Common best proximity points in complex valued b-metric spaces

Abstract: In this paper, we obtain some results on common best proximity points for non-self mappings between two subsets of complex valued b-metric spaces. For this purpose, first we generalize some well-known results that were proved in classic metric spaces on complex valued b-metric space by some new definitions. Second we present a type of contractive condition and develop a common best proximity point theorem for non-self mappings in complex-valued b-metric spaces. Our results are supported by some examples.


Introduction and preliminaries
Fixed point theory focuses on solving the equation Tx = x, where T is a self-mapping defined on a subset of a metric space or other suitable spaces. If it is assumed that, T is not a self-mapping then the equation Tx = x does not have a solution. Consequently, the significant aim is determining an element x that is in close proximity to Tx in some sense. Eventually, the target is finding an element x in a metric space satisfying the following conditions that d(x, Tx) = d(A, B) and d(x, Sx) = d (A, B) where d is a metric function and d(A, B): = inf{d(x, y): x ∈ A, y ∈ B}. Now, if T, S:A → B are two non-self mappings, then the equations Sx = x and Tx = x are likely to have no solution, the solution known as a common fixed point of the mappings S and T (see, Ahmad, Azam, & Saejung, 2014;Klineam & Suanoom, 2013;Mukheimer, 2014;Rouzkard & Imdad, 2012;Sintunavarat & Kumam, 2012).
So, the purpose is finding an element x in A such that d(x, Sx) = d(A, B) and d(x, Tx) = d (A, B) where x is called the common best proximity point of mappings S and T in a metric space (see,

PUBLIC INTEREST STATEMENT
Fixed point theory focuses on solving the equation Tx = x , where T is a self-mapping defined on a subset of a metric space. If it is assumed that, T is not a self-mapping then the equation Tx = x is likely to have no solution. Consequently, the significant aim is determining an element x that is in close proximity to Tx in some sense. Eventually, the target is finding an element x in a metric space, that satisfy the following condition, The purpose of this article is generalizing some well-known results about common best proximity points that were established in the classic metric space to the complex-valued b-metric spaces. https://doi.org/10.1080/23311835.2017.1329887 Amini-Harandi, 2014;Sadiq Basha, 2012, 2013. Azam, Fisher, and Khan (2011) introduced the notion of complex-valued metric space which is a generalization of the classical metric space and established the existence of common fixed point theorems for mappings satisfying contraction conditions (see Azam et al., 2011, Theorem 4). Rao, Swamy, and Prasad (2013) introduced the notion of complex-valued b-metric spaces. The purpose of this article is generalizing some well-known results about common best proximity points that were established in the classic metric space to the complex valued b-metric space by some new definitions, presenting a type of contractive condition and developing a common best proximity point theorem for non-self mappings in the complex valued b-metric space.
Let ℂ be the set of complex numbers and z 1 , z 2 ∈ ℂ. Define a partial order ⪯ on ℂ as follows: It follows that z 1 ⪯ z 2 if and only if one of the following conditions is satisfied: In particular, we will write z 1 ⋨ z 2 if z 1 ≠ z 2 and one of (i), (ii), and (iii) is satisfied where we denote z 1 ≺ z 2 if only (iii) is satisfied. Note that Definition 1 (Azam et al., 2011) Let X be a nonempty set. Suppose that the mapping d:X × X → ℂ, satisfies: (a) 0 ⪯ d(x, y), for all x, y ∈ X and d(x, y) = 0 if and only if x = y; Then d is called a complex-valued metric on X, and (X, d) is called a complex-valued metric space.
Example 1 Let X = ℂ. Define the mapping d:X × X → ℂ for all z 1 , z 2 ∈ X, by where z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 . Clearly, the pair (X, d) is a complex-valued metric space.
Definition 2 (Rao et al., 2013) Let X be a nonempty set and s ≥ 1 be a given real number. Suppose that the mapping d:X × X → ℂ, satisfies: (a) 0 ⪯ d(x, y), for all x, y ∈ X and d(x, y) = 0 if and only if x = y; Then d is called a complex-valued b-metric on X , and (X, d) is called a complex-valued b-metric space (with constant s).
Example 2 Let X = ℂ. Define the mapping d:X × X → ℂ for all z 1 , z 2 ∈ X, by where z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 . Clearly, the pair (X, d) is a complex valued b-metric space with s = 2.
Definition 3 (Rao et al., 2013) Let (X, d) be a complex valued b-metric. Consider the following.
(d) A subset A ⊆ X is called closed whenever each limit point of A belongs to A.
(e) A sub-basis for a Hausdorff topology on X is a family F = {B(x, r):x ∈ X and 0 ≺ r}.
Definition 4 (Choudhury, Metiya, & Maity, 2014) Let A be a subset of ℂ. If there exists u ∈ ℂ such that z ⪯ u for all z ∈ A, then A is bounded above and u is an upper bound. Similarly, if there exists l ∈ ℂ such that l ⪯ z, for all z ∈ A, then A is bounded below and l is a lower bound. Similarly, for a subset A ⊆ ℂ which is bounded below if there exists a lower bound t of A such that for every lower bound l of A, l ⪯ t, then the lower bound t is called inf A.
Suppose that A ⊆ ℂ is bounded above. Then there exists q = u + iv ∈ ℂ such that z = x + iy ⪯ q = u + iv, for all z ∈ A. It follows that x <= u and y <= v, for all z = x + iy ∈ A; i.e. S = {x:z = x + iy ∈ A} and T = {y:z = x + iy ∈ A} are two sets of real numbers which are bounded above. Hence both sup S and sup T exist. Let x = sup S and ȳ = sup T. Then Any subset A ⊆ ℂ which is bounded above has supremum. Equivalently, any subset A ⊆ ℂ which is bounded below has infimum.
(iii) If every Cauchy sequence is convergent in (X, d), then (X, d) is called a complete complexvalued b-metric space.
Remark 1 In a b-metric space (X, d), the following assertions hold: (i) a convergent sequence has a unique limit; (ii) each convergent sequence is Cauchy; The following Lemmas prove like Lemmas 3 and 2 in Azam et al. (2011), respectively.
Lemma 1 Let (X, d) be a complex valued b-metric space and let {x n } be a sequence in X. Then {x n } is a Cauchy sequence if and only if |d(x n , x n+m )| → 0 as n → ∞.
Lemma 2 Let (X, d) be a complex-valued b-metric space and let {x n } be a sequence in X. Then {x n } converges to x if and only if |d(x n , x)| → 0 as n → ∞. Definition 9 Let A and B be two non-empty subsets of a complex-valued b-metric space (X, d). The mappings S:A → B and T:A → B are said to commute proximally if they satisfy the condition that Definition 10 Let A and B be two non-empty subsets of a complex-valued b-metric space (X, d) with s ≥ 1. Non-self mappings S, T:A ⟶ B are said to satisfy a L-contractive condition if there exist nonnegative numbers i where i = 1, … , 4 and s( 1 + 2 ) + 3 + s(s + 1) 4 < 1, then for each x, y ∈ A,

Given nonempty subsets
Definition 12 Let (X, d) be a complex-valued b-metric space. A mapping T:A → B is said to weakly dominate a mapping S:A → B proximally if there exists a non-negative real number < 1 s such that for all u 1 , If T dominates S then T weakly dominates S. But the converse is not true.
Example 3 Let us consider the complex valued b-metric space (X, d) with s = 2, where X = ℂ and let d:X × X ⟶ ℂ be given as where z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 . Let A and B be two subsets of X given by i for each non-negative real number < 1.
But obviously, we have that for = 1 64 , T weakly dominates S proximally.

Common best proximity point by weakly dominate proximally property
Theorem 1 Let (X, d) be a complete complex-valued b-metric space with s ≥ 1, A and B be two nonempty subsets of X. Assume that A 0 and B 0 are nonempty and A 0 is closed. Let S:A → B and T:A → B be two non-self mappings that satisfy the following conditions: (a) T weakly dominates S proximally (b) S and T commute proximally (c) S and T are continuous

Then there exists a unique element x ∈ A such that
Proof Let x 0 be a fixed element in A 0 . Since S(A 0 ) ⊆ T(A 0 ), then there exists an element x 1 ∈ A 0 such that Sx 0 = Tx 1 . Then by continuing this process we can choose x n ∈ A 0 such that there exists x n+1 ∈ A 0 satisfying since S(A 0 ) ⊆ B 0 , there exists an element u n ∈ A such that By choosing x n and u n , it follows that Since T weakly dominates S proximally then we have where < 1 and and We focus on Re d(u n , u n+1 ) and conclude for Im d(u n , u n+1 ) and finally for d(u n , u n+1 ), We will prove that {u n } is a Cauchy sequence. We distinguish two cases.
Case I. Suppose that so we get that
Let m, n ∈ N and m > n, we have By (4) and s < 1, we conclude that

Case II. Assume that
Put h = ∕2 1− ∕2 < 1, (note that sh < 1), so we have that Like above, for any m > n where m, n ∈ N we have Similarly, we can conclude that for any m > n where m, n ∈ N This implies that for any m > n, where m, n ∈ N Then {u n } is a Cauchy sequence and since X is complete and A 0 is closed, theRe exists u ∈ A 0 such that u n → u. By hypothesis, mappings S and T are commuting proximally and by (3) we have that Since T and S are continuous it implies that As Su ∈ S(A 0 ) ⊆ B 0 , there exists an x ∈ A 0 such that Since S and T commute proximally, Sx = Tx. Also, Sx ∈ S(A 0 ) ⊆ B 0 , there exists a z ∈ A 0 such that Re d(u n , u n+1 ) ≤ h n Re d(u 0 , u 1 ).
Re d(u n , u m ) → 0 as m, n → ∞.  Since T weakly dominates S then from (5) and (6), we can conclude that It follows that x = z, therefore we have that We now show that S and T have unique common best proximity point. For this, assume that x * in A is a second common best proximity point of S and T, then Since T weakly dominate S proximally then from (7) and (8), we have Consequently, x = x * and S and T have a unique common best proximity point. ✷ Example 4 Let us consider the complex-valued b-metric space (X, d) with s = 2, where X = ℂ and let d:X × X ⟶ ℂ be given as where z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 . Let A and B be two subsets of X given by Then A and B are closed and bounded subsets of X such that Let T, S:A ⟶ B be defined as and Therefore T and S satisfy the properties mentioned in Theorem 1. Hence the conditions of Theorem 1 are satisfied and 1 + 0i is the unique common best proximity point of S and T. By Theorem 1, we obtain the following results in the fixed point theorem.
Corollary 1 Let (X, d) be a complex-valued b-metric space with s ≥ 1. Let S, T:X → X be continuous mappings and T commutes with S. Further let S and T satisfy S(X) ⊆ T(X) and there exists a constant < 1 s such that for every x, y ∈ X where Then S and T have a unique common fixed point.
If T is assumed to be identity mapping in Corollary 1, then we have the following result.
Corollary 2 Let (X, d) be a complex-valued b-metric space with s ≥ 1, S be a continuous self-mapping on X and there exists a constant < 1 s such that for every x, y ∈ X where Then S has a unique fixed point.
Let m, n ∈ N and m > n, we have By (11) and sh < 1, we conclude that Therefore, {u n } is a Cauchy sequence and there exists u ∈ A 0 such that u n → u as n → ∞. Also, we have that Since S and T commute proximally, we get that Thus, it follows that Tu = Su, because S and T are continuous. Since Su ∈ S(A 0 ) ⊆ B 0 , there exists x ∈ A 0 such that Therefore, Tx = Sx, because S and T commute proximally. Since Sx ∈ S(A 0 ) ⊆ B 0 , there exists z ∈ A 0 , it implies that By L-contractive condition, we get that   Therefore, Su = Sx. From (12) and (13) Suppose that x * is another common best proximity point of the mappings S and T so that Since S and T commute proximally, then Sx = Tx and Sx * = Tx * . So we have which implies that Sx = Sx * . Since the pair (A, B) satisfies weak P-property, from (15) and (16) we have that Eventually, we have that x = x * . Hence S and T have a unique common best proximity point. It is verified that the (A, B) satisfies the weak P-property. Also T and S satisfy the properties mentioned in Theorem 2. Hence the conditions of Theorem 2 are satisfied and it is seen that 0 = 0 + i0 is the unique common best proximity point of S and T. If we suppose that S and T are self-mappings on X, then Theorem 2 implies the following common fixed point theorem, that generalizes and complements the results of Hardy and Rogers (1973), Jungck (1976), Reich (1971aReich ( , 1971b and others in complex-valued b-metric spaces.   T(0 + iy) = 1 + iy for each 0 ≤ y ≤ 1 S(0 + iy) = 1 + i y 4 for each 0 ≤ y ≤ 1.