Fixed point theorem with contractive mapping on C∗-Algebra valued G-metric space

Banach-Caccioppoli Fixed Point Theorem is an interesting theorem in metric space theory. This theorem states that if T: X → X is a contractive mapping on complete metric space, then T has a unique fixed point. In 2018, the notion of C∗-algebra valued G-metric space was introduced by Congcong Shen, Lining Jiang, and Zhenhua Ma. The C∗-algebra valued G-metric space is a generalization of the G-metric space and the C∗-algebra valued metric space, meanwhile the G-metric space and the C∗-algebra valued metric space itself is a generalization of known metric space. The G-metric generalized the domain of metric from X×X into X×X×X, the C∗-algebra valued metric generalized the codomain from real number into C∗-algebra, and the C∗-algebra valued G-metric space generalized both the domain and the codomain. In C∗-algebra valued G-metric space, there is one theorem that is similar to the Banach-Caccioppoli Fixed Point Theorem, called by fixed point theorem with contractive mapping on C∗-algebra valued G-metric space. This theorem is already proven by Congcong Shen, Lining Jiang, Zhenhua Ma (2018). In this paper, we discuss another new proof of this theorem by using the metric function d(x, y) = max{G(x, x, y),G(y, x, x)g.