1. Introduction
The fixed point theory is a beautiful mixture of analysis, topology, and geometry. Over several decades the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena. In particular, fixed point theory has been applied to cope with the solutions to problems in functional equations, ordinary differential equations, integral equations, fractional equations, and more (see [
1,
2,
3,
4,
5,
6,
7,
8,
9]). It has been applied in such diverse fields as biology, chemistry, economics, engineering, game theory, physics, and logic programming. One of the most celebrated fixed point theorems is the Banach contraction mapping principle (see [
10]) or Banach fixed point theorem, which is stated as follows.
Let
be a complete metric space. Suppose that the mapping
is a Banach-type contraction, i.e., it satisfies
for all
, where
is a constant. Then, the mapping
f has a unique fixed point in
X.
This principle has subsequently been developed further, including the presentation of the iteration sequence. In 1975, Kramosil and Michalek [
11] considered fuzzy metric space, which is a generalization of typical metric space, and extended the relevant topological concepts, leading to a great many applications in different areas; readers may refer to [
9] and the references therein. In 2007, Huang and Zhang [
12] introduced cone metric space, which greatly generalizes metric space. Moreover, they obtained fixed point theorems for Banach-type contraction, Kannan-type contraction, and Chatterjea-type contraction. Afterwards, a large number of fixed point results in cone metric spaces were presented (see [
13,
14,
15]). In 2015, cone metric properties were combined with fuzzy sets in metric space to deduce a new space called fuzzy cone metric space. This developmental contribution was established by Oner et al. [
16], who discussed topological properties and studied fixed point results with applications under certain conditions in such spaces. Utilizing this concept, several different authors (see [
8]) have considered various mappings, such as compatible and weakly compatible mappings, coupled contractive type mappings, quasi-contraction mappings, and rational contraction mappings, along with their applications, to study the existence of solutions for a number of different integral equations in fuzzy cone metric spaces.
In 2012 Rawashdeh et al. [
17] defined an ordered space called
E-metric space, which is similar to cone metric space, and proved that the contractive sequence is a Cauchy sequence in
E-metric spaces. In 2013, Pales and Petre [
18] introduced the concept of strict positivity in Riesz spaces and presented a multi-valued nonlinear fixed point theorem in
E-metric spaces, generalizing the fixed point theorems obtained by Wegrzyk [
1], Cevik and Altun [
19], Critescu [
20], and Matkowski [
21]. In 2019, Huang [
7] used semi-interior points in cones to generalize the fixed point theorems of Hardy–Rogers type contraction in
E-metric spaces.
At present, there are few research results on fixed point theorems in
E-metric spaces. In this paper, we obtain the existence and uniqueness of fixed points for Ćirić-type contraction [
22] in
E-metric spaces. In addition, we demonstrate the existence and uniqueness of fixed points for
-
-type contraction in
E-metric spaces. We consider these to be new results, as thus far there have been no fixed point results presented for Ćirić-type contraction in
E-metric spaces. In addition, it is well known that
E-metric spaces greatly generalize metric spaces, cone metric spaces, and certain other spaces. From this viewpoint, our fixed point results in
E-metric spaces have profound and far-reaching significance. Furthermore, for the sake of application, we provide the solutions to a class of differential equations.
2. Preliminaries
In this paper, without special explanations, , , , and denote the set of all real numbers, the set of all nonnegative real numbers, the set of positive integers, and the set of all nonnegative integers, respectively.
In this section, we recall several basic concepts which are needed in the following sections.
Definition 1 ([
12]).
Suppose that E is a Banach space, is the zero element of E, and P is a non-empty closed subset of E. If:
then P is called a geometrical cone in E (in short, a cone). If int, then P is said to be a solid cone, that is, intP denotes the set of all interior points of P.
We say that “⪯” and “≪” are two partial orders in
E if
and
If there is a constant
such that
implies
then
P is called a normal cone in
E (see [
12]), where the least constant satisfying the above inequality is called the normal constant of
P.
As an example, take and ; then, P is a cone in E, as it satisfies Definition 1, where (“⪯” is exactly “≤”) if and only if .
Definition 2 ([
12]).
Let E be a Banach space, be the zero element of E, and be a non-empty closed convex subset of E. Then, is called a positive cone if:(1) ;
(2) .
Let
. If there exists
such that
, then
is called a semi-interior point in
(see [
23]). Denote
as the closed unit ball of
E and
as the positive part of
U.
Definition 3 ([
17]).
Let E be a real normed space with a norm . If the following conditions hold:(1) for all , ;
(2) for any , , ,
then E is called a real ordered vector space.
Definition 4 ([
17]).
Let X be a nonempty set and E be a real normed space. The mapping is said to be an E-metric if, for all , it satisfies(i)
(ii)
(iii) .
In this case, the pair is called an E-metric space.
Remark 1. With regard to the topology of E-metric spaces, especially for the properties of countability, Hausdorffness, and nets, readers may refer to [
17,
23].
Both here and subsequently, we denote by
the set of all semi-interior points of
. We say ⋘ is a partial order on
if
Definition 5 ([
24]).
Let be an E-metric space, be a sequence in X, and , . We then say:(i) is e-convergent to ξ if for any , there exists such that for all . We denote as ;
(ii) is an e-Cauchy sequence if for any , there exists such that for all ;
(iii) is e-complete if every e-Cauchy sequence is e-convergent to some point in X.
Theorem 1 ([
24]).
Suppose that is an e-complete E-metric space and . If the mapping satisfieswhere , then f has a unique fixed point in X. Definition 6 ([
7]).
A sequence in is said to be an e-sequence if for each there exists such that for all . Lemma 1 ([
7]).
Let and be two sequences in E such thatThen, is an e-sequence. Lemma 2 ([
7]).
Let and be e-sequences in E and let be constants. Then, is an e-sequence in E. Lemma 3 ([
7]).
Let and ; then, . Lemma 4 ([
7]).
If for any , then . Lemma 5 ([
15]).
If is a constant and , then . Lemma 6 ([
12]).
Let be an E-metric space with a normal cone and let , , be sequences in X such that and , as . Then, , as . Lemma 7 ([
7]).
Let and for each ; then, . Definition 7 ([
25]).
For a nonempty set X, let be a function and be a mapping. Then, f is said to be an α-admissible function if, for any , it satisfies Definition 8 ([
2]).
For a nonempty set X, is a sequence in X, . Suppose that is a function. Then, X is said to be α-regular if for any it satisfies Definition 9 ([
26]).
Let X be a nonempty set, be a constant, and be a mapping. If, for any , the following conditions hold:(i) ;
(ii) ;
(iii) ,
then d is called a b-metric and the pair is called a b-metric space.
Definition 10 ([
27]).
Suppose that is a b-metric space, is a sequence in X, and . We then say that:(1) is convergent to ξ if , i.e., or as ;
(2) is a Cauchy sequence if ;
(3) is complete if every Cauchy sequence is convergent to some point in X.
Theorem 2 ([
5]).
Suppose that is a complete b-metric space with the parameter , is a function, is a nondecreasing function, and is an α-admissible function such thatwhere , , is a mapping. If there exists such that , and one of the following conditions holds:(1) f is continuous, or
(2) X is α-regular,
then f has a fixed point in X.
3. Main Results
First, motivated by Theorem 1, we aim to consider the existence and uniqueness of fixed points in
E-metric space if the following Ćirić-type contractive condition is satisfied:
where
Theorem 3. Let be an e-complete E-metric space and let and P be a cone in E. If the mapping satisfies the following Ćirić-type contractive condition:where and are the same as in (
1),
then f has a unique fixed point in X. Proof. Choose
and construct the Picard iterative sequence
by
,
, ⋯,
, ⋯. If there exists
such that
, then
is a fixed point of
f. Thus, the proof is completed. Without loss of generality, we assume that
for any
. Taking advantage of (
2), we can conclude that
where
We discuss (
3) as follows:
(A) If
, we have
which follows that
Subsequently, according to (
4) and Condition (iii) in Definition 4, for any
,
, we have
Using (
5) and Lemma 1, we can be sure that
is an
e-Cauchy sequence in
X.
(B) If
, we obtain
and furthermore, we have
Take
, then
. Thus, from the proof of (A), we know that
is an
e-Cauchy sequence.
(C) If
, then by combining (
3) and Condition (i) in Definition 4 we have
, which contradicts our hypothesis.
(D) If
, then
which means that
On account of
,
. This result conflicts with our hypothesis.
In summary, we claim that is an e-Cauchy sequence. Because is an e-complete E-metric space, there exists such that as , which is to say that is an e-sequence in E.
In the following, we prove that f has a fixed point.
Combining (
2) and Condition (iii) in Definition 4, we conclude that
where
In the following, we divide the above into five cases.
(i) If
, then by (
6), we have
Making the most of Lemma 2 and the fact that
is an
e-sequence, we deduce that
is an
e-sequence. Hence, from Lemmas 3 and 4, it is obvious that
, i.e.,
. That is,
is a fixed point of
f.
(ii) If
, then from (
6), we have
from which it follows that
Because
is an
e-sequence, from Lemmas 3 and 4 we have
. Therefore,
, i.e.,
. That is,
is a fixed point of
f.
(iii) If
, then from (
6), we can speculate that
Because
is an
e-sequence, from Lemma 2, it follows that
is an
e-sequence. Accordingly, based on Lemmas 3 and 4, we claim that
, i.e.,
. That is,
is a fixed point of
f.
(iv) If
, then by (
6) we arrive at
which means that
Because
is an
e-sequence, from Lemma 2 it is easy to see that
is an
e-sequence. Consequently, from Lemmas 3 and 4 we have
. Thus,
, i.e.,
. That is,
is a fixed point of
f.
(v) If
, then from (
6) we obtain
Note
is an
e-Cauchy sequence, implying that
is an
e-sequence as well. Because
is an
e-sequence, per Lemma 2 it is valid that
is an
e-sequence. Now, via Lemmas 3 and 4, we have
, i.e.,
. Thus,
is a fixed point of
f.
Finally, we prove that
f has only one fixed point. To this end, suppose that
and
are two fixed points of
f. According to (
2), we have
where
We discuss two cases concerning (
7) as follows:
If
, then
In view of
and Lemma 5, we have
. Hence,
.
If
, then
Making use of Condition (i) in Definition 4, we infer that
. Thus,
. □
From the proof of Theorem 3, we reach the following conclusion.
Corollary 1. Suppose that is an e-complete E-metric space, and P is a cone in E. If is a mapping satisfyingwhere and , then f has a unique fixed point in X. Example 1 ([
24]).
Suppose that is the subset of equipped with the pointwise partial order including the unit disk, while is the polygon of with verticesWe take a Minkowski functional (see [
23])
with respect to . We can define the norm byTake a sequence in E, whereand , which depends on ξ. Here, let E be an ordered space. We can define the cone byequipped with the normWe assume that , P is a subspace of E and is a mapping defined bySetting and , we establish thatBecause of , we take . Then,That is to say, f satisfies the condition (
8)
in Corollary 1,
meaning that f has a unique fixed point. Example 2. Put . Because in Theorem 3, we know that Theorem 3 is unsuitable for Example 1.
Example 3. Let with . Put and . Define by for all , where such that . Then, P is a non-normal cone (see [15]) and is an e-complete E-metric space. Define a mapping bywhere . Let . Note thatwhere . It is obvious that and f is a Ćirić-type contraction and not a Banach-type contraction. Thus, all conditions of Theorem 3 are satisfied. Then, using Theorem 3, it follows that f has a unique fixed point in X. Stimulated by Theorem 2, we obtain the following theorem.
Theorem 4. Let be an e-completeE-metric space, P be a normal cone with normal constant M, , be a function, and be a nondecreasing function. Suppose that is an α-admissible function satisfying the following α-ψ type contractive condition:where , and is a constant. If there exists such that and one of the following conditions is satisfied: (I) f is continuous, or
(II) X is α-regular,
then f has a fixed point in X. Moreover, if the following condition is satisfied:
(III) for each , there exists a such that and ,
then f has a unique fixed point in X.
Proof. Based on the assumption that there exists
such that
, we define an iterative sequence
by letting
,
, ⋯,
, ⋯. Because
and
f is an
-admissible function, we have
. By induction, we obtain
for any
. If there exists
such that
, then
is a fixed point of
f. Thus, the proof is completed. Now, suppose that
for any
. Making use of (
9), we have
where
We consider (
10) as follows:
(i) If
, then
Because
is nondecreasing, from (
11) we obtain
which establishes that
From (
12) and Condition (i) in Definition 4, it follows that
as
. Thus, for any
,
, we have
Because
P is a normal cone in
E, this implies that
Note that (
13) means
as
. As a consequence, per Lemma 1 we can confirm that
is an
e-sequence. In other words,
is an
e-Cauchy sequence. Because
is
e-complete, there exists
such that
as
.
(ii) If
, then
Since
is nondecreasing, from (
14) we obtain
Owing to
, we know that
, i.e.,
. It is obvious that
, which conflicts with the previous hypothesis.
Next, we prove that is a fixed point of f.
(I) If
f is continuous, then
i.e.,
is a fixed point of
f.
(II) If
X is
-regular, then from (
9) we have
Since
is nondecreasing, via (
15) we obtain
where
We can then discuss the above as follows:
(i) If
, then
Passing to the limit from both sides of (
16) and noting that
as
and
P is a normal cone, from Lemma 6 we have
. Thus,
.
(ii) If
, we note that
it then immediately follows from the normality of the cone that
therefore,
Passing to the limit from both sides of (
17) and noting that
, as
and
P is a normal cone, per Lemma 6 we have
. Thus,
.
(iii) If
, then
Passing to the limit from both sides of (
18) and noting that
, as
and
P is a normal cone, per Lemma 6 we can claim that
. In view of
, we have
. Hence,
. That is to say,
f has a fixed point
.
Assume that Condition (III) is satisfied. If
f has two fixed points
, then per (III) there exists a
in
X such that
Due to (
19) and the
-admissibility of
f, for any
we can obtain
As a consequence of (
9) and (
20), it is easy to see that
Because
is nondecreasing, from (
21) we obtain
where
Finally, we can show that
To this end, we discuss the following:
(i) If
, then from (
22) we have
On account of
, if we take the limit as
from both sides of (
24), we have (
23).
(ii) If
, then per (
22) we have
Via the above proof, it is not hard to verify that
is an
e-Cauchy sequence, meaning that
. Thus, from (
25) we have (
23).
(iii) If
, then from (
22) we can obtain
which implies (
23).
Similar to the proof of (
23), using (
9) and (
20) we can easily show that
By combining (
23) and (
26), we can claim that
. □
Remark 2. In Theorem 4, we prove the fixed point results for α-ψ type contraction in E-metric space, followed by Theorem 2.1 in [5] and Theorem 2.9 in [3], obtaining the fixed point theorem in ordered vector spaces.