Research Journal of Applied Sciences, Engineering and Technology 2(3): 239-244, 2010
ISSN: 2040-7467
© M axwell Scientific Organization, 2010
Submitted Date: March 09, 2010
Accepted Date: March 30, 2010
Published Date: May 10, 2010
A Common Fixed Point Theorem Satisfying Integral Type for
Occasionally Weakly Compatible Maps
1
Deo Brat Ojha, 2 Manish Kumar Mishra and 1 Udayana K atoch
1
Department of Mathematics,
2
Departm ent of Pharmacy, R.K .G.I.T . Delhi-M eerut Road G haziabad, 20100 3-U P, India
Abstract: The aim of this paper is to establish a unique common fixed-point theorem for two pair of maps
satisfying a new contractive condition of integral type by using the conc ept of occasionally weakly com patible
single and multi-valued maps in a metric space. This result improves the result of Bouhadjera, Djoudi and
Fisher.
Key w ords: Compatible maps, contractive condition of integral type, occasionally weakly compatible maps
INTRODUCTION
In this study, (X,d) denotes a metric space and
CB(X) the family of all nonempty closed and bounded
subset of X . let J be the Housdorff metric on CB(X)
induc ed by the metric d : i.e.,
Obviously, weakly commuting maps are compatible,
but the converse is not true in general (Jungck, 19 86).
Many authors have obtained a lot of fixed-point theorems
for com patible map pings satisfying contractive type
conditions and assuming continuity of at least one of
mappings.
Kaneko and S essa (1 989), extended the concept of
com patibility for single valued maps to the setting of
s i n g le a n d m u l t i- va lu ed m ap s a s f ol lo w s,
are said to be
for all X, and x0X and
comp atible if
for A, B in CB(X), where
Whenever {x n } is a sequence in
Let f, g is two self maps of a metric space (X,d) . In
his study, Sessa (1982) defin ed f and g to be weakly
com muting if for all x0X.
d(fgx,gfx)#d(gx,fx)
It can be seen that commuting maps (fgx= gfxœ x0X)are
weakly commuting, but converse is false in general
(Sessa, 198 2).
Then after, Jungck (1986) extended the concepts of
com mutativity and weak commutativity by giving the
notion of com patibility. Maps f an d g com patible if
X such that
and
.
Turkoghe et al. (2002) gave another generalization of
com mutativity and weak commutativity for single valued
maps by introducing the definition:
and
are called com patible if
W henever{x n }and {y n } are sequences in X such that
for some
, where
for n=1,2,...........
W henever {x n } is a sequence in X su ch that
for some .t0X
Jungck and Rhoades (1998) weakened the notion of
com patibility for single and multi-valued maps by giving
the concept of weak compatibility. They defined the maps
f and F above to be weakly com patible if they co mm ute
at their coincidence points; i.e., if fFx = Ffx whenever
fx0Fx.
Corresponding Author: Deo Brat Ojha, Department of Mathematics, R.K.G.I.T. Delhi-Meerut Road Ghaziabad, 201003U.P, India
239
Res. J. Appl. Sci. Eng. Technol., 2(3): 239-244, 2010
Abbas and Rhoades (2007), generalized the concept
of weak compatibility in the setting of single and
multi–valued maps by introducing the notion of
occa sionally weak comp atibility (owc) maps f and F are
said to be occasionally weak comp atibility if and only if
there exits some p oint x in X such that fx0Fx and fFxfFfx
Pathak el al. (2007) gave the common fixed-point
theorem satisfying Integral type implicit relations.
Hakima et al., (2008) have given the common fixed-point
theorem for occasionally weak compatibility maps.
Gairola and Rawat (2008, 2009) have given common
fixed-point theorem for integral type inequality. Chugh
et al. (2005) gave common fixed-point theorem for
weakly compatible mapping of integral type in D-m etric
spaces and gave an example for a pair of maps
f,g; (x,d)v(x,d) is weakly compa tible pair if they
commute at coinciden ce po ints i.e. fx = gx iff fgx = gfx
Ex1- let x = [0,3] be equipp ed w ith the usual metric
space d(x,y) = # x-y#, defined f ,g ; [0,3]v[0,3] by
Our aim here is to establish and prove a unique
common fixed-point theorem, by using the concept of
com mutativity and weak comm utativity m aps. For ma in
results, we ne ed the follow ing lemm a whose proof is
obvious.
Lemm a: let A, B in CB (X), then for any a0A. We have
RESULTS AND DISCUSSION
Theorem 2: let (X,d) be a m etric space, let f.g:XvX and
F.G:XvCB(X) be single and multi-valued maps,
respectively such that (f,F) are (g,G) owc and satisfy
inequality
and
(1)
Then for any , x0[1,3],fgx= gfx, show ing that f, gare
weakly
compatible maps on [0,3] (Gairola and
Raw at, 2008).
In this study a unique common fixed point theorem
for two pair of maps satisfying a new contractive
condition of integral type by using the concept of
occa sionally weakly compatible single and multi-valued
maps in a metric space.
For all x, y in X, where p $2, is an integer. a $ 0,
0 < c < 1. W here, n;
is a lebesgue integrable
mapping which is summable, nonnegative and such, for
each
MATERIALS AND METHODS
Then f, g, F and G has a unique
Theorem 1: Let (X,d) be a complete metric space,
"0[0,1],f;XvX a mapping such that for each x,y0X,
common fixed point in X.
Proof: Since the pair {f, F} and {g, G} are comm utativity
and weak co mm utativity. Then there exit tw o elem ent u
and v in X su ch that fug Fu,fFufFfu and gvg
Gv,gGvfGgv
W here n;
is a Lebesg ue integrable
mapping which is summable, nonnegative and such, for
each
By lemma and the triangle inequality we have
(2)
Then f has a unique common fixed point zsuch that
for each x0X,
Rhoades (2003) extended
Suppose that
this result by replacing the above condition by the
following
inequ ality (1). W e get,
240
> 0.Then by
Res. J. Appl. Sci. Eng. Technol., 2(3): 239-244, 2010
=
(3)
and
(4)
By lemma and then
(5)
W hich is a contradiction. Hence J(Fu ,Gv )=0, which implies that fu =gv. Again by lemm a and the triang le inequality
we have
(6)
W e claim that
, suppose not then
and using inequality (1) we obtain
=
(7)
But
and
(8)
Equation (8) lemma and so
(9)
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Res. J. Appl. Sci. Eng. Technol., 2(3): 239-244, 2010
A contradiction: This implies that J(Ffu ,Gv ) = 0, thu s f 2 u = fu =gv.
Similarly, we can prove that g 2 v= gv. Putting,fu = gv = z then ,fz = gz = z, z , Fz and z , Gz. Therefore, z is a common
fixed point of maps f,g,F and G. Now suppose that f,g,F and G have anoth er com mon fixed point z!…z . Then by lemma
and the triangle inequality we have
(10)
assume that
. Then by using inequality (1) gives
=
Then since and
(11)
and
(12)
W e have
(13)
A contradiction: Then J(Fz ,Gz’ ) = 0. and hence z!=z. If we put theorem (2) f = g and F = G. we obtain the following
result.
Corollary: Let (X,d) be a m etric spa ce and let f:XvX,F:XvCB(X) , be a single and a multi-valued m ap respectively,
suppose that f and F are owc and satisfy the inequality
(14)
For all x, y in X, where p$2 is an integer. a$0 and 0<c<1. Then f and F have unique common fixed point in X.
Now, letting f = g. We get next corollary.
Corollary: Let (X,d) be a m etric spa ce, f : XvX be a simple map and F,G: X v CB(X) be two multi-valued maps such
that;
C
The pairs {f ,F} and {f ,G} are owc .
C
The inequality
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Res. J. Appl. Sci. Eng. Technol., 2(3): 239-244, 2010
(15)
Holds for all x, y in X, where p$2 is an integer, a$0. and 0<c<1. Then f , F and G have a unique common fixed point
in X . Now , we g ive an exam ple, w hich illustrate our main result.
Examp le: let X = [0,3] endowed with the Euclidean metric d, define f, g: XvCB(X) as follows:
First we have , f(2) = 2 fF(2) ={2} and fF(2) = {2} = Ff(2) and g(2) = 2 f G(2) ={2} and gG(2) = {2} = Gg(2) i.e. f and
F as well as g and G are ow c . Also, for all x and y in X, inequality (1) is satisfied for a large enough a. So all hypotheses
of theorem (2) are satisfied and is the unique comm on fixe d point f, g, F and G are disc ontinu ous at t = 2. Further, we
have F(X) = {0 ,2},d f(x) = [0,2] c{3} but
G(X) =
So, this exam ple illustrates the g enera lity of our result.
Gairola, U.C. and A.S. Rawat, 2008. A fixed-point
theorem for integral type inequality. Int. J. Math.
Anal., 2(15): 709-712.
Gairola, U.C. and A.S. Rawat, 2009. A fixed-point
theorem for two pair of maps satisfying a new
contractive condition of integral type. Int. Math.
Forum, 4(4): 177-183.
Hakima, B., D. Ahcene and B. Fisher, 2008. A unique
common fixed point theorem for occasionally weak
com patibility maps. Surreys Math. Applications, 3:
177-182.
Jungck, G., 1986. Compatible mappings and common
fixed points. Int. J. Math. Sci., 9(4): 771-779.
Jungck, G. and B.E. Rhoades, 1998. Fixed point for set
valued functions w ith out continuity. Ind. J. Pure
Appl. Math., 29(3): 227-238.
Kaneko, H. an d Sessa S., 1989. Fixed point theorems for
compatible multi-valued and single-valued mappings.
Internate J. Math. Sci., 12(2): 257-262.
Pathak, H.K., K.T. Rakesh and S. Khan, 2007 . A common
fixed point theorem satisfying Integral type Im plicit
relations. Appl. Math. E-Notes, 7: 22-228.
CONCLUSION
W e establish a unique common fixed-point theorem
for two pair of maps satisfying a new contractive
condition of integral type by using the concept of
occa sionally weakly compatible single and multi-valued
maps in a metric space.
REFERENCES
Abbas, M. and B.E. Rhoades, 2007. Common fixed point
theorems for hybrid pa irs of occasionally w eakly
com patible map ping satisfying generalized
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type in D-metric spaces. Universitatea Din Bac|u
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