Assad-Kirk-Type Fixed Point Theorems for a Pair of Nonself Mappings on Cone Metric Spaces

Abstract

New fixed point results for a pair of non-self mappings defined on a closed subset of a metrically convex cone metric space which is not necessarily normal are obtained. By adapting Assad-Kirk's method the existence of a unique common fixed point for a pair of non-self mappings is proved, using only the assumption that the cone interior is nonempty. Examples show that the obtained results are proper extensions of the existing ones.

New common fixed point theorem for a family of non-self mappings in cone metric spaces

“…Sumitra et al [24] generalized the fixed point theorems of Ciric and Ume [6] for a pair of non-self mappings to non-normal cone metric spaces. In the same time, Sumitra et al's [24] results extended the results of Jankovic et al [15] and Radenovic and Rhoades [17]. Motivated by Sumitra et al [24], we prove a common fixed point theorem for a family of non-self mappings on cone metric spaces in which the cone need not be normal and the condition is weaker.…”

“…Recently, Huang and Zhang [9] generalized the concept of a metric space, replacing the set of real numbers by ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. Subsequently, the study of fixed point theorems in such spaces is followed by some other mathematicians, see [1,2,11,14,15,17,20,25,26]. The study of fixed point theorems for non-self mappings in metrically convex metric spaces was initiated by Assad and Kirk [4].…”

“…Jankovic et al [15] proved new common fixed point results for a pair of non-self mappings defined on a closed subset of metrically convex cone metric space which is not necessarily normal by adapting Assad-Kirks method. Huang et al [10] proved a fixed point theorem for a family of non-self mappings in cone metric spaces which generalizes the result of Jankovic et al [15]. Sumitra et al [24] generalized the fixed point theorems of Ciric and Ume [6] for a pair of non-self mappings to non-normal cone metric spaces.…”

Common fixed point theorems for weakly compatible non-self mappings in metric spaces of hyperbolic type

“…Some authors studied common fixed point theorems for non-self mappings in metric spaces of hyperbolic type [See: 8,9]. Motivated by Jankovic et al [7], we prove some common fixed point theorems for a pair of weakly compatible non-self mappings satisfying a generalized contraction condition in the setting of metric space of hyperbolic type. Throughout our consideration, we suppose that (X, d) is a metric space which contains a family L of metric segments (isometric images of real line segment) such that a) each two points x, y ∈ X are endpoints of exactly one number seg[x, y] of L, and…”

“…The study of fixed point theorems for multivalued non-self mappings in a metric space (X, d) was initiated by Assad [2] and Assad and Kirk [3]. Many authors have studied the existence and uniqueness of fixed and common fixed points result for nonself contraction mappings in cone metric spaces [see; 4,5,6,7 ]. Some authors studied common fixed point theorems for non-self mappings in metric spaces of hyperbolic type [See: 8,9].…”