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In this research, we first check that the abstract cone b-metric is discontinuous in the case of normal cone by a counter example. We obtain several meaningful results about generalized b-quasicontraction and Ćirić-type b-quasicontraction in cone b-metric spaces over Banach algebras, weakening certain important conditions of the spaces and the mappings. Meanwhile, several valid examples are given to demonstrate the new notions and fixed point results, when the existing theorems in the literature are not applicable.