The authors obtain several theorems of the existence and uniqueness of fixed point for contractive mappings under c-distance in cone metric spaces over Banach algebras without the assumptions of normality of cones and the continuity of mappings. The results greatly improve and complement some previous results. Moreover, a supportive example to illustrate the main assertions is also given. Otherwise, by solving the problem of existence and uniqueness for an elementary equation, a significant application for the obtained results is presented.