Variable Dimension Complexes Part Ii a Unified Approach to Some Combinatorial
Variable Dimension Complexes Part Ii a Unified Approach to Some Combinatorial
Robert Michael Freund
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Extend f in a piecewise linear (PWL) fashion on all of s"~ . F is continuous and maps S into S ~ . Thus there is a fixed point v of f. Let a be the smallest real simplex In C containing v, and let x be the set of vertices of ■ IR be continuous. A point x in D is said to be a stationary point of the pair (f, D ) (see Eaves [3]) if and only if there exists z € IR and y € IR such that i) y>0, z>0 ii) f(x) = y - ee iii) x. Y = iv) z(l - e*x) = Figure 2(a) Figue 2Cb) - 10 - We have the following: Le...mma (Hartman and Stampacchia [7], and Karamardian [8a], [8b], [9]). There exists a stationary point x* of (f, D^). PROOF : Our proof is based on the Generalized Covering Lemma. For 1 ■ 1 n, define C = {(x, w) e s"|f(x);fO and f^(x)0}. Note that each C^ (1 ■= 1, .... N+1) is closed, and that U. _. C D S'^. Thus by the Generalized Covering Lemma, there exists (x*, w*) in S such that i) X* > implies (x*, w*) e C, 1 = 1, ... , n, and ii) w*>0 implies (x*, w*) e c""*"-"-. We now show that x* is a stationary point of (f, D ).
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