Advances in Mathematical Economics Volume 20 by Shigeo Kusuoka, Toru Maruyama

By Shigeo Kusuoka, Toru Maruyama

The sequence is designed to assemble these mathematicians who're heavily drawn to getting new demanding stimuli from monetary theories with these economists who're looking potent mathematical instruments for his or her study. loads of monetary difficulties might be formulated as restricted optimizations and equilibration in their ideas. a variety of mathematical theories were offering economists with fundamental machineries for those difficulties coming up in monetary idea. Conversely, mathematicians were encouraged by means of numerous mathematical problems raised by way of fiscal theories.

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T// is integrable, for each x 2 E, and jXj is integrable. t/ for all x 2 E and for all t 2 Œ0; 1. The following establishes a sharp relationship between multivalued AumannPettis integral and multivalued fractional Aumann-Pettis integral. A; B/ denotes the Hausdorff distance between two convex weakly compact subsets A and B in E. 2. Let X W Œ0; 1 ,! E be a convex weakly compact valued Pettisintegrable multimapping. Let ˛n 21; 2 such that ˛n ! 1. 1 0 Proof. ˛ 1j ! 0 when ˛n !

5. Let X W Œ0; 1 ,! Œ0; 1/-solutions set X to the (FDI) multifunction. Œ0; 1/ of all continuous mappings from Œ0; 1 into E endowed with the topology of uniform convergence. Proof. Step 1. t/ dt W f 2 SXPe g is convex and compact in E (see [13, 17]). ˛;1 Step 2. Œ0; 1/-solutions set X to (27) is characterized by X D fuf W Œ0; 1 ! s/ds f 2 SXPe g 48 C. Castaing et al. Œ0; 1/. Œ0; 1/. Indeed, let x 2 BE and let tn < t with tn ! t in Œ0; 1. s//jds so that the second member goes to 0 when tn !

Let H W Œ0; 1 properties Œ0; 1 ! t; s/ 2 Œ0; 1 Œ0; 1. 30 C. Castaing et al. Let f W Œ0; 1 ! E be a Pettis-integrable mapping. Then the mapping Z uf W t 7! Œ0; 1/. Proof. tn / be a sequence in Œ0; 1 such that tn ! t 2 Œ0; 1 . L1 ; L1 /, see also [12] for a more general result concerning the Mackey topology for bounded sequences in L1 E . Œ0; 1/, the second term in the above estimation goes to 0 as tn ! t showing that uf is continuous on Œ0; 1 with respect to the strong topology of E. The following lemma is crucial for our purpose.

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