A strict minimax inequality criterion and some of its consequences

作者：Ricceri Biagio^*

来源：Positivity, 2012, 16(3): 455-470.

DOI：10.1007/s11117-012-0164-x

• Strict minimax inequality
• Global minimum
• Uniqueness
• Multiplicity
• Convexity

In this paper, we point out a very flexible scheme within which a strict minimax inequality occurs. We then show the fruitfulness of this approach presenting a series of various consequences. Here is one of them: Let Y be a finite-dimensional real Hilbert space, J : Y -%26gt; R a C (1) function with locally Lipschitzian derivative, and a C (1) convex function with locally Lipschitzian derivative at 0 and . Then, for each for which J%26apos;(x (0)) not equal 0, there exists delta %26gt; 0 such that, for each , the restriction of J to B(x (0), r) has a unique global minimum u (r) which satisfies %26lt;br%26gt;J(u(r)) %26lt;= J(x) - phi(x - u(r)) %26lt;br%26gt;for all x is an element of B(x(0), r), where B(x(0), r) = {x is an element of Y : parallel to x - x(0)parallel to %26lt;= r}.

• 出版日期2012-9