By Castro B. Alfonso (auth.), Djairo Guedes de Figueiredo, Chaim Samuel Hönig (eds.)
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Extra resources for Differential Equations: Proceedings of the 1st Latin American School of Differential Equations, Held at São Paulo, Brazil, June 29–July 17, 1981
L=I 2 ~i : I . (@, h i ) II~11 Now = 0 = 1 > ~n" which implies (14) sup [ ( T x , and consequently x):llxtl : l, x- Fn_l] > ~n' V Fn_ 1 . D. 8. For each result. n positive sup (Tx, dimension. This x) has to situation 40 (15) Xn : sup i n f Fn where the dimension supremum i s n. n x 6 Fn] subspaces holds the r i g h t of Fn for x):llxll ~'-n : i n f sup [ ( T x , Fn Proof. -- I , H with -n : l, side of x e Fn]. F (15). Taking the n n subspace generated by llxll 2 : ~ : I, @l . . . 3. @I . .
Let Bj n ~+, orthogonal f u n c t i o n s Fn muj > 0. L 2 r ' - a p p r o x i m a t i o n to the c h a r a c t e r i s t i c normalize them so as to have that functions where I/r' muj = 1 for + I/r j = l]. Let us = 1 . . . n. We claim be the subspace generated by the m u t u a l l y uj. So f o r I mu 2 = ~j 2 u = ~juj I muj2 @ Fn = ~j 2 and aEu, < we have 44 where c = max j =I , . . ,n a[uj, I mu2 The r e s u l t follows uj]. Hence 1 V u ~ F . n >_ ~ aFu, u-l , from t h i s and ( 2 3 ) . A similar proof for (d).
S c i . 1 (1978) 187-201. Clark, D. A V a r i a n t Indiana Univ. Thews, K. M u l t i p l e S o l u t i o n s f o r E l l i p t i c Boundary Value Problems w i t h Odd N o n l i n e a r i t i e s Math. Z. 163 (1972) 163-175. D~ I] per Problemi Acad. Naz. On the E x i s t e n c e of M u l t i p l e S o l u t i o n s f o r a Class of N o n l i n e a r Boundary Value Problems. Rend. Sem. Mat. U n i v . Padova 49 (1973) 195-204. V a r i a t i o n a l Methods f o r N o n l i n e a r E i g e n v a l u e Problems.