# Download Ordinary Differential Equations in Banach Spaces by Klaus Deimling (auth.) PDF

By Klaus Deimling (auth.)

**Read or Download Ordinary Differential Equations in Banach Spaces PDF**

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**Extra resources for Ordinary Differential Equations in Banach Spaces**

**Example text**

1 - ~Xn,Xn) + (Tx n - T ( O ) , X n ) + = (y - T(O) -- since exists in X T is a c c r e t i v e , in b o t h + . is d e n s e 0 ~ hence case T(O)l too. If, imply , for all however, Tx n + y If (ii) 0 _> llxnl 2 therefore . e. d. For example, condition ( T x , x ) + / I x I ÷ ~ as able to a p p l y had been U(t) Ixl ÷ ~ Banach's strict is o n l y (ii) is s a t i s f i e d . In the p r o o f fixed contractions. e. e. e. 3. bounded TI: one this convex, proof Lemma Theorem (ii) the following simple Let X be a real T(X) = X an x e K O r fixed point Banach space ; Cc X closed of n o n e x p a n s i v e maps point.

Then, 0 = t o < t I < ... < tp : b e w i t h u(O) = x o , u ( t i ) 6 D r , u is linear in and lu(t)-u(~)j ~ ( c + e ) I t - ~ I on [0,be] (it) lu'(t)-f(ti,u(ti)) (iii) if(t,x)-f(ti,u(ti))I Ix-u(ti)l has been Proof. +6 hold. < b 1 [f(t,x) - f ( t i , u ( t i ) ) I J e for s a t i s f y i n g Ix - u(ti) I ~ (c+e)6 with implicitely the tj up to t i (for some such that (e) and x e [r(Xo) we a s s u m e to JxX defined number (1) has an There (ti,ti+ 1) extended already be the largest [ti,ti+l] ! s for t E [ti,ti+l] ( e + s ) ( t i + l - t i) type: ti+l-t i _< E such that (i) I ~ e in problem of the f o l l o w i n g -- Then, i ~ O) we let and the c o n d i t i o n s g t e [ti,ti+6] and X ~ [ r ( X o ) e hold simultaneously.

O b v i o u s l y , we h a v e t ~ (ti,n,ti+l,n] lu(t) < r and c o n s i d e r ~ , as n ÷ D r and D r is closed. d. 3. 1. = D~r(X (8) lim I-~C+ such Let o) to look for approximate conditions X be a B a n a c h closed Let b < m i n { a , r / c } Then range one in D r p r o v i d e d on f w h i c h un We on the w h o l e space, , f: JXDr l-lp(x+lf(t,x),D) conditions solutions DCX range for problem of the and t~ J (1) has folZowing guarantee with un , x oE D and If(t,x)l , _< c , X~[r(Xo)m~D a solution extra situations of e v e r y , J = [O,a] c R ÷ X continuous = 0 start on [O,b] conditions with is s a t i s - fied.