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Common fixed point and coincidence point of generalized contractions in ordered metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 229 (2012)
Abstract
Let be a partially ordered set and d be a complete metric on X. The notion of f-contractive for a set-valued mapping due to Latif and Beg is extended through an implicit relation. Coincidence and fixed point results are obtained for mappings satisfying generalized contractions in a partially ordered metric space X. Our results improve and extend several known results in the existing literature.
MSC:47H10, 47H04, 47H07.
1 Introduction and preliminaries
Let be a metric space, be the class of all non-empty subsets of X, be the class of all non-empty closed subsets of X and be the class of all non-empty closed and bounded subsets of X. For , let
where
D is called the Hausdorff metric induced by d.
Let be any single-valued mapping. A mapping is f-contractive if there exists a real number κ with such that
By introducing the notion of f-contractiveness for set-valued mappings, Kaneko [1] obtained a result which gives the existence of a coincidence point in a metric space. An improved version of Kaneko’s result was obtained by Latif and Beg [2]. They proved the following.
Theorem 1.1 [[2], Theorem 2.6]
Let be a continuous map with complete. Suppose that is a f-contractive map such that . Then there exists such that .
In [3], Kaneko and Sessa proved the following coincidence point result.
Theorem 1.2 [3]
Let be a complete metric space and be compatible continuous mappings such that and
for all , where .
Then there exists such that .
A mapping is called K-mapping if there exists a real number κ with such that
Definition 1.3 Let be a mapping. A point is said to be a fixed point of F if .
Definition 1.4 Let and be mappings. A point is said to be a coincidence point of F and f if .
Definition 1.5 A partial order is a binary relation ⪯ over a set X which satisfies the following conditions:
-
1.
(reflexivity);
-
2.
If and , then (antisymmetry);
-
3.
If and , then (transitivity)
for all x, y and z in X.
A set with a partial order ⪯ is called a partially ordered set.
Let be a partially ordered set and . Elements x and y are said to be comparable elements of X if either or .
Recently, there have been so many exciting developments in the field of existence of a fixed point in partially ordered sets (see [4–20] and the references cited therein). This trend was started by Ran and Reurings in [11] where they extended the Banach contraction principle in partially ordered sets with some application to a matrix equation. Ran and Reurings [11] proved the following seminal result.
Theorem 1.6 [11]
Let be a partially ordered set such that every pair has an upper and lower bound. Let d be a complete metric on X and be a continuous monotone (either order-preserving or order-reversing) mapping. Suppose that the following conditions hold:
-
1.
There exists with
-
2.
There exists with or .
Then f is a Picard operator (PO), that is, f has a unique fixed point and for each ,
Theorem 1.6 was further extended and refined in [7–9, 12, 15, 19]. These results are hybrid of the two fundamental classical theorems: Banach’s fixed point theorem (see [21]) and Tarski’s [22] fixed point theorem. Our aim in this paper is to introduce a generalization of f-contractiveness through an implicit relation. This implicit relation is then used to obtain the existence of coincidence and common fixed points for a pair of single-valued mapping and set-valued mapping on a partially ordered metric space. In Section 2, we prove a coincidence point theorem where we use an implicit relation only for comparable elements of a partially ordered set X. Our result generalizes/extends [1–3, 23, 24] work to partial ordered sets. Section 3 deals with the existence of a common fixed point by using the notion of set-valued mapping, which improves the results of Latif and Beg [2] to partially ordered sets.
We will make use of the following lemma in the proof of our result in the next section.
Lemma 1.7 [25]
Let and . Then for , there exists an element such that .
Throughout the next two sections, we take as a partially ordered set with a complete metric d.
2 Implicit relation and coincidence points
Implicit relations in metric spaces have been considered by several authors in connection with solving nonlinear functional equations (see, for instance, [16–18, 26] and the references cited therein). First, we give some implicit relations for subsequent use.
Let be the set of nonnegative real numbers and be the set of continuous real-valued functions satisfying the following conditions:
: is non-increasing in .
: there exists a real number κ with and such that the inequalities
and
imply
: implies .
Next, we give some examples for T satisfying -.
Example 2.1 , where .
: is obvious.
: Let , then choose such that (this is possible since ). As , therefore . Now let . If , then . Hence a contradiction. Thus and . If , then . Thus is satisfied.
: Since , therefore . It further implies that .
Example 2.2 , where .
: is obvious.
: Let , then choose so that . Since , therefore . Now let . If , then . Hence a contradiction. Thus and . If , then . Thus is satisfied.
: Since , therefore . It further implies that .
Example 2.3 , where , .
: is obvious.
: Let , then choose so that (this is possible since ). Since , therefore . Now let . If , then , hence a contradiction. Thus and . Thus is satisfied with .
: Since , therefore . It further implies that .
Theorem 2.4 Let and satisfying
for all comparable elements x, y of X and for some . If the following conditions are satisfied:
-
1.
and is closed;
-
2.
If , then ;
-
3.
If is such that , then for all n,
then there exists p with .
Proof Let , then by using assumptions 1 and 2, we can choose with such that . Since , then for any , from Lemma 1.7, there exists such that
Using assumptions 1 and 2, since , there exists such that and so .
Now using (A), we have
Using the facts that , ,
and by , we have
that is,
where , , . By using , we have (),
Using (2.2) in (2.1), we have
Since , then for , from Lemma 1.7, there exists such that
Using assumptions 1 and 2, since , there exists such that and so .
Now, since , by using (A), we have
by we have
that is,
where , , . Therefore, by using , we have ()
And so from (2.3) and (2.4), we have
Again, since , then for , from Lemma 1.7, there exists such that
Using assumptions 1 and 2, since , there exists such that and so .
Now, since , by using (A) we have
by we have
that is,
where , , . Therefore, by using , we have ()
Now, using (2.5), (2.6) and (2.7), we have
Continuing in this way, we obtain a sequence with such that for and
Hence, is a Cauchy sequence. So, there exists a point y in the complete metric space X such that
Now, since is closed, there exists such that and by assumption 3, for all n.
Now, using (A), we have
Now, taking limit as and using , also the facts that , , , we have
that is,
By using , we get
Next, since ,
taking limit as , we obtain
From above, we have , a contradiction. So, . Therefore, and . □
Remark 2.5 In assumptions 2 and 3 of Theorem 2.4, we need only comparability of the elements. Theorem 2.4 with Example 2.2 partially improve [[27], Theorem 3.10].
Corollary 2.6 Let and satisfy
for some κ with and for all comparable elements x, y of X.
Also, assume that the following conditions are satisfied:
-
1.
and is closed.
-
2.
If , then .
-
3.
If is such that , then for all n.
Then there exists p such that .
Proof Let , then it is obvious that . Therefore, the proof is complete from Theorem 2.4. □
Corollary 2.7 Let and satisfy
for some κ with and for all comparable elements x, y of X. Also, assume that the following conditions are satisfied:
-
1.
and is closed.
-
2.
If , then .
-
3.
If is such that , then for all n.
Then .
Remark 2.8 Corollary 2.6 extends Latif and Beg [[2], Theorem 2.6], the result of Kaneko and Sessa [1] in a partially ordered set. Corollary 2.7 also extends the results of [1–3, 24] to a partially ordered set.
3 Common fixed points
In this section, we define set-valued mappings on partially ordered metric spaces, and by using the definitions, we obtain the existence of common fixed points. Let A and B be two non-empty subsets of , the relations between A and B are denoted and defined as follows:
Definition 3.1 Let M be a non-empty subset of X and . A mapping F is said to be set-valued if there exists , and for any , there exists a with such that
for all with .
For set-valued mappings, we just required comparability of the elements, but order does not matter.
Theorem 3.2 Let M be a non-empty closed subset of X and be a set-valued mapping satisfying:
-
1.
There exists in M such that ;
-
2.
If is a sequence in M whose consecutive terms are comparable, then for all n.
Then there exists with .
Proof Let . Then by assumption 1, there exists such that . Now, since F is a set-valued mapping, there is with such that
which gives
and consequently
Continuing in this way, we obtain a sequence whose consecutive terms are comparable and
Take , then we have
Next, we will show that is a Cauchy sequence in X. Let . Then
because , .
Therefore, as , which further implies that is a Cauchy sequence. So, there exists some point (say) x in the complete metric space X such that . By using assumption 2, for all n.
Further, since M is closed, . Now we want to show that .
Since with also for all n and F is a set-valued mapping, there exists with such that
Now
and using (3.2), we have
Letting , we obtain .
As and Fx is closed, so . □
Example 3.3 Let be a subset of with usual order defined as follows: for , if and only if . Let d be a metric on X defined as follows:
for all , so that is a complete metric space. Define as follows:
and .
Consider for , there exists such that .
Next .
Similarly, for other comparable elements of M, one can see that F is a set-valued mapping.
Further is such that . Also, assumption 2 of Theorem 3.2 is satisfied and is the fixed point of F.
Theorem 3.4 Let M be a non-empty closed subset of X and be a sequence of mappings satisfying the following:
(B): For any two mappings , and for any , , there exists a with such that
for all with and for some .
Assume that the following conditions also hold:
-
1.
For each , .
-
2.
If is a sequence in X whose consecutive terms are comparable, then for all n.
Then there exists with .
Proof Let . Then by assumption 1, there exists such that . Now, using (B), there is with such that
which gives
Next, for this , there exists with such that
Using (3.3), we obtain
Continuing in this way, we obtain a sequence whose consecutive terms are comparable and
Take , then we have
Therefore, is a Cauchy sequence in a complete metric space X, so . By using assumption 2, for all n.
Further, since M is closed, . Let be any arbitrary member of . Now, since with . Also, for all n. By using (B), there exists such that
Now
which gives
Letting , we have .
As and is closed, so , i.e., . □
Remark 3.5 Theorems 3.2 and 3.4 improve/extend [[2], Theorems 4.1 and 4.2].
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The present version of the paper owes much to the precise and kind remarks of the learned referees.
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IB gave the idea and ARB wrote the initial draft. Both authors read and agreed upon the draft and finalized the manuscript. Correspondence was mainly done by IB. All authors read and approved the final manuscript.
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Beg, I., Butt, A.R. Common fixed point and coincidence point of generalized contractions in ordered metric spaces. Fixed Point Theory Appl 2012, 229 (2012). https://doi.org/10.1186/1687-1812-2012-229
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DOI: https://doi.org/10.1186/1687-1812-2012-229