On Some Common Fixed Point Results for Weakly Contraction Mappings with Application

In this paper, we introduce a new class of generalized weakly contractive mappings and prove common fixed point results by using different algorithms involving this new class of mappings in the framework of b-metric spaces, which generalize the results of Cho. We also provide two examples to show the applicability and validity of our results. As an application of our result, we obtain a solution to an integral equation. Our results extend and improve several comparable results in the existing literature.


Introduction
The Banach fixed point theorem [1] popularly known as the Banach contraction mapping principle is a rewarding result in fixed point theory. It has widespread applications in both pure and applied mathematics and has been extended in many different directions. One of the most popular and interesting topics among them is the study of new classes of spaces and their fundamental properties.
In 1993, Czerwik [2] introduced firstly the concept of b -metric space and proved some fixed point theorems of contractive mappings in b-metric space. After that, some authors have researched on the fixed point theorems of various new types of contractive conditions in b-metric space. Aydi et al. in [3] proved common fixed point results for single-valued and multivalued mappings satisfying a weak ϕ-contraction in b-metric spaces. Starting from the results of Berinde [4], Pacurar [5] proved the existence and uniqueness of the fixed point of ϕ-contractions and Zada et al. [6] established fixed point results satisfying contractive conditions of rational type. In 2019, Hussain et al. studied the existence and uniqueness of periodic common fixed point for pairs of mappings via rational type contraction in [7]. After that, in [8], the authors obtained fixed point theorems for cyclic ðα, βÞ − ðψ, ϕÞ s -rational type contractions and discussed the existence of a unique solution to nonlinear fractional differential equations. Also using rational type contractive conditions, Hussain et al. [9] got the existence and uniqueness of a common n-tupled fixed point for a pair of mappings. Using a contraction condition defined by means of a comparison function, [10] established results regarding the common fixed points of two mappings. In 2014, Abbas et al. obtained the results on common fixed point of four mappings in b -metric space in [11]. Iqbal et al. [12] introduced a generalized multivalued ðα, LÞ-almost contraction and proved the existence and uniqueness of the fixed point for a specific mapping in the b-metric space.
Inspired by Czerwik's results, Hussain and Shah in [13] introduced the notion of a cone b-metric space, which means that it is a generalization of b-metric spaces and cone metric spaces; they considered topological properties of cone b -metric spaces and obtained some results on KKM mappings in the setting of cone b-metric spaces. Younis et al. [14] studied the existence of fixed points of a new class of generalized F-contraction in partial b-metric space. In [15], some fixed point results for weakly contractive mappings in ordered partial metric space were obtained. Recently, Samet et al. [16] introduced the concept of α-admissible and α − ψ-contractive mappings and presented fixed point theorems for them. In [17,18], Zoto et al. studied generalized α s p contractive mappings and ðα − ψ, ϕÞ-contractions in b-metric-like space. In 2020, Isik et al. [19] firstly introduced the structure of extended quasi b-metric-like spaces as a generalization of both quasi metric-like spaces and quasi b-metric-like spaces. Also, they presented the notion of JSR-contractive mappings in the setup of extended quasi b-metric-like spaces and investigated the existence of fixed point for such mappings. Abu-Donia et al. [20] proved the uniqueness and existence of the fixed points for five mappings from a complete intuitionistic fuzzy 3-metric space into itself under weak compatible of type ðαÞ and asymptotically regular. In 2015, Ege [21] introduced complex valued rectangular b-metric space and proved an analogue of the Banach contraction principle in this space. Recently, Younis et al. [22] provided much simpler and shorter proofs of some new results in rectangular metric spaces, and Mitrovic et al. [23] gave a proof of the results of Miculescu and Mihail [24] and Suzuki [25] in extended b -metric spaces. In graphical b-metric spaces, Younis et al. presented fixed point results for Kannan-type and Reichtype mappings in [26,27]. Lately, Gholidahneh et al. [28] introduced the notion of a modular p-metric space (an extended modular b-metric space) and established some fixed point results for α-v-Meir-Keeler contractions in this new space.
In particular, Choudhury et al. [36] obtained a generalization of the weak contraction principle in metric spaces by using altering distance functions as follows: Theorem 1 (see [36]). Suppose that a mapping g : X ⟶ X, where X is a metric space with metric d, satisfies the following condition: for all x, y ∈ X, where φ : ½0,+∞Þ ⟶ ½0,+∞Þ is a continuous function and ψ : ½0,+∞Þ ⟶ ½0,+∞Þ is an altering function, that is, ψ is a nondecreasing and continuous function, and ψ ðtÞ = 0 if and only if t = 0: Then, g has a unique fixed point.
Let X be a metric space with metric d, let T : X ⟶ X, and let φ : X ⟶ ½0,+∞Þ be a lower semicontinuous function. Then, T is called a generalized weakly contractive mapping if it satisfies the following condition: where ψ ∈ Ψ, ϕ ∈ Φ, and Cho [44] extended the results of Choudhury et al. [36] to generalized weakly contractive mappings in the setting of metric spaces and obtained the following result: Theorem 2 (see [44]). Let X be complete. If T is a generalized weakly contractive mapping, then there exists a unique z ∈ X such that z = Tz and φðzÞ = 0: Motivated and inspired by Theorem 2.1 in [44], in this paper, our purpose is to introduce a new class of generalized weakly contractive mappings and obtain a few of common fixed point results by using different algorithms involving generalized weakly contractive conditions in the framework of b-metric space, which generalize the results of Cho. Furthermore, we provide examples that elaborated the useability of our results. Meanwhile, we present an application to the existence of solutions to an integral equation by means of one of our results.

Preliminaries
In this section, in order to get our main results, we will introduce some definitions and lemmas first.
Definition 3 (see [2]). Let X be a nonempty set and s ≥ 1 be a given real number. A mapping d : X × X ⟶ ½0,+∞Þ is said to be a b-metric if and only if, for all x, y, z ∈ X, the following conditions are satisfied: In general, ðX, dÞ is called a b-metric space with parameter s ≥ 1.
Remark 4. We should note that a b-metric space with s = 1 is a metric space. We can find several examples of b-metric spaces which are not metric spaces (see [45]).
Definition 6 (see [11]). Let ðX, dÞ be a b -metric space with parameter s ≥ 1. Then, a sequence fx n g in X is said to be (i) b-convergent if and only if there exists x ∈ X such that dðx n , xÞ ⟶ 0 as n ⟶ +∞, (ii) a Cauchy sequence if and only if dðx n , x m Þ ⟶ 0 when n, m ⟶ +∞: In addition, a b-metric space is called complete if and only if each Cauchy sequence in this space is b-convergent.
Definition 7 (see [47]). Let f and g be two self-mappings on a nonempty set X. If w = f x = gx, for some x ∈ X, then x is said to be the coincidence point of f and g, where w is called the point of coincidence of f and g. Let Cðf , gÞ denote the set of all coincidence points of f and g.
Definition 8 (see [47]). Let f and g be two self-mappings defined on a nonempty set X. Then, f and g is said to be weakly compatible if they commute at every coincidence point, that is, The following lemma plays an important role to obtain our main results: Lemma 9 (see [46]). Let ðX, dÞ be a b -metric space with parameter s ≥ 1. Assume that fx n g and fy n g are b-convergent to x and y, respectively. Then, we have In particular, if x = y, then we have lim n⟶+∞ dðx n , y n Þ = 0. Moreover, for each z ∈ X, we have

Main Results
In this section, we will establish common fixed point theorems for generalized weakly contractive mappings in complete b-metric space. Furthermore, we also provide two examples to support our results. A function f : X ⟶ ½0,+∞Þ, where ðX, dÞ is a b-metric space, is called lower semicontinuous if, for all x ∈ X and fx n g are b-convergent to x, we have We shall consider that the contractive conditions in this section are constructed via auxiliary functions defined with the families Ψ, Φ, respectively: Þis a nondecreasing and continuous function f g , Þis a nondecreasing and lower semicontinuous function and ϕ t ð Þ = 0 if and only if t = 0g: ð9Þ Theorem 10. Let ðX, dÞ be a complete b-metric space with parameter s ≥ 1, and let f , g : X ⟶ X be given selfmappings satisfying g as injective and f ðXÞ ⊂ gðXÞ where gðXÞ is closed. Suppose φ : X ⟶ ½0,+∞Þ is a lower semicontinuous function and p ≥ 2 is a constant. If there are functions ψ ∈ Ψ and ϕ ∈ Φ such that where then f and g have a unique coincidence point in X. Moreover, f and g have a unique common fixed point provided that f and g are weakly compatible.
Proof. Let x 0 ∈ X. As f ðXÞ ⊂ gðXÞ, there exists x 1 ∈ X with f x 0 = gx 1 . Now we define the sequences fx n g and fy n g in X by y n = f x n = gx n+1 for all n ∈ ℕ. If y n = y n+1 for some n ∈ ℕ; then, we have y n = y n+1 = f x n+1 = gx n+1 and f and g have a coincidence point. Without loss of generality, we assume that y n ≠ y n+1 for all n ∈ ℕ. Applying (10) with x = x n and y = x n+1 , we obtain If dðy n , y n+1 Þ + φðy n Þ + φðy n+1 Þ > dðy n , y n−1 Þ + φðy n Þ + φ ðy n−1 Þ, for some n ∈ ℕ, in view of (12), (13), and (14), we have which implies ϕðdðy n , y n+1 Þ + φðy n Þ + φðy n+1 ÞÞ = 0. Hence, y n = y n+1 , a contradiction. Thus, we have It follows from (16) that fdðy n , y n+1 Þ + φðy n Þ + φðy n+1 Þg is a nonincreasing sequence, and so there exists r ≥ 0 such that By virtue of (12), (17), and (18), one can obtain Now assume that r > 0: Taking the upper limit as n ⟶ ∞ in (20), we have which implies that ψðrÞ ≤ ψðrÞ − ϕðrÞ, a contradiction. This yields that It follows that lim n⟶+∞ dðy n , y n+1 Þ = 0 and lim n⟶+∞ φð y n Þ = 0: Now we shall prove that fy n g is a Cauchy sequence in X: Suppose on the contrary that fy n g is not Cauchy. It follows that there exists ε > 0 for which one can find sequences f y m k g and fy n k g of fy n g satisfying n k is the smallest index for which n k > m k > k, d y m k , y n k −1 < ε: By the triangle inequality in b-metric space and (23) and (24), we have ε ≤ d y m k , y n k ≤ sd y m k , y n k −1 + sd y n k −1 , y n k < sε + sd y n k −1 , y n k : Taking the upper limit as k ⟶ +∞ in the above inequality, we have Also, d y m k , y n k ≤ sd y m k , y n k −1 + sd y n k −1 , y n k , Journal of Function Spaces d y m k , y n k ≤ sd y m k , y m k −1 + sd y m k −1 , y n k , ð28Þ d y m k −1 , y n k ≤ sd y m k −1 , y m k + sd y m k , y n k : From (23), (24), and (27), we obtain Using (23), (28), and (29), we get Similarly, so there is Using the same method, one can obtain that In view of the definition of mðx, y, d, f , g, φÞ, we deduce Taking the upper limit as k ⟶ +∞ in (35), we obtain Also, we have It follows that Applying (10) with x = x m k and y = x n k , one can get which implies that a contradiction to (38). It follows that fy n g is a Cauchy sequence in X: Following from the definition of φ, we get That is, φðgzÞ = φðuÞ = 0:

Journal of Function Spaces
If f z ≠ gz, taking x = x n k and y = z in contractive condition (10), we deduce that where By simple calculation, we obtain lim inf By taking the upper limit as k ⟶ +∞ in (44) and using (46) and (47), one can get Hence, dð f z, gzÞ + φð f zÞ = 0, which implies that f z = gz and φðf zÞ = 0: Now we claim that z is the unique coincidence point of f and g: If not, there exist z, z ′ ∈ Cðf , gÞ and z ≠ z ′ ; applying (10) with x = z and y = z′, we obtain that Here, It follows from (49) that Hence, we get that dðgz, gz ′ Þ + φðgz ′ Þ = 0, which implies that gz = gz ′ and φðgz ′ Þ = 0: Since g is an injective mapping, then z = z′; that is, z is a unique coincidence point of f and g: Further, if f and g are weakly compatible, then it is easy to show that z is a unique common fixed point of f and g: This completes the proof.
It is clear that f ðXÞ ⊂ gðXÞ. For all x, y ∈ X, we have x + x 2 À Á 2 + y + y 2 À Á 2 , According to above inequalities, we get that It follows that all conditions of Theorem 10 are satisfied with s = 2,p = 2: It is easy to obtain that 0 is the unique common fixed point of f and g: Note that, for x = 0, y ∈ ð0,+∞Þ, one can calculate that which implies that Theorem 1 of [10] cannot be applied to testify the existence of common fixed points of the mappings f and g in X.
If φ = 0 in Theorem 10, we can get the following result: Corollary 12. Let ðX, dÞ be a complete b-metric space with parameter s ≥ 1, and let f , g : X ⟶ X be given selfmappings satisfying g as injective and f ðXÞ ⊂ gðXÞ where gð XÞ is closed. Suppose p ≥ 2 is a constant. If there are functions ψ ∈ Ψ and ϕ ∈ Φ such that then f and g have a unique coincidence point in X. Moreover, f and g have a unique common fixed point provided that f and g are weakly compatible.
If we consider the corresponding problem in the setting of metric space, that is, s = 1 in Theorem 10, one can obtain the following: Corollary 13. Let ðX, dÞ be a complete metric space, and let f , g : X ⟶ X be given self-mappings satisfying g as injective and f ðXÞ ⊂ gðXÞ where gðXÞ is closed. Suppose φ : X ⟶ ½0, +∞Þ is a lower semicontinuous function. If there are functions ψ ∈ Ψ and ϕ ∈ Φ such that where and lðx, y, d, f , g, φÞ is the same as Theorem 10, then f and g have a unique coincidence point in X. Moreover, f and g have a unique common fixed point provided that f and g are weakly compatible.

Theorem 14.
Let ðX, dÞ be a complete b-metric space with parameter s ≥ 1 and let f , g : X ⟶ X be given self-mappings, and one of f and g is continuous. Suppose φ : X ⟶ ½0,+∞Þ is a lower semicontinuous function and p ≥ 3, 0 < λ ≤ 1/4 are two constants. If there are functions ψ ∈ Ψ and ϕ ∈ Φ such that where then f and g have a unique common fixed point in X.
Proof. Let x 0 ∈ X be an arbitrary point. Define a sequence f x n g in X by x 2i+1 = f x 2i ,x 2i+2 = gx 2i+1 for i = 0, 1, 2, ⋯. Firstly, we prove that f and g have at most one common fixed point.