On nonsymmetric theorems for coincidence of multi-valued map

Abstract

Given a finite group G which acts freely on $\mathbb {S}^{n}$, H a normal cyclic subgroup of prime order, in de Mattos and dos Santos (Topol. Methods Nonlinear Anal. 33:105–120, 2009) have defined and estimate the cohomological dimension of the set $A_{\varphi }(f, H, G)$ of (H, G)-coincidence points of a continuous map $f: X \rightarrow Y$ relative to an essential map $\varphi : X \rightarrow \mathbb {S}^{n}$, where X is a compact Hausdorff space and Y is a k-dimensional CW-complex. The first aim of this work is to extend this result to a multi-valued map $F: X \multimap Y$. The second aim of this work is to estimate the cohomological dimension of the set

$$\begin{aligned} \displaystyle A(F, h)=\left\{ (x_{1},\ldots , x_p) \in X^{p} \ | \ h(x_{i+1})=T^{i}h(x_{1}), \cap _{i=1}^{p} F(x_{i})\ne \emptyset \right\} , \end{aligned}$$

where $F: X \multimap M$ is an acyclic multi-valued map and $h: X \rightarrow N$ is a continuous map such that $h^{*}: H^{n}(N) \rightarrow H^{n}(X)$ is a non trivial homomorphism, where X is a Hausdorff compact space and N is a connected closed manifold homology $n-$sphere and equipped with a free action of the cyclic group $\mathbb {Z}_{p}$ generated by a periodic homeomorphism $T: N \rightarrow N$ of prime period p and M is a connected manifold( which M and N suppose orientable if $p > 2$).