Download e-book for iPad: Advances in mathematical economics. Vol.18 by Shigeo Kusuoka, Toru Maruyama

By Shigeo Kusuoka, Toru Maruyama

ISBN-10: 4431548335

ISBN-13: 9784431548331

ISBN-10: 4431548343

ISBN-13: 9784431548348

A lot of financial difficulties could be formulated as limited optimizations and equilibration in their ideas. a variety of mathematical theories were providing economists with essential machineries for those difficulties bobbing up in fiscal concept. Conversely, mathematicians were motivated by way of a number of mathematical problems raised by way of fiscal theories. The sequence is designed to collect these mathematicians who're heavily attracted to getting new not easy stimuli from fiscal theories with these economists who're looking potent mathematical instruments for his or her research.

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Extra resources for Advances in mathematical economics. Vol.18

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T z∈Z t0 ∂t Proof. Assume by contradiction that there exist a ϕ ∈ C 1 ([0, 1] × E) such that VJ reaches a local maximum at (t0 , x0 ) ∈ [0, η1 [×E for which ∂ϕ (t0 , x0 ) + max{J (t0 , x0 , z)} + δ ∗ (∇ϕ(t0 , x0 ), e˙t0 ,x0 (t0 ) + z∈Z ∂t 1 t0 ∂Gt0 (t0 , s)Zds) ∂t ≤ −η < 0 for some η > 0. 1) 26 C. Castaing et al. where ut0 ,x0 ,f is the trajectory solution associated with the control f ∈ SZ1 starting from x0 at time t0 to ⎧ u¨ t ,x ,f (t) + γ u˙ t0 ,x0 ,f (t) = f (t), t ∈ [t0 , 1] ⎪ ⎪ ⎨ 0 0 m−2 (SODE) ⎪ u (t ) = x , u (1) = αi ut0 ,x0 ,f (ηi ).

20, 249–277 (2012) 11. : Bolza, relaxation and viscosity problems governed by a second order differential equation. J. Nonlinear Convex Anal. 14(2), 451–482 (2013) Optimal Control Problems Governed by a Second Order Ordinary . . 59 12. : Convex analysis and measurable multifunctions. In: Lecture Notes in Mathematics, vol. 580. Springer, Berlin (1977) 13. : Control problems governed by functional evolution inclusions with Young measures. J. Nonlinear Convex Anal. 5(1), 131–152 (2004) 14. : Young Measures on Topological Spaces.

We denote by PE ([0, 1], dt) the space of all Pettis-integrable functions f : [0, 1] → E endowed with the 1 Pettis norm ||f ||P e = supx ∗ ∈B E∗ 0 | x ∗ , f (t) |dt. A mapping f : [0, 1] → E is Pettis-integrable iff the set { x ∗ , f : ||x ∗ || ≤ 1} is uniformly integrable in the space L1R ([0, 1], dt). )) : ||x ∗ || ≤ 1} is uniformly integrable in the space L1R ([0, 1], dt). 2; or [14], Cor. 3] Optimal Control Problems Governed by a Second Order Ordinary . . 29 the set S P e of all Pettis-integrable selections of a convex compact valued Pettis-integrable mapping Γ : [0, 1] ⇒ E is sequentially σ (PE1 , L∞ ⊗ E ∗ )compact.

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