Fixed points of non-self almost contractions

Abstract

Let X be a convex metric space, K a non-empty closed subset of X and T : K → X a non-self almost contraction. Berinde and Pacurar [Berinde, V. and P ˘ acurar, M., Fixed point theorems for nonself single-valued almost contractions, Fixed Point Theory, 14 (2013), No. 2, 301–312], proved that if T has the so called property (M) and satisfies Rothe’s boundary condition, i.e., maps ∂K (the boundary of K) into K, then T has a fixed point in K. In this paper we observe that property (M) can be removed and, hence, the above fixed point theorem takes place in a different setting.