Download PDF by Kaufman R. M.: A. F. Lavriks truncated equations

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Convexity of

(y-x^'(x)}, which proves ip'(x) G dtp(x). Conversely choose w* G dip(x). Then we get, for any y G X and t > 0, 46 Chapter 1. Dissipative and Maximal Monotone Operators Again using Gateaux differentiability of ip at x we get, for t \, 0, (y,(y,w*) for all y G X, which implies w* =

Let Q, be a bounded open set in Rd with sufficiently smooth boundary. 70) 0. We only need to consider the case a < 00. We choose a sequence (W„)„ 6 N with \u — UU\L2 —• 0 and 00. By definition of

00((^, un)Hi = (4>,v)Hi for all G HQ(Q).

Proof. We choose y G X and define the operator C by Cx = Ax — x + y, x £ dom A. , C is wdissipative with w — — 1. The operator C is also m-dissipative. Indeed, the equation z € (/ — C)x is equivalent to 2 _ 1 (z + y) g (7 — 2~lA)x, which by m-dissipativity of A has a solution x e dom A. The operator C + B = A + B — I + y is w-dissipative with to = — 1, because A + B is dissipative. 28 holds for C (and any B). 44) l i m i n f - d i s t ( r a n g e ( J - X(C + B)),x) =0 for all x G dom(C + B) = dom A.

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A. F. Lavriks truncated equations by Kaufman R. M.


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