Journal of Modern Mathematics Frontier
Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
for all x, y, z, w U . Then, for s 2 if i 0 and for s 1 if i 1 we get
Hence the (IV.3) holds either, L 2s 1 for s 1 if i 0 and L
in x, in y, in z , in w
in
1 for s 1 if i 1 . 2s 1
Now from (IV.4), we prove the following cases for (i).
s s s n n n s n n i x i y i z i w , i n s n s n s n s n i x i y i z i w , i n x s n y s n z s nw s i i i i n i 4s 4s 4s in x in y in z in w 0 as n , 0 as n , 0 as n ,
Case (i): L 2s 1 for s 1 if i 0 . f (2 x, 2 x) 8 f ( x, x) A( x, x)
2
s 1 1 0
1 2
18 2 s 1 s || x || 2
s 1
2 18 2 s 1
4s
,
1 2
s 1
s 1
2
2 2 18 2 s 1
22
s
Thus (IV.1) is holds. But we have x x 12 x , has
s 1
2
s || x ||
2 18 2 || x || s 1 1
s || x ||
s 1
s
2 2s
1 for s 1 if i 1 . 2s 1 f (2 x, 2 x) 8 f ( x, x) A( x, x)
the property x L i i x , for all x U . Hence
Case (ii): L
1 2 1 (4 ( x, x, x, x) ( x, 2 x, x, 2 x)) 2 s s 2 (18 || x || 2 || 2 x || ) (4 2 2 s ) || x ||4 s 2 2s 4s 4s 2 (22 2 2 2 ) || x ||
x x
11
1 s 1 18 2 s 1 2 s || x || 1 2 1 s 1 2 s 1 18 2 s 1 2 s s 1 || x || 2 1 2
2 2 18 2 s 1
2s 2
(18 || i x ||s 2 || 2 i x ||s ) 2 i 1 i x (4 22 s ) || i x ||4 s i 2 i (22 22 s 2 24 s ) || i x ||4 s 2i
2
s || x ||
2 18 2 || x || s 1
Now
s 1
s 1
s
2s 2
In similar manner we can prove the following cases 1 for s 1 if i 1 , 2 4 s 1 1 for s 1 if i 0 and L 4 s 1 for s 1 if 2
L 24s 1 for s 1 if i 0 and L
and L 24s 1
i 1 for (ii) and (iii) respectively. complete.
s i (18 || x ||s 2 || 2 x ||s ) 2 i 4 s i (4 22 s ) || x ||4 s 2 i 4s i (22 22 s 2 24 s ) || x ||4 s 2i s 1 s s 2 i (18 || x || 2 || 2 x || ) i4 s 1 (4 22 s ) || x ||4 s 2 4 s 1 2s 4s 4s 2 i (22 2 2 2 ) || x ||
Hence the proof is
Theorem 4.4: Let f : U 2 V be a mapping for which there exist a function : U 4 (0, ] with the condition lim
n
1
i3n
( in x, in y, in z , in w) 0
with i 2 if i 0 and i
1 2
(IV.10)
if i 1 such that the
functional inequality F ( x, y, z, w) ( x, y, z, w)
(IV.11)
for all x, y, z, w U . If there exists L L i 1 such that the function
is 1 ( x) i4 s 1 ( x) 4 s 1 i ( x )
x x 12 x ,
has the property x L i3 i x .
for all x U .
23
(IV.12)