1. Introduction
With a view to enhance the domain of applicability, Matthews [
1] initiated the idea of a partial metric space by weakening the metric conditions and also proved an analogue of Banach contraction principle in such spaces. Thereafter, many well-known results of metric fixed point theory were extended to partial metric spaces (see [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16] and references therein).
On the other hand, Turinici [
17] initiated the idea of order theoretic metric fixed point results, which was put in more natural and systematic forms by Ran and Reurings [
18], Nieto and Rodríguez-López [
19,
20], and some others. Very recently, Alam and Imdad [
21] extended the Banach contraction principle to complete metric space endowed with an arbitrary binary relation. This idea has inspired intense activity in this theme, and by now, there exists considerable literature around this result (e.g., [
6,
21,
22,
23,
24,
25]).
Proving new results in metric fixed point theory by replacing contraction conditions with a generalized one continues to be the natural approach. In recent years, several well-known contraction conditions such as Kannan type, Chatterjee type, Ciric type, phi-contractions, and some others were introduced in this direction.
In this paper, we introduce some useful notions, namely, -precompleteness, -g-continuity and -compatibility, and utilize the same to establish common fixed point results for generalized weak -contraction mappings in partial metric spaces endowed with an arbitrary binary relation . We also derive several useful corollaries which are either new results in their own right or sharpened versions of some known results. Finally, an application is provided to validate the utility of our result.
3. Relation Theoretic Notions and Auxiliary Results
Let M be a non-empty set. A binary relation on M is a subset of . For , we write if is related to under . Sometimes, we denote it as instead of . Further, if such that and are distinct, then we write (sometimes as ). It is observed that is also a binary relation on M. and ∅ are trivial binary relations on M, specifically called a universal relation and empty relation, respectively. The inverse, transpose or dual relation of is denoted by and is defined as . We denote by the symmetric closure of , which is defined as .
Throughout this manuscript, M is a non-empty set, stands for a binary relation on M and denotes an identity mapping, and S and g are self-mappings on M.
Definition 4. [26] For a binary relation : - (a)
Two elements are said to be -comparative if or . We denote it by .
- (b)
is said to be complete if , for all .
Proposition 1. [21] For a binary relation on M, we have (for all ): Definition 5. [21] A sequence is said to be -preserving if , for all . Here, we follow the notion (of
-preserving) as used by Alam and Imdad [
21]. Notice that Roldán and Shahzad [
27] and Shahzad et al. [
28] used the term “
-nondecreasing” instead of “
-preserving”.
Definition 6. [29] Let . If for each , there exists a point such that and , then N is said to be -directed. Definition 7. [30] For , a path of length in from to is a finite sequence such that , and , for each . Definition 8. [31] Let . If for each , there exists a path in from to , then N is said to be -connected. Definition 9. [21] is said to be S-closed if implies that , for all . Definition 10. [31] is said to be -closed if implies that , for all . Observe that on setting , Definition 10 reduces to Definition 9.
Proposition 2. [31] If is -closed, then is also -closed. Definition 11. [23] is said to be locally S-transitive if for each -preserving sequence with range , the binary relation is transitive. Motivated by Alam and Imdad [
31], we introduce the notion of
-continuity and
-
g-continuity in the context of partial metric space as follows:
Definition 12. Let be a partial metric space endowed with a binary relation . A self-mapping S on M is said to be -continuous at a point if for any -preserving sequence such that , we have . S is -continuous if it is -continuous at each point of M.
Definition 13. Let be a partial metric space endowed with a binary relation . A self mapping S is said to be -continuous at a point if for any sequence with -preserving and , we have . S is -g-continuous if it is -g-continuous at each point of M.
Remark 2. Notice that for , Definition 13 reduces to Definition 12.
In the next definition, we introduce -compatibility.
Definition 14. Let be a partial metric space endowed with binary relation and . S and g are said to be -compatible if for any sequence such that and are -preserving and , we have: Inspired by Imdad et al. [
24], we introduce the following notions in the setting of partial metric spaces in the similar way.
Definition 15. Let be a partial metric space endowed with a binary relation . A subset is said to be -precomplete if each -preserving Cauchy sequence converges to some .
Remark 3. Every -complete subset of M is -precomplete.
Proposition 3. Every -closed subspace of an -complete partial metric space is -complete.
Proposition 4. An -complete subspace of a partial metric space is -closed.
Next, we introduce the notion of -self closedness in the setting of partial metric spaces.
Definition 16. Let be a partial metric space endowed with binary relation . Then is said to be ρ-self closed if for each -preserving sequence with , there exists a subsequence of such that , for all .
We now state the following lemma needed in our subsequent discussion.
Lemma 3. Let M be a non-empty set and . Then there exists a subset with and is one–one.
We use the following notations in our subsequent discussions:
: Set of all coincidence points of S and g;
: The collection of all points such that .
4. Main Results
Let denote the set of all mappings satisfying the following:
- ()
is non-decreasing;
- ()
iff and if .
Notice that Reference [
32] used the condition that
is continuous. Inspired by Reference [
33], we replace their condition by a more weaker condition
. In fact, this condition is also weaker than that
is lower semi-continuous. Indeed, if
is a lower semi-continuous function, then for a sequence
with
, we have
Before presenting our main result, we define the following.
Definition 17. Let M be a non-empty set endowed with an arbitrary binary relation and . Then, N is said to be -directed if for each , there exists a point such that , for and .
Definition 18. Let M be a non-empty set endowed with an arbitrary binary relation and . Then, N is said to be -connected if for each , there exists a path between and such that , for .
Remark 4. For , Definitions 17 and 18 reduce to -directed and -connected.
Now, we state and prove our first main result, which runs as follows:
Theorem 1. Let be a partial metric space equipped with a binary relation , , an -precomplete subspace in M and . Assume that the following conditions are satisfied:
;
is -closed;
;
is locally S-transitive;
S satisfies generalized Ćirić-type weak -contraction, i.e.,for all with and , where: S and g are -compatible;
S and g are -continuous;
or alternatively:
- ()
;
either S is -continuous or S and g are continuous or is ρ-self closed.
Then, S and g have a coincidence point.
Proof. Choose
as in
and construct a sequence
in
M as follows:
If there is some
such that
, then
is the coincidence point of the pair
and we are done. Henceforth, assume that
, for all
. In view of condition
, we have
, for all
. Employing condition
, we have:
which implies:
where:
Now, if
, then Equation (
2) becomes:
a contradiction. Hence, we have
and Equation (
3) implies that
is non-decreasing (also bounded below by 0). Thus, there exists
such that
. Next, we show that
. Suppose, by contrast, that it is not so, i.e.,
. Passing the limit
in Equation (
2), we get:
which is a contradiction. Hence:
We also have:
which, on letting
and applying Equation (
4), yields that:
Now, our claim is that
is a Cauchy sequence in
. Otherwise, there exist two subsequences
and
of
such that
is the smallest integer for which:
Since
, for all
, Equation (
5) gives:
Now, using triangular inequality, we have:
Letting
in the above inequality, we obtain:
Again, the triangle inequality yields the following:
and:
which together give rise to:
Now, on taking
, the above inequality gives:
In a similar manner, one can show that:
Using
, we have
and hence, Equation (
1) implies:
Using Equations (
6) and (
7) and letting
in the above inequality, we get:
a contradiction. Hence,
is Cauchy in
(as
) which is also
-preserving. Lemma 1 ensures that it is also Cauchy in
. Thus, the
-precompleteness of
N in
M ensures the existence of a point
such that:
Now, by Equation (
9) and Lemma 1, we get:
Further, by the definition of
and Equation (
8), we have:
Finally, to prove the existence of coincidence point of
S and
g, we make use of conditions
and
. Firstly, assume that
holds. Now, as
, so using assumption
and Equation (
8), we obtain:
By the definition of
, we have
is also
-preserving (i.e.,
, for all
n), so using assumption
and Equation (
11), we get:
By using Equation (
8) and
-continuity of
S, we obtain:
As
and
are
-preserving and
, by the condition
, we have:
Now, from Equations (
13)–(
15) and continuity of
, it follows that:
i.e.,
and we are done. Secondly, suppose that
is satisfied. Then, by
, there exists some
such that
. Hence, Equations (
8) and (
11) respectively reduce to:
and:
Next, to accomplish that
z is a coincidence point of
S and
g, we utilize
. Thus, suppose that
S is
-
g-continuous, then using Equation (
16), we obtain:
Now, by virtue of uniqueness of limit, Equations (
17) and (
18) give
.
Next, assume that
S and
g are continuous. Then owing to Lemma 3, there exists
such that
and
is injective. Now, define a mapping
by:
As
is injective and
,
is well-defined. Further, due to the continuity of
S and
g,
is continuous. The fact that
, assumptions
and
imply that:
Thus, without loss of generality, we can construct
, satisfying Equation (
16) with
. On using Equations (
16), (
17), and (
19) with continuity of
, we obtain:
and we are done. Alternatively, if
is
-self closed, then for any
-preserving sequence
in
N with
, there exists a subsequence
of
such that
, for all
. Suppose
, then we have:
Letting
and using Equation (
8), we get:
Now, applying
and
, condition
gives:
which, on letting
and using Equations (
8) and (
20) and Lemma 2, yields that:
a contradiction. Hence
, i.e.,
. This completes the proof. □
Now, we present a corresponding uniqueness result.
Theorem 2. In addition to the assumptions of Theorem 1, if we assume that the following condition is satisfied:
is -connected,
then S and g have a unique point of coincidence. Moreover, if:
S and g are weakly compatible,
then S and g have a unique common fixed point.
Proof. Firstly, Theorem 1 ensures that
. Let
. Then, there exists
such that
and
. Our claim is that
. Now, owing to hypothesis
, there exists a path, say
of some finite length
l in
from
to
with:
and:
Define constant sequences
and
, then we have
and
, for all
. Further, set
, for each
and define sequences
by:
By mathematical induction, we will prove that:
In view of Equation (
21), the result holds for
. Now, suppose it holds for
, i.e.:
By
-closedness of
and Proposition 2, we have:
i.e., the result holds for
and hence, it holds for all
. Also from Equation (
22), we have
and
is
-closed, so by Proposition 2 and Equation (
4), we have:
Now, for all
and for each
, define
. Our claim is that:
Suppose, by contrast, that
. Since
,
or
, for all
and for each
. Making use of Equation (
1), we have:
or:
where:
Now, letting
and using Equation (
23), we obtain:
which, on applying Equation (
24) after taking limit, yields that:
a contradiction. Therefore,
.
Hence, , i.e., . Thus, S and g have a unique point of coincidence.
Secondly, to justify the existence of a unique common fixed point, we consider
, i.e.,
, for some
. By the condition
,
S and
g commute at their coincidence points, i.e.,
thereby yielding
, i.e.,
. Thus, by the uniqueness of point of the coincidence point, we have:
The uniqueness of the common fixed point is a direct consequence of the uniqueness of the coincidence point. This finishes the proof. □
We present the following example to support our result.
Example 1. Let with partial metric defined by: Define a binary relation . Clearly, is a complete partial metric space. Define by: It is clear that is -closed and S and g are continuous. Next, define by: Clearly, . Observe that all the conditions of Theorems 1 and 2 are fulfilled (with ). Hence, S and g have a unique common fixed point (namely 0).
Next, we present the following corollaries.
Corollary 1. The conclusion of Theorem 2 remains valid if we replace the condition by any one of the following:
is complete;
is -directed.
Proof. If holds true, then for any , we have and , for some (as ). In view of , we have , i.e., is a path of length 1 in from to . Hence, condition of Theorem 2 is fulfilled and the result is concluded by Theorem 2.
On the other hand, if condition holds, then for each (such that and , for ), there exists such that , , i.e., is a path of length 2 in from to and . Hence, condition of Theorem 2 is fulfilled and again by Theorem 2, the conclusion follows. □
Corollary 2. The conclusions of Theorems 1 and 2 remain true if we replace assumption by the following one:
S satisfiesfor all with and .
Proof. As
, we have:
for all
with
. Thus, all the assumptions of Theorems 1 and 2 are satisfied and the conclusions hold. □
Following Reference [
32], it can be easily seen that in a partial metric space
, for all
, the conditions:
and:
are more weaker than:
and:
respectively. However, the converse need not be true in general (even the above assertion is true for any
). This leads us to our next corollary.
Corollary 3. The conclusions of Theorems 1 and 2 remain true if we replace assumption by the following one:
S satisfies:or:for all with and .
By setting , with and in Corollary 3, we deduce the following corollaries:
Corollary 4. The conclusions of Theorems 1 and 2 remain true if we replace assumption with the following one:
there exists such that:for all with and .
We see that the above corollary is a relatively new and somewhat refined version of Alam and Imdad [
31] type result in partial metric space with some refinement, e.g.:
We use -precompleteness of subspace in place of -completeness.
We use -analogous of compatibility, continuity, closedness and -self closedness instead of their -analogous.
Corollary 5. The conclusions of Theorems 1 and 2 remain true if we replace assumption with the following one:
S satisfies:for all with and .
By considering , the following fixed point result can be deduced easily from Theorems 1 and 2.
Corollary 6. Let be a partial metric space equipped with a binary relation , an -precomplete subspace in M and . Assume that the following assumptions are satisfied:
There exists such that ;
is S-closed;
;
is locally S-transitive;
S satisfies generalized Ćirić-type weak -contraction, i.e.:for all with and , where: either S is -continuous or is ρ-self closed.
Then, S has a fixed point. In addition, if:
N is -connected,
then the fixed point is unique.
In place of -precomplete of N, if we use the -completeness of the whole space M, then we find a particular version of Theorem 1.
Corollary 7. Let be an -complete partial metric space and satisfy the following assumptions:
;
is -closed;
;
is locally S-transitive;
S satisfies generalized Ćirić-type weak -contraction, i.e.,:for all with and , where: S and g are -compatible;
S and g are -continuous;
or alternatively:
- ()
there exists an -closed subspace N of M such that ;
either S is -g-continuous or S and g are continuous or is ρ-self closed.
Then, S and g have a coincidence point.
Proof. The result follows by Proposition 3 and Remark 3. □
Moreover, in Corollary 7, if we assume g to be surjective, then assumption as well as assumption can be removed trivially since .