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Loads of financial difficulties can formulated as limited optimizations and equilibration in their options. quite a few mathematical theories were delivering economists with quintessential machineries for those difficulties coming up in monetary idea. Conversely, mathematicians were encouraged by way of a number of mathematical problems raised by means of financial theories. The sequence is designed to compile these mathematicians who have been heavily attracted to getting new difficult stimuli from fiscal theories with these economists who're looking for powerful mathematical instruments for his or her researchers.

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**Advances in Mathematical Economics. Vol, 10**

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Jx ^ 0 for each x e E. y is a monotone operator. If £• is strictly convex, then J is one to one, that is, X y/^y =^ Jxr\Jy = &. 54 4. 5. 6. 7. 8. 9. T. Ibaraki and W. Takahashi If E is reflexive, then 7 is a mapping of E onto £"*. If £" is smooth, then the duality mapping J is single valued. If E has a Frechet differentiable norm, then J is norm to norm continuous. If £^ has a uniformly Gateaux differentiable norm, then J is norm to weak* uniformly continuous on each bounded subset of E. E is strictly convex if and only if 7 is a strictly monotone operator.

By our assumption, the sequence (Aun) is bounded in L^([0, 1], dt) and converges in measure to Au with respect to the norm topology of F\ By virtue of Castaing [10], see also Grothendieck [8] for the one dimensional case, we conclude that l i m ( ^ M „ , Un)ijoo / I \ = {Au, u),joo j \ \. As J(u) := /Q j(t, u(t))dt is convex lower semicontinuous on L\^([0, 1], dt), when L ) , ( [ 0 , 1], dt) is endowed with the topology a (L),, Lf,), see ([18], Theorem VII-7), we have lim

Z) being the compact metrizable space of all probability Radon measures defined on Z, endowed with the vague topology and let Hbe a convex weakly compact subset 6>/L^([0, 1], dt). Let us consider the integral fiinctional J{u, k) := \ Jo Uz j(t, u(t), z)Xt(dz) dt; J V(M, k) eHxIZ. Then the following hold: (a) there is a pair (u,X) eH x7Z such that maxmin/(M, X) < J(u, X) < XeTlueH vmnm3xJ(u,X). ueH veU (b) max min J(u, k) = min max 7(M, k). Proof (a) follows the same line of the proof of the preceding result.