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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Application of the extremum principle to investigating certain extremal problems

Authors: L. Mikołajczyk and S. Walczak
Journal: Trans. Amer. Math. Soc. 259 (1980), 147-155
MSC: Primary 49B21
MathSciNet review: 561829
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Abstract: Denote by C, K, X, respectively, a complex plane, the disc $ \{ z\, \in \,{\textbf{C:}}\,\left\vert z \right\vert\, < \,1\} $ and any compact Hausdorff space. Denote by P a set of probabilistic measures defined on Borel subsets of the space X. For $ \mu \, \in \,P$, let $ f(z)\, = \,\int_X {q(z,\,t)\,d\mu } ,\,z\, \in \,K$, and $ {\mathcal{F}}\, = \,\{ f:\,\mu \, \in \,P\} $. Consider a finite sequence of real functions $ {F_0},\,{F_{1,}}\, \ldots ,\,{F_m}$ defined in the space $ {R^{2n}}$. Let $ {\zeta _1},\, \ldots ,\,{\zeta _k}$ be fixed points of the disc K and $ \eta (f)\, = \,[\operatorname{re} \,{f^{(0)}}(\zeta ),\,\operatorname{im} \,{f... ...e} \,{f^{({n_k})}}({\zeta _k}),\,\operatorname{im} \,{f^{({n_k})}}({\zeta _k})]$, where $ f\, \in \,{\mathcal{F}},\,n\, = \,{n_1}\, + \, \cdots \, + \,{n_k}\, + \,k$. Let $ {F_j}(f)\, = \,{F_j}(\eta (f)),\,j\, = \,0,\,1,\, \ldots ,\,m$. We consider the following extremal problem. Determine a minimum of the functional $ {F_0}(f)$ under the conditions $ {F_j}(f)\, \leqslant \,0,\,j\, = \,1,\,2,\, \ldots ,\,m,\,f\, \in \,{\mathcal{F}}$. We apply the extremum principle to solve this problem. In the linear case this problem was investigated in [11].

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Article copyright: © Copyright 1980 American Mathematical Society