The Brouwer Fixed Point Theorem states that any continuous mapping from a closed ball in Euclidean space to itself has at least one fixed point. This theorem has a wide variety of applications in areas such as differential equations, economics, and game theory. Although this is fundamentally an analytic and topological statement, there exists an elegant combinatorial proof using Sperner’s Lemma...

We study the Tychonoff Compactness Theorem for several different definitions of a compact space.

The Four Functionals Fixed Point Theorem is a generalization of the original, as well as the functional generalizations, of the Leggett–Williams Fixed Point Theorem. In the Four Functionals Fixed Point Theorem, neither the upper nor the lower boundary of the underlying set is required to map below or above the boundary in the functional sense. As an application, the existence of a positive solu...

The Lefschetz Fixed Point Theorem generalizes a collection of fixed point theorems for different topological spaces, including maps on the n-sphere and the n-disk. Although the theorem is easily written in terms of compact manifolds, in this paper we will work entirely with topological spaces that are simplicial complexes or retracts of simplicial complexes. After developing the fundamentals of...

1. INTRODUCTION. The proof of the Brouwer fixed-point Theorem based on Sperner's Lemma [S] is often presented as an elementary combi-natorial alternative to advanced proofs based on algebraic topology. See, for example, Section 6.3 of [P1]. One may ask if this proof is really based on ideas completely different from the ideas of algebraic topology (and, in particular, the ideas of Brouwer's own...

A. Abian 1] proved the following theorem: Let f be a mapping from a nite set D. Then f has a xed point if and only if D is not a union of three mutually disjoint sets A, B and C such that A \ fA] = B \ fB] = C \ fC] = ;: (The range of f is not necessarily the subset of its domain). The proof of the suuciency is by induction on the number of elements of D. A. M akowski and K. Wi sniewski 13] sho...

Michael Handel proved in [7] the existence of a fixed point for an orientation preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments [9]. These methods per...

In this paper, we prove some common fixed point results for two self mappingsf and g on S-metric space such that f is a g.w.c.m with respect to g.