1. Introduction
The Banach contraction principle (for short, BCP), the most celebrated theorem in metric fixed point theory, has undergone many extensions and generalizations due to its simple nature and wide range of applicability. Some significant generalizations of BCP are seen in [
1,
2,
3,
4]. Generally speaking, these generalizations usually contain two sides. On the one hand, BCP is extended from metric space to generalized metric space, such as
b-metirc space [
1], partial metric space [
5], ordered Banach space [
6],
b-metric-like space [
7], etc. One of the most prevalent spaces is
b-metric-like space, which was introduced by Alghmandi et al. [
7] in 2013. In 2014, Hussain et al. [
8] obtained fixed point results for Ćirić type contraction and
-contraction in
b-metric-like spaces. Afterwards, Joshi et al. [
9] in 2017 presented fixed point theorems for generalized
F-contractions in
b-metric-like spaces. In the same year, Zoto et al. [
10] offered some generalizations for
-
-contractions in
b-metric-like spaces. Whereafter, Zoto et al. [
11] in 2018 obtained fixed point theorems for
-contractive mappings in
b-metric-like spaces. Subsequently, Zoto et al. [
12] in 2019 investigated common fixed point theorems for a class of
,
-contractive mappings in
b-metric-like spaces. In the meanwhile, De la Sen et al. [
13] in 2019 gave fixed point results for
-
-graphic contraction in
b-metric-like spaces. Later in 2020, Fabiano et al. [
14] discussed fixed point theorems for
-Dass–Gupta–Jaggi type contraction in
b-metric-like spaces. Recently in 2021, Mitrović et al. [
15] established fixed point theorems for Jaggi-
W-contraction in
b-metric-like spaces.
On the other hand, BCP is extended for different contractive mappings. One of the most important contractions, is
F-contraction on metric space, which was introduced by Wardowski [
16] in 2012. Shukla et al. in 2014 established ordered
F-contraction in [
17] from metric spaces to partial metric spaces. In 2018, Kadelburg and Radenović in [
18] extended
F-contraction in [
16] from metric spaces to
b-metric spaces. In 2019, Hammad and De la Sen [
19] considered fixed point theorem for the generalized almost
-Jaggi
F-contraction-type in
b-metric-like spaces. As followed by them, Huang et al. in [
20] introduced the notion of convex
F-contraction and proved fixed point theorems for both continuous and discontinuous mappings.
Among these extensions cited above, throughout this paper, first and foremost, we initiate several generalized
F-contractions, such as
-contraction, general
-contraction and
r-order
-contraction. We give several fixed point theorems for such contractions in
b-metric-like spaces. As compared with previous contractions from [
8,
9,
10,
12,
13,
14,
19], our contractions mainly aim at generalized
F-contractions, which are the sharp generalizations of
F-contraction introduced by Wardowski [
16]. As we know,
F-contraction is one of the generalizations of Banach type contraction, whereas generalized
F-contractions greatly extend
F-contractions. As a result, our conclusions related to generalized
F-contractions have strong theoretical significance and practical influence. It is worth mentioning that we demonstrate our assertions by much fewer conditions and more straightforward proofs than the counterpart from previous results. In addition, we illustrate the vitality of our conclusions by some supportive examples. As an application, we obtain the existence and uniqueness of solution to a class of integral equations. Regarding finding the solutions for such equations, there have emerged numerous versions in the existing literature; our method used in this paper is very easy to be understood since it contains simple conditions and comes straight to the point with short proof.
2. Preliminaries
It is customary for a paper to firstly list some useful definitions, lemmas and other contributed results.
In the following, unless otherwise specified, we always assume as the set of all real numbers, the set of all nonnegative integers, and the set of all positive integers.
Definition 1 ([
7]).
Let M be a nonempty set and a constant. The mapping b: is called a b-metric-like if for all , the following conditions are satisfied:(b1) implies ;
(b2) ;
(b3) .
The pair is called a b-metric-like space with parameter .
In a b-metric-like space , if and , then ; however, the converse need not be true, since may be positive for some .
Example 1 ([
11]).
Let and be a constant. Define a function by or . Then, is a b-metric-like space with parameter . Clearly, is neither a b-metric (see [1]), nor a metric-like space (see [5]), nor a partial b-metric space (see [21]). Definition 2 ([
7]).
Let be a b-metric-like space with parameter , a sequence in M and . We say:(i) is said to be a b-convergent sequence if ;
(ii) is said to be a b-Cauchy sequence if exists and is finite;
(iii) is called b-complete, if, for every b-Cauchy sequence in M, there exists such that .
Definition 3 ([
7]).
Let be a b-metric-like space with parameter and a function. We say that f is b-continuous if for each sequence in M with as , then as . Remark 1 ([
14]).
In a b-metric-like space, if and the limit of exists, then the limit is unique. Lemma 1 ([
22]).
Let be a b-metric-like space with parameter and a sequence in M such thatfor some and each . Then, is a b-Cauchy sequence with . In 2012, Wardowski [
16] defined the
F-contraction in metric spaces as follows:
Definition 4 ([
16]).
Let be a metric space. The mapping is called an F-contraction if there exists a function such that(F1) F is strictly increasing on ;
(F2) for each sequence of positive numbers, ;
(F3) there exists such that ;
(F4) there exists such that for all with .
Huang et al. [
20] modified Definition 4 and defined the notion of convex
F-contraction in the framework of
b-metric spaces.
Definition 5 ([
20]).
Let be a b-metric space and f a self-mapping on M. We say that f is a convex F-contraction if there exists a function such that (F1) holds and:(i) for each sequence of positive numbers, if , then ;
(ii) there exists such that ;
(iii) there exist and such that for all where .
Remark 2. Condition (iii) yields that for all . Hence, the sequence is a decreasing sequence.
Definition 6 ([
23]).
Let be a b-metric-like space, f a self-mapping on M and a sequence in M. We say is a Picard iterative sequence generated by f if for any , holds for all . Inspired by the above notions, we provide some new definitions and theorems in the sequel.
3. Main Results
In this section, we introduce the notion of -contraction, general -contraction and r-order -contraction and give some fixed point theorems based on them. We also provide three examples to support our conclusions.
First of all, motivated by Definitions 4 and 5, we present the following concept.
Definition 7. Let f be a self-mapping on b-metric-like space with parameter , be a Picard iterative sequence generated by f and be an increasing function. We say that f is an -contraction if: for all , where , are constants with .
Remark 3. Definition 7 improves the corresponding definitions given in [13,20,24]. It contains a fewer conditions compared with the previous ones. It covers many contractive conditions since the set of all increasing functions F is a very broad set. Remark 4. If , then we get the case in metric spaces. If , then we get the definition of -contraction in [11]. Remark 5. Although the conditions of Definition 7 look strong since it involves the Picard iteration, to the best of our knowledge, the Picard iteration is one of the most frequently used iterations in fixed point theory. Hence, our object is more targeted for the convenience of applications. Otherwise, since and F is increasing, then by (1), we speculate: Then by the monotonicity of F, we get:which implies that Clearly, . As a consequence, -contraction generalizes usual contractions in general.
Example 2. Let and define a mapping on by . Then, is a b-complete b-metric-like space with parameter . Suppose that is a function on and is a mapping by . It is easy to see that f is an -contraction. Indeed, it is easy to show that (1) is satisfied for all , where , , , and is a Picard iterative sequence generated by f.
The following lemma will be used in our main results.
Lemma 2. Let be a b-metric-like space with parameter , f an -contraction on M and a Picard iterative sequence generated by f. Then, is a b-Cauchy sequence with .
Proof. The proof is clear if there exists such that . Without loss of generality, we assume that for all . Thus, for all . Notice (2) and , via Lemma 1, the sequence is a b-Cauchy sequence in M with . □
Now, our first theorem becomes valid for presentation, which generalizes many recent results.
Theorem 1. Let be a b-complete b-metric-like space with parameter and f a b-continuous -contraction on M. Then, f has a fixed point in M provided that for all .
Proof. For any
, by Lemma 2, we can obtain that the Picard iterative sequence
generated by
f is a
b-Cauchy sequence with
. Since
is
b-complete, then there exists
such that:
Now that
f is
b-continuous, one has:
By virtue of
letting
in the above inequality, we obtain
. Therefore,
. That is to say,
is a fixed point of
f. □
Example 3. Under the hypotheses of Example 2, it is not hard to verify that for all . Therefore, all the conditions of Theorem 1 are satisfied and hence f has a fixed point in M.
The following definition is the extension of
-Jaggi
F-contractions related to [
8,
11,
13,
14,
19,
20,
25,
26].
Definition 8. Let be a b-metric-like space with parameter , f be a self-mapping on M, be an increasing mapping and be a function such that for all . Then, the mapping f is said to be a general -contraction if:for all and , where and are constants with . Theorem 2. Let be a b-complete b-metric-like space with parameter and f a b-continuous general -contraction on M. Then, f has a unique fixed point provided that for all .
Proof. Let
and define the Picard iterative sequence
as
. If
for some
, then
is a fixed point of
f because of
. So we always assume that
, i.e.,
for all
. By using (3) and the monotonicity of
F, we have:
which shows that (1) holds. Accordingly,
f is an
-contraction. Thus, by virtue of Theorem 1,
f has a fixed point.
We show that
f has a unique fixed point in
M. Indeed, first of all, we prove that
if
is a fixed point of
f. On the contrary, assume
, i.e.,
, then by (3) and the monotonicity of
F, we get:
which implies that:
As a consequence of , then . This is a contradiction with and . Accordingly, .
Let
and
be two distinct fixed points of
f. By the above statement, we have
and
. In view of
, that is,
, i.e.,
, then from (3) and the monotonicity of
F, we speculate:
which establishes that
In view of , then . This is a contradiction with and . Therefore, , i.e., . In other words, the fixed point is unique. □
Remark 6. Theorem 2 generalizes the previous theorems from [9,10,12,15,19,20,24,25,26] and some of them used rational expressions under the contractive conditions. If we take the function as , we can obtain from the above results in case of Jaggi and Gupta contractions. Corollary 1. Let be a b-complete b-metric-like space with parameter , f be a b-continuous self-mapping on M, be an increasing mapping. If for all and , where and are constants with . Then, f has a unique fixed point in M.
Proof. Use the function on in (3). By Theorem 2, we obtain the proof. □
Corollary 2. Let be a b-complete b-metric-like space with parameter , f be a b-continuous self-mapping on M, and be an increasing mapping. If for all and , where and are constants with . Then, f has a unique fixed point in M.
Proof. Use the function on in (3). Via Theorem 2, we get the desired result. □
Example 4. Let and for all . It is not hard to verify that is a b-complete b-metric-like space with parameter . Suppose that is a function on and is a mapping by for all . Then f is a b-continuous general -contraction on M. Indeed, it is easy to show that (3) is satisfied for all and , where , , , , is a function such that for all . Therefore, f has a unique fixed point .
Now we show that (3) is satisfied for all and . As a matter of fact, by for all , using the mean value theorem of differentials, we have: for all and .
Theorem 3. Let be a b-complete b-metric-like space with parameter , f be a b-continuous self-mapping on M, be an increasing mapping. If for all with , where and are constants with . Then, f has a unique fixed point provided that for all .
Proof. Let
be a Picard iterative sequence as
initiated on each point
. Assume the general case that
, i.e.,
for all
. Considering (4) and the monotonicity of
F, we have:
Hence, f is an -contraction. By Theorem 1, f has a fixed point.
Now we show the fixed point of
f is unique. To this end, assume that there exist two distinct fixed points
and
, then
, i.e.,
. By (4) and the monotonicity of
F, we have:
which establishes that:
In view of , then . This is a contradiction with and . Therefore, . It leads to . That is to say, the mapping f has a unique fixed point in M. □
Definition 9. Let f be a self-mapping on b-metric-like space with parameter , and let be a Picard iterative sequence generated by f, be an increasing function. We say that f is a r-order -contraction if for all , it satisfieswhere and are constants with . Remark 7. Clearly, -contraction is 1-order -contraction. Hence, -contraction is the special case of r-order -contraction. In other words, r-order -contraction greatly generalizes -contraction. In addition, by replacing , we obtain the notion of r-order F-contraction in the setting of metric spaces.
Theorem 4. Let be a b-complete b-metric-like space with parameter and f a b-continuous r-order -contraction. Then, f has a fixed point provided that for all .
Proof. Let
be a Picard iterative sequence as
initiated on each point
. Without loss of generality, we assume that
, i.e.,
for all
. Taking advantage of (5), we obtain:
By the monotonicity of
F, we have:
which follows that
Note that
, then by Lemma 1,
is a
b-Cauchy sequence in
M such that
. Since
is
b-complete, then there exists some
such that
Following the same argument as in Theorem 1, we claim that f has a fixed point. □
Theorem 5. Let be a b-complete b-metric-like space with parameter , f be a b-continuous self-mapping on M, be an increasing mapping and be a function such that for all . If for all and , where and are constants with . Then, the mapping f has a unique fixed point provided that for all .
Proof. Let
be a Picard iterative sequence as
initiated on each point
. Assume the general case that
, i.e.,
for all
. By using (6), we have
which shows that (5) holds, and hence
f is an
r-order
-contraction. Thus, by Theorem 4,
f has a fixed point.
We prove that
if
is a fixed point of
f. Indeed, by supposing the contrary, that is,
, i.e.,
. It follows immediately from (6) that:
Making full use of the monotonicity of
F, we claim that
As a consequence of , then . This is a contradiction with and . Accordingly, .
Let
and
be two distinct fixed points of
f. By the above statement, we have
and
. In view of
, that is,
, i.e.,
. then from (6) and the monotonicity of
F, we speculate:
which follows from the monotonicity of
F that
Notice that implies . This is a contradiction with and . Thus, . Therefore, . That is to say, the fixed point of f is unique. □
Remark 8. It can be easily shown that our new approach of r-order -contraction covers many classical types of contractions such as Kannan, Reich, Chatteria, Hardy, Ćirić, etc. Consequently, it could be developed as a prospective work in the future. Kindly see the reference from [27].