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Selection theorem in L¹

Andrzej Nowak, Celina Rom (2006)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Let F be a multifunction from a metric space X into L¹, and B a subset of X. We give sufficient conditions for the existence of a measurable selector of F which is continuous at every point of B. Among other assumptions, we require the decomposability of F(x) for x ∈ B.

Solving variational inclusions by a multipoint iteration method under center-Hölder continuity conditions

Catherine Cabuzel, Alain Pietrus (2007)

Applicationes Mathematicae

We prove the existence of a sequence ( x k ) satisfying 0 f ( x k ) + i = 1 M a i f ( x k + β i ( x k + 1 - x k ) ) ( x k + 1 - x k ) + F ( x k + 1 ) , where f is a function whose second order Fréchet derivative ∇²f satifies a center-Hölder condition and F is a set-valued map from a Banach space X to the subsets of a Banach space Y. We show that the convergence of this method is superquadratic.

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