COMMON FIXED POINT THEOREMS FOR GENERALIZED [620037]

COMMON FIXED POINT THEOREMS FOR GENERALIZED
F-( ;')-CONTRACTIONS IN MODULAR SPACES
MUJAHID ABBAS, ARSLAN HOJAT ANSARI, LILIANA GURAN, AND WISSAM KASSAB
Abstract. Using a new class of functions, in this paper we establish the existence
of xed points of almost generalized contractions on modular spaces. Also we obtain
common xed point results for a pair of mappings satisfying generalized F-( ;')-
contractive condition on a modular space. Our results generalize and extend various
results in the existing literature.
1.Introduction
The Banach's contraction principle ensures the existence and uniqueness of the xed
point of nonlinear operators satisfying the contraction condition. Banach contraction
principle has been generalized in many directions with respect to its applications in
mathematics and other related disciplines. Extensions of Banach contraction principle
have been obtained either by generalizing the domain of the mapping (see, e.g., [1, 6,
10, 39, 40]) or by extending the contractive conditions [14, 15, 17, 19].
Existence of xed points in ordered metric spaces has been studied by Ran and
Reuring [37]. Nieto and Rodriguez-Lopez [34] extended the results in [37] for non-
decreasing mappings and applied to obtain a unique solution for a rst order ordinary
dierential equation with periodic boundary conditions; for recent development, see
[7, 41, 42].
In 2008, Jachymski [20] investigated a new approach in metric xed point theory
by replacing order structure with graph structure on a metric space. In this way, the
results proved in ordered metric spaces are generalized (see for detail [20] and the
reference therein).
Abbas and Nazir [3] obtained some xed point results for graphic contraction pair
endowed with a graph. Beg and Butt [9] proved xed point theorems for set-valued
mappings on a metric space with a graph. Tiammee and Suantai [43] introduced graph-
preserving multi-valued mapping and a new type of multi-valued weak G-contraction
on a metric space endowed with a directed graph, see [35, 38]).
Nakano [33] initiated the concept of a modular space. Later, it was redened and
generalized by Musielak and Orlicz [32] but, the most important development of this
theory is due to Luxemburg [27], Mazur and Orlicz [28], and Turpin [44].
2010 Mathematics Subject Classication. 47H10, 54H25, 54E50.
Key words and phrases. modular space, almost contraction, C-class function, generalized F-( ;')-
contractive operator, xed point, Banach G-Contraction.
1

2 M. ABBAS, A. H. ANSARI, L. GURAN, AND W. KASSAB
The xed point theory in modular function spaces has recently explored by re-
searchers as Khamsi et al. [12], [22], [23]. Kuaket and Kumam [24] and Mongkolkeha
and Kumam [29, 30, 31] obtained some xed and common xed point results for gen-
eralized contraction mappings in modular spaces. Also, Kumam [25] proved some xed
point results for nonexpansive mappings in arbitrary modular spaces. Very recent,
Kutbi and Latif [26] obtained some xed points results for multivalued maps in mod-
ular function spaces.
In this paper, we dene a new class of functions, F-( ;')-contractions, and we
establish the existence of xed points for this new type of functions on a modular
spaces. Also we obtain common xed point results for a pair of mappings satisfying
generalized F-( ;')-contractive condition on a modular space. Our results generalize
and extend various results in the existing literature.
2.Preliminaries
In this paper we will denote by letters R,R+andNthe set of all real numbers,
the set of all nonnegative real numbers and the set of all positive integer numbers,
respectively.
Next, let us recall some crucial denitions for xed point theory area.
Let (X;d) be a metric space and T:X!X. A pointx2Xis called a xed point
ofTif and only if Tx=x:
A mapping T:X!Xis called a Picard operator (PO), if
(1)F(T) =fx2X:Tx=xg=fzg,
(2) for any x02X;the Picard iteration xn=Tnx0converges to z.
In the above denition, the sequence is called a sequence of successive approximations
ofT, starting from x0:
Banach contractive condition make the mappings to be continuous. Kannan obtained
in [21] the existence of a xed point for a mapping that can have a discontinuity,
considering a weaker contractive conditions. Later, in many papers was proved xed
point or common xed point results for various classes of contractive type conditions
that do not require the continuity of the mappings (see [18], [45]).
Berinde introduced in [14] the concept of weak contraction as follows.
Denition 2.1. Let (X;d) be a metric space. A map T:X!Xis called an almost
contraction or a -weak contraction If there exist a constant 2(0;1) and some L0
such that
d(Tx;Ty )d(x;y) +Ld(x;Ty );
for anyx;y2X.
In [15], Berinde replace weak contraction with almost contraction which is more
appropriate.

COMMON FIXED POINT THEOREMS FOR F-( ;')-CONTRACTIONS IN MODULAR SPACES 3
Recently Babu, Sandhya and Kameswari [8] considered a new class of mappings-the
mappings that satisfy condition (B)
Denition 2.2. Let (X;d) be a metric space. A map T:X!Xis said to satisfy
`condition (B)' if there exist a constant 2]0;1[ and some L0 such that
d(Tx;Ty )d(x;y) +Lminfd(x;Tx );d(y;Ty );d(x;Ty );d(y;Tx )g;
for allx;y2X:
They proved the following xed point theorem.
Theorem 2.3 ([8], Theorem 2.3)) .Let(X;d)be a complete metric space and T:X!
Xbe a map satisfying condition (B). Then T has a unique xed point.
In [13] Berinde introduced the concept of generalized almost contraction as follows:
Denition 2.4. Let (X;d) be a metric space. A map T:X!Xis called generalized
almost contraction if there exist a constant 2]0;1[ and some L0 such that
d(Tx;Ty )M1(x;y) +Lminfd(x;Tx );d(y;Ty );d(x;Ty );d(y;Tx )g;
where
M1(x;y) = max
d(x;y);d(x;Tx );d(y;Ty );1
2[d(x;Ty ) +d(y;Tx )]
:
Theorem 2.5. Let(X;d)be a complete metric space and T:X!Xa generalized
almost contraction. Then Thas a unique xed point.
A pointy2Xis called a point of coincidence of two self-mappings fandTonXif
there exists a point x2Xsuch thaty=Tx=fx. The point xis called coincidence
point of a pair ( f;T).
Concerning this condition (B)in [2] Abbas et al. introduced a generalization for a
pair of self maps and obtained a unique point of coincidence .
Further, let us recall crucial notions for modular spaces.
Denition 2.6. LetXbe an arbitrary vector space. A functional :X![0;1) is
called a modular if for any x;y2X, the following conditions hold:
(m1)(x) = 0 if and only if x= 0;
(m2)(x) =(x) for every scalar withjj= 1;
(m3)(x+y)(x) +(y) whenever += 1;and;0.
If axiom (m3) is replaced with (x+y)s(x) +s(y) wheres+s= 1,
;0, ands2(0;1], thenis calleds-convex modular. If s= 1, then we say that
is convex modular.
Some consequences of condition ( m3) are the followings:
Remark 2.7.[16]

4 M. ABBAS, A. H. ANSARI, L. GURAN, AND W. KASSAB
(r1) Fora,b2Rwithjaj<jbjwe have(ax)<(bx) for allx2X;
(r2) Fora1;:::;an2R+withnP
i=1ai= 1, we have
 nX
i=1aixi!
nX
i=1(xi);for anyx1;:::;xn2X:
Proposition 2.8 ([31]).LetXbe a modular space. If a;b2R+withba;then
(ax)(bx).
A mapping :R![0;1] dened by (x) =p
jxjis a trivial example of a modular
functional.
The vector space Xgiven by
X=fx2X;(x)!0 as!0g
is called a modular space. Generally, the modular is not sub-additive and therefore
does not behave as a norm or a distance. One can associate to a modular an F-norm.
Proposition 2.9. The modular space Xcan be equipped with an F-norm dened by
kxk= inff>0;(x
)g:
Whenis convex modular, then
kxk= inff>0;(x
)1g
denes a norm on the modular space X, and is called the Luxemburg norm.
Dene the-ball,B(x;r), centered at x2Xwith radius ras
B(x;r) =fh2X;(xh)rg:
A function modular is said to satisfy:
a:
2-type condition if there exists K > 0 such that for any x2X, we have
(2x)K(x)
b:
2-condition if (2xn)!0 asn!1 , whenever (xn)!0 asn!1 .
Denition 2.10. A sequencefxngin modular space Xis said to be:
(t1)-convergent to x2Xif(xnx)!0 asn!1
(t2)-Cauchy if(xnxm)!0 asn,m!1:
Xis called-complete if any -Cauchy sequence is -convergent. But, the -
convergence does not imply -Cauchy since does not satisfy the triangle inequality.
In fact, one can show that this will happen if and only if satises the
2-condition.
In [11] we nd that the norm and modular convergence are also the same when we deal
with the
2-type condition.
Then suppose the modular function is convex and satises the
2-type condition.
Mongkolkeha and Kumam gave in [31] the existence of xed points generalized weak
contractive mapping in modular space as follows.

COMMON FIXED POINT THEOREMS FOR F-( ;')-CONTRACTIONS IN MODULAR SPACES 5
Theorem 2.11. LetXbe a-complete modular space and T:X!X. Suppose
that there exists continuous and monotone nondecreasing functions ;':R+!R+
such that (t) ='(t) = 0 if and only if t= 0. If for any x;y2X, the following
condition hold:
((TxTy)) (m(x;y))'(m(x;y)); (2.1)
where
m(x;y) = max(
(xy);(xTx);(yTy);1
2(xTy)

+1
2(yTx)

2)
:
ThenThas a unique xed point.
Denition 2.12. LetXbe a modular space and T:X!Xbe self map. We say
thatTis a-continuous if (xnx)!0, then(TxnTx)!0 asn!1 .
Now, we recall the notion of C-class function introduced by Ansari in [4], see also
[5].
Denition 2.13 ([4]).A mapping F: [0;+1)2!Ris calledC-class function if it is
continuous and the following conditions hold:
(1)F(s;t)sfor alls;t2[0;+1);
(2)F(s;t) =simplies that either s= 0 ort= 0.
DenoteCthe family of C-class functions.
Example 2.14 ([4]).The following functions F: [0;+1)2!Rdened for all s;t2
[0;+1) by the formulas are of C-class:
(1)F(s;t) =st,F(s;t) =s)t= 0;
(2)F(s;t) =ms, 0<m< 1,F(s;t) =s)s= 0;
(3)F(s;t) =s
(1+t)r,r2(0;+1),F(s;t) =s)s= 0 ort= 0;
(4)F(s;t) = loga[(t+as)=(1 +t)],a>1,F(s;t) =s)s= 0 ort= 0;
(5)F(s;t) = ln[(1 + as)=2],e>a> 1,F(s;t) =s)s= 0;
(6)F(s;t) = (s+l)(1=(1+t)r)l,l>1;r2(0;+1),F(s;t) =s)t= 0;
(7)F(s;t) =slogt+aa,a>1,F(s;t) =s)s= 0 ort= 0;
(8)F(s;t) =s(1+s
2+s)(t
1+t),F(s;t) =s)t= 0;
(9)F(s;t) =s(s),: [0;+1)![0;1) a continuous function, F(s;t) =s)s= 0;
(10)F(s;t) =st
k+t;F(s;t) =s)t= 0;
(11)F(s;t) =s'(t),F(s;t) =s)t= 0;here': [0;+1)![0;+1) is a
continuous function such that '(t) = 0,t= 0;
(12)F(s;t) =sh(s;t),F(s;t) =s)s= 0;hereh: [0;+1)[0;+1)![0;+1) is
a continuous function such that h(t;s)<1 for allt;s> 0;
(13)F(s;t) =s(2+t
1+t)t,F(s;t) =s)t= 0;
(14)F(s;t) =np
ln(1 +sn),F(s;t) =s)s= 0;

6 M. ABBAS, A. H. ANSARI, L. GURAN, AND W. KASSAB
(15)F(s;t) =(s),F(s;t) =s)s= 0;here: [0;+1)![0;+1) is a continuous
function such that (0) = 0 and (t)<tfort>0;
(16)F(s;t) =s
(1+s)r;r2(0;+1),F(s;t) =s)s= 0
The following lemma is useful to show that a given sequence is Cauchy.
Lemma 2.15. Suppose that X!is a non-Archimedean modular metric space. Let fxng
be a sequence in Xsuch that!1(xn;xn+1)!0asn!+1. Iffxngis not a Cauchy
sequence, then there exist an " > 0and sequences of positive integers fm(k)gand
fn(k)gwithm(k)>n(k)ksuch that!1(xn(k);xm(k))"and!1(xn(k);xm(k)1)<"
for allk2Nand
(1) lim
k!+1!1(xn(k);xm(k)) =";
(2) lim
k!+1!1(xn(k)1;xm(k)1) =";
(3) lim
k!+1!1(xn(k)1;xm(k)) =";
(4) lim
k!+1!1(xn(k);xm(k)1) =".
3.Common Fixed Points of F-( ;')-Contractions
We recall the denitions for usual sets of functions. = f : [0;1)![0;1) :
is continuous nondecreasing function and (t) = 0 if and only if t= 0g,  =
f': [ 0;1)![0;1) :'is lower-semi continuous function and '(t) = 0 if and only if
t= 0g, u=f': [0;1)![ 0;1) :'is lower-semi continuous function and '(0)
0g:
In this section, we obtain common xed point results for F-( ;')-contractive map-
pings in the framework of a modular space. Let us dene the notion of generalized
F-( ;')-contractive mapping.
Denition 3.1. LetXbe a-modular space and T:X!X. We say that Tis a
generalized F-( ;')-contractive mapping if there exists F2C, 2 ,'2u, and
L0 such that for any x;y2Xwe have
((SxTy))F( (M(x;y));'(M(x;y))) +L (N(x;y)); (3.1)
where
M(x;y) = max(
(xy);(xTx);(yTy);1
2(yTx)

+1
2(xTy)

2)
and
N(x;y) = minf(xTx);(yTy);(yTx);(xTy)g:
Theorem 3.2. LetXbe a-complete modular space and S; T :X!X. Suppose
thatTis a generalized F-( ;')-contractive mapping, then SandThave a unique
common xed point provided that one of the mappings SorTis-continuous.

COMMON FIXED POINT THEOREMS FOR F-( ;')-CONTRACTIONS IN MODULAR SPACES 7
Proof. Letx0be a given point in X:We construct a sequence fxngforn0 by two
step iterative process as:
x2n+2=Tx2n+1; x 2n+1=Sx2n: (3.2)
Step 1.Prove that (xnxn+1)!0 asn!1 .
Using Remark 2.7, properties of functions and'and substituting x=x2nand
y=x2n+1in (3.1), we have
((x2n+1x2n+2)) = ((Sx2nTx2n+1))
F( (M(x2n;x2n+1));'(M(x2n;x2n+1))) +L (N(x2n;x2n+1))
where
M(x2n;x2n+1) = max
(x2nx2n+1);(x2nSx2n);(x2n+1Tx2n+1);
(1
2(x2n+1Sx2n))+(1
2(x2nTx2n+1))
2
= max
(x2nx2n+1);(x2nx2n+1);(x2n+1x2n+2);
(1
2(x2n+1x2n+1))+(1
2(x2nx2n+2))
2
max
(x2nx2n+1);(x2n+1x2n+2);
(x2nx2n+1)+(x2n+1x2n+2)
2
= max
(x2nx2n+1);(x2n+1x2n+2)
and
N(x2n;x2n+1) = min
(x2nSx2n);(x2n+1Tx2n+1);
(x2nTx2n+1);(x2n+1Sx2n)
= min
(x2nx2n+1);(x2n+1x2n+2);
(x2nx2n+2);(x2n+1x2n+1)
= min
(x2nx2n+1);(x2n+1x2n+2);
(x2nx2n+2);0
= 0:
Then we have
((x2n+1x2n+2))F( (maxf(x2nx2n+1);(x2n+1x2n+2)g);
'(maxf(x2nx2n+1);(x2n+1x2n+2)g)):(3.3)
We consider the following case:
If maxf(x2nx2n+1);(x2n+1x2n+2)g=(x2n+1x2n+2) for some n;then
using the denition of ', (3.3) we obtain
((x2n+1x2n+2))F( ((x2n+1x2n+2));'((x2n+1x2n+2)))
 ((x2n+1x2n+2));

8 M. ABBAS, A. H. ANSARI, L. GURAN, AND W. KASSAB
so, ((x2n+1x2n+2)) = 0;'((x2n+1x2n+2)) = 0 therefore (x2n+1x2n+2) = 0,
a contradiction. Consequently
maxf(x2nx2n+1);(x2n+1x2n+2)g=(x2nx2n+1):
Then, from (3.3), we have
((x2n+1x2n+2))F( ((x2nx2n+1));'((x2nx2n+1)))
 ((x2nx2n+1)):(3.4)
Then, we obtain that
((x2n+1x2n))F( ((x2n1x2n));'((x2n1x2n)))
 ((x2n1x2n)):(3.5)
From (3.4) and (3.5), it follows that f(xnxn+1)gis monotone decreasing and
bounded below. Therefore, there is r0 such that
lim
n!1(xnxn+1) =r:
On taking limit as n!1 on both sides of the inequality (3.4), we get
(r)F( (r);'(r));
which implies that (r) = 0 or'(r) = 0, that is, r= 0. Hence
lim
n!1(xnxn+1) = 0: (3.6)
Step 2.We show thatfxngis a-Cauchy sequence.
Then it is sucient to prove that fx2ngis a-Cauchy sequence.
We assume on contrary. As in the hypothesis of Lemma 2.15, there exists " > 0
such that we can nd two subsequences fmkgandfnkgof positive integers satisfying
nk>mkksuch the following inequalities hold:
(x2nkx2mk)";  (2 (x2nkx2mk))<": (3.7)
From (3.7) and Remark 2.7, it follows that
"(x2nkx2mk)
=(x2nkx2nk1+x2nk1x2mk)
(2 (x2nkx2nk1)) +(2 (x2nk1x2mk))
<"+(2 (x2nkx2nk1)):
Taking limit as k!1 , we obtain that
lim
k!1(x2nkx2mk) =" (3.8)
Usingx=x2nkandy=x2mk1in (3.1), we have
((x2nk+1x2mk)) = ((Sx2nkTx2mk1))
F( (M(x2nk;x2mk1));'(M(x2nk;x2mk1))) +L (N(x2nk;x2mk1))(3.9)

COMMON FIXED POINT THEOREMS FOR F-( ;')-CONTRACTIONS IN MODULAR SPACES 9
where
M(x2nk;x2mk1) = maxn
(x2nkx2mk1);(x2nkSx2nk);(x2mk1Tx2mk1);
(1
2(x2mk1Sx2nk))+(1
2(x2nkTx2mk1))
2o
= maxn
(x2nkx2mk1);(x2nkx2nk+1);(x2mk1x2mk);
(1
2(x2mk1x2nk+1))+(1
2(x2nkx2mk))
2o
(3.10)
and
N(x2nk;x2mk1) = min
(x2nkSx2nk);(x2mk1Tx2mk1);
(x2mk1Sx2nk);(x2nkTx2mk1)
= min
(x2nkx2nk+1);(x2mk1x2mk);
(x2mk1x2nk+1);(x2nkx2mk)
:(3.11)
From (3.7) and Remark 2.7, it follows that
(x2nk+1x2mk) =(x2nk+1x2nk+x2nkx2nk1+x2nk1x2mk)
(2 (x2nk+1x2nk+x2nkx2nk1)) +(2 (x2nk1x2mk))
(4 (x2nk+1x2nk)) +(4 (x2nkx2nk1)) +(2 (x2nk1x2mk))
<(4 (x2nk+1x2nk)) +(4 (x2nkx2nk1)) +":(3.12)
Also, from (3.10) we have
(x2nkx2mk1) =(x2nkx2nk1+x2nk1x2mk+x2mkx2mk1)
(2 (x2nkx2nk1+x2mkx2mk1)) +(2 (x2nk1x2mk))
(4 (x2nkx2nk1)) +(4 (x2mkx2mk1)) +(2 (x2nk1x2mk))
<(4 (x2nkx2nk1)) +(4 (x2mkx2mk1)) +":(3.13)
Then we have,
1
2(x2mk1x2nk+1)

=1
2(x2mk1x2mk+x2mkx2nk+1)

(x2mk1x2mk) +(x2mkx2nk+1)
=(x2mk1x2mk) +(x2mkx2nk1+x2nk1x2nk+x2nkx2nk+1)
(x2mk1x2mk) +(2 (x2nk1x2mk))
+(2 (x2nk1x2nk+x2nkx2nk+1))
(x2mk1x2mk) +(2 (x2nk1x2mk)) +(4 (x2nk1x2nk))
+(4 (x2nkx2nk+1))
<"+(x2mk1x2mk) +(4 (x2nk1x2nk)) +(4 (x2nkx2nk+1)):(3.14)
Using Remark 2.7 and Proposition 2.8, we obtain that
1
2(x2nkx2mk)

=1
2(x2nkx2nk1+x2nk1x2mk)

(x2nkx2nk1) +(x2nk1x2mk)
(x2nkx2nk1) +(2 (x2nk1x2mk))
<(x2nkx2nk1) +":(3.15)

10 M. ABBAS, A. H. ANSARI, L. GURAN, AND W. KASSAB
Using (3.13), (3.14), (3.15) and arranging (3.10), we get that
M(x2nk;x2mk1) = maxn
(x2nkx2mk1);(x2nkx2nk+1);
(x2mkx2mk1);(1
2(x2mk1x2nk+1))+(1
2(x2nkx2mk))
2o
<maxn
(4 (x2nkx2nk1)) +(4 (x2mkx2mk1)) +";
(x2nkx2nk+1);(x2mkx2mk1);
"+(x2mk1x2mk)+(4(x2nk1x2nk))+(4(x2nkx2nk+1))+(x2nkx2nk1)+"
2o
:
(3.16)
Taking limit as k!1 on both sides of (3.9) and by using, (3.11), (3.12), (3.16),
Proposition 2.8, we get
(")F( (");'("))
which implies that (") = 0;or,'(r) = 0, that is, "= 0. Contradiction. Hence, using
Lemma 2.15 we obtain that fx2ngis a-Cauchy sequence.
Step 3.Next we prove the existence of a xed point of one mapping.
AsXis a-complete, there exists a z2Xsuch that(xnz)!0 asn!1 . As-
sume thatSis-continuous. By (x2nz)!0 asn!1 , we have(Sx2nSz)!0,
that is,(x2n+1Sz)!0 asn!1 . By the uniqueness of limit we obtain that
Sz=z. Thuszis a xed point of S.
Step 4.We prove that zis a xed point of a mapping T.
From (3:1) and Remark 2.7, we get
((Sx2nTz))
F( (M(x2n;z));'(M(x2n;z))) +L (N(x2n;z))(3.17)
where
M(x2n;z) = maxn
(x2nz);(x2nSx2n);(zTz);
(1
2(x2nTz))+(1
2(zSx2n))
2o
= maxn
(x2nz);(x2nx2n+1);(zTz);
(1
2(x2nTz))+(1
2(zx2n+1))
2o
maxn
(x2nz);(x2nx2n+1);(zTz);
(x2nz)+(zTz)+(1
2(zx2n+1))
2o(3.18)
and
N(x2n;z) = minf(x2nSx2n);(zTz);(x2nTz);(zSx2n)g
= minf(x2nx2n+1);(zTz);(x2nTz);(zx2n+1)g:(3.19)
Taking limit n!1 by (3.17)-(3.19), we get
((zTz))F( ((zTz));'((zTz)))

COMMON FIXED POINT THEOREMS FOR F-( ;')-CONTRACTIONS IN MODULAR SPACES 11
which implies that ((zTz)) = 0, or '((zTz)) = 0 and so (zTz) = 0.
Then, we get that Tz=z. Thuszis a xed point of T. Hence,zis a common xed
point ofSandT.
Similarly, if we suppose that Tis-continuous, then we get the same result.
Step 5.We prove the uniqueness of a common xed point of two mappings.
Letwbe another common xed point, that is, w=Sw,w=Tw, andw6=z.
((zw)) = ((SzTw))
F( (M(z;w));'(M(z;w))) +L (N(z;w))(3.20)
where
M(z;w) = maxn
(zw);(zSz);(wTw);
(1
2(wSz))+(1
2(zTw))
2o (3.21)
and
N(z;w) = minf(zSz);(wTw);(wSz);(zTw)g: (3.22)
Using (3.20)-(3.22) we have
((zw))F( ((zw));'((zw))) ((zw));
which implies that ((zw)) = 0 or'((zw)) = 0 and so (zw) = 0. Con-
tradiction, then z=w. 
Further, let us present some results directly obtained from Theorem 3.2.
Corollary 3.3. LetS;T be self mappings on a -complete modular space Xsuch that
for anyx;y2Xfollowing condition holds:
((SxTy))F( (M(x;y));'(M(x;y))) (3.23)
whereF2C, 2 ,'2u, and
M(x;y) = maxn
(xy);(xSx);(yTy);1
2(ySx)

+1
2(xTy)

2o
:
ThenSandThave a unique common xed point provided that one of the mappings S
orTis-continuous.
Corollary 3.4. LetS;T be self mappings on a -complete modular space Xsuch that
for anyx;y2Xthe following condition hold:
 ((SxTy))F(M(x;y);'(M(x;y))): (3.24)
whereF2C,'2u;and
M(x;y) = maxn
(xy);(xSx);(yTy);1
2(ySx)

+1
2(xTy)

2o
:
ThenSandThave a unique common xed point provided that one of the mappings S
orTis-continuous.

12 M. ABBAS, A. H. ANSARI, L. GURAN, AND W. KASSAB
Corollary 3.5. [36]LetS;T be self mappings on a -complete modular space X,F2C,
2 and'2such that for any x;y2X;following condition hold:
(SxTy)F(M(x;y);'(M(x;y))): (3.25)
and
M(x;y) = maxn
(xy);(xSx);(yTy);1
2(ySx)

+1
2(xTy)

2o
:
ThenSandThave a unique common xed point provided that one of the mappings S
orTis-continuous.
Corollary 3.6. [36]LetS;T be self mappings on a -complete modular space X. Sup-
pose that there exists F2C, 2 ,'2,k2[0;1)andL0such that for any
x;y2X, the following condition hold:
(SxTy)kM(x;y) +LN(x;y); (3.26)
where
M(x;y) = maxn
(xy);(xSx);(yTy);1
2(ySx)

+1
2(xTy)

2o
andN(x;y) = minf(xSx);(yTy);(ySx);(xTy)g:ThenSandT
have a unique common xed point provided that one of the mappings SorTis-
continuous.
Denez=f:R+!R+:is a Lebesgue integrable mapping which is summable,
nonnegative and satisesR
0(t)dt> 0, for each >0g.
Corollary 3.7. LetS;T be self mappings on a -complete modular space X. Suppose
that there exists k2[0;1)andL0such that for any x;y2X, the following condition
hold:
(SxTy)Z
0(t)dtF(M(x;y)Z
0(t)dt;'(M(x;y)Z
0(t)dt) +LN(x;y)Z
0(t)dt); (3.27)
whereF2C,'2u;and
M(x;y) = maxn
(xy);(xSx);(yTy);1
2(ySx)

+1
2(xTy)

2o
andN(x;y) = minf(xSx);(yTy);(ySx);(xTy)g:ThenSandT
have a unique common xed point provided that one of the mappings SorTis-
continuous.
Proof. Take (t) =t :The result then follows from Theorem 3.2. 

COMMON FIXED POINT THEOREMS FOR F-( ;')-CONTRACTIONS IN MODULAR SPACES 13
Corollary 3.8. LetTbe a self mappings on a -complete modular space Xwhich
satises the following inequality
((TxTy))F( (m(x;y));'(m(x;y))) +L (n(x;y)) (3.28)
for allx;y2X, whereF2C, 2 ,'2u,
m(x;y) = maxn
(xy);(xTx);(yTy);1
2(yT)

+1
2(xTy)

2o
andn(x;y) = minf(xTx);(yTy);(yTx);(xTy)g. ThenThas a
unique xed point.
Example 3.9. LetX=R,(x) =x2for allx2X. Let us prove that :X![0;1)
is a modular. Then we have:
(m1)(x) =x2= 0 if and only if x= 0;
(m2)(x) = (ax)2=(x) for every scalar withjj= 1;
(m3)(x+y) = (x+y)2= (x)2+ (y)2+ 2xyx2+y2=(x) +(y)
whenever+= 1;and;0.
We dene the mappings S;T:X!Xby the rules:
Sx=x
4; Tx =x2
16; x2X:
TakeF(s;t) =5
6sfor allt;s > 0 andL1000:It is veried that all the
conditions of Theorem 3.2 are satised. Moreover, 0 is a common xed point
ofSandT.
Authors' contributions: All authors contributed equally to the writing of this
paper. All authors read and approved the nal manuscript.
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(M. Abbas) Department of Mathematics and Applied Mathematics, University of Pre-
toria, Lynnwood road, Pretoria 0002, South Africa, &Department of Mathematics Uni-
versity of Management and Technology, C-II, Johar Town, Lahore Pakistan.
Email address :abbas@kau.sa
(A. H. Ansari) Department of Mathematics, Karaj Branch, Islamic Azad University,
Karaj, Iran
Email address :analsisamirmath2@gmail.com, mathanalsisamir4@gmail.com
(L. Guran) Department of Pharmaceutical Sciences, "Vasile Goldis " Western Univer-
sity of Arad,, Liviu Rebreanu Street, no. 86, 310045, Arad, Romania.
Email address :lguran@uvvg.ro, gliliana.math@gmail.com
(W. Kassab)Corresponding author. Department of Mathematics and Computer Sci-
ence, University Politehnica of Bucharest, 060042, Romania, &International University
of Beirut -BIU- Lebanon.
Email address :kassab w@hotmail.com