# Variational and Non-variational Methods in Nonlinear by Dumitru Motreanu, Vicentiu D. Radulescu

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 -----+ .

18] G. Lebourg, Valeur moyenne pour gradient generalise, C. R. Acad. Sci. Paris 281 (1975), 795-797.  D. Motreanu and V. V. Motreanu, Coerciveness Property for a Class of Nonsmooth Functionals, Z. Anal. Anwend. 19 (2000), 1087-1093.  D. Motreanu, V. V. Motreanu and D. Paf_lca, A version of Zhong's coercivity result for a general class of nonsmooth functionals, Abstr. Appl. , to appear.  D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Q1wlitative Properties of the Solutions of Hemivariational Ineq1talities and Applications, Kluwer Academic Publishers, Nonconvex Optimization and Its Applications, Vol.