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On some optimization problems in not necessarily locally convex space. (English) Zbl 1199.46008
Let \(X\) be a Hausdorff (not necessarily locally convex) topological vector space with a fundamental system \(\mathcal U\) of neighbourhoods of \(0\). A subset \(L\) of \(X\) is said to be of \(Z\)-type if for every \(V\in\mathcal U\) there exists \(U\in\mathcal U\) such that \(\operatorname{conv}(U\cap(L-L))\subset V\). Let \(K\) be a nonempty convex subset of \(X\) and let \(D\) be a nonempty compact subset of \(K\). Suppose that \(\phi:X\times X\to\mathbb R\) is a continuous function and \(S:K\to 2^D\) a continuous set-valued map such that: (1) for each \(x\in K\), \(S(x)\) is a nonempty closed subset of \(D\); (2) \(S(K)\) is a subset of \(Z\)-type; (3) for each \(x\in K\), the mapping \(y\mapsto\phi(x,y)\) is quasi-convex on \(S(x)\). The author proves that under these conditions there exists a point \(\bar x\in D\) such that \(\bar x\in S(\bar x)\) and \(\phi(\bar x,y)\geq\phi(\bar x,\bar x)\) for all \(y\in S(\bar x)\). Further, a generalization of Kaczynski-Zeidan’s existence result [T. Kaczynski and V. Zeidan, Nonlinear Anal., Theory Methods Appl. 13, No. 3, 259–261 (1989; Zbl 0685.49009)] is proved for noncompact infinite optimization problems in not necessarily locally convex spaces.
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
49J27 Existence theories for problems in abstract spaces
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