Generalized Reich–Ćirić–Rus-Type and Kannan-Type Contractions in Cone b -Metric Spaces over Banach Algebras

Abstract

In this paper, we firstly introduce the generalized Reich‐Ćirić‐Rus-type and Kannan-type contractions in cone $b$ -metric spaces over Banach algebras and then obtain some fixed point theorems satisfying these generalized contractive conditions, without appealing to the compactness of $X$ . Secondly, we prove the existence and uniqueness results for fixed points of asymptotically regular mappings with generalized Lipschitz constants. The continuity of the mappings is deleted or relaxed. At last, we prove that the completeness of cone $b$ -metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in $X$ . Our results greatly extend several important results in the literature. Moreover, we present some nontrivial examples to support the new concepts and our fixed point theorems.