Some new theorems of expanding mappings without continuity in cone metric spaces

Abstract

In this paper, fixed point theorems for one mapping and common fixed point theorems for two mappings satisfying generalized expansive conditions are obtained. The mappings are not necessarily continuous and the cone is not normal. These results improve and generalize several well-known comparable results in (Aage and Salunke, in Acta Mathematica Sinica, English Series 27(6):1101-1106, 2011). Moreover, examples are given to support our new results. MSC: 54H25; 47H10

A hybrid class of expansive-contractive mappings in cone b-metric spaces

“…The theorem extends and generalizes the results of Han and Xu [8] to cone b-metric spaces and to four maps. Proof…”

“…s = 1), and if S = T , f = g, a 4 = a 5 = 0 in (2.1), with condition (2.13) satisfied, we obtain Theorem 2.1 in [25]. With s = 1, S = T = I d X in (2.1), and condition (2.13) satisfied, we obtain the following corollary (of Theorem 2.1), which is the properly stated second main result of Han and Xu [8]: ( f x, y) + a 5 d(gy, x) for all x, y ∈ X , where the real numbers a i satisfy a i ≥ 0 for i ≥ 2, and such that (2.13) holds, i.e., a 1 + a 2 + a 3 > 1, a 2 ≤ 1 + a 4 and a 3 ≤ 1 + a 5 . Then f and g have a common fixed point.…”

“…[1,23,25] The most recent and perhaps most general theorems on expanding mappings in cone metric spaces were initially proposed by Han and Xu [8] who considered a pair of surjective expansion mappings. [27] was an erratum of the results in [8] and a valid generalization of the results of Aage and Salunke [1]. Even though the maps considered in [27] are not necessarily continuous, the coefficients a i in the expanding conditions are taken to be positive.…”

“…However, with the generalization of metric spaces to b-metric spaces [3], cone metric spaces [10,15,16,20] and recently, cone b-metric spaces [11], fixed point theorems for expanding mappings have been proved in cone metric spaces (e.g. [1,23,25] The most recent and perhaps most general theorems on expanding mappings in cone metric spaces were initially proposed by Han and Xu [8] who considered a pair of surjective expansion mappings. [27] was an erratum of the results in [8] and a valid generalization of the results of Aage and Salunke [1].…”

Tripled random coincidence point and common fixed point results of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras

“…They greatly expanded the famous Banach contraction principle in such setting. Since then, a lot of papers have appeared on cone metric spaces and b-metric spaces (see [1,5,11,15,21,28,33,35]). Hussain and Shah [20] introduced cone b-metric space and generalized both cone metric space and b-metric space.…”