$$|g(r,s)|\le \frac{\epsilon +f(pr,qs)+f(ps,qr)}{k|f(p,q)|}.$$

Let g be unbounded. Then f is unbounded by a similar method to the proof of Theorem 1; g satisfies ($CDM$). Now by a similar method to the calculation in Theorem 1 with the unboundedness of g, we have

$$f(p,q)=\underset{n\to \mathrm{\infty }}{lim}\frac{f(p{x}_n,q{y}_n)+f(p{y}_n,q{x}_n)}{\theta (pq,x_n{y}_n)g(x_n,y_n)}$$

for any $r,s,x_n,y_n\in G$. Since g satisfies ($CDM$), we have

$$\begin{array}{r}|f(pr,qs)+f(ps,qr)-\theta (pq,rs)f(p,q)g(r,s)|\\ =\underset{n\to \mathrm{\infty }}{lim}|\frac{f(pr{x}_n,qs{y}_n)+f(pr{y}_n,qs{x}_n)+f(ps{x}_n,qr{y}_n)+f(ps{y}_n,qr{x}_n)}{\theta (prqs,x_n{y}_n)g(x_n,y_n)}\\ -\theta (pq,rs)f(p,q)g(r,s)|\\ \le \underset{n\to \mathrm{\infty }}{lim}\left|\frac{1}{\theta (pqrs,x_n{y}_n)}\right|\cdot |\frac{f(pr{x}_n,qs{y}_n)+f(ps{y}_n,qr{x}_n)}{g(x_n,y_n)}\\ -f(p,q)\theta (pq,r{x}_{n}s{y}_n)\frac{g(r{x}_n,s{y}_n)}{g(x_n,y_n)}|\\ +\underset{n\to \mathrm{\infty }}{lim}\left|\frac{1}{\theta (pqrs,x_n{y}_n)}\right|\cdot |\frac{f(pr{y}_n,qs{x}_n)+f(ps{x}_n,qr{y}_n)}{g(x_n,y_n)}\\ -f(p,q)\theta (pq,r{y}_{n}s{x}_n)\frac{g(r{y}_n,s{x}_n)}{g(x_n,y_n)}|\\ +|f(p,q)|\underset{n\to \mathrm{\infty }}{lim}|\frac{\theta (pq,r{x}_{n}s{y}_n)\theta (rs,x_n{y}_n)}{\theta (pqrs,x_n{y}_n)}\cdot \frac{g(r{x}_n,s{y}_n)+g(r{y}_n,s{x}_n)}{\theta (rs,x_n{y}_n)g(x_n{y}_n)}\\ -\theta (pq,rs)g(r,s)|\\ =|f(p,q)|\underset{n\to \mathrm{\infty }}{lim}|\frac{\theta (pq,r{x}_{n}s{y}_n)\theta (rs,x_n{y}_n)}{\theta (pqrs,x_n{y}_n)}\cdot \frac{g(r{x}_n,s{y}_n)+g(r{y}_n,s{x}_n)}{\theta (rs,x_n{y}_n)g(x_n{y}_n)}\\ -\theta (pq,rs)g(r,s)|\\ =|f(p,q)||\theta (pq,rs)g(r,s)-\theta (pq,rs)g(r,s)|=0.\end{array}$$

Thus f and g imply the required ($G{M}_{fffg}$). The same procedure implies that the above inequalities change to

$$\begin{array}{r}|f(pr,qs)+f(ps,qr)-\theta (pq,rs)f(p,q)f(r,s)|\\ \le |f(p,q)|\underset{n\to \mathrm{\infty }}{lim}|\frac{\theta (pq,r{x}_{n}s{y}_n)\theta (rs,x_n{y}_n)}{\theta (pqrs,x_n{y}_n)}\cdot \frac{f(r{x}_n,s{y}_n)+f(r{y}_n,s{x}_n)}{\theta (rs,x_n{y}_n)g(x_n{y}_n)}-\theta (pq,rs)f(r,s)|\\ =|f(p,q)||\theta (pq,rs)f(r,s)-\theta (pq,rs)f(r,s)|=0,\end{array}$$

as desired. □

The proof of the following theorem is the same procedure as in the proof of Theorem 3.

Theorem 4 Let $f,g:G^2\to \mathbb{R}$, $\varphi :G^2\to {\mathbb{R}}_{+}$ be functions and a function $\theta :G^2\to {\mathbb{R}}_k$ be a cocycle satisfying

$$|f(pr,qs)+f(ps,qr)-\theta (pq,rs)g(p,q)f(r,s)|\le \epsilon$$

for any $p,q,r,s\in G$. Then f (or g) is bounded, or f and g satisfy ($CDM$) and also f, g, θ satisfy ($G{M}_{fffg}$).

Example 1 Let

$$f(x,y)=a^{lnxy}+\frac{\epsilon }{2},g(x,y)=a^{lnxy},\theta (x,y)=2.$$

Then we have

$$|f(p,q)-g(p,q)|\le \frac{\epsilon }{2}$$

and

$$\begin{array}{r}|f(pr,qs)+f(ps,qr)-\theta (pq,rs)g(p,q)g(r,s)|\\ =|a^{lnprqs}+a^{lnpsqr}+\epsilon -2a^{lnpq}{a}^{lnrs}|\\ =\epsilon .\end{array}$$

Thus g satisfies ($CDM$). But f, g, θ being nonzero functions do not satisfy ($G{M}_{ffgg}$).

Let ($S;\diamond$) and ($\tilde{S};\diamond$) be a semigroup and a group with semigroup operation ⋄, respectively.

Theorem 5 Let $f,g,h:S^2,{\tilde{S}}^2\to \mathbb{R}$ and $\varphi :S^2,{\tilde{S}}^2\to \mathbb{R}$ be a nonzero function satisfying

$$\begin{array}{r}|f(p\diamond r,q\diamond s)+f(p\diamond s,q\diamond r)-\theta (pq,rs)f(p,q)g(r,s)|\\ \le \{\begin{array}{ll}\phantom{i}(\mathrm{i})\varphi (r,s)& \mathrm{\forall }p,q,r,s\in \tilde{S},\\ (\mathrm{ii})\varphi (p,q)& \mathrm{\forall }p,q,r,s\in S.\end{array}\end{array}$$

(2.9)

• (a) In case (i), let $|f(p,q)-g(p,q)|\le M$ for all $p,q\in S$ and some constant M.

  Then either g is bounded or h satisfies ($CDM$).

• (b) In case (ii), let $|f(p,q)-h(p,q)|\le M$ for all $p,q\in G$ and some constant M.

Then either h is bounded or g satisfies ($CDM$).