Common Fixed Point Results for a Pair of Multivalued Mappings in Complex-Valued b-Metric Spaces

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa Department of Mathematics, Usmanu Danfodiyo University Sokoto, P.M.B., 2346 Sokoto State, Nigeria Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, P.O. Box 60, 0204, South Africa


Introduction
Fixed point theory is a well-researched area of mathematics; in particular, results concerning fixed points of contractive type mappings are found useful for determining the existence and uniqueness of solutions of various mathematical models. In this field, Banach [1] introduced the notion of contraction mapping in a complete metric space and gave a fixed point result for finding the fixed point of the contraction mapping. Later in 1969, Kannan [2] gave another contractive type mapping that demonstrated the fixed point theorem. However, in the Kannan contraction result, the continuity property required for the result of Banach was shown to be not necessary. Other authors have also studied several contraction mappings with differing properties (see, for instance, Chaterjea [3]). Since then, the theory of fixed points has been developed regarding results on finding fixed points of self and nonself mappings which are single-valued in a metric space.
Moreover, the study of fixed points for multivalued type contractive mappings was pioneered by Nadler [4] and further studied by Markin [5]. Since then, many researchers have generalized and extended various fixed point results from single-valued contractive mappings to multivalued contractive type mappings. For more literature concerning such extensions and generalizations, see, for instance, [6][7][8][9][10][11][12] and other references therein.
On the other hand, the axiomatic development of metric spaces was started by M. Fréchet, a French mathematician in the year 1906. The importance of metric spaces in the natural growth of functional analysis is huge. Several authors have drawn inspirations from the impact of this natural idea to mathematics and functional analysis in particular. Therefore, there have been several generalizations of this notion in the forms of rectangular metric spaces, semimetric spaces, quasimetric spaces, quasisemimetric spaces, D-metric spaces, cone metric spaces, and more recently the graphical rectangular b-metric spaces. We refer the reader to the following references for surveys on these generalizations [1,[13][14][15][16][17][18].
One of these generalizations in the last decade is that of Azam et al. [19,20]. They introduced the notion of complex-valued metric spaces, and some fixed point theorems for mappings with some rational inequalities were established. The central and core idea is to define rational expressions which are not well posed in the cone metric spaces, and thus, such results of analysis cannot be extended to cone metric spaces but to complex-valued metric spaces. Complex-valued metric spaces find interesting applications in many branches of mathematics such as algebraic geometry and number theory as well as in field of studies such as physics, thermodynamics, and electrical engineering.
Furthermore, the idea of b-metric was introduced in 1989 by Bakhtin [21]. Based on this presentation, Rao et al. [22] introduced the concept of fixed point theorems on complex-valued b-metric spaces which is a natural generalization of the complex-valued metric spaces. The relationship between the complex-valued b-metric and cone metric space is well known. Inspired by [22], many authors have proven the existence of fixed points of different mappings satisfying rational inequalities in the framework of complex-valued b -metric spaces (see [19,23] for more details).
In 2016, Singh et al. [24] introduced a contractive type mapping satisfying some rational inequalities. They obtained the existence of common fixed point for a pair of singlevalued mappings satisfying more general contraction conditions in the framework of complex-valued metric spaces. Since then, fixed point theory has been the center of extensive research for many authors (see, e.g., [25,26]).
Our motivation in this work is in twofolds: we extend the results of [19,24,27] from complex-valued metric spaces to complex-valued b-metric spaces; we also generalize the contraction mappings used therein by introducing a multivalued contraction mapping satisfying a general condition in complex-valued b-metric spaces. Our result also improves and strengthens the results of [20,22,28,29] and many other related results in the literature.

Preliminaries
In this section, we give some basic definitions and results which will be useful in establishing our main result. Let ℂ be the set of complex numbers and z 1 , z 2 ∈ ℂ: Also, we define partial order ≺ and ⪯ on ℂ as follows: Then, a set X satisfying such metric d c written in pair as ðX, d c Þ is called a complex-valued b-metric space.
Example 2 (see [23]). Let X = ½0, 1, define a mapping d c : X × X → ℂ by d c = jx − yj 2 + ijx − yj 2 for all x, y ∈ X: Then, ðX, d c Þ is a complex-valued b-metric space with τ = 2: Definition 3 (see [19]). Let ðX, d c Þ be a complex-valued b -metric space, and then, a point x ∈ X is Definition 4 (see [22]). Let fx n g be a sequence in a complexvalued b -metric space ðX, d c Þ and x ∈ X, then (i) x is the limit point of fx n g if for every c ∈ ℂ with 0 ≺ c, there exits n 0 ∈ ℚ such that d c ðx n , xÞ ≺ c for all n > n 0 , we write lim x n+m Þ ≺ c for all n > n 0 and n, m ∈ ℚ, then fx n g is a cauchy sequence in ðX, d c Þ (iii) ðX, d c Þ is complete if every cauchy sequence is convergent in ðX, d c Þ Lemma 5 (see [28]). Let ðX, d c Þ be a complex-valued b -metric space and fx n g be a sequence in ðX, d c Þ: Then, fx n g converges to x ∈ X if and only if |d c ðx n , xÞ | → 0 as n → ∞: Lemma 6 (see [28]). Let ðX, d c Þ be a complex-valued b -metric space and fx n g be a sequence in ðX, d c Þ: Then, fx n g is a Cauchy sequence if and only if |d c ðx n , x n+m Þ | → 0 as n → ∞: We denote by CBðXÞ the family of closed and bounded subsets of the set X: Definition 7 (see [19]). Let ðX, d c Þ be a complex-valued b -metric space and fx n g be a sequence in ðX, d c Þ: Denote sðuÞ = fz ∈ ℂ : u⪯zg and for a ∈ X and B ∈ CBðXÞ: For A, B ∈ CBðXÞ, we denote 2 Abstract and Applied Analysis Definition 9 (see [22]). Let ðX, d c Þ be a complex-valued b -metric space.
(i) Let T : X → CBðXÞ be a multivalued mapping. For x ∈ X and A ∈ CBðXÞ, define Thus, for x, y ∈ X, Definition 10 (see [23]). Let ðX, d c Þ be a complex-valued b -metric space and S, T : X → CBðXÞ be multivalued mappings.

Main Result
In this section, we state and prove our main findings in the sequel.
Theorem 11. Let ðX, d c Þ be a complete complex-valued b -metric space and let S, T : X → CBðXÞ be multivalued mappings with g.l.b. property. Let λ, μ, γ, δ : X × X × X → ½0, 1Þ be mappings such that ∀x, y ∈ X, Then, S and T have a unique common fixed point.
Proof. Let x 0 be an arbitrary point in X, then Tx 0 ≠ ∅: Pick (6). Then, we get This implies

Abstract and Applied Analysis
Since Thus, there exists some x 2 ∈ Tx 1 such that By using the g.l.b. property of S and T, we obtain that which implies that That is, from which we get Let ρ = ðλðx 0 , x 1 , aÞ + δðx 0 , x 1 , aÞÞ/ð1 − μðx 0 , x 1 , aÞÞ: Clearly, ρ < 1 ; then, we can inductively define a sequence fx n g ∈ X such that |d c ðx n , x n+m Þ | ≤ρ n | d c ðx 0 , x 1 Þ | , for n = 0, 1, ⋯,x 2n+1 ∈ Sx 2n , and x 2n+2 ∈ Tx 2n+1 : Now, for m > n and the fact that X is complex-valued b-metric space, we get That is, Letting m, n → ∞ in (21), we get |d c ðx n , x m Þ | → 0: This implies by Lemma 6 that fx n g is a cauchy sequence in X: Since X is complete, there exists u ∈ X such that x n → u as n → ∞: Next, we show that u ∈ Su and u ∈ Tu: Again, from (6) Now, since x 2n+1 ∈ Sx 2n , we have From which we obtain that This implies that there exists some u n ∈ Tu such that That is, By using the g.l.b. property of S and T, we get From d c ðu, u n Þ⪯τðd c ðu, x 2n+1 Þ + d c ðx 2n+1 , u n ÞÞ, we have Therefore, we get By letting n → ∞ in the above inequality, we get |d c ðu, u n Þ | → 0 as n → ∞: By Lemma 5, we get that lim n→∞ u n = u: Since Tu is closed, we have u ∈ Tu: Proceeding in a similar 5 Abstract and Applied Analysis fashion, it follows easily that u ∈ Su: Therefore, S and T have a common fixed point.

Corollary 12.
Let ðX, d c Þ be a complete complex-valued b -metric space and let S, T : X → CBðXÞ be multivalued mappings with g.l.b. property. Let λ, μ, γ, δ : X × X → ½0, 1Þ be mappings such that ∀x, y ∈ X: Then, S and T have a unique common fixed point.
By setting μ = γ = 0 in Theorem 11, we have the following corollary.
Corollary 13. Let ðX, d c Þ be a complete complex valued b -metric space and let S, T : X → CBðXÞ be multivalued mappings with g.l.b. property. Let λ, δ : X × X × X → ½0, 1Þ be mappings such that ∀x, y ∈ X, Then, S and T have a unique common fixed point. Also, by setting γ = δ = 0 in Theorem 11, we have the following consequence.

Corollary 14.
Let ðX, d c Þ be a complete complex-valued b -metric space and let S, T : X → CBðXÞ be multivalued mappings with g.l.b. property. Let λ, μ, γ, δ : X × X × X → ½0, 1Þ be mappings such that ∀x, y ∈ X, Then, S and T have a unique common fixed point.

Corollary 15.
Let ðX, d c Þ be a complete complex-valued b -metric space and let S, T : X → CBðXÞ be multivalued mappings with g.l.b property. Let λ, μ : X × X × X → ½0, 1Þ be mappings such that ∀x, y ∈ X, Then, S and T have a unique common fixed point. By putting S = T in our main theorem, we obtain the following corollary.