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American Journal of Engineering Research (AJER)
e-ISSN : 2320-0847 p-ISSN : 2320-0936
Volume-3, Issue-9, pp-167-170
www.ajer.org
Research Paper Open Access
Common Fixed Point Theorems for Hybrid Pairs of Occasionally
Weakly Compatible Mappings in b-Metric Space.
1
Priyanka Nigam,
2S.S. Pagey
1
Sagar Institute of Science and Technology, Bhopal (M.P.) India.
2
Institute for Excellence in Higher Education, Bhopal (M.P.) India.
ABSTRACT: The objective of this paper is to obtain some common fixed point theorems for hybrid pairs of single and multi-valued occasionally weakly compatible mappings in b-metric space.
KEYWORDS: Occasionally weakly compatible mappings, single and multi—valued maps, common fixed point theorem, b- metric space.
2000 Mathematics Subject Classification: 47H10; 54H25.
I. INTRODUCTION
The study of fixed point theorems, involving four single-valued maps, began with the assumption that all of the maps are commuted. Sessa [8] weakened the condition of commutativity to that of pairwise weakly commuting. Jungck generalized the notion of weak commutativity to that of pairwise compatible [5] and then pairwise weakly compatible maps [6]. Jungck and Rhoades [7] introduced the concept of occasionally weakly compatible maps.
Abbas and Rhoades [1] generalized the concept of weak compatibility in the setting of single and multi-valued maps by introducing the notion of occasionally weakly compatible (owc).
The concept of was introduced by Czerwik[3]. Several papers deal with fixed point theory for single and multi- valued maps in
In this paper we extend the result of Hakima Bouhadjera [2] from to
II. PRELIMINARY NOTES
Let denotes a metric space and the family of all nonempty closed and bounded subsets of . Let be the Hausdorff metric on induced by the metric d; i.e.,
for in where
Definition2.1.[3] Let X be a nonempty set and a given real number. A function (nonnegative real numbers) is called a provided that, for all
The pair is called with parameter s.
It is clear that the definition of is an extension of usual metric space. Also, if we consider in above definition, then we obtain definition of usual metric space.
Definition2.2.[1] Maps and are said to be occasionally weakly compatible (owc) if and only if there exist some point in such that and
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Lemma2.3. Let be any and let then for any we have
III. MAIN RESULTS
Theorem3.1 Let be a b-metric space with parameter . Let and be single and multi-valued maps, respectively such that the pairs and are owc and satisfy inequality
(3.1)
for all where . Then have a unique common fixed point in X.
Proof: Since the pairs and are owc, then there exist two elements such that and
First we prove that By Lemma [2.3] and by we have . Suppose
that . Then by (3.1) we get
Since and by Lemma [2.3], and then
This inequality is false as unless which implies that
Again by Lemma [2.3] and we have
. We claim that . Suppose not. Then and using inequality (3.1) we get
.
But and by Lemma [2.3] and so
,
which is false as unless , thus
Similarly, we can prove that
Putting then and Therefore is the common fixed point of
maps .
Now suppose that have another common fixed point Then by lemma [2.3] and we have .
Assume that . Then the use of (3.1) gives
.
Since and ,
we have .
which is false as Then and hence
Corollary3.2 Let be a b-metric space with parameter . Let and be single and multi-valued maps, respectively such that the pairs and are owc and satisfy inequality
(3.2)
for all where . Then have a unique common fixed point in X.
Proof: Clearly the result immediately follows from Theorem 3.1.
Theorem3.3 Let be a b-metric space with parameter . Let and be single and multi-valued maps, respectively such that the pairs and are owc and satisfy inequality
(3.3)
for all with also . Then have a unique common
w w w . a j e r . o r g Page 169 Proof: Since the pairs and are owc, then there exist two elements such that
and
First we prove that By Lemma [2.3] and by we have . Suppose
that . Then by (3.3) we get
=
Since and by Lemma [2.3], and then
= =
, as .
This inequality is false as unless which implies that
Again by Lemma [2.3] and we have
. We claim that . Suppose not. Then and using inequality (3.3) we get
=
But and by Lemma [2.3] and so
= =
, as . ,
which is false as unless , thus
Similarly, we can prove that
Putting then and Therefore is the common fixed point of
maps .
Now suppose that have another common fixed point Then by lemma and we have .
Assume that . Then the use of (3.3) gives
.
Since and ,
we have .
which is false as Then and hence
Theorem3.4 Let be a b-metric space with parameter . Let and be single and multi-valued maps, respectively such that the pairs and are owc and satisfy inequality
(3.4)
for all with also and Then have a unique common fixed point
in X.
w w w . a j e r . o r g Page 170
First we prove that By Lemma [2.3] and by we have . Suppose
that . Then by (3.4) we get
Since and by Lemma [2.3], and then
This inequality is false as unless which implies that
Again by Lemma [2.3] and we have
. We claim that . Suppose not. Then and using inequality (3.4) we get
.
But and by Lemma [2.3] and so
,
which is false as unless , thus
Similarly, we can prove that
Putting then and Therefore is the common fixed point of
maps .
Now suppose that have another common fixed point Then by lemma [2.3] and we have .
Assume that . Then the use of (3.4) gives
.
Since and ,
we have .
which is false as Then and hence
If we put in above Theorem and we obtain the following result.
Corollary3.5 Let be a b-metric space with parameter . Let and be single and multi-valued maps, respectively such that the pairs are owc and satisfy inequality
(3.5)
for all with also and Then have a unique common fixed point in X.
Now, letting we get the next corollary.
Corollary3.6 Let be a b-metric space with parameter . Let and be single and multi-valued maps, respectively such that the pairs and are owc and satisfy inequality
(3.6)
for all with also and Then have a unique common fixed point
in X.
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(3.7)
for all where . Then have a unique common fixed point in X.
Proof: Clearly the result immediately follows from Theorem 3.1.
REFERENCES
[1]. M. Abbas and B.E. Rhoades, “Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized contractive condition of integral type”, Fixed Point Theory Appl. 2007, Art. ID 54101, 9pp. (2008i:540).
[2]. H. Bouhadjera, A. Djoudi and B. Fisher,”A unique common fixed point theorem for occasionally weakly compatible maps”,
Surveys in Mathematical and its Applications, Vol 3, 2008 ISSN 1842-298(electronic), 1843-7265 (print), 177-182.
[3]. S. Czerwik,” Contraction mappings in b-metric spaces”, Acta Math Inf Univ Ostraviensis”, 1, 1993, 5 –11.
[4]. S.Czerwik,” Nonlinear set-valued contraction mappings in b-metric spaces”, Atti Sem Mat Fis Univ Modena. Vol 46, No 2,
1998, 263–276.
[5]. G.Jungck,” Compatible mappings and common fixed points”, International Journal of Mathematics and Mathematical Sciences,
Vol 9, No. 4, 1986, 771-779. (87m:54122)
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[7]. G.Jungck and B. E. Rhoades,” Fixed Point Theorems for Occasionally Weakly Compatible Mappings”, Fixed Point Theory, Vol
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[8]. S.Sessa,” On a weak commutativity condition of mappings in fixed point considerations”, Publications de l’ Institute
