On fixed point theory for generalized contractions in cone metric spaces via scalarizing

Abstract

In this paper, some fixed point theorems for generalized contractions in cone metric spaces are provided. The normal condition on the underling cone is omitted. Moreover, the equivalency between the ordered boundedness and topologically boundedness, without using normality on the cone, for a subset of an ordered topological vector space is presented. The results of this article can be considered as the extension of [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc., 136(5) (2008), 1861-1869], [M. Kikkawa and T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal., 69(9) (2008), 2942-2949] and [A.P. Farajzadeh, A. Amini-Harandi, D. Baleanu, Fixed point theory for generalized contractions in cone metric spaces, Commun. Nonlinear. Sci. Numer. Simulat. 17(2)(2012) 708-712].