[Untitled]
In: Differential Equations, Jg. 37 (2001), S. 141-143
Online
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Zugriff:
In the present paper, which is a continuation of [1{5], we give an algorithm for solving an extremal problem in a Hilbert space. The algorithm is based on the method of auxiliary positionally controlled models [6]. Consider Hilbert spaces (H;jj )a nd (V;kk )( Vis densely and continuously embedded in H), a uniformly convex Banach space (U;jjU), a convex proper lower semicontinuous function ’ : H ! R + = fr 2 R : r 0g[f +1g, and two linear continuous operators B : U ! H and A : V ! V. We assume that A is a symmetric operator satisfying the coerciveness condition hAx;xiVV + !jxj 2 kxk 2 , x 2 V ,f or some > 0a nd ! 2 R .B y (x;u )w e denote the functional (x;u )=( 1 =2)hAx;xi +’(x) (Bu;x), which is convex in x. We consider the following minimization problem. The element x (u0 )=a rg minf (x;u0 ): x2 Vg corresponding to some unknown u0 2 P (where P U is a given convex bounded closed set) is measured with some error h. Namely, the measurement produces an element h 2 V such that jh x (u0)j h. The problem is to construct a procedure for nding an element uh2 P such that uh! u in U as h! 0+; u =a rg minfjujU : u2 U (x (u0))g; U (x (u0)) =fu2 P :a rg min f(x;u ): x2 Vg = x ( u 0 )g : The minimization problem for the functional (x;u0) is equivalent to the elliptic variational inequality [7, p. 39]
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Autor/in / Beteiligte Person: | Maksimov, Vyacheslav |
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Zeitschrift: | Differential Equations, Jg. 37 (2001), S. 141-143 |
Veröffentlichung: | Springer Science and Business Media LLC, 2001 |
Medientyp: | unknown |
ISSN: | 0012-2661 (print) |
DOI: | 10.1023/a:1019288803057 |
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