Superstability of the functional equation related to distance measures

Abstract

The functional equation related to a distance measure

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

can be generalized as follows:

$$\sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=f(P)f(Q), $$

where f is an information measure, P and Q are in the set of n-ary discrete complete probability, and $\sigma_{i}$ is a permutation for each $i=0, 1, \ldots, n-1$.

In this paper, we investigate the superstability of the above functional equation and also four generalized functional equations:

$$\begin{aligned}& \sum_{i=1}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=f(P)g(Q),\qquad \sum _{i=0}^{n-1} f\bigl(P\cdot\sigma_{i} (Q) \bigr)=g(P)f(Q), \\& \sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=g(P)g(Q),\qquad \sum _{i=0}^{n-1} f\bigl(P\cdot\sigma_{i} (Q) \bigr)=g(P)h(Q). \end{aligned}$$

1 Introduction

Baker et al. in [1] introduced that if f satisfies the stability inequality $|E_{1}(f)-E_{2}(f)|\leq \varepsilon$, then either f is bounded or $E_{1}(f)=E_{2}(f)$. This is now frequently referred to as superstability. Baker [2] also proved the superstability of the cosine functional equation (also called the d’Alembert functional equation).

In this paper, let $(G, \cdot)$ be a commutative group and I denote the open unit interval $(0, 1)$. Also let $\mathbb{R}$ denote the set of real numbers and ${\mathbb{R}_{+}} = \{ x \in\mathbb{R} \mid x > 0 \}$ be a set of positive real numbers. Further, let

$$ \Gamma_{n}^{0} = \Biggl\{ P = ( p_{1} , p_{2} , \ldots, p_{n} ) \Bigm| 0 < p_{k} < 1 , \sum_{k=1}^{n} p_{k} = 1 \Biggr\} $$

denote the set of all n-ary discrete complete probability distributions (without zero probabilities), that is, $\Gamma_{n}^{0}$ is the class of discrete distributions on a finite set Ω of cardinality n with $n \geq2$. Almost all similarity, affinity or distance measures $\mu_{n} : \Gamma_{n}^{0} \times \Gamma_{n}^{0} \to\mathbb{R}_{+}$ that have been proposed between two discrete probability distributions can be represented in the sum-form

$$ \mu_{n} (P, Q ) = \sum_{k=1}^{n} \phi( p_{k} , q_{k} ) , $$

(1.1)

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

$$ \mu_{n} (P, Q ) = \psi \Biggl( \sum_{k=1}^{n} \phi( p_{k} , q_{k} ) \Biggr) , $$

(1.2)

where $\psi: \mathbb{R} \to\mathbb{R}_{+}$ is an increasing function on $\mathbb{R}$. 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 $\Gamma_{n}^{0}$, the symmetric divergence of degree α is defined as

$$ J_{n, {\alpha}} (P, Q) = {1 \over {{2^{\alpha-1} - 1}}} \Biggl[ \sum _{k=1}^{n} \bigl( p_{k}^{\alpha} q_{k}^{1-\alpha} + p_{k}^{1- \alpha} q_{k}^{\alpha} \bigr) - 2 \Biggr] . $$

For all $P, Q \in\Gamma_{n}^{0}$, we define the product

$$ P\cdot R = ( p_{1} r_{1} , p_{1} r_{2} , \ldots, p_{1}r_{m}, p_{2} r_{1} , \ldots, p_{2} r_{m} , \ldots, p_{n} r_{m} ). $$

In [3], Chung et al. characterized all symmetrically compositive sum-form distance measures with a measurable generating function. The following functional equation

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

(DM)

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

Suppose $f : I^{2} \to\mathbb{R}$ satisfies (DM) for all $p,q, r, s \in I$. Then

$$f(p, q) = M_{1} (p) M_{2} (q) + M_{1} (q) M_{2} (p), $$

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.

In [4] and [5], Kim (second author) and Sahoo obtained the superstability results of the equation (DM), its stability and four generalizations of (DM), namely

$$\begin{aligned}& f(pr,qs)+f(ps,qr)=f(p,q) g(r,s), \end{aligned}$$

(DMfg)

$$\begin{aligned}& f(pr,qs)+f(ps,qr)=g(p,q) f(r,s), \end{aligned}$$

(DMgf)

$$\begin{aligned}& f(pr,qs)+f(ps,qr)=g(p,q) g(r,s), \end{aligned}$$

(DMgg)

$$\begin{aligned}& f(pr,qs)+f(ps,qr)=g(p,q) h(r,s) \end{aligned}$$

(DMgh)

for all $p, q, r, s \in G$.

The above equation (DM) characterized by distance measures can be considered by characterization of a symmetrically compositive sum-form information measurable functional equation.

The functional equation (DM) can be generalized as follows. Let $f:\Gamma_{n}^{0} \rightarrow R$ be a function and

$$ \sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=f(P)f(Q) $$

(IM)

for all $P=(p_{1} , p_{2} , \ldots, p_{n} ), Q =(q_{1} , q_{2} , \ldots, q_{n} ) \in\Gamma_{n}^{0}$, where $\sigma_{i}:I^{n} \rightarrow I^{n}$ is a permutation defined by

$$\sigma_{i}(x_{1} , x_{2} , \ldots, x_{n} ):=(x_{i+1}, x_{i+2}, \ldots, x_{n} , x_{1} , x_{2} , \ldots, x_{i} ) $$

for each $i \in N$, and define $P\cdot Q :=(p_{1} q_{1}, p_{2} q_{2} , \ldots, p_{n} q_{n} )$.

For other functional equations with the information measure, the interested reader should refer to [6–9] and [10–12].

This paper aims to investigate the superstability of (IM) and also four generalized functional equations of (IM) as well as that of the following type functional equations:

$$\begin{aligned}& \sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=f(P)g(Q), \end{aligned}$$

(GIMfg)

$$\begin{aligned}& \sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=g(P)f(Q), \end{aligned}$$

(GIMgf)

$$\begin{aligned}& \sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=g(P)g(Q), \end{aligned}$$

(GIMgg)

$$\begin{aligned}& \sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=g(P)h(Q) \end{aligned}$$

(GIMgh)

for all $P, Q \in G$.

2 Results

In this section, we investigate the superstability of the pexiderized equation related to (IM).

Theorem 1

Let $f,g,h: G^{n} \rightarrow\mathbb{R}$ and $\phi: G^{n} \rightarrow \mathbb{R}_{+}$ be functions satisfying

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-g(X)h(Y) \Biggr\vert \leq\phi(Y) $$

(2.1)

and $|f(X)-g(X)| \leq M$ for all $X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}$ and some constant M. Then either g is bounded or h is a solution of (IM).

Proof

Let g be an unbounded solution of inequality (2.1). Then there exists a sequence $\{(Z_{m})=(z_{1m} , z_{2m} , \ldots, z_{nm}) \mid m \in N \}$ in $G^{n}$ such that $0 \neq|g(Z_{m})|\rightarrow\infty$ as $m\rightarrow\infty$.

Letting $X=Z_{m}$, i.e., $x_{i} =z_{im}$ in (2.1) for each i and dividing $|g(Z_{m} )|$, we have

$$ \biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{i} (Y) )}{g(Z_{m} )}-h(Y)\biggr\vert \leq\frac{\phi(Y)}{|g(Z_{m} )|}. $$

Passing to the limit as $m\rightarrow\infty$, we obtain that

$$ h(Y)=\lim_{m\to\infty}\frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma _{i} (Y))}{g(Z_{m} )}. $$

(2.2)

By (2.1), we have

$$\begin{aligned}& \biggl\vert \frac{\sum_{i=0}^{n-1} f((Z_{m} \cdot X) \cdot\sigma_{i} (Y))-g(Z_{m} \cdot X )h(Y)}{g(Z_{m} )}\biggr\vert \\& \quad \leq\frac{\phi(Y)}{|g(Z_{m} )|}\rightarrow0 \end{aligned}$$

(2.3)

as $m\rightarrow\infty$. Also, for each j,

$$\begin{aligned}& \biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{j}(X) \cdot \sigma_{i} (Y))-g(Z_{m} \cdot\sigma_{j}(X) )h(Y)}{g(Z_{m} )}\biggr\vert \\& \quad \leq\frac{\phi(Y)}{|g(Z_{m} )|}\rightarrow0 \end{aligned}$$

(2.4)

as $m\rightarrow\infty$. Note that $\sigma_{i}(X\cdot Y)=\sigma_{i}(X)\cdot\sigma _{i}(Y)$, $\sigma_{i}(\sigma_{j}(Y))=\sigma_{i+j}(Y)$, $\sigma _{n+j}(Y)=\sigma_{j}(Y)$ and $\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{1} (X) \cdot\sigma_{i+1}(Y) )=\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{1} (X)\cdot\sigma_{i}(Y) )$. Thus, from (2.2), (2.3) and (2.4), we obtain

$$\begin{aligned}& \Biggl\vert \sum_{i=0}^{n-1} h\bigl(X\cdot \sigma_{i} (Y) \bigr)-h(X)h(Y) \Biggr\vert \\& \quad =\lim_{m\to\infty}\Biggl\vert \sum _{i=0}^{n-1} \frac{\sum_{j=0}^{n-1} f(Z_{m} \cdot\sigma_{j}(X \cdot\sigma_{i} (Y)))}{g(Z_{m} )} -h(X)h(Y) \Biggr\vert \\& \quad =\lim_{m\to\infty}\biggl\vert \frac{\sum_{j=0}^{n-1} f(Z_{m} \cdot\sigma _{j}(X\cdot \sigma_{0}(Y)))+\sum_{j=0}^{n-1} f(Z_{m} \cdot\sigma _{j}(X\cdot\sigma_{1} (Y)))}{g(Z_{m} )} \\& \qquad {} + \cdots+\frac{\sum_{j=0}^{n-1} f(Z_{m} \cdot\sigma_{j}(X\cdot \sigma_{n-1} (Y)))}{g(Z_{m} )} -h(X)h(Y) \biggr\vert \\& \quad =\lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma _{0}(X\cdot\sigma_{i}(Y)))+\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma _{1}(X\cdot\sigma_{i} (Y)))}{g(Z_{m} )} \\& \qquad {} + \cdots+\frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{n-1}(X\cdot \sigma_{i} (Y)))}{g(Z_{m} )} -h(X)h(Y) \biggr\vert \\& \quad =\lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma _{0}(X)\cdot\sigma_{i}(Y))+\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma _{1}(X)\cdot\sigma_{i} (Y))}{g(Z_{m} )} \\& \qquad {} + \cdots+\frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{n-1}(X)\cdot \sigma_{i} (Y))}{g(Z_{m} )} -h(X)h(Y) \biggr\vert \\& \quad \leq\lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot \sigma_{0}(X)\cdot\sigma_{i} (Y))-g(Z_{m} \cdot\sigma _{0}(X))h(Y)}{g(Z_{m} )}\biggr\vert \\& \qquad {} +\lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot \sigma_{1}(X)\cdot\sigma_{i} (Y))-g(Z_{m} \cdot\sigma _{1}(X))h(Y)}{g(Z_{m} )}\biggr\vert \\& \qquad {} + \cdots+\lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{n-1}(X)\cdot\sigma_{i} (Y))-g(Z_{m} \cdot\sigma _{n-1}(X))h(Y)}{g(Z_{m} )} \biggr\vert \\& \qquad {} + \lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot \sigma_{i}(X))\cdot h(Y)}{g(Z_{m} )}-h(X)h(Y) \biggr\vert \\& \qquad {} + \lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} (g-f)(Z_{m} \cdot \sigma_{i}(X))\cdot h(Y)}{g(Z_{m} )}\biggr\vert \\& \quad = \bigl\vert h(X)h(Y)-h(X)h(Y)\bigr\vert =0. \end{aligned}$$

 □

Theorem 2

Let $f,g,h: G^{n} \rightarrow\mathbb{R}$ and $\phi: G^{n} \rightarrow \mathbb{R}_{+}$ be functions satisfying

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-g(X)h(Y) \Biggr\vert \leq\phi(X) $$

(2.5)

and $|f(X)-h(X)| \leq M$ for all $X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}$ and some constant M. Then either h is bounded or g is a solution of (IM).

Proof

Assume that there exists a sequence $\{(Z_{m})=(z_{1m} , z_{2m} , \ldots, z_{nm}) \mid m \in N \}$ in $G^{n}$ such that $\lim_{m to \infty} |h(Z_{m})|=\infty$ with $|h(Z_{m})|\neq0$ for each m.

Letting $Y=Z_{m}$, i.e., $x_{i} =z_{im}$ in (2.5) for each i and dividing $|h(Z_{m} )|$, we have

$$ \biggl\vert \frac{\sum_{i=0}^{n-1} f(X \cdot\sigma_{i} (Z_{m} ) )}{h(Z_{m} )}-g(X )\biggr\vert \leq\frac{\phi(X)}{|h(Z_{m} )|}. $$

Passing to the limit as $m\rightarrow\infty$, we obtain that

$$ g(X)=\lim_{m\to\infty}\frac{\sum_{i=0}^{n-1} f(X \cdot\sigma_{i} (Z_{m} ))}{h(Z_{m} )}. $$

(2.6)

By (2.5), we have

$$\begin{aligned}& \biggl\vert \frac{\sum_{i=0}^{n-1} f(X \cdot\sigma_{i} (Y \cdot Z_{m} ))-g( X )h(Y\cdot Z_{m})}{h( Z_{m} )}\biggr\vert \\& \quad \leq\frac{\phi(X)}{|h(Z_{m} )|}\rightarrow0 \end{aligned}$$

(2.7)

as $m\rightarrow\infty$. Also, for each j,

$$\begin{aligned}& \biggl\vert \frac{\sum_{i=0}^{n-1} f(X \cdot\sigma_{i}(Y \cdot\sigma _{j}(Z_{m} )))-g(X) h(Y \cdot\sigma_{j}( Z_{m}))}{h(Z_{m} )}\biggr\vert \\& \quad \leq\frac{\phi(X)}{|h(Z_{m} )|}\rightarrow0 \end{aligned}$$

(2.8)

as $m\rightarrow\infty$.

By using (2.5), (2.6) and (2.7), let us go through the same procedure as in Theorem 1, then we arrive at the required result. □

Corollary 1

Let $f: G^{n} \rightarrow\mathbb{R}$ and $\phi: G^{n} \rightarrow \mathbb{R}_{+}$ be functions satisfying

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-f(X)f(Y) \Biggr\vert \leq\max \bigl\{ \phi(X),\phi(Y)\bigr\} $$

(2.9)

for all $X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}$. Then either f is bounded or f is a solution of (IM).

Proof

By Theorems 1 and 2, it is trivial. □

Corollary 2

Let $f: G^{n} \rightarrow\mathbb{R}$ and $\phi: G^{n} \rightarrow \mathbb{R}_{+}$ be functions satisfying

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-f(X)g(Y) \Biggr\vert \leq\min \bigl\{ \phi(X),\phi(Y)\bigr\} $$

(2.10)

for all $X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}$. Then either f (or g) is bounded or g satisfies (IM). And also $\{f,g\}$ satisfies (GIMfg).

Proof

By Theorem 1, we have that either f is bounded or g satisfies (IM). Also, it follows from (2.10) that

$$\bigl\vert g(Y)\bigr\vert \leq\frac{\phi(X)+\sum_{i=0}^{n-1}|f(\sigma_{i}(Y) )|}{|f(X)|}. $$

Thus if f is bounded, then g is bounded. Hence, by Theorem 1, in the case g is unbounded, g also is a solution of (IM).

Let g be unbounded. By a similar method as the calculation in Theorem 2 with the unboundedness of g, we have

$$ f(X)=\lim_{m\to\infty}\frac{\sum_{i=0}^{n-1} f(X \cdot\sigma_{i} (Z_{m} ))}{g(Z_{m} )} $$

(2.11)

for all $X, Z_{m} \in G^{n}$ and $0 \neq|g(Z_{m} )|\rightarrow\infty$ as $m\rightarrow\infty$.

From a similar calculation as that in Theorem 1 and Theorem 2, we obtain the required result. □

Corollary 3

Let $f, g: G^{n} \rightarrow\mathbb{R}$ and $\phi: G^{n} \rightarrow \mathbb{R}_{+}$ be functions satisfying

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-g(X)f(Y) \Biggr\vert \leq\min \bigl\{ \phi(X),\phi(Y)\bigr\} $$

(2.12)

for all $X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}$. Then either f (or g) is bounded or g satisfies (IM). And also $\{f,g\}$ satisfies (GIMgf).

Proof

By Theorem 2, we have that either f is bounded or g is a solution of (IM). Suppose that g be unbounded, then f is unbounded. Hence, by Theorem 2, g also is a solution of (IM). By a similar method as the calculation in Theorem 1 with the unboundedness of g, we have

$$ f(X)=\lim_{m\to\infty}\frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma _{i} (X ))}{g(Z_{m} )} $$

(2.13)

for all $X, Z_{m} \in G^{n}$ and $0 \neq|g(Z_{m} )|\rightarrow\infty$ as $m\rightarrow\infty$.

From a similar calculation as that in Corollary 2 we obtain the required result. □

Corollary 4

Let $f, g, h: G^{n} \rightarrow\mathbb{R}$ and $\phi: G^{n} \rightarrow\mathbb{R}_{+}$ be functions satisfying

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-g(X)h(Y) \Biggr\vert \leq\min \bigl\{ \phi(X),\phi(Y)\bigr\} $$

(2.14)

and $\max\{ |f(X)-g(X)|, |f(X)-h(X)|\} \leq M$ for all $X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}$ and for some M. Then either f (or g, or h) is bounded or g and h are solutions of (IM).

Corollary 5

Let $f, g: G^{n} \rightarrow\mathbb{R}$ and $\phi: G^{n} \rightarrow \mathbb{R}_{+}$ be functions satisfying

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-g(X)g(Y) \Biggr\vert \leq\min \bigl\{ \phi(X),\phi(Y)\bigr\} $$

(2.15)

and $\{ |f(X)-g(X)|\} \leq M$ for all $X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}$ and for some M. Then either f (or g) is bounded or g satisfies (IM).

3 Discussion

We consider the functional equation

$$ \sum_{i=0}^{n-1} f\bigl(X\cdot \sigma_{i} (Y) \bigr)=f(X)f(Y) $$

for all $X, Y \in G^{n}$, where $f: G^{n} \rightarrow\mathbb {R}$ is the unknown function to be determined, and $\sigma_{i} (X)=(x_{i+1},x_{i+2}, \ldots, x_{n}, x_{1} , x_{2} , \ldots, x_{i} )$. If $n=2$, the solution of the above functional equation is known on the semigroup $S=(0,1)$ when the semigroup operation is multiplication [3]. It is not known when $n\geq3$, but there is a special solution of it.

For example, let $X=(x _{1} ,x _{2} , \ldots,x _{n} )$ and $Y=(y _{1}, y _{2} , \ldots,y _{n} )$. And define $f(X)=f(x _{1}, x _{2}, \ldots,x _{n} ):= \sum_{i=1} ^{n} \frac{1}{ x _{i}}$. Then f is a solution of the above equation. Thus our results are not limited. We expect to know the general solution of it.

4 Conclusions

In the present paper we considered generalized functional equations related to distance measures and investigated the stability of them. We extended for two-variables in (DM) to n-variables in (IM). That is, the following functional equation satisfies the property of superstability

$$\sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=f(P)f(Q), $$

where f is an information measure, P and Q are in the set of n-ary discrete complete probability, and $\sigma_{i}$ is a permutation for each $i=0, 1, \ldots, n-1$.

Also the pexiderized functional equation of the above equation satisfies the property of superstability.