Journal of Mathematical Sciences, Vol. 120, No. 1, 204 ARCWISE CONNECTEDNESS OF SETS OF SOLUTIONS TO DIFFERENTIAL INCLUSIONS V. Staicu UDC 517.977 Dedicated to the memory of Aristide Halanay 1. Introduction Consider the Cauchy problem x ′ ∈ F (t, x), x(0) = ξ, where F : [0,T ] × R n →K(R n ) is a compact-valued multifunction. For ξ ∈ R n , denote by S F (ξ ) the set of its solutions. If F is Lipschitzian with respect to x, then the set S F (ξ ) possesses a lot of interesting properties. One of them is the arcwise connectedness in the space AC ([0,T ], R n ), which was proved in [14] (see also [6, 13]). The main tool used in these papers is a continuous selection result for the solution mapping ξ →S F (ξ ) owing to Cellina [4] and improved by many authors (see [5,7,12,13]). Another approach based on the Baire category was used by De Blasi and Pianigiani [2] for proving the arcwise connectedness of S ext F (ξ ), where ext F (t, x) is the set of extreme points of F (t, x), where F (t, x) denotes a compact convex subset in R n and F is assumed to be Lipschitzian with respect to x. The case where the set of values of the right-hand side is not necessarily closed was considered in [2] and, in a more general framework, in [11]. Similar questions in infinite-dimensional spaces were studied in [8]. The aim of this paper is to prove the arcwise connectedness of the set of solutions to Lipschitzian differential inclusions on the interval [0, ∞). Consider the Cauchy problem x ′ ∈ F (t, x), x(0) = ξ, ((P ξ )) where F : [0, ∞) × R n →K(R n ) is a compact-valued multifunction Lipschitzian with respect to x. Denote by X the space of continuous functions x : [0, ∞) → R n with derivatives x ′ ∈ L 1 loc ([0, ∞), R n ) endowed with the distance d defined by the formula d(x, y)= ‖x(0) − y(0)‖ + ∞  n=1 1 2 n  n 0 ‖x ′ (t) − y ′ (t)‖dt 1+  n 0 ‖x ′ (t) − y ′ (t)‖dt . (1) Let S F (ξ ) be the set of solutions to problem (P ξ ), i.e., S F (ξ )=  x ∈ X : x(0) = ξ, x ′ (t) ∈ F (t, (x(t))) for a.e. t ∈ [0, ∞)  . We prove that S F (ξ ) is arcwise connected in (X, d). Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 107, Aveiro Seminar on Control, Optimization, and Graph Theory, 2002. 1006 1072–3374/4/1201–1006 c  204 Plenum Publishing Corporation