By Jean-Pierre Aubin

ISBN-10: 0471036501

ISBN-13: 9780471036500

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**Extra resources for Approximation of Elliptic Boundary-value Problems**

**Example text**

Is 6 > 0 such that whenever x, y 6 S(X) and suppose Let r > O. Since X* is uniformly convex, there II% + f~l > 2 _ 8, ~ have llfx- f~l < , llx - Hi < 6. I1%+ 91+ Then IIx- HI 22. Therefore, Ilfx + f~l > 2 _ 6 forcing I I ~ x - f~l < ' . x, y e S(X) ~th Let Thus, we've shown that for IIx - MI < ~, II~x - ~1 < ' as pro~sea. The complete duality of uniform smoothness and uniform convexity is a consequence of Theorem 1 and the following: Theorem 2: Every uniformly convex Banach space is reflexive. Proof: If X is a uniformly convex Banach space, f 6 S(X*) and (xn) is a sequence of members of S(X) such that f(Xn) -- i then (Xn) is Cauchy.

K is not compact. An easy exercise for the interested reader is the following Theorem 2: Closed bounded convex subsets of uniformly convex Banach spaces have normal structure. Definition: U: Let C be a subset of the Banach space X. A map C - X is said to be non-expansive whenever for x, y E C IIUx " UYll-----II x " Yll holds. Theorem 3: Let K be a weakly compact convex subset of the Banach space X. Suppose K possesses normal structure. U: K -~ K has a fixed point. Then each non-expansive Proof: We introduce some useful notation: rx(K) = sup{llx " YlI: r(K) = fnf {rx(K): K c = {x E K: I~ K c Y E x E K] (radius of K) rCK) = rx(K) ] is a non-empty closed convex subset of K.

Notes and Remarks Theorems I and 2 of w are quite classical and can be found in V. Klee [36] and V. L. ~mulian ([52], [53], [54]). follows J. R. Giles [28]. Our presentation Related references include J. A. Clarkson [I0], D. F. Cudia ([ll], [12]), M. M. Day ([17], [18]) and V. Klee ~37], [38]). D. F. Cudia [12] contains an extensive study of the Gaussian spherical image mepping (support mapping) for domains other than the unit ball. It seems somewhat curious that for a long period the only known strictly convex spaces (in their natural norm) were already uniformly convex.

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