A Remark to the Schauder Fixed Point Theorem

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 [18], L. Górniewicz and D. RozpłochNowakowska in [12], S. Szufla in [21] and D. Bugajewski in [6]. The famous Schauder Fixed Point Theorem [20] has been generalized in various directions by using different methods. For references, see [4]–[6], [9], [10], [12], [18], [19], [21] and [24]. Closely related with a generalization of that theorem is a long-standing conjecture in the fixed point theory which was formulated by R. D. Nussbaum in 1972 in [18] and which reads as follows: Let M be a closed, bounded convex set in a Banach space and T : M → M a continuous map. Assume that there exists an integer n 1 such that T n is compact. Then T has a fixed point. R. D. Nussbaum proved this conjecture with the additional assumption that T restricted to an appropriate open set is continuously Fréchet differentiable. Using algebraic topology methods, especially the generalized Lefschetz number, he proved a series of asymptotic fixed point theorems, that is, theorems in which 2000 Mathematics Subject Classification. 47H10.