Generalized Kannan-type contraction and fixed point theorems

Abstract

In this paper, the generalized Kannan-type contraction in cone metric spaces over Banach algebras is introduced. The fixed point theorems satisfying generalized contractive conditions are obtained, without appealing to completeness of X or normality of the cone. The continuity of the mapping is relaxed. Furthermore, we prove that the completeness in cone metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in X. These results greatly generalize several well-known comparable results in the literature.