Continuous selectors of fixed point sets of multifunctions with decomposable values (Q1273434)

Let \(X\) be a separable Banach space, \(M\) a separable metric space, \(T\) a metric space with a Radon measure. Let \(\Gamma : M \times L_1(T,X)\multimap L_1(T,X)\) be a multifunction with nonempty closed and decomposable values, lower semicontinuous (l.s.c.) in the first argument and satisfying a Lipschitz type condition with respect to the Hausdorff distance in the second one. Then \(\text{Fix } \Gamma(\xi,\cdot)\) is a nonempty absolute retract for every \(\xi \in M\), a multimap \(\xi\multimap \text{Fix } \Gamma(\xi,\cdot)\) is l.s.c. and admits a continuous selection. As application, the author considers the l.s.c. dependence of the solutions set for a partial differential inclusion on a parameter and initial data and existence of a continuous selection for this multimap.