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 [Some asymptotic fixed point theorems, Trans. Amer. Math. Soc. 171 (1972), 349–375], L. Górniewicz and D. Rozpłoch-Nowakowska in [On the Schauder Fixed Point Theorem, Topology in Nonlinear Analysis, Banach Center Publications, vol. 35, Polish Academy of Sciences, Warszawa, 1996], S. Szufla in [On the application of measure of noncompactness to existence theorems, Rend. Sem. Math. Univ. Padova 75 (1986), 1–14] and D. Bugajewski in [Weak solutions of integral equations with weakly singular kernel in Banach spaces, Comm. Math. 34 (1994), 49–58].