New maximum theorems with strict quasi-concavity (Q2756102)

Let \(X\) be a non-empty convex subset of a vector space \(E\), and let \(f:X \to R\). If for each \(x_1, x_2\in X\), \(x_1\neq x_2\), and every \(\lambda \in (0,1)\), NEWLINE\[NEWLINEf\bigl(\lambda x_1+(1-\lambda)x_2 \bigr)>\min \bigl\{f(x_1), f (x_2) \bigr\},NEWLINE\]NEWLINE then \(f\) is called strictly quasi-convex on \(x\).NEWLINENEWLINENEWLINEThe author proves the following theorem:NEWLINENEWLINENEWLINETheorem: Let \(X\) be a non-empty convex subset of a Hausdorff topology vector space, \(Y\) be a non-empty subset of a Hausdorff topological vector space and let \(f:X\times Y\to R\) be an upper semicontinuous and strictly quasi-concave mapping on \(X\times Y\). If \(C:X \to 2^Y\) is an upper semicontinuous convex correspondence such that each \(C(x)\) is non-empty compact and \(\Phi(x_0)\) is compact for some \(x_0\in X\), then there exists a point \(\widehat x\in X\) such that NEWLINE\[NEWLINE\max_{y\in C(x)}f(x,y) \leq\max_{y \in C(\widehat x)} f(\widehat x,y), \quad\text{for all }x\in X,NEWLINE\]NEWLINE where \(\varphi :X\to 2^X\) defined by \(\Phi(x)= \{y\in X\mid g(x) \leq g(y)\}\), and \(g(x)= \max_{y\in C(x)} f(x,y)\).