1. Introduction
In 1922, Banach [1] proved a famous theorem named the Banach contraction principle. Many scholars developed it. In 2012, Wardowski [2] introduced F-contraction which is an amusing development of Banach contraction. In 2014, Wardowski and Dung [3] extended F-contraction to F-weak contraction. In the same year, several fixed point results of the F-Suzuki contraction were got by Piri and Kumam [4], and an F-contraction of Hardy-Rogers type was raised by Cosentino and Vetro [5]. In 2018, Ali et al. [6] presented an (α, F)-contraction which is a generalization of the Wardowski type contraction. Qawaqneh et al. [7] posed (α-β-F)-Geraghty contraction. Several authors gained some interesting extensions and generalizations of the F-contraction (see [8] [9] [10] [11] and references therein). In 2015, Khojasteh et al. [12] projected the Z-contraction via simulation function, this kind of contraction was generalized to the Banach contraction and several known types of nonlinear contractions. Recently, a number of researchers have studied these contractive conditions (see [13] - [18] and references therein). For more recent results related to fixed point theory, refer to [19] [20] [21] [22]. In 2018, Isik et al. [23] acquired the existence and uniqueness of fixed point of almost contraction via simulation functions in metric spaces.
Inspired by the researches of F-contractions, Z-contractions, and almost contraction, in this article, by combining the ideas of these contractions, we propose a new almost type α-F-Z-weak contraction in complete metric spaces. Some sufficient conditions for the existence and uniqueness of fixed points in complete metric spaces were provided. Furthermore, some related fixed point results can derive from our main results.
2. Preliminaries
In the section, we firstly list some useful definitions and results. we denote by R the set of all real numbers, by R+ the set of all non-negative real numbers, by N the set of all non-negative integers and by X the set of all nonempty.
In 2012, α-admissible mapping was firstly introduced by Samet et al. [24].
Definition 1. [24] If there exists a function
such that a mapping
satisfies
, for all
.
Then mapping T is an α-admissible mapping.
Example 1. Let
,
and
by
,
, and
Then T is an α-admissible mapping.
In 2013, Karapınar et al. [25] presented triangular α-admissible mapping.
Definition 2. [25] If there exists a function
such that a mapping
satisfies
1)
, for all
;
2)
and
, for all
.
Then mapping T is a triangular α-admissible mapping.
Definition 3. [2] If
satisfies the following conditions:
(F1) F is strictly increasing, i.e.,
, for all
;
(F2) for each sequence
of positive numbers,
if and only if
;
(F3) there exists
such that
.
Then we say that F is a F-function.
We denote the set of all functions by F.
Example 2. The following functions
are elements of
.
1)
, where
;
2)
, where
;
3)
, where
.
In 2012, Wardowski [2] advanced F-contraction and obtained the existence and uniqueness of the fixed point of F-contraction in complete metric spaces. Without special explanation, F later in the article belongs to
.
Definition 4. [2] Let
be a metric space. If there exists a
such that a mapping
satisfies
, for all
.
Then T is said to be an F-contraction.
In 2014, Wardowski and Dung [3] extended F-contraction to F-weak contraction.
Definition 5. [3] Let
be a metric space. If there exists a
such that a mapping
satisfies
, for all
,
where
Then T is said to be an F-weak contraction.
Remark 1. If
, then F-weak contraction becomes F-contraction. It indicated that F-contraction is a special form of F-weak contraction.
In 2018, Ali et al. [6] projected (α, F)-contraction. When
for all
, then (α, F)-contraction reduces to F-contraction.
Definition 6. [6] Let
be a metric space. If there exists a
and
such that a mapping
satisfies
, for all
,
Then T is said to be an (α, F)-contraction.
In 2015, Khojasteh et al. [12] defined Z-contraction and gained the existence and uniqueness of the fixed point.
Definition 7. [12] Let
be a metric space. If there exists a
such that a mapping
satisfies
, for all
.
Then T is said to be a Z-contraction, where
is a mapping satisfying the following conditions:
(ζ1)
;
(ζ2)
, for all
;
(ζ3) if
are sequences with
such that
, then
Now we take some examples.
Example 3. [12] (1)
,
;
(2)
, where
are self-mappings on
such that
if and only if
and
, for all
;
(3)
, where
is a self-mapping on
with
and
.
Theorem 1. [12] Every Z-contraction in complete metric spaces has a unique fixed point.
In 2018, Isik and Gungor et al. presented almost Z-contraction and obtained the following fixed point theorem.
Definition 8. [23] Let
be a metric space. We say that
is an almost Z-contraction, if there exists a constant
such that
, for all
.
where
Theorem 2. [23] Let
be a complete metric space and
be an almost Z-contraction. Then, T has a unique fixed point, for arbitrary initial point
, the Picard sequence
converges to the fixed point.
3. Main Results
Firstly, we put forward almost type α-F-Z-weak contraction in metric spaces.
Definition 9. Let
be a metric space and
be a given mapping. We say that T is said to be an almost type α-F-Z-weak contraction if there exist
,
,
,
and
such that
, (1)
for all
, where
Remark 2. If T is an almost type α-F-Z-weak contraction, then
, for all
. (2)
Example 4. Let
and d be the usual metric on X. Define a mapping
by
Also define
and
.
Then T is an almost type α-F-Z-weak contraction with
and
,
. But T is not an F-weak contraction. Indeed, when
,
,
, this is a contradiction.
Remark 3. By the definition of T and Remark 2, we notice that an (α, F)-contraction must be an almost type α-F-Z-weak contraction, but the converse is not true (see Example 4). The converse holds only if
.
Example 5. Let
and
. Define
by
Also define
, and
Then T is an almost type α-F-Z-weak contraction for all
and
,
.
Now we prove our main results.
Theorem 3. Let
be a complete metric space. Suppose that T is an almost type α-F-Z-weak contraction. If T satisfies the following conditions:
1) There exists
such that
;
2) T is triangular α-admissible;
3) T or
satisfy one of the following conditions:
a) T is continuous;
b)
is continuous and if
, such that
;
c) If
is a sequence in X such that
for all n and
, then
, for all
, then T has at least a fixed point.
Proof. Define a sequence
by
,
. By (1), (2) and Mathematical induction, it easily follows that
, for all
, (3)
and
, for all
with
. (4)
If there exists
such that
, then
, so
is a fixed point of T, the proof is completed. If
for all
, i.e.,
for all
. Set
,
in (1), by (2) and (3), then
(5)
where
Now if
, by (5), it deduce that
(6)
this is a contradiction. So
, we get
So
By (F1), we have
. Thus
is a strictly non-increasing sequence with
. so assume that
. If
, take the right limits on the both sides of (6), it follows that
this is a contradiction. So
, that is
(7)
Now we claim that
is a Cauchy sequence. If
is not a Cauchy sequence, then there exist
, and two sequences
,
, where
,
are two positive integers and
such that
,
. By the triangle inequality, it follows that
(8)
Take the limits on the both sides of (8), we obtain
(9)
By the triangle inequality, we have
(10)
and
(11)
Let
in (10) and (11), hence
(12)
In the similar way, therefore
(13)
(14)
Set
,
in (1), by (2) and (4), it follows that
(15)
where
(16)
and
(17)
Let
in (16), (17) and by (9), (12), (13), (14), it shows
So
(18)
Take the right limits on the both sides of (15) and by (14) and (18),
this is a contradiction. So
is a Cauchy sequence in complete metric space
. Thus there exists a
such that
, as
. Furthermore
Case I: (1) holds;
Then we obtain
that is
.
Case II: (2) holds;
We have
that is
(19)
If
, set
,
in (1), by (2) and (19), it follows that
(20)
where
So (20) can be simplified to
, this is a contradiction. So
.
Case III: (3) holds;
If there exists
such that
,
, then
, that is
. On the contrary, set
,
in (1), by (2) and
, then
(21)
where
(22)
(23)
Let
in (22) and (23), it shows
So
(24)
Take the right limits on the both sides of (21) and by (24), we get
this is a contradiction. So
. Hence, T has a fixed point.
Remark 4. In the proof of Theorems 3, we only use (F1), (ζ2), it shows Theorems 3 is true as long as F and ζ satisfy (F1) and (ζ2), respectively.
Remark 5. Example 4 satisfies all the hypothesis of Theorem 3, so T has a fixed point. Indeed,
and
are two fixed points of T.
Theorem 3 shows that T has a fixed point, but it can’t guarantee the uniqueness of fixed point of T. Now in order to assure the uniqueness of fixed point of T, we consider the following condition:
4) For all s,
, where
denotes the set of fixed points of T.
Theorem 4. Adding (4) to the conditions of Theorem 3, we can assure the uniqueness of fixed point of T.
Proof. We argue by contradiction, assume that there exist
such that
with
. From (4), we have
. Therefore, if follows from the definitions of T and
(25)
where
So (25) can be simplified to
, it is a contraction. So
.
Remark 6. Example 5 satisfies all the hypothesis of Theorem 4, so T has a unique fixed point. In fact,
is the unique fixed point of T.
Corollary 5. Let
be a complete metric space. Suppose T satisfies the following conditions:
, for all
,
1) There exists
such that
;
2) T is triangular α-admissible;
3) T is continuous or T2 is continuous and if
is a sequence in X such that
, then
or if
is a sequence in X such that
and
, then
for all
, then T has a fixed point.
Proof. Take
in Theorem 3.
Corollary 6. Let
be a complete metric space. If there exists
is a lower semi-continuous function with
if and only if
such that for all
,
, for all
, (26)
then T has a unique fixed point.
Proof. From (26), so there exists
such that
.
Let
,
,
,
, so T is an almost type α-F-Z-weak contraction. By Theorem 4, the proof is completed.
Corollary 7. [1] Let
be a complete metric space. If there exists
such that for all
,
then T has a unique fixed point.
Proof. Let
, so by Corollary 6, the proof is completed.
4. Consequences
4.1. Fixed Point Theorems in Partially Ordered Metric Spaces
Definition 10.
is said to a complete partially ordered metric space, if
is a complete metric space and X is a nonempty set endowed with a partial order
.
Definition 11.
is non-decreasing endowed with a partial order
if
.
Theorem 8. Let
be a complete partially ordered metric space. If there exist
Such that T satisfies the following conditions:
1) There exists
such that
;
2) T is non-decreasing;
3) for all
,
;
4) T is continuous or if
is a sequence in X such that
and
for all
, then
, then T has a fixed point.
Proof. Let
. Then T satisfies all the conditions of Theorem 3, so the proof is completed.
Now in order to assure the uniqueness of fixed point of T, we considesr the following condition:
4') For all
such that
or
.
Theorem 9. Adding (4') to the conditions of Theorem 8, we can assure the uniqueness of fixed point of T.
Proof. Let
. Then T satisfies all the conditions of Theorem 4, so the proof is completed.
4.2. Fixed Point Theorems of Cyclic Mappings
Definition 12. Let A and B be two nonempty subsets of a metric
and
be a mapping. If
and
, then T is a cyclic mapping.
Theorem 10. Let
be a complete partially ordered metric space, A and B be two nonempty subsets of X such that
and
. If there exist
,
such that T satisfies the following conditions:
1) There exists
such that
;
2) T is a cyclic mapping;
3) For all
,
;
4) T is continuous or if there exists a sequences
such that
and
for all
, then
, thenT has a fixed point, that is, there exists a
such that
.
Proof. Let
,
. Then T satisfies all the conditions of Theorem 3, so T has a fixed point, the proof is completed.
Now in order to ensure the uniqueness of fixed point of T, we consider the following condition:
4") for all
such that
.
Theorem 11. Adding (4") to the conditions of Theorem 10, we can assure the uniqueness of fixed point of T.
5. Conclusion
In this paper, we investigate a new type of contraction named almost type α-F-Z-weak contraction, which is produced by the combination of F-contraction, Z-contraction, and almost contraction. In Section 3, sufficient conditions for the existence and uniqueness of the fixed point of such contraction in complete metric spaces are provided. There are some related fixed point results that can derive from our results. In Section 4, we propose the cases of partially ordered metric spaces and cycle mappings, some corresponding fixed point results are obtained.
Acknowledgements
This work is partially supported by the National Natural Science Foundation of China (Grant No.11872043), Central Government Funds of Guiding Local Scientific and Technological Development for Sichuan Province (Grant No.2021ZYD0017), Zigong Science and Technology Program (Grant No.2020YGJC03), the Opening Project of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (Grant No.2020WYJ01), 2020 Graduate Innovation Project of Sichuan University of Science and Engineering (Grant No.y2020078), 2021 Innovation and Entrepreneurship Training Program for College Students of Sichuan University of Science and Engineering (Grant No.cx2021150).