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
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 \(p > 2\)).
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Notes
A map \(\varphi : X \rightarrow \mathbb {S}^{n}\) is said to be an essential map if \(\varphi \) induces nonzero homomorphism \(\varphi ^{*}: H^{n}(\mathbb {S}^{n},\mathbb {Z}_{p}) \rightarrow H^{n}(X,\mathbb {Z}_{p})\)
A function \(g: K \rightarrow \mathbb {R}\) is said to be quasiconcave if, for all \(c \in \mathbb {R}\), the set \(K_{c} = \left\{ y \in K \ | \ g(y) \ge c\right\} \) is convex.
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de Andrade, A.E.R., Penteado, N.C.L. & Ura, S.T. On nonsymmetric theorems for coincidence of multi-valued map. J. Fixed Point Theory Appl. 25, 65 (2023). https://doi.org/10.1007/s11784-023-01064-w
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DOI: https://doi.org/10.1007/s11784-023-01064-w