Fixed points of n-valued maps, the fixed point property and the case of surfaces—A braid approach

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

“…Let Spl(X, Y, n) denote the set of split n-valued maps between X and Y . A priori, φ : X ⊸ Y is just an n-valued function, but if it is split then it is continuous by [22,Proposition 42], which justifies the use of the word 'map' in the definition. Partly for this reason, split n-valued maps play an important rôle in the theory.…”

“…In a recent paper [22], we studied some aspects of fixed point theory of n-valued maps from X to Y by introducing an equivalent and natural formulation in terms of singlevalued maps from X to the n th unordered configuration space D n (Y ) of X, where D n (Y ) is the quotient of the n th (ordered) configuration space F n (X) of X, defined by:…”

“…Configuration spaces play an important rôle in several branches of mathematics and have been extensively studied, see [15,16] for example. As in [22], a map Φ : X −→ D n (Y ) will be called an n-unordered map, and a map Ψ : X −→ F n (Y ) will be called an n-ordered map. For such an n-ordered map, for i = 1, .…”

“…There is an obvious bijection between the set of npoint subsets of Y and D n (Y ) that induces a bijection between the set of n-valued functions from X to Y and the set of functions from X to D n (Y ). If we suppose in addition that X and Y are metric spaces, then D n (Y ) may be equipped with a certain Hausdorff metric [22,Appendix], in which case this bijection restricts to a bijection between the set of n-valued maps from X to Y and the set of (continuous) maps from X to D n (Y ) [22,Theorem 8]. If φ : X ⊸ Y is an n-valued map then we shall say that the map Φ : X −→ D n (Y ) obtained by this bijection as the map associated to φ.…”

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.…”