1. Introduction and Preliminaries
In the context, the symbols , , and denote the set of all real, positive real, and natural numbers, respectively.
Many authors have generalized the Banach fixed point theorem [
1], which states: Let
be a complete metric space and
be a self-mapping on it, if for all
and
then,
has a unique fixed point and the sequence
is convergent to the same fixed point, for all
.
Some authors have worked on the right side of the inequality (
1) by replacing
with mappings, and others, instead of the underlying space, took more general spaces (see, for example [
2,
3,
4,
5] and references therein).
In 2012, Wardowski [
6] presented a new type of contraction called an
F-contraction, where
, and showed new fixed point results related with the
F-contraction. He investigated some examples to obtain a different type of contraction in the literature.
Definition 1. ([
6])
. Let be a metric space. A mapping is said to be an F-contraction if there exists and such thatwhere Σ
is the set of functions satisfying the following conditions:() 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 the elements of
. Furthermore, substituting these functions in (
2), we obtain the following contractions known in the literature, for all
with
and
Remark 1. The Equation (2) implies thatthat is, Γ
is a contractive for all such that . Hence, every F-contraction mapping is continuous. Remark 2 ([
7])
. Consider , where and . Then . By using the concept of
F-contraction, Wardowski [
6] established a fixed point theorem that improves the Banach contraction principle in a different way than in the known results from the literature.
Theorem 1 ([
6])
. Let Γ
be a self-mapping on a complete metric space satisfying the condition (2). Then Γ
has a unique fixed point Moreover, for any the sequence is convergent to In 2014, Isik [
8] extended this theorem for two mappings by introducing the following lemma:
Lemma 1 ([
8])
. Let be a complete metric space. Let Υ
and Γ
be self-mappings on it, and suppose . If there exists such thatfor all satisfying Then the mappings Υ
and Γ
have a unique fixed point. The notion of
F-contraction is generalized by Abbas et al. [
9] to obtain certain fixed point results, Batra et al. [
10,
11] to extend it on graphs and alter distances, and Cosentino and Vetro [
12] to introduce some fixed point consequences for Hardy–Rogers-type self-mappings in ordered and complete metric spaces.
Another direction, the coupled fixed point, was introduced and studied by Bhaskar and Lakhsmikantham [
13]. They studied coupled fixed point results by using suitable contraction mappings and applied their results to show the existence of solutions for a periodic boundary value problem, so it has been a subject of interest by many researchers in this direction, (see, for example [
14,
15,
16,
17,
18,
19,
20,
21]).
Definition 2. Let Ω be a nonempty set and be given mappings:
- (i)
An element is said to be a coupled fixed point of the nonlinear mapping if and
- (ii)
An element is said to be a coupled common fixed point of nonlinear mappings Υ and Γ if and .
Consistent with Jachymski [
22], let
be a metric space and
ℜ the diagonal of
. Let
be a directed graph, where the set
contains all vertices coinciding with
and the set
contains all edges of the graph containing all loops, that is,
. In addition, we assume that the graph
G has no parallel edges. We denote by
the graph obtained from
G by reversing the direction of edges. Thus,
A set of coupled fixed points of a nonlinear mapping
is denoted by
, that is,
Definition 3 ([
23])
. A mapping is called edge preserving if , then, and . Definition 4 ([
23])
. The mapping is called G-continuous if for all , and for any positive integers sequence , with , , as and , we have that Definition 5 ([
23])
. Let G be a directed graph and be a complete metric space. The triple satisfies the following:- (A1)
If for any sequence such that and , for , we have
- (A2)
If for any sequence with and , for , we have
As in paper [
23], we define
as:
Proposition 1 ([
23])
. Consider a nonlinear edge preserving mapping , then:- (i)
and implies and
- (ii)
implies and for all ;
- (iii)
implies for all
The structure of the article is as follows. In
Section 2, we obtain some coupled fixed point results under
F-contraction mappings in complete metric space without and with a directed graph. In
Section 3, some application to find analytical solutions for the coupled systems of functional and nonlinear integral equations are presented. In the final section,
Section 4, illustrative examples are discussed to support some of our results.
2. Coupled Fixed Point Results
Now we are going to prove our main results. We begin with the following known lemma:
Lemma 2. Let be a complete metric space, be a cartesian product, and be defined by Then, the pair is a complete metric space.
Here, we’ll prove the first main theorem of our paper.
Theorem 2. Let be nonlinear continuous mappings on a complete metric space such that there exists and satisfyingfor all Then Υ
and Γ
have a unique coupled common fixed point. Proof. Consider the mappings
such that
Next, we check if
and
satisfy the contractive condition (
3) appearing in Lemma 1 on a complete metric space
(see Lemma 2). For
suppose that
We can distinguish two cases:
Case 1..
Since
by using the contractive condition (
4), we have
Case 2..
Since
by using our assumption, we get
Therefore, in both cases, the contractive condition (
3) appearing in Lemma 1 is satisfied. So
and
have a unique common fixed point
this means that
Hence, is a coupled common fixed point of the mappings and .
Suppose that there exists another coupled common fixed point
such that
or equivalently,
The uniqueness of the fixed points of and completes the proof. □
Taking in Theorem 2, we can get the pivotal following result:
Corollary 1. Let be continuous mappings on a complete metric space such that there exist and satisfyingfor all Then Γ
has a unique coupled common fixed point. Now, we shall define an F-G-rational contraction mapping and some related results in a directed graph.
Definition 6. A nonlinear mapping is said to be an F-G-rational contraction if:
- (1)
Γ is edge preserving;
- (2)
there exists a number such thatfor all with
Lemma 3. Let be an F-G-rational contraction on a metric space with a directed graph G. Then for all we get Proof. Suppose that
Since
is edge preserving, we can write
Proposition 1 (i), leads to
and
By definition of
, we get
Therefore, by mathematical induction, we reach the conclusion. □
Lemma 4. Let be an F-G-rational contraction on a complete metric space with a directed graph G. Then, for each , there exist such that and as
Proof. Consider
so
and
by Lemma 3 and putting
,
and
, then we can write
that is,
and
As
in (
6), we obtain that
So, by (
), we obtain
It follows from the axiom (
) that
By (
6), for all
we obtain
Take in a count (
7), (
8), and passing
in (
9), one observes that
By (
10), there exists
such that
for all
or
Using (
11), for
we have
The convergence of the series leads to Similarly, we can write Therefore and are Cauchy sequences in The proof is terminated by the completeness of □
Theorem 3. Let be an F-G-rational contraction on a complete metric space with a directed graph G. Consider that
- (i)
Γ is G-continuous; or
- (ii)
the triple satisfies the conditions that , , and F are continuous. Then iff
- (iii)
if with and then
Proof. Consider . Then there exists such that and Hence and This yields
On the other hand, let Then there exists ; this mean that and
Let a sequence of positive integers
. Proposition 1 (ii) gives
Applying Lemma 4 on (
12), there exists
and
such that
Now, we claim that the mapping has a unique coupled fixed point.
(i) Consider that
is
G-continuous. Then we have
Applying the triangle inequality, we get
By the continuity of
and (
13), one can write
, which yields
By the same manner, we can prove that
Therefore a nonlinear mapping
has a coupled fixed point
and
(ii) Consider the triple
satisfying the properties
,
. Then, we have
Applying the mapping
F, we obtain that
Passing
in (
14), we conclude that
, that is,
i.e.,
Similarly, one can prove that
Therefore,
(iii) Suppose that
since
,
, and
is an
F-
G-rational contraction mapping, we get
A contradiction. Hence □
3. Applications
Fixed point theory is one of the cornerstones in the development of mathematics since it plays a basic role in applications of many branches of mathematics, especially in differential and integral equations (see, for example [
24,
25,
26,
27,
28]).
In this section, we study the existence of solutions for functional and nonlinear integral equations using the results proved in the previous section.
Before we present applications of our results, we need the following lemma:
Lemma 5 ([
29])
. Suppose , and let be a function defined byThen, is strictly increasing;
and is a concave function;
for ,
3.1. System of Functional Equations
Dynamic programming is one of the most important tools in studying dynamic economic models, especially in optimization problems and others. One of these trends uses the fixed point technique to find solutions of a system of functional equations arising in dynamic programming (for example, see [
30,
31,
32,
33,
34,
35,
36,
37]).
Consider the following system of functional equations:
appearing in the study of dynamic programming [
38,
39,
40,
41], where
and
S is a state space,
D is a decision space,
,
, and
.
Let
denote the set of all bounded real-valued functions on a nonempty set
S, and for any
, define
It is well known that
endowed with the sup metric
for all
, is a complete metric space.
System (
15) will be considered under the following conditions:
- (i)
and are bounded functions;
- (ii)
for arbitrary points
,
and
such that
Theorem 4. Under assumptions and , system (15) has a unique bounded common solution in . Proof. Define the operator
on
as
for all
and
. Since functions
and
are bounded, then
is well-defined.
Now, we will show that
satisfies the condition (
5) appearing in Corollary 1 with the sup metric
Let
. Then, by
, we get
where we have used the nondecreasing character of
(Lemma 5). Therefore,
This yields that
or equivalently,
This leads to
and it follows that
This tells us that
satisfies the contractive condition (
5) with
(Remark 2). By Corollary 1, there exists a unique coupled fixed point of
That is, the Equations (
15) have a unique bounded solution in
□
3.2. Nonlinear Integral Equations
In this part, we deal with Volterra-type integral equations. This part is divided into two parts: in the first part, we will apply the result of Corollary 1 to prove the existence and uniqueness of solutions of a system of nonlinear integral equations in complete metric space. In the second part, we will discuss at least one solution for a different system of nonlinear integral equations by using the results of Theorem 3 in a directed graph under the same space.
Now, consider the following first system of nonlinear integral equations:
where
and
are unknown variables and
is the deterministic free term defined for all
.
System (
16) will be considered under the following conditions:
- (1)
where is the space of all continuous functions on .
- (2)
is a continuous function satisfying
for any
and
where
and
The following results (see [
25]) are important in the sequel.
Suppose that
and let
be a function such that
and for
,
Theorem 5. Under conditions and , system (16) has a unique solution Proof. For
and
we define
by
In virtue of
and
and the fact that the operator
G is continuous on
, it is clear that if
, then
. Therefore,
Next, we check that
satisfies the contractive condition (
5) appearing in Corollary 1.
In fact, suppose that
then for all
one can write
where we have used the nondecreasing character of
(Lemma 5) and the fact that
. Therefore,
This yields that
or equivalently,
or
or
This tells us that the contractive condition is satisfied with (Remark 2).
By Corollary 1, there exists a unique coupled fixed point of a mapping
i.e., there exists a unique
such that for any
,
and this completes the proof. □
Next, consider the second nonlinear system of integral equations as the form:
where
with
Let
with the norm,
for all
. Suppose that
G is a directed graph defined by the following:
Then is a complete metric space equipped with G.
If we take , then the diagonal ℜ of is included in . On the other hand, Moreover, has the properties and . In this case and
Theorem 6. The system (17) has at least one solution as long as the following conditions are satisfied: - (i)
and are continuous functions such that - (ii)
for all with , we have ∀ ;
- (iii)
there exists such thatfor any and - (iv)
there exists such thatwhere
Proof. Define the mapping
as
Now we prove that
is
G-edge preserving. Let
with
Then we have
Similarly,
Next, the condition (iv) follows that
Finally,
where we have used the nondecreasing character of
(Lemma 5). Therefore,
This yields that
or equivalently,
or
This says us that
is an
F-
G-rational contraction with
(Remark 2). By Theorem 3, there exists at least the coupled fixed point of a mapping
that is the solution of the integral system (
17). □