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Abstract
In this paper, motivated by the recent work [Journal of Nonlinear Sciences and Applications, 10(4):1544–1537] some generalized nonlinear contractive conditions via implicit functions and α-admissible pairs of multi-valued mappings in the setting of b-metric like spaces have been introduced. Some common fixed point results for such mappings in this framework have been provided. Then, some corollaries and consequences for our obtained results are given. Our results are the multi-valued versions of [Journal of Nonlinear Sciences and Applications, 10(4):1544–1537]. An example also is provided to support our obtained results. The presented results generalize and extend some earlier results in the literature.
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Fixed point theory is one of mathematical research fields which deals with existence and uniqueness of fixed points, coincidence points and common fixed points of 1 or more than 1 mappings defined on metric spaces and generalized metric spaces such as b-metric spaces, partial metric spaces, metric like spaces etc. The obtained results will apply to find a solution of integral equations, differential equations and matrix equations.
1. Introduction
Fixed point theory is one of applicable mathematical research fields. The obtained results in this field will apply to find solutions for integral equations, differential equations and matrix equations. Before starting the topic of this paper it is worth to recall some recent works in fixed point theory. For instance, Xu, Tang, Yang, and Srivastava (Citation2016) obtained sharp estimates of the classical boundary Schwarz lemma involving the boundary fixed points for holomorphic functions which map the unit disk in C into itself. Also, Srivastava, Bedre, Khairnar, and Desale (Citation2014a, Citation2014b) obtained some hybrid fixed point theorems of Krasnoselskii type, which involve product of two operators in partially ordered normed linear spaces and applied their results to proving the existence of solutions for fractional integral equations under certain monotonicity conditions.
The concepts of -metric spaces (initiated by Bakhtin, Citation1989, see also Czerwik, Citation1998) and partial metric spaces (introduced by Matthews, Citation1994) are two of the most important generalizations of the notion of a metric space. Amini-Harandi (Citation2014) introduced the metric like space as an extension of the concept of a partial metric space. Earlier this notion was known as dislocated metric space studied by Hitzler and Seda (Citation2000). The concept of b-metric like space which is given by Alghamdi, Hussain, and Salimi (Citation2013) (see also Hussain, Roshan, Parvaneh, & Kadelburg, Citation2014) generalized the notions of metric space, b-metric space, partial metric space, metric like space and partial b-metric space.
2. Preliminaries
In this section, we give some notions and results that will be needed in the sequel.
Definition 4.1. (Bakhtin, Citation1989) Let be a nonempty set and
be a constant real number. The function
is called a
-metric on
if the following conditions hold:
iff
for all
;
for all
;
for all
.
In this case, is called a
-metric space with parameter
.
Definition 4.2. (Alghamdi et al., Citation2013) Let be a nonempty set and
be a constant real number. The function
is called a
-metric like on
if the following conditions hold:
implies that
for all
;
for all
;
for all
.
In this case, is called a
-metric like space with parameter
.
Definition 4.3. (Alghamdi et al., Citation2013) Let be a
-metric like space and let
be any sequence in
and
. Then:
The sequence
is said to be convergent to
, if
The sequence
is said to be a Cauchy sequence in
if
exists and is finite.
is said to be complete if any Cauchy sequence
in
be convergent to some
.
Lemma 4.4. (Hussain et al., Citation2014) Let be a
-metric like space and
be a sequence in
which is converges to some point
with
. Then, for each
,
Recall that is said to be sequentially continuous at
if for any sequence
in
converging to
, we have
, that is,
. Also,
is called sequentially continuous on
if it is sequentially continuous at each
.
Now, take and denote by
the set of all natural numbers. Let
be the set of all functions
satisfying
(ѱ1) is strictly increasing,
(ѱ4) for each
.
It is easy to check that for any , then
for all
.
Now, we introduce the following set of functions.
Definition 4.5. Let be a constant real number. Denote by
the set of all functions
such that
(F1) is nondecreasing in variable
and nonincreasing in variables
,
(F4) there exists such that for all
,
implies that
and
implies that
.
In the following examples we give two examples of functions in .
Example 4.6. , where
such that
.
Example 4.7. , where
.
Samet, Vetro, and Vetro (Citation2013) introduced an important notion of -admissible mappings as follows.
Definition 4.8. Let be a nonempty set and
be a given mapping. Then,
is said to be
-admissible whenever there exists
such that
for all .
The above notion used to prove many important results in literature (see Ali, Kamran, & Karapnar, Citation2014; Jleli, Karapnar, & Samet, Citation2013b, Citation2013a; Karapnar & Samet, Citation2014; Pansuwan, Sintunavarat, Parvaneh, & Cho, Citation2015). Mohammadi, Rezapour, and Shahzad (Citation2013) extended the -admissible notion to multi-valued mappings as follows.
Definition 4.9. Let be a nonempty set,
be a given multi-valued mapping and
. Then,
is said to be
-admissible whenever for any
and
with
, one has
, for all
.
Recently, Aydi (Citation2015) generalized Definition (4.8) for a pair of mappings as follows.
Definition 4.10. Let be a nonempty set,
be two given mappings and
. Then, the pair
is said to be a generalized
-admissible pair whenever
for all .
Also, Aydi, Velhi, and Sahmim (Citation2017) introduced a pair of -implicit contractive mappings as follows and proved common fixed point results for such mappings.
Definition 4.11. Let be a
-metric like space,
be two given mappings and
. Then,
is said to be an
-implicit contractive pair of mappings whenever there exists
such that
for all .
In this paper, we introduce -implicit contractive pair of multi-valued mappings on
-metric like spaces and establish common fixed point results for our presented mappings.
Denote by the set of all nonempty closed bounded subsets of
. Define the Pompeiu-Hausdorff metric
induced by
on
as follows
for all where
. An element
is said to be a fixed point of a multi-valued mapping
whenever
.
3. Main results
In this section, we provide our main results. Firstly, we give the following definitions.
Definition 5.1. Let be a nonempty set,
be two given multi-valued mappings and
. The pair
is said to be a generalized
-admissible pair whenever
(α1)
(α4)
Definition 5.2. Let be a
-metric like space,
be two given multi-valued mappings and
. Then,
is said to be an
-implicit contractive pair of multi-valued mappings whenever there exists
such that
for all .
Theorem 5.3. Let be a complete
-metric like space and
be two given
-implicit multi-valued mappings. Suppose that
is a generalized
-admissible pair,
there exist
and
such that
and
,
and
are sequentially continuous on
.
Then, there exists such that
Assume in addition that
for all
satisfying (3.4),
satisfies
, where
is as follows:
Fγ if for all
, then there exists
such that
.
Then, is a common fixed point of
and
, that is,
and
.
Proof. By assumption, there exist and
such that
and
. Applying (5.1), we have
From , we get
From ,
. If
, then
and so
is a fixed point of
and
which yields that
. Thus,
is a common fixed point of
and
. Thus, we may assume that
. Now, we have
. Thus, there exists
such that
. From (
1), we get
and
. Applying (5.1), with
and
, we have
From , we get
From and since
is strictly increasing, we obtain that
. Thus, there exists
such that
. From (
4), we get
and
. Continuing this process, we obtain a sequence
in
such that
and
for all . Thus, for any
with
, we have
as . Thus,
is a Cauchy sequence. Since
is complete, there exists
such that
, that is,
Now, we show that is a common fixed point of
and
. We have
and
Using the sequential continuity of , we have
and using the sequential continuity of , we have
Now, we have
and
Using (5.3), (5.4) and (5.5), we see that (5.2) holds. Thus, by hypothesis (iv), . Applying (5.1) for
, we get
On the other hand, from (5.4) and (5.5),
and
Applying (F1) in the fifth and sixth variables, we obtain that
From , we get
The above inequality holds unless which yields that
. Thus,
. From (5.4) and (5.5), we get
, that is,
and
.
Taking in Theorem 3.3, we derive the following result.
Corollary 5.4. Let be a complete
-metric like space and
be a given multi-valued mapping. Assume that there exists
and
such that
for all . Suppose that
is a generalized
-admissible pair,
There exist
and
such that
and
,
is sequentially continuous on
Then, there exists such that
Assume in addition that
for all
satisfying (5.6),
satisfies
.
Then, is a fixed point of
, that is,
.
Considering the -metric case in Theorem (5.3) we have the following results.
Corollary 5.5. Let be a complete
-metric space and
be two multi-valued mappings such that there exists
such that
for all . Suppose that
is a generalized
-admissible pair,
There exist
and
such that
and
,
and
are sequentially continuous on
.
Then, there exists such that
and
.
Proof. Following the proof of Theorem (5.3), we find a sequence in
and
such that (5.2) holds. Since in a
-metric space, we have
and
, thus, from (5.2), we get
, that is,
and
.
Corollary 5.6. Let be a complete
-metric space and
be a multi-valued mapping such that there exists
such that
for all . Suppose that
is a generalized
-admissible pair,
There exist
and
such that
and
,
is sequentially continuous on
.
Then, there exists such that
.
Now, let the function used in Theorem (5.3) be the function presented in Example (4.7). Then, we have the following results.
Corollary 5.7. Let be a complete
-metric like space and
be two multi-valued mappings such that
for all , where
. Suppose that
is a generalized
-admissible pair,
There exist
and
such that
and
,
and
are sequentially continuous on
.
Then, there exists such that (5.2) holds. Assume in addition that
for all
satisfying (5.2).
Then, and
.
In the following result, we omit the continuity condition of and
and replace it by the following assertion.
(H) For each sequence in
with
and
for all
and
, then, we have
and
for all
.
in this case, we say that is double-sided Regular.
Also, we replace Definition (4.5) by the following.
Definition 5.8. Let be a constant real number. Denote by
the set of all continuous functions
such that
(G1) is nondecreasing in variable
and nonincreasing in variables
,
(G4) there exists such that
for all
, implies that
, and
implies that
.
Remark 5.9. In the case that , the continuity condition is not a necessary condition, but in Definition (5.8) the continuity condition of
is essential. Also, the property
implies
, but the reverse is not true.
Theorem 5.10. Let be a complete
-metric like space and
be two multi-valued mappings such that there exists
such that
for all . Suppose that
is a generalized
-admissible pair,
There exist
and
such that
and
,
be double-sided Regular.
Then, there exists such that
and
. We also have
.
Proof. Following the proof of Theorem (5.3), there exists a sequence in
such that
for all , which is convergent to some point
and
From double-sided Regularity assumption, we have and
for all
. Taking
in (5.1), we have
Since , from (F1), we get
On the other hand, we have
Taking and using the continuity of
, we get
Using (G4), we obtain that which yields that
. Similarly, Taking
and
in (3.1), we have
Since , from (F1), we get
On the other hand,
Taking and using the continuity of
, we get
Using (G4), we obtain that which yields that
.
Corollary 5.11. Let be a complete
-metric like space and
be a multi-valued mapping such that there exists
such that
for all . Suppose that
is a generalized
-admissible pair,
There exist
and
such that
and
,
be double-sided Regular.
Then, there exists such that
. We also have
.
Example 5.12. Let and
. It is easy to check that
is a complete b-metric like space with
. Define
by
and
Then, for any , we have
Therefore, we have
for all , where
. It is easy to check that
is a generalized
-admissible pair,
There exist
and
such that
and
,
(H) holds.
Thus, all of the conditions of Corollary (5.7) holds and so and
have a common fixed point. Here,
and
.
Corollary 5.13. Let be a complete
-metric like space and
be two multi-valued mappings such that
for all , where
. Suppose that
is a generalized
-admissible pair,
there exist
and
such that
and
,
be double-sided Regular.
Then, there exists such that
and
.
Corollary 5.14. Let be a complete
-metric like space and
be a multi-valued mapping such that
for all , where
. Suppose that
is a generalized
-admissible pair,
there exist
and
such that
and
,
be double-sided Regular.
Then, has a fixed point.
Corollary 5.15. Let be a complete
-metric like space and
be two multi-valued mappings such that
for all , where
. Then, there exists
such that
and
.
Corollary 5.16. Let be a complete
-metric like space and
be a multi-valued mapping such that
for all , where
. Then,
has a fixed point.
Corollary 5.17. Let be a complete metric like space and
be a multi-valued mapping such that
for all , where
. Then,
has a fixed point.
Additional information
Funding
Notes on contributors
Vahid Parvaneh
The research field of all authors of this paper is the fixed point theory with its applications. The authors have published some papers in this field in recent years. They are assistant professor and faculty member in Islamic Azad university. In this paper, the authors deal with existence theorems for common fixed points of multi-valued mappings in the framework of metric like spaces. Their obtained results generalize and extend some results in this field.
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