Research Journal of Applied Sciences, Engineering and Technology 2(4): 387-395, 2010
ISSN: 2040-7467
© M axwell Scientific Organization, 2010
Submitted Date: April 22, 2010
Accepted Date: May 08, 2010
Published Date: July 10, 2010
Some Results on Common Fixed Point of Multivalued Generalized n -Weak
Contractive Mappings in Integral type Inequality
1
1
Deo Brat Ojha and 2 Manish Ku mar Mishra
Departm ent of Mathematics, R.K.G .I.T., Delhi-M eerut Road G haziabad, 20100 3 UP, India
2
R.K .G.I.T . Delhi-M eerut road Ghaziabad, 2010 03 U P, India
Abstract: The aim of this study is to establish unique Common Fixed Point of Multivalued Generalized nW eak Contractive Mappings in Integral type inequality. This result improves the result of Nadler and DaTerKaneko’s and references there in.
Key w ords: Common fixed point, contractive mapping, fixed point
INTRODUCTION
Fixed point and coincidence results are presented for
multivalued generalized n -weak contractive mappings on
complete metric spaces, where N: [0, +4) ÷ [0, +4) is a
lower semicontinous function with N (0) = 0 and N (t) >
0 for all t > 0. Our results extend previous results by
Zhang and Song (2009), as well as by Rhoades (2001),
Nadler (1969) and DaTer and K aneko (1995). Branciari
(2002) gave a fixed-point result for a single mapping
satisfying an analogue of Banach’s contraction principle,
which is stated as follows;
If (X, d) be a metric space. We denote the family of
all nonempties closed and bounded subsets of X by CB
(X). A mapping T: X÷X is said to be N -weak contractive
if there exists a map n: [0, +4) ÷ [0, +4) with N(0) = 0
and N(t) > 0 for all t > 0, such that
(1)
For all x, y , X.
Also two mappings T, S: X÷X are called generalized
N-weak contractions if there exists a map N: [0, +4) ÷ [0,
+4) with N(0) = 0 and N(t) > 0 for all t > 0, such that:
Theorem 1: Let (x, d) be a complete metric space. c , [0,
1], T: X ÷X a mapping such that , for each x, y , X.
(2)
For all x, y , X, where
W here f : R + ÷ R + is a lebesgue -integrable mapping which
is summab le, non-negative an d such that for each s > 0.
(3)
then T has a uniqu e fixed point z , X such
that, for each z , X , lim n T n x = z.
This result was further generalized by Abbas and
Rhoades (2007), Alionche (2006), Gairola and Rawat
(2008), Kum ar et al. (2007) and Rhodes (2003). How ever,
Suzuki (2007) has shown that Meir-Keeler contraction of
Integral type is still Meir-Keeler contraction and Alionche
and Fisher (2007) extends function for three complete and
compact metric spaces. In this study, we prove the
following fixed-point theorem for complete metric spaces.
Now, Behzad and Moradi (2010), extends the
common fixed point of Multivalued Generalized n-Weak
Contractive Mappings.
A mapping T : X÷CB(X) is said to be a weak contraction
if there exists 0 # " < 1 su ch that:
(4)
for all x, y , X, where H denotes the Hausdorff metric on
CB(X) induced by d, that is:
(5)
Corresponding Author: Deo Brat O jha, D epartm ent of M athem atics, R.K.G.I.T., Delhi-Meerut Road Ghaziabad,
201003 UP, India
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Res. J. Appl. Sci. Eng. Technol., 2(4): 387-395, 2010
for all A, B , CB (X), and where,
(9)
for all x, y , X. Then there exists a point x , X such that
x , Tx (i.e., x is a fixed point of T ).
DaTer and Kaneko (1995) proved the existence o f a
fixed point for a multivalued weak contraction mapping
of a complete metric space X into CB (X).
A mapping T : X÷CB (X) is said to be -weak
contractive if there exists a map , n: [0, +4) ÷ [0, +4)
with n(0) = 0 and n(t) > 0 for all t > 0 su ch that:
Theorem 3: Let (X, d) be a comp lete metric space and T
: X ÷ CB (X) be such that:
(6)
(10)
for all x, y , X.
The concepts of weak and n -weak contractive
mappings were defined by DaTer and Kaneko (19 95).
Many authors have studied fixed points for multivalued
mapp ings. Interested reade r may co nsult (2010).
for some 0 # " < 1 and for all x, y , X (i.e., weak
contraction). If x ÷ d (x, Tx) is lower semicontinous ,
then there exits x 0 , X such that x 0 , Tx 0 .
Now extend Nadler and DaTer-Kaneko’s theorems
(2010) to multivalued generalized weak contraction
mappings in (Definition 1). Rhoades (2001).Theorem 1
proved the following fixed point theorem for N-weak
contractive single valued mappings, giving another
generalization of the Banach Contraction Principle.
MATERIALS AND METHODS
In this study, (X, d) denotes a complete metric space
and H denotes the Hausdorff metric on CB(X).
Definition 1: Two mappings T, S : X ÷ CB (X) are called
generalized weak contractions if there exists 0 # " < 1
such that:
Theorem 4: Let (X , d) be a comp lete me tric space and T
: X ÷ CB (X) be a m apping such that
all x, y , X. (7)
(11)
Definition 2: Two m appings T, S : X ÷ CB (X) are called
generalized N-weak contractive if there exists a map N:
[0, +4) ÷ [0, +4) with N(0) = 0 and N(t) > 0 for all t > 0,
such that:
for all x, y , X.
for every x, y , X (i.e., weak contraction). where N: [0,
+4) ÷ [0, +4) is a continuous and nondecreasing function
with N(0) = 0 and N(t) > 0 for all t > 0. Then T has a
uniqu e fixed point.
By choosing R (t) = t - N(t) N-weak contractions
become mappings of Boyd and Wong (1969) type and by
defining k(t) = (t – N(t))/t, for t > 0 and k(0) = 0, then Nweak contractions become mappings of Reich type
(1974). Recently Zhang and Song (2009) proved the
following theorem on the existence of a common fixed
point for two single valued generalized N-weak
contraction mappings.
(8)
In the proof of our main results, we will use the
following well-known lemma, and refer to Nadler (1969)
or Assad and K irk (1972) for its proof.
Theorem 5: Let (X , d) be a complete metric space and
T: X ÷ CB (X) be a two mapping such that for all
x, y , X.
RESULTS AND DISCUSSION
Theorem 2: Let (X , d) be a complete metric space.
Suppose T : X ÷ CB (X) is a contraction mapping in the
sense that for some 0 # " < 1,
(12)
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Res. J. Appl. Sci. Eng. Technol., 2(4): 387-395, 2010
(i.e., generalized N-weak contractions), wh ere N: [0, +4) ÷ [0, +4) is an lower semi continuous function with N(0)
= 0 and N(t) > 0 for all t > 0. Then there exists a unique point x , X such that x = Tx = Sx.
we extend Theorem 4 by assuming N to be on ly low er sem i continuous, and extend Theorem 5 to multivalued
mappings.
Lemm a: If A, B , CB(X) and a , A, then for each g > 0, there exists b , B such that:
(13)
Theorem 6: Let (X, d) be a comp lete me tric spac e and let T : X ÷ CB (X) and S : X ÷ CB (X) be two mappings such
that for all x, y , X:
(14)
(i.e., generalized n -weak contractive), where n: [0, +4) ÷ [0, +4) is an lower semi continuous function with n(0)
= 0 and n(t) > 0 for all t > 0. Then there exists a unique point x , X such that x = Tx = Sx.
Proof: Unicity of the common fixed point follows from Eq. (14) and (16). Obviously M (x, y) = 0 if and only if x = y
is a com mon fixed point of T and S . Let x 0 , X and x 1 , Sx 0 . Let x 2 = Sx 1 . By Lemm a, there exists x 3 , Sx 2 such that:
(15)
W e let x 4 =Tx 3 induc tively an d we let
and by Lemma we choose
such that:
(16)
W e break the argum ent into four steps.
Step 1:
Proof: Using (14) and (16),
(17)
whe re
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Res. J. Appl. Sci. Eng. Technol., 2(4): 387-395, 2010
(by 17)
(18)
So
Hence by (17)
(19)
also
(20)
where,
(21)
(by (20)).
So
. Hence by (20),
(22)
Therefore, by (14) and (1 6), we con clude that:
for all k , N.
390
(23)
Res. J. Appl. Sci. Eng. Technol., 2(4): 387-395, 2010
Therefore, the sequence
is monotone non-increasing and bounded below.
So there exists r $ 0 such that
(24)
Since L is l.s.c.,
(25)
and so N (r) =0. Hence r = 0.
By 17, w e con clude that
Step 2: {x n } is a bounded sequence.
Proof. If {x n } were un bounded, then by Step 1, {x 2 n } and {x 2n-1 } are unbounded. W e choose the sequence
such that n(1) = 1,n(2) > n(1) is even an d min imal in the sense that
,
and similarly n(3)>n(2) is odd and minimal in the sense that ,
and
is
even
and
m inimal
in
, and
the sen se that,
the
sense
that
and
is odd and minim al in
and
.
Obviously n(k) $ k for every k , N. By Step 1, there exists N 0 (----) N such that for all k $ N 0 we have
. So for every k $ N 0 , we have n(k+1) $ n(k) $ 2 and
:
(26)
Hence
also
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Res. J. Appl. Sci. Eng. Technol., 2(4): 387-395, 2010
(27)
and this shows that
. So if n (k+1) is odd, then:
(28)
where ,
(29)
and this shows that
. Since N is l.s.c. and (28) holds, we have 1 # 1 – L(1). So L(1) = 0
and this is a contradiction.
Step 3: {x n } is Cauchy.
Proof: Let
So there exists C $ 0 such that
. Since {x n } is bounded, C n < +4 for all n , N Obviously {C n } is decreasing.
. W e need to show that C = 0.
For every k , N, there exists n(k), m(k) , N su ch that m(k) > n(k) $ k and:
(30)
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Res. J. Appl. Sci. Eng. Technol., 2(4): 387-395, 2010
By 30, w e con clude that:
(31)
From Step 1 and Eq. (31), we have:
(32)
So we may assume that for every k , N, m(k) is odd and n(k) is even. Hence:
(33)
where,
(34)
This inequ ality sho ws that,
Since n is lower semi continuous and (33) holds, we have C
# C – N(C). Hence N(C) = 0 and so C = 0 T herefo re, {x n } is a Cauchy sequence.
Step 4: T and S have a com mon fixed point.
Proof: Since (x, d) is complete and {x n } is Cauchy, there exists x , X such that
For every n , N:
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(35)
where.
(36)
Since L is l.s.c and 35 holds, letting n ÷ 4 in (35) w e get:
and this shows that,
(37)
So
and hence
Since Sx , CB(X), Also
(38)
(39)
So from (38), we have:
(40)
Thus
and hence
Therefore, x = Tx.
Remark: In the proof of Zhang and Son g (2009), the boundedness of the sequence {C n } is used, but not proved. Also,
for the proof tha t {x n } is a Cau chy sequence, the m onotonicity of L is used, without being explicitly mentioned.
In our proof of Theorem 6, w hich is different from Zhang an d Song (2009) L is not assumed to be non decreasing
The following theorem extends Rhoades’ theorem by assuming to be only l.s.c.
Theorem 7: Let (X , d) be a comp lete me tric spac e and let T : X ÷ X be a m appings su ch that:
(41)
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Res. J. Appl. Sci. Eng. Technol., 2(4): 387-395, 2010
for every x, y , X. (i.e., L weak contractive), where N: [0,+4) ÷ [0,+4) is an lower semi continuous function with N(0)
= 0 and N(t) > 0 for all t > 0. Then T has a unique fixed po int.
Proof. The proof is similar to the proof of Theorem 6, by taking S = T, and replacing M(x, y) with (x, y),
Remark: With a similar proof as in Theorem 2 in Theorem 6 we can replace the inequality (41) by the following
inequality (42) for two single valued mappings T, S : X ÷ X.
(42)
Branciari, A., 2002. A Fixed point theorem for mappings
satisfying a general contractive condition of Integral
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Gairola, U.C. and A.S. Raw at, 2008. A fixed point
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Kumar, S., R. Chug and R. Kumar, 2007. Fixed point
theorem for compatible mappings satisfying a
contractive cond ition of Integral type. Sochow . J.
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Nadler, S.B., 1969. Multi-valued contraction mappings.
Pacific J. Math., 30: 475-488.
Reich, S., 1974. Some fixed point problems. Atti della
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CONCLUSION
W e establish a unique Common Fixed Point of
Multivalued Generalized n-Weak Contractive Mappings
in Integral type inequality.
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