Quasi Nonexpansive Sequences In Dislocated Quasi – Metric Spaces

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 8-28-8/1 , Tamil Street , Chinna Waltair, Visakhapatanam-530 017, 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, Al-Aman 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 × 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{(, ), (, )} , , .

1. 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 .

2. 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.

3. 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 .

4. 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 .

5. 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 dq-limit 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

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

2. { } 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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