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Metric spaces admitting only trivial weak contractions

Richárd Balka (2013)

Fundamenta Mathematicae

If (X,d) is a metric space then a map f: X → X is defined to be a weak contraction if d(f(x),f(y)) < d(x,y) for all x,y ∈ X, x ≠ y. We determine the simplest non-closed sets X ⊆ ℝⁿ in the sense of descriptive set-theoretic complexity such that every weak contraction f: X → X is constant. In order to do so, we prove that there exists a non-closed F σ set F ⊆ ℝ such that every weak contraction f: F → F is constant. Similarly, there exists a non-closed G δ set G ⊆ ℝ such that every weak contraction...

Minimax theorems without changeless proportion

Liang-Ju Chu, Chi-Nan Tsai (2003)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds: i n f y Y s u p x X f ( x , y ) s u p x X i n f y Y g ( x , y ) . We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: s u p x X f ( x , y ) s u p x X g ( x , y ) , ∀y ∈ Y. However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some properties...

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