Common Best Proximity Point Theorems for Generalized Proximal Weakly Contractive Mappings in b-Metric Space

In this paper, common best proximity point theorems for weakly contractive mapping in b-metric spaces in the cases of nonself-mappings are proved; we introduced the notion of generalized proximal weakly contractive mappings in b-metric spaces and proved the existence and uniqueness of common best proximity point for these mappings in complete b-metric spaces. We also included some supporting examples that our ﬁ nding is more generalized with the references we used.


Introduction
The metric fixed point theory gained impetus due to its wide range of applicability to resolve diverse problems emanating from the theory of nonlinear differential equations, theory of nonlinear integral equations, game theory, mathematical economics, and so forth. The first fixed point theorem was given by Brouwer [1], but the credit of making concept useful and popular goes to polish mathematician, Banach [2] who proved the famous contraction mapping theorem in 1922 in the setting of metric space. This principle guarantees the existence and uniqueness of fixed point of certain selfmaps of metric spaces and provides a constructive method to find those fixed points. This principle includes different directions in different spaces adopted by mathematicians for example metric spaces, G-metric spaces, partial metric spaces, and cone metric spaces.
A classical best approximation theorem was introduced by Fan [3], which states that "if A is a non-empty compact convex subset of a Hausdorff locally convex topological vector space B and T : A ⟶ B is a continuous mapping, then there exists an element x ∈ A such that dðx, TxÞ = dðTx, AÞ ." Afterwards, Prolla [4], Reich [5], and Sehgal and Singh [6] have derived extensions of Fan Theorem in many directions. The common fixed point theorem insists to the authors to investigation on common best proximity point theorem for nonself-mappings. The common best proximity point theorem assures a common optimal solution at which both the real valued multiobjective functions x ⟶ dðx, SxÞ and x ⟶ dðx, TxÞ attain the global minimal value dðA, BÞ. A number of authors have improved, generalized, and extended this basic result either by defining a new contractive mapping in the context of a complete metric space or extend best proximity results from fixed point theory (see [7][8][9][10][11][12]). Definition 1. Let X be a nonempty set and T : X ⟶ X a selfmap. A point x ∈ X is said to be fixed point of T if Tx = x.
Tx = x ⇒ x/2 = x, and we get x = 0 ∈ X, which is a fixed point of T.
So T is weakly contractive.
Definition 14 (see [17]). 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: Remark 15 (see [18]). 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.
Example 16 (see [19]). Let ðX, ρÞ be a metric space, and dðx, yÞ = ðρðx, yÞÞ p , where p > 1 is a real number. Then, d ðx, yÞ is a b-metric space with Definition 17 (see [20]). Let ðX, dÞ be a b-metric space with parameter s ≥ 1. Then, a sequence {x n } 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 b-Cauchy sequence if and only if dðx n , x m Þ ⟶ 0 as n, m ⟶ ∞, for all n, m ∈ ℕ In addition, a b-metric space is called complete if and only if each Cauchy sequence in this space is b-convergent.
Definition 19 (see [21]). 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 Cð f , gÞ denote the set of all coincidence points of f and g.
Definition 20 (see [21]). Let f and g be two self-mappings defined on a nonempty set X. Then, f and g are said to be weakly compatible if they commute at every coincidence point, that is, f x = gx ⇒ f gx = gf x, for every x ∈ Cð f , gÞ.

Example 21.
(i) f , g : R ⟶ R defined by f ðxÞ = x/3 and gðxÞ = x 2 , x ∈ ℝ. In this example, f and g have coincidence 2 Abstract and Applied Analysis point at x = 0, and x = 1/3 but f and g are not weakly compatible (ii) X = ½0, 3 equipped with the usual metric space dðx , yÞ = jx − yj Define f , g : X ⟶ X by the following: This example shows, for any x ∈ ½1, 3, f gx = gf x. Therefore, f and g are weakly compatible maps on ½0, 3.
In this study, motivated and inspired by Yan Hao and Hongyan Guan [22], we introduce the notion of generalized proximal weakly contractive mappings in b-metric spaces and prove a common best proximity point theorem for generalized proximal weakly contractive mapping defined on complete b-metric spaces.

Preliminaries
Definition 22 (see [23]). Let A and B be nonempty subsets of a metric space ðX, dÞ. We denote by A 0 and B 0 the following sets: where dðA, BÞ = inf fdðx, yÞ: x ∈ A, y ∈ Bg is the distance between A and B.
Definition 23 (see [24]). Let A, B be nonempty subset of metric space ðX, dÞ. Given a nonself-mapping T : A ⟶ B, then an element x * ∈ A is called best proximity point of the mapping if Definition 24 (see [25]). Let f , g : A ⟶ B be nonselfmappings. An element x ⋆ ∈ A is said to be a common best proximity point of the pair ðf , gÞ if this condition is satisfied: Definition 25 (see [26]). Let f , g : Lemma 26 (see [19]). Let ðX, dÞ be a b-metric space with parameter s ≥ 1. Assume that x n and y n are b-convergent to x and y, respectively. Then, we have the following: In particular, if x = y, then we have lim n⟶∞ dðx n , y n Þ = 0. Moreover, for each z ∈ X, we have the following: Definition 27 (see [22]). A function f : X ⟶ ½0,∞Þ, where ðX, dÞ is a b-metric space, is called lower semicontinuous if for all x ∈ X, and a sequence fx n g is b-convergent to x, and we have Consider the following: Ψ = fψ : ½0,∞Þ ⟶ ½0,∞Þ such that ψ is continuous and nondecreasing function}.
Also, we denote Φ = fϕ : ½0,∞Þ ⟶ ½0,∞Þ such that ϕ is nondecreasing and lower semicontinuous, and Hao and Guan [22] proved the following common fixed point results for generalized weakly contractive mapping in b-metric spaces: Theorem 28 (see [22]). Let ðX, dÞ be a complete b-metric space with parameter s ≥ 1, 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 and p ≥ 2 is a constant. If there are functions where then f and g have a unique coincidence point in X.

Abstract and Applied Analysis
Moreover, f and g have a unique common fixed point provided that f and g are weakly compatible.

Result and Discussion
Definition 29. Let ðX, dÞ be a b-metric space and A and B be two nonempty subset of a b-metric space ðX, dÞ with parameter s ≥ 1 and p ≥ 2 is a constant. A pair of map f , g : A ⟶ B is said to be a generalized proximal weakly contractive mapping, if for all x, y, h, t, r, m ∈ A, then where l d ðx, y, h, t, r, m, d, φÞ = max fdðr, mÞ + φðrÞ + φðmÞ, dð t, mÞ + φðtÞ + φðmÞg,ψ ∈ Ψ, ϕ ∈ Φ, and φ : X ⟶ ½0,∞Þ is a lower semicontinuous function.
Theorem 30. Let ðA, BÞ be a pair of nonempty subsets of a complete b-metric space ðX, dÞ, and assume that A 0 and B 0 are nonempty such that A 0 is closed. Define a pair of mapping f , g : A ⟶ B satisfying the following conditions: (iv) f and g are commute proximity Then, f and g have a unique common best proximity point.
Proof. We prove the existence of common best proximity point. Also, Continuing this process in a similar fashion, obtain the sequence fx n g and fx n+1 g in A 0 such that for each n ≥ 0.
Since f ðA 0 Þ ⊆ B 0 and A 0 is nonempty set, there exists u n ∈ A 0 such that for all n ≥ 0: Further, we obtain that for all n ≥ 0: Our first goal is to show that f u = gu, for some u ∈ A 0 . Suppose that u n = u n+1 , for some n ≥ 0, by (2) and (3), we get that Since f and g commute proximally, f u n = u n+1 = gu n , and so we are done.
Assume that u n ≠ u n+1 , for all n ≥ 0. From (3), note that for all n ≥ 1. Since a pair ðf , gÞ is generalized proximal weakly contractive map with x = x n , y = x n+1 , we have that Abstract and Applied Analysis If dðu n , u n+1 Þ + φðu n Þ + φðu n+1 Þ > dðu n−1 , u n Þ + φðu n−1 Þ + φðu n Þ, for some n ∈ ℕ, in view of (5)-(8), we have which implies ϕðdðu n , u n+1 Þ + φðu n Þ + φðu n+1 ÞÞ = 0. Hence, u n = u n+1 , a contradiction. Thus, we have It follows from (10) that {dðu n , u n+1 Þ + φðu n Þ + φðu n+1 Þ} is a nonincreasing sequence, and so there exists r ≥ 0 such that By (5), (11), and (12), we can obtain Now assume that r > 0. Taking the upper limit as n ⟶ ∞ in (15), we have which implies that ψðrÞ ≤ ψðrÞ − ϕðrÞ, a contradiction. Thus, we have It follows that Now, we claim that {u n } is a Cauchy sequence. Suppose contradiction, that is, {u n }, is not a Cauchy sequence. Then, there exists ε > 0 such that there are subsequences {u m k } and {u n k } of {u n } so that for all k ∈ ℕ with n k > m k > k, we obtain By triangular inequality in b-metric space and (19) and (20), we have Taking the upper limit as k ⟶ ∞ in the above inequality, we have Also, we have Then, by taking the upper limit as k ⟶ ∞ in (42), we have 5 Abstract and Applied Analysis which implies It is from By taking the upper limit as k ⟶ ∞ in (43), we have In similar fashion by taking the lower limit, we can obtain Since fu m k g and fu n k g satisfy equations (26) and (27), we obtain that for each k ∈ ℕ. Since f and g are generalized proximal weakly contractive mapping with x = x n k and y = x m k , we have From the definition, we have Taking the upper limit as k ⟶ ∞, we obtain Also, we have By taking the lower limit as k ⟶ ∞, we have By applying generalized proximal weakly contractive mapping with x = x n k and y = x m k , we have which implies that a contradiction to (53). Hence, the sequence fu n g is Cauchy. Since A 0 be a closed subset of the complete bmetric space X, there exists u ∈ A 0 such that By the definition of φ, we have Consider, by (2) and (3), that Since f and g are commute proximally, Now, we claim the existence of common best proximity point of f and g. Since By the assumption that f and g commute proximally, According to the assumption that f ðA 0 Þ ⊆ B 0 , there exists z ⋆ ∈ A 0 such that Next, we claim that x ⋆ = z ⋆ . Suppose that x ⋆ ≠ z ⋆ , that is, dðx ⋆ , z ⋆ ÞÞ > 0. By applying generalized proximal weakly contractive mapping with x = u and y = x ⋆ , we observe that where φ : X ⟶ ½0,∞Þ, defined by φðx, 0Þ = x 2 , and define a mapping ψ, ϕ : ½0,∞Þ ⟶ ½0,∞Þ with ψðtÞ = t, and ϕðtÞ = 35t/98. Clearly, φ is lower semicontinuous function, and ψ is continuous and nondecreasing function. Further, ϕ is nondecreasing and lower semicontinuous, and ϕðtÞ = 0 ⇔ t = 0.
Notice that f and g are continuous. Now, we check that f and g are generalized proximal weakly contractive mapping.