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 Authors : K. P. R. Sastry, S. Kalesha Vali , Ch. Srinivasa Rao, M. A. Rahamatulla
 Paper ID : IJERTV2IS4365
 Volume & Issue : Volume 02, Issue 04 (April 2013)
 Published (First Online): 16042013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Quasi Nonexpansive Sequences In Dislocated Quasi – Metric Spaces
K. P. R. Sastry 1, S. Kalesha Vali 2, Ch. Srinivasa Rao 3 and M.A. Rahamatulla 4
1 8288/1 , Tamil Street , Chinna Waltair, Visakhapatanam530 017, India ,
2 Department of Mathematics, JNTUK University College of Engineering , Vizianagaram – 535 003, A.P. , India ,
3 Department of Mathematics, Mrs. A.V.N. College, Visakhapatnam 530 001, India
4 Department of Mathematics, AlAman College of Engineering, Visakhapatnam 531 173, India
Abstract :
We introduce the notion of a quasi – nonexpansive sequence with respect to a non – empty subset of a dislocated quasi – nonexpansive metric space and extend the results of M.A.Ahmed and F.M.Zeyada [1] to such sequences.
Mathematical Subject Classification : 47 H 10, 54H25 . Key Words :
Dislocated quasi – nonexpansive w.r.to , quasi metric spaces , asymptotically regular , decreasing sequences and metric spaces .
1 . INTRODUCTION
M.A.Ahmed and F.M.Zeyada [1] established the convergence of a sequence { } ,
in a dislocated – quasi metric space (X ,d ) if a map : X X is quasi nonexpansive
with respect to { } . We observe that the role playe by the map in proving the convergence , is meagre . Consequently , we introduce the notion of a quasi nonexpansive sequence with respect to a non empty subset of a dislocated quasi metric space and establish the convergence of such sequences under certain conditions .
These results extend the results of [1] .
We begin with various definitions
Definition 1.1:
Let X be a non – empty set and let : X Ã— X [0,) be a function called a distance function satisfying one or more of 1.1.1 (1.1.5) .
(1.1.1) : d(, ) = 0 .
(1.1.2) : d(, ) = , = 0 = , . (1.1.3) : d(, ) = , , .
(1.1.4) : d(, ) (, ) + , , , .
(1.1.5) : d(, ) max{(, ), (, )} , , .

If d satisfies (1.1.2) and (1.1.4) then d is called a dislocated quasi metric (or) dq – metric and ( X ,d ) is called a dq – metric space .

If d satisfies (1.1.2) , (1.1.3) and (1.1.4) then d is called a dislocated metric and ( X ,d ) is called a dislocated metric space.

If d satisfies (1.1.1) , (1.1.2) and (1.1.4) then d is called a quasi metric (or) q metric and ( X , d ) is called a quasi metric space (or) q metric space .

If d satisfies (1.1.1) , (1.1.2) , (1.1.3) and (1.1.4) then d is called a metric and ( X , d ) is called a metric space .

If d satisfies (1.1.1) , (1.1.2) , (1.1.3) and (1.1.5) then d is called an ultra metric and ( X , d ) is called an ultra metric space .
We observe that every ultra metric is a metric .
Let be a subset of a quasi metric space ( X , d ) and : D be any mapping . Assume that () is the set of all fixed points of . For a given 0 ,
the sequence of iterates { } is defined by
( I ) : = (1 ) = ( 0) , where and is the set of all positive integers .
Definition 1.2 : ( F.M.Zeyada , G.H.Hassan and M.A.Ahmed [11] )
A sequence { } in a dislocated quasi metric space ( X , d ) is called Cauchy ,
if to each > 0 , there exists 0 , such that for all , 0 , ( , ) < .
Definition 1.3 :
A sequence { } in a dislocated quasi metric space ( X , d ) is said to be dislocated quasi – convergent (or) dq – convergent to , if
lim , = lim , = 0 .
In this case is called a dislocated quasi limit (or) dq – limit of { } and
we write . It can be shown that dqlimit of a sequence { } , if exists is unique .
Note : In a dislocated quasi metric space , when we talk of dq convergence or dq limit , we conveniently drop the prefix dq , in the absence of any ambiguity .
Definition 1.4 :
A dislocated quasi metric space ( X , d ) is complete , if every Cauchy sequence in it is dq – convergent .
Definition 1.5 :
Let ( X , d ) be a dislocated quasi metric space . Let .
Then , = , , , .
Definition 1.6 : ( M.A.Ahmed and F.M.Zeyada [1] , definition 2.1 )
Let ( X , d ) be a quasi – metric space and . The mapping
is said to be quasi – nonexpansive w.r.to a sequence { } of D , if for all 0 and for every ,
+1, , , where = the fixed point set of
( we assume that ) .
The following results are proved in ( F.M.Zeyada , G.H.Hassan and M.A.Ahmed [11] ) Lemma 1.7 : ( F.M.Zeyada , G.H.Hassan and M.A.Ahmed [11] )
Let ( X , d ) be a dislocated quasi metric space .
Then every dq convergent sequence in is Cauchy .
It may be noted that the converse of lemma 1.7 is not true .
Lemma 1.8 : ( F.M.Zeyada , G.H.Hassan and M.A.Ahmed [11] )
Let ( X , d ) be a dislocated quasi metric space . If { } is a sequence in
dq – converging to , then every subsequence of { } dq – converges to .
Lemma 1.9 : ( F.M.Zeyada , G.H.Hassan and M.A.Ahmed [11] )
Dislocated quasi limits in a dq metric space are unique .
( M.A.Ahmed and F.M.Zeyada [1] ) proved the following results .
Theorem 1.10 : ( M.A.Ahmed and F.M.Zeyada [1] , Theorem 2.1 )
Let { } be a sequence in a subset of a q metric space ( X , d ) and
: D be a map such that . Then
(a) lim , ( ) = 0 if { } converges to a unique point in ;
(b) { } converges to a unique point in ( ) if lim , ( ) = 0 ,
is a closed set , is quasi – nonexpansive w.r.to { } and is complete .
Theorem 1.11 : ( M.A.Ahmed and F.M.Zeyada [1] , Theorem 2.2 )
Let { } be a sequence in a subset of a complete q metric space ( X , d ) and : D be a map such that is a closed set . Assume that ( i ) is quasi – nonexpansive w.r.to { } ;
( ii ) lim , +1 = 0
(iii) if the sequence { } satisfies lim , +1 = 0 , then lim inf , = 0 or lim sup , = 0 . Then { } converges to a unique point in .
Note : The presence of conditions (ii) and (iii) guarantees that lim , ( ) = 0 .
We show in Example 2.6 that condition(ii) alone may not guarantee that
lim , ( ) = 0 .
In this section , we introduce the notion of a quasi nonexpansive sequence with respect to a non – empty subset of a dislocated quasi nonexpansive metric space and extend the results in [1] to such spaces .
Definition 2.1 :
Let ( X , d ) be a dislocated quasi metric space , and { }
such that = 1,2,3, ,
Then { } is said to be quasi – nonexpansive w.r.to , if
+1, ,
and
, +1 , and = 1,2,3, ,
Lemma 2.2 :
Let ( X , d ) be a dislocated quasi metric space , and { } ,
. Suppose that { } is quasi – nonexpansive w.r.to . Then
lim , = 0 , +1 0 and +1 , 0 .
Proof :
Let > 0 . Then there exists and positive integer such that
,
<
2
and ,
<
2
.
and
{ } is quasi – nonexpansive w.r.to ,
2
2
+1 , , , <
, +1
, /p>
,
<
2
Now
, +1 , + , +1
< 2
+ 2
and
= .
+1 , +1 , + ,
< 2
+ 2
= .
, +1 0 and +1 , 0 .
Lemma 2.3 :
Suppose { } is quasi – nonexpansive w.r.to . Then
lim , = 0 { } is a Cauchy sequence .
Proof :
Let > 0 . Then there exists positive integer such that
2
2
2
2
Now
, <
.
2
2
2
2
, < , <
and , <
2
2
and
, , <
,
,
<
2
.
Now suppose , . Then
, , + ,
<
<
2
=
+ 2
and
, , + ,
< 2
=
+ 2
{ } is a Cauchy sequence .
Lemma 2.4 :
Let ( X , d ) be a dislocated quasi metric space and { } be a sequence in . Assume that
be a non empty subset of . If { } is quasi – nonexpansive w.r.to ,
then , is a monotonically decreasing sequence in [0, ) .
Proof :
Since { } is quasi – nonexpansive w.r.to ,
+1, , ( 2.4.1 ) , for all 0 and for every .
From (2.4.1) , taking the infimum over , we get that
+1, , for all 0 .
Hence { , } is a monotonically decreasing sequence in [0,) .
Lemma 2.5 :
Let ( X , d ) be a dislocated quasi metric space and { } be a sequence in .
Suppose { } is quasi – nonexpansive w.r.to satisfying lim , = 0 .
Then { } is a Cauchy sequence .
Proof :
Since { } is a quasi – nonexpansive w.r.to , to each > 0 ,
there exists and positive integer such that
2
2
, <
and , <
, .
2
2
Suppose , . Then
, , + ,
and
2
<
<
=
+
2
, , + ,
<
2
+
2
=
{ } is a Cauchy sequence .
The following example shows that converse of Lemma 2.5 is not true . Example 2.6 :
= { ( 1,0) , (1,0) and the segment [ (0,1) ,(0,2)] of the – axis }
is the usual Euclidean distance in 2 .
= { (1, 0) , (1,0)} , = (0, 1+ 1 ) , = 1,2,3
Then { } is dislocated quasi – nonexpansive w.r.to
, +1 , ,
, +1 0 and 0 ,1 .
Now we state and prove our first main result , which is an extension of Theorem 1.10 to quasi nonexpansive sequences .
Theorem 2.7 :
Let { } be a sequence in a subset of a dislocated quasi metric space ( X , d ) and F ( ) . Then

lim , = 0 , if { } converges to a point in

{ } converges to a unique point in , if lim , = 0 ,
is a closed set , { } is quasi – nonexpansive w.r.to and is complete .
Proof of (a) :
Since { } converges to a point in , there exists a point such that
lim , = 0 and lim , = 0
Given > 0 , there exists a positive integer such that
2
2
, <
and , <
.
2
2
, , + ,
<
<
+
+
2 2
Now
= .
, < for every > 0 ,
, = 0 .
2
2
, , <
.
lim , = 0
(a) holds .
Proof of (b) :
Let ( X , d ) be a complete dislocated quasi – metric space and { } be a sequence in and F . Assume that { } is quasi – nonexpansive w.r.to , is closed and lim , = 0 . Then { } is a Cauchy sequence by lemma 2.5 ,
hence there exists such that { } converges to . Let > 0 . There exists a positive integer such that
2
2
, <
2
2
, <
for every and and , <
.
2
2
There exists such that
2
2
2
2
, < and , <
, , + ,
<
<
+
+
2 2
=
, <
and similarly we have
+1 , +1 , ,
, , + ,
<
<
+
+
2 2
=
, <
is a limit point of
, since is closed
Since limits are unique ( by lemma 1.9 ) , { } converges to a unique point .
Hence ( b) holds .
The following theorem which is an analogue of Theorem 1.11 establishes the convergence of the sequence .
Theorem 2.8 :
Let ( X , d ) be a complete dislocated – quasi metric space . Assume that { } is a sequence in and . Further assume that there is a
mapping 0, [0,1) such that is monotonically increasing and
+1 , ( , ) , for = 1,2,3 ( 2.8.1)
Then { } is Cauchy and { } converges to a point . If further is closed then .
Proof :
By hypothesis
+1 , ( , ) , , ,
so that { , } is decreasing and hence { ( , )} is decreasing since is increasing .
+1 , ( , ) , ( , ) ( 1 , ) 1 ,
( , )( 1 , ) . ( 1 , ) 1 ,
( 1 , )( 1 , ) . ( 1 , ) 1 ,
= (( 1 , )) . 1 , 0 as .
( ( , < 1) Thus , 0 as .
By lemma 2.5 , { } is Cauchy sequence and hence converges to a point
since X is complete . If is closed by ( Theorem (2.7) (b) ) follows that .
The following Example shows that
Theorem 2.8 may not hold good if ( 2.8.1) is replaced by
+1 , , for = 1,2,3, ( 2.8.2) even if we assume that (even in a metric space ) lim , +1 = 0 ( 2.8.3)
Example 2.9 :
Let be the subset of x consisting of the points (1,0) , (1,0) and the segments of the axis joining the two points (0,1) and (0,2) .
Hence = { 1,0 , 1,0 and ( 0, )/ 1 2 }
Let be the Euclidean metric in 2 .
Take = { 1,0 , 1,0 } and = {( 0, 1 + 1 )/ = 1,2, . . }
Then { } is quasi nonexpansive w.r.to to ,
+1 , , for = 1,2,3,
, +1 0 , is closed but { , } does not converges to 0 .
Acknowlegements :
The fourth author ( M.A.Rahamatulla) is grateful to the authorities of Al – Aman College of Engineering ,Visakhapatnam and I.H.Farooqui Sir
for granting permission to carry on this research . The fourth author is deeply indebted to the authorities of SITAM College of Engineering , Vizianagaram for permitting to use the facilities in their campus while doing the research.
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