By E. S. Venttsel'

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**Extra resources for An introduction to the theory of games**

**Example text**

Let (X , d) be a complete metric space and let : X → F(X ) be a set-valued mapping. Suppose that there exists x0 ∈ X and constants r > 0, α ∈ (0, 1), ∈ (0, r) such ¯ 0 , r) the sets (x1 ), (x2 ) are nonempty, that for all x1 , x2 in the closed ball B(x dH ( (x1 ), (x2 )) ≤ α d(x1 , x2 ), and d(x0 , (x0 )) ≤ (1 − d)(r − ). ¯ 0 , r) such that x¯ ∈ Then there exists x¯ ∈ B(x If moreover (x) ¯ and d(x0 , x) ¯ ≤ (1−α)−1 d(x0 , (x0 ))+ . 6) ¯ 0 , r), then the sequence Sk = (Sk−1 ), for all nonempty closed subsets S1 and S2 of B(x ¯ with S0 = {x0 }, converges in the Hausdorff metric to a set S¯ satisfying S¯ = (S).

6. 8 (Stability). Let x0 ∈ (p0 ) be a regular point. 9) > 0. Proof. Let F be the closed convex set-valued mapping from X into Y given by F (x) = g (x0 , p0 )(x − x0 ) + g(x0 , p0 ) − K {} for x ∈ C, for x ∈ / C. The regular point condition implies that there exists an η > 0 such that η BY ⊂ F (x0 + BX ). 5. For δ > 0 and r > 0 let ρ = (η − 2δ r)−1 (1 + r) where we assume that 2δ r < η and 2ρδ < 1. Let h(x, p) = g(x0 , p0 ) + g (x0 , p0 )(x − x0 ) − g(x, p) ✐ ✐ ✐ ✐ ✐ ✐ ✐ 38 ItoKunisc 2008/6/12 page 38 ✐ Chapter 2.

12. 15) where x1 − x22 − x3 . 3 is e here, and K = {0}. 15) and that ∇e(x ∗ ) is not surjective. Hence x ∗ is not a regular point for the constraint e(x) = 0. 15). 1 becomes f (x ∗ ) + e (x ∗ )∗ λ∗ = 0 in X ∗ , where e (x ∗ )∗ : Z ∗ → X ∗ is the adjoint of e (x ∗ ). Hence if e (x ∗ ) is not surjective and if it has closed range, then ker e (x ∗ )∗ is not empty and the Lagrange multiplier is not unique. 13. 16) ∂y = 0 on and y = 0 on ∂ \ , ∂n where is a bounded domain in Rn , n ≤ 3, with smooth boundary ∂ and is a connected, strict subset of ∂ .

### An introduction to the theory of games by E. S. Venttsel'

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