Some New Theorems on c-Distance without Continuity in Cone Metric Spaces over Banach Algebras

Abstract

The fixed point theorems for one mapping and the common fixed point theorems for two mappings satisfying generalized Lipschitz conditions are obtained, without appealing to continuity for mappings or normality for cone in the conditions. Furthermore, we not only get the existence of the fixed point but also get the uniqueness. These results greatly improve and generalize several well-known comparable results in the literature. Moreover, example is given to support our new results.

Fixed Point Results under Generalized c-Distance in Cone b-Metric Spaces Over Banach Algebras

“…Now, in Theorem 1, set s = 1. We obtain the same Theorem 13 of Han and Xu [17] as follows: for all x, y ∈ X, where α, β, γ ∈ P such that γ commutes with α + β + γ + 2δ and…”

“…Our results are useful, since a generalized c-distance in cone b-metric spaces over a Banach algebra is not equivalent to a wt-distance in b-metric spaces. In addition, we remove the continuity condition of the mapping f and the normality condition of the cone P. Moreover, if we consider s = 1, then we can obtain same results with respect to a c-distance in cone metric spaces over a Banach algebra (see [17]). Further, two examples are considered for support our definitions and theorems.…”

“…Let m, n ∈ N with m > n ≥ 1. Using Equation (16) and (q 2 ), we deduce q(x n , x m ) sq(x n , x n+1 ) + s 2 q(x n+1 , x n+2 ) + · · · + s m−n q(x m−1 , x m ) (17) (sh n + s 2 h n+1 · · · + s m−n h m−1 )q(x 0 , x 1 ) (e − sh) −1 sh n q(x 0 , x 1 ).…”

“…In addition, they considered some examples to support their results. Recently, in this manner, Han and Xu [17] proved some common fixed point results and fixed point theorems without the hypothesis of continuity of the mappings and the normality of the cone.…”

Common Fixed Points of Two Mappings regarding a Generalizedc-Distance over a Banach Algebra

“…In 2013, Liu and Xu [14] offered a cone metric space over Banach algebra by replacing a Banach space E with a Banach algebra E. After this definition, some other researchers suggested new several various spaces over a Banach algebra and extended available results in [15][16][17] and their references. In 2015, Huang et al [18] proposed a c-distance in cone metric spaces over a Banach algebra E. In 2018, Han and Xu [19] proved some common fixed point results by removing the assumption of continuity of the mappings and by deleting the hypothesis of the normality of the cone. Recently, Arabnia et al [20] suggested a generalized c-distance in cone b-metric spaces over a Banach algebra.…”