Common fixed point theorems in C∗ -algebra-valued metric spaces

Abstract

In this paper, we establish coincidence fixed point and common fixed point theorems for two mappings in complete C * -algebra-valued metric spaces which satisfy new contractive conditions. Some applications of our obtained results are given. c 2016 all rights reserved.

Remarks on Common Fixed Point Results in C^∗-Algebra-Valued Metric Spaces

“…Hence, the Theorem 1.1 coincides with the result of Ma et al [5]. Also, the authors of [7] proved the following theorem as a corollary. In next section, we show that every point x ∈ X , T n x is a fixed point of T whenever m > n.…”

“…We conclude that the common fixed point results of Xin et al [7] are not new in literature. Infact, many fixed point results as well as common fixed point results in C * -algebra-valued metric spaces can be easily deduced from their metric counterparts.…”

“…Remark 2.6. In a similar fashion, one can see that the Theorem 2.9 and Theorem 2.10 in Xin et al [7] are equivalent to Theorem 1.3 with (C2) and (C3) contraction principles, respectively.…”

“…In this section, we show that the results presented by Xin et al [7] do not produce any new idea in literature. for any x, y ∈ X and a ∈ A with a < 1.…”

“…Many researchers have already done their research in this structure but surprisingly Kadelburg and Radenović [4] and Alsulami et al [2] observed that the all these results in this structure can be directly deduced as consequences of different fixed point results in standard metric and other related structures of metric counterpart. Recently, Xin et al [7] presented common fixed point results on C * -algebra-valued metric spaces. They established the following result.…”