1. Introduction and Preliminaries
In 1922, Banach laid the foundations of metric fixed point theory by proposing his prominent fixed point result. To put a finer point on it, Banach observed that if a self-mapping
T, defined on complete metric space
, fulfills the contraction inequality, i.e., there exists a constant
such that
then it possesses a unique fixed point in
X. In 1968, Kannan [
1] proposed a new fixed point result. He considered the following contraction type:
where
. Very recently, in [
2], the acclaimed theorem of Kannan was revisited by taking the interpolation theory into account. For a metric space
, the self-mapping
is said to be an
interpolative Kannan type contraction, if there are constants
and
such that
for all
with
. The main result in [
2] via an interpolative Kannan type contraction is
Theorem 1 ([
2])
. In the framework of a complete metric space , if a mapping forms an interpolative Kannan type contraction, then it possesses a unique fixed point in X. Note that when the inequality (
2) holds for all
, and if
T possesses a fixed point (say
), then
for each
, that is,
T is a constant mapping, which is the trivial case, so the fixed point of
T is unique. The appropriate condition on
and
in (
2) should be
, where
. In this case, the author [
2] ensured the existence of a unique fixed point. There is a gap, that is, such fixed point is not necessarily unique. The following example illustrates our concern.
Example 1. Set that is endowed with the Euclidean metric . ConsiderLet . Then, . Thus, (2) is satisfied for all and . It is evident that both 0 and 1 are fixed points for the self-mapping T. As a correction of Theorem 1, we should state
Theorem 2. Let be a complete metric space. A self-mapping possesses a fixed point in X, if there exist constants and such thatfor all . The following theorem was proved by Reich, Rus and Ćirić [
3,
4,
5,
6,
7] independently to combine and improve both Banach and Kannan fixed point theorems.
Theorem 3. In the framework of a complete metric space , if forms a Reich–Rus–Ćirić contraction mapping, i.e.,for all , where , then T possesses a unique fixed point. Notice that several variations of Reich contractions (
3) can be stated. We may state the following:
where
such that
.
In this paper, we shall investigate the validity of the interpolation approach for Reich contractions in the context of partial metric spaces that was introduced by Matthews [
8].
Definition 1. Let X be a non-empty set. A function is said to be a partial metric, if the following conditions are fulfilled for each ,In this case, is said to be a partial metric space. The function
defined as
is a standard metric on
X. It is natural to define the basic topological concepts, in particular, convergence of a sequence, fundamental (Cauchy) sequence criteria, continuity of the mappings, and completeness of the topological space in the framework of partial metric spaces; see, e.g., [
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18].
Definition 2. In the framework of a partial metric space , we say that
- (i)
A sequence converges to the limit ξ if ;
- (ii)
A sequence is fundamental or Cauchy if exists and is finite;
- (iii)
A partial metric space is complete if each fundamental sequence converges to a point such that ;
- (iv)
A mapping is continuous at a point if for each , there exists such that .
For what follows, we shall recall the following lemma that can be derived easily (see [
8]).
Lemma 1. Let p be a partial metric on a non-empty set X and be the corresponding standard metric space on the same set X.
- (a)
A sequence is fundamental in the framework of a partial metric if and only if it is a fundamental sequence in the setting of the corresponding standard metric space .
- (b)
A partial metric space is complete if and only if the corresponding standard metric space is complete. Moreover, - (c)
If as in a partial metric space with , then we have
In this paper, we initiate the notion of interpolative Reich–Rus–Ćirić type contractions on partial metric spaces. We also present two examples illustrating our approach.
2. Main Results
We start this section by introducing the notion of interpolative Reich–Rus–Ćirić type contractions.
Definition 3. In the framework of a partial metric space , a mapping is called an interpolative Reich–Rus–Ćirić type contraction, if there are constants and such thatfor all , Theorem 4. In the framework of a partial metric space , if is an interpolative Reich–Rus–Ćirić type contraction, then T has a fixed point in X.
Proof. We take an arbitrary point
and build an iterative sequence
by
for each positive integer
n. If there exists
such that
, then
is a fixed point of
T. The proof is completed. Henceforwards, assume that
for each
. By substituting the values
and
in (
7), we find that
By a calculation, we derive
from the inequality (
8). We conclude that
is a non-increasing sequence with non-negative terms. Thus, there is a nonnegative constant
ℓ such that
Note that
Indeed, from (
9), we deduce that
Regarding
, and by taking
in the inequality (
10), we deduce that
For what follows, we shall prove that
is a fundamental (Cauchy) sequence by employing standard tools. More precisely, starting with the triangle inequality, we shall get the following estimation:
Letting
in the inequality (
11), we ascertain that
is a fundamental sequence.
Hence,
, that is,
is a fundamental sequence in
. By Lemma 1,
is also Cauchy in
. More particularly, since
is complete,
is also complete. Hence, there exists
such that
which implies that
As a next step, we make evident that the limit
of the iterative sequence
is a fixed point of the given mapping
T. Assume that
, so
. Recall that
for each
. By letting
and
in (
7), we determine that
Letting
in the inequality (
14), we find out
, so
, which is a contradiction. Thus,
. □
The following examples illustrate Theorem 4.
Example 2. Let be a set endowed with the classical partial metric , that is, | 1 | 3 | 4 | 7 |
1 | 1 | 3 | 4 | 7 |
3 | 3 | 3 | 4 | 7 |
4 | 4 | 4 | 4 | 7 |
7 | 7 | 7 | 7 | 7 |
We define a self-mapping T on X by . It is clear that T is not a Reich–Rus–Ćirić contraction. Indeed, there is no such that the following inequality is fulfilled: On the other hand, choose , and . Let ; then, . Without loss of generality, we have
Case 1: . Here, Case 2: . we have Case 3: and . Here, Thus, the self-mapping T is an interpolative Reich–Rus–Ćirić type contraction and are the desired fixed points. Note that, in the setting of interpolative Reich–Rus–Ćirić type contractions, the constant lies between 0 and 1, although in the classical version it is restricted by 1/3.
Example 3. Following Example 1, let . Consider Consider . Clearly, (7) holds for all [by taking , and ]. Note that T has two fixed points, which are 0 and 1. On the other hand, taking and , we have, for any ,that is, Corollary 4 in [9] is not applicable. The following is an immediate consequence of our main result.
Corollary 1. In the framework of a standard metric space , if is an interpolative Reich–Rus–Ćirić type contraction, that is,for all , then T possesses a fixed point in X. Proceeding as Theorem 4, we shall extend Theorem 2 to partial metric spaces.
Theorem 5. Let be a complete partial metric space and be such thatfor all , where and . Then, T possesses a fixed point in X.