$C^{*}$-Valued contractive type mappings

Abstract

In this paper we generalize the notion of C * -valued contraction mappings, recently introduced by Ma et al., by weakening the contractive condition introduced by them. Using the new notion of C * -valued contractive type mappings, we establish a fixed point theorem for such mappings. Our result generalizes the result by Ma et al. and those contained therein except for the uniqueness.

Fixed Point Theorems for Cyclic Contractions inC⁎-Algebra-Valuedb-Metric Spaces

“…, ∈ ( ) satisfy ∑ ∞ =1 ‖ ‖ 2 < 1. Later, many authors extend and improve the result of Ma et al For example, in [23], Batul and Kamran generalized the notation of * -valued contraction mappings by weakening the contractive condition introduced by Ma et al (the mapping is called * -valued contractive type mappings) and establish a fixed point theorem for such mapping which is more generalized than the result of Ma et al; in [24], Shehwar and Kamran extend and improve the result of Ma et al [22] and Jachymski [25] by proving a fixed point theorem for self-mappings on * -valued metric spaces satisfying the contractive condition for those pairs of elements from the metric space which form edges of a graph in the metric space. In 2015, Ma and Jiang [26] introduced a concept of *algebra-valued -metric spaces which generalize an ordinary * -algebra-valued metric space and give some fixed point theorems for self-map with contractive condition on such spaces.…”

An equivalence of results in $C^*$-algebra valued b-metric and b-metric spaces

“…[21], [7] and [26] can be derived from the existing corresponding fixed point theorems in the setting of the standard metric space in the literature.…”

“…Similar to the relation between a metric and a b-metric, Ma and Jiang [20] then generalised the notion of a C * -algebra-valued metric to a C * -algebra-valued b-metric. The classes of C * -algebra-valued metric spaces and C * -algebra-valued b-metric spaces were then studied by Batul and Kamran [7], Shehwar and Kamran [26], Klin-eama and Kaskasem [19], Qiaoling et al [24], Shehwar et al [25], Tianqing [27].…”

C*-Algebra Valued Fuzzy Soft Metric Spaces and Results for Hybrid Pair of Mappings

“…In the year 2014, Ma et al [7] introduced the concept of C * -algebra valued metric space and established some fixed point results. Later, Alsulami et al [32] suggested some remarks on C * -algebras and proved Banach type contraction result, this line of research was continued in (see [8,[10][11][12]31,34,35]).…”