1. Introduction and Preliminaries
In 1977, Bernfeld et al. [
1] introduced the concept of a fixed point for mappings that have different domains and ranges, which is called PPF-dependent fixed point or the fixed point with PPF-dependence. Also, they introduced the notion of Banach type contraction and proved some important results under this contraction. Recently, some authors have established existence and uniqueness of PPF-dependent fixed point for different types of contraction mappings (see [
2,
3,
4,
5,
6]), and others interested in the applications can find PPF-dependent solutions of a periodic boundary value problem and functional differential equations which may depend upon past, present and future considerations (see [
7,
8,
9]).
A new contraction, called
F-contraction, was originally raised by Wardowski [
10] in 2012. He proved a fixed point theorem under this contraction and extended many fixed point results in a different aspect. After that, a generalization of the notion of
F-contraction to obtain certain fixed point results was given by Abbas et al. [
11]. Batra et al. [
12,
13] provided a remarkable generalization of
F-contraction on graphs and altered distances. Recently, some fixed point results for Hardy-Rogers-type self mappings on abstract spaces have been discussed by Cosentino and Vetro [
14].
A generalized multi-valued
F-contraction mapping to discuss results of fixed point theory in a complete metric space was announced by Acar et al. [
15,
16]. This idea seemed to be a very useful and powerful method in the study of functional and integral equations (see [
17]). We refer the reader to, for example [
18,
19,
20,
21,
22,
23,
24], and references therein for more information on different aspects of fixed point theorems via
F-contractions.
Definition 1 ([
10])
. A nonlinear self-mapping T on a metric space is said to be an F-contraction, if there exist and such thatwhere Γ
is the set of functions such that the following axioms hold: F is strictly increasing, i.e., for all such that
for every sequence of positive numbers iff
there exists such that
The following functions
for
are all the elements of
. Furthermore, substituting in Condition (
1) these functions, we obtain the following contractions known in the literature, for all
with
and
From the axiom
and Condition (
1), one can conclude that every
F-contraction
T is a contractive mapping and hence automatically continuous.
Theorem 1 ([
10])
. Let be an F-contraction on a complete metric space , then it has a unique fixed point Moreover, for any the sequence converges to 2. Preliminaries
Let
E be a real Banach space with the norm
; given a closed interval
in
we consider a Banach space
of continuous
E-valued functions defined on
I, endowed with the supremum norm
defined by
for all
For a fixed element
the Razumikhin or minimal class of functions in
is defined by
It’s obvious that every constant function from I to E belongs to
Definition 2. Let A be a subset of Then
(i) A is said to be topologically closed with respect to the norm topology if for each sequence in A with as implies
(ii) A is said to be algebraically closed with respect to the difference if when
Definition 3 ([
1])
. A mapping is said to be a PPF-dependent fixed point or a fixed point with PPF-dependence of mapping if for some Example 1 ([
25])
. Let be defined byHence, T is a contraction with a constant Let for all Since we have: ξ is a PPF fixed point with dependence of T.
Definition 4 ([
1])
. Let be two operators. A point is called a PPF-dependent common fixed point or a common fixed point with PPF-dependence of T and S if for some Clearly, if we take then a PPF-dependent common fixed point of T and S collapses to a PPF-dependent fixed point.
Definition 5 ([
26])
. Let and A point is called a PPF-dependent coincidence point or coincidence point with PPF-dependence of P and Q if for some Let
be a collection of all non-empty closed bounded subsets of
E, and
H be the Hausdorff metric determined by
Then, for all
,
where
In 1989, Mizoguchi and Takahashi [
27] extended Banach fixed point theorem in a complete metric space. After that, Farajzadeh et al. [
28] extended the above results by introducing the following definitions:
Definition 6. Let . A point is called a PPF fixed point of T if for some
Please note that if is a single-valued mapping, then a multivalued mapping can be obtained by , for all Hence, the set of PPF-dependent fixed points of S coincides with the set of PPF-dependent fixed point of T.
Definition 7. A point is called a PPF-dependent coincidence point of g and T if for some where is a single valued mapping and is a multi-valued mapping.
Notice that, the Definitions 6 and 7 are coincide if we take g equal to the identity mapping.
3. PPF-Dependent Fixed Point
In this section, we begin with introducing our new concept of a multi-valued generalized F-contraction and some important results in the setting of Banach spaces are given by using it.
Definition 8. The mapping is called a multivalued generalized F-contraction if and there exists such thatfor all The following example shows that a multivalued generalized F-contraction is not necessary in a multivalued contraction.
Example 2. Let be a real Banach space with usual norm and let Define the mapping by We prove that T is a multi-valued generalized F-contraction with respect to with It’s clear that for all , we consider the following two cases:
Case 1.For and , we have Case 2.For, we get This implies that T is a multi-valued generalized F-contraction.
Then T is not a multi-valued contraction.
Now, we present our first theorem concerning with the existence of a PPF-dependent fixed point for a multi-valued generalized F-contraction in a Banach space.
Theorem 2. Suppose that is a multivalued generalized F-contraction. Then, T has a PPF-dependent fixed point in
Proof. Let
, since
and
is closed, there exists
such that
. Choose
such that
If
then
is a PPF-dependent fixed point, so the proof is complete. Let
then there exists
such that
On the other hand, from
and Condition (
1), we can write
Also, since
there exists
such that
and
Then from Condition (
2), we have
If we continue recursively, then we obtain a sequence
in
such that
Since
is algebraically closed with respect to the difference, we have
So, by Condition (
2), one can write
If
for all
, then
is a PPF-dependent fixed point of
T, so the proof is complete. Thus, suppose that for every
Denote
for
. Then
using Inequality (
4) we prove the following:
From Inequality (
5), we have
So by
one can write
Applying there exists such that
By Inequality (
5), we get for all
Letting
in Inequality (
6), we obtain that
From Equation (
7), we observe that
for all
So, we have
To prove that is a Cauchy sequence in consider such that
Using the triangular inequality and Formula Inequality (
8), we have
Since the series is convergent, so the limit as we get This yields that is a Cauchy sequence in Completeness of yields that converges to a point that is
From Condition (
2), for all
with
we get
and so
for all
Then
Passing to limit
we obtain that
that is
Hence
T has a PPF-dependent fixed point in
☐
Please note that Theorem 2, is algebraically closed, so is closed for all If we choose to be compact, thus, we can present the following problem: Let be the set of all continuous E-valued functions and be a multi-valued generalized F-contraction. Does T has a PPF-dependent fixed point in By adding a condition of we can give a partial answer to this problem as follows:
Theorem 3. Suppose that is a multi-valued generalized F-contraction. Assume that is topologically closed and algebraically closed with respect to the difference. Assume also that F satisfies
for all with
Then, T has a PPF-dependent fixed point in
Proof. Let
since
and
is compact, there exists
such that
Choose
such that
If
then
is a PPF-dependent fixed point, so the proof is complete. Let
then there exists
such that
On the other hand, from
and Condition (
1), we have Inequality (
3). Applying
we can write (note
and so from Inequality (
3), we have
Then from Inequality (
9) there exists
such that
If
we are finished. Otherwise, by the same way we can find
such that
We continue recursively, then we obtain a sequence
in
such that
for some
and
for all
. We can finish the proof by a similar technique of Theorem 2. ☐
We know that, F satisfies if it satisfies and is right-continuous.
4. PPF-Dependent Coincidence Point
In this section, we prove the existence of PPF-dependent coincidence points for a pair of mappings (single and multivalued) under the Condition (
2) by replacing the condition of
is topologically closed with equivalent conditions in a Banach space.
Theorem 4. Let be a single valued mapping and be a multi-valued mapping satisfying the following conditions:
(i)
(ii) is complete,
(iii)for all and for some Assume that is algebraically closed with respect to the difference. Then T and f have a PPF-dependent coincidence point in
Proof. Let
since
and
we can choose
such that
If
then
is a PPF-dependent coincidence point of
f and
T, so let
then there exists
such that
On the other hand, from
and
we have
From Inequality (
10), we can write
Also, since
there exists
such that by Inequality (
11), we get
By the same above technique, we can get a sequence
such that
in
for all
Since
is algebraically closed with respect to the difference, it follows that
So by Inequality (
10), we have
As in the proof Theorem 2, by taking
for all
, we obtain
is a Cauchy sequence in
The completeness of
leads to
is a convergent sequence. Suppose that
for some
So, there exists
such that
that is
Hence, for each
and for all
with
we get
Taking the limit as we have Hence, the proof is completed. ☐
In the following theorem, we prove the existence and uniqueness of a PPF-dependent common fixed point for two multi-valued generalized F-contraction in Banach space.
Definition 9. Let be a Banach space and be multi-valued mappings. The pair is called a pair of new multivalued generalized F-contractions if and there exists such thatwherefor all and for some Theorem 5. Let be a Banach space and be a pair of new multi-valued generalized F-contractions (12). Assume that is algebraically closed with respect to the difference. Then T and S have a PPF-dependent fixed point in Moreover, if T or S is a single-valued mapping, then a fixed point with PPF-dependence is unique. Proof. Let
be arbitrary, since
is nonempty-closed, there exists
such that
Choose
such that
and
Again, taking
let
Choose
such that
and
Continuing in this way, by induction, we obtain
such that
Then from Condition (
1), with
we have
for all
where
If
then
which is a contradiction due to
. Therefore
By using Inequalities (
13) and (
14), we get
Repeating these steps, we can write
Similarly, we obtain that
Inequalities (
15) and (
16) can jointly by written as
Taking limits with
on both sides of Inequality (
17), we have
since
then
By Inequality (
17), for all
we obtain
Considering Equalities (
18) and (
19) and letting
in Inequality (
20), we get
Since Equality (
21) holds, there exist
such that
or,
From Inequality (
22) we get that
is a Cauchy sequence in
Since
is complete, there exists
such that
By the Condition (
12), for all
and for some
with
where
Taking limit
and using Equality (
23), we can write
Since
F is strictly increasing, Equality (
24) implies
Taking limit
and using Equality (
24), we have
which is a contradiction, hence
or
Similarly, using Equality (
23) and the inequality
we can show that
or
. Hence
S and
T have a PPF-dependent fixed point in
We next prove that if
T is a single-valued mapping, the PPF-dependent fixed point of
S and
T is a unique. Assume that
is another PPF-dependent fixed point of
S and
By using Condition (
12), we have
Hence,
where
this yields,
which is a contradiction. Therefore
or
This completes the proof. ☐
In the following example, we justify requirements of Theorem 5.
Example 3. Let with respect to the norm and . Define the multi-valued mappings as follows: Define the function by for all
By Condition (12) with we havealso, Now, we present two cases as follows:
Case 1.If and then we getwhich implies that Case 2.Similarly if and we have Hence, all the axioms of Theorem 5 are satisfied, so have a PPF-dependent fixed point.
Please note that Theorem 5 remains valid if we substitute
defined in Definition 9 by any of the following formulas:
5. Application to A System of Integral Equations
No one can deny that fixed point theory has become the most wide spread in functional analysis because of its great applications, especially in differential and integral equations (see [
29,
30,
31]). Accordingly, we will apply the results we have obtained to find the existence and uniqueness of a solution of nonlinear integral equations.
Let
and
be two closed bounded intervals in
for reals
and
ℵ denote the space of continuous real-valued functions defined on
. We define the supremum norm
by
It is known that ℵ is a Banach space with this norm.
For fixed
define a function
by
where the argument
a represents the delay in the argument solution.
Consider the following nonlinear integral equations:
for all
Now, we prove the following theorem to ensure the existence of a common solution of our problems (
26).
Theorem 6. System (26) has only one common solution defined on if the following conditions hold:
,
suppose thatfor all . Proof. We define a norm on
by
we obtain that,
Next we show that
is a Banach space. Let
be a Cauchy sequence in
It is easy to see that
is a Cauchy sequence in
This implies that
is a Cauchy sequence in
for each
Then
converges to
for each
Since
is a sequence of uniformly continuous functions for a fixed
is also continuous in
So the sequence
converge to
Therefore
is complete, hence,
is a Banach space.
After that, we define the multi-functions
by
for all
. Suppose there exist
such that
for all
and
where
From the assumptions
and Functions (
27), we can write
This implies that
hence,
this is equivalent to
or
which further implies that
This implies that
is a pair of multi-valued generalized
F-contraction for
Moreover, the Razumikhin
is
which is topologically closed and algebraically closed with respect to difference. Now all hypotheses of Theorem 5 are automatically satisfied with
. Therefore, there exists a PPF-dependence coincidence point
of
T and
S that is,
. Hence, the integral Equation (
26) has a solution. This completes the proof. ☐
Questions
(i) Are the results in Theorems 2 and 3 still true when the norm closedness for is replaced by weak closedness or weak closedness (for dual Banach spaces)?
(ii) Is there some way to improve the results of Theorems 4 and 5 to more than two or a family of mappings?