In the present paper, measure differential equations involving the distributional Henstock-Kurzweil integral are investigated. Theorems on the existence and structure of the set of solutions are established by using Schauder$^prime s$ fixed point theorem and Vidossich theorem. Two examples of the main results paper are presented. The new results are generalizations of some previous results in t...

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...

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...