Juliusz Schauder Center Winter School Topological Methods in Nonlinear Analysis Lecture Notes in Nonlinear Analysis Volume 11, 2010, 1–1 ON THE WAŻEWSKI RETRACT METHOD Grzegorz Gabor Abstract. The original version of the Ważewski theorem as well as its newer formulations are presented. Several applications of the Ważewski method in differential equations are given. In the second part some multi- valued Ważewski type theorems are provided with open problems finishing this note. 1. Ważewski’s retract method in dynamical systems Consider the following Cauchy problem: (1.1)  ˙ x(t)= f (x(t)) for t ≥ 0, x(0) = x 0 ∈ Ω, where Ω ⊂ R n and f :Ω → R n is so regular that the problem has a unique local solution for every x 0 ∈ Ω, which depends continuously on the initial condition (we can think about a locally Lipschitz continuous map). This implies that a local semiflow π: D → R n , where D is an open subset of Ω × [0, ∞) containing 0, is given. It means that, for every x ∈ Ω, the set {t ≥ 0 | (x,t) ∈ D} is an interval [0,ω x ) for some 0 <ω x ≤∞ and (i) ω π(x,t) = ω x − t for each x ∈ Ω and t ∈ [0,ω x ), (ii) π(x, 0) = x for every x ∈ Ω, (iii) π(x,s+t)= π(π(x,t),s) for each x ∈ Ω and s,t> 0 such that s+t<ω x . 2010 Mathematics Subject Classification. Primary 54H20; Secondary 34C25, 37B30. Key words and phrases. homotopy index, differential inclusions, multivalued maps, equi- librium, exit sets, single-valued approximations. c 2010 Juliusz Schauder Center for Nonlinear Studies