Abstract
The purpose of this paper is to prove some fixed point theorems in a complete metric space equipped with a partial ordering using w-distances together with the aid of altering functions.
MSC:54H25, 47H10.
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1 Introduction with preliminaries
The concept of a w-distance was initiated by Kada, Suzuki and Takahashi [1] and was primarily utilized to improve Caristi’s fixed point theorem [2], Ekeland’s variational principle [3] and nonconvex minimization theorems whose descriptions and details are available in Takahashi [4]. Proving existence results of fixed points on partially ordered metric spaces has been a relatively new development in metric fixed points theory. In [5], an analogue of Banach’s fixed point theorem in a partially ordered metric space has been proved besides discussing some applications to matrix equations. Ran and Reurings have further weakened the usual contraction condition but merely up to monotone operators.
Branciari [6] established a fixed point result for an integral-type inequality, which is a generalization of the Banach contraction principle. Vijayaraju et al. [7] obtained a general principle, which made it possible to prove many fixed point theorems for pairs of maps satisfying integral-type contraction conditions.
Several fixed point and common fixed point theorems in metric and semi-metric spaces for compatible, weakly compatible and owc mappings satisfying contractive conditions of integral type were proved in [7–9] and in other papers. Later on, Suzuki [10] proved that integral-type contractions are Meir-Keeler contractions. He also showed that Meir-Keeler contractions of integral type are still Meir-Keeler contractions. Jachymski [11] also proved that most contractive conditions of integral type given recently by many authors coincide with classical ones. But he gave a new contractive condition of integral type which is independent of classical ones. Recently Popa and Mocanu [9] obtained integral-type contractions via an altering distance function and proved general common fixed point results for integral-type contractive conditions.
In [12], Razani et al. proved a fixed point theorem for -contractive mappings on [i.e., for each , ], which is a new version of the main theorem in [6], by considering the concept of a w-distance. In fact, he proved the following result.
Theorem [12]
Let p be a w-distance on a complete metric space , ϕ be nondecreasing, continuous and for each and ψ be nondecreasing, right continuous and for all . Suppose is a -contractive map on , then has a unique fixed point in . Moreover, is a fixed point of for each .
In [13] Lakzian and Lin obtained some generalizations of fixed point theorems by Kada et al. [1], Hicks and Rhoades [14] and several other results with respect to -contractive maps on a complete metric space.
In this paper, we use the concept of a w-distance to prove the fixed point theorems in partially ordered metric spaces. Our results do not only generalize some fixed point theorems, but also improve and simplify the previous results.
Before presenting our results, we collect relevant definitions and results which will be needed in our subsequent discussion.
Definition 1 Let be a nonempty set. Then is called a partially ordered metric space if
-
(i)
is a partially ordered set and
-
(ii)
is a metric space.
Definition 2 Let be a partially ordered set. Then
-
(a)
elements are called comparable with respect to ‘⪯’ if either or ;
-
(b)
a mapping is called nondecreasing with respect to ‘⪯’ if implies .
Definition 3 [1]
Let be a metric space. Then the function is called a w-distance on if the following conditions are satisfied:
-
(a)
for any ,
-
(b)
for any , is lower semi-continuous, i.e., if and in , then ,
-
(c)
for any , there exists such that and imply .
Example 1 [15]
Let be a metric space and let g be a continuous mapping from into itself. Then a function defined by
is a w-distance on .
Clearly, every metric is a w-distance but not conversely. The following example substantiates this fact.
Example 2 Let be a metric space. A function defined by for every is a w-distance on , where k is a positive real number. But p is not a metric since for any .
Example 3 Let be a metric space with a metric
Let be defined by
and for every .
Clearly, conditions (b) and (c) from the definition of w-distance are satisfied (for every , put ), while condition (a) is not satisfied as
Definition 4 Let be a function.
-
(a)
(i.e., a set of fixed points of ).
-
(b)
The function is called a Picard operator (briefly, PO) if there exists such that and converges to for all .
-
(c)
The function is called orbitally -continuous for any if the following condition is satisfied:
For any , as and for any imply that as .
Let be a partially ordered set. Let us denote by the subset of defined by
Definition 5 A map is said to be orbitally continuous if and as implies that as .
Suppose
Moreover, let
Also, let
Example 4 Let and be two non-negative sequences such that is strictly decreasing and converging to zero, and (for each ) , where . Define by , , if , if , then ψ is in Ψ.
Now, we prove the following two lemmas.
Lemma 1 If , then for each .
Proof Owing to the monotonicity of ψ, for each , is non-increasing and also non-negative. Thus, there exists such that . Suppose on the contrary that . As ψ is right continuous, therefore
which is a contradiction as . Thus . □
Lemma 2 If , and (), then .
Proof If there exists and such that
then
yielding thereby . □
The following two lemmas are crucial in the proofs of our main results.
Let be a metric space equipped with a w-distance p. Let and be sequences in whereas and be sequences in converging to zero. Then the following conditions hold (for ):
-
(a)
if and for , then . In particular, if and , then ;
-
(b)
if and for , then ;
-
(c)
if for with , then is a Cauchy sequence;
-
(d)
if for , then is a Cauchy sequence.
Lemma 4 [1]
Let p be a w-distance on a metric space and be a sequence in such that for each , there exists such that implies (or ). Then is a Cauchy sequence.
2 -contractive maps
Now, we present our main result as follows.
Theorem 1 Let be a complete partially ordered metric space equipped with a w-distance p and be nondecreasing mapping. Suppose that
-
(a)
there exists such that ,
-
(b)
there exist and such that
for all ,
-
(c)
either is orbitally continuous at or
(c′) is orbitally -continuous and there exists a subsequence of converging to such that for any .
Then . Moreover, if , then .
Proof If for some , then there is nothing to prove. Otherwise, let there be such that , and . Owing to the monotonicity of , we can write . Continuing this process inductively, we obtain
for any . Now, we proceed to show that
By using condition (b) and the properties of ϕ, ψ, we get
Now, using Lemma 1, , which due to Lemma 2 gives rise to
so that (1) is established.
Similarly, we can show
Next, we proceed to show
Suppose (4) is untrue. Then we can find a with sequences , such that
wherein . By (1) there exists such that implies
Notice that in view of (5) and (6), . We can assume that is a minimum index such that (5) holds so that
Now (1), (5) and (7) imply
so that
If , then there exists such that
Since ϕ is continuous and nondecreasing and also , by using condition (b) and (8), one gets
Notice that
and ψ is right continuous, therefore . This is indeed a contradiction and
so we have
which is a contradiction. Thus, (4) is proved.
Owing to Lemma 4, is a Cauchy sequence in . Since is a complete metric space, there exists such that as .
Now, we show that is a fixed point of . If (c) holds, then (as ). By the lower semi-continuity of , we have
By using (4), we have as . Now, in view of Lemma 3, we conclude that
Next, suppose that (c′) holds. Since converges to , and is -continuous, it follows that converges to . As earlier, by the lower semi-continuity of , we conclude that .
If , we have
This is a contradiction which amounts to say that so that . This completes the proof. □
The following example substantiates the fact that the condition of a partial ordering on the underlying metric space is necessary in Theorem 1.
Example 5 Consider which is indeed a complete metric space under a usual metric (for all ) wherein by defining , we are in the receipt of a w-distance p on . We consider as follows:
where ‘⪯’ is the usual ordering.
Let be given by
Obviously, is a nondecreasing map. Also, there is in such that , i.e., , and satisfies condition (c′). We now show that satisfies (b) wherein are defined as
Clearly, and . If , condition (b) is satisfied.
Let and , then
as for any , .
Next, let and with . Then we have
as for any , . Hence condition (b) is satisfied.
Thus, all the conditions of Theorem 1 are satisfied implying thereby the existence of a fixed point of the map , which are indeed two in number, namely: .
Here, it can be pointed out that this example will not work in a metric space equipped with a w-distance without a partial ordering as condition (b) of Theorem 1 will not be satisfied. To substantiate this claim, choosing and in condition (b), we get
which is a contradiction.
In Theorem 1, if is a continuous map, we deduce the following corollary.
Corollary 1 Let be a complete partially ordered metric space equipped with a w-distance p and be a continuous and nondecreasing mapping. Suppose that
-
(a)
there exists such that ,
-
(b)
there exist and such that
for all .
Then . Moreover, if , then .
In Theorem 1, setting , the identity mapping, we deduce the following corollary.
Corollary 2 Let be a complete partially ordered metric space equipped with a w-distance p and be a nondecreasing mapping. Suppose that
-
(a)
there exists such that ,
-
(b)
there exists such that
for all ,
-
(c)
either is orbitally continuous at or
(c′) is orbitally -continuous and there exists a subsequence of which converges to such that for any .
Then . Moreover, if , then .
Choosing , the identity mapping and (for all and ) in Theorem 1, we deduce the following corollary.
Corollary 3 Let be a complete partially ordered metric space equipped with a w-distance p and be a nondecreasing mapping. Suppose that
-
(a)
there exists such that ,
-
(b)
if
for all , where and
-
(c)
either is orbitally continuous at or
(c′) is orbitally -continuous and there exists a subsequence of which converges to such that for any .
Then . Moreover, if , then .
Suppose, is a Lebesgue-integrable mapping which is summable and for each . Now, in the next corollary, set and , where in Theorem 3. Clearly, and . Hence, we can derive the following corollary as a special case.
Corollary 4 Let be a complete partially ordered metric space equipped with a w-distance p and be a nondecreasing mapping. Suppose that
-
(a)
there exists such that ,
-
(b)
for all ,
-
(c)
either is orbitally continuous at or
(c′) is orbitally -continuous and there exists a subsequence of which converges to such that for any .
Then . Moreover, if , then .
The following simple example demonstrates Theorem 1.
Example 6 Consider , which is a complete metric space with . Define by and , which is a continuous map. Moreover, by defining
p is a w-distance on . For and in , define ‘⪯’ as follows: and (‘≤’ is the usual ordering) so that
Let be given by
Clearly, is a nondecreasing and continuous map. Also, , i.e., .
We now show that satisfies (b) with which are defined as
Clearly, and . Let in
and for , we have
and
Therefore, for every , we have
Thus, all the conditions of Theorem 1 are satisfied implying thereby the existence of a fixed point of which is indeed .
Theorem 2 Let be a complete partially ordered metric space equipped with a w-distance p and be a nondecreasing mapping. Suppose that
-
(a)
there exists such that ,
-
(b)
there exist and such that
for all , and
(c″) for every with ,
Then . Moreover, if , then .
Proof Observe that the sequence is a Cauchy sequence (in view of the proof of Theorem 1), and so there exists a point in such that . Since , therefore for each , there exists such that implies . Since and is lower semi-continuous, therefore
Therefore, . Set , so that
Now, assume that . Then due to hypothesis (c″), we have
as . This is a contradiction. Hence .
If , we have
which is a contradiction implying thereby so that . This completes the proof. □
Corollary 5 Replacing condition (c″) by hypothesis (c) or (c′) of Corollary 2 (also of Corollary 3 or Corollary 4) the fixed point of continues to exist.
In what follows, we give a sufficient condition for the uniqueness of a fixed point in Theorem 1 which runs as follows:
-
(A)
for every , there exists a lower bound or an upper bound.
In [18], it is proved that condition (A) is equivalent to the following one:
-
(B)
for every , there exists for which and .
Theorem 3 With the addition of condition (B) to the hypotheses of Theorem 1 (or Theorem 2), the fixed point of turns out to be unique. Moreover,
for every provided , i.e., the map is a Picard operator.
Proof Following the proof of Theorem 1, . Suppose there exist two fixed points and of in . We prove that
We distinguish two cases.
Case 1: If . Suppose that , then by using condition (b) and the property of ψ, we get
which is a contradiction. Therefore we have (9).
Also, in view of Theorem 1, we have
On using (9), (10) and Lemma 3, we have , i.e., the fixed point of is unique.
Case 2: If , then owing to condition (B), there exists such that and . As and , proceeding along the lines of the proof of Theorem 1, we can prove
By using Lemma 3, we infer that , i.e., the fixed point of is unique. Now, we prove
for every provided .
Let and . Proceeding along the lines of the proof of Theorem 1, we can prove , and owing to and p is a w-distance (lower semi-continuous), then , by Lemma 3, we get
Suppose and . Owing to condition (B), there exists some z in such that and .
Since and , by using condition (b) (proceeding along the lines of the proof of Theorem 1), we can prove and .
By the triangular inequality, we can write
Letting , we get , and also p is a w-distance (lower semi-continuous), we have , which due to Lemma 3 implies
This completes the proof. □
The following example demonstrates Theorem 3.
Example 7 Let , which is a complete partially ordered metric space with the usual metric d and the usual order ‘⪯’. Clearly, condition (B) holds in . We define by . Let ϕ and ψ be the mappings with defined by
and . Obviously, and . Assume that by for any . It is easy to see that is nondecreasing. Also, there is in such that , and satisfies (c′). Also, for any , we have . So, for arbitrary , we have
Now, we show that satisfies (b).
Thus, all the conditions of Theorem 3 are satisfied and is the unique fixed point for . Moreover, .
Corollary 6 With the addition of condition (B) to the hypotheses of Corollary 1 (or Corollaries 2, 3, 4, 5) the fixed point of turns out to be unique. Moreover,
for every provided , i.e., the map is a Picard operator.
Our next example highlights the role of condition (c″) in Theorem 2.
Example 8 Consider , which is a complete metric space with the usual metric for all . Moreover, by defining , p is a w-distance on . We consider as follows:
where ⪯ is the usual ordering.
Let be given by
Obviously, is a nondecreasing map. Also, there is in such that .
We now show that satisfies (b) with which are defined as
Clearly, and . If , condition (b) is satisfied.
Now, let and (). Then we have
so that condition (b) is satisfied.
Since , we have
Thus, all the conditions of Theorem 2 are satisfied except (c″).
Clearly, has got no fixed point in .
3 -contractive maps
In this section we state some results in a partial ordered metric space with -contractive maps. In Section 2 we considered the condition nondecreasing for the function , but in this section we will prove some theorems by replacing the condition nondecreasing to monotonicity for the function.
Theorem 4 Let be a complete partially ordered metric space equipped with a w-distance p and be a monotone mapping. Suppose that
-
(a)
there exists such that ,
-
(b)
there exist and such that
for all ,
-
(c)
either is orbitally continuous at or
(c′) is orbitally -continuous and there exists a subsequence of converging to such that for any .
Then . Moreover, if , then .
Proof If for some , then there is nothing to prove. Otherwise, let there be such that , and . Owing to monotonicity of , we can write . Continuing this process inductively, we obtain
for any . Now, we proceed to show that
On using condition (b) and the properties of γ, ψ, we get
By using Lemma 1, we have so that by Lemma 2, we have
which establishes (11).
Similarly, we can show
Now, we proceed to show that is a Cauchy sequence. By the triangle inequality, the continuity of γ and (11), we have
as so that , which amounts to say that
By induction, for any , we have
So, by Lemma 3, is a Cauchy sequence and due to the completeness of , there exists such that .
If (c) or (c′) holds, then proceeding along the lines of the proof of Theorem 1, we can show that
If , we have
which is a contradiction so that , implying thereby . This completes the proof. □
Theorem 5 Let be a complete partially ordered metric space equipped with a w-distance p and be a monotone mapping. Suppose that
-
(a)
there exists such that ,
-
(b)
there exist and such that
for all , and
(c″) for every with ,
Then . Moreover, if , then .
Proof Proceeding along the lines of the proof of Theorem 4, the sequence is a Cauchy sequence, and so there exists a point in such that . Since , therefore for each , there exists such that implies . As and is lower semi-continuous, therefore
Therefore . Setting , , we have
Now, assume that . Then by hypothesis (c″), we have
as . This is a contradiction so that .
If , we have
which is a contradiction so that , yielding thereby . This completes the proof. □
Our next example demonstrates Theorem 5 which exhibits the utility of this theorem over Theorems 1 and 2.
Example 9 Consider , which is a complete metric space with the usual metric for all . Moreover, by defining , p is a w-distance on . We consider as follows:
where ⪯ is the usual ordering.
Let be given by
Obviously, is a monotone map. Also, there is in such that , i.e., .
We now show that satisfies (b) with which are respectively defined as and
Clearly, and . If as , it is easy to show that condition (b) is satisfied.
Suppose as , one gets
as for any , we have . Hence condition (b) is satisfied.
If , we have so that
Thus, all the conditions of Theorem 5 are satisfied and is a fixed point of .
But if is chosen, then and so that . Hence, condition (b) of Theorems 1 and 2 does not hold.
Theorem 6 Let be a complete partially ordered metric space equipped with a w-distance p and be a monotone mapping. Suppose that
-
(a)
there exists such that ,
(b1) there exist and such that
for all ,
(c″) for every with ,
Then . Moreover, if , then .
Proof For , set and . Then we have and . On using condition (b1), we get
or
Therefore, by choosing , all the conditions of Theorem 5 are satisfied ensuring the conclusions of the theorem. □
The set of all subadditive functions γ in Γ is denoted by .
Theorem 7 Let be a complete partially ordered metric space equipped with a w-distance p and be a monotone mapping. Suppose that
-
(a)
there exists such that ,
(b2) there exist and such that
for all , and
for every with . Then . Moreover, if , then .
Proof Set , then . On using condition (b2) (as ), we have
Thus, .
Therefore, by choosing , all the conditions of Theorem 5 are satisfied ensuring the conclusions of the theorem. □
Our final example demonstrates Theorem 4.
Example 10 Consider , which is a complete metric space with . Define by and , and , which is a continuous map. Moreover, by defining
p is a w-distance on . For and in , define ‘⪯’ as follows: and and , where ‘≤’ is the usual ordering and
Define as
Clearly, is a monotone and continuous map. Also, , i.e., .
We now show that satisfies (b) with which are defined as
Notice that and . If in , then
and for , we have
Therefore, for every , we have
Thus, all the conditions of Theorem 4 are satisfied implying thereby the existence of fixed points of the map which are indeed and .
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Acknowledgements
The authors are thankful to the learned referees for their deep observations and pertinent suggestions, which greatly helped us to improve the paper significantly. The third author also thanks for the support of CSIR, Govt. of India, Grant No - 25(2882)/NS/12/EMR-II.
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All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
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Rouzkard, F., Imdad, M. & Gopal, D. Some existence and uniqueness theorems on ordered metric spaces via generalized distances. Fixed Point Theory Appl 2013, 45 (2013). https://doi.org/10.1186/1687-1812-2013-45
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DOI: https://doi.org/10.1186/1687-1812-2013-45