### A remark to the Schauder fixed point theorem

DOI: http://dx.doi.org/10.12775/TMNA.2000.005

#### 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].

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].

#### Keywords

Schauder fixed point theorem; admissible couple; asymptotic fixed point theorem; proper mapping

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