Open Journal of Discrete Mathematics
Vol.08 No.02(2018), Article ID:83858,13 pages
10.4236/ojdm.2018.82004
Some Common Fixed Point Theorems Satisfying Meir-Keeler Type Contractive Conditions
Dinesh Panthi
Department of Mathematics, Valmeeki Campus, Nepal Sanskrit University, Kathmandu, Nepal
Copyright © 2018 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: February 20, 2018; Accepted: April 17, 2018; Published: April 20, 2018
ABSTRACT
In this article, we establish some common fixed point results for two pairs of compatible mappings satisfying Meir-Keeler type contractive conditions in metric space and dislocated metric space which extend and improve some similar fixed point results in the literature.
Keywords:
Common Fixed Point, Compatible Maps, Cauchy Sequence, Continuous Function
1. Introduction
The word metrics plays a dominant role in most of the fields like, geometry, economics, statistics, graph theory, probability theory, coding theory, game theory, pattern recognition, computer graphics, theory of information and computer semantics, molecular biology etc. M Frechet in 1906, first time introduced the notion of metric space which is now a very useful topic in mathematical analysis. In 1912, L. E. J. Brouwer [1] established a topological fixed point theorem. In 1922, S. Banach [2] established a contraction mapping theorem in a complete metric space, is a primary result of functional analysis. After the establishment of contraction mapping theorem, various authors generalized the theorem and established a huge number of fixed point results in the literature.
In 1969, A. Meir and E. Keeler [3] obtained a remarkable generalization of Banach Contraction principle with the notion of weakly uniformly strict contraction which is famous as contraction principle. This theorem has also been generalized by various authors for single, pairs and even for sequence of mappings.
There exists a vast literature which generalizes the result of Meir and Keeler. Maiti and Pal [4] established a fixed point theorem for a self map T of a metric space satisfying the following condition which is the generalization of weakly uniformly strict contraction (1). For every there exists a such that
Park-Rhoades [5] and Rao-Rao [6] extended this result for two self mapping S and T in a metric space satisfying the condition
In 1986, Jungck [7] and Pant [8] extended the results for four mappings.
The study of common fixed points satisfying contractive type conditions has been a very active field of research activity. The most general common fixed point theorems for four mappings, say A, B, S and T of a metric space use either a.
Banach type contractive condition of the form
1)
where
a Meir-Keeler type contractive definition
2) given there exists a such that
or -contractive condition of the form
3)
where is such that for each
The contractive condition (2) ensures the existence of a fixed point, only when f satisfies some additional conditions.
The following conditions on the function have introduced and employed by various authors for the establishment of fixed point
1) is non decreasing and is non increasing [9] .
2) is non decreasing and for each [10] [11] .
3) is upper semi-continuous [11] [12] [13] .
4) is non decreasing and continuous from the right [14] .
In 1985, S. G. Matthwes [15] generalized Banach Contraction Mapping Theorem under metric domains in domain theory. In 2000, P. Hitzeler and A. K. Seda [16] provided a generalization on the notion of topology and gave a name as dislocated topology. He presented variants of Banach Contraction Principle for various modified forms of metric space including dislocated metric space. Since then, many authors have established fixed point theorems in dislocated metric space. In the literature one can find many interesting recent articles in the field of dislocated metric space (See for examples [17] [18] [19] [20] [21] ). The study of dislocated metric plays very important role in Topology, semantics of logic programming and in electronics engineering.
In this paper, we establish some common fixed point theorems for two pairs of compatible and weakly compatible mappings satisfying Meir-Keeler type contractive condition in dislocated metric space.
2. Preliminaries
We start with the following definitions and theorems.
Definition 1. [16] Let X be a non empty set and let be a function satisfying the following conditions:
1)
2) implies .
3)
4) for all .
If d satisfies the conditions 1 - 4 Then d is called the metric on X and the pair (X, d) is called the metric space. If d satisfies the conditions 2 - 4, then d is called the d-metric on X and the pair (X, d) is called the dislocated metric space.
Definition 2. [3] A self mapping T of a metric space (X, d) is called a weakly uniformly strict contraction or simply an contraction if for each there exists such that for all
(1)
Theorem 1. [3] Let (X,d) be a complete metric space and is weakly uniformly strict contraction then T has a unique common fixed point,say z and for any , .
Definition 3. [7] Two mappings S and T from a metric space (X, d) into itself are called compatible if
whenever is a sequence in X such that for some
Definition 4. [22] Let A and S be mappings from a metric space into itself. Then, A and S are said to be weakly compatible if they commute at their coincident point; that is, for some implies .
Definition 5. [23] Two self mappings A and S of a metric space (X, d) are called reciprocally continuous if and whenever is a sequence such that for some .
If A and S both are continuous they are obviously reciprocally continuous but the converse is not true.
3. Main Results
Now we establish the following lemma in dislocated metric space.
Lemma 1. Let (X, d) be a dislocated metric space. Let be mappings satisfying the conditions
and (2)
Assume further that given for each there exists such that for all
(3)
and
whenever (4)
where
then for each , the sequence in X defined by the rule
is a Cauchy sequence.
Proof. let , then by condition (2) we can define a sequence
such that
and for (5)
Now we have
But,
and,
Hence,
which implies that,
[ is impossible.]
Similarly, we can show that
Consequently, we conclude that . For n even or odd, the last inequality implies that is a Cauchy sequence.
Now, we establish a common fixed point theorem for two pairs of compatible mappings in metric space.
Theorem 2. Let (X, d) be a complete metric space. Let such that the pairs (A, S) and (B, T) be compatible mappings which satisfy the following conditions
(6)
Given, , such that
where
(7)
(8)
. Suppose that the mappings in one of the pairs (A, S) or (B, T) are
reciprocally continuous, then A, B, S and T have a unique common fixed point.
Proof:
let be any point in X. Define sequences and in X given by the rule
and for (9)
then by Jachymski’s lemma [11] , is a Cauchy sequence. Sine X is complete, there exists a point such that . Also the sequences
and (10)
Suppose that the pair is reciprocally continuous, then and . Since the pair is compatible so
and (11)
This implies that . Hence .
Since, there exists a point such that .We claim that .If ,then by using condition (8) we get,
which is a contradiction. So . Hence .
Since the compatible maps commute at their coincidence point, we get . This further implies that . If , by condition (8) we get
which is a contradictions, so . Hence, . Thus Az is the common fixed point of the mappings A and S. Similarly we obtain is the common fixed point of the mappings B and T.
Uniqueness:
If possible, let u and v are two common fixed points of the maps A, B, S and T. Now by virtu of (8)
which is a contradiction. This shows that
The proof is similar when the mappings B and T are assumed compatible and reciprocally continuous. This completes the proof of the theorem.
Now, on the light of above theorem, one can establish the following corollaries easily.
Corrollary 1. Let (X, d) be a complete metric space. Let such that the pairs (A, S) and (B, S) be compatible mappings which satisfy the following conditions
Given, , such that
where
suppose that the mappings in one of the pairs (A, S) or (B, S) are
reciprocally continuous, then A, B and S have a unique common fixed point.
Corrollary 2. Let (X, d) be a complete metric space. Let such that the pairs (A, S) and (A, T) be compatible mappings which satisfy the following conditions
Given, , such that
where
. Suppose that the mappings in one of the pairs (A, S) or (A, T) are
reciprocally continuous, then A, S and T have a unique common fixed point.
Corrollary 3. Let (X, d) be a complete metric space. Let such that the pair (A, S) be compatible mappings which satisfy the following conditions
Given, , such that
where
. Suppose that the pair (A, S) is reciprocally continuous, then A, and
S have a unique common fixed point.
Corrollary 4. Let (X, d) be a complete metric space. Let such that the pairs and be compatible mappings which satisfy the following conditions
Given, , such that
where
. Suppose that the mappings in one of the pairs or
are reciprocally continuous, then A, B, IX have a unique common fixed point.
Now, we establish a common fixed point theorem for two pais of compatible mappings in dislocated metric space.
Theorem 3. Let (X, d) be a complete dislocated metric space. Let such that the pairs (A, S) and (B, T) be compatible mappings which satisfy the following conditions
(12)
Given, , such that
where
(13)
(14)
where, and is such that for each . If
one of the mappings A, B, S or T be continuous then A, B, S and T have a unique common fixed point.
Proof. Let be any point in X. Define sequences and in X given by the rule
and for (15)
then by above lemma (1), is a Cauchy sequence. Sine X is complete there exists a point such that . Also the sequences
and (16)
Suppose that S is continuous, then and . Since the pair is compatible so
(17)
Since , corresponding to each value of n, there exists a sequence such that .
Thus and .
We assert that
If not, there exists a subsequence of , a number and a positive number N such that for each , we have
and
Moreover, by virtu of (14) for large m we obtain,
now taking limit as we have
which is a contradiction, Hence
This represents that,
We claim that
If , then by virtu of (14) for large n, We obtain
Now letting we obtain,
which is a contradiction. Therefore, .
Since, there exists a point such that . We claim that . If by using condition (14) we get,
which is a contradiction. So . Hence .
Sice the compatible maps commute at their coincidence point, we get . This further implies that .
If , by condition (14) we get
which is a contradictions, so . Hence, . Thus Az is the common fixed point of the mappings A and S.
Similarly we obtain is the common fixed point of the mappings B and T when T is supposed to be continuous.
Uniqueness:
Let u and v are two common fixed points of the mappings A, B, S and T. Now by virtu of (14),
which is a contradiction. Hence .
Moreover, the proof follows on similar lines when A or B is assumed to be continuous since and .
This completes the proof of the theorem.
Now, in the light of the above theorem one can establish the following corollaries easily.
Corrollary 5. Let (X, d) be a complete dislocated metric space. Let such that the pairs (A, S) and (B, S) be compatible mappings which satisfy the following conditions
and (18)
Given, , such that
where
(19)
where, and is such that for each . If
one of the mappings A, B or S be continuous then A, B and S have a unique common fixed point.
Corrollary 6. Let (X, d) be a complete dislocated metric space. Let such that the pairs (A, S) and (A, T) be compatible mappings which satisfy the following conditions
and (20)
Given, , such that
where
(21)
where, and is such that for each . If
one of the mappings A, S or T be continuous then A, S and T have a unique common fixed point.
Corrollary 7. Let (X, d) be a complete dislocated metric space. Let such that the pair (A, S) be compatible mappings which satisfy the following conditions
(22)
Given, , such that
where
(23)
where, and is such that for each . If
one of the mappings A or S be continuous then A and S have a unique common fixed point.
Corrollary 8. Let (X, d) be a complete dislocated metric space. Let such that the pairs (A, IX) and (B, IX) be compatible mappings which satisfy the following conditions
and (24)
Given, , such that
where
(25)
where, and is such that for each . If
one of the mappings A, B, or IX be continuous then A, B and IX have a unique common fixed point.
Remarks: Our results extend and improve the results of Meir-Keeler [3] , Bouhadjera and Djoudi [24] , Jha, Pant and Singh [25] , Pant and Jha [26] in dislocated metric space.
Acknowledgements
This work is carried out under Small Research Development and Innovation Grant SRDIG-73/74-S&T-08 supported by University Grants Commission, Nepal.
Cite this paper
Panthi, D. (2018) Some Common Fixed Point Theorems Satisfying Meir-Keeler Type Contractive Conditions. Open Journal of Discrete Mathematics, 8, 35-47. https://doi.org/10.4236/ojdm.2018.82004
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