Generalized coincidence theory for set-valued maps

Abstract

This paper presents a coincidence theory for general classes of maps based on the notion of a Φ-essential map (we will also discuss Φ-epi maps).

A note on the topological transversality theorem for the admissible maps of Gorniewicz

“…(ii) It is very easy to replace "compact maps" with "k-set contractive maps" (here 0 k < 1) [9] or even "k-set countably contractive maps" (here 0 k < 1) [1] in Definition 2.1 and 2.4 and obtain the obvious analogue of the results in this paper. (iii) It is also very easy to replace "essential" with "Φ-essential" [9][10][11] and obtain the analogue of the results in this paper (the obvious details are left to the reader).…”

“…The topological transversality theorem was established by Granas [6] for single valued maps. It was extended by many authors for Kakutani maps [9,12], acyclic maps [2] and other general classes of maps [3,10,11]. In this paper we consider the topological transversality theorem for the admissible maps of Gorniewicz [4] and we obtain an "almost" topological transversality theorem.…”

A Topological Coincidence Theory for Multifunctions via Homotopy

“…The topological transversality theorem of Granas [1] states that if F and G are continuous compact single valued maps and F ∼ = G then F is essential if and only if G is essential. These concepts were generalized to multimaps (compact and noncompact) and for Φ-essential maps in a general setting (see [2,3] and the references therein). In this paper we approach this differently and we present a very general topological transversality theorem for coincidences.…”

Generalized essential maps and coincidence type theory for compact multifunctions

“…The topological transversality theorem [4] for continuous compact maps states that for continuous compact maps F and G with F ∼ = G then F is essential if and only if G is essential. The essential map theory was extended to set valued maps and to d-essential maps [6][7][8]. In this paper we consider admissible maps (see below) and we establish a very general topological transversality theorem.…”