$C^{\ast}$-Algebra-valued b-metric spaces and related fixed point theorems

Abstract

Based on the concept and properties of C * -algebras, the paper introduces a concept of C * -algebra-valued b-metric spaces which generalizes the concept of C * -algebra-valued metric spaces and gives some basic fixed point theorems for self-map with contractive condition on such spaces. As applications, existence and uniqueness results for a type of operator equation and an integral equation are given. MSC: 47H10; 46L07

Fixed Point Theory Using ψ Contractive Mapping in C ∗ -Algebra Valued B-Metric Space

“…In addition, the notion of C * -algebra-valued metric space is generalized to that of C * -algebravalued b-metric space, where b is an element of C * -algebra greater than 1 and the triangle inequality is modified into d(x, y) ≤ b ( d(x, z) + d(z, y)). Then, various fixed point theorems are obtained for self-map with contractive condition [2]. Besides, though Alsulami et al [3] investigated that fixed point results in C * -algebra-valued metric space can be obtained by using the classical Banach fixed point theorems, C * -algebra valued b-metric space is still an interesting and challenging topic with promising applications, which has not received sufficient consideration in fixed point theory [4,5].…”

Quasi (s,r)-Contractive Multi-Valued Operators and Related Fixed Point Theorems

“…As it is well known to all, the proverbial Banach contraction mapping principle is a very useful, simple and classical tool in modern mathematics, and has been widely used in many branches of mathematics and physics. Many mathematicians have researched and generalized the Banach contraction mapping principle along different directions, such as the fixed point theorem of fuzzy metric spaces, C * -algebra valued metric spaces and so on [1][2][3][4][5]. In general the theorem has been extended in two directions.…”

Determination of the Some Results for Coupled Fixed Point Theory in C*-Algebra Valued Metric Spaces

“…Here, we study on C * -algebra valued metric space and give some examples.The idea of this metric is to replace the set of real numbers by the positive cone C *algebra.Notation of the set of positive elements on the C * -algebras was introduced in [5]. There is a new version of the C * -algebra valued metric space in [6]. Readers also have a look references [1], [7] and [10]- [14] for C * -algebras and fixed point theory.…”