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Fixed point theorems in ordered metric spaces via w-distances
Fixed Point Theory and Applications volume 2012, Article number: 222 (2012)
Abstract
The purpose of this paper is to prove some fixed point theorems in a complete metric space equipped with a partial ordering employing generalized distances together with altering distance functions.
MSC:54H25, 47H10.
1 Introduction
The existing literature of metric fixed point theory contains numerous noted generalizations of the Banach contraction mapping principle (e.g., [1] and [2]). One variety of such generalizations is the contractive fixed point theorems contained in Khan et al. [1] wherein the authors utilized altering functions to alter the distance between two points in a metric space. Such altering functions are also sometimes referred to as control functions.
The following altering distance function is instrumental in our forthcoming results.
Definition A (cf. [1])
A map is said to be an altering distance function if
-
(a)
Ï• is continuous and nondecreasing and
-
(b)
if and only if .
Using the function Ï•, Khan et al. [1] proved the following result.
Theorem A (cf. [1])
Let be a self-mapping defined on a complete metric space satisfying the condition
for and , where Ï• is the earlier described altering distance function. Then has a unique fixed point.
In the recent past, the idea of altering function has been utilized by many researchers (e.g., [1–12]). Quite recently, Alber and Guerre-Delabriere [13] initiated the study of weakly contractive mappings which were initially confined to Hilbert spaces. Rhoades [11] utilized this idea in the context of complete metric spaces and proved the following interesting theorem.
Theorem B (cf. [11])
Let be a self-mapping defined on a complete metric space satisfying the condition
for , where Ï• is the earlier described altering distance function. Then has a unique fixed point.
In fact, Alber and Guerre-Delabriere assumed the additional assumption (on Ï•). But Rhoades [11] proved his theorem without this requirement on Ï•.
In [5], Dutta and Choudhury presented a generalization of Theorem B by proving the following result.
Theorem C (cf. [5])
Let be a self-mapping defined on a complete metric space satisfying the condition
where ψ and ϕ are altering distance functions. Then has a unique fixed point.
The purpose of this paper is to prove some fixed point theorems in ordered metric spaces employing a w-distance as well as altering functions. Recall that the concept of w-distance was initiated by Kada, Suzuki, and Takahashi [7] and was primarily utilized to improve Caristi’s fixed point theorem [4], Ekeland’s variational principle [6], and the nonconvex minimization theorems whose descriptions and details are available in Takahashi [14]. The existence of a fixed point on partially ordered metric spaces has been a relatively new development in metric fixed point theory. In [10], Ran and Reurings proved an analogue of Banach’s fixed point theorem in a partially ordered metric space besides discussing some applications to matrix equations. In fact, Ran and Reurings have weakened the usual contraction condition but merely up to monotone operators. Proving new fixed point theorems in an ordered metric space setting to improve earlier stated theorems have been a subject of vigorous research interest; for the literature of this kind one can be referred to [5, 8, 9, 15]. Our results, in this paper, not only generalize the analogous fixed point theorems but are relatively simpler and more natural than the related ones. Our improvements in this paper are indeed four-fold:
-
(i)
a generalized distance is used instead of metric,
-
(ii)
a relatively more general contraction condition is used,
-
(iii)
the continuity of the involved mapping is weakened to orbital continuity, and
-
(iv)
the comparability conditions used by earlier authors are also sharpened.
2 Preliminaries
Before presenting our results, we collect relevant definitions and results which will be needed in the proof of our main results.
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 .
Let be a metric space. Then a 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 .
Clearly, every metric is a w-distance but not conversely. The following example substantiates this fact.
Example 1 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 .
Definition 4 Let be a function. Then
-
(a)
(i.e., is the 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 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 imply that as .
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 ):
-
(i)
if and for , then . In particular, if and , then ,
-
(ii)
if and for , then ,
-
(iii)
if for with , then is a Cauchy sequence,
-
(iv)
if for , then is a Cauchy sequence.
Lemma 2 [7]
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.
3 Main results
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 a nondecreasing mapping. Suppose that
-
(a)
there exists such that ,
-
(b)
there exist two altering distance functions ψ, ϕ such that
for all , where
-
(c)
either is orbitally continuous at or
(c′) is orbitally -continuous, and there exists a subsequence of converging to such that for any .
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
Write and for any .
By using condition (b), we have
so that
for any .
Also,
Therefore, for every (owing to monotonicity of ψ), i.e., the sequence is decreasing so that for the nonnegative decreasing sequence , there exists some such that
Assume that . On letting in (3.2) besides using (3.3), we get
which amounts to say that . As Ï• is an altering (distance) function, , which is a contradiction to nonzeroness of r yielding thereby
which establishes (3.1).
Proceeding with earlier lines, we can also show that
Write and for any .
Now, using (b), we get
so that
which amounts to say that , i.e., a nonnegative sequence is decreasing. As earlier, we have
which proves (3.4). Now, we proceed to show
Suppose (3.5) is untrue. Then we can find a with sequences , such that
wherein . By (3.1) there exists such that implies
Notice that in view of (3.6) and (3.7), . We can assume that is a minimum index such that (3.6) holds so that
which in view of (3.6) gives rise to
so that
Next, we show that
If , then there exists such that
Since ψ is continuous and nondecreasing and also , on using condition (b), one gets
with
implying thereby
Letting in (3.8) and using (3.9), we get
so that implying thereby , which is a contradiction. Hence,
and we have
Therefore, owing to (3.1) and (3.4), we have
which is a contradiction. Hence, (3.5) holds. Owing to Lemma 2, is a Cauchy sequence in . Since is a complete metric space, there exists such that .
Now, we show that is a fixed point of . If (c) holds, then (as ). By lower semi-continuity of , we have

On using (3.5), we have . Now, in view of Lemma 1, we conclude that
Next, suppose that (c′) holds. Since converges to , and is -continuous, it follows that converges to . As earlier, by lower semi-continuity of , we conclude that . This completes the proof. □
Setting (the identity mapping) in Theorem 1, we deduce the following corollary.
Corollary 1 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 an altering distance function Ï• such that
for all , where
-
(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 .
Choosing (the identity mapping) and (for all ) in Theorem 1, 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)
if for all and ,
where
-
(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 .
As an application of Corollary 2, we can also prove the following related result.
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)
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 .
Proof On using condition (b), we can write
where . Therefore, all the conditions of Corollary 2 are satisfied, which ensures the conclusion. □
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)
for all ,
where , and is a Lebesgue integrable mapping which is summable and (for each ),
-
(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 .
Proof Choose and (for all ). Clearly, ψ and ϕ are altering distance functions. Now, in view of Theorem 1, result follows. □
Remark 1 In Theorem 1, let , and (identity) and (). Then Theorem 1 is the classical Banach fixed point theorem.
Lemma 3 Let be a complete partially ordered metric space and be a map wherein p is a w-distance on . If
-
(a)
,
-
(b)
there exist two altering distance functions ψ, ϕ such that
for all , with
then .
Proof Suppose . As and
Therefore,
which amounts to say that . As ϕ is an altering distance function, we infer that . This completes the proof. □
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 [9], it is proved that condition (A) is equivalent to the following one:
(B): for every , there exists for which and .
4 Results with uniqueness
Theorem 3 With the addition of condition (B) to the hypotheses of Theorem 1, 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 distinguish two cases.
Case 1: If , owing to condition (b) (of Theorem 1) and Lemma 3, we have
As
therefore
which amounts to say that . As Ï• is an altering distance function, therefore, for every ,
Also, in view of Lemma 3, we get , and by using Lemma 1, we have , i.e., the fixed point of is unique.
Case 2: If , then owing to condition (B), there exists such that and . As , due to monotonicity of , we get for any , and henceforth
with

therefore
Since ψ is a nondecreasing function, therefore , i.e., the nonnegative sequence is decreasing. As earlier, we have
Also, since , therefore proceeding as earlier, we can prove that
By using this and Lemma 1, we infer that , i.e., the fixed point of is unique.
Now, we proceed to show
for every provided . We distinguish two cases.
Case 1: Let and . As earlier, we have
Also, in view of Lemma 3, we have
and by using Lemma 1, we get
Case 2: Let and . Owing to condition (B), there exists some z in such that and . As earlier, we can prove and . By the triangular inequality,
one gets
Since , due to monotonicity of , we can write . Continuing this process inductively, we obtain
Now, we proceed to show that
Suppose . Since , then for arbitrary ϵ (), there exists such that for every , we have . Also, since , then there exists such that for every , we have . Therefore, for every , we have

Now, on using (b), for every , we get
Therefore, as ψ is an altering distance function, we get the nonnegative sequence is decreasing. As earlier, we can prove , which is indeed a contradiction to nonzeroness of δ, implying thereby
Also, since , therefore using the arguments of the earlier case, we can prove
and by lower semi-continuity , we have

As , thus, in view of Lemma 1, we conclude that
This completes the proof. □
Corollary 4 With the addition of condition (B) to the hypotheses of Corollary 1 (or Corollary 2, Corollary 3) the fixed point of turns out to be unique. Moreover,
for every provided , i.e., the map is a Picard operator.
Corollary 5 With the addition of condition (B) to the hypotheses of Theorem 2, the fixed point of turns out to be unique. Moreover,
for every provided , i.e., the map is a Picard operator.
5 Illustrative examples
In what follows, we furnish two illustrative examples wherein one demonstrates Theorem 1 on the existence of a fixed point, while the other one exhibits the uniqueness of the fixed point in respect of Theorem 3.
Example 2 Consider equipped with the usual metric for all , and to be a w-distance on . Define an order relation ⪯ on as
where ≤ is usual order. Then it is clear that
Let be given by
Obviously, is a complete partially ordered metric space. It is easy to see that is nondecreasing. Also, there is in such that , i.e., , and satisfies (c′).
We now show that satisfies (b) with which are defined as
If and , then . Otherwise, if with , then either , or , (), which evolve into two cases as follows.
Case 1. If () and , then
and
Case 2. Next, if and (), then
For , we have
which is equivalent to
or
The preceding inequality holds as
so that
Also (with ),
or
or
or
or
Therefore,
Also, we can write
which amounts to say that the inequality (I) holds and so does the inequality (b) (of Theorem 1).
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 0 and . Here, it is worth pointing out that condition (B) does not hold in respect of this example.
We give another example that illustrates Theorem 3.
Example 3 Let , where is a complete partially ordered metric space with a metric d and usual order ≤. Clearly, condition (B) holds in .
We define by . Let and . Assume that by for any . Obviously, ϕ and ψ are altering distance functions, it is easy to see that is nondecreasing and self-map. Also, there is in such that , and satisfies (c′) (of Theorem 3). Now, we show that satisfies (b) (of Theorem 3). If , clearly, condition (b) is satisfied. Now, suppose that , then we have
By making use of condition (b), one gets
so that
or
The preceding inequality holds and so does the inequality (b) (of Theorem 3).
Thus, all the conditions of Theorem 3 are satisfied. We note that is a unique fixed point for . Moreover .
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Both authors are grateful to two anonymous referees for their fruitful suggestions and observations.
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Imdad, M., Rouzkard, F. Fixed point theorems in ordered metric spaces via w-distances. Fixed Point Theory Appl 2012, 222 (2012). https://doi.org/10.1186/1687-1812-2012-222
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DOI: https://doi.org/10.1186/1687-1812-2012-222