Internat. J. Math. & Math. Sci.
Vol. 9 No. 3 (1986) 545-550
545
ON THE GENERAL SOLUTION OF A
FUNCTIONAL EQUATION CONNECTED TO SUM
FORM INFORMATION MEASURES ON OPEN DOMAIN
III
PL. KANNAPPAN and P.K. SAHOO
Faculty of Mathematics
University of Waterloo
Waterloo, Ontario, N2L 3GI
(Received August 6, 1985 and in revised form April 30, 1986)
In this series, this paper is devoted to the study of a functional equation
ABSTRACT.
connected with the characterization of weighted entropy and weighted entropy of degree
8.
Here, we find the general solution of the functional equation (2) on an open domain,
without using O-probability and 1-probability.
KEY WORDS AND PHRASES.
Functional equation, weighted entropy, weighted entropy of
degree 8, open domain, sum form.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE:
1.
of
INTRODUCTION.
0
{P
Let F
n
F
0.
n
(,A,B)
Let
R
n
(Pl P2
{x
+
94A15, 94A17, 39B40.
Pn
x
<
0
> 0},
pj
where
<
IR
{AI,A
}’ (n > l) of
2
experiment is defined by Belis and Guiasu [i] as
n
HI(p’U)n
I
k=l
Pk ()
(Pl P2
Pn
where
degree
B (B
F
n
R-{I})
.
n
I
i=l
P F
n
ukP klOg Pk
(u
U
and
(i-21-) -I
u
I
2
u
n
IR
n
+
,
The weigthed entropy of
n
I uk(pk-pSk )-
k=l
satisfy the following functional equation (see Kannappan [3])
m
n
F
m
u.,v
n
m
i=l j=l
i
m
n
I qjvj I f(Pi’Ui
j=l
+
(l.i)
i
A generalization of (I) is the following:
+
l
m
I PiU--" j=l
I f(qj’vj)
i=l
[ f(piqj,uivj)
j=l
Q
n
The weighted entropy of such an
of an experiment is defined by Emptoz [2] as
HI(p,u)
n
The measures
An
is the objective probability of the event
Hn(P’U)
for all
Pk
be a probability space and let us consider an experiment that is a finite
measurable partition
P
I
i} and F
be the closure
n
k=l
is the set of real numbers. Let
i,
m
m
n
j=l
j=l
j=l
(1.2)
i
PL. KANNAPPAN and P. K. SAHO0
546
Fn,
P
where
Fm,
Q
(u
Rn+, (Vl,V2,.
,Un
u2,.
I
=
measurable solution of (1.2) for
m+,
Vm)
-{0,i}.
e,B
was given by Kannappan in [3].
1
The
In a recent
paper of Kannappan and Sahoo [4], measurable solution of a more general functional
equation than (1.2) was given using the result of this paper.
[1.2)
where
-{0 i}
and
mine the general solution of
(v
I
2.
m
Vm
v2’"
FOn"
p
>
m,n (fixed and)
-
In this paper, we deter-
FOm" (Ul,U2
Q
3
n+’
Un
on an open domain
SOLUTION OF (1.2) ON AN OPEN DOMAIN
We need the following result in this sequel.
Result 1 [5].
f,g: ]0,i[
Let
n
n
m
i=l j=l
P
for
F
0
n’
F
Q
R-{O.I}
e,8
m’
m
1
i
0
be real valued functions and satisfy
m
n
8
.= 1
j=l
m,n (> 3)
and
are arbitrary but fixed integers
Then the general solutions of (2.1) are given by
ape+bp
f(p)
A(p) +
g(p)
A’(p) +
,
6
for
a(p-pS)+c
and
A(p)+D(p)pe+dPS’]
f(p)
=
for
A’(p)+D(p)pe+c
g(p)
A,A’
are arbitrary constants,
where
a,b,c,d
A(1)
O, A’(1)+mc
0
D(pq)
D
and
8
are additive functions on
with
is a real valued function satisfying
D(p)+D(q),
p,q
]0,i[.
(2.2)
Now we proceed to determine the general solution of (1.2) on ]0,I[.
Let
]0,I[
be a real valued function and satisfy the functional equation
+
(1.2) for an arbitrary but fixed pair of positive integers m,n (> 3), for P F
f:
O,
Q
F
v
0
m
v
n
1,2
for
u,v
where
-{0, I}.
,8
with
m f
i=l
i
n
[
Putting
v
n
u,v
+
n
i
[ f(piqj,u)
u
for
u
+
and
P
1
(piqj,u)
and
Q
I Pi
i=l "
F
0
m
Ij=lq." i=l f(Pi,U)
U
,I)+
[lqSj
j=
(2.3)
f(Pi,U)
u
i=l
in "’z.), we get
n
u
FOn
n
m
+
m
m
[ Pi" I f(q
i=l
j=l
v
Putting
i=l j=l
.
1 2
i
in (2.3), we get
n
m
m f
for all
u
m
[ Pi [ f(qj,v)
v
i=l
j=l
i=l j=l
where
u.1
in (1.2), we obtain
m
uv)
[ (PiqJ
uv
?
+
Letting
m
m
[ f (qj ,I)+ [ lqj6.
i=l
1
j=
3"__
For fixed
u
f(Pi ,u)
u
+, (2.4) is
of the form
(2.1) and hence its general solutions cna be obtained from Result i.
(2.4)
GENERAL SOLUTION OF A FUNCTIONAL EQUATION
First we consider the case
f’(p, u)
where
+
a,b:
A
Al(l,u)
Result i, we have
Then from
(P,U)u+a(u)up+b(u)up 8
(2.5)
are real valued functions of
-]R
variable, with
1
8.
e
--0.
(a(uv)-a(v))
u
A
and
is additive in the first
1
Letting
(2.5) into (2.3), we get
m
n
n
m
I PiB. j=l
I Pie .__I 1qj+(b(uv)-b(u)) i=l
I q.
i=l
(2.6)
m
n
I Pi" j=llq j
(b(v)+a(u))
0
i=l
(
8,
e
Noting
547
I)
i, 6
n
n
I Pi
i=l
equating the coefficients of
and
I
i=l
Pi
(then
m
using the same for
[lq)
and
in (2.6), we get
j=
a(v),
a(uv)
b(u)
b(uv)
and
b(v)
-a(u).
From these it is easy to see that
a(u)
u,v
for all
]R
-b(v)
Now putting (2.7) into (2.5), we get
+
f(p,u)
with
A
l(l,u)
l
i=l
A
l(p,u)u+au(p-p )
m
m
[
i=
Al(Pi,Ui)U -i
i),
+
luiPi
llvjqj"
(2.9)
3-=
is additive in the first variable, by putting
1
@
that
n
v
m
i=ll
A
!IAI
1 (Piqj ,uvuv
i
i
j=l
n
Since
(2.8)
Again letting (2.8) into (1.2), we get
0.
n
(2.7)
constant
a,
u.l
1
and
1
Pi
(note
we have
m
[ A l(qj
v )v
(2.10)
0.
j=l
We let
v
I
v
Vm_ 1
v
l(qj,v)v
+ A
2
and
v
m
v’
where
v,v’
+
into (2.10) and
obtain
m-I
[
j=l
Since
A
A
0.
is additive in the first variable, and
1
Al(qm ’v)v
for all
l(qm,v’)v’
qm
]0, i[,
and
Al(l,v)
0,
we get
(2.11)
Al(’v’ )v’
v,v’
+.
From equation (2.11) it is clear that
PL. KANNAPPAN and P. K. SAHO0
548
A l(x,y)y
A
where
(2.12)
is an additive function with
f(p,u)
where
A(x)
A
A(1)
au(pe-pB),
A(p) +
]0, i[, u m
p
is an additive function on
Now using (2.12) in (2.8), we obtain
0.
A(1)
with
0
(2.13)
+
and
a
is an arbitrary
constant.
B.
e
Next we consider the case
Again the general solution of (2.4) from Result 1
can be obtained as
uA2(P,U)
f(p,u)
where
+
d:
Dl(P,u)pu
+
+
d(u)pu
is a real valued function of
]R
A2(1,u)
the first variable with
0
u
A
and
]0,1[
DI:
and
(2.14)
is an additive function in
2
lR+
(2.14) into (2.4), we get by equating the coefficient of
IR
satisfies (2 2)
l:tting
n
I
i=l
Pi
(note
I)
m
[D
Using
u
l(qj,u)-D l(qj,l)-dl]q.
in (2.15), gives
1
d
(2.15)
Hence (2.15) with
O.
I
O.
d
I
O,
by the use of the
Result 1 of [5], yields
(DI( x u)_DI( x
for all
D
1
]0,i[
x
A
and
A3(x --m’U) y
I
1
--m
Y
A
Since
x
A
3
+ A
m’ u)
(2 16)
I
3(y- --,U)m x
0
A
i
3(xy- --,U).m
and all
in (2.17), we get
ClA 3(l,u).
u
]R
(2.18)
Thus,
+
Dl(X,u)-Dl(X,l)
From (2.19), we see that
Dl(X,y)
D
1
D
1
A
3
for
(2.19)
is independent of
u,
i.e.
]0,i[
x
satisfies (2.2),
0
(2.16) reduces to
O.
D(x),
Since
(2.17)
is additive in the first variable we obtain from (2.18) that
]0,I[,
and since
i
A3(x
A3(O,u)
and using
3(x,u)
e
is an additive function in the first variable.
3
(2.2), we get
satisfies
Putting
l))x
(2.20)
also satisfies (2.2).
D
Using (2.20) in (2.14),
we get
f(p,u)
where
A
2
uA
2(p,u)+D(p)up+d(u)up
is additive with
A2(l,u)
n
(d(uv)-d(u)-d(v))i:l[
O.
(2.21)
Letting (2.21) into (2.3), we get
m
j=ll (Piqj )
0
(2.22)
GENERAL SOLUTION OF A FUNCTIONAL EQUATION
n
for all
u,v
m
ilj (piqj
Since
+
d(u) + d(v),
d(uv)
549
we obtain
0
u,v
]R
(2.23)
+
Again putting (2.21) into (1.2) and using (2.23) and (2.2), we get
n
u.
1
u v )u v
i
i
i
m
" ZiA2(qj,v
Z siPi
)v. +
(2.24)
n
m
ZlV jqj" IZiA2(Pi
+
Putting
n
m
Z Z A2(Piq
i=l j=l
and
1
n
D-i
1
u )u
i i"
in (2.4), we obtain
m
[A 2(qj,v
)v
(2.25)
0.
j=l
Note that (2.25) is of the form of (2.10) and hence by a similar argument we get
A
where
A
is additive with
A
is additive on
]R
Using (2.26) in (2.21), we obtain
A(1) --0.
A(p) + D(p)up
f(p.u)
where
(2.26)
A(q)
2(q,u)u
e
A(1)
with
(2.27)
+ d(u)up
0
and
]0,I[- 3R,
D:
d:
]R
+
IR,
are
functions satisfying (2.2) and (2.23) respectively.
Thus we have proved the following theorem.
Let
Theorem.
]0,I[
f:
arbitrary but fixed pair of
is given by (2.13) when
If
Corollary.
f
e
be a real valued function satisfying (1.2) for
0
{0,I}, P E F 0 Q
m,n (> 3) and e B
Then f
F
m
n
B and by (2.27) when e B.
+
is measurable in the Theorem then
f(p,u)
a(p-p 8)
e
B
f(p u)
buplog
p + cp u log u, e
B
and
where
a,b,c
Remark.
e
are arbitrary constants.
Because of the occurrence of the parameters
independent of
ACKNOWLEDGEMENT.
m
and
e,B
as powers,
f
is
n.
This work is partially supported by a NSERC of Canada grant.
REFERENCES
I.
BELIS, M. and GUIASU, S.- A quantitative-qualitative measure of information in
cybernetic systems, IEEE Trans. Inform. Theory IT-14, 593-594 (1968),
2.
EMPTOZ, H.; Information de type
Paris 282 A, 911-914 (1976).
B
integrant un concept d’utilite, C.R. Acad. Sci.
PL. KANNAPPAN and P. K. SAHO0
550
3.
KANNAPPAN, PL.; On some functional equations from additive and nonadditive measures
-III, Stochastica, 4, 15-22 (1980).
4.
KANNAPPAN, PL. and SAHO0, P.K.; On a functional equation connected to sum form
nonadditive information measures on open domain -III, to appear in Stochastica,
(1985).
5.
KANNAPPAN, PL. and SAHO0, P.K.; On the general solution of a functional equation
connected to sum form information measures on open domain
II, (submitted), 1985.