Fixed Point Results for Generalized F -Contractions in b -Metric-like Spaces

: The purpose of this paper is to introduce several generalized F -contractions in b -metric-like spaces and establish some ﬁxed point theorems for such contractions. Moreover, some nontrivial examples are given to illustrate the superiority of our results. In addition, as an application, we ﬁnd the existence and uniqueness of a solution to a class of integral equations in the context of b -metric-like spaces.


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 forĆirić type contraction and φ-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 (α-ψ, φ)-contractions in b-metric-like spaces. Whereafter, Zoto et al. [11] in 2018 obtained fixed point theorems for (s, p, α)-contractive mappings in b-metric-like spaces. Subsequently, Zoto et al. [12] in 2019 investigated common fixed point theorems for a class of (s,q)-contractive mappings in b-metric-like spaces. In the meanwhile, De la Sen et al. [13] in 2019 gave fixed point results for (s-q)-graphic contraction in b-metric-like spaces. Later in 2020, Fabiano et al. [14] discussed fixed point theorems for (s, q)-Dass-Gupta-Jaggi type contraction in b-metriclike 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 (s, q)-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 (s, q, F)-contraction, general (s, q, F)-contraction and r-order (s, q, F)-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.

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 R as the set of all real numbers, N the set of all nonnegative integers, and N * the set of all positive integers. Definition 1 ([7]). Let M be a nonempty set and s ≥ 1 a constant. The mapping b: M × M → [0, +∞) is called a b-metric-like if for all ξ, η, ζ ∈ M, the following conditions are satisfied: The pair (M, b, s) is called a b-metric-like space with parameter s ≥ 1.
In 2012, Wardowski [16] defined the F-contraction in metric spaces as follows: (F3) there exists c ∈ (0, 1) such that lim Huang et al. [20] modified Definition 4 and defined the notion of convex F-contraction in the framework of b-metric spaces. (ii) there exists c ∈ 0, 1 1+log 2 s such that lim Remark 2. Condition (iii) yields that d n < d n−1 for all n ∈ N. Hence, the sequence {d n } is a decreasing sequence.

Definition 6 ([23]
). Let (M, b, s) be a b-metric-like space, f a self-mapping on M and {η n } a sequence in M. We say {η n } is a Picard iterative sequence generated by f if for any η 0 ∈ M, η n+1 = f η n holds for all n ∈ N. Inspired by the above notions, we provide some new definitions and theorems in the sequel.

Main Results
In this section, we introduce the notion of (s, q, F)-contraction, general (s, q, F)-contraction and r-order (s, q, F)-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 (M, b, s) with parameter s ≥ 1, {η n } be a Picard iterative sequence generated by f and F : (0, +∞) → R be an increasing function. We say that f is an (s, q, F)-contraction if: 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 s = 1, then we get the case in metric spaces. If α = 0, then we get the definition of (s, q, λ)-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 τ > 0 and F is increasing, then by (1), we speculate: Then by the monotonicity of F, we get: Clearly, β s q −α ∈ [0, 1). As a consequence, (s, q, F)-contraction generalizes usual contractions in general.
The following lemma will be used in our main results. Proof. The proof is clear if there exists n 0 ∈ N such that η n 0 +1 = η n 0 . Without loss of generality, we assume that η n+1 = η n for all n ∈ N. Thus, b n := b(η n , η n+1 ) > 0 for all n ∈ N. Notice (2) and β s q −α ∈ [0, 1), via Lemma 1, the sequence {η n } is a b-Cauchy sequence in M with lim n,m→∞ b(η n , η m ) = 0. Now, our first theorem becomes valid for presentation, which generalizes many recent results.
Proof. For any η 0 ∈ M, by Lemma 2, we can obtain that the Picard iterative sequence {η n } generated by f is a b-Cauchy sequence with lim for all η ∈ M. Therefore, all the conditions of Theorem 1 are satisfied and hence f has a fixed point The following definition is the extension of (s, q)-Jaggi F-contractions related to [8,11,13,14,19,20,25,26].
Definition 9. Let f be a self-mapping on b-metric-like space (M, b, s) with parameter s ≥ 1, and let {η n } be a Picard iterative sequence generated by f , F : (0, +∞) → R be an increasing function.

Remark 7.
Clearly, (s, q, F)-contraction is 1-order (s, q, F)-contraction. Hence, (s, q, F)-contraction is the special case of r-order (s, q, F)-contraction. In other words, r-order (s, q, F)-contraction greatly generalizes (s, q, F)-contraction. In addition, by replacing s = 1, we obtain the notion of r-order F-contraction in the setting of metric spaces. Proof. Let {η n } be a Picard iterative sequence as η n+1 = f η n initiated on each point η 0 ∈ M. Without loss of generality, we assume that η n+1 = η n , i.e., b n := b( f η n−1 , f η n ) = b(η n , η n+1 ) > 0 for all n ∈ N * . Taking advantage of (5), we obtain: By the monotonicity of F, we have: which follows that s qr b r n < αb r n + βb r n−1 . This leads to Note that β s qr −α 1 r < 1, then by Lemma 1, {η n } is a b-Cauchy sequence in M such that lim n,m→∞ b(η n , η m ) = 0. Since (M, b, s) is b-complete, then there exists some η * ∈ M such that Following the same argument as in Theorem 1, we claim that f has a fixed point.

Remark 8.
It can be easily shown that our new approach of r-order (s, q, F)-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].

Application
Stimulated by the work in [6,7,28,29], we investigate the existence of solution to a class of nonlinear integral equations utilizing the results proved in the previous section.
As a consequence, all the conditions of Theorem 2 are satisfied. Therefore, by Theorem 2, T has a unique fixed point in M. That is to say, the integral Equation (7) has a unique solution in C([0, T]).

Remark 9.
As compared with Theorem 6.1 of [7], Theorem 6 has much fewer conditions and more straightforward proof. We consider the existence and uniqueness of the solution to the integral Equation (7) in the setting of b-metric-like spaces, which are different from other counterparts in the existing literature. Our conditions are not complicated and our proof is quite forthright.

Remark 10.
It has been more widely used for the fixed point theory in b-metric-like spaces. It is not only for all kinds of integral equations (see [7,[9][10][11][12]), but also for other types of equations. As an example, it has been applied to an electric circuit equation (see [19]), the conversion of solar energy to electrical energy (see [9]), impulsive differential equations (see [30]) and fractional differential equations (see [31,32]). As a result, our results may have wide applications in the future.

Conclusions
Nowadays, fixed point theory plays an important role in natural science and in solving different social problems. In this work, a technique is furnished, based on generalized F-contractions, such as the (s, q, F)-contraction, the general (s, q, F)-contraction, and the r-order (s, q, F)-contraction. We establish several fixed point theorems on such contractions with illustrative examples in the framework of b-metric-like spaces. As has been observed in studies, the class of b-metric-like spaces contains the other classes of generalized metric spaces (e.g., b-metric spaces, metric-like spaces, etc.), then our results in this paper generalize and improve many known results in the existing literature. Additionally, we have applied our results to obtain the existence of a solution for a class of integral equations. We believe that the idea of further elaborating our method, which is presented in the main result section, is very useful and can be applied to impulsive differential and nonlinear fractional differential equations in the future.

Data Availability Statement:
The data presented in this study are available upon request from the corresponding author.