# Read e-book online Advances in mathematical economics. Vol.18 PDF

By Shigeo Kusuoka, Toru Maruyama

ISBN-10: 4431548335

ISBN-13: 9784431548331

ISBN-10: 4431548343

ISBN-13: 9784431548348

A lot of financial difficulties might be formulated as restricted optimizations and equilibration in their options. a variety of mathematical theories were offering economists with crucial machineries for those difficulties coming up in fiscal concept. Conversely, mathematicians were influenced through a number of mathematical problems raised via financial theories. The sequence is designed to compile these mathematicians who're heavily drawn to getting new not easy stimuli from fiscal theories with these economists who're looking powerful mathematical instruments for his or her research.

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Proving that Graph is compact. By applying the Kakutani–Ky Fan fixed point theorem to , we find u ∈ X such that u = (u) which is a WP2,1 ([τ, 1])-solution of the (SODE) under consideration. ,E The compactness in CE ([τ, 1]) of X := {uτ,x,f : [τ, 1] → E : uτ,x,f (t) = eτ,x (t) 1 + Gτ (t, s)f (s)ds, t ∈ [τ, 1], f ∈ S P e } τ Y := {u˙ τ,x,f : [τ, 1] → E : u˙ τ,x,f (t) = e˙τ,x (t) 1 + τ ∂Gτ (t, s)f (s)ds, t ∈ [τ, 1], f ∈ S P e } ∂t are of importance and rely on some delicate arguments in the pioneering work of [1, 2] involving the Pettis uniformly integrable condition, Grothendieck lemma characterizing the Mackey topology for bounded sets in L∞ R [24] and other compactness results.

Here is an application to the existence of WP2,1 ,E ([τ, 1])-solution of a (SODE) with m-point boundary condition. 2. Let F : [0, 1] × (E × E) → E be a Carath´eodory mapping satisfying F (t, x, y) ∈ (t) 36 C. Castaing et al. for all (t, x, y) ∈ [0, 1] × E × E where : [0, 1] ⇒ E is a convex compact valued Pettis-integrable mapping. Then the (SODE) ⎧ ¨ + γ u(t) ˙ = F (t, u(t), u(t)), ˙ t ∈ [τ, 1] ⎪ ⎨ u(t) m−2 ⎪ ⎩ u(τ ) = x, u(1) = αi u(ηi ) i=1 has a WP2,1 ,E ([τ, 1])-solution. Proof. Let us set X := {uτ,x,f : [τ, 1] → E : uτ,x,f (t) = eτ,x (t) 1 + Gτ (t, s)f (s)ds, t ∈ [τ, 1], f ∈ S P e }.

Castaing et al. where vτ +σ,u τ,x,f 1 (τ +σ ),g 2 (t) denotes the trajectory solution on [τ + σ, 1] associated with the control g 2 ∈ SZ1 starting from uτ,x,f 1 (τ + σ ) at time τ + σ to ⎧ v¨τ +σ,u 1 (τ +σ ),g 2 (t) + γ v˙τ +σ,u 1 (τ +σ ),g 2 (t) = g 2 (t), ⎪ ⎪ τ,x,f τ,x,f ⎪ ⎪ ⎪ t ∈ [τ + σ, 1] ⎪ ⎪ ⎨ (SODE) vτ +σ,uτ,x,f 1 (τ +σ ),g 2 (τ + σ ) = uτ,x,f 1 (τ + σ ), ⎪ ⎪ m−2 ⎪ ⎪ ⎪ ⎪ (1) = αi vτ +σ,u 1 (τ +σ ),g 2 (ηi ). v ⎪ τ +σ,u 1 (τ +σ ),g 2 ⎩ τ,x,f τ,x,f i=1 Let us set f := 1[τ,τ +σ ] f 1 + 1[τ +σ,1] f 2 .

### Advances in mathematical economics. Vol.18 by Shigeo Kusuoka, Toru Maruyama

by William

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