Advances in Pure Mathematics, 2012, 2, 280-284
http://dx.doi.org/10.4236/apm.2012.24036 Published Online July 2012 (http://www.SciRP.org/journal/apm)
A Family of Non-Self Maps Satisfying
i
i
-Contractive
Condition and Having Unique Common Fixed Point in
Metrically Convex Spaces*
Yongjie Piao, Dongzhe Piao
Department of Mathematics, College of Science, Yanbian University, Yanji, China
Email: pyj6216@hotmail.com
Received February 17, 2012; revised March 19, 2012; accepted March 26, 2012
ABSTRACT
Class of 5-dimensional functions was introduced and a convergent sequence determined by non-self mappings sa-
tisfying certain
-contractive conditio n was constructed , and then that the limit of the sequence is the unique common
fixed point of the mappings was proved. Finally, several more general forms were given. Our main results generalize
and unify many same type fixed point theorems in references.
Keywords: Metrically Convex Space; 5-Dimensional Fu nc t io ns
; i
-Contractive Condition ; Common Fixed Point;
Complete
1. Introduction
There have appeared many fixed point theorems for sin-
gle-valued self-map of closed subset of Banach space.
However, in many applications, the mapping under con-
siderations is a not self-mapping of closed sets. 1976,
Assad [1] gave sufficient condition for such single-val-
ued mapping to obtain a fixed point by proving a fixed
point theorem for Kannan mappings on a Banach space
and putting certain boundary conditions on the mapping.
Similar results for multi-valued mappings were respect-
tively given by Assad [2] and Assad and Kirk [3]. Later,
some authors generalized the same type results on com-
plete metrically convex metric spaces, see [4-8]. Those
above results were discussed under some contractive
conditions or certain boundary condition. Recently, the
author discussed unique common fixed point theorems
for a family of contractive or quasi-contractive type
mappings on metrically convex spaces or 2-metric spaces,
see [9-13], these results improve many known common
fixed point theorems. In order to generalize and unify
further these results, in this note, we shall discuss and
obtain some unique common fixed point theorems for a
family of more general non-self maps satisfying
-
contractive condition on closed subset of a complete
metrically convex metric space.
We need the following definition and lemma in the
sequel.

,
Definition 1.1. ([4,5]) A metric space
X
d
,,
is said
to be metrically convex if, for any
x
yX
with
x
y
, there exists zX
, , such that zxzy
,, ,dxz dzydxy.
Lemma 1.2. ([3,4]) If K is a nonempty closed subset
of a complete metrically convex metric space
,
X
d,
then for any
x
K
y
KzK, there exists
and
such that

,, ,.dxz dzydxy
Let
denotes a family of mappings such that each

5
:

 
,,2,,ttt tttt
is continuous and increasing in
,
each coordinate variable, and also

0t
0,t
for every , where .
0,0,0,0,0 0
. Obviously,
which belongs to
: There exist many functions

5
:
Example 1.3. Let


be defined by

12345 12345
1
,,,, .
7
ttttt ttttt

Then obviously ,

5
:
Example 1.4. Let


be defined by
 



12345
11111
12345 1
,,,, ln
7ttttt
ttttt
   
.
Then obviously,
is continuous and increasing in
each coordinate variable, and
*This paper is supported by the Foundation of Education Ministry, Jilin
Province, China (No. 2 01 1 [ 343]).
C
opyright © 2012 SciRes. APM
Y. J. PIAO, D. Z. PIAO 281
 
 

111211tt ttt 
.
1
1
,,2,, ln
7
66
.
ln
77
t
ttt ttt
tt



Hence


5


Example 1.5. Let be defined by
:

12345 1
1
, ,, ,arctanarctan
7
arctan arctan
ttttt t


2 3
45
arctan
.
t t
tt
Then obviously,
is continuous and increasing in
each coordinate variable, and
 



,,2,,
1arctanarctanarctan 2arcta
7
11
4arctanarctan 242
77
ttt ttt
tt
ttt



n arctan
.
ttt
tt

.
Hence

2. Unique Common Fixed Point
Here, we will discuss unique existence problems of
common fixed points for a family of non-self mappings
satisfying certain i
-contractive condition and certain
boundary condition in complete metrically convex
spaces.
Theorem 2.1. Let K be a nonempty closed subset of a
complete metrically convex metric space

,
X
d

:i
TK X
, and
i a family of non-self maps such that for
each ,
x
yK and with i, ,ijj

,,,,
j j
y dxTy




,,,
,,,,
ij ii
i
dTxTyqdxTx dyT
dyTx dxy
(1)
where 1
0i
2
q and


TKii
T
0
for each i. Further,
if i for each , then

has an
unique com mon fixed point in K.
Ki
Proof. Take an
x
K
. We will construct two se-
quences
n
x
and

n
x
. in the following manner:
Define 110
x
Tx
If 1,
x
K
put 11
x
x; if 1
x
K
,
then by Lemma 1.2 there exists 1
x
K

1
 .

.
such that
0 Define 221

01
x
1
dxx
,
1
,,dx xdx
x
Tx
If
2,
x
K
put 22
x
x
; if 2,
x
K
then by Lemma 1.2
there exists 2
x
K


2
,dxx such that

x

12
dx x
,
12 2. Continuing in this way,
we obtain ,dx

n
x
and

n
x
:
i) 1;
nnn
x
Tx
ii) if n
x
K
, then ;
nn
x
x
iii) if ,
n
x
K
then there exists n
x
K
,
n n
x
 such that
by Lemma 1.2.

,,
nnn
x dxx

11n
dx
dx
Let
:
inii
Pxxxx


and
:
inii
xxxQx
 ,
then since
i
TK K
i
n
for all , it is easy to show
that
x
Q
implies 11
,
nn
x
xP


,dxx
1
,
nn
(2)
Now, we wish to estimate 1nn. We can divide
the proof into three cases in view of (2).
x
xP
, we have Case I.
 








1111
11 1
111 1
11111
1111
1
,, ,
,,,,
,,,,,
,,,, ,,0,,
,,, ,,,,
0,, .
nnn nnnn n
nnnn nnn
nnn nnnnn
nnn nnnnnn
nnnnnnnnn
nn
dxxdxxdTxT x
qdxTx dxTx
dxT xdxTxdxx
qdxxdxx dxxdxx
qdxxdxxdxx dxx
dx x

 
 

 




(3)
11
,,,
nn nn
dx xdxx

then (3) becomes If



111
111
1
,,,,,
2,,,,,
<,.
nnn nnnn
nn nn nn
nn
dxxqdxxdxx
dxx dxxdxx
qd xx


Which is a contradiction since 1
2
q, hence we have
that
11
,,
nnn n
dxxdx x

. In this case, (3) further
becomes the following



111
111
1
,,,,,
2,,,,,
,.
nnnn nn n
nn nn nn
nn
dxxqdx xdx x
dxxdxxdxx
qd xx


,
n
(4)
x
P
Case II.
1n
x
Q
, then by iii) and (2), we
have
 
 



 



1111
1111
11 1
111 1
11111
111
,, ,
,, ,
,,,,
,,,,,
,, , ,, ,0,,
,, , ,,
nnnnn n
nnnnnnnn
nnnn nnn
nnn nnnnn
nnnnnnnnn
nnn nnn
dx xdx xdxx
dx xdx xdTxTx
qdxTx dxTx
dxTxdxTxdxx
qdx xdxxdx xdx x
qdxxdxx dxx


 
 








1
1
,,
0,, .
nnn
nn
dx x
dx x


(5)
11
,,,
nn nn
dx xdxx

If then we obtain from (5)
that
 


111
111
1
, (,,,,
2,,,,,
<,.
nnn nnnn
nn nn nn
nn
dxxqdxxdxx
dxx dxxdxx
qd xx




Copyright © 2012 SciRes. APM
Y. J. PIAO, D. Z. PIAO
282
Which is a contradiction since 1
2
q

1
,,.
nn
dxx


111
,,
,,
nn
nn
x
dx x

,
, hence

dx x
1nn In this case, we obtain from (5)
that





11
11
1
,,
,,
2,,,
,.
nn nn
nnn
nnnn
nn
dxx dxx
qdxxdx
dx xdxx
qd xx


(6)
Case III. n
x
Q 1.
n
x
P
By (2) and iii), we know
1n
x
P
, and we obtain that




 
 



11
11
11
11
111
11
11 1
1
,,
,,
,,
,,
,,
,,
,,,,
,
nnnn nn
nn nn nn
nn nn
nn nn
nn nnnn
nnn nnn
nnn nnn
nn
dxxdxx dxx
dxxdxx dxx
dx xdxx
dx xdxx
dxxdTxT x
dxxqdx Tx
dxT xdxTxd
dx x






 














111
,,
,,,,,
nnn
nnnn nn
qdxxd
dxxdxxdxx



11
1
,
,
,,,
,
nnn
n n
dxT x
x x


11
, ,
.
nn
xx


,dx x
(7)
Since

,,
nnn
dx xdxx
11nnn

 
1
,,,
n n
dx x




1
11
11
,,
,,,
,,,
n
nn
nn
dx x
dx x



11
,,
nn
dxx

,

,,dx xdx x
11nn nn
and
hence (7) can be restated the following

nn
dx x

 


 


 

1
11
11 11
11
11 1
11
11
,
,,),
,, ,, ,
,,,
,, ,,,
,,,
,, ,
nn
nnn nnn
nnnn nn
nnn nnnn
nnn nn n
nnn nnnn
nn n
dx x
dx xqdx xdxx
dx xdx xdx x
dx xqdx xdxx
dx xdxxdxx
dx xqdx xdxx
dx xdx

 

 









1
,,
nnn
xdxx

(8)
If , then (8) becomes

nn
dx x
 


1 1
111
,,,,
,,
,
nn
nn
dxx
xx
 



11
11
, ,
2,,,
,,
nnn nnnn
nn nn
nn nn
dxxdxxq dxx
dxx dxxd
dx xqdxx





hence
 
11
,,
,
n n
dx x
q


1
1
nn
dxx

and therefore, by (6) in Case II, we obtain

11 21
1
,,,.
11
nnn nnn
q
dxxdx xdxx
qq
 


(9)
11
,,,
nnn n
dxxdxx

then (8) becomes If
 





1
111
111
11
1
,
,,,,,
2,,,,,
,,
1,.
nn
nnn nnnn
nn nn nn
nn nn
nn
dxx
dx xqdxxdxx
dxx dxxdxx
dx xqdxx
qdxx








By (6) in Case II again, we obtain that


11
21
,1 ,
(1 ),.
nnn n
nn
dxxqdx x
qqdxx





(10)
Thus in two situations, we obtain from (9) and (10)
that


121
,max,1 ,.
1
nnn n
q
dxxqqdx x
q





Hence in all three cases (see, (4), (6), (9) and (10)), we
find that



1
211
,
max,, 1max,,,
1
,>2.
nn
nn nn
dxx
q
qqqdxxdxx
q
nn
 





,
Let

max,, 1
1
q
M
qqq
q


01M
, then
since 1
02
q, and we have that

1211
,max,,,
forallwith2.
nnnnn n
dxxMdxxdx x
nn




And therefore,



1
2
10112
,max,,,
forallwith2.
n
nn
dx xMdxxdxx
nn




Let 1
01 12
max, ,,,Mdxxdxx


then
2
1
,n
nn
dx xM
n2n
mnN
 

for all with . Hence
for ,
2
1
,, 0
i
mn ii
iN iN
dx xdxxM
 



N

as . Which means that n
x
is a Cauchy se-
quence. Let p be a limit of
n
x
, then since K is
closed and n
pK
x
K
for all n. From (2), we are
easy to know that there exists an infinite subsequence
Copyright © 2012 SciRes. APM
Y. J. PIAO, D. Z. PIAO 283

k
n
x
of

n
x
such that 1.
k
n
x
P
1
k k
n n
x x

 n
kk
nn





1
11
1
,
,,
,
,, ,
,.
kk
kk
k
k
k
nn
n nn
n
n
n
Txp
dpTx
p
dpx
p

,k
Hence
1
kk
nn
Tx . For any fixed , we can take
such that , then











1
,,
,,
,,,,
,,,
,, ,
,,,
kk
kk k
kkk
kk
kk
kk
nn nn
nnnn
nnnn
nn n
nnnn
nn n
dTppdTpTxd
dTpTxdx p
qdpTpdxTx
dTpxd pxdx
qdpTpdxx
dTpxd pxdx


Let then by the continuity of
, the above
becomes





,, ,,
,
,,
,,
n
nnn
pd pp
d pp
p
p dTpp
dT





(,),,
,, ,
,,
2,,,
,.
nnn
n
nn
n
dTppqdTppdp
dTppd pp
qdTppdTp
dTpp dTp
qdTpp
Since , n
01q

,0pp
, i.e., n. This
completes that p is the common fixed point of
Tp p
n
pT n
If u and v are common fixed points of .
nn
T




1
,,,
,
Tud uv
d uv
du 0qv
, then
 



 

12
11 22
1
1
,,
,,,,,,
0,0, ,,,, ,
,,,,2 ,,,,
,
duvdTuTv
qduTudvTvd uTvd v
qd uvd uvd uv
qd uvd uvd uvd uv
qdu v
Hence since ; this is, u

,0v1
.
This completes that
n
Tn

,
has an unique common
fixed poi nt p.
The following is the very particular form of Theorem
2.1:
Corolla ry 2 .2. Let K be a nonempty closed subset of a
complete metrically convex metric space
X
d

:i
TK X
, and
i a family of non-self maps such that for
each ,
x
yK and with i, ,ijj




1
,arctan ,arctan
15
arctan ,arctan,ar
ij i
ji
d TxTydxTx
dxTydyTx




,
ctan, .
j
d yTy
dxy

TK
Further, if i for each i, then
K
ii
T
has an unique common fixed poi nt in K.
Proof. Let 7
15
q and
i

i
be that in Example 3
i
, then q and for each
satisfy all conditions of
Theorem 1, hence
ii
T

,
has an unique common fixed
point in K by Theorem 2.1.
From Theorem 2.1, we can obtain more general forms
than Theorem 2. 1.
Theorem 2.3. Let K be a nonempty closed sub set of a
complete metrically convex metric space
X
d, and
:
ii
TKX
ii
m
,
a family a family of maps and
x
yK
and of positive integers such that for each
,ij ij with
,




,,,,,
,,,,,
jj
ii
ji
mm
mm
iji ij
mm
ji
dT xTyqdxT x dyTy
dxTy dyT xdxy
where 1
02
qi for each . Further,
i
and
i
m
TKK
i for all i
if

ii
T
i
m
STi
, then has unique
common fixed point in K.
Proof. Let ii
for each , then
i
Si
satisfy all conditions of Theorem 2.1, hence

ii
S
pK has
an unique common fixed point .
Next, we prove that p is also an unique common fixed
point of
ii
T
i. In fact, for any fixed ,



1
i
m
ii iii i
STp TpTSp Tp
 ,
i
Tp i is a fixed point of Si for each hence
j. For
any
ji
 
with ,





 


 


 


 


 






 


 


 












,,
,,,,
,,,,
,
,,,,
2, ,, ,
,
,
ijiii ji
ii iiiji
iji iii
ii
iijii ji
iji iji
iji
iji
dTpSTpdSTp STp
q dTpSTpdTpSTp
dT p S T pdTp STp
dT pT p
q dTpSTpdTpSTp
dT p S T pdTp S T p
dT pST p
qdT pST p
01q
 

,
Since
,0
ij
dT p S T p
i, which im-
plies that
Tp STpj
iji
. Similarly, for any
ji
with
, we can obtain that
 
 

,,
iji iji
dTpSTp qdTpSTp, hence
. Therefore is a common fixed
iji
Tp STp

i
Tp
point of
jj
S
i for all . By the uniqueness of
Copyright © 2012 SciRes. APM
Y. J. PIAO, D. Z. PIAO
Copyright © 2012 SciRes. APM
284

j
S

i
Tp p
common fixed point of j
, we have
for all , this means that p is a common fixed point
of i. If u and v are common fixed points of
, then u and v are also common fixed points of
ii, hence again by uniqueness of common fixed
points of

ii, we have that , This com-
pletes that p is the unique common fixed point of
.
i

i
T

ii
T

SSuvp

,

i
Ti
Theorem 2.4. Let K be a nonempty clo sed subset of a
complete metrically convex metric space
X
d

,
ij
TK X

,,
ij ij
m
, and
,:
ij a family of maps and
,
a
family of positive integers such that for each
x
yK
and with ,
12
ii 12
ii


,
2
2
,
,,
,,,
i j
j
m
ij
yT y
x dxy
,,j


,, ,
12 1
121 1
,,
21
21
,,, ,
,,
,,
,,,
ijijij
ij i
mm m
ij ijijij
mm
ij ij
dT xTyqdxTxd
dxTydyT
,
where 1
2
q01,ij

1,ij and for each


,ij
m
ij
TK
.
Further, if 1) for all ; 2)


,
12 2
,, ,iii
TT
K,ij
1
,i
TT

for all 12
,, ,ii
with
, then
has an unique common fixed point in K.

,
ij
,ij
T
Proof For any fixed , has an unique
j

,iji
T
j
pKcommon fixed point by 1) and Theorem 2.3.
Now, we prove that pp
for all ,
. In fact, for
each 12
,, ,ii
with
, since Tp

1,ip

2,i
and

Tp p

, hence

p
1
,i
T p
12
,,ii
TT



211 2
,,, ,ii ii
TT pTT




1
,i
pT p
, and
therefore
 
 

Tp
by
2). This means that 1,i
1
i
is a common fixed point
of for all , But
i
T
22
,i
22
,ii
T
has an unique
common fixed p
, hence 1,i

pTp

for all 1
i
,
which implies that p
is a common fixed point of
11
,i
i
Tp
, hence p

since p
is the unique com-
mon fixed point of . Let

11
,i
i
T*
j
pp*
p

,ij
, then is
the unique fixed point of . This completes our
proof.
,ij
T
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