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Applied Mathematics E-Notes, 14(2014), 69-85 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
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Fixed Points Of ( ; ')-Almost Weakly Contractive
Maps In G-Metric Spaces
Gutti Venkata Ravindranadh Babuy, Dasari Ratna Babuz,
Kanuri Nageswara Raox, Bendi Venkata Siva Kumary
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Received 26 August 2013
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Abstract
In this paper, we introduce ( ; ')-almost weakly contractive maps in G-metric
spaces and prove the existence of …xed points. Our Theorem 4 generalizes the
result of Aage and Salunke (Theorem 2, [1]). We also extend it to a pair of weakly
compatible maps and prove the existence of common …xed points. We provide
examples in support of our results.
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Introduction and Preliminaries
The development of …xed point theory is based on the generalization of contraction
conditions in one direction or/and generalization of ambient spaces of the operator
under consideration on the other. Banach contraction principle plays an important role
in solving nonlinear equations, and it is one of the most useful results in …xed point
theory. In the direction of generalization of contraction conditions, in 1997, Alber
and Guerre-Delabriere [3] introduced weakly contractive maps which are extensions
of contraction maps and obtained …xed point results in the setting of Hilbert spaces.
Rhoades [16] extended this concept to metric spaces. In 2008, Dutta and Choudhury
[12] introduced ( ; ')-weakly contractive maps and proved the existence of …xed points
in complete metric spaces. In 2009, Doric [11] extended it to a pair of maps. For more
literature in this direction, we refer to Choudhury, Konar and Rhoades [9], Babu,
Nageswara Rao and Alemayehu [4], Sastry, Babu and Kidane [17], Babu and Sailaja
[5] and Zhang and Song [19]. In continuation to the extensions of contraction maps,
Berinde [7] introduced ‘weak contractions’ as a generalization of contraction maps.
Berinde renamed ‘weak contractions’as ‘almost contractions’in his later work [8]. For
more works on almost contractions and its generalizations, we refer to Babu, Sandhya
and Kameswari [6], Abbas, Babu and Alemayehu [2] and the related references cited
in these papers.
Throughout this paper, we denote R+ = [0; 1) and
= f = : R+ ! R+ is continuous on R+ ;
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is nondecreasing,
(t) > 0 for t > 0; (0) = 0g :
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Mathematics Subject Classi…cations:47H10, 54H25.
of Mathematics, Andhra University, Visakhapatnam-530 003, India
z Department of Mathematics, PSCMRCET, Vijayawada-520 001, India
x Department of Mathematics, GIET, Rajahmundry, India
y Department
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Fixed Points of ( ; ')-Almost Weakly Contractive Maps
In the metric space setting Dutta and Choudhury [12] introduced ( ; ')-weakly contractive maps as follows:
DEFINITION 1 ([12]). Let (X; d) be a metric space. Let T : X ! X be a map. If
there exist ; ' 2 such that
(d(T x; T y))
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'(d(x; y))
for all x; y 2 X, then T is said to be a ( ; ')-weakly contractive map.
Dutta and Choudhury [12] proved that every ( ; ')-weakly contractive map has a
unique …xed point in complete metric spaces. On the other hand, Berinde [7] introduced
‘weak contractions’as a generalization of contraction maps.
DEFINITION 2 ([7]). Let (X; d) be a metric space. A selfmap T : X ! X is said
to be a weak contraction if there exist 2 (0; 1) and L 0 such that for all x; y 2 X,
d(T x; T y)
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(d(x; y))
d(x; y) + Ld(y; T x):
Berinde [7] proved that every weak contraction has a …xed point in complete metric
spaces and provided an example to show that this …xed point need not be unique. In
order to obtain the uniqueness of …xed point, Berinde [7] used the following condition:
there exist 2 (0; 1) and L1 0 such that
d(T x; T y)
d(x; y) + L1 d(x; T x) for all x; y 2 X
(1)
and proved that every weak contraction together with (1) has a unique …xed point in
complete metric spaces, and further posed the following problem: “Find a contractive
type condition di¤erent from (1), that ensures the uniqueness of …xed point of weak
contractions".
In this context Babu, Sandhya and Kameswari [6] answered the above problem by
introducing ‘condition (B)’as follows:
DEFINITION 3 ([6]). Let (X; d) be a metric space. A map T : X ! X is said to
satisfy condition (B) if there exist 0 < < 1 and L 0 such that for all x; y 2 X,
d(T x; T y)
d(x; y) + L minfd(x; T x); d(y; T y); d(x; T y); d(y; T x)g:
Babu, Sandhya and Kameswari [6] proved that every selfmap T of a complete
metric space satisfying condition (B) has a unique …xed point. On the other hand,
in the direction of generalization of ambient spaces, in 2005, Mustafa and Sims [15]
introduced a new notion namely generalized metric space called G-metric space and
studied the existence of …xed points of various types of contraction mappings in Gmetric spaces.
DEFINITION 4 ([15]). Let X be a nonempty set and let G : X 3 ! R+ be a
function satisfying:
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(G1) G(x; y; z) = 0 if x = y = z,
(G2) 0 < G(x; x; y) for all x; y 2 X, with x 6= y;
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(G3) G(x; x; y)
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(G4) G(x; y; z) = G(x; z; y) = G(y; z; x) =
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(G5) G(x; y; z)
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G(x; a; a) + G(a; y; z) for all x; y; z; a 2 X (rectangle inequality).
EXAMPLE 1 ([15]). Let (X; d) be a metric space. The mapping Gs : X 3 ! R+
de…ned by
Gs (x; y; z) = d(x; y) + d(y; z) + d(x; z)
for all x; y; z 2 X is a G-metric and so (X; Gs ) is a G-metric space.
EXAMPLE 2 ([15]). Let (X; d) be a metric space. The mapping Gm : X 3 ! R+
de…ned by
Gm (x; y; z) = max fd(x; y); d(y; z); d(x; z)g
for all x; y; z 2 X is a G-metric and so (X; Gm ) is a G-metric space.
EXAMPLE 3. Let X be a nonempty set. We denote the class of all real valued
bounded functions on X by B(X). For f 2 B(X), we de…ne
kf k = sup fjf (x)j =x 2 Xg :
Then (B(X); jj:jj) is a normed linear space. We de…ne metric d on B(X) by d(f; g) =
kf gk for f; g 2 B(X). Now we de…ne generalized metric G on B(X) by
G(f; g; h) = kf
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(symmetry in all variables) and,
Then the function G is called a generalized metric, or, more specially a G-metric on
X, and the pair (X; G) is called a G-metric space.
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G(x; y; z) for all x; y; z 2 X with z 6= y
gk + kg
hk + kh
fk
for all f; g; h 2 B(X). Then clearly G is a generalized metric on B(X). The space
(B(X); G) is a generalized metric space.
DEFINITION 5 ([15]). Let (X; G) be a G-metric space and let fxn g be a sequence
of points of X. We say that fxn g is G-convergent to x if limn;m!1 G(x; xn ; xm ) = 0;
that is, for any > 0, there exists N 2 N such that G(x; xn ; xm ) < for all n; m N .
We refer to x as the limit of the sequence fxn g.
PROPOSITION 1 ([15]). Let (X; G) be a G-metric space. Then for any x; y; z; a 2
X we have that:
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(1) if G(x; y; z) = 0; then x = y = z:
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(2) G(x; y; z)
G(x; x; y) + G(x; x; z):
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Fixed Points of ( ; ')-Almost Weakly Contractive Maps
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(3) G(x; y; y)
2G(y; x; x):
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(4) G(x; y; z)
G(x; a; z) + G(a; y; z):
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(5) G(x; y; z)
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PROPOSITION 2 ([15]). Let (X; G) be a G-metric space. Then the following
statements are equivalent:
(1) fxn g is G-convergent to x:
(2) G(xn ; xn ; x) ! 0 as n ! 1:
(3) G(xn ; x; x) ! 0 as n ! 1:
DEFINITION 6 ([15]). Let X be a G-metric space. A sequence fxn g is called GCauchy if given > 0, there is an N 2 N such that G(xn ; xm ; xl ) < for all n; m; l N ;
that is, if G(xn ; xm ; xl ) ! 0 as n; m; l ! 1:
PROPOSITION 3 ([15]). In a G-metric space X, the following two statements are
equivalent:
(1) The sequence fxn g is G-Cauchy.
(2) For every > 0, there exists N 2 N such that G(xn ; xm ; xm ) < for all n; m
PROPOSITION 4 ([15]). Let X be a G-metric space. Then the function G(x; y; z)
is jointly continuous in all three of its variables.
PROPOSITION 5 ([15]). Every G-metric space X de…nes a metric space (X; dG )
by
dG (x; y) = G(x; y; y) + G(y; x; x) for all x; y 2 X:
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Mustafa, Obiedat and Awawdeh [14] proved the following result.
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THEOREM 1 ([14]). Let (X; G) be a complete G-metric space, and let T : X ! X
be a mapping satisfying one of the following conditions:
G(T x; T y; T z)
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N:
DEFINITION 7 ([15]). A G-metric space X is said to be G-complete (or a complete
G-metric space) if every G-Cauchy sequence in X is G-convergent in X.
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(G(x; y; a) + G(x; a; z) + G(a; y; z)) :
aG(x; y; z) + bG(x; T x; T x) + cG(y; T y; T y) + dG(z; T z; T z)
or
G(T x; T y; T z)
aG(x; y; z) + bG(x; x; T x) + cG(y; y; T y) + dG(z; z; T z)
for all x; y; z 2 X where 0 a + b + c + d < 1. Then T has a unique …xed point (say
u, i.e., T u = u), and T is G-continuous at u.
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In 2011, Aage and Salunke [1] introduced weakly contractive maps in G-metric
spaces and proved the existence of …xed points in G-metric spaces.
DEFINITION 8. Let (X; G) be a G-metric space. Let T : X ! X be a selfmap of
X. T is said to be a weakly contractive map in G if, there exists ' 2 such that
G(T x; T y; T z)
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(2)
DEFINITION 9 ([10]). Let f and g be two selfmaps on a G-metric space (X; G).
The mappings f and g are said to be compatible if limn!1 G (f gxn ; gf xn ; gf xn ) = 0
whenever fxn g is a sequence in X such that limn!1 f xn = limn!1 gxn = z for some
z 2 X.
DEFINITION 10 ([10,13]). Two maps f and g on a G-metric space (X; G) are said
to be weakly compatible if they commute at their coincidence point.
Here we note that every pair of compatible maps is weakly compatible but its
converse need not be true (Example 1.4, [10]). Shatanawi [18] proved the following
common …xed point theorem for a pair of weakly compatible maps.
THEOREM 3 ([18]). Let X be a G-metric space. Suppose the maps f; g : X !
X satisfy the following condition: there exists a nondecreasing function
: R+ !
R+ satisfying limn!1 n (t) = 0 for all t 2 (0; 1) such that either
G(f x; f y; f z)
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'(G(x; y; z)) for each x; y; z 2 X:
THEOREM 2 ([1]). Let (X; G) be a complete G-metric space and let T : X ! X
be a weakly contractive map in G. Then T has a unique …xed point in X.
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G(x; y; z)
(maxfG(gx; gy; gz); G(gx; f x; f x); G(gy; f y; f y); G(gz; f z; f z)g);
(3)
or
G(f x; f y; f z)
(maxfG(gx; gy; gz); G(gx; gx; f x); G(gy; gy; f y); G(gz; gz; f z)g)
for all x; y; z 2 X. If f (X)
g(X) and g(X) is a G-complete subspace of X, then
f and g have a unique point of coincidence in X. Moreover, if f and g are weakly
compatible, then f and g have a unique common …xed point.
Unfortunately, the example given in support of Theorem 2 by Aage and Salunke
(Example 2, [1]) is false in the sense that the maps T and ' de…ned in this example
do not satisfy the inequality (2). For, the example considered by Aage and Salunke is
the following.
EXAMPLE 4 ([1]). Let X = [0; 1]. De…ne G : X 3 ! R+ by
G(x; y; z) = jx
yj + jy
zj + jz
xj for all x; y; z 2 X:
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Fixed Points of ( ; ')-Almost Weakly Contractive Maps
Then (X; G) is a complete G-metric space. The authors de…ned T on X by T x = x
2
and '(t) = t2 ; t 0. Let us choose x = 1; y = 12 and z = 14 . Then G(T x; T y; T z) =
G(x; y; z) = 32 and '(G(x; y; z)) = 98 . Hence
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= G(T x; T y; T z)
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G(x; y; z)
'(G(x; y; z)) =
The following is a suitable example in support of Theorem 2.
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EXAMPLE 5. Let X = [0; 1]. We de…ne G : X 3 ! R+ by
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3
:
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Also the inequality (2) fails to hold at x = 21 ; y = 13 and z = 0. Hence T and ' do not
satisfy the inequality (2) so that T is not a weakly contractive map with this ', even
though T has a …xed point 0.
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x2
2
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16 ,
G(x; y; z) =
0;
if x = y = z
maxfx; y; zg; otherwise:
Then (X; G) is a complete G-metric space. Let T x =
of generality, we assume that x > y > z. Then
x2
2
and '(t) =
G(T x; T y; T z) = maxfT x; T y; T zg =
t2
4.
Without loss
x2
;
2
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G(x; y; z) = maxfx; y; zg = x and '(G(x; y; z)) =
2
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x2
:
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2
2
Now, G(x; y; z) '(G(x; y; z)) = x x4 . Therefore G(T x; T y; T z) = x2 < x x4 =
G(x; y; z) '(G(x; y; z)). Hence T satis…es the inequality (2) so that T is a weakly
contractive map. Thus by Theorem 2, we have T has a unique …xed point and it is 0
in X.
Motivated by the ‘( ; ')-weakly contractive maps’introduced by Dutta and Choudhury [12], ‘almost weak contractions’of Berinde [7, 8] and ‘condition (B)’of Babu, Sandhya and Kameswari [6] in metric space setting, in this paper we introduce ‘( ; ')-almost
weakly contractive maps’in G-metric spaces and prove the existence of …xed points in
complete G-metric spaces. The importance of the class of ( ; ')-almost weakly contractive maps is that this class properly includes the class of weakly contractive maps
studied by Aage and Salunke [1] so that the class of ( ; ')-almost weakly contractive
maps is larger than the class of weakly contractive maps, which is illustrated in Example 6. Hence, the results obtained on the existence of …xed points of ( ; ')-almost
weakly contractive maps generalize the results of Aage and Salunke [1].
In the following, we introduce ( ; ')-almost weakly contractive maps.
DEFINITION 11. Let (X; G) be a G-metric space and let T be a selfmap of X. If
there exist and ' in and L 0 such that
(G(T x; T y; T z))
(G(x; y; z))
'(G(x; y; z)) + L m(x; y; z)
(4)
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for all x; y; z 2 X, where
m(x; y; z) = minfG(T x; x; x); G(T x; y; y); G(T x; z; z); G(T x; y; z)g;
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then we call T is a ( ; ')-almost weakly contractive map on X.
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We observe that if
is the identity map and L = 0 in (4) then T is a weakly
contractive map. Hence the class of all weakly contractive maps is contained in the
class of all ( ; ')-almost weakly contractive maps. Further, every ( ; ')-almost weakly
contractive map need not be a weakly contractive map (Example 6).
In Section 2, we prove the existence of …xed points of ( ; ')-almost weakly contractive maps in G-metric spaces. Our main result (Theorem 4) generalizes the result of
Aage and Salunke (Theorem 2, [1]). We also extend it to a pair of weakly compatible
maps and prove the existence of common …xed points. Corollaries and examples in
support of our results are provided in Section 3.
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The following is the main result of this paper.
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Main Results
THEOREM 4. Let (X; G) be a complete G-metric space and let T be a ( ; ')almost weakly contractive map. Then T has a unique …xed point in X.
PROOF. Let x0 2 X. We de…ne the sequence fxn g by xn = T (xn 1 ); n = 1; 2; ::::
If xn+1 = xn for some n 2 N, then trivially xn a …xed point of T . Suppose xn+1 6= xn
for all n 2 N. We now consider
(G(xn ; xn+1 ; xn+1 ))
=
(G(T xn
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(G(xn
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+Lm(xn
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where m(xn
1 ; xn ; xn )
1 ; T xn ; T xn ))
1 ; xn ; xn ))
'(G(xn
1 ; xn ; xn ))
1 ; xn ; xn );
= 0 so that
(G(xn ; xn+1 ; xn+1 ))
(G(xn
1 ; xn ; xn ))
'(G(xn
1 ; xn ; xn )):
(5)
By using the property of ', we have
(G(xn ; xn+1 ; xn+1 )) < (G(xn
1 ; xn ; xn ))
Now, by applying the nondecreasing property of
G(xn ; xn+1 ; xn+1 )
G(xn
for n = 1; 2; : : : :
(6)
, it follows that
1 ; xn ; xn )
for n = 1; 2; : : : :
Therefore fG(xn ; xn+1 ; xn+1 )g is a monotone decreasing sequence of nonnegative reals
and hence there exists r
0 such that G(xn ; xn+1 ; xn+1 ) ! r as n ! 1: Now, on
letting n ! 1 in the inequality (5), we have (r)
(r) '(r) so that '(r)
0:
Since '(r) 0; it follows that '(r) = 0 so that r = 0.
i.e., G(xn ; xn+1 ; xn+1 ) ! 0 as n ! 1:
(7)
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Fixed Points of ( ; ')-Almost Weakly Contractive Maps
We now prove that the sequence fxn g is Cauchy.
On the contrary, if fxn g is not Cauchy, then there exists an > 0 for which we can
…nd subsequences fxnk g; fxmk g of fxn g with nk > mk k such that
G(xnk ; xmk ; xmk )
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(8)
Corresponding to each mk , we can choose nk such that it is the smallest integer with
nk > mk and satisfying (8). Then, we have
G(xnk ; xmk ; xmk )
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:
and G(xnk
1 ; xmk ; xmk )
< :
(9)
We now prove the following three identities:
(i) limk!1 G(xnk ; xmk ; xmk ) = :
(ii) limk!1 G(xnk
1 ; xmk 1 ; xmk 1 )
(iii) limk!1 G(xnk ; xmk
1 ; xmk 1 )
= :
= :
From (9), we have G(xnk ; xmk ; xmk )
so that
lim inf G(xnk ; xmk ; xmk ):
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(10)
k!1
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Also,
G(xnk ; xmk ; xmk )
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G(xnk ; xnk
1 ; xnk 1 )
+ G(xnk
G(xnk ; xnk
1 ; xnk 1 )
+ ;
lim sup G(xnk ; xmk ; xmk )
:
<
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and hence
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k!1
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lim inf G(xnk ; xmk ; xmk )
lim sup G(xnk ; xmk ; xmk )
k!1
k!1
so that limk!1 G(xnk ; xmk ; xmk ) exists and
= lim inf G(xnk ; xmk ; xmk )
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k!1
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(11)
From (10) and (11), we have
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1 ; xmk ; xmk )
lim sup G(xnk ; xmk ; xmk ) = :
k!1
Hence
lim G(xnk ; xmk ; xmk ) = :
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(12)
k!1
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Therefore (i) holds. Also,
G(xnk
1 ; xmk ; xmk )
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G(xnk
1 ; xnk ; xnk ) + G(xnk ; xmk ; xmk )
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+G(xmk ; xmk
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G(xnk
1 ; xmk 1 ; xmk 1 )
+ G(xmk ; xmk
1 ; xmk 1 ):
1 ; xmk 1 )
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On taking limit superior as k ! 1 and using (7) and (12), we get
lim sup G(xnk
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1 ; xmk 1 ; xmk 1 )
k!1
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G(xnk ; xmk ; xmk )
G(xnk ; xnk
+G(xmk
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Hence, we have that
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G(xnk
1 ; xmk 1 ; xmk 1 )
1 ; xnk 1 )
lim inf G(xnk
lim inf G(xnk
k!1
1 ; xmk 1 ; xmk 1 ):
1 ; xmk 1 ; xmk 1 )
lim G(xnk
k!1
lim sup G(xnk
G(xnk ; xmk ; xmk )
G(xnk ; xmk
1 ; xmk 1 ; xmk 1 )
G(xnk ; xmk
1 ; xmk 1 )
1 ; xmk 1 )
G(xmk
lim inf G(xnk ; xmk
G(xnk ; xmk
1 ; xmk 1 )
G(xnk ; xnk
1 ; xmk ; xmk )
:
(16)
1 ; xnk 1 )
+ G(xnk
1 ; xmk 1 ; xmk 1 )
and hence using (7) and (ii), we get
lim sup G(xnk ; xmk
k!1
1 ; xmk 1 )
:
(17)
Now, from (16) and (17), we have
lim inf G(xnk ; xmk
k!1
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1 ; xmk ; xmk ):
Now,
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(15)
+ G(xmk
1 ; xmk 1 )
k!1
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= :
and
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:
This implies that
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1 ; xmk 1 ; xmk 1 )
k!1
Therefore (ii) holds. Let us now prove (iii). From (8), we have
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(14)
Hence it follows that
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1 ; xnk 1 )
Thus from (13) and (14), we have
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G(xnk ; xnk
1 ; xmk ; xmk ):
Now on taking limit inferior both sides, and using (7) and (12), we get
k!1
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1 ; xmk 1 ; xmk 1 )
1 ; xmk ; xmk ):
G(xmk
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+ G(xnk
G(xnk ; xmk ; xmk )
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(13)
Now,
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:
so that G(xnk ; xmk
1 ; xmk 1 )
1 ; xmk 1 )
!
lim sup G(xnk ; xmk
k!1
1 ; xmk 1 )
as k ! 1. Therefore (iii) holds.
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Fixed Points of ( ; ')-Almost Weakly Contractive Maps
Now
(G(xnk ; xmk ; xmk ))
=
(G(T xnk
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(G(xnk
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+L m(xnk
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( )
(G(xn ; T p; T p))
=
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1 ; xmk 1 ; xmk 1 ))
1 ; xmk 1 ; xmk 1 ):
( )
(G(T xn
(G(xn
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'(G(xnk
'( ) < ( );
which is a contradiction. Therefore fxn g is a G-Cauchy sequence. Since X is complete,
there exists p 2 X such that fxn g is G-convergent to p. We now consider
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293
1 ; xmk 1 ; xmk 1 ))
On letting k ! 1 and using (i)–(iii) and (7), we get
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288
1 ; T xmk 1 ; T xmk 1 ))
1 ; T p; T p))
1 ; p; p))
'(G(xn
1 ; p; p))
+ Lm(xn
1 ; p; p):
On letting n ! 1, we have '(G(p; T p; T p))
0 so that we must have T p = p.
Therefore p is a …xed point of T in X.
Uniqueness: Suppose T has two …xed points p and q in X with p 6= q. Now, we
consider
(G(p; q; q))
296
=
297
=
298
(G(T p; T q; T q))
(G(p; q; q))
'(G(p; q; q)) + Lm(p; q; q)
(G(p; q; q))
'(G(p; q; q)) < (G(p; q; q));
300
which is a contradiction. Therefore (G((p; q; q)) = 0 so that G(p; q; q) = 0 and hence
that p = q. Thus, p is the unique …xed point of T in X. Hence the theorem follows.
301
We now prove a common …xed point theorem for a pair of weakly compatible maps.
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302
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304
THEOREM 5. Let (X; G) be a complete G-metric space and let T and S be two
selfmaps on (X; G). Assume that T (X)
S(X); S is continuous, and there exist
; ' 2 and L 0 such that
(G(T x; T y; T z))
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311
(M (x; y; z))
'(M (x; y; z)) + Lm(x; y; z);
(18)
where
M (x; y; z) = maxfG(Sx; Sy; Sz); G(Sx; T x; T x); G(Sy; T y; T y); G(Sz; T z; T z)g
and
m(x; y; z) = minfG(T x; Sx; Sx); G(T x; Sy; Sy); G(T x; Sz; Sz); G(T x; Sy; Sz)g
for x; y; z 2 X. Then T and S have a unique common …xed point in X provided T and
S are weakly compatible maps.
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313
314
79
PROOF. Let x0 2 X be arbitrary. Since T (X) S(X), we can choose fxn g X
such that T (xn ) = S(xn+1 ) = yn (say); n = 0; 1; 2; :::: Let n 1 be an integer. Then
by using the inequality (18) we have
(G(yn ; yn+1 ; yn+1 ))
315
=
(G(T xn ; T xn+1 ; T xn+1 ))
316
(M (xn ; xn+1 ; xn+1 ))
317
+Lm(xn ; xn+1 ; xn+1 );
318
where
M (xn ; xn+1 ; xn+1 )
319
=
= G(Sxn ; Sxn+1 ; Sxn+1 )
321
and
m(xn ; xn+1 ; xn+1 )
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324
=
minfG(T xn ; Sxn ; Sxn ); G(T xn ; Sxn+1 ; Sxn+1 );
G(T xn ; Sxn+1 ; Sxn+1 ); G(T xn ; Sxn+1 ; Sxn+1 )g
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326
=
minfG(yn ; yn
327
=
0
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
maxfG(Sxn ; Sxn+1 ; Sxn+1 ); G(Sxn ; T xn ; T xn );
G(Sxn+1 ; T xn+1 ; T xn+1 ); G(Sxn+1 ; T xn+1 ; T xn+1 )g
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322
'(M (xn ; xn+1 ; xn+1 ))
1 ; yn 1 ); G(yn ; yn ; yn ); G(yn ; yn ; yn ); G(yn ; yn ; yn )g
since G(yn ; yn ; yn ) = 0. This implies that
(G(yn ; yn+1 ; yn+1 ))
(G(yn
1 ; yn ; yn ))
'(G(yn
1 ; yn ; yn )):
(19)
Hence, from the inequality (18), if ym = ym+1 for some m, then it follows that yn = ym
for all n m so that fyn g is Cauchy. Therefore, without loss of generality, we assume
that yn 6= yn+1 for all n = 0; 1; 2; : : : : Now, from (18), we have
(G(yn ; yn+1 ; yn+1 )) < (G(yn
Hence by the nondecreasing nature of
G(yn ; yn+1 ; yn+1 )
1 ; yn ; yn )):
, it follows that
G(yn
1 ; yn ; yn )
for all n = 1; 2; : : : :
Therefore fG(yn ; yn+1 ; yn+1 )g is a monotone decreasing sequence of nonnegative reals.
So, there exists r
0 such that G(yn ; yn+1 ; yn+1 ) ! r as n ! 1. Now, from the
inequality (19), we have
(G(yn ; yn+1 ; yn+1 ))
On letting n ! 1, we have (r)
follows that '(r) = 0 so that r = 0.
(G(yn
(r)
1 ; yn ; yn ))
'(G(yn
'(r) so that '(r)
i.e., G(yn ; yn+1 ; yn+1 ) ! 0 as n ! 1:
1 ; yn ; yn )):
0. Since '(r)
0, it
(20)
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Fixed Points of ( ; ')-Almost Weakly Contractive Maps
We now prove that the sequence fyn g is Cauchy. If we suppose that fyn g is not
Cauchy, then there exists an > 0 and there exist subsequences fynk g; fymk g of fyn g
with nk > mk k such that
G(ynk ; ymk ; ymk )
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347
348
351
352
353
354
G(ynk ; ymk ; ymk )
and G(ynk
1 ; ymk ; ymk )
(ii) limk!1 G(ynk
1 ; ymk 1 ; ymk 1 )
(iii) limk!1 G(ynk ; ymk
1 ; ymk 1 )
= :
= :
We now consider
(G(ynk ; ymk ; ymk ))
=
(G(T xnk ; T xmk ; T xmk ))
(M (xnk ; xmk ; xmk ))
357
+Lm(xnk ; xmk ; xmk );
M (xnk ; xmk ; xmk )
=
maxfG(Sxnk ; Sxmk ; Sxmk ); G(Sxnk ; T xnk ; T xnk );
G(Sxmk ; T xmk ; T xmk ); G(Sxmk ; T xmk ; T xmk )g
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361
=
G(Sxnk ; Sxmk ; Sxmk ) = G(ynk
=
minfG(T xnk ; Sxnk ; Sxnk ); G(T xnk Sxmk ; Sxmk );
m(xnk ; xmk ; xmk )
G(T xnk ; Sxmk ; Sxmk ); G(T xnk ; Sxmk ; Sxmk )g:
Therefore,
366
(G(ynk ; ymk ; ymk ))
367
(G(ynk
368
369
370
371
372
373
1 ; ymk 1 ; ymk 1 )
and
364
365
'(M (xnk ; xmk ; xmk ))
where
359
363
(22)
(i) limk!1 G(ynk ; ymk ; ymk ) = :
356
362
< :
Now the following identities follow as in the proof of Theorem 4.
355
358
(21)
Corresponding to each mk , we can choose nk such that it is the smallest integer with
nk > mk and satisfying (21). Then, we have
349
350
:
1 ; ymk 1 ; ymk 1 ))
+L min fG(ynk ; ynk
G(ynk ; ymk
'(G(ynk
1 ; ymk 1 ; ymk 1 ))
1 ; ynk 1 ); G(ynk ; ymk 1 ; ymk 1 );
1 ; ymk 1 ); G(ynk ; ymk 1 ; ymk 1 )g:
On letting k ! 1 and using (i)–(iii) and (20), we get
( )
( )
'( ) < ( );
which is a contradiction. Therefore fyn g is a G-Cauchy sequence. Since X is complete,
there exists z 2 X such that limn!1 yn = limn!1 T xn = limn!1 Sxn+1 = z. We
Babu et al.
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375
376
377
81
now prove that z is a common …xed point of T and S. Since S is continuous, we have
limn!1 ST xn = limn!1 SSxn = Sz. Further, since S and T are weakly compatible,
we have limn!1 G(T Sxn ; ST xn ; ST xn ) = 0, which implies limn!1 T Sxn = Sz. Now,
from (18), we have
(G(T Sxn ; T xn ; T xn ))
378
(M (Sxn ; xn ; xn ))
+Lm(Sxn ; xn ; xn );
379
380
M (Sxn ; xn ; xn )
=
= G(SSxn ; Sxn ; Sxn )
383
and
m(Sxn ; xn ; xn )
385
=
minfG(T Sxn ; SSxn ; SSxn ); G(T Sxn ; Sxn ; Sxn );
G(T Sxn ; Sxn ; Sxn ); G(T Sxn ; Sxn ; Sxn )g:
386
388
maxfG(SSxn ; Sxn ; Sxn ); G(SSxn ; T Sxn ; T Sxn );
G(Sxn ; T xn ; T xn ); G(Sxn ; T xn ; T xn )g
382
387
(23)
where
381
384
'(M (Sxn ; xn ; xn ))
Therefore, from (23), we have
(G(T Sxn ; T xn ; T xn ))
(G(SSxn ; Sxn ; Sxn ))
'(G(SSxn ; Sxn ; Sxn ))
389
+L minfG(T Sxn ; SSxn ; SSxn ); G(T Sxn ; Sxn ; Sxn );
390
G(T Sxn ; Sxn ; Sxn ); G(T Sxn ; Sxn ; Sxn )g:
391
On letting n ! 1, we get
(G(Sz; z; z))
392
393
which implies that '(G(Sz; z; z))
(G(T xn ; T z; T z))
394
395
396
(G(Sz; z; z))
'(G(Sz; z; z));
0 so that we must have Sz = z. Now, we consider
(M (xn ; z; z))
'(M (xn ; z; z)) + Lm(xn ; z; z);
(24)
where
M (xn ; z; z) = maxfG(Sxn ; Sz; Sz); G(Sxn ; T xn ; T xn ); G(Sz; T z; T z); G(Sz; T z; T z)g
397
and
398
m(xn ; z; z) = minfG(T xn ; Sxn ; Sxn ); G(T xn ; Sz; Sz); G(T xn ; Sz; Sz); G(T xn ; Sz; Sz)g:
399
Also, we have
400
401
402
403
404
405
406
407
lim M (xn ; z; z) = G(z; T z; T z) and lim m(xn ; z; z) = 0;
n!1
n!1
(25)
since limn!1 G(T Sxn ; SSxn ; SSxn ) = 0. Now, on letting n ! 1 in (24) and using
(25), we get
(G(z; T z; T z))
(G(z; T z; T z))
'(G(z; T z; T z)):
Then '(G(z; T z; T z)) 0. Therefore, '(G(z; T z; T z)) = 0 so that G(z; T z; T z) = 0.
Therefore, T z = z. Thus z is a common …xed point of T and S. Uniqueness of common
…xed point of T and S follows from the inequality (18). This completes the proof of
the theorem.
82
408
409
410
411
412
3
Fixed Points of ( ; ')-Almost Weakly Contractive Maps
Corollaries and Examples
In this section, we draw some corollaries from the main results of Section 2 and provide
examples in support of our results. The following is an example in support of Theorem
4.
EXAMPLE 6. Let X = [0; 1]. We de…ne G : X 3 ! R+ by
G(x; y; z) =
413
414
415
0;
if x = y = z
maxfx; y; zg; otherwise:
Then (X; G) is a complete G-metric space.
8 1
< 2
2x
T (x) =
:
1
We de…ne T : X ! X by
if x = 0;
if 0 < x < 12 ;
if 12 x 1:
2
416
417
418
419
420
421
422
2
= G(T x; T y; T z)
3
423
424
425
426
427
428
429
430
431
432
433
436
437
438
439
G(x; y; z)
'(G(x; y; z)) =
1
3
1
'( ) for any ' 2
3
:
Hence T does not satisfy the inequality (2) for any ' 2
so that T is not a weakly
contractive map in G-metric space. Thus Theorem 2 is not applicable.
Further, this example suggests that the class of ( ; ')-almost weakly contractive
maps is larger than the class of weakly contractive maps in G-metric spaces.
REMARK 1. Theorem 2 follows as a corollary to Theorem 4 by choosing as the
identity map and L = 0. Hence Example 6 suggests that Theorem 4 is a generalization
of Theorem 2.
COROLLARY 1. Let (X; G) be a complete G-metric space and let T and S be
two selfmaps on (X; G). Assume that T (X) S(X), S is continuous, and there exist
; ' 2 such that
(G(T x; T y; T z))
434
435
2
We de…ne and ' on R+ by (t) = t3 and '(t) = t2 . Then, it is easy to verify that
T satis…es the inequality (4) with L = 1. i.e., T is a ( ; ')-almost weakly contractive
map. Thus T satis…es all the hypothesis of Theorem 4 and 1 is the unique …xed point
of T . Here we observe that T is not a continuous map.
Further, we observe that T is not a weakly contractive map. For, let us choose x =
1
and
y = z = 0. Then G(T x; T y; T z) = 32 ; G(x; y; z) = 13 and '(G(x; y; z)) = '( 31 ).
3
Hence,
(M (x; y; z))
'(M (x; y; z));
(26)
where
M (x; y; z) = maxfG(Sx; Sy; Sz); G(Sx; T x; T x); G(Sy; T y; T y); G(Sz; T z; T z)g
for x; y; z 2 X. Then T and S have a unique common …xed point in X provided T and
S are weakly compatible maps.
PROOF. Follows from Theorem 5 by choosing L = 0.
Babu et al.
83
440
The following is an example in support of Corollary 1.
441
EXAMPLE 7. Let X = [0; 1]. We de…ne G : X 3 ! R+ by
443
Then (X; G) is a complete G-metric space. We de…ne T; S : X ! X and , ' on R+ by
T (x) =
444
445
446
449
450
451
x2
x
, S(x) = (5
2
4
x
G(Sx; Sy; Sz) = maxfSx; Sy; Szg = maxf (5
4
S(X).Without loss of generality, we
x2
x2 y 2 z 2
; ; g=
:
2 2 2
2
y
x); (5
4
z
y); (5
4
z)g =
x
(5
4
x):
Now, we consider
(G(T x; T y; T z))
=
x4
3
x2
(5
16
x)2 = [G(Sx; Sy; Sz)]2
4
[M (x; y; z)]2
3
(M (x; y; z)) '(M (x; y; z)):
[M (x; y; z)]2 =
=
1
[M (x; y; z)]2
3
Therefore,
(G(T x; T y; T z))
455
457
4t2
t2
and '(t) =
3
3
Also
453
456
(t) =
G(T x; T y; T z) = maxfT x; T y; T zg = maxf
452
454
x),
for all x 2 X and t 2 R+ . Then clearly T (X)
assume that x > y > z. Then
447
448
0;
if x = y = z;
maxfx; y; zg; otherwise:
G(x; y; z) =
442
(M (x; y; z))
'(M (x; y; z))
so that the inequality (26) of Corollary 1 holds. Thus, T and S satisfy all the hypotheses
of Corollary 1 and 0 is the unique common …xed point of T and S.
461
REMARK 2. We observe that the ' that is used in Theorem 5 is di¤erent from
that is used in Theorem 3.
Acknowledgment. The authors are sincerely thankful to the anonymous referee
for his/her valuable suggestions which helped us to improve the quality of the paper.
462
References
458
459
460
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464
465
466
467
[1] C. T. Aage and J. N. Salunke, Fixed points for weak contractions in G-metric
spaces, Appl. Math. E-Notes, 12(2012), 23–28.
[2] M. Abbas, G. V. R. Babu and G. N. Alemayehu, On common …xed points of weakly
compatible mappings satisfying ‘Generalized condition (B)’, Filomat 25(2011), 9–
19.
84
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
Fixed Points of ( ; ')-Almost Weakly Contractive Maps
[3] Ya. I. Alber and Guerre-Delabriere, Principles of weakly contractive maps in Hilbet
spaces, New results in Operator Theory and its applications (I. Gohberg, and Yu.
Lyubich, Eds.), Oper. Theory Adv. Appl., 98(1997), 7–22.
[4] G. V. R. Babu, K. Nageswara Rao and G. N. Alemayehu, Common …xed points of
two pairs of generalized weakly contractive maps, Adv. Studies in Contemporary
Math., 20(2010), 575–594.
[5] G. V. R. Babu and P. D. Sailaja, A …xed point theorem of generalized weakly
contractive maps in orbitally complete metric spaces, Thai J. Math., 9(2011),
1–10.
[6] G. V. R. Babu, M. L. Sandhya and M. V. R. Kameswari, A note on a …xed point
theorem of Berinde on weak contractions, Carpath. J. Math., 24(2008), 8–12.
[7] V. Berinde, Approximating …xed points of weak contractions using the Picard
iteration, Nonlinear Anal. Forum, 9(2004), 43–53.
[8] V. Berinde, General constructive …xed point theorems for Ciric-type almost contractions in metric spaces, Carpath. J. Math., 24(2008), 10–19.
[9] B. S. Choudhury, P. Konar, B. E. Rhoades and N. Metiya, Fixed point therorems
for generalized weakly contractive mappings, Nonlinear Anal. Theory, Methods
and Appl., 74(2011), 2116–2126.
[10] B. S. Choudhury, Sanjay Kumar, Asha and Krishnapada Das, Some …xed point
theorems in G-metric spaces, Math. Sci. Lett., 1(2012), 25–31.
[11] D. Doric, Common …xed point for generalized ( ; ')-weak contractions, Appl.
Math. Lett., 22(2009), 1896–1900.
[12] P. N. Dutta and B. S. Choudhury, A generalisation of contraction principle in
metric spaces, Fixed Point Theory and Appl., (2008), Article ID 406368, 8 pages.
[13] G. Jungck and B. E. Rhoades, Fixed points for set valuaed functions without
continuity, Indian J. Pure and Appl. Math., 29(1998), 227–238.
[14] Z. Mustafa, H. Obiedat and F. Awawdeh, Some …xed point theorem for mapping
on complete G-metric spaces, Fixed Point Theory and Appl., (2008) Article ID
189870.
[15] Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J. Nonlinear
and Convex Anal., 7(2006), 289–297.
[16] B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal.,
47(2001), 2683–2693.
[17] K. P. R. Sastry, G. V. R. Babu and K. T. Kidane, A common …xed point of generalized ( ; )-weakly contractive maps where is nondecreasing (not necessarily
continuous or lower semicontinuous), Advances in Fixed Point Theory, 2(2012),
203–223.
Babu et al.
505
506
507
508
85
[18] W. Shatanawi, Common …xed point result for two self-maps in G-metric spaces,
Mat. Vesnik, 65(2013), 143–150.
[19] Q. Zhang and Y. Song, Fixed point theory for generalized '-weak contractions,
Appl. Math. Lett., 22(2009), 75–78.