A Common Fixed Point Theorem for Two Pairs of Mappings in Dislocated Metric Space ()
1. Introduction
In 1922, S. Banach [2] established a fixed point theorem for contraction mapping in metric space. Since then a number of fixed point theorems have been proved by many authors and various generalizations of this theorem have been established. In 1982, S. Sessa [3] introduced the concept of weakly commuting maps and G. Jungck [4] in 1986, initiated the concept of compatibility. In 1998, Jungck and Rhoades [5] initiated the notion of weakly compatible maps and pointed that compatible maps were weakly compatible but not conversely.
The study of common fixed point of mappings satisfying contractive type conditions has been a very active field of research activity. In 1986, S. G. Matthews [6] introduced the concept of dislocated metric space under the name of metric domains in domain theory. In 2000, P. Hitzler and A. K. Seda [7] generalized the famous Banach Contraction Principle in dislocated metric space. The study of dislocated metric plays very important role in topology, logic programming and in electronics engineering.
The purpose of this article is to establish a common fixed point theorem for two pairs of weakly compatible mappings in dislocated metric spaces which generalize and improve similar results of fixed point in the literature.
2. Preliminaries
We start with the following definitions, lemmas and theorems.
Definition 1. [7] Let X be a non empty set and let be a function satisfying the following conditions:
1).
2) implies
3) for all.
Then d is called dislocated metric (or d-metric) on X and the pair is called the dislocated metric space (or d-metric space).
Definition 2. [7] A sequence in a d-metric space is called a Cauchy sequence if for given, there corresponds such that for all , we have.
Definition 3. [7] A sequence in d-metric space converges with respect to d (or in d) if there exists such that as
In this case, x is called limit of (in d)and we write
Definition 4. [7] A d-metric space is called complete if every Cauchy sequence in it is convergent with respect to d.
Definition 5. [7] Let be a d-metric space. A map is called contraction if there exists a number with such that
We state the following lemmas without proof.
Lemma 1. Let be a d-metric space. If is a contraction function, then is a Cauchy sequence for each
Lemma 2. [7] Limits in a d-metric space are unique.
Theorem 1. [7] Let be a complete d-metric space and let be a contraction mapping, then T has a unique fixed point.
Definition 6. Let A and S be two self mappings on a set X. Mappings A and S are said to be commuting if .
Definition 7. Let A and S be two self mappings on a set X. If for some, then x is called coincidence point of A and S.
Definition 8. [5] 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 9. A function is said to be contractive modulus if for
Definition 10. A real valued function defined on is said to be upper semicontinuous if
for every sequence with as
It is clear that every continuous function is upper semicontinuous but converse may not be true.
In 1983, B. Fisher [1] established the following theorem in metric space.
Theorem 2. Suppose that S, P, T and Q are four self maps of a complete metric space (X, d) satisfying the following conditions
1) and.
2) Pairs (S, P) and (T, Q) are commuting.
3) One of S, P, T and Q is continuous.
4) where for all and
Then S, P, T and Q have a unique common fixed point. Also, z is the unique common fixed point of pairs (S, P) and (T, Q).
3. Main Results
Theorem 3. Let (X, d) be a complete d-metric space. Suppose that A, B, S and T are four self mappings of X satisfying the following conditions
i)
ii) where is an upper semicontinuous contractive modulus and
iii) The pairs and are weakly compatible, then A, B, S and T have an unique common fixed point.
Proof. Let be an arbitrary point of X and define a sequence in X such that
Now by condition ii), we have
where
is not possible since is a contractive modulus, so
(1)
Since is upper semicontinuous, contractive modulus the Equation (1) implies that the sequence is monotonic decreasing and continuous.
Hence there exists a real number such that
Taking limit in (1) we obtain which is possible if, sice is contractive modulus. therfore
We claim that is a cauchy sequence.
if possible, let is not a cauchy sequence. Then there exists a real number and subsequences and such that and
(2)
so that
Hence
Now
Taking limit as we have
So by contractive condition ii) and (2)
(3)
where
Now taking limit as we get
Therefore from (3) we have which is a contradiction, since is contractive modulus.
Hence is a cauchy sequence.
Since X is complete, there exists a point u in X such that. So,
Hence,
Since there exists a point such that. Now by condition ii)
where
Taking limit as we have
Thus implies which is a contradiction, since is a contractive modulus. Thus. Hence which represents that v is the coincidence point of A and S.
Since the pair are weakly compatible, so
Again, since there exists a point such that. Then by condition ii) we have,
where
If then which implies
a contradiction, since is a contractive modulus.
Again if then
a contradiction. Hence, Which implies. Therefore. Thus w is the coinci- dence point of B and T.
Since the pair are weakly compatible, so. Now we show that u is the fixed point of S.
By condition ii), we have
where,
If then,
a contradiction since is contractive modulus.
If or, one can observe that there are contradictions for both cases. Hence we conclude that which implies that
Therefore,
Now we show that u is the fixed point of T. Again by condition ii),
where,
If then,
a contradiction.
If or one can observe that there are contradictions for both cases. Hence we conclude that which implies that
Therefore
Hence, i.e. u is the common fixed point of the mappings and T.
Uniqueness:
If possible let u and z are two common fixed points of the mappings and T. By condition ii) we have,
where,
If then,
a contradiction, since is a contractive modulus.
Again if or one can observe that there are contradictions for both cases. Hence we conclude that which implies that
Therefore, u is the unique common fixed point of the four mappings and T. This completes the proof of the theorem.
Now we have the following corollaries:
Corollary 1. Let (X, d) be a complete dislocated metric space. Suppose that A, S and T are three self map- pings of X satisfying the following conditions:
1) and.
2) where is an upper semicontinuous contractive modulus and
.
3) The pairs and are weakly compatible, then A, S and T have an unique common fixed point.
Proof. If we take in theorem (3) and follow the similar proof we get the required result.
Corollary 2. Let (X, d) be a complete dislocated metric space. Suppose that A and S are two self mappings of X satisfying the following conditions.
1).
2) where is an upper semicontinuous contractive modulus and
.
3) The pair is weakly compatible, then A and S have an unique common fixed point.
Proof. If we take and in theorem (3) and follow the similar proof we get the required result.
Corollary 3. Let (X, d) be a complete dislocated metric space. Suppose that S and T are two self mappings of X satisfying the following conditions
1) where is an upper semicontinuous contractive modulus and
.
2) The pairs and are weakly compatible, then S and T have an unique common fixed point.
Proof. If we take in theorem (3) and follow the similar proof we get the required result.
Corollary 4 Let (X, d) be a complete dislocated metric space. Let be a map satisfying the following conditions
where is an upper semicontinuous contractive modulus and
then the map S has a unique fixed point.
Proof. If we take in corollary (3) and follow the similar proof we get the required result.