We introduce the notion of a C * -algebra-valued b-metric space. We generalize the Banach contraction principle in this new setting. As an application of our result, we establish an existence result for an integral equation in a C * -algebra-valued b-metric space.

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.

We present the extension of Caristi's fixed point theorem for mappings defined on C * -algebra valued metric spaces. We prove the existence of fixed point using the concept of minimal element in C * -algebra valued metric space by introducing the notion of partial order on X.

This article is focused on the generalization of some fixed point theorems with Kannan-type contractions in the setting of new extended b -metric spaces. An idea of asymptotic regularity has been incorporated to achieve the new results. As an application, the existence of a solution of the Fredholm-type integral equation is presented.

<abstract><p>In this article, the concept of a Hausdorff fuzzy $ b $-metric space is introduced. The new notion is used to establish some fixed point results for multivalued mappings in $ G $-complete fuzzy $ b $-metric spaces satisfying a suitable requirement of contractiveness. An illustrative example is formulated to support the results. Eventually, an application for the existence of a solution for an integral inclusion is established which involves showing the materiality of the obtained results. These results are more general and some theorems proved by of Shehzad et al. are their special cases.</p></abstract>