1. Introduction
Fixed-point techniques in complete metric spaces (CMSs) became popular in 1922 after Banach presented their principle [
1]. This technique has particular resonance in many important disciplines, such as topology, dynamical systems, differential and integral equations, economics, game theory, biological sciences, computer science and chemistry [
2,
3]. As of the importance of this approach, it became the main controller in the study of the existence and uniqueness of the solution to many differential and integral equations [
4,
5,
6].
Bhaskar and Lakshmikantham [
7] introduced the concept of mixed monotone maps and coupled fixed points. Consequently, many authors established coupled fixed point results for contractive mappings under suitable conditions in partially ordered metric spaces (POMSs) with some important applications. For more details, see [
8,
9].
In 2011, Berinde and Borcut [
10] initiated the notion of triple fixed points (TFPs) and established some TFP results for contractive mappings in POMSs. Afterward, many investigators established TFP theorems for contraction mappings in various spaces. For more contributions in this regard, see [
11,
12,
13,
14,
15].
In 1973, Geraghty [
16] extended the Banach contraction principle [
1] by replacing a contraction coefficient with a function satisfying certain conditions. Later, the results of [
16] were extended by Hamini and Emami [
5] in POMSs. In 2013, the idea of
-Geraghty contraction type mappings and some nice fixed point consequences were established in a CMS by Cho et al. [
17]. As many scholars are interested in this regard, it is sufficient to mention [
18,
19].
In 1989, Blassi and Myjak [
20] defined another approach to fixed points, the so-called well-posedness of a fixed point problem. The concept of well-posedness of a fixed point problem for a single valued mapping has evoked much interest for several authors, see [
21,
22].
In this manuscript, the existence and uniqueness of a TFP are established for Geraghty-type contraction mappings under appropriate conditions. Furthermore, an example is given to support our results. Moreover, the well-posedness of a TFP problem and dominated mappings are obtained. As an application, the existence solution to a system of differential equations is derived.
2. Preliminaries
In this section, we present some notations and basic definitions that are useful in the sequel. Assume that and are two non-empty sets and ℜ is a relation from to i.e., Here, the pair or means is ℜ related to the domain of ℜ defined by the set for some , the range of ℜ defined by the set for some and the inverse of ℜ is which is defined by
A relation ℜ from to is said to be a relation on Suppose that ℜ is a relation on . The relation ℜ is called directed if for given there is so that and . If the relation ℜ is reflexive, anti-symmetric and transitive, then it is called a partial order relation on .
Definition 1. Assume that is a CMS with a binary relation ℜ on it. We say that Ω has regular property if for each sequence converging to with we have for each ϖ or if then for each ϖ.
Definition 2 ([
10]).
Let Ω be a non-empty set. A trio is said to be a TFP of the mapping if and Example 1. Assume that and is a mapping described as Then, there is a unique TFP of whenever
Now, in order to achieve the goal of this paper, we present the following auxiliary functions.
Assume that is a class of all functions so that, for each sequence implies
Let be a class of all functions so that, for any sequence implies
Clearly, the class is more extended than , that is contains .
The example below illustrates this containment.
Example 2. Let be a function defined by It is clear that and does not exist (the value of the limit is not unique). So The sequence is bounded and has two subsequences and Thus, the limits are and Therefore, and whenever Hence,
3. Main Results
In this part, we obtain some TFP results under certain conditions. Furthermore, illustrative examples are given to support the theoretical results. Now, we begin with the following definition:
Definition 3. Suppose that with a binary relation ℜ on it. A mapping is called an dominated mapping if for all Example 3. Suppose that and the mapping is defined by Assume that a binary relation ℜ on Ω
is described by Hence, r is an dominated mapping.
Problem Suppose that
is a metric space. We consider the problem of finding a TFP
of the mapping
so that
Definition 4. The problem is said to be well-posed if the following conditions hold:
- (W1)
the point is a TFP of
- (W2)
and as whenever is any sequence in such that is finite and
Now, we establish our first main result as follows:
Theorem 1. Assume that is a CMS with a transitive relation ℜ on it so that Ω
has regular property. Let be the dominated mapping and there is . If for so that or or , and the inequality below holds Then, r has a TFP.
Proof. Let
be an arbitrary point. Define three sequences
and
in
by
As
r is
dominated, we find
From (
2)–(
5), we obtain
where
Applying (
6) in (
7), we have
Analogously, we can write
and
By adding (
8)–(
10), we have
Assume that
Thus, by (
11), one obtains
a contradiction. Thus,
for each
which means that a sequence of positive real numbers
is decreasing. Hence, there is
so that
Based on (
11), we obtain
If possible, suppose that
Taking the limit supremum on both sides of (
12), we have
which leads to
which implies that
Using the property of
ℑ, we have
Hence,
which is contrary to our assumption. Based on the foregoing, we can write
and
Now, we shall show that
and
are Cauchy sequences. Assume on the contrary that either
or
or
is not a Cauchy sequence. Then, either
Therefore,
which implies that, for each
, we can find subsequences
and
of positive integer with
>
so that, for every
and
As
in the above inequality and by (
13), we find
Taking the limit as
in (
17) and (
18) and by (
13) and (
16), we obtain
Once again, letting
in the above inequality and using (
13), (
16) and (
19), we conclude that
From (
4) and the transitivity hypothesis of
ℜ, one finds
Applying (
2), we find
where
Similarly, one can write
and
Combining (
21), (
23) and (
24), we can write
When
in (
22) and by (
13), (
19) and (
20), we have
Taking the limit supremum in (
25) and applying (
14) and (
26), we find
Based on the property of
ℑ, we can write
which leads to
Again, the property of
ℑ implies that
that is,
This contradicts our assumption. Therefore,
and
are Cauchy sequences in
. As
is complete, there are
so that
It follows from (
27)–(
29) and (
2) that
Taking the limit as
in (
30), we have
Analogously, we obtain
since
Then, we find
which implies that
Therefore, a trio is a TFP of This finishes the proof. □
The following result is released, if we take ℜ as a partially ordered relation:
Corollary 1. Assume that is a CMS with a partial order ⪯ on it so that ⅁ has regular property (means if is a monotone convergent sequence with limit then or according to the sequence is increasing or decreasing). Let be a dominated map (means for each and and there is so that the condition (2) of Theorem 1 is fulfilled for all with ( and ) or ( and ). Then, r has a TFP. If we take ℜ is the universal relation, that is, in Theorem 1, then we obtain the result below:
Corollary 2. Suppose that is a mapping defined on a CMS Suppose also there is so that (2) of Theorem 1 is verified for all . Then, r has a TFP. In order to obtain the uniqueness of a TFP of r, we present the following theorem:
Theorem 2. In addition to the stipulations of Theorem 1, assume that both ℜ and are directed. Then, r has a unique TFP.
Proof. Based on
Theorem 1, the set of TFPs of r is non-empty. Suppose that
and
are two TFPs of
i.e.,
Our goal is to prove
and
Using the directed property of
ℜ and
there are
and
so that
and
this implies that
and
. Take
and
Therefore,
and
Assume that
In the same way as the proof of Theorem 1, we built three sequences
and
as follows:
for all
. As
r is
dominated, we have
Again, following the same mechanism used in Theorem 1, the sequences
and
are Cauchy sequences in
and there are
,
so that
Now, we show that
and
which implies that
Suppose, on the contrary, that
We claim that
Since (
), (
) and (
, by the transitivity property of
we find
and
Hence, our assumption holds for
Suppose that (
34) is true for some
which implies that
and
By (
32),
and
The transitivity property of
ℜ implies that
and
. Hence, our claim is proved. Using (
2) and (
34), we find
where
Adding (
35), (
37) and (
38), we find
Passing limit in (
36) as
and applying (
33), we obtain
Taking the limit supremum in (
39) as
, using (
33) and (
40), one can write
that is
which implies that
From the property of
ℑ, we can write
which contradicts with our assumption that
Hence,
that is
With the same manner, we can show that
By (
43) and (
44), we find
and
Therefore, the TFP is unique. □
We support our study by the example below.
Example 4. Let and be a usual metric. Define a function by if and if Define the mapping by and a binary relation ℜ on Ω as follows: It is easy to see that Ω is regular with respect to ℜ and the mapping is an dominated. Suppose that so that , ) or , or , ). Therefore, and Hence, we have Therefore, all requirements of Theorem 1 are satisfied and (0,0,0) is a TFP of
4. Well-Posedness
We begin this part with the following assumption:
If
is any solution of the problem
—that is, by (
1) and
is any sequence in
for which
then
and
for all
Theorem 3. In addition to the assumption of Theorem 2, the TFP problem is well-posed, provided that the hypothesis is satisfied.
Proof. Theorem 2 says that the point
is a TFP of
This means the point
is a solution of (
1), that is
and
Let
be any sequence in
such that
is finite, and
Then, there is
so that
and also by the hypothesis
and
for all
By (
2), we find
where
Similarly, we can obtain
and
Adding (
45), (
47) and (
48), we have
Taking the limit supremum as
in (
46), we have
Hence,
Taking the limit supremum as
in (
49) and using (
50), we can write
which implies that
Therefore,
Using the property of
we obtain that
that is
which is a contradiction. Hence, we find
Then, we have
which implies that
It follows that
which leads to
,
and
as
Hence, the TFP problem
is well-posed. □
5. Some Results for -Dominated Mappings
In this section, we introduce -dominated mappings and discuss the extension of mappings equipped with admissibility conditions in the TFP theory.
Definition 5. Let Ω
be a non-empty set and be a given mapping. A mapping is called an dominated mapping if for all we have Definition 6. Let Ω
be a non-empty set and be a given mapping. We say that α has triangular property if for each Definition 7. Let be a metric space and be a mapping. We say that Ω has regular property if for each convergent sequence with limit for all ϖ implies for all
Theorem 4. Suppose that is a CMS and is a mapping so that Ω
fulfills regular property and α has triangular property. Let be an dominated mapping and there is so that (2) of Theorem 1 is fulfilled for all with and Then, r has a TFP. Proof. Define a binary relation
ℜ on
by
Then
- (i)
, and leads to and
- (ii)
and leads to and for all
- (iii)
and leads to and whenever is a convergent sequence with and
Therefore, all assumptions boil down to the hypotheses of Theorem 1. Hence, according to Theorem 1, the map r has a TFP in . □
6. Solving a System of Differential Equations
In this section, we apply Theorems 1 and 2 to discuss the existence and uniqueness solution for the following differential equation:
for each
Problem (
51) is equivalent to the following integral system:
for all
where ℧ is the Green’s function defined by
Suppose that
is the space of all real valued continuous functions defined on
Define a metric
⅁ by
Clearly,
is a CMS. Let
be equipped with the universal relation
that is,
for all
Define a mapping
by
Solving system (
51) is equivalent to finding a unique solution to the mapping (
53). Now, system (
51) will be considered under the following postulates:
- (H1)
The function is continuous;
- (H2)
where is a fixed number;
- (H3)
For all
, we have
where
Our main theorem in this part is as follows:
Theorem 5. Under assumptions –, system (51) has a unique solution in Proof. Since
U is the universal relation on
, from the definition of
r, we have
for all
This means that
r is
dominated mapping. Furthermore, every universal relation is a binary relation, so
has
regular property.
Now, from our hypotheses
and
, for each
we have
Applying the condition
we find
It follows that
where
and
Therefore, all hypotheses of Theorems 1 and 2 are fulfilled. Hence, the problem (
51) has a unique solution on
□
7. Conclusions
Fixed-point techniques are considered the backbone of mathematical analysis because of their many applications. Hence, this method has attracted many authors who are interested in this direction. Amongst the interesting applications is the study of algorithms and what they mean by convergence and divergence in the field of optimization, game theory, ordinary and fractional differential equations, differential and integral equations, and many other applications.
In our manuscript, we investigate the existence and uniqueness of TFPs for Geraghty-type contraction maps under appropriate assumptions. Furthermore, the main results are supported by an example. In addition, well-posed and
-dominated mappings for the TFP problem are presented. Finally, the existence solution to a system of differential equations is derived. As future work, motivated by the work of [
23,
24], the main results of this article can be generalized to
tuple fixed point theorems.