1. Introduction
The idea of the theory of modular spaces was first put forward by Nakano [
1] and was later reconsidered in detail by Musielak and Orlicz [
2]. In 2010, Chistyakov [
3,
4] generalized modular spaces and complete metric spaces by introducing modular metric spaces. In the last two decades, the modular metric space has been an interesting abstract space for nonlinear functional analysis, and hence, it has been investigated densely by many researchers. For more features of the concepts of modular metric space, see e.g., [
5,
6,
7].
The metric fixed point theory is a very rich area for research that was initiated by Banach. One of the most interesting and early characterizations of the Banach theorem was given by Kannan [
8]. It was later understood that Kannan contraction is independent from the Banach contractions [
9]. Another crucial contraction, weakly uniformly strict contraction, was observed by Meir–Keeler [
10]. It was later called the Meir–Keeler contraction. As it is expected, it is a proper generalization of Banach’s principle, see e.g., [
11,
12]. In addition to all such linear extensions of the contraction mapping, we underline the notion of admissible auxiliary function that plays one of the key roles in initiating many interesting contractions. In particular, Samet et al. [
13] used admissible functions to extend the renowned Banach fixed point theorem. Following this paper, a huge number of papers appeared in which the admissible functions are used to improve well-known existing results in the metric fixed point theory.
In addition to all these advances on the metric fixed point theory, we need to mention the interpolative contractions. Very recently, the concept of interpolative contraction was suggested by the first author [
14] by revising the Kannan contraction [
8]. Following this paper [
14], many research papers have been published on interpolative contractions in the setting of distinct abstract spaces [
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25] and for differently combined well-known contractions [
26,
27,
28,
29,
30,
31,
32].
The aim of the paper is to examine the existence and uniqueness of interpolative Meir–Keeler contraction types via admissible mappings for fixed point theory in the context of the modular metric spaces. For this purpose, we reserve the first section for the introduction. The aim of the second section is to collect and clarify the mentioned notions above as well as to give the fundamentals of the metric fixed point theory. In the third section, we shall highlight the main results for interpolative Meir–Keeler contraction types in modular metric spaces and non-Archimedean modular metric spaces.
2. Preliminaries
We start this section by presenting well-known notations, collecting the basic definitions and fundamental results.
2.1. Concepts Related to Modular Metric Spaces
In this subsection, we shall present some basic concepts and properties in modular metric spaces. First of all, we recall the definition of the modular space:
Definition 1 ([2]).Let be a vector space over (
or ).
A functional is called a modular if for any y and z in , it satisfies the following conditions: - (n1)
iff;
- (n2)
for every scalar α with;
- (n3)
, forand.
Let be a nonempty set, and because of the disparity of the arguments, functionwill be written byfor everyand
In what follows, we state the definition of modular metric spaces (hereinafter referred to as “MMS”).
Definition 2 ([3,
4]).Letbe a set. A functionis said to be a metric modular onif it satisfies the following, for all,
- ()
for all ⇔ ;
- ()
for every;
- ()
for every.
If instead of (), we have only the condition
- ()
, for each, then w is said to be a (metric) pseudomodular on.
If we replace (
) by
for all
and all
, then
is called a non-Archimedean modular metric space (hereinafter referred to “non-AMMS”) [
33]. Since (
) implies (
), each non-AMMS is an MMS.
Remark 1. If w is a pseudomodular metric on a set,
then the functionis nonincreasing onfor all.
Indeed, ifλ, then Definition 3 ([34]).A pseudomodular w on is said to satisfy the-condition (on)
if the following condition holds: Given a sequenceand, if there exists a number, possibly depending onand y, such that if, then.
Next, we recollect the basic topological notions in the context of modular spaces.
Definition 4 ([3,
35]).Letbe an MMS. - (i)
The sequenceinis notified to be convergent toif, asfor every.
- (ii)
The sequenceinis notified to be Cauchy if, asfor all.
- (iii)
A subsetofis notified to be closed if each limit of a convergent sequence ofis contained in.
- (iv)
A subsetofis notified to be complete if any Cauchy sequence inis a convergent sequence and its limit is in.
- (v)
A subsetofis notified to be bounded if for every,
2.2. Basic Definitions and Theorems
We shall start this section by stating the renowned Meir–Keeler fixed point theorem.
Theorem 1 ([10]).Letbe a complete metric space. Ifforms a Meir–Keeler contraction on M, that is, "for any given,
there is asuch thatfor every",
thenpossesses a unique fixed point.
Let us recall the -admissible functions:
Definition 5 ([13]).Let be a nonempty set,
and.
We notify thatis α-admissible if Karapınar et al. [
36] defined the concept of triangular
-admissible mapping as follows.
Definition 6 ([36]).Let be a nonempty set,
be a self mapping defined on M andbe a function.
is said to be a triangular α-admissible mapping if the following conditions: - (1)
implies;
- (2)
implies;
hold for all.
Lemma 1 ([13]).Let be a nonempty set andbe a triangular α-admissible mapping. Suppose that there existssuch thatIf,
thenfor allwith By
, we denote the family of altering distance functions [
37], that is, function
, such that the following conditions fulfill:
- (1)
is continuous and nondecreasing;
- (2)
if and only if .
Lemma 2 ([37,
38,
39]).Letbe a nondecreasing and continuous function. Then, the following two conditions are equivalent: - (1)
for all;
- (2)
for all.
Next, we state the definition of the interpolative contraction:
Definition 7 ([14]).Letbe a metric space. A mappingis said to be an interpolative Kannan-type contraction if we have two constantsand αsuch thatfor everywithand.
Theorem 2 ([14]).Letbe a complete metric space andbe an interpolative Kannan-type contraction mapping. Then,
possesses a fixed point. Inspired by interpolative and the Meir–Keeler, the notion of the interpolative Kannan–Meir–Keeler [
30] was defined in 2021:
Definition 8 ([30]).Letbe a complete metric space. A mappingis said to be an interpolative Kannan–Meir–Keeler contraction on M if there existssuch that given,
there existssuch that for every,
where Let
The aim of this paper is to introduce a new contraction, namely, interpolative Meir–Keeler contraction, in the context of modular metric space. Consequently, we shall examine the existence and uniqueness of the fixed point for such mapping in the mentioned setting. In order to indicate the validity, an illustrative example is considered.
3. Main Results
We start this section by stating the definition of the interpolative Meir–Keeler contraction in MMS via admissible mappings:
Definition 9. Letbe an MMS. A self-mappingis defined as aninterpolative Meir–Keeler contraction of type I if there exist the functions,
and constants,
whenever for everythere existssuch thatfor everyand.
The following lemma is an immediate consequence of Definition 9.
Lemma 3. Ifis aninterpolative Meir–Keeler contraction of type I mapping, then for every.
Theorem 3. Letbe a complete MMS, where the modular w satisfies thecondition. Letbe aninterpolative Meir–Keeler contraction of type I mapping and assume that
is a triangular α-admissible mapping;
there existssuch that;
eitheris continuous; or,
- ()
ifis a sequence insuch thatfor eachandas, thenfor all.
Then, there existssuch that
Proof. Let in be such that . Define the Picard sequence , starting at , that is, for . Using the conditions and , we obtain , which implies , and also, using Lemma 1 for every , .
Furthermore, clearly if
for some
, then evidently,
has a fixed point. Thus, we assume that
. Hence, we have
Using Lemma 3, it follows that for every
, we have
and using the property of function
,
Consequently, we obtain that the sequence
is strictly decreasing. In addition, from
, for all
, we obtain that the sequence
is convergent and so there exists a point
such that
. We claim that
. On the contrary, if
, then taking
, by (
2), we deduce that there exists
such that
Moreover, using
(for
), we write
such that for each
, we have
or, using the properties of the function
,
for any
. Therefore, by (
5), we obtain that
which is a contradiction. Accordingly, we prove that
Now, we will indicate that
is a Cauchy sequence. Let
and we consider that
, with
As we have
and from (
)
we choose
such that
and
for
, and we assert that
for any
. Indeed for
and (
6) holds. Assume that the above inequality holds for
Then, we show that the above inequality holds for
Indeed, by property
, Lemmas 3 and 1, we obtain
Consequently, the sequence
is Cauchy. As the completeness of the space
, there exists
such that
. Since
is continuous,
so
; that is,
is a fixed point of
.
We can obtain that there is a fixed point of
without any continuity assumption for the mapping
by property
. Using the condition
and (
4), we have
for every
. We claim that
. On the contrary, if
, by Picard sequence, we have
, and then
Letting in the above inequality and by the right continuity of at 0, we obtain that , so . Consequently, is a fixed point of . □
If in Theorem 3, we take where , we have the following corollary:
Corollary 1. Letbe a continuous self mapping on a complete MMS.
Suppose that there exist a functionand constants,
,
such that for every,
there existssuch thatfor everyand.
Moreover,
is a triangular α-admissible mapping,
there existssuch that;
then there existsuch that, that is, possesses a fixed point.
If in Theorem 3, we obtain for every , we obtain the following corollary:
Corollary 2. Letbe a continuous self mapping on a complete MMS.
If there exist a function,
and a constant,
such that for allthere existssuch thatfor everyand.
Then,
has a fixed point. If in Corollary 2, we consider where , we have the following corollary:
Corollary 3. Letbe a continuous self mapping on a complete MMS.
If there existand,
such that for all,
there existssuch thatfor every,
and.
Then,
has a fixed point. Example 1. Let,
where,
,
and,
be a metric modular on M, with,
where,
,
and.
Let,
be defined bywhere.
Let also,
,
and,
Since it is easy to check that–hold, we will verify that the mappingis aninterpolative Meir–Keeler contraction of type I, by choosingfor any.
For,
,
we have,
and then (2) holds for every.
Forand,
we have,
respectively.
We obtainand Thereupon, (2) holds for every.
(The other cases are not interesting, taking in account the definition of the function α.) Consequently, all the assumptions of Theorem 3 being satisfied, it follows that the mapping has fixed points; there are , respectively .
Now, we investigate fixed point results for interpolative Meir–Keeler contraction in non-AMMS via admissible mappings.
Definition 10. Letbe a non-AMMS. A mappingis called aninterpolative Meir–Keeler contraction of type II if there exist two functions,
and constants,
withsuch that for allthere existssuch thatfor everyand.
Lemma 4. Ifis aninterpolative Meir–Keeler contraction type II mapping, then for every.
Theorem 4. Letbe a complete non-AMMS andbe aninterpolative Meir–Keeler contraction of type II. Suppose that
is a triangular α-admissible mapping;
there existssuch that,
ifis a sequence insuch thatfor eachandas, thenfor all.
Then there existssuch that
Proof. Let be such that . Let be a Picard sequence starting at , that is, for . By and , we obtain implies , and taking Lemma 1 into account, we obtain for all with .
In addition, understandably, if there exists
such that
, then
and openly
has a fixed point. So, we assume that
. So, we have
Keeping Lemma 3 in mind, it follows that for every
, we write
and thus, we obtain
Accordingly, we show that sequence
is strictly decreasing. From
, for every
, we provide that the sequence
is convergent; at the time, we have a point
such that
. We pretense that
. If not
, then letting
, from (
10), we deduce that there exists
such that
Again, since
, (for
) we can find
such that
for any
. By using (
14), we obtain that
which is a contradiction. Correspondingly, we show that
Now, we indicate that
is a Cauchy sequence. Let
and choose
with
there exists
so that
Evidently, condition (
16) is true with
replaced by
Moreover, by using (
15), we have
so that
for all
Let
Apparently,
since
. We will prove that
If
, then
in (
18). If
, then we obtain the following two cases:
Then from (
17)–(
19), we obtain
Then, using Lemma 1 we obtain
so,
and then, we deduce that
In this way, we obtain .
Then, we write
which shows that
. Therefore, by Case 1 and Case 2, we show that
for all
Now, for
by (
20), we show that
which indicates that
is Cauchy. As the completeness of the space
, there exists
such that
.
Using condition
and (
12), we have
for every
. We show that
. Inversely, let
, using the Picard sequence, we obtain
. Moreover,
Taking , we obtain the above inequality, , so . Thus, is a fixed point of . □
If in Theorem 4, we obtain where , then we have the following corollary:
Corollary 4. Letbe a continuous self-mapping on a complete non-AMMS.
Assume that there exists a function,
and constantswith,
such that for allthere existssuch thatfor every.
Furthermore, assume that is a triangular α-admissible mapping;
there existssuch that,
Then, there existssuch that, that is, possesses a fixed point.
If in Theorem 4, we obtain for every , then we have the following corollary:
Corollary 5. Letbe a continuous self mapping on a complete non-AMMS.
Suppose that there exist a functionand constantswith,
such that for allthere existssuch thatfor every.
Then, there existssuch that;
that is,
possesses a fixed point. If in Corollary 5, we obtain where , then we have the following corollary:
Corollary 6. Letbe a continuous self-mapping on a complete non-AMMS.
Suppose that there existand constantswith,
such that for allthere existssuch thatfor every.
Then, the mappingpossesses a fixed point.