A remark to the Schauder fixed point theorem

Abstract

In the paper some sufficient conditions are established in order that a continuous map have a fixed point. The results are related to those obtained by R. D. Nussbaum in [< i> Some asymptotic fixed point theorems< /i> , Trans. Amer. Math. Soc. < b> 171< /b> (1972), 349–375], L. Górniewicz and D. Rozpłoch-Nowakowska in [< i> On the Schauder Fixed Point Theorem< /i> , Topology in Nonlinear Analysis, Banach Center Publications, vol. 35, Polish Academy of Sciences, Warszawa, 1996], S. Szufla in [< i> On the application of measure of noncompactness to existence theorems< /i> , Rend. Sem. Math. Univ. Padova < b> 75< /b> (1986), 1–14] and D. Bugajewski in [< i> Weak solutions of integral equations with weakly singular kernel in Banach spaces< /i> , Comm. Math. < b> 34< /b> (1994), 49–58].