Fixed points of n-valued maps on surfaces and the Wecken property—a configuration space approach

Abstract

In this paper, we explore the fixed point theory of n-valued maps using configuration spaces and braid groups, focussing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane (resp. the 2-sphere S 2 ) has the Wecken property for n-valued maps for all n ∈ N (resp. all n ≥ 3). In the case n = 2 and S 2 , we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split n-valued map φ : X ⊸ X of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering q : X −→ X with a subset of the coordinate maps of a lift of the n-valued split map φ • q : X ⊸ X.

The 2-sphere is Wecken for n-valued maps

“…, then N (φ) = n by Corollary 7.3 of [8], so φ has the Wecken property. Gonçalves and Guaschi established the Wecken property for n-valued maps of the projective plane in [4]. As its title states, the present paper proves that the two-sphere is also Wecken for n-valued maps.…”

“…We include a proof so that the proof of the Wecken property for all n-valued maps of S 2 will be self-contained in this paper. Our proof is quite different from that in [4]. Proof.…”

“…Proposition 3 below is Proposition 4(c)(i) of [4]. We include a proof so that the proof of the Wecken property for all n-valued maps of S 2 will be self-contained in this paper.…”

Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses

“…The topological fixed point theory for multivalued maps has been developed in two main directions: (i) for admissible maps (in the sense of Górniewicz) and their particular cases like acyclic maps, R δ -maps, and so forth (see e.g., References [22,25,26], and the references therein), and (ii) for n-valued maps (see e.g., References [20][21][22][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]) and their generalizations like n-acyclic maps (see e.g., Reference [49]) and weighted maps (see e.g., Reference [50]). In the present paper, we will be exclusively interested in the second class of n-valued maps whose research made a big progress in the recent years.…”