Fixed points of non-self almost contractions

Description

Let $X$ be a convex metric space, $K$ a non-empty closed subset of $X$ and $T: K \rightarrow X$ a non-self almost contraction. Berinde and Păcurar [Berinde, V. and Păcurar, 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 $\partial K$ (the boundary of $K$ ) into $K$ , then $T$ has a fixed point in $K$ . In this paper we observe that property $\left( M\right)$ can be removed and, hence, the above fixed point theorem takes place in a different setting.