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

By Shigeo Kusuoka, Toru Maruyama

The sequence is designed to compile these mathematicians who're heavily drawn to getting new tough stimuli from fiscal theories with these economists who're looking powerful mathematical instruments for his or her examine. loads of monetary difficulties should be formulated as restricted optimizations and equilibration in their strategies. quite a few mathematical theories were offering economists with necessary machineries for those difficulties bobbing up in fiscal concept. Conversely, mathematicians were motivated through a variety of mathematical problems raised through fiscal theories.

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5 Multivalued Integral Inclusion The results stated in the preceding section lead naturally to the study of multivalued Pettis integral inclusion and the multivalued fractional Pettis integral inclusion. For simplicity we begin with an example. 1. Assume that E is a reflexive separable Banach space and F W Œ0; 1 E ! t; x/ 2 0; 1 E where h W Œ0; 1 ! RC is a positive integrable function. Let H W Œ0; 1 Œ0; 1 ! t; :/ is measurable for every fixed t 2 Œ0; 1. Œ0; 1/. Proof. t/BE for all t 2 Œ0; 1.

0; 1/-solutions to (16). s/ds, t 2 Œ0; 1. L1E ; L1 E /-compact, by Eberlein-Smulian theorem, we may assume that . s/ds t 2 Œ0; 1. Œ0; 1/. Remark. 1, we have proven the continuous dependence SX1 ! Œ0; 1/ of the mappings f 7! uf and f 7! L1E ; L1 E /-compact set SX . This fact has some importance in further applications. 36 C. Castaing et al. 2. Let X W Œ0; 1 ,! E be a convex weakly compact valued measurable and integrably bounded mapping. Let F W Œ0; 1 E E ,! t; x; y/ 2 Œ0; 1 E E. Œ0; 1/.

Castaing et al. t; s/ D Œ0; 1 ! ˛/ for all t; s 2 Œ0; 1. 2. Let G as above. s/ds; t 2 Œ0; 1. s/ds, t 2 Œ0; 1. Proof. Let x 2 E . 1/it˛Cˇ 1 : Letting t ! 1, I ˛ f is continuous, so is uf . 1/ (5) On a Fractional Differential Inclusion in Banach Space Under Weak. . s/ds: (9) ˛;1 From (9) we note that w-D˛ 1 uf is continuous on Œ0; 1. Œ0; 1/ and the proof is complete. 28 C. Castaing et al. Remark. It is worth to mention that Z . 2 (iv) . ˛/ < 2. Some arguments and notations given above will be used in the next section.

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