Fixed point results for two pairs of non-self hybrid mappings in metric spaces of hyperbolic type

This research paper proves some interesting results on common fixed point for two pairs of non-self hybrid (single valued and multivalued) contractive mappings in metric spaces of hyperbolic type. The results are established without employing the weakly commutativity and continuity assumptions. We adopted an existing method of proof to obtain our results. The results generalize and improve some results proved in related works in literature. An example is given to validate our claim.


Introduction
*Fixed point theorem for single valued selfmappings in metric space was first proved by Banach (1992). Later Nadler (1969) introduced fixed point results for multivalued mappings in metric spaces. Takahashi (1970) introduced the property of convexity in metric spaces and established some fixed point theorems that generalized some results in Banach spaces. Assad and Kirk (1972) discovered that in convex metric spaces some maps are not selfmapping and proved the existence and uniqueness of the fixed point for non-self multivalued mapping in metric spaces. Kirk (1982) further introduced the concepts of metric spaces of hyperbolic type by placing Krasnoselskii's result (for = (1 − ) + for some (0, 1)) in the setting of convex metric spaces.
Definition 1.1: Let ( , ) be a metric space where X is a non-empty set and d is a mapping : × → such that for every , , (Frechet, 1906) 1 ( , ) ≥ 0, 2 ( , ) = 0 if and only if = , 3 ( , ) = ( , ), 4 ( , ) ≤ ( , ) + ( , on X is an operator : × × → which satisfies the following axioms (Takahashi, 1970), for every and . ( , ) is equipped with a convex structure, then X is known as convex metric space. Definition 1.3: Let ( , ) be a metric space and L a family of metric segment. X is called a metric space of hyperbolic type if the following axioms are satisfied (Kirk, 1982); (1.2) Some authors worked on the convergence theorems of contractive maps in metric spaces and its generalizations with applications (Okeke and Abbas, 2015;Okeke and Kim, 2015;Bishop et al., 2017). Huang et al. (2014) established a common fixed point theorem for two pairs of non-self mappings satisfying certain generalized contractive conditions of Ciric type in cone metric spaces. Ahmed and Khan (1997) established the existence and uniqueness of some common fixed point of a pair of hybrid non-self mapping in metrically convex metric spaces. The authors in Ahmed and Khan (1997) gave the following definition. Ahmed and Imdad (1998) further generalized the result of Ahmed and Khan (1997) to two pairs of hybrid non-self-mappings in the same setting. Ciric and Cakić (2009) introduced new non-self contractive mappings and proved the coincidence and common fixed point for the two pairs of hybrid mappings in complete convex metric spaces. Ciric et al. (2007) established common fixed point theorems for two pairs of non-self hybrid operators fulfilling certain generalized contraction conditions without employing the compatibility and continuity of the mappings in metrically convex metric spaces. Eke (2016) proved the existence and uniqueness of common fixed point for a pair of weakly compatible non-self operators fulfilling more general contractive conditions in metric spaces of hyperbolic type. Eke et al. (2018) introduced a new class of nonlinear contraction operators in metric spaces and proved common fixed point theorem for a pair of non-self mappings fulfilling the new contraction conditions in metric spaces of hyperbolic type.
The purpose of this research is to prove the coincidence and common fixed point theorems for two pairs of non-self hybrid mappings fulfilling certain generalized contraction conditions in metric space of hyperbolic type.

Main results
Theorem 2.1: Suppose ( , ) is a metric space of hyperbolic type and K a nonempty closed subset of X. If is a nonempty boundary of , , : → ( ) and , : → such that and ( ) and ( ) are complete then E and N have a coincidence, and F and M have a coincidence in K. Moreover if there exist u and w such that = = , then E, F, M and N have a common fixed point.
This choice is possible because c2 ϵ δ K ⊆ MK ∩ NK. Hence c2 ϵ δ K ∩ seg [b1, b2]. We can choose b3 ϵ Fa2 ⊆ K such that Continuing in the process, we develop sequence (d) bn ∉cn whenever cn ϵ δ K ∩ seg . This proves that E, F, M and N are non-self-mappings.

Remark 2.3:
Theorem 2.1 is proved in the setting of metric spaces of hyperbolic type without compatibility and continuity of the functions. Thus, Theorem 2.1 generalized Theorem 3.1 of Ahmed and Khan (1997). Theorem 2.1 is independent of Theorem 2.1 of Ciric et al. (2007) in the setting of metric spaces of hyperbolic type.