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Extra info for Mathematical Theories of Optimization: Proceedings of the International Conference Held in S. Margherita Ligure (Genova) November 30 – December 4, 1981
HIo ( ~ ) ~ H 2 ( and denote by ~loc,~ ~)); y'E ~oc(R+;H) ) where R+= [o,+ ~K and y' is the strong derivative of y. Given a Banach space X and shall denote by functions on ~a,b] a compact interval, we C([a,b];X) the space of all continuous X-valued [a,b] and by BV(Ka,b] ;X) the space of X-valued functions of bounded variation on ~a,bB . By wl'2(a,b;X) we shall denote the space ( YE L2(a,b;X);Y'E L2(a,b;Xi) and by AC(~a,bB;X) the space of all absolutely continuous functions y:~a,b~ ) X. By ACIoc(R+;X) we shall denote the space of functions y:R + ~ X which are absolutely continuous on every compact of R +.
Hypo-convergence equivalence it s p e c i a l i z e s are univariate. 10. bivariate  or h y p o - c o n v e r g e n c e We e x e m p l i f y We p r o c e e d limit classes. limit since by their the con- Theorem whose setting and it is p o s s i b l e of convergence. cf. by R o c k a T e l l a r the two v a r i a b l e s possibility. 1 s F ~ e/h~s F h/~-li ~ 1 ~ e~-~i F9 F~ (u,v) 29 Thus : e/h-ls F ~ = e/h-li F ~ h/e-li F ~ ~ h/e-ls F ~ and, a f o r t i o r i , h/e-ls F ~ = e/h-li F ~ imply each epi/hypo-convergence.
By a locally Lipschitz function we mean a function which is Lipschitz on bounded sets. 5) and by @°:H~H ~@(y) If If @ @ ~ R we shall ~ R the function @°(y,v) = lira sup ( ~(z+ ~ v ) - @ ( z ) ) ~--~o z--~y ~,:H :H ~-I ~ 2H the generalized gradient of @ (~7~,~8~,K13~) =(wEH;(w,v) ~@°(y,v) is convex then ~ @ for all v£ H). is just the subdifferential of @ • admits a continuous G~teaux derivative ~ @ then ~ =~ In the sequel we shall denote by H2'I(Qt), Q t = Q X 3t,T~ usual Sobolev space ( y~ L2(t,T;HI(Q )i'~ H2(~ ));Ys~L2(t,T;H)) .