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Superstability of the functional equation with a cocycle related to distance measures
Journal of Inequalities and Applications volume 2014, Article number: 393 (2014)
Abstract
In this paper, we obtain the superstability of the functional equation for all , where G is an Abelian group, f a functional on , and θ a cocycle on . This functional equation is a generalized form of the functional equation , 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: , , , .
MSC:39B82, 39B52.
1 Introduction
Let be an Abelian group. Let I denote the open unit interval . Let ℝ and ℂ denote the set of real and complex numbers, respectively. Let be a set of positive real numbers and for some .
Further, let
denote the set of all n-ary discrete complete probability distributions (without zero probabilities), that is, is the class of discrete distributions on a finite set Ω of cardinality n with . 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 that have been proposed between two discrete probability distributions can be represented in the sum form
where is a real-valued function on unit square, or a monotonic transformation of the right side of (1.1), that is,
where is an increasing function on ℝ. The function ϕ is called a generating function. It is also referred to as the kernel of .
In information theory, for P and Q in , the symmetric divergence of degree α is defined as
It is easy to see that is symmetric. That is, for all . Moreover, it satisfies the composition law
for all and where and
In view of this, symmetrically compositive statistical distance measures are defined as follows. A sequence of symmetric measures is said to be symmetrically compositive if for some ,
for all , , where
Chung, Kannappan, Ng and Sahoo [1] characterized symmetrically compositive sum-form distance measures with a measurable generating function. The following functional equation:
(FE)
holding for all 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 satisfies the functional equation (FE), that is,
for all . Then
where are multiplicative functions. Further, either and are both real or is the complex conjugate of . The converse is also true.
The stability of the functional equation (FE), as well as the four generalizations of (FE), namely,
() ,
() ,
() ,
()
for all , 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 is a cocycle if it satisfies the equation
For example, if for a function , then F is a cocycle. Also if for a function , then θ is a cocycle, that is, , and in this case, it is well known that is represented by where B is an arbitrary skew-symmetric biadditive function and M is some function [11]. If , then is a cocycle and in this case, is represented by .
Let us consider the generalized characterization of a symmetrically compositive sum form related to distance measures with a cocycle:
()
for all and where f, θ are functionals on , which can be represented as exponential functional equation in reduction.
In fact, if , then , and also if , and then f, θ satisfy the equation .
This paper aims to investigate the superstability of four generalized functional equations of (), namely, as well as that of the following type functional equations:
() ,
() ,
() ,
() .
2 Superstability of the equations
In this section, we investigate the superstability of () and four generalized functional equations (), (), (), and ().
Theorem 1 Let , be functions and a function be a cocycle satisfying
and for all and some constant M.
Then either g is bounded or h satisfies ().
Proof Let g be an unbounded solution of inequality (2.1). Then there exists a sequence in such that as .
Letting , in (2.1) and dividing by , we have
Passing to the limit as , we obtain
Letting , in (2.1) and dividing by , we have
as .
Letting , in (2.1) and dividing by , we have
as .
Note that for any a, b, c in G, by the definition of the cocycle. Letting , , and we have
for any p, q, r, s, , in G. Thus, from (2.2), (2.3), and (2.4), we obtain
□
Theorem 2 Let , be functions and a function be a cocycle satisfying
and for all and some constant M.
Then either h is bounded or g satisfies ().
Proof For h to be an unbounded solution of inequality (2.5), we can choose a sequence in such that as .
Letting , in (2.5) and dividing by , we have
Passing to the limit as , we obtain
Replacing , in (2.5) and dividing by , we have
as .
Replacing , in (2.5) and dividing by , we have
as .
Thus from (2.6), (2.7), and (2.8), we obtain
□
Corollary 1 Let , be functions and a function be a cocycle satisfying
for any and for all and some constant M. Then either g is bounded or g satisfies ().
Corollary 2 Let , be functions and a function be a cocycle satisfying
for any . Then either g is bounded, or f satisfies () and also f and g satisfy ().
Corollary 3 Let , be functions and a function be a cocycle satisfying
for any . Then either f is bounded, or g satisfies () and also g and f satisfy
() .
Corollary 4 Let , be functions and a function be a cocycle satisfying
for any . Then either f is bounded, or g satisfies () and also f and g satisfy ().
Corollary 5 Let , be functions and a function be a cocycle satisfying
for any . Then either g is bounded, or f satisfies () and also f and g satisfy ().
Corollary 6 Let , be functions and a function be a cocycle satisfying
for any . Then either f is bounded, or g satisfies () and also f and g satisfy ().
Corollary 7 Let and , be functions satisfying
for any . Then either f is bounded or f satisfies the following equation:
Corollary 8 Let , be functions satisfying
for any . Then either f is bounded or f satisfies (FE).
Theorem 3 Let , be functions and a function be a cocycle satisfying
for any . Then f (or g) is bounded, or f and g satisfy () and also f, g, θ satisfy ().
Proof Replacing by and by for all in Theorem 1, we find that f is bounded or g satisfies (). Note that f is bounded iff g is bounded. Namely, for all
Let g be unbounded. Then f is unbounded by a similar method to the proof of Theorem 1; g satisfies (). Now by a similar method to the calculation in Theorem 1 with the unboundedness of g, we have
for any . Since g satisfies (), we have
Thus f and g imply the required (). The same procedure implies that the above inequalities change to
as desired. □
The proof of the following theorem is the same procedure as in the proof of Theorem 3.
Theorem 4 Let , be functions and a function be a cocycle satisfying
for any . Then f (or g) is bounded, or f and g satisfy () and also f, g, θ satisfy ().
Example 1 Let
Then we have
and
Thus g satisfies (). But f, g, θ being nonzero functions do not satisfy ().
Let () and () be a semigroup and a group with semigroup operation ⋄, respectively.
Theorem 5 Let and be a nonzero function satisfying
-
(a)
In case (i), let for all and some constant M.
Then either g is bounded or h satisfies ().
-
(b)
In case (ii), let for all and some constant M.
Then either h is bounded or g satisfies ().
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Acknowledgements
The first author and the second author of this work were supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2013-0154000) and (Grant number: 2010-0010243), respectively.
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Lee, Y.W., Kim, G.H. Superstability of the functional equation with a cocycle related to distance measures. J Inequal Appl 2014, 393 (2014). https://doi.org/10.1186/1029-242X-2014-393
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DOI: https://doi.org/10.1186/1029-242X-2014-393