A remark on directional contractions

Abstract:

Let X be a Banach space and D a convex subset of X. A mapping $T:D \to X$ is called a directional contraction if there exists a constant $\alpha \in (0,1)$ such that corresponding to each $x,y \in D$ there exists $\varepsilon = \varepsilon (x,y) \in (0,1]$ for which $\left \| {T(x + \varepsilon (y - x)) - T(x)} \right \| \leqslant \alpha \varepsilon \left \| {x - y} \right \|$. Tests for lipschitzianness are obtained which yield the fact that if a closed mapping is a directional contraction, then it must be a global contraction, and sufficient conditions are given under which a nonclosed directional contraction $T:D \to D$ always has a fixed point.