1. Introduction
The advancement of fixed point theory in the last few decades has mainly been related to introducing a new kind of generalized metric space and extensions of the Banach Contraction Principle. This principle has become known as a useful tool for establishing the existence and uniqueness of a fixed point for contractive mappings. In this direction, in 2012, Wardowski [
1,
2] introduced a new type of contraction named
-contraction as a generalization of this important principle in metric spaces. Subsequently, many researchers [
3,
4,
5,
6,
7,
8,
9,
10] further developed this new category by improving its properties and extending it in a more generalized setting. In the meantime, other recently defined concepts such as
-admissible mapping in [
11] promoted in [
12,
13,
14,
15,
16], Suzuki contraction widely used in [
17,
18,
19,
20,
21], and formulations in partial metric spaces, metric-like spaces,
-metric spaces, and
-metric -like spaces underline their significance and offer a broader understanding in various contexts of the fixed point theory. For an extended introduction, we could mention many new theorems and corresponding classical results with applications in the above spaces, resulting in notions of interpolative and hybrid contractions; see [
22,
23,
24].
In this paper, we introduce the notion of generalized Suzuki-type -contraction via a set of implicit relations in the setting of -metric-like spaces. It strictly extends the known generalizations of metric and -metric spaces. Moreover, it presents a new approach and includes many types of contractions such as Suzuki -contraction, interpolative, hybrid, and -order hybrid contractions, exploring diverse fixed point theorems, implementations, and deduction for earlier and recent results.
2. Preliminaries
Definition 1. Ref. [
16].
Let be a nonempty set and
be a given real number. A mapping
is called a
-metric-like if for all
the following conditions are satisfied: implies
The pair
is called a
-metric-like space (for short
-m.l.s).
Note that by the first axiom of a definition, the self-distance of an arbitrary point may be positive.
There are various examples of
-metric-like space in the reference literature. To illustrate them, we selected some from [
16].
Example 1. Let
and
defined by
for all
Then,
is a
-m.l.s with parameter
and
is not a
-metric on
.
Example 2. Let
and
defined by
for all
Then,
is a
-metric-like on
with parameter
and it is not a
-metric or a metric-like on
.
Definition 2. Ref. [
16].
Let be a
-m.l.s with parameter
Then, for any sequence
in
the following applies:- (a)
is said to be convergent to if ;
- (b)
is said to be a Cauchy sequence in
if
exists and is finite;
- (c)
is called a complete
-m.l.s if, for every Cauchy sequence
in
, there exists
such that
.
Remark 1. In a
-metric-like-space:
- -
The limit of a convergent sequence is not necessarily unique.
- -
A convergent sequence need not be a Cauchy sequence.
The following example characterizes and supports Definition 2.
Example 3. Let
and
defined by
for all
Also let be the sequence
as For any we have . Therefore, the sequence is convergent where for each , that is the limit of sequence, is not unique. Also, it is noted that does not exist, so it is not Cauchy.
Definition 3. Ref. [
16].
Let be a
-m.l.s with parameter
, and a function
. We say that the function
is continuous if for each sequence
the sequence
whenever
as
that is if
yields
.
Remark 2. In a
-m.l.s with parameter
, if
then the limit of the sequence
is unique if it exists.
Lemma 1. Ref. [
14].
Let be a
-m.l.s with parameter
. Then, the following applies:- (a)
If
then
- (b)
If
is a sequence such that
, then we have
;
- (c)
If
, then
Lemma 2. Ref. [
16].
Let be a
-metric-like space with parameter
, and suppose that
is
-convergent to
with
. Then, for each
we have Definition 4. Ref. [
1].
Let be a metric space and
be a mapping. Then,
is called an-contraction if there exists a function
such that
. is strictly increasing on
;
For each sequence
of positive numbers, There exists
such that
;
There exists
such thatfor all
with
For examples that show the class of
-contraction, the reader can confront the extended literature in [
1,
2,
3,
6,
7,
9,
12].
Definition 5. Ref. [
11].
Let be a non-empty set. Let and be given functions. We say that is an -admissible mapping if implies that for all .
Definition 6. Ref. [
18].
Let and be a function, be a sequence in and . Then, is called -regular if for any : and converges to then Lemma 3. Ref. [
14].
Let be complete -m.l.s with parameter , let be a sequence such that If for the sequence , , then there exists and sequences and of natural numbers with (positive integers) such that and .
In the sequel, let we represent some notations and properties.
Suppose that satisfied the conditions:
- (a)
F is continuous and non-decreasing;
- (b)
For any sequence of positive real numbers, if then
This family of all functions will be denoted by .
And is the set of all continuous functions , satisfying the conditions:
is non-decreasing in respect to each variable;
Let be the mappings and . We say that satisfies admissible convergence property (for short -property), if for every sequence in such that for all and converges to then
3. Results
In this main section, among an enormous work presented for types of Suzuki contractions and interesting generalizations established in various spaces, we propose our new general definitions and related theorems concerning such contractive mappings.
Definition 7. Let be a complete -m.l.s with parameter , be a self-mapping, and there exist , and . Then, is called a generalized -admissible Suzuki-type -contraction if the following condition is satisfied:impliesfor all , and , where for some
Definition 8. Let be a -m.l.s with parameter , be a self-mapping and .
Then, is an -admissible Hardy–Rogers Suzuki-type interpolative -contraction, if there exist and with such thatimpliesfor all
, and , where Definition 9. Let be a -m.l.s with parameter , be a self-mapping and .
Then, is an -admissible Hardy–Rogers Suzuki-type -order hybrid -contraction, if there exist such that:impliesfor all
,
,
such that
and
, where Remark 3. Some notable cases of introduced definitions include the following:
corresponds to all in metric setting.
can be fixed regarding its condition, and it turns to a generalized Suzuki-type
-contractions.
leads to types of Suzuki
-contraction defined in a
-m.l.s.
All cases above taken simultaneously.
Theorem 1. Let be a complete -m.l.s with parameter and a generalized -admissible Suzuki-type -contraction. Assume that
- J1.
there exists with
- J2.
is -admissible and satisfies -property.
Then, has a fixed point in , and it is unique if , for all
Proof. Let be for the Picard sequence induced by function with initial point with Let we perform the general case where for each (that is the same with ). Since is -admissible, then implies Repeating this process we obtain Hence, we have , and implies .□
Therefore, we apply Condition (1) of theorem
If suppose that
then from Inequality (4), we obtain
that implies
And from property of
F have
That is a contradiction based with property of
So we have that
Then, using the result (5), Inequality (4) yields
Repeating this in general we obtain
As per above we have
since
Hence, in view of property of F we obtain
Next, we show that
Let we suppose that
Then, by Lemma 3, there exist
and sequences
and
of positive integers with
such that
and
,
,
From (7) and (8) we observe and
Hence, applying Condition (1), we obtain
that implies
By taking the limit superior as
along (10), we write
From (11), by using Lemmas 3 and result (7), we conclude
Hence, the acquired inequality
is a contradiction since
and
The sequence
is a
-Cauchy sequence such that
Since
is a b-complete
-m.l.s, there is
, such that
On the contrary, there exists
with
Using (14) and (17), we conclude
and inequality above implies that
that is a contradiction due to (16). Hence, Inequality (15) is true.
So, we are in conditions to apply (1), and we have
Taking limit superior as
in Inequality (18) and keeping in mind (7), (14), and Lemma 3, we obtain
that implies
Inequality (19) leads to a contradiction, so and is a fixed point of function . Also, the uniqueness of the fixed point can be proved easily from Condition (1) of the theorem.
Let there be two different fixed points named with
Hence, being in the conditions of theorem and in view of (14), Inequality (1) implies
From (20), we obtain
which implies
The above inequality implies that is a contradiction due to property of . Therefore, the fixed point is unique.
Example 4. Let be a complete -m.l.s with parameter , where the -m.l. distance is given as . Define the mappings and by Clearly, the mapping
is
-admissible. And for
, we have
Take
such as
,
and
. We see that for
that
, we have
and one can compute:
Then, is a generalized Suzuki-type -contraction, and assumptions of theorem 1 are satisfied. Hence, is the unique fixed point for .
Theorem 2. Let be a complete -m.l.s with parameter and a generalized -admissible Suzuki-type -contraction. Assume that
- J1.
is -admissible and there exists with
- J1.
is regular and for every sequence in S such that for all we have for all with
Then, has a fixed point, and it is unique if , for all
Proof. Using J1, we define the Picard sequence
for
induced by function
with initial point
with
From the theorem above, it is concluded that the sequence
is a
-Cauchy sequence such that
And there is
, such that
In the same, we can show that
Since
S is regular, there exists a subsequence
of
such that
for all
k ∈ N. Similarly holds
Therefore, applying (1), we have
Taking limit superior in Inequality (22) and keeping in mind lemma 3, we obtain
which implies
Inequality (23) leads to a contradiction, so and is a fixed point of function . □
Let be two different fixed points named with .
Hence, Inequality (1) can be written as
From (24), we obtain
that is
which implies
that is a contradiction due to property of
. Therefore, the fixed point is unique.
Corollary 1. Let be a complete -m.l.s with parameter and If there exist such thatimpliesfor all
,
where for some
then has a unique fixed point in
Proof. Corollary can be obtained from Theorem 1 by taking □
Now we will propose some new results belonging to the class of Suzuki -contractions.
Theorem 3. Let we have a -m.l.s with parameter , and If is an Hardy–Rogers Suzuki- type interpolative -contraction, satisfying conditions , then has a fixed point in . And it is unique if , for all
Proof. It is a consequence of Theorem 1 by taking as ; where and □
Theorem 4. Let we have a -m.l.s with parameter , and If there exist such that hold condition and:impliesfor all , ,
and .
Proof. It comes from Theorem 1 by taking as ; where □
Theorem 5. Let be a -m.l.s with parameter , and mappings . If the following conditions are satisfied
- -
There exists with
- -
is -admissible mapping and satisfies -property;
- -
is a Hardy–Rogers Suzuki –type
-order hybrid
-contraction.
Then, has a fixed point in . And it is unique if , for all
Proof. The
can be represented as
where
,
with
. Hence, the proof can be classified in Theorem 1. □
Corollary 2. Let be a -m.l.s with parameter , and . If there exist such thatimpliesfor all
,
,
and
such that
whereThen, has a fixed point in .
Proof. It can be derived from Theorem 5 by taking □
Corollary 3. Let be a -m.l.s with parameter , and , . If there exist such that hold condition andimpliesfor all , and such that Proof. The proof is called done by Theorem 1, if we take
as
and
Corollary 4. Let be a -m.l.s with parameter , and . If there exist such that:impliesfor all , and such that Proof. It is generated by corollary 3 by taking □
4. Application
The study of the existence and finding of the solution of differential and integral equations is a longstanding problem, so one of the main tools of the solution is developed and consists of the application of the fixed point method. Many researchers have employed various contractions in different metric spaces to define the necessary conditions for a variety of types of linear and nonlinear integral equations
In this supported section, by employing
-m.l.s, the purpose is to prove the existence and uniqueness of the solution for the following integral equation of the form:
where
and given continuous functions
,
.
Consider the set of real continuous functions defined on with the -metric-like for all . The pair is a complete -m.l.s with parameter .
Define the mapping
for all
, by
Associated with the following hypotheses:
- (i)
The mapping is continuous;
There exists constant such that satisfies for ;
- (ii)
The constants and function satisfy condition for .
Theorem 6. If for the integral Equation (26) assume the assertions: (i), (ii), (iii) then, the integral Equation (26) has a unique solution in .
Proof. Solving Equation (26) is equivalent to find
which is a fixed point of function
. And for all
, and
we have
Further, taking
as
we can obtain:
Consequently, by choosing:
,
we deduce
which implies that
is a generalized Suzuki-type
-contraction. Thus, Corollary 1 is applicable and
is the fixed point of
which is the solution of the integral Equation (26). □