Applications of the proximity map to random fixed point theorems in Hilbert spaces (Q1910077)

The authors prove a random version of the well-known minimal displacement theorem due to Ky Fan. Among other results the following one is proved: if \((\Omega,\Sigma)\) is a measurable space, \(S\) is a closed convex and separable subset of a Hilbert space \(H\), \(T:\Omega\times S\to H\) is a continuous 1-set-contractive random operator such that, for any \(\omega\in\Omega\), \(T(\{\omega\}\times S)\) is bounded and \(I-P\circ T(\omega,\cdot)\) is demiclosed where \(P\) stands for the proximity map onto \(S\), then there is a measurable \(\xi:\Omega\to S\) such that \(|\xi(\omega)- T(\omega,\xi(\omega))|= d(T(\omega, \xi(\omega)),S)\). Some corollaries and applications are provided and discussed.