Advances in Pure Mathematics, 2011, 1, 33-41
doi:10.4236/apm.2011.13009 Published Online May 2011 (http://www.SciRP.org/journal/apm)
A Note on Convergence of a Sequence and its
Applications to Geometry of Banach Spaces
Hemant Kumar Pathak
School of Studies in Mathematics, Pandit Ravishankar Shukla University, Raipur, India
Received January 9, 2011; revised April 22, 2011; accepted April 30, 2011
Abstract
The purpose of this note is to point out several obscure places in the results of Ahmed and Zeyada [J. Math.
Anal. Appl. 274 (2002) 458-465]. In order to rectify and improve the results of Ahmed and Zeyada, we in-
troduce the concepts of locally quasi-nonexpansive, biased quasi-nonexpansive and conditionally biased qu-
asi-nonexpansive of a mapping w.r.t. a sequence in metric spaces. In the sequel, we establish some theorems
on convergence of a sequence in complete metric spaces. As consequences of our main result, we obtain
some results of Ghosh and Debnath [J. Math. Anal. Appl. 207 (1997) 96-103], Kirk [Ann. Univ. Mariae Cu-
rie-Sklodowska Sec. A LI.2, 15 (1997) 167-178] and Petryshyn and Williamson [J. Math. Anal. Appl. 43
(1973) 459-497]. Some applications of our main results to geometry of Banach spaces are also discussed.
Keywords: Locally Quasi-Nonexpansive, Biased Quasi-Nonexpansive, Conditionally Biased
Quasi-Nonexpansive, Drop, Super Drop
1. Introduction
In the last four decades of the last century, there have
been a multitude of results on fixed points of nonexpan-
sive and quasi-nonexpansive mappings in Banach spaces
(e.g., [5-7, 9-11]) .
Our aim in this note is to point out several obscure
places in the results of Ahmed and Zeyada [J. Math.
Anal. Appl. 274 (2002) 458-465]. In order to rectify and
improve the results of Ahmed and Zeyada, we introduce
the concepts of locally qu asi-nonexpansive, biased quasi-
nonexpansive and conditionally biased quasi-nonexpan-
sive of a mapping w.r.t. a sequence in metric spaces.
Let
X
be a metric space and D a nonempty subset
of
X
. Let T be a mapping of D into X and let


F
T be
the set of all fixed points of T. For a given 0
x
D

, the
sequence of iterate

n
x
is determined by



10
,1,2,3
n
nn
xTx Txn

  (I)
Let
X
be a normed space,

0, 1   and


0,1  ,
the sequence of iterates

n
x
are defined by
 

10
,
1,1,2,3
n
nn
xTx Tx
TI Tn





  (II)


 
,1,0
,
,
1[1 ],
1,2, 3....
n
nn
xT xT x
TITIT
n
 

 


 

(III)
The iteration scheme (I) is called Teoplitz iteration
and the iteration scheme (II) was introduced by Mann
[12] while the iteration scheme (III) was introduced by
Ishikawa [9].
The concept of quasi-nonexpansive mapping was ini-
tiated by Tricomi in 1941 for real functions. It was fur-
ther studied by Diaz and Metcalf [5] and Doston [6,7] for
mappings in Banach spaces. Recently, this concept was
given by Kirk [10] in metric spaces as follows:
Definition 1.1. The mapping T is said to be quasi-
nonexpansive if for each
x
D

and for every


pFT,






,,.dT xpd xp A mapping T is conditionally
quasi-nonexpansive if it is quasi-nonexpansive whenever


FT

.
We now introduce the following definition :
Definition 1.2. The mapping T is said to be locally
quasi-nonexpansive at

pFT if for each
x
D

,






,,dT xpdxp.
Obviously, quasi-nonexpansive locally quasi-nonex-
pansive at each


pFT but the reverse implication

H. K. PATHAK 34
may not be true. To this end, we observe the following
example.
Example 1.1. Let


0,1X and 3 0, 4 D 


be en-
dowed with the Euclidean metric d. Define the mapping
:TD
X
 by

2
3
2
Tx x for each .
x
D Then we
observe that

2
0, 3 FT  

, for all
x
D and 0p




TF

, we have that
  
2
3
,00,
2
dT xpxxdxp,
i.e., T is locally quasi-nonexpansive at


0pFT .
However, one can easily see that T is not locally quasi-
nonexpansive at

2
3
pFT . Indeed, for all 2
0, 3 x





and

2
3
pFT we have
  
2
32 2
,,
23 3
dT xpxxdxp .
Hence we conclude that T is no t quasi-n onexpansive,
although it is locally quasi-nonexpansive at


0
p
FT .
The concept of asymptotic regularity was formally in-
troduced by Browder and Petryshyn [3] for mappings in
Hilbert spaces. Recently, it was defined by Kirk [11] in
metric spaces as follows:
Definition 1.3. The mapping T is said to be asymp-
totically regular if



1
lim ,0
nn
ndTx Tx

  for each
x
D.
2. Main Results
Let N denote the set of all positive integers and


0N Ahmed and Zeyada [1] introduce-ed the fol-
lowing:
Definition 2.1. The mapping T is said to be quasi-
nonexpansive w.r.t. a sequence

n
x
if for all n


and for each

pFT,

1,, nn
dxpdx p
.
The following lemma was quoted by Ahmed and
Zeyada [1] without proof.
Lemma A. If T is quasi-nonexpansive, then T is
quasi-nonexpansive w.r.t. a sequence

0
n
Tx (respec-
tively,
 
0,0
,
nn
TxT x

) for each 0
x
D.
Remark 2.1. We notice that the above lemma is valid
if

0
n
Tx D for each n

 and a given 0
x
D

(or D is
T
-invarient). So the correct version of Lemma
A should be read as follows:
Lemma 2.1. If T is quasi-nonexpansive and for a
given 0
x
D

and each n

,

0
n
Tx D, then T
is quasi-nonexpansive w.r.t. a sequence

0
n
Tx (re-
spectively,




0,0
,
nn
TxT x

) for each 0
x
D.
Further, they claimed that the reverse implication in
Lemma A may not be true in their Example 2.1. We
again notice that there are several obscure places in this
example. We now quote Example 2.1 of Ahmed and
Zeyada [1] in the following:
Example A. Let


0,1X and 4
0, 5 D 


be en-
dowed with the Euclidean metric d We define the
mapping :TD X by

2
2Tx x for each
x
D

.
For a given 01
4
x
D

 we have




1
21 21
10
0
11
,00
22
,
nn
n
n
dTx p
dT xp

 

 
 
 

where

21
14 120
n
n
TDn

N and


FT


0, i.e., T is quasi-nonexpansive w.r.t. a sequence


14
n
T Furthermore, the map T is quasi-nonexpan-
sive w.r.t. a sequence



12 12
n
T and



12,12 12
n
T.
They found that T is neither conditionally quasi-non-
expansive nor quasi-nonexpansive, for 3
4
x
D and




0, 34,034,0
p
Fd dT and D is not
closed.
Remark 2.2. We notice that the following claims
made in Example A were false:
1) :TDX is a mapping. In fact,
  
32
0, 0,1
25
TD X





 .
2)




FT 0, In fact,

1
0, 2 FT  

.
3) T is quasi-nonexpansive w.r.t. a sequence

n
T



14 .
4) T is quasi-nonexpansive w.r.t. a sequence




12 12
n
T and




12,12 12
n
T.
However, (i) can be rectified by taking
X
as
32
0, 25





 or any superset of 32
0, 25




in


0, Even if
this correction is made we find that the remaining state-
ments 2) - 4) will remain false. Consequently, the claim
of Ahmed and Zeyada [1] that the reverse implication in
Lemma 2.1 may not be true seems false.
We now introduce the following definition .
Definition 2.2. The mapping T is said to be locally
quasi-nonexpansive at


pFT w.r.t. a sequence


n
x

H. K. PATHAK 35
if for all n

,

1,, nn
dxpdx p
.
Obviously, locally quasi-nonexpansiveness at p


F
T locally quasi-nonexpansiveness at p



F
T
w.r.t. a sequence


n
x
.
We now state the followin g le m ma without proof.
Lemma 2.2. If T is quasi-nonexpansive w.r.t. a se-
quence


n
x
then T is locally quasi-nonexpansive at
each

pFT w.r.t. the sequence

n
x
.
The reverse implication in Lemma 2.2 may not be true
as shown in the following example:
Example 2.1. Let


0, 1X and 2 0, 3 D



be en-
dowed with the Euclidean metric d Define the map- ping
:TD
X
 by


2
2Tx x for each
x
D Then we
observe that

1
0, 2 FT  

. For a given 01
4
x
D


and

0pFT we have that




1
21 21
10
0
11
,00
22
,
nn
n
n
dTx p
dT xp

 

 
 



*
where
21
1
2
1
4
n
n
TD







 i.e., T is locally quasi-
nonexpansive at


0pFT w.r.t. a sequence
1
4
n
T





However, one can easily see that T is not
locally quasi-nonexpansive at

1
2
pFT w.r.t. the
sequence .
1
4
n
T





Indeed, we ha ve




11
2121
10
0
1111
,2222
,
nn
n
n
dTx p
dT xp


 

 
 



**
for all n

 Consequently, T is neither quasi-nonex-
pansive nor quasi-nonexpansive w.r.t. the sequence
1
4
n
T





.
We now introduce the following:
Definition 2.3. The mapping :TD X is said to
be biased quasi-nonexpansive (b.q.n) w.r.t. a sequence


n
xX
 if for all n

 and at each




cond
p
F
T,

1,, nn
dxpdx p

where
  


 

lico msup
l
nd: ,
,iminf
n
n
n
n
pF dFT T
T
x
p
dx F




A mapping T is conditionally biased quasi-nonex-
pansive (c.b.q.n) w.r.t. a sequence


n
x
if




cond FT

.
Remark 2.3. We observe that the following implica-
tions are obvious:
(a) Conditional biased quasi-nonexpansiveness w.r.t. a
sequence


n
x biased quasi-nonexpansiveness w.r.t.
a sequence


n
x
but the reverse impl ication may not be
true (Indeed, any mapping :TDX for which




cond FT

 is a biased quasi-nonexpansive w.r.t.
a sequence


n
x
but not conditionally biased quasi-
nonexpansive w.r.t. a sequence

n
x
. However, under
certain conditions a biased quasi-nonexpansive map w.r.t.
a sequence


n
x
may be a conditional biased quasi-
nonexpansive w.r.t. a sequence

n
x
(see Lemma 2.6
below).
(b) If T is conditionally biased quasi-nonexpansive
w.r.t. a sequence


n
x
and




cond TFTF
then
T
is locally quasi-nonexpansive at each


pTF
w.r.t. a sequence


n
x
.
(c) If T is biased quasi-nonexpansive w.r.t. a se-
quence


n
x
and



cond FTFT Ö then T
is locally quasi-nonexpansive at each




cond
p
TF
w.r.t. a sequence


n
x
.
(d) Quasi-nonexpansivenes  locally quasi-nonex-
pansiveness at


pFT

 locally quasi-nonexpan-
siveness at


pFT w.r.t. a sequence

n
x
.
In Example 2.1 above, we observe that
1) for


0pFT, we have

21
21
1
,sup 0
2
10
2
lim suplim
lim
n
n
n
n
n
ndxp














2) for

1
2
pFT , we have

21
21
l11 ,sup
22
111
imsup lim
li 2 m22
n
n
nn
n
n
dx p

















21 21
1
liminfliminfl 1 ,0
22
im
nn
n
nnn
dx TF

 
 


 


H. K. PATHAK 36
Here
  
cond 0FT  and in view of (*) and (**),
it is evident that T is conditionally biased quasi-non-
expansive (c.b.q.n.) w.r.t. a sequence 1
4
n
T





and
hence it is biased quasi-nonexpansive (b.q.n.) w.r.t. a
sequence 1
4
n
T





.
We now show in the following example that


cond FT need not be a singleton set.
Example 2.2. Let


0,2X and
 


0,1 1,2D
be endowed with the Euclidean metric d Define the
mapping :TDX by Txx for


0,1x

1, 2 and

2Tx for 2x. Clearly,


FT


0,2 Consider the sequence



1
n
x in
X
then
we observe that
1) for


0pFT, we have

lim suplim supli,10m11
nn
nn
dx p
 
;
2) for


2pFT , we have

lim suplim supli,12m11
nn
nn
dx p
 
;
and
 
,lim inf1lim 1
n
nn
dx FT
.
Thus we have



cond F0,2T and it is evident
that T is conditionally biased quasi nonexpansive
(c.b.q.n.) w.r.t. the sequence



1
n
x in
X
, and
hence it is biased quasi-nonexpansive (b.q.n.) w.r.t. the
sequence



1
n
x in
X
.
However, interested reader can check that if we con-
sider the sequence

n
x
such that 1
n
x

 then



condFT2 Further, we observe that for2p

 
Fcond T and for all n


we have

1,, nn
dxpdx p

Thus, T is conditionally biased quasi-nonexpansive
(c.b.q.n .) w.r.t. the sequ ence


n
x
in
X
.
On the other hand, if we consider the sequence


n
x
such that 1
n
x  then



cond F0T and T is
conditionally biased quasi-nonexpansive (c.b.q.n.) w.r.t.
the sequence

n
x
in
X
.
Remark 2.4. Example 2.2 above also shows that


condFT is a closed set even though T is discon-
tinuous at 2p.
We need the following lemmas to prove our main
theorem:
Lemma 2.3. Let T be locally quasinonexpansive at


pFT w.r.t.

n
x
and
 
lim ,0
n
ndx TF 

.
Then

n
x
is a Cauchy sequence.
Proof. Since



lim ,0
n
ndx TF
  then for any given
0

 there exists 1 n


such that for each 1,nn


,2
n
dx FT


So, there exists


qFT such that
for all

1,,2 n
nndxq

.
Thus, for any 1 ,mn n we have

,,,
22
mn mn
dxxdxqdx q



,


qFT,
Hence


n
x
is a Cauchy sequence.
Lemma 2.4. Let T be conditionally biased quasi-
nonexpansive w.r.t.


n
x
,and

 
lim inf,0
n
ndx TF


Then:
1)


n
x
converges to a point
p
in

 
cond FT and
T
is locally quasi-nonexpansive at




dFcon
p
T w.r.t.


n
x
.
2)


n
x
is a Cauchy sequence.
Proof. 1) Since T is conditionally biased quasi-
nonexpansive w.r.t.


n
x
, it follows that




cond FT
.

As



lim inf,0
n
ndxTF
  we have that


lim sup,0
n
ndx p 

for some
 
dFconpT.So, we
have


lim, 0
n
ndx p 

for some



dFconpT; i.e.,


n
x
converges to a point p in



condF T and
T
is locally quasi-nonexpansive at




dFcon
p
T w.r.t.


n
x
.
2) From


lim, 0
n
ndx p 

it follows that for any
given 0

 there exists 1 nN such that for each

1,,2 n
nndxp

. Thus, for any 1 ,mn n, we have

,,,
22
mn mn
dxxdxqdx q






qFT,
Hence


n
x
is a Cauchy sequence.
The following lemma follows easily.
Lemma 2.5. Let T be biased quasi-nonexpansive
w.r.t.


n
x
, and let


n
x
converges to a point p in


TF Then:
1)


n
x
converges to a point p in




cond FT
and T is conditionally biased quasi-nonexpansive w.r.t.


n
x
;
2)


n
x
is a Cauchy sequence.
We now state our main theorem in the present paper.
Theorem 2.1. Let


FT be a nonempty closed set.
Then
1)




lim ,0
n
ndx TF 

if

n
x
converges to a po-
int p in


FT;
2)


n
x
converges to a point in

FT if

H. K. PATHAK 37


lim ,0
n
ndx TF
 , T is locally quasi-nonexpansive
at
 
w.r.t. n FpT
x
 and
X
is complete.
Proof. 1) Since

FT is closed,

FpT and the
mapping
 
,xTdxF is continuo us (see [1, p. 13]),
then
  

 
lim,lim ,,0
nn
nn
dx Fdx FdpFTTT
 

2) From Lemma 2.3,

n
x
is a Cauchy sequence.
Since
X
is complete, then


n
x
converges to a point,
say q in
X
. Since


FT is closed, then
  

 
0 lim,lim ,,
nn
nn
dx Fdx FTTTdpF
 

implies that

FqT.
As consequences of Theorem 2.1, we have the fol-
lowing:
Corollary 2.1. Let


FT a nonempty closed set and
for a given 0
x
D and each

0
,n nTxD   Then
1)


0
lim ,0
n
nTdTx F   if

0
n
Tx converges to
a point p in


FT;
2)

0
n
Tx co n verges to a poi nt in

FT if,


0
lim ,0,
n
ndTx FT  T is locally qusi-nonexpansive
at

FpT w.r.t.

0
n
Tx and
X
is complete.
Corollary 2.2. Let
X
be a normed linear space,

FT a nonempty closed set and for a given 0
x
D

and each n

,

0
n
Tx D

.
(1) If the sequence

0
n
Tx

converges to a point p
in


FT , then



0
lim ,0
n
ndTx FT

 
(2) If


0
lim ,0
n
ndTx FT



T is locally quasi-
nonexpansive at

FpT w.r.t.

0
n
Tx

and
X
is
complete, then

0
n
Tx

converges to a point p in

FT.
Corollary 2.3. Let
X
be a normed linear space,


FT a nonempty closed set and for a given 0
x
D

and each n

,

,0
n
Tx D

 Then
(1)


,0
lim ,0
n
ndTx FT

  if the sequence


,0
n
Tx

converges to a point p in


FT ;
(2)

,0
n
Tx

converges to a point p in


FT if


,0
lim ,0
n
ndTx FT

 , T is locally quasi-nonexpan-
sive at

FpT w.r.t.

,0
n
Tx

and
X
is complete.
Note that the continuity of T implies that


FT is
closed but the converse need not be true. To effect this
consider the following example.
Example 2.3. Let


0,X and


0, 1D be
endowed with the Euclidean metric d. Define the map-
ing :TD
X
 by


Tx x

if 1 0, 2 x 

and


Tx
2
3
x
 if 1,1
2
x 

Obviously,
 

0,1 2FT  is a
nonempty closed but T is not continuous at 12x

.
Remark 2.5. (a) In order to support the above fact
Ahmed and Zeyada [1] stated wrongly in their Example
2.2, where


0,1X,

 

0,1 41 2,5 6D ,


Tx x

.
If


0,1 4X and

2Tx x if


12,56x
that T is not continuous. In fact, we observe that in this
example T is continuous.
(b) From Lemma 2.1, Examples 2.1 and 2.3, the con-
tinuity of T implies that


FT is closed but the con-
verse may not be true; then we have that Corollaries 2.1,
2.2 and 2.3 are improvement of Theorem 1.1 in [13,
p.462], Theorem 1.1

in [13, p. 469], and Theorem 3.1
in [8, p. 98], respectively.
(c) Since every quasi-nonexpansive map w.r.t. a se-
quence


n
x
is locally quasi-nonexpansive at each


p
F
T w.r.t. a sequence

n
x
, but the converse
may not be true; we have that Theorem 2.1, Corollaries
2.1, 2.2 and 2.3 are improvement of corresponding
Theorem 2.1, Corollary 2.1, 2.2 and 2.3 of Ahmed and
Zeyada [1].
(d) By considering the closedness of

F
T in lieu of
the continuity of T and :TD X instead of
:TX X we have that our Corollary 2.1 improves
Proposition 1.1 of Kirk [10, p. 168].
(e) The closedness condition of D in Theorem 1.1
and 1.1

of Petryshyn and Williamson [12, p. 462, 469]
and Theorem 3.1 in [8, p. 98] is superfluous.
(f) The convexity condition of D in Theorem 1.1

of Petryshyn and Williamson [12, p. 469] is superfluous
because the author assumed in their theorem that


0
n
Tx D


for each n


and a given 0
x
D

in
condition


1.3

.
Theorem 2.2. Let
 
cond FT be a nonempty
closed set. Then


n
x
converges to a point in




cond FT if
 


inf ,cond0lim n
ndx TF


, T is
condionally biased quasi-nonexpansive w.r.t.


n
x
and
X
is complete.
Proof. Since



 
cond FT TF we have that





inf ,cond0lim n
ndx TF 

implies

,iminfln
ndx 



0FT

Now using the technique of the proof of
Theorem 2.1 the conclusion follows from Lemma 2.3.
The following results follows easily from Lemma 2.5.
Theorem 2.3. Let


FT be a nonempty closed set.
Then


n
x
converges to a point in




cond FT if


n
x
converges to a point p in


FT , T is biased
quasi-nonexpansive w.r.t.


n
x
and
X
is complete.

H. K. PATHAK 38
Theorem 2.4. Let
X
be a complete metric space and
let
 
cond FT be a nonempty closed set. Assume that
1) T is biased quasi-nonexpa ns ive w.r.t.

n
x
;
2)

1
lim ,0
nn
ndx x

  or

n
x
is a Cauchy se-
quence;
3) if the sequence


n
y
satisfies

1
lim ,0
nn
ndy y



then




lim in,n0fcod
nn
dy FT
 
or




lim su,n0pcod
nn
dy FT
 .
Then

n
x
converges to a point in



cond FT .
Proof. Since
 
cond FT

 it follows from (i)
that T is condionally biased quasi-nonexpansive w.r.t.


n
x
and the sequence
 


,cond
n
dxFT is mono-
tonically decreasing and bounded from below by zero.
Then
 

inf ,cim ndlo
n
ndx TF  exists.
From 2) and 3), we have that




lim in,n0fcod
nn
dx TF
 
or




lim su,n0pcod
nn
dx TF
 .
Then



,lcim ond0
n
ndx TF   Therefore, by The-
orem 2.2, the sequence

n
x
converges to a point in


cond FT .
As consequences of Theorem 2.4, we obtain the
following:
Corollary 2.4. Let
X
be a complete metric space
and let


cond FT be a nonempty closed set. Assume
that
1) T is biased quasi-nonexpan sive w.r .t.

n
x
;
2) T is asymptoticc regular at 0
x
D( or


0
n
Tx is a Cauchy sequence );
3) if the sequence


n
y
satisfies

1
lim ,0
nn
ndy y



then




limin, n0fcod
nn
dy FT
 
or




lim su,n0pcod
nn
dy FT
 .
Then


0
n
Tx converges to a point in




cond FT .
Corollary 2.5. Let
X
be a Banach space and let


cond FT be a nonempty closed set. Assume that
1) T is biased quasi-nonexpan sive w.r .t.




0
n
Tx ;
2)
T
is asymptoticc regular at 0
x
D (or




0
n
Tx
is a Cauchy sequence );
3) if the sequence


n
y
satisfies lim 0
nn
nyTy



,
then




lim in,n0fcod
nn
dy FT
 
or




lim su,n0pcod
nn
dy FT
 .
Then




0
n
Tx converges to a point in




cond FT .
Corollary 2.6. Let
X
be a Banach space and let




cond FT be a nonempty closed set. Assume that
1)
T
is biased quasi-none xpansive w.r.t.



,0
n
Tx

;
2) T is asymptoticc regular at 0
x
D

(or




,0
n
Tx

is a Cauchy sequence );
3) if the sequence


n
y
satisfies , lim 0
nn
nyTy

 ,
then




limin, n0fcod
nn
dy FT
 
or
 


lim su,n0pcod
nn
dy FT
 .
Then




,0
n
Tx

converges to a point in




cond FT.
Remark 2.6. From Lemmas 2.1 and 2.2, Examples 2.1
and 2.3, Remark 2.3, the continuity of T implies that


F
T is closed but the converse may not be true; we
obtain that Corollary 2.4 include Theorem 1.2 in [12, p.
464] and Theorem 3.2 in [7, p. 99] as special cases.
As another consequence of Theorem 2.1, we establish
the following theorem:
Theorem 2.5. Let
X
be a complete metric space and
let




cond FT be a nonempty closed set. Assume that
1) T is biased quasi-no ne xpansive w.r.t.


n
x
;
2) for every
 
condDTxF there exists x p





cond FT such that

 
1,, nx nx
dxpdx p
;
3) the sequence


n
x
contains a subsequence


j
n
x
converging to
x
D


.
Then


n
x
converges to a point in



cond FT .
Proof. Since




cond FT  it follows from (i)
that T is condionally biased quasi-nonexpansive w.r.t.


n
x
and the sequence
 



,cond
n
dxFT is mono-
tonically decreasing and bounded from below by zero.
Then
 


 

l,conimdlim ,cond
nn
nn
FT FTdxd x
 


H. K. PATHAK 39
0r exists. We now apply Theorem 2.4. It suf fices to
show that r=0. If


lim cond
n
nxx
F
T

  then
=0r. If


cond
F
Tx then




condDTxF

Thus there exists



cond
x
F
Tp  such that






11
,lim ,lim,
lim,lim ,,
x
x
nn
xx
nn
n
xx
nn
dx pdxpdxp
dx pdx pdx p






 

 


This is a contradiction. So,


cond
F
Tx.
Corollary 2.7. Let
X
be a complete metric space,
 
cond
F
T a nonempty closed set and for a given
0
x
D and each n

,

0
n
Tx D. Assume that
1) T is biased quasi-no ne xpansive w.r.t.



0
n
Tx ;
2) for every
 
condDTxF there exists


cond
xTpF such that




100
,,
nn
x
x
dTxpdTxp
;
3) the sequence


0
n
Tx contains a subsequence

 0
j
n
Tx converging to
x
D


.
Then




0
n
Tx converges to a point in
 
cond FT .
Corollary 2.8. Let
X
be a Banach space,


cond
F
T a nonempty closed set and for a given
0
x
D and each n

,



,0
nDTx

 Assume that
1) T is biased quasi-no ne xpansive w.r.t.



,0
n
Tx

;
2) for every



condDTxF there exists x p



cond FT such that
 
100 nn
x
x
Txp Txp

 ;
3) the sequence


0
n
Tx contains a subsequence

 0
j
n
Tx

converging to
x
D


.
Then



0
j
n
Tx

converges to a point in



cond FT .
Corollary 2.9. Let
X
be a Banach space,
 
cond
F
T a nonempty closed set and for a given
0
x
D and each n

,



,0
nDTx

 Assume that
1) T is biased quasi-nonexpa nsive w.r .t.




,0
n
Tx

;
2) for every
 
condDTxF there exists


cond
xTpF such that
 
1
,0 ,0
nn
x
x
Txp Txp
 
 ;
3) the sequence


0
n
Tx contains a subsequence



,0
j
n
Tx

converging to
x
D
.
Then




,0
n
Tx

converges to a point in




cond FT .
Remark 2.7. From Lemmas 2.1 and 2.2, Examples
2.1 and 2.3, Remark 2.3, the continuity of T implies
that


FT is closed but the converse may not be true ;
we obtain that Corollary 2.7 is an improvement of Theo-
rem 1.3 in [13, p. 466].
3. Applications to Geometry of
Banach Spaces
Throughout this section, let R denote the set of real
numbers. Let


,
K
Kzr be a closed ball in a Banach
space
X
. For a sequence

0
nn
x
K

Ú converging to
X
we define

lim DSD,
n
n
x
K
 
where



00
Dconv
x
K
and




1
Dconv D
nnn
xn




and


SD ,
x
K is called a superdrop .
Clearly, for a constant sequence



n
x
x con-
verging to
x
we have 1 DD
nn
n

 so that






D ,conv
x
kxK and is called a drop Thus
the concept of a drop is a special case of superdrop
It is also clear that if

D,
y
xK then




D, D,
y
KxK and if 0z then
y
x.
Recall that a function :X

R is called a lower
semicontinuous whenever



:
x
Xxa   is closed
for each a

R.
Caristi [4] proved the following:
Theorem A. Let


,
X
d be complete and
:X

R a lower semicontinuous function with a
finite lower bound. Let :TXX be any function
such that



   
,dxTxx Tx   for each
.
x
X

Then T has a fixed point.
We now state and prove some applications of our
main results in section 2 to geometry of Banach Spaces.
Theorem 3.1. Let C be a closed subset of a Banach
space
X
let zXC

 and let

,
K
Kzr be a
closed ball of radius


,rdzCR

 Let
x
be an
arbitrary element of C let

n
x
be a sequence in C
converging to
X
and let :TC X be any continuous
function defined implicitly by




SD ,Tx CxK
for each
x
C

in the sense that

D
nn
Tx C for
each n


. Then

H. K. PATHAK 40
1)


lim ,0
n
ndx TF
  if


n
x
converges to a point
p in

FT ;
2)


n
x
converges to a point in


FT if
 
lim ,0
n
ndx TF
 , T is locally quasi-nonexpansive
at

p
F
T w.r.t.

n
x
.
Proof. Without loss of generality we may assume that
0z. Let
x
R   and let


SD ,
X
AxK
Then it is clear that T maps
X
into itself. For given
y
X and a sequence

n
y
converging to y, we
shall estimate

yTy on
X
.
For given
y
X and the corresponding sequence

n
y
there is a sequence

n
b in
X
with
 
1,01
nn n
Ty tbtyt Now

nn
Ty tb

1n ty, we have


nn nn
ty by Ty
so because nn
ybR


, we find that

nn
yTy
tR


.
Thus,

 


nnnn
nn
nn
yTytyb
ty br
ryTy
Rr








Define

,,dxyx yxyX  and


y

ry
Rr
 

 then
X
is complete as a metric space and
:X

R is a continuous function. So,

is a
lower-semicon tinuous function. Also, the above inequal-
ity takes the form








,
nn nn
dyTyy Ty   .
Proceeding to the limit as n we obtain


,dyTy




yTy   for each
y
X

.
There- fore, applying the theorem of Caristi we obtain
that T has a fixed point

ppx for each ,
x
C

i.e.,

FT . By continuity of T,

FT is closed.
Hence the conclusion follows from Theorem 2.1.
Since drop is a special case of super drop, we have the
following:
Corolla ry 3 .1. Let C be a closed subset of a Banach
space
X
let zXC and let

,
K
Kzr be a
closed ball of radius


,rdzCR Let
x
be an
arbitrary element of C, and let :TC X be any (not
necessarily continuous) function defined implicitly by
 
D,Tx CxK for each
x
C. Then
(1)


lim ,0
n
ndx TF
  if


n
x
converges to a point
p in


FT ;
(2)


n
x
converges to a point in


FT if




lim ,0
n
ndx TF 

, T is locally quasi-nonexpansive
at


p
F
T w.r.t.


n
x
.
We now prove the following result for biased quasi-
nonexpansi ve m a pping w. r.t. a seque nce

n
x
.
Theorem 3.2. Let C be a closed subset of a Banach
space
X
let zXC

 and let

,
K
Kzr be a
closed ball of radius

,rdzCR

 Let
x
be an
arbitrary element of C,


n
x
a sequence in C con-
verging to
X
, and let :TCX be any con- tinuous
function defined implicitly by

Tx C


SD ,
x
K
for each
x
C

in the sense that

n
Tx Dn C for
each n


. If


n
x
converges to a point in


,FTT
is biased quasi-nonexpansive w.r.t.


n
x
then


n
x
converges to a point in
 
cond FT .
Proof. Using Theorem 2.3. instead of Theorem 2.1 the
conclusion fo llows on the lines of the proof tech nique of
Theorem 3.1.
As a consequence of Theorem 3.2, we obtain the
following:
Corolla ry 3 .2. Let C be a closed subset of a Banach
space
X
let zXC

 and let

,
K
Kzr be a
closed ball of radius

,.rdzC R

 Let
x
be an
arbitrary element of C, and let :TC X be any (not
necessarily continuous) function defined implicitly by




D,Tx CxK for each
x
C. If


n
x
conver-
ges to a point in


FT,T is biased quasi-nonexpansive
w.r.t.


n
x
then


n
x
converges to a point in




cond FT .
Open Question. To what extent can the continuity
hypothesis on T be muted in Theorems 3.1 and 3.2?
4. References
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Sequence in Complete Metric Spaces and its Applications
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[2] J.-P. Aubin, “Applied Abstract Analysis,” Wiley-Inter-
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[3] F. E. Browder and W. V. Petryshyn, “The Solution by
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