X satisfying d(H(w, t), w) ~ t, f(H(w, t)) ~ f(w)- (}t, whenever wE B 0 (u) and t E [0, 8]. Proof. (e)), t) for some small 8' > 0. It is easy to observe that K has the properties required in the statement of the proposition. Conversely, let K: Bf>(u) x [0, 8]---+ X be a map as in the statement of the proposition. (e)- CYt:::::; JL- CYt. The proof is complete. • Remark 1.
Hence there exists a continuous extension w : K -----+ X of v such that w = 0 on K* and, for any t E K, llw(t) II :::; 1. 1) we choose small variations of the path p: qh(t) = p(t)- hw(t), where h > 0 is sufficiently small. 2) In what follows, c > 0 is fixed, while h-----+ 0. Let th E K be such that f(qh(th)) = 7/J(qh)· We can choose hn-----+ 0 such that the sequence {thn} converges to some to, and it is obvious that to E B(p). 2) yields -"" < f(p(th)- hw(th))- f(p(th)) < f(p(th)- hw(to))- f(p(th)) ~h h + f(p(th)- hw(th))- f(p(th)- hw(to)) h Since f is locally Lipschitz and thn -----+ .
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