International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 11, November 2017)
403
Necessary and Sufficient Conditions for Solution of the Fourth
order Cauchy Difference Equation on Finite Cyclic Groups
K. Thangavelu
1, M. Pradeep
21Principal, C Kandaswami Naidu College for Men, Chennai-600102, Tamilnadu, India
2Department of Mathematics, Arignar Anna Government Arts College, Cheyyar-604407, Tamilnadu, India
Abstract— Let f: G
H be a function, where (G,.) is a group and (H,+) is an abelian group. In this paper, thefollowing Fourth Order Cauchy difference of
G x x x x x x C f x C f x C f
x C f x C f x x x x x f C f
i i i
i i
i
i i i
i
5 4 3 2 1 5
1 = 1 5
1 = 2 5
1 = 3
5 1 = 4 5
1 = 5 5 4 3 2 1 (4)
, , , , )) ( ( )) ( ( )) ( (
)) ( ( )) ( ( = ) , , , , ( :
is studied. Where
(
(
))
1 = i
n i
r
x
C
f
is defined as function ofcombination
r
at a time fromn
objects. Then Necessary and Sufficient conditions on finite cyclic groups are obtained.Keywords— Cauchy difference equation; finite cyclic group
I. INTRODUCTION
It is well known from [1] that Jenson’s functional equation
f
(
x
y
)
f
(
x
y
)
=
2
f
(
x
)
(1.1)with additional condition
f
(0)
=
0,
is equivalent toCauchy’s equation
f
(
x
y
)
=
f
(
x
)
f
(
y
)
on the realline. Let (G, .) be a group, (H,
) be an abelian group. Lete
G and 0
H denote identity elements. The study of (1.1) was extended groups for f maps G into H in [2], where the general solution for a free group H with twogenerators and
G
=
GL
n(
z
),
n
3
(see[3]). Since thefunctional equations involve Cauchy difference, which made it become much more interesting [4–7]. For a
function f: G
H, its cauchy differenceC
(m)f
, isdefined by
,
=
(0)
f
f
C
(1.2)) ( ) ( ) ( = ) ,
( 1 2 1 2 1 2
(1)
x f x f x x f x x f
C (1.3)
) , , , ( =
) , , ,
( 1 2 3 2
) ( 2 2
1 1) (
m m
m m
x x x x f C x
x x f
C
) , , , ( )
, , ,
( 2 3 2
) ( 2 3
1 ) (
m
m m
m
x x x f C x
x x f
C (1.4)
The first order cauchy difference
C
(1)f
will beabbreviated as Cf. In [9], by using the reduction formulas and relations, as given in [2,3], the general solution of third order Cauchy difference equation was provided on symmetric groups.
In this paper, we consider the following functional equation:
)) ( ( )) ( ( )) ( (
5
1 = 3 5
1 = 4 5
1 =
5 i
i i
i i
i
x C f x C f x C
f
G x x x x x x
C f x C
f i
i i
i
1 2 3 4 55
1 = 1 5
1 =
2( )) ( ( ))=0 , , , , (
(1.5)
It follows from (1.4) that (1.5) is equivalent to the vanishing fourth order cauchy difference equation
0
=
(4)f
C
The purpose of this paper is to determine thesolutions of equation (1.5). The solution of equation (1.5) will be denoted by
:
|
(1.5)
=
)
,
(
(4)
fsatisfies
H
G
f
H
G
KerC
(1.6)Remark 1 1.
KerC
(4)(
G
,
H
)
is an abelian group under the pointwise addition of functions;2.
Hom
(
G
,
H
)
KerC
(4)(
G
,
H
)
II. PROPERTIES OF SOLUTIONS
Lemma 1 Suppose that (4)( , ). H G KerC
f Then
0,
=
)
(
e
f
(2.1)e
y
or
e
x
when
y
x
Cf
(
,
)
=
0,
=
=
(2.2)e
z
or
e
y
or
e
x
when
z
y
x
f
C
(2)(
,
,
)
=
0,
=
=
=
(2.3)e u or e z or e y or e x when u
z y x f
C(3) ( , , , )=0, = = = =
(2.4)
variable each
t r w sm homomorphi a
is f
C(3) ...
(2.5)
) , , ( )
, ( )
( = )
(x nf x nC2 Cf x x nC3 C(2)f x x x
f n
) , , , (
(3)
4 C f xx xx nC
(2.6)
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 11, November 2017)
404
Proof 1 Puttingx
1=
e
in (1.5) we get (2.1).) ( ) ( ) ( ) ( )
(x2x3x4x5 f x2x3x4 f x2x3x5 f x2x4x5 f x3x4x5
f
)
(
)
(
)
(
)
(
)
(
x
2x
3x
4x
5f
x
2x
3f
x
2x
4f
x
2x
5f
x
3x
4f
)
(
)
(
)
(
)
(
)
(
x
3x
5f
x
4x
5f
x
2x
3x
4f
x
2x
3x
5f
x
2x
4x
5f
) ( ) ( ) ( ) ( ) ( )
(x3x4x5 f x2 f x3 f x4 f x5 f x2x3
f
)
(
)
(
)
(
)
(
)
(
x
2x
4f
x
2x
5f
x
3x
4f
x
3x
5f
x
4x
5f
0
=
)
(
)
(
)
(
)
(
)
(
e
f
x
2f
x
3f
x
4f
x
5f
therefore f(e)
=
0.Then from (2.1) we obtain (2.2)-(2.4)
)
(
)
(
)
(
=
)
,
(
x
e
f
xe
f
x
f
e
Cf
)
(
)
(
=
f
x
f
x
0
=
obtain
can
we
Similarly
0,
=
)
,
(
e
y
Cf
)
(
)
(
)
(
)
(
)
(
=
)
,
,
(
(2)e
f
yz
f
ez
f
ey
f
eyz
f
z
y
e
f
C
)
(
)
(
y
f
z
f
0,
=
obtain
can
we
Similarly
0,
=
)
,
,
(
(2)z
e
x
f
C
0,
=
)
,
,
(
(2)
e
y
x
f
C
0,
=
)
,
,
,
(
(3)u
z
y
e
f
C
0,
=
)
,
,
,
(
(3)u
z
e
x
f
C
0,
=
)
,
,
,
(
(3)
u
e
y
x
f
C
0.
=
)
,
,
,
(
(3)e
z
y
x
f
C
Furthermore, by the definition of
C
(3)f
, we have) ( ) ( ) ( ) ( = ) , , , ( (3)
xzu f xywu f xywz f xywzu f u z yw x f
C
f(ywzu) f(xyw) f(xz) f(xu) f(ywz) f(ywu) f(zu) f(x)
f(yw) f(z) f(u)
and
)
,
,
,
(
)
,
,
,
(
(3)(3)
u
z
w
x
f
C
u
z
y
x
f
C
)
(
)
(
)
(
)
(
)
(
)
(
=
f
xyzw
f
xyz
f
xzu
f
xyu
f
yzu
f
xy
)
(
)
(
)
(
)
(
)
(
)
(
)
(
xz
f
xu
f
yz
f
yu
f
zu
f
x
f
y
f
)
(
)
(
)
(
)
(
)
(
)
(
z
f
u
f
xwzu
f
xwz
f
xwu
f
xzu
f
)
(
)
(
)
(
)
(
)
(
)
(
wzu
f
xw
f
xz
f
xu
f
wz
f
wu
f
)
(
)
(
)
(
)
(
)
(
zu
f
x
f
w
f
z
f
u
f
One can easily check that
)
,
,
,
(
)
,
,
,
(
)
,
,
,
(
(3) (3)(3)
u
z
w
x
f
C
u
z
y
x
f
C
u
z
yw
x
f
C
=
(
,
,
,
,
)
=
0
(4)
u
z
w
y
x
f
C
)
,
,
,
(
)
,
,
,
(
)
,
,
,
(
(3) (3)(3)
u
z
w
x
f
C
u
z
y
x
f
C
u
z
yw
x
f
C
)
(
)
(
)
(
)
(
=
f
xywzu
f
xywz
f
xywu
f
xzu
)
(
)
(
)
(
)
(
)
(
ywzu
f
xyw
f
xz
f
xu
f
ywz
f
)
(
)
(
)
(
)
(
)
(
)
(
ywu
f
zu
f
x
f
yw
f
z
f
u
f
)
(
)
(
)
(
)
(
)
(
xyzw
f
xyz
f
xzu
f
xyu
f
yzu
f
)
(
)
(
)
(
)
(
)
(
)
(
xy
f
xz
f
xu
f
yz
f
yu
f
zu
f
)
(
)
(
)
(
)
(
)
(
)
(
x
f
y
f
z
f
u
f
xwzu
f
xwz
f
) ( ) ( ) ( ) ( ) ( )
(xwu f xzu f wzu f xw f xz f xu
f
)
(
)
(
)
(
)
(
)
(
)
(
)
(
wz
f
wu
f
zu
f
x
f
w
f
z
f
u
f
)
(
)
(
)
(
)
(
)
(
=
f
xywzu
f
xywz
f
xywu
f
ywzu
f
xyzu
)
(
)
(
)
(
)
(
)
(
)
(
xwzu
f
xyw
f
ywz
f
ywu
f
xyz
f
xyu
f
)
(
)
(
)
(
)
(
)
(
)
(
yzu
f
xwz
f
xwu
f
xzu
f
wzu
f
yw
f
)
(
)
(
)
(
)
(
)
(
)
(
)
(
xy
f
yz
f
yu
f
xw
f
xz
f
xu
f
wz
f
)
(
)
(
)
(
)
(
)
(
)
(
)
(
wu
f
zu
f
y
f
x
f
w
f
z
f
u
f
)
,
,
,
,
(
=
C
(4)f
x
y
w
z
u
(1.5)
0
=
by
Hence, the above relations imply the
C
(3)f
(
x
,.,
z
,
u
)
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 11, November 2017)
405
)
,
,
(.,
(3)u
z
y
f
C
,C
(3)f
(
x
,
y
,.,
u
)
and,.)
,
,
(
(3)z
y
x
f
C
. This proves (2.5).We now consider (2.6). Actually, it is trivial for n
=
0,1,2,3 by (2.1) and by the definition of Cf. Suppose that(2.6) holds for all natural numbers smaller than n
5, then)
(
=
)
(
x
f
x
3xxx
f
n n)
(
)
(
)
(
)
(
)
(
=
f
x
n3xx
f
x
n3xx
f
x
n3xx
f
xxx
f
x
n3x
)
(
)
(
)
(
)
(
)
(
x
3x
f
x
3x
f
xx
f
xx
f
xx
f
n
n
)
,
,
,
(
)
(
)
(
)
(
)
(
x
3f
x
f
x
f
x
C
(3)f
x
3x
x
x
f
n
n
)
(
)
(
)
(
)
(
)
(
)
(
=
f
x
n1
f
x
n1
f
x
n1
f
x
3
f
x
n2
f
x
n2)
(
)
(
)
(
)
(
)
(
)
(
)
(
x
2f
x
2f
x
2f
x
3f
x
f
x
f
x
f
n
n
)
,
,
,
(
3(3)
x
x
x
x
f
C
n
)
(
3
)
(
)
(
3
)
(
3
)
(
)
(
3
=
f
x
n1
f
x
3
f
x
n2
f
x
2
f
x
n3
f
x
C
(3)f
(
x
n3,
x
,
x
,
x
)
(
1)
(
)
(
1)
(
,
)
(
1)
(
,
,
)
3
=
n
f
x
n
C
2Cf
x
x
n
C
3C
(2)f
x
x
x
)
,
,
,
(
1)
(
n
C
4C
(3)f
x
x
x
x
3
f
(
x
)
3
C
2Cf
(
x
,
x
)
3
C
3C
2f
(
x
,
x
,
x
)
(
2)
(
)
(
2)
(
,
)
(
2)
(
,
,
)
3
n
f
x
n
C
2Cf
x
x
n
C
3C
(2)f
x
x
x
3
2
(
)
2
(
,
)
)
,
,
,
(
2)
(
n
C
4C
(3)f
x
x
x
x
f
x
C
2Cf
x
x
(
n
3)
f
(
x
)
(
n
3)
C
2Cf
(
x
,
x
)
(
n
3)
C
3C
(2)f
(
x
,
x
,
x
)
3
(
)
(
3)
(
,
,
,
)
)
,
,
,
(
3)
(
n
C
4C
(3)f
x
x
x
x
f
x
n
C
(3)f
x
x
x
x
)
,
,
(
)
,
(
)
(
=
nf
x
nC
2Cf
x
x
nC
3C
(2)f
x
x
x
)
,
,
,
(
(3)4
C
f
x
x
x
x
nC
where the definition of
C
(3)f
and (2.5) are used in thesecond equation. This gives (2.6) fora all n
0. On the other hand, for any fixed integer n>
0, by (1.4) and (2.1), we have) ( ) ( ) ( ) ( ) ( = ) , , ,
( 2 3
(3)f xn xn xn xn f xn f xn f xn f xn f xn
C
)
(
)
(
)
(
)
(
)
(
e
f
x
2f
x
2f
e
f
e
f
n
n
)
(
)
(
)
(
)
(
)
(
x
2nf
x
nf
x
nf
x
nf
x
nf
)
(
)
(
)
(
6
)
(
4
=
f
x
2n
f
x
n
f
x
3n
f
x
n)
(
)
(
6
)
(
4
=
)
(
x
nf
x
2nf
x
nf
x
3nf
)
,
,
,
(
(3) n n n n
x
x
x
x
f
C
2
(
)
2
2(
,
)
2
34
=
nf
x
nC
Cf
x
x
nC
)
,
,
,
(
2
)
,
,
(
4 (3)(2)
x
x
x
x
f
C
nC
x
x
x
f
C
(
)
2(
,
)
36
nf
x
nC
Cf
x
x
nC
)
,
,
,
(
)
,
,
(
4 (3)(2)
x
x
x
x
f
C
nC
x
x
x
f
C
3
nf
(
x
)
3
nC
2Cf
(
x
,
x
)
3
nC
3
)
,
,
,
(
3
)
,
,
(
4 (3)(2)
x
x
x
x
f
C
nC
x
x
x
f
C
)
,
,
,
(
(3) 4x
x
x
x
f
C
n
(
)
2(
,
)
3=
nf
x
nC
Cf
x
x
nC
)
,
,
,
(
)
,
,
(
4 (3)(2)
x
x
x
x
f
C
nC
x
x
x
f
C
From (2.5) and the above claim for n
>
0. This confirms(2.6) for n
<
0.Remark 2 For any function f:G
H, the following statements are pairwise equivalent:The function
f
KerC
(4)(
G
,
H
)
;
C
(3)f
(.,
y
,
z
,
u
)
is a homomorphism;
C
(3)f
(
x
,.,
z
,
u
)
is a homomorphism;
C
(3)f
(
x
,
y
,.,
u
)
is a homomorphism;International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 7, Issue 11, November 2017)
406
Before presenting Proposition 1, we first introduce the following useful lemma, which was given in [8]
Lemma 2 (Lemma 2.4 in [8]) The following identity is
valid for any function f:G
H andl
N
;)
,
,
,
(
=
)
(
2 1 1) (
< < 2 < 1 1 2
1 i i im
m l m i i i l m
l
C
f
x
x
x
x
x
x
f
(2.7)Proposition 1 Suppose that
f
KerC
(4)(
G
,
H
)
. Then)
(
22 1 1
l n l n n
x
x
x
f
(
)
(
,
)
(
,
,
)
=
2 3 (2)1
i i i i
i i i
i i l i
x
x
x
f
C
C
n
x
x
Cf
C
n
x
f
n
(
,
)
)
,
,
,
(
22 1 1 2 < 1 1 (3)
4
i n i i n i l i i i i i i
i
C
C
f
x
x
x
x
Cf
x
x
n
)
,
,
(
3 2 1 (2) 3 2 1 3 < 2 < 1 1
i i i i
i i l i i i
x
x
x
f
C
n
n
n
)
,
,
,
(
4 3 2 1 (3) 4 3 2 1 4 < 3 < 2 < 1 1
i i i i i
i i i l i i i i
x
x
x
x
f
C
n
n
n
n
(2.8)
for
n
i
Z
and allx
i
G
,i
=
1,2,
l
such that1
jj
x
x
,j
=
1,2,
,
l
1
Proof 2 Replacing
x
i in (2.7) byx
ini, we have)
,
,
,
(
=
)
(
22 1 1 1) (
< < 2 < 1 1 2
2 1 1
m i n m i i n i i n i m l m i i i l m l n l n n
x
x
x
f
C
x
x
x
f
The vanishing of
C
(m1)f
for m
5 yields)
,
(
)
(
=
)
(
22 1 1 2 < 1 1 1
2 2 1 1
i n i i n i l i i i n i l i l n l n n
x
x
Cf
x
f
x
x
x
f
)
,
,
(
33 2 2 1 1 (2)
3 < 2 < 1 1
i n i i n i i n i l
i i i
x
x
x
f
C
)
,
,
,
(
44 3 3 2 2 1 1 (3)
4 < 3 < 2 < 1 1
i n i i n i i n i i n i l
i i i i
x
x
x
x
f
C
Therefore, by (2.6) and (2.5), we have
)
,
,
(
)
,
(
)
(
=
)
(
ni i i i 2 i i i 3 (2) i i ii
n
f
x
n
C
Cf
x
x
n
C
C
f
x
x
x
x
f
)
,
,
,
(
(3)4 i i i i
i
C
C
f
x
x
x
x
n
)
,
,
(
=
)
,
,
(
3 2 1 (2) 3 2 1 3 3 2 2 1 1 (2)
i i i i
i i i n i i n i i n
i
x
x
n
n
n
C
f
x
x
x
x
f
C
)
,
,
,
(
=
)
,
,
,
(
4 3 2 1 (3) 4 3 2 1 4 4 3 3 2 2 1 1 (3)
i i i i i
i i i i n i i n i i n i i n
i
x
x
x
n
n
n
n
C
f
x
x
x
x
x
f
C
which is (2.8). This completes proof.
Remark 3 In particular, if
l
=
1
, then Proposition 1 holds.III. SOLUTION ON THE FINITE CYCLIC GROUP
C
nLet
C
a
a
ne
n
=
|
=
be a cyclic group of order nwith generator a. In this section, we study the general solution on the finite cyclic group
C
n.Theorem 1 Assume that
n
is odd andnCf
(
a
,
a
)
=
0
and
nC
(2)f
(
a
,
a
,
a
)
=
0
. Thenf
KerC
(4)(
C
n,
H
)
if and only if it is given by
)
,
,
(
)
,
(
)
(
=
)
(
a
pf
a
pC
2Cf
a
a
pC
3C
(2)f
a
a
a
f
p
Z
p
a
a
a
a
f
C
pC
4 (3)(
,
,
,
)
(3.1)where
f
(
a
)
andC
(3)f
(
a
,
a
,
a
)
satisfy0
=
)
,
,
,
(
(3)a
a
a
a
f
nC
(3.2)0
=
)
,
,
,
(
)
(
a
nC
4C
(3)f
a
a
a
a
nf
(3.3)Proof 3 Necessity. Let
f
:
C
n
H
be a functionsatisfying (1.5). Then by (2.6), we see that f also satisfies (3.1) i.e.,
)
,
,
(
)
,
(
)
(
=
)
(
a
pf
a
pC
2Cf
a
a
pC
3C
(2)f
a
a
a
f
p
Z
p
a
a
a
a
f
C
pC
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407
Let
p
=
n
in (3.1), sincenCf
(
a
,
a
)
=
0
and0
=
)
,
,
(
(2)a
a
a
f
nC
andnC
2 andnC
3 are integers, thesummand
nC
2Cf
(
a
,
a
)
=
0
and0
=
)
,
,
(
(2)3
C
f
a
a
a
nC
and by the fact thata
ne
=
,0
=
)
(
e
f
, we obtain0
=
)
,
,
,
(
)
(
a
nC
4C
(3)f
a
a
a
a
nf
Furthermore, by using (2.5),(2.4), and
a
n=
e
, we get)
,
,
,
(
=
)
,
,
,
(
(3) (3)a
a
a
a
f
C
a
a
a
a
f
nC
n)
,
,
,
(
=
C
(3)f
e
a
a
a
=
0
This proves (3.2)-(3.3). Sufficiency. We claim that
(3.1)-(3.3) defines a function on
C
n. Indeed, for eachp
Z
, by (3.1) and(3.3) we have
( ) ( ) ( ) 2 ( , ) ( ) 3= ) ( )
(a f a p n f a p nC Cf aa p nC
f pn p
)
,
,
,
(
)
(
)
,
,
(
4 (3)(2)
a
a
a
a
f
C
C
n
p
a
a
a
f
C
pf
(
a
)
pC
2Cf
(
a
,
a
)
pC
3C
(2)f
(
a
,
a
,
a
)
)
,
,
,
(
(3)4
C
f
a
a
a
a
pC
(
)
(
,
)
)
(
)
(
=
p
n
p
f
a
p
n
C
2
pC
2Cf
a
a
4(2) 3
3
(
,
,
)
(
)
)
(
p
n
C
pC
C
f
a
a
a
p
n
C
(3)(
,
,
,
)
4C
f
a
a
a
a
pC
)
(
=
f
a
n)
(
=
f
e
=
0
Where the last identity is obtained because
0
=
)
,
(
a
a
nCf
,0
=
)
,
,
(
2a
a
a
f
nC
, (3.2), andn
is odd.Finally, for any
n r l q p m
C
a
u
a
w
a
z
a
y
a
x
=
,
=
,
=
,
=
,
=
we have ) ( ) ( ) ( ) ( ) ( )
(xyzwu f xyzw f xyzu f xywu f xzwu f yzwu
f
) ( ) ( ) ( ) ( ) ( )
(xyz f xyw f xyu f xzw f xzu f xwu
f
) ( ) ( ) ( ) ( ) ( )
(yzw f yzu f ywu f zwu f xy f xz
f
) ( ) ( ) ( ) ( ) ( ) ( )
(xw f xu f yz f yw f yu f zw f zu
f
)
(
)
(
)
(
)
(
)
(
)
(
wu
f
x
f
y
f
z
f
w
f
u
f
) ( ) ( ) ( ) (= f ampqlr f ampql f ampqr f amplr
)
(
)
(
)
(
)
(
a
m q l rf
a
p q l rf
a
m p qf
a
m p lf
)
(
)
(
)
(
)
(
a
m p rf
a
m q lf
a
m q rf
a
m l rf
)
(
)
(
)
(
)
(
a
p q lf
a
p q rf
a
p l rf
a
q l rf
)
(
)
(
)
(
)
(
)
(
a
m pf
a
m qf
a
m lf
a
m rf
a
p qf
)
(
)
(
)
(
)
(
)
(
a
p lf
a
p rf
a
q lf
a
q rf
a
l rf
)
(
)
(
)
(
)
(
)
(
a
mf
a
pf
a
qf
a
lf
a
rf
Let ) , , , ( ) , , ( ) , ( ) ( = (3) 4 1 (2) 3 1 2 11f a NCCf aa NCC f aaa NCC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 2 (2) 3 2 2 2
2f a NCCf aa NCC f aaa NCC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 3 (2) 3 3 2 3
3f a NCCf aa NCC f aaa NCC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 4 (2) 3 4 2 4
4f a NCCf aa NCC f aaa NCC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 5 (2) 3 5 2 5
5f a NCCf aa NCC f aaa NCC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 6 (2) 3 6 2 6
6f a NCCf aa NCC f aaa NCC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 7 (2) 3 7 2 7
7f a NCCf aa NCC f aaa NCC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 8 (2) 3 8 2 8
8f a NCCf aa NCC f aaa NCC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 9 (2) 3 9 2 9
9f a NCCf aa NCC f aaa NCC f aaaa
N
) , , , ( ) , , ( ) , ( )
( 10 4 (3)
(2) 3 10 2
10
10f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 11 (2) 3 11 2 11
11f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 12 (2) 3 12 2 12
12f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 13 (2) 3 13 2 13
13f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 14 (2) 3 14 2 14
14f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 15 (2) 3 15 2 15
15f a N CCf a a N CC f aaa N CC f a aaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 16 (2) 3 16 2 16
16f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 17 (2) 3 17 2 17
17f a N CCf a a N CC f aaa N CC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 18 (2) 3 18 2 18
18f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 19 (2) 3 19 2 19
19f a N CCf aa N CC f aa a N CC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 20 (2) 3 20 2 20
20f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( ) , , ( ) , ( ) ( (3) 4 21 (2) 3 21 2 21
21f a N CCf aa N CC f aaa N CC f aaa a
N
International Journal of Emerging Technology and Advanced Engineering
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408
) , , , ( )
, , ( )
, ( )
( (3)
4 22 (2)
3 22 2
22
22f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( )
, , ( )
, ( )
( 23 4 (3)
(2) 3 23 2
23
23f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( )
, , ( )
, ( )
( 24 2 24 3 (2) 24 4 (3)
24f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( )
, , ( )
, ( )
( (3)
4 25 (2)
3 25 2
25
25f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( )
, , ( )
, ( )
( (3)
4 26 (2)
3 26 2
26
26f a N CCf aa N CC f aaa N CC f aaaa
N
) , , , ( )
, , ( )
, ( )
( 4 (3)
(2) 3
2Cf aa mCC f aaa mCC f aaaa mC
a
mf
) , , , ( )
, , ( )
, ( )
( (3)
4 (2)
3
2Cf aa pCC f aaa pCC f aaaa
pC a
pf
) , , , ( )
, , ( )
, ( )
( (3)
4 (2)
3
2Cf aa qCC f aaa qCC f aaa a
qC a
qf
)
,
,
,
(
)
,
,
(
)
,
(
)
(
4 (3)(2) 3
2
Cf
a
a
lC
C
f
a
a
a
lC
C
f
a
a
a
a
lC
a
lf
)
,
,
,
(
)
,
,
(
)
,
(
)
(
a
rC
2Cf
a
a
rC
3C
(2)f
a
a
a
rC
4C
(3)f
a
a
a
a
rf
which, after a long and tedious computation, gives
0
.Consequently, (4)( , )
H C C
f n . This completes the
proof.
REFERENCES
[1] Aczel, J: Lectures on Functional Equations and Their Applications.Academic Press New York (1966).
[2] Ng, CT: Jensen’s functional equation on groups, Aequ. Math. 39
(1990), 85–99.
[3] Ng, CT: Jensen’s functional equation on groups II, Aequ. Math. 58
(1999), 311–320.
[4] Baron, K. Kannappan,P: On the Cauchy difference,Aequ. Math. 46
(1993), 112–118.
[5] Brzdek, J: On the Cauchy difference on normed spaces, Abh. Math. semin. Univ. Hamb. 66 (1996) 143–150
[6] Ebanks, B: Generalized Cauchy difference functional equations, Aequ. Math. 70 (2005), 154–176.
[7] Ebanks, B: Generalized Cauchy difference functional equations,II Proc. Am. Math. Soc. 136 (2008), 3911–3919.
[8] Ng, CT, Zhao, H: Kernel of the second order Cauchy difference on groups, Aequ. Math. 86 (2013), 155–170.
