By Klaus Deimling (auth.)
ISBN-10: 3540082603
ISBN-13: 9783540082606
ISBN-10: 3540373381
ISBN-13: 9783540373384
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Example text
Let X be a real , f: X + X c o n t i n u o u s Banaeh ~: [+ + ~ solution Then, of the (1) has Proof. 3 lim x(t) t÷~ %(0) to and problem Obviously, solution interval such on J U1 that p' p~(t) = ~(p) , for every with ~ ~ a , p(O) Xoe X . Therefore, [0,~) E 0 is the m a x i m a l in J (1) has . As a unique in the p r o o f Let h > 0 a n d %(t) = Ix(t+h)-x(t)I for t < ~-h ! ~(%(t))%(t) for t > 0 since %(t)D-%(t) exists 6> 0 such x is c o n t i n u o u s , %(0) < 6 and we therefore ~ < a implies that %(0) find 61(s) %(t) the existence < ~ implies > 0 such < s in = 0 .
E. the dual w e d g e K~ = {x ~ • X* x~(z) to , and x e ~D t h e n K and x 6 ~K If x * ~ K e a n d x~(x) To show this (4) is e q u i v a l e n t ~ > 0 . 1 are of l i t t l e use set, the a p p r o x i - since all of t h e m of D . Since we are g o i n g to i m p o s e on f at p o i n t s of D only, we h a v e no m e a n s to 52 prove that a subsequence It will t u r n out that appropriate, since, this m e t h o d of these approximate the c l a s s i c a l by m e a n s solutions Euler-Cauchy of the b o u n d a r y in such a way that at least is convergent.
D. 3. 2 like is t a k e n . ,t p only which of complicated . We w o u l d of T h e o r e m D for theorem X be a B a n a c h closed, lim k÷O+ x-lp(x+~f(t,x),D) (10) (f(t,x)-f(t~y),x-y)+ b < min{a,r/c} hand . Then sides space, D C X , J = [0,~ f: J x D r ÷ X c o n t i n u o u s ( 2 ) For r i g h t proof (1) a condition this . However, ~14] ; see . 4. Let X be a B a n a c h Dr = D~r(Xo) f satisfy the (11) condition ~(f(JxB)) where value on J is c o n t i n u o u s problem = ~(p) . 2 ~ such (it) It like is p o s s i b l e (12) for = 0 has on I ~ = , x o • D and If(t,x) I < c all [O~b] Bc D r and .