International Journal of Technical Research and Applications e-ISSN: 2320-8163,
www.ijtra.com Volume 3, Issue 4 (July-August 2015), PP. 87-96
COUPLED FIXED POINT THEOREMS FOR  - 
CONTRACTIVE TYPE MAPPINGS IN
PARTIALLY ORDERED PARTIAL B - METRIC
SPACES
K. P. R. Sastry1, K. K. M. Sarma2, Ch. Srinivasa Rao3, Vedula Perraju4
1
Tamil Street, Chinna Waltair, Visakhapatnam-530 017, India.
Professor, Department of Mathematics, Andhra University, Visakhapatnam-530 003, India.
3, 4
Associate Professor, Department of Mathematics, Mrs.A.V.N. College, Visakhapatnam - 530 001, India
[email protected], [email protected], [email protected], [email protected]
2
Abstract— Sastry et.al.[28] used the concept of  
contraction on a complete partially ordered partial b - metric
space to study the existence and uniqueness of coupled fixed
points for nonlinear contractive mappings controlled by a
generalized contractive type condition. In this connection, an
open problem is posed. In this paper, we partially answer the
open problem in the affirmative. Still the open problem awaits
complete answer. Supporting example is also provided.
Index Terms—Component, formatting, style, styling, insert.
Coupled fixed point,   contractive mappings, mixed
monotone property,  admissible, partial b - metric, complete
partially ordered partial b - metric space.
I.
INTRODUCTION
Most of the fixed point theorems in nonlinear analysis
usually start with Banach [7] contraction principle. A huge
amount of literature is witnessed on applications,
generalizations and extensions of this principle carried out by
several authors in different directions like weakening the
hypothesis and considering different mappings. Fixed point
theory is an essential tool in the study of various varieties of
problems in control theory, economic theory, nonlinear
analysis and global analysis. But all the generalizations may
not be from this principle. In 1989, Bakktin [6] introduced the
concept of a b - metric space as a generalization of a metric
space. In 1993, Czerwik[9] extended many results related to
the b - metric space. In 1994, Matthews [17] introduced the
concept of partial metric space in which the self distance of
any point of space may not be zero. In 1996, O'Neill [26]
generalized the concept of partial metric space by admitting
negative distances. In 2013, Shukla [32] generalized both the
concepts of b - metric and partial metric space by introducing
the notation of partial b - metric spaces. Many authors recently
studied the existence of fixed points of self maps in different
types of metric spaces [14,33,20,30,35]. Some authors
[4,18,22,28,29] obtained some fixed point theorems in b metric spaces. After that some authors started to prove
  ~ versions of certain fixed point theorems in different
types of metric spaces [3,14]. Recently Samet et.al [27] and
Jalal Hassanzadeasl [12] obtained fixed point theorems for
  contractive mappings. Mustafa [22] gave a
generalization of Banach contraction principle in complete
ordered partial b - metric space by introducing the notion of a
generalized   weakly contractive mapping. Bhaskar and
Lakshmikantham [5] introduced the notation of mixed
monotone property and proved coupled fixed point theorems.
In 2012, Mohammad Mursaleen et.al [19] proved coupled
fixed point theorems for  -  contractive type mappings in
partially ordered metric spaces. Sastry et.al.[28] discussed
about   contraction on a complete partially ordered
partial metric space to study the existence and uniqueness of
coupled fixed points and posed an open problem to discuss the
same on a complete partially ordered partial b- metric space.
II. TYPE STYLE AND FONTS
In this paper we take up this open problem of Sastry
et.al.[28] to study the existence and uniqueness of coupled
fixed points for contractive mappings controlled by a
generalized contractive type condition on a complete partially
ordered partial b- metric space under  - contraction for
s  1 and gave a partial answer for this problem. In fact, we
show that coupled fixed point theorems on a complete partially
ordered partial b - metric space with coefficient of partial b metric space s  1 by using mixed monotone property under
 - contraction. A supporting example is also given.
Further an open problem is also given at the end of this paper.
Definition 1.1.(H.Aydi et al [4]) Let X be a non empty set
and let s  1 be a given real number. A function
d : X  X  [0, ) is called a b - metric if for all
x, y, z  X the following conditions are satisfied.
(i) d ( x, y)  0 if and only if x  y .
(ii) d ( x, y)  d ( y, x)
(iii) d ( x, y )  s{d ( x, z )  d ( z, y)}
Definition 1.2. (S.G.Matthews [17]) Let X be a non empty
set. A function p : X  X  [0, ) is called a partial
metric, if for all x, y, z  X the following conditions are
satisfied.
(i) x  y if and only if
p( x, x) = p( x, y) = p( y, y) .
p( x, x)  p( x, y)
(iii) p( x, y)  p( y, x)
(iv) p( x, y )  p( x, z ) + p( z, y) - p( z, z )
The pair ( X , p) is called a partial metric space.
(ii)
Shukla [32] introduced the notation of a partial b - metric
space as follows.
87 | P a g e
International Journal of Technical Research and Applications e-ISSN: 2320-8163,
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Definition 1.3.(S.Shukla[32]) Let X be a non empty set and
let s  1 be a given real number. A function
p : X  X  [0, ) is called a partial b - metric if for all
x, y, z  X the following conditions are satisfied.
(i) x  y if and only if p( x, x) = p( x, y ) = p( y, y)
(ii) p( x, x)  p( x, y )
(iii) p( x, y ) = p( y, x)
(iv) p( x, y )  s{ p( x, z )  p( z, y)} - p( z, z )
The pair ( X , p) is called a partial b - metric space. The
number s  1 is called a coefficient of ( X , p) . We note that
if s  1 , we get definition 1.2
Definition 1.4. A sequence {xn } in a partial b - metric
space ( X , p) is said to be:
(i) convergent to a point x  X if lim p( xn , x) =
n
p( x, x)
(ii) a Cauchy sequence if
lim p( xn , xm ) exists and is
n , m
finite
(iii) a partial b - metric space
( X , p) is said to be
complete
if
every
Cauchy
sequence {xn } in
X converges to a point x  X
such that lim p( xn , xm ) = lim p( xn , x) = p( x, x) .
n
n , m
Definition 1.5.(E.Karapinar and B.Samet [15]) Let
a partially ordered set. A sequence
non decreasing, if
( X , ) be
{xn }  X is said to be
xn  xn1 n  N
In Sastry et.al[28], the notion of a partially ordered partial
metric space is introduced.
Definition 1.6.(Z.Mustafa[20]) A triple ( X , , p) is called
Definition 1.9.( T.G.Bhaskar and V.Lakshmikantham [5] ) An
element ( x, y)  X  X is said to be a coupled fixed point
of the mapping
F : X  X  X if F ( x, y)  x and F ( y, x)  y
( X , p) be a
partial b - metric space,  : X  X  [0, ) and
T : X  X be a given mappings. We say that T is  admissible if x, y  X ,  ( x, y)  1 implies that
 (Tx, Ty)  1 . Also we say that T is L - admissible, or
Definition 2.0.(Aiman Mukheimer[18]) Let
R - admissible if x, y  X ,  ( x, y)  1 implies that
 (Tx, y)  1 or  ( x, Ty)  1 respectively.
Definition
2.1.(Mohammad
Murasaleen
mappings. Then
Using the definition 1.6, the following theorems are
established in Sastry et.al.[28].
Theorem 2.2.(Sastry et.al.[28] Theorem 2.7)Let ( X , , p)
be a complete partially ordered partial metric space. Let
F : X  X  X be a mapping having the mixed monotone
X .Suppose   and  : X 2  X 2  [0, )
such that for x, y, u, v  X , the following holds :
 (( x, y),(u, v)) p( F ( x, y), F (u, v)) 
(2.2.1)
 (max{ p( x, u), p( y, v)})  x  u and y  v
property of
Suppose that
(i) F is  - admissible.
(ii)there
exists x0 , y0  X
such
 (( x0 , y0 ),( F ( x0 , y0 ), F ( y0 , x0 )))  1 ,
 (( y0 , x0 ),( F ( y0 , x0 ), F ( x0 , y0 )))  1
definiteness sake Sastry et.al[28](Definition 2.1) adopting the
definition 1.2 for partial metric and defined the triple
( X , , p) as partially ordered partial metric space. A
and
partially ordered partial metric space ( X , , p) is said to be
complete if every Cauchy sequence is convergent.
Definition 1.7.(Mohammad Mursaleen. et. al. [19] ) Define
 = { /  :[0, )  [0, ) is non-decreasing and
(1.7.1)}
lim  (r )  t  t  0
satisfies
r t
1.7.1)
Definition 1.8.(T.G.Bhaskar and V.Lakshmikantham [5])
Let ( X , ) be a partially ordered set and F : X  X  X
be a mapping. Then F is said to have the mixed monotone
property if F ( x, y) is monotone non - decreasing in x and
y ; i.e., for any x, y  X ,
x1 , x2  X , x1  x2  F ( x1 , y)  F ( x2 , y)
(1.8.1)
and y1 , y2  X y1  y2  F ( x, y1 )  F ( x, y2 )
(1.8.2)
is monotone non - increasing in
[19])
2
( X , ) is a partially
ordered set and p is a partial b - metric on X . For
an ordered partial b - metric space if
et.al.
 : X  X  [0, ) be two
F is said to be
 - admissible if , {( x, y),(u, v)}  1 
 ({F ( x, y), F ( y, x)},{F (u, v), F (v, u)})  1
for all x, y, u, v  X
Let F : X  X  X and
2
that
x0  F ( x0 , y0 ) and y0  F ( y0 , x0 ) .
(iii) F is continuous Then F has a coupled fixed point, that is
 x, y  X such that F ( x, y)  x and F ( y, x)  y
Theorem 2.3.(Sastry et.al.[28] Theorem 2.8) Let ( X , , p)
be a complete partially ordered partial metric space. Let
F : X  X  X be a mapping having the mixed monotone
property
of
Suppose
and
 
X.
 : X 2  X 2  [0, )
such that for
x, y, u, v  X , the
following holds :
 (( x, y),(u, v)) p( F ( x, y), F (u, v)) 
 (max{ p( x, u), p( y, v)})  x  u, y  v .
(2.3.1)
Suppose that
(i) F is  - admissible.
88 | P a g e
International Journal of Technical Research and Applications e-ISSN: 2320-8163,
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(ii)
there
exist
x0 , y0  X
such
,  (( y0 , x0 ),( F ( y0 , x0 ), F ( x0 , y0 )))  1 .and
that  (( x0 , y0 ),( F ( x0 , y0 ), F ( y0 , x0 )))  1 ,
x0  F ( x0 , y0 ) and y0  F ( y0 , x0 ) .
 (( y0 , x0 ),( F ( y0 , x0 ), F ( x0 , y0 )))  1.
(iii) F is continuous
(iv)If sequence {xn } is non-increasing and sequence
and x0
 F ( x0 , y0 ) and y0  F ( y0 , x0 )
{xn } is a non-decreasing sequence and { yn } is a nonincreasing sequence, then lim xn  x  xn  x and lim
(iii)If
n
n
yn  y  yn  y
(iv)
Further
If
 (( xn , yn ),
( xn1 , yn1 ))  1 and
then
F has a coupled fixed point, that is  x, y  X such
that F ( x, y)  x and F ( y, x)  y .
Theorem 2.4. (Sastry et.al.[28] Theorem 2.9)Let ( X , , p)
be a complete partially ordered partial metric space. Let
F : X  X  X be a mapping having the mixed monotone
property
of
Suppose
and
 
X.
 : X  X  [0, )
x, y, u, v  X , the
such that for
following holds :
 (( x, y),(u, v)) p( F ( x, y), F (u, v)) 
 (max{ p( x, u), p( y, v)})
 x  u and y  v . Suppose that
(i) F is  - admissible.
x0 , y0  X
(ii)
there
exist
(1.14.1)
such
that
 (( x0 , y0 ),( F ( x0 , y0 ), F ( y0 , x0 )))  1 ,
 (( y0 , x0 ),( F ( y0 , x0 ), F ( x0 , y0 )))  1
lim yn  y  yn  y ;
n
x0  F ( x0 , y0 ) and y0  F ( y0 , x0 ) .
( x, y) and ( s, t ) in X  X , there
exists (u, v)
such
that  (( x, y),(u, v))  1
and
 ((s, t ),(u, v))  1.
Then F has a unique coupled fixed point.
et.al.[28]
Theorem
2.10)Let
( X , , p) be a complete partially ordered partial metric
space. Let F : X  X  X be a mapping having the mixed
monotone property of X . Suppose   
and
 : X 2  X 2  [0, )
such that for x, y, u, v  X , the
following holds:
 (( x, y),(u, v)) p( F ( x, y), F (u, v)) 
 (max{ p( x, u), p( y, v)})
 x  u and y  v . Suppose that
(i) F is  - admissible.
x0 , y0  X
(ii)
there
exist
 (( x0 , y0 ),( F ( x0 , y0 ), F ( y0 , x0 )))  1
 (( xn , yn ),( xn1 , yn1 ))  1
and
F has a coupled fixed point, that is  x, y  X such
that F ( x, y)  x and F ( y, x)  y .
In this connection, the following open problem is posed in
Sastry et.al.[28]. Are theorems 2.2, 2.3, 2.4, 2.5 hold good in
partially ordered partial b - metric spaces ( s  1 )?
In the next section we partially answer the open problem by
imposing a link between coefficient s>1of a partially ordered
partial b - metric space ( X , , p) and the function   .
As theorems 1.12, 1.13, 1.14, 1.15 proved when s  1 in
Sastry et.al [28], we assume, without laws of generality, that
s>1.
Main Result In this section we continue our study of the open
problem of Sastry et.al.[28] under   contractions. We
study the existence and uniqueness of coupled fixed point
theorems by considering maps on complete partially ordered
partial b - metric space under  -  contraction. This paper
is a sequel of Sastry et.al.[28]. We begin this section with the
following definition.
Definition 2.6.(Sastry et.al.[29]) Suppose ( X , ) is a
is a partial b  metric
p asin definition1.3with coefficient s  1 . Then we say
that the triplet ( X , , p) is a partially ordered partial b -
(iii) F is continuous
Suppose for every
2.5.(Sastry
If
 (( yn , xn ),( yn1 , xn1 ))  1  n ,then
 (( xn , yn ),( x, y))  1 and  (( yn , xn ),( y, x))  1
partially
and
Theorem
xn  x
lim
Then
Then
2
then
n
Further
 (( yn , xn ),( yn1 , xn1 ))  1
n ,
 (( xn , yn ),( x, y))  1 and  (( yn , xn ),( y, x))  1
2
non-decreasing
{ yn } is
 xn  x and
(2.5.1)
such
that
ordered
set
and
metric space. A partially ordered partial b - metric space
( X , , p) is said to be complete if every Cauchy sequence in
X is convergent.
( X , , p) be a partially ordered partial b
metric space, with coefficient s  1 .
Define  s = { /  :[0, )  [0, ) is non-decreasing
and satisfies (2.7.1)}
t
lim  (r )   t  0
(2.7.1)
r t
s
We observe that 1 =  . In what follows, we assume,
unless otherwise specified, that ( X , , p) is a partially
Definition 2.7.Let
ordered partial b - metric space with s > 1 .
Now we state the following useful lemmas, whose proofs can
be found in Sastry. et. al [29].
Lemma 2.8.Let ( X , , p) be a complete partially ordered
partial b - metric space.
89 | P a g e
International Journal of Technical Research and Applications e-ISSN: 2320-8163,
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{xn } be a sequence in X such that
lim p( xn , xn1 )  0 . Suppose lim xn  x and lim
Let
n
n
n
xn  y Then lim p( xn , x) = lim p( xn , y ) = p( x, y)
n
n
Now we state our first main result:
Theorem 3.1.Let ( X , , p) be a complete partially ordered
partial b - metric space with coefficient $ s > 1 $. Let
F : X  X  X be a mapping having the mixed monotone
X.
 : X 2  X 2  [0, )
Suppose
  s
and
such that for x, y, u, v  X , the
following holds :
 (( x, y),(u, v))
p( F ( x, y), F (u, v))
 (max{ p( x, u), p( y, v)})
 x  u and y  v .Suppose that
(i) F is  - admissible.
x0 , y0  X
(ii)
there
exist
 (( x0 , y0 ),( F ( x0 , y0 ), F ( y0 , x0 )))  1 ,
 (( y0 , x0 ),( F ( y0 , x0 ), F ( x0 , y0 )))  1
x0  F ( x0 , y0 ) and y0  F ( y0 , x0 ) .
(3.1.1)
that
and
(iii) F is continuous
Then F has a coupled fixed point, that is
F ( x, y)  x and F ( y, x)  y
x0 , y0  X
Proof:
Let
be

such
 x, y  X such
that
as
in
(ii ) .
Then  (( x0 , y0 ),( F ( x0 , y0 ), F ( y0 , x0 )))  1 ,
 (( y0 , x0 ),( F ( y0 , x0 ), F ( x0 , y0 )))  1, x0  F ( x0 , y0 )
and
y0  F ( y0 , x0 ) .
F ( x0 , y0 )  x1 and F ( y0 , x0 )  y1 .
Then clearly x0  x1 and y0  y1 .
Write
Define
x2 , y2  X
as
x2  F ( x1 , y1 )
xn1  F ( xn , yn )  F ( xn1 , yn )  F ( xn1 , yn1 )  xn2
and
Hence sequence
t
 t  0 and s  1 is the coefficient of ( X , p)
s
of
property,
 xn1  xn 2 and yn1  yn 2
n
property
xn  xn1 and yn  yn1 . Then by mixed monotone
yn1  F ( yn , xn )  F ( yn1 , xn )  F ( yn1 , xn1 )  yn2
x y
Lemma 2.9. p( x, y)  0  x  y
n
Lemma3.0. If    s then (i) lim  (t )  0  t  0
and hence
(ii)  (t ) 
Suppose
and
y2  F ( y1 , x1 ) . Now from (ii) ,
 (( x0 , y0 ),( x1 , y1 ))  1
  (( F ( x0 , y0 ), F ( y0 , x0 )),( F ( x1, y1 ), F ( y1, x1 )))  1
  (( x1 , y1 ),( x2 , y2 ))  1
By mixed monotone property x0  x1 and y0  y1
 F ( x0 , y0 )  F ( x1 , y0 )  F ( x1, y1 )
and
F ( y0 , x0 )  F ( y1 , x0 )  F ( y1, x1 )
y1  y2 . Inductively, define

x1  x2 and
xn1  F ( xn , yn ) and yn1  F ( yn , xn ) , n  0,1, 2,...
{xn } is non-decreasing and sequence { yn }
is non-increasing.
Suppose ( xn , yn )  ( xn1 , yn1 ) for some
Then
n.
xn  xn1  F ( xn , yn ) and yn  yn1  F ( yn , xn )
 F has a coupled fixed point. In fact ( xn , yn ) is a coupled
fixed point.
Hence we may assume that
( xn1 , yn1 )  ( xn , yn )  n  0 .
F is  - admissible, we have



(( x0 , y0 ),( x1 , y1 ))
1
(( F ( x0 , y0 ), F ( y0 , x0 )),( F ( x1 , y1 ), F ( y1, x1 )))  1
  (( x1 , y1 ),( x2 , y2 ))  1
By
Mathematical
Induction


(3.1.2)
(( xn , yn ),( xn1 , yn1 ))  1  n  0
Similarly  (( yn , xn ),( yn1 , xn 1 ))  1  n  0 (3.1.3)
Now p ( xn , xn1 )  p ( F ( xn1 , yn1 ), F ( xn , yn ))
  (( xn1 , yn1 ),( xn , yn )) p ( F ( xn1 , yn1 ), F ( xn , yn ))
  ( max { p ( xn1 , xn ), p ( yn1 , yn )})
Similarly p ( yn , yn1 ) 
 ( max { p ( yn1 , yn ), p ( xn1, xn )})
max { p ( xn , xn1 ), p ( yn , yn1 )}


 ( max { p ( xn1 , xn ), p ( yn1, yn )})
1
 n   (n 1 )  n 1 ( by (ii) of lemma 3.0 )
s
where n = max { p ( xn , xn1 ), p ( yn , yn1 )}
Since
 n   (n1 )   2 (n2 )  ...   n (0 )
where 0 $=$ p ( x0 , x1 )  0
{n }
sequence
is
decreasing
and

lim n  lim n (0 )  0 ( by (i ) lemma 3.0)
n 
n
 sequence {n } is convergent and converges to 0
 lim n  0
n
 lim max { p ( xn , xn1 ), p ( yn , yn1 )}  0 (3.1.4)
n
 lim p ( xn , xn1 )  0
n
and
lim p ( yn , yn1 )  0
(3.1.5)
n
Let us show that the sequences
{xn } and { yn } are Cauchy
sequences.
90 | P a g e
International Journal of Technical Research and Applications e-ISSN: 2320-8163,
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If possible let
{xn } or { yn } fails to be a Cauchy sequence,
s max { p ( xmk 1 , xmk ), p ( ymk 1 , ymk )}  ò
Allowing k   and using (3.1.4)
ò  max { p ( xmk , xnk ), p ( ymk , ynk )}  ò
Then either
lim p ( xm , xn )  0 or lim p ( ym , yn )  0
m , n
m , n
 max{ lim p ( xm , xn ) , lim p ( ym , yn )}  0
m , n 
m , n
ò  0 for which we can find sub sequences {mk } and
{nk } of positive integers with
max { p ( xmk , xnk ), p ( ymk , ynk )}  ò .
 ò  max { p ( xmk , xnk ), p ( ymk , ynk )} 
nk  mk  k å

l i m(m a x { p m(k x
k 
, x ) ,pmk y(
nk
equal to ò
Therefore (I) follows.
Again we have p ( xmk , xnk )
nk
y, exists) }and
)
 s( p ( xmk , xnk 1 ) $+$
nk to be the smallest positive integer
and
{mk }
{nk }
å
satisfy max { p ( xmk , xnk ), p ( ymk , ynk )}  ò and
 s(s( p ( xmk , xmk 1 ) + p ( xmk 1 , xnk 1 )) -
max p ( xmk , xnk 1 ), p ( ymk , ynk 1 ) }  ò
Now we explore the properties of the sequences
 s 2 p ( xmk , xmk 1 ) + s 2 p ( xmk 1 , xnk 1 ) + sp ( xnk 1 , xnk )
Further we can choose
p ( xnk 1 , xnk ))
p ( xmk 1 , xmk 1 )) + sp ( xnk 1 , xnk )
Similarly
which use in thesequ el.
I. lim (max { p ( xmk , xnk ), p ( ymk , ynk )})  ò
p ( ymk , ynk )  s 2 p ( ymk , ymk 1 ) +
s 2 p ( ymk 1 , ynk 1 ) $+$ sp ( ynk 1 , ynk )
k 

II. ò 
ò
max { p ( xmk , xnk ), p ( ymk , ynk )} 
s 2 lim inf (max { p ( xmk 1 , xnk 1 ), p ( ymk 1 , ynk 1 )})
s 2 max { p ( xmk , xmk 1 ), p ( ymk , ymk 1 )} 
 s 2 lim sup(max { p ( xmk 1 , xnk 1 ), p ( ymk 1 , ynk 1 )})
s 2 max { p ( xmk 1 , xnk 1 ), p ( ymk 1 , ynk 1 )} 
k 0
k 
s max { p ( xnk , xnk 1 ), p ( ynk , ynk 1 )}
Allowing k   and using (3.1.4)
2
We have ò  s lim inf
sò
III.ò
 s lim inf (max { p ( xmk , xnk 1 ), p ( ymk , ynk 1 )})
3
k 
k 
(max { p ( xmk 1 , xnk 1 ), p ( ymk 1 , ynk 1 )})
 s lim sup(max { p ( xmk , xnk 1 ), p ( ymk , ynk 1 )})
On the other hand,
k 
 sò
Now
p ( xmk 1 , xnk 1 )
 s( p ( xmk 1 , xmk ) + p ( xmk , xnk 1 )) - p ( xmk , xnk )
p ( xmk 1 , xnk )  p ( F ( xmk , ymk ), F ( xnk 1 , ynk 1 ))
  (( xmk , ymk ),( xnk 1 , ynk 1 ))
 s( p ( xmk 1 , xmk ) $+$ p ( xmk , xnk 1 ))
p ( F ( xmk , ymk ), F ( xnk 1 , ynk 1 ))
 sp ( xmk 1 , xmk ) + sò
  (max{ p( xmk , xnk 1 ), p( ymk , ynk 1 ) })
Similarly
(3.1.6)

p ( ymk 1 , ynk )
 (max{ p( ym , yn 1 ), p( xm , xn 1 ) }
k
k
k
(3.1.7)
k
from
(3.1.6)
and
(3.1.7)

ò  max{ p ( xmk 1 , xnk ), p ( ymk 1 , ynk )} 
we
have
k
k
ò
s
Now,
k
k
(3.1.8)
p ( xmk , xnk )

s( p ( xmk , xmk 1 )
p ( xmk 1 , xnk )) - p ( xmk 1 , xmk 1 )
 s( p ( xmk , xmk 1 ) + p ( xmk 1 , xnk ))
 sp ( xmk , xmk 1 ) + ò ( by (3.1.8))
Similarly,
p ( ymk , ynk )  sp ( ymk 1 , ymk ) + ò
( by (3.1.8))
Similarly
p ( ymk 1 , ynk 1 )  sp ( ymk 1 , ymk ) + sò
 max { p ( xmk 1 , xnk 1 ), p ( ymk 1 , ynk 1 )} 
s max { p ( xmk 1 , xmk ), p ( ymk 1 , ymk )} + sò
(3.1.10)
Allowing k   and using (3.1.4)
lim sup(max { p ( xmk 1 , xnk 1 ), p ( ymk 1 , ynk 1 )})
k 
 (max{ p( xm , xn 1 ), p( ym , yn 1 ) })   (ò)
<
(3.1.9)
+
(3.1.1)
 sò
 From (3.1.9) and (3.1.11)
ò
s 2 lim inf (max { p ( xmk 1 , xnk 1 ), p ( ymk 1 , ynk 1 )})
k 0
 s lim sup(max { p ( xmk 1 , xnk 1 ), p ( ymk 1 , ynk 1 )})
2
k 
sò
3
Therefore ( II ) follows.
Finally we have ,
p ( xmk , xnk )  s( p ( xmk , xnk 1 ) + p ( xnk 1 , xnk ))
91 | P a g e
International Journal of Technical Research and Applications e-ISSN: 2320-8163,
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 sò + sp ( xnk 1 , xnk ))
tk
 tk
s
sò
ò  lim  (tk )  lim   (tk )   ò ,
k 
tk ( sò)
s
From (3.1.14) again,
Similarly
p ( ymk , ynk )  s( p ( ymk , ynk 1 ) + p ( ynk 1 , ynk ))
 sò + sp ( ynk 1 , ynk ))
 ò
max { p ( xmk , xnk ), p ( ymk , ynk )}

ò   (tk ) 
is a contradiction.
 sequences {xn },{ yn } are Cauchy sequences.

smax { p ( xmk , xnk 1 ), p ( ymk , ynk 1 )}
 by(3.1.4), lim p ( xm , xn )  0
m , n
lim p ( ym , yn )  0
smax { p ( xnk 1 , xnk ), p ( ynk 1 , ynk )}
and
 sò  s max { p ( xnk 1 , xnk ), p ( ynk 1 , ynk )}
Allowing k   and using (3.1.4)
ò
 s lim inf (max { p ( xmk , xnk 1 ), p ( ymk , ynk 1 )})
{xn }  x and { yn }  y as n  
 lim p ( xn , x)  0  p ( x, x)
 s lim sup(max { p ( xmk , xnk 1 ), p ( ymk , ynk 1 )})
We have
m , n
Let sequence
n
and
lim p ( yn , y)  0  p ( y, y)
n
k 
k 
Now
 sò
xn1  F ( xn , yn ) and yn1  F ( yn , xn )
lim xn1  lim F ( xn , yn )  F ( x, y)
n
and also
Therefore (III) follows.
Now
from
p( xmk , xnk ) 
n
lim xn1  x
n
p ( F ( xmk 1 , ymk 1 ), F ( xnk 1 , ynk 1 ))
 By lemma 2.8 and (3.1.4), we have x  F ( x, y)
Similarly y  lim yn1  lim F ( yn , xn )  F ( y, x)
  (( xmk 1 , ymk 1 ),( xnk 1 , ynk 1 ))
 x and y are coupled fixed points of F in X .
(3.1.1)
n 
n
p ( F ( xmk 1 , ymk 1 ), F ( xnk 1 , ynk 1 ))
  (max{ p( xmk 1 , xnk 1 ), p( ymk 1 , ynk 1 ) })
(3.1.12)
Similarly
p ( ymk , ynk )  p ( F ( ymk 1 , xmk 1 ), F ( ynk 1 , xnk 1 ))
property
 p ( F ( ymk 1 , xmk 1 ), F ( ynk 1 , xnk 1 ))
 (( ym 1 , xm 1 ),( yn 1 , xn 1 ))
  (max{ p( ym 1 , yn 1 ), p( xm 1 , xn 1 ) }
k
k
k
k
k
k
from
(3.1.10)
and
(3.1.11)

ò  max{ p ( xmk , xnk ), p ( ymk , ynk )}
(3.1.13)
we
have
every
But by (II), lim inf (tk )  lim sup (tk )  sò
k 
Hence
k 
sò  lim inf (tk )  lim sup (tk )  sò
k 
k 
t
 lim (tk )  sò  lim ( k )  ò
k 
k  s
t
Hence from (3.1.14), ò  lim  (tk )  lim ( k )  ò
k 
k  s
 lim  (tk )  ò
k 
such that for
and
x, y, u, v  X , the
 (( x, y),(u, v))
p( F ( x, y), F (u, v))
 (max{ p( x, u), p( y, v)})  x  u, y  v

(3.2.1)
Suppose that
(i) F is  - admissible.
(ii)
1
(3.1.14)
  (tk )  tk
s
where tk  max{ p( xmk 1 , xnk 1 ), p( ymk 1 , ynk 1 )
for
  s
Suppose
following holds :
  (max{ p( xmk 1 , xnk 1 ), p( ymk 1 , ynk 1 )
 sò  tk
X.
of
 : X 2  X 2  [0, )
k
k
Now we state and prove our second main result:
Theorem 3.2.
Let ( X , , p) be a complete partially ordered partial
b - metric space with coefficient
s > 1 . Let
F : X  X  X be a mapping having the mixed monotone
there
x0 , y0  X
exist
such
that
 (( x0 , y0 ),( F ( x0 , y0 ), F ( y0 , x0 )))  1 ,
 (( y0 , x0 ),( F ( y0 , x0 ), F ( x0 , y0 )))  1.
k
and
x0  F ( x0 , y0 ) and y0  F ( y0 , x0 )
(iii)If {xn } is a non-decreasing sequence and { yn } is a nonincreasing sequence, then lim xn  x  xn  x and
n
lim yn  y  yn  y  n  N
n
(iv)
Further
If
 (( xn , yn ),( xn1 , yn1 ))  1
 (( yn , xn ),( yn1 , xn1 ))  1
n ,
 (( xn , yn ),( x, y))  1 and  (( yn , xn ),( y, x))  1
and
then
92 | P a g e
International Journal of Technical Research and Applications e-ISSN: 2320-8163,
www.ijtra.com Volume 3, Issue 4 (July-August 2015), PP. 87-96
F has a coupled fixed point, that is~  x, y  X such
that F ( x, y)  x and F ( y, x)  y .
Proof: Using (ii ) , define the sequences {xn } and { yn } as
Then
in Theorem 3.1. Then it can be shown, as in Theorem 3.1,
sequences {xn } and { yn } are Cauchy sequences. Suppose
lim xn  x and lim yn  y
n
n
{xn } is non-decreasing and sequence { yn } is
non-increasing, by (iii ) , lim xn  x  xn  x and
Since sequence
n
lim yn  y  yn  y
n
and also from (3.1.2) and (3.1.3) of Theorem 3.1
 (( xn , yn ),( x, y))  1 and  (( yn , xn ),( y, x))  1
 p( xn1 , F ( x, y)) = p( F ( xn , yn ), F ( x, y ))
  (( xn , yn ), ( x, y ))
p( F ( xn , yn ), F ( x, y))
  (max{( p( xn , x), p( yn , y ))})
 xn  x and yn  y
Similarly
p( yn1 , F ( y, x))   (max{( p( yn , y), p( xn , x))})
 xn  x and yn  y
 max { p( xn1 , F ( x, y)), p( yn1 , F ( y, x))}
  (max{( p( xn , x), p( yn , y))})
 lim max { p( xn1 , F ( x, y)), p( yn1 , F ( y, x))}
n
 lim  (max{( p( xn , x), p( yn , y))})
n
 lim max{( p( xn , x), p( yn , y))})  0 (from (3.1.5) of
n
theorem 3.1 )
 lim p( xn1 , F ( x, y))  0 and lim p( yn1 , F ( y, x))  0
n
n
 lim xn1  F ( x, y)) and lim yn1  F ( y, x))
n
n
{xn } and sequence { yn } are Cauchy
sequences and lim xn  x ;
Since sequence
n
lim yn  y  lim xn1  x and lim yn1  y
n
n
n
p( x, F ( x, y))
 s( p( x, xn1 ) + p( xn1 , F ( x, y))) - p( xn1 , xn1 )
 p( x, F ( x, y))  0 and similarly p( y, F ( y, x))  0
Hence by lemma 2.9 , x  F ( x, y) and y  F ( y, x)
 F has coupled fixed point in X .
Further
Now we state and prove our third main result.
In this result, we obtain a sufficient condition which assumes
the uniqueness of coupled fixed point.
Theorem 3.3. Let ( X , , p) be a complete partially ordered
partial b - metric space with coefficient S > 1. Let
F : X  X  X be a mapping having the mixed monotone
property
of
X.
 : X 2  X 2  [0, )
  s
Suppose
such that for
and
x, y, u, v  X , the
following holds :
 (( x, y),(u, v))
p( F ( x, y), F (u, v))
 (max{ p( x, u), p( y, v)})
 x  u and y  v . Suppose that
(i) F is  - admissible.
(ii)
there
exist
x0 , y0  X

(3.3.1)
such
that
 (( x0 , y0 ),( F ( x0 , y0 ), F ( y0 , x0 )))  1 ,
 (( y0 , x0 ),( F ( y0 , x0 ), F ( x0 , y0 )))  1and
x0  F ( x0 , y0 ) and y0  F ( y0 , x0 ) .
( x, y) and ( w, t ) in
X  X , there exists (u, v) such that  (( x, y),(u, v))  1
and  (( w, t ),(u, v))  1 .
Then F has a unique coupled fixed point.
(iii) F is continuous Suppose for every
Proof: We have by theorem 3.1 the set of coupled fixed points
is non-empty. Suppose ( x, y ) and ( w, t ) are coupled points
of the mapping F : X  X  X , that is
x  F ( x, y) ,
y  F ( y, x) and w  F (w, t ) , t  F (t , w) . By
assumption, there exists (u, v) in X  X such that (u, v) is
comparable to ( x, y ) and ( w, t ) . Let u  u0 and v  v0 .
Write
u1  F (u0 , v0 ) and v1  F (v0 , u0 ) . Thus inductively
{un } and {vn } as
un1  F (un , vn ) and vn1  F (vn , un ) Since (u, v) is
comparable
to
therefore
( x, y) ,
x  u  u0  F ( x, y)  F (u0 , y)  F (u0 , v0 ) ( y  v0 )
 x  F (u0 , v0 )  u1 and similarly y  v1
 x  u1 and y  v1 and by induction x  un and y  vn
n  1.
Since u  u0 and v  v0 , we get  (( x, y),(u0 , v0 ))  1

 (( F ( x, y), F ( y, x)),( F (u0 , v0 ), F (v0 , u0 )))  1 
 (( x, y),(u1 , v1 ))  1
By
mathematical
induction,
we
obtain

 (( x, y),(un , vn ))  1  n  1
(3.3.2)
Similarly  (( y, x),(vn , un ))  1  n  1
(3.3.3)
 from (3.3.1),(3.3.2),(3.3.3) we have
p ( x, un1 )  p ( F ( x, y), F (un , vn ))
  (( x, y),(un , vn )) p ( F ( x, y), F (un , vn ))
(3.3.4)
  (max{( x, un ),( y, vn )})
we
can
define
sequences
Similarly
p ( y, vn1 )   (max{( y, vn ),( x, un )})
 from (3.3.4),(3.3.5) we have
(3.3.5)
93 | P a g e
International Journal of Technical Research and Applications e-ISSN: 2320-8163,
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max{ p ( x, un1 ), p ( y, vn1 )}
{xn } is non-increasing and sequence { yn } is
non-decreasing then lim xn  x  xn  x and lim
(iv)If sequence
  (max{( x, un ),( y, vn )})  n 1   ( n )
where n 1 = max { p ( x, un1 ), p ( y, vn1 )}
 n 1   ( n ) 
where
0

sequence
=
n 1
( 0 )
max { p ( x, u0 ), p ( y, v0 )}
n1  lim
n
 (n1 )  ...  
2
n 1
{n }
is
decreasing
lim
and
n
(0 )  0 (by (i ) lemma 3.0)
 sequence {n } is convergent and converges to 0
 lim n1  0  lim max { p ( x, un1 ), p ( y, vn1 )}
n
n
(3.3.6)
0
 lim { p ( x, un1 )}  0 and lim { p ( y, vn1 )}  0
s2
lim { p (w, un1 )}  0 and lim { p (t , vn1 )}  0
Define F : X  X  X by
n
1
F ( x, y)   x(1  y ) if  x, y  X
2
2
2
Define  : X  X  [0, ) by
1 if x  y, u  v
 (( x, y), (u, v)))  
0 otherwise
t
t
t
Define  (t ) 
  (t )    (t ) 
2
3
s
Now
p ( x, w)
 s( p ( x, un1 )  p (un1 , w))  p (un1 , un1 )
 sp ( x, un1 )  sp (un1 , w)
 p( x, w)
 s lim p ( x, un1 ) + s lim p (un1 , w)  0
n 
n 
 p ( x, w)  0
Similarly p ( y, t )  0
Hence by lemma 2.9, x  w and y  t
 F has unique coupled fixed point.
Without
and
 : X 2  X 2  [0, )
such that for
x, y, u, v  X , the following holds :
 (( x, y),(u, v))
p( F ( x, y), F (u, v))
 (max{ p( x, u), p( y, v)})
 x  u and y  v . Suppose that
(i) F is  - admissible.
x0 , y0  X
(ii)
there
exist
that  (( x0 , y0 ),( F ( x0 , y0 ), F ( y0 , x0 )))  1
,  (( y0 , x0 ),( F ( y0 , x0 ), F ( x0 , y0 )))  1 .and
x0  F ( x0 , y0 ) and y0  F ( y0 , x0 ) .
(iii) F is continuous ( Or )
of
generality,
we
may
assume
that
 p( x, y)  x 2 ,
partial b - metric space with $ s > 1$. Let F : X  X  X
be a mapping having the mixed monotone property of X .
  s
loss
0  y  x 1
The following theorem which is the fourth main result can
be established on the lines of the proof of theorems 3.1 and
3.2
Theorem 3.4.Let ( X , , p) be a complete partially ordered
Suppose
Proof: follows as in the theorems 3.1 and 3.2
Now we give an example in support of theorem 3.1
3.5 Example: Let X  [0,1] with usual ordering. Define
is a partially ordered partial b - metric space with coefficient
Similarly
n
yn  y  yn  y  n  N ;
Further
If
~
and
 (( xn , yn ),( xn1 , yn1 ))  1
then
 (( yn , xn ),( yn1 , xn1 ))  1
n ,
 (( xn , yn ),( x, y))  1 and  (( yn , xn ),( y, x))  1
Then F has a coupled fixed point, that is  x, y  X such
that F ( x, y)  x and F ( y, x)  y .
p( x, y)  (max{x, y})2  x, y  X . Clearly, ( X , , p)
n
n
n
n

(3.4.1)
such
p( F ( x, y), F (u, v)) = (max {F ( x, y), F (u, v)})2
1
1
2
2
= max({ x(1  y )} ,{ u (1  v)} )
2
2
2
2
2
p( x, u)  max( x , u ), p( y, v)  max( y , v 2 ) ,
 (max{ p( x, u), p( y, v)})

1
2
2
2
2
2
2
= (max{x , u }, max{ y , v })  (max{x , u })
3
(since x  y and u  v )
Now
 (( x, y),(u, v))
p( F ( x, y), F (u, v)) = p( F ( x, y), F (u, v))
1
1
2
2
= max({ x(1  y )} ,{ u (1  v)} )
2
2
1
2
2
and  (max{ p( x, u), p( y, v)}) = (max{x , u })
3
Now we show that
 (( x, y),(u, v)))
p( F ( x, y), F (u, v))
(3.5.1)
 (max{ p( x, u), p( y, v)})

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International Journal of Technical Research and Applications e-ISSN: 2320-8163,
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That is, we show that
1
1
max({ x(1  y )}2 ,{ u (1  v)}2 )
2
2
1
 (max{x 2 , u 2 })
3
(3.5.2)
The following are the cases
Case(1):
Let
1
{ x(1  y )}2
2

1
{ u (1  v)}2
2
and
1 2 1 2
x  u
3
3
1
1
{ x(1  y )}2  x 2  (3.5.2) holds  (3.5.1) holds
3
2
1
1
2
Case(2): Let { x(1  y )}
 { u (1  v)}2 and
2
2
1 2 1 2
u  x
3
3
1 2
1 2
1
{ x(1  y )}2 
u  (3.5.2) holds
x 
3
3
2
 (3.10.1) holds
1
1
2
Case(3): Let { x(1  y )}
 { u (1  v)}2 and
2
2
1 2 1 2
x  u
3
3
1 2
1 2
1
{ u (1  v)}2 
u 
x  (3.5.2) holds
3
3
2
 (3.5.1) holds
1
1
2
Case(4): Let { x(1  y )}
 { u (1  v)}2 and
2
2
1 2 1 2
u  x
3
3
1
1
{ u (1  v)}2  u 2  (3.5.2) holds  (3.5.1) holds
3
2
Hence
 (( x, y),(u, v)))
p( F ( x, y), F (u, v))

 (max{ p( x, u), p( y, v)})
1
(i)
Let
and
x'  F ( x, y )  x(1  y )
2
1
y '  F ( y, x)  y (1  x)
2
1
Further
let
and
u '  F (u, v)  u (1  v)
2
1
v'  F (v, u )  v(1  u )
2
'
'
'
'
'
'
'
'
Since x , y , u , v  X and x  y , u  v
  (( x' , y ' ),(u ' , v' ))  1  F is  - admissible.
0  x0 ,0  y0
(ii)
Taking
F ( x0 , y0 )  0, F ( y0 , x0 )  0

then
 (( x0 , y0 ),( F ( x0 , y0 ), F ( y0 , x0 )))  1
and
 (( y0 , x0 ),( F ( y0 , x0 ), F ( x0 , y0 )))  1
(iii) F is continuous.
 F satisfies the hypothesis of theorem 2.6
Now F (0,0)  0 and F (0,0)  0
Hence (0,0) is a coupled fixed point of F in X +
Note It can be shown that this example also supports
Theorem 3.2 .
Open Problem: Are the theorems 3.1, 3.2, 3.3 and 3.4 true for
partial b - metric spaces with coefficient s  1 when  is
defined independent of s ?
III.
ACKNOWLEDGEMENTS
The fourth author is grateful to the management of
Mrs.AVNCollege,
Visakhapatnam
and
Mathematics
Department of Andhra University for giving necessary
permission and providing necessary facilities to carry on this
research.
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