1. Introduction and Preliminaries
Banach contraction principle is a pivotal result of metric-fixed point theory. In subsequent years, this classical result has been generalized and improved in numerous ways and by now there exists extensive literature on this theme. In 1997, Alber and Guerre-Delabriere [
1] introduced the notion of weak contraction and utilized the same to prove the existence and uniqueness of a fixed point of a self-mapping, satisfying a weak contraction condition on Hilbert spaces. In 2001, Rhoades [
2] showed that this result remains true for complete metric spaces too. In recent years, the idea of weak contraction has been exploited by several researchers (e.g., [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13]).
On the other hand, in 2004, Ran and Reurings [
14] proved an order-theoretic analogue of Banach contraction principle which marks the beginning of a vigorous research activity. This noted-paper of Ran and Reurings is well followed by two very useful articles from Nieto and Rodríguez-López [
15,
16]. Presently, proving an order-theoretic analogue of metric-fixed point results is an area of active research and by now there exists considerable literature on this topic (e.g., [
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27]). Our work in this paper is on similar lines wherein our results are proved using
-generalized weakly contractive mappings.
To present our main results, the following definitions, basic results and relevant historical overviews are needed.
We denote by the set of natural numbers including zero, i.e., . As usual, stands for the identity mapping defined on X. For brevity, we write instead of .
Definition 1. [28] A function is called an altering distance function if it is continuous, increasing and satisfies if and only if . We denote the set of all altering distance functions by Ψ
. Definition 2. [7] A self-mapping f on a metric space is said to be -weakly contractive mapping if for all where . Remark 1. In Definition 2, if we set , then f is known as φ-weakly contractive mapping (see [1]). Definition 3. [29] A self-mapping f on a metric space is said to be -generalized weakly contractive mapping if for all where and φ: is a continuous function with if and only if Definition 4. [27] A triple is called an ordered metric space if is a metric space and is an ordered set. Moreover, two elements are said to be comparable if either or . For brevity, we denote it by . Remark 2. With a view to emphasize the order-theoretic analogue of Definition 2 (resp. Definition 3), it can be pointed out that the inequality (1) (resp. (2)) is required to hold merely for comparable elements, i.e., for all such that (rather than for every pair of elements in X). Definition 5. [21] Let be a pair of self-mappings on an ordered set . Then the mapping- (i)
f is said to be g-increasing if , for all ,
- (ii)
f is said to be g-decreasing if , for all ,
- (iii)
f is said to be g-monotone if f is either g-increasing or g-decreasing.
Definition 6. [30] Let be a pair of self-mappings on a metric space and . We say that f is g-continuous at x if , for any sequence . Moreover, f is called g-continuous if it is g-continuous at every point of X. Let be a sequence in an ordered metric space . If is an increasing (resp. decreasing, monotone) and converges to x, we denote it by (resp. , ).
Definition 7. [20] Let be a pair of self-mappings on an ordered metric space and . Then f is called -continuous (resp. -continuous, -continuous) at if for every sequence with (resp. , ). Moreover, f is called -continuous (resp. -continuous, -continuous) if it is -continuous (resp. -continuous, -continuous) at every point of X. On setting , Definition 7 reduces to the usual definition of -continuity (resp. -continuity, -continuity) of self-mapping f on X.
Remark 3. In an ordered metric space, g-continuity ⇒ -continuity ⇒ -continuity (as well as -continuity).
Definition 8. Let be a pair of self-mappings on an ordered metric space . Then the pair is said to be[31] compatible if whenever is a sequence in X such that [20] -compatible (resp. -compatible, O
-compatible) if whenever is a sequence in X such that and are increasing (resp. decreasing, monotone) sequences with . [32] weakly compatible if , for every coincidence point of f and g.
Remark 4. In an ordered metric space, compatibility ⇒ O-compatibility ⇒ -compatibility (as well as -compatibility) ⇒ weak compatibility.
Definition 9. [20] An ordered metric space is called -complete (resp. -complete, O
-complete ) if every increasing (resp. decreasing, monotone) Cauchy sequence in X converges to a point of X. Remark 5. In an ordered metric space, completeness ⇒ O-completeness -completeness (as well as -completeness).
Definition 10. [20] Let be a pair of self-mappings on an ordered metric space . Then- (i)
is said to have g-ICU-property (Increasing-Convergence-Upper-Bound) if g-image of every increasing convergent sequence in X is bounded above by g-image of its limit, i.e., - (ii)
is said to have g-DCL-property (Decreasing-convergence-Lower-Bound) if g-image of every decreasing convergent sequence in X is bounded below by g-image of its limit, i.e., - (iii)
is said to have g-MCB-property (Monotone-Convergence-Boundedness) if it has both g-ICU as well as g-DCL-property.
On setting , Definition 10(i) (resp. 10(ii), 10(iii)) reduces to the definition of the ICU-property (resp. DCL-property, MCB-property).
Definition 11. [24] Let D be a subset of an ordered set and g a self-mapping on X. We say that D is g-directed if for every pair of elements there is such that and . Notice that, on setting
in Definition 11,
D is said to be directed due to [
24].
The following three lemmas are needed to prove our results:
Lemma 1. [33] Let be a pair of self-mappings defined on an ordered set . If f is g-monotone and , then . Lemma 2. [33] Let be a pair of weakly compatible self-mappings defined on non-empty set X. Then every point of coincidence of the pair is also a coincidence point. Proof. Let
x be a point of coincidence of
f and
g such that
for some
. On using the weak compatibility of
f and
g, we have
which implies that
is a coincidence point of
f and
g. ☐
The following lemma was proved as a part of the proof of Theorem 2.1 of [
23].
Lemma 3. [23] Let be an ordered metric space and a sequence in X such that If is not a Cauchy sequence, then there exist and two subsequences and of such that- (i)
- (ii)
- (iii)
- (iv)
the sequences tend to ϵ when
Alber and Guerre-Delabriere [
1] proved that every
-weakly contractive mapping defined on a Hilbert space possesses a unique fixed point. Thereafter, Rhoads [
2] proved that this result is also true for complete metric spaces.
Theorem 1. [2] (Theorem 1) Let be a complete metric space. If the mapping is a φ-weakly contractive mapping, then f has a unique fixed point. It is worth noting that, Alber and Guerre-Delabriere [
1] assumed that the altering distance function
satisfies an extra condition (which is
), but Rhoades [
2] obtained the above result without using this condition.
Thereafter, Dutta and Choudhury [
7] proved a generalization of Theorem 1 as follows:
Theorem 2. [7] (Theorem 2.1) Let be a complete metric space and a -weakly contractive mapping. Then f has a unique fixed point. Choudhury et al. [
29] proved a generalization of the above two theorems as follows:
Theorem 3. [29] (Theorem 3.1) Let be a complete metric space and a -generalized weakly contractive mapping on X. Then f has a unique fixed point. On the other hand, in the setting of ordered metric spaces, Harjani and Sadarangani [
22] proved an order-theoretic analogue of Theorem 1 as follows:
Theorem 4. [22] (Theorems 2 and 3) Let be a complete ordered metric space and f an increasing self-mapping on X. Suppose that the following conditions hold:- (i)
f is a φ-weakly contractive mapping with ,
- (ii)
either f is a continuous mapping or enjoys ICU-property.
Then f has a fixed point provided there exists such that .
Subsequently, Harjani and Sadarangani [
23] proved the following result which is an order-theoretic analogue of Theorem 2 as well as a generalization of Theorem 4.
Theorem 5. [23] (Theorems 2.1 and 2.2) Let be a complete ordered metric space and f an increasing self-mapping on X. Suppose that the following conditions hold:- (i)
f is a -weakly contractive mapping,
- (ii)
either f is a continuous mapping or enjoys ICU-property.
Then f has a fixed point provided there exists such that .
Here, it can be pointed out that Harjani and Sadarangani [
22,
23] proposed the following sufficient condition for the uniqueness of the fixed point in Theorems 4 and 5:
The aim of this article is to prove an order-theoretic analogue of Theorem 3 so as to improve and generalize Theorems 4 and 5. The improvement realized in our results is three-fold which we describe as under:
- (a)
relatively weaker notions of the continuity and completeness are employed,
- (b)
the -weak contractive condition is replaced by a -generalized weak contractive condition (defined later) involving a pair of self mappings,
- (c)
a weaker uniqueness condition is utilized.
We demonstrate the genuineness of our results by a suitable example. As an application, we prove a result for mappings satisfying integral type -generalized weak contractive condition.
2. Results on Coincidence Point
In the sequel, we use the following definition:
Definition 12. Let be a pair of self-mappings on an ordered metric space . Then f is said to be a -generalized weakly contractive mapping if for all such that we havewhere , and is a lower-semi continuous function with if and only if Observe that, on setting , Definition 12 remains relatively weaker than the order-theoretic analogue of Definition 3 as the class of lower-semi continuous functions is larger than the class of continuous functions.
Now, we prove our main result as follows:
Theorem 6. Let be an ordered metric space and Y an -complete subspace of X. Let be a pair of self-mappings on X such that the mapping f is g-increasing. Suppose the following conditions hold:- (i)
f is a -generalized weakly contractive mapping,
- (ii)
- (a)
and
- (b)
either f is -continuous or f and g are continuous or has ICU-property.
Then the pair has a coincidence point provided there exists such that .
Proof. Choose
such that
. As the mapping
f is
g-increasing and
, we can define increasing mapping sequences
and
in
X such that for all
Observe that, and are in Y. Moreover, if for some , then is the required coincidence point and we are done. Henceforth, we assume that for all .
We assert that
On setting
in (
4), we get
for all
, where
By the triangular inequality,
. If possible, assume
, then
so that (
6) reduces to
a contradiction. Thus,
and (
6) becomes
As
is an increasing function,
is a decreasing sequence of positive real numbers so that
On taking the limit superior as
in inequality (
6), we obtain
which implies that
, a contradiction. Therefore,
i.e.,
Now, we assert that
is a Cauchy sequence in
Y. For if it is not Cauchy, owing to Lemma 3, there exist
and two subsequences
and
of
such that
and
Since
, on putting
and
in (
4), we have (for all
)
where
Taking limit superior as
in (
7), we have
a contradiction. Thus,
is a Cauchy sequence in
Y. Therefore, there exists some
such that
Due to the condition (ii)a, there exists some
such that
, so that
Now, using the condition (ii)b, we show that
z is a coincidence point of the pair
. Firstly, assume that
f is
-continuous. In view of (
9), we have
which (in view of (
5)) by the uniqueness of the limit implies
.
Secondly, let
f and
g be continuous mappings. Then, the proof can be outlined on the lines of the proof of Theorem 1 in [
20].
Lastly, assume that
enjoys
-property. Then,
and on setting
in (
4), we have (for all
)
where
On using (
5), (
9) and taking limit superior in (
10) as
, we have
a contradiction unless
. This concludes the proof. ☐
Theorem 7. Theorem 6 remains true if assumptions embodied in the condition (ii) are replaced by the following (besides retaining the rest of the hypotheses).- (ĩi)
- (a)
,
- (b)
g is -continuous,
- (c)
is -compatible pair and
- (d)
either f is -continuous or has g-ICU-property.
Proof. The proof runs on the lines of the proof of Theorem 6 except wherever we used conditions in (ii), which can be altered as follows: Owing to (
5) and (
8), we have
where
. In view of the condition (ĩi)b, we have
Also, in view of the condition (ĩi)c, we have
so that,
Now, on using the condition (ĩi)d, we show that
x is a coincidence point of
f and
g. Let
f be
-continuous. Then, from (
11), we have
Combining last two equations, we get and hence we are done.
Alternately, let
enjoy
g-ICU-property. By (
11), we have
for all
. On putting
in (
4), we get
for all
, where,
On taking the limit of (
12) as
, we arrive at a contradiction unless
. This concludes the proof. ☐
Remark 6. Observe that the condition (ĩi)a utilized in Theorem 7 is relatively weaker than the condition (ii)a of Theorem 6.
On setting in Theorems 6 and 7, we deduce the following:
Corollary 1. Let be an ordered metric space, Y an -complete subspace of X and f an increasing self-mapping on X such that . Suppose the following conditions hold:- (i)
f is a -generalized weakly contractive mapping,
- (ii)
either f is -continuous or has ICU-property.
Then, f has a fixed point provided there exists such that .
Remark 7. - (a)
If , then Corollary 1 reduces to a sharpened version of Theorem 5, as the increasing condition on the altering distance function φ is found unnecessary and a weaker notion of the continuity of φ is utilized.
- (b)
If and in Corollary 1, we get Theorem 4 without the assumption .
- (c)
The completeness in Theorems 4 and 5 is merely required on any subspace rather than the whole space X such that this subspace contains . Further, these results can be obtained utilizing a relatively weaker notion of the continuity and completeness.
Example 1. Consider endowed with the usual metric d. Then, is an -complete ordered metric space wherein the partial order ‘⪯’
is defined by: iff for and . Define by and . Consider f and g two self-mappings on X defined by: and . Then, the left hand side of the inequality (4) is To compute the right hand side of the inequality, we haveand Thus, the right hand side of (4) is By a routine calculation, we can see that inequality (4) is satisfied, that is, f is a -generalized weakly contractive mapping and the pair has a coincidence point (namely ) supporting Theorems 6 and 7. On setting in Example 1, we create a situation wherein neither Theorem 4 nor Theorem 5 can be used, as the whole space is not complete while our Corollary 1 works. This substantiates the genuineness of our results proved in this paper.
Definition 13. Let be a pair of self-mappings on an ordered metric space . Then, f is said to be a lean -generalized weakly contractive mapping if for all such that we havewhere , and is a continuous function with if and only if As , Definition 12 is weaker than Definition 13.
Corollary 2. Theorem 6 remains true if the condition (i) is replaced by the following condition (besides retaining the rest of the hypothesis).- (ï)
f is a lean -generalized weakly contractive mapping.
Corollary 3. Theorem 7 remains true if the condition (i) is replaced by the condition (ï) (besides retaining the rest of the hypothesis).
3. Results on Common Fixed Points
Theorem 8. In addition to the hypotheses of Corollary 2, if is g-directed, then the pair has a unique point of coincidence.
Proof. Let
be such that
We assert that . By the hypothesis, there exists such that is comparable to both and . For , we may assume (other case is similar).
Set
Since
and
f is a
g-increasing mapping, one can define a sequence
such that
To establish the assertion, we distinguish two cases:
Firstly, if for some . Then by Lemma 1, , that is, On using induction on m, for all establishing the assertion in this case.
Secondly, if
for all
, then on setting
and
in (
13), we get
for all
, where
Obviously,
. Assume that
. Then
Therefore, from (
15), we have
As
is increasing, we have
a contradiction to our assumption. Hence,
so that
and (
15) reduces to
Now,
is a decreasing sequence of strictly positive real numbers which must posses a limit
. Letting
in (
15), we get
which is a contradiction unless
. Thus, in all, our assertion is established.
Similarly, when
, one can show that
On using triangular inequality, (
14) and (
16), we have
which shows that the pair
has a unique point of coincidence. ☐
Theorem 9. In addition to the hypotheses of Theorem 8, if the pair is weakly compatible, then the pair has a unique common fixed point.
Proof. Let be an arbitrary coincidence point of the pair . Due to Theorem 8, there exists a unique point of coincidence (say) such that . By Lemma 2, w itself is a coincidence point, i.e., . Now, again, Theorem 8 ensures that i.e., w is a unique common fixed point of f and g. ☐
Theorem 10. In addition to the hypotheses of Corollary 3, if is g-directed, then the pair has a unique common fixed point.
Proof. On the lines of the proof of Theorem 8, one can show that the pair has a unique point of coincidence. In view of the hypothesis (condition 1c of Theorem 7), is an -compatible pair and hence is a weakly compatible pair (by Remark 4). Now, the proof can be completed on the lines of the proof of Theorem 9. ☐
Remark 8. On setting , the uniqueness condition utilized in Theorem 8 (also in Theorem 10) remains slightly weaker than the condition (3). Remark 9. One can obtain dual type results corresponding to all results in Section 2 and Section 3 by replacing “-analogues” with “-analogues” and “ICU-property” with “DCL-property” provided the existence of such that is replaced by the existence of such that . Remark 10. One can obtain companied type results corresponding to all results in Section 2 and Section 3 by replacing “-analogues” with “-analogues” and “ICU-property” with “MCU-property” provided the existence of such that is replaced by the existence of such that . Remark 11. By using Zermelo’s well-ordering Theorem, the set X can be well ordered and the contraction conditions in all above results of Section 2 and Section 3 are valid for each . Therefore, each of Theorems 9 and 10 covers Theorems 1, 2, 3 and Theorem 2.1 of [4]. As an application of Theorem 6 (resp. Theorem 7), we have the following result on coincidence point for mappings satisfying integral type -weakly contraction in ordered metric space.
Let Λ be the set of functions
satisfying the following:
- (a)
is a Lebesgue-integrable mapping on each compact subset of
- (b)
for all
Theorem 11. Let be an ordered metric space and Y an -complete subspace of X. Let be a pair of self-mappings on X such that f is g-increasing. Suppose that for every with and , we havewhere ψ and φ are as in Definition 12. If there exists such that and the condition (ii) of Theorem 6 (resp. condition (ĩi) of Theorem 7) is satisfied, then the pair has a coincidence point. Proof. Define
by
, then (
17) can be written as
Since is an altering distance function and is a lower semi-continuous function with if and only if The desired result follows from Theorem 6 (resp. Theorem 7). ☐