Remarks on Generalized Contractions

Abstract

The main result concerns an extension of Schauder's fixed point theorem to not necessarily continuous maps $T$ which satisfy

∥ T x - T y ∥ \leq a ∥ x - y ∥ + b max {∥ T x - x ∥, ∥ T y - y ∥} with a \geq 0, b \in (0, 1) .

Dietrich Göhde, Remarks on Generalized Contractions. Z. Anal. Anwend. 27 (2008), no. 1, pp. 119–124