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Some variational convergence results for a class of evolution inclusions of second order using Young measure. (English) Zbl 1145.49005

Kusuoka, S. (ed.) et al., Advances in mathematical economics. Vol. 7. Tokyo: Springer (ISBN 978-4-431-24332-8/hbk). Advances in Mathematical Economics 7, 1-32 (2005).
Summary: This paper has two main parts. In the first part, we discuss the existence and uniqueness of a \(W_E^{2,1}\)-solution \(u_{\mu,\nu}\) of a second-order differential equation with two boundary points conditions in a finite dimensional space, governed by controls \(\mu,\nu\) which are measures on a compact metric space. We also discuss the dependence on the controls and the variational properties of the value function \(V_h(t,\mu):= \sup_{\mu\in{\mathcal R}} h(u_{\mu,\nu}(t))\), associated with a bounded lower semicontinuous function \(h\). In the second main part, we discuss the limiting behaviour of a sequence of dynamics governed by second order evolution inclusions with two boundary points conditions. We prove that (up to extracted sequences) the solutions stably converge to a Young measure \(\nu\) and we show that the limit measure \(\nu\) satisfies a Fatou-type lemma in mathematical economics with variational-type inclusion property.
For the entire collection see [Zbl 1142.91009].

MSC:

49J40 Variational inequalities
49J45 Methods involving semicontinuity and convergence; relaxation
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
34G25 Evolution inclusions
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