Edelstein-type fixed point theorems in compact TVS-cone metric spaces

Abstract

In this paper we prove two fixed point theorems in compact cone metric spaces over normal cones. The first theorem generalizes Edelstein theorem [8] and is different from the generalization obtained in [11]. The second theorem generalizes the main result in [10] and the first theorem. However, the two theorems fail in different categories. Moreover, different versions of the two theorems are proved in TVS-cone metric spaces by making use of the nonlinear scalarization function used very recently by Wei-Shih Du in [A note on cone metric fixed point theory and its equivalence, Nonlinear Analysis, 72(5), 2259-2261 (2010).] to prove the equivalence of the Banach contraction principle in cone metric spaces and usual metric spaces.