Fixed point theorems for $b$-generalized contractive mappings with weak continuity conditions

The purpose of this paper was to introduce several $b$-generalized contractive mappings in the framework of cone $b$-metric spaces over Banach algebras. The obtained contractions generalized and extended the counterparts in metric spaces, cone metric spaces, and $b$-metric spaces. Moreover, via weakening the completeness of the spaces, we gave some fixed point theorems for asymptotically regular mappings without considering the orbital continuity and $k$-continuity of the mappings. Those who need a specification is our results do not rely on the continuity of $b$-metric and the normality of cones. In addition, some nontrivial examples were presented to illustrate the superiority of our fixed point theorems.