Product measurability with applications to a stochastic contact problem with friction

Abstract

A new product measurability result for evolution equations with random inputs, when there is no uniqueness of the omega-wise problem, is established using results on measurable selection theorems for measurable multi-functions. The abstract result is applied to a general stochastic system of ODEs with delays and to a frictional contact problem in which the gap between a viscoelastic body and the foundation and the motion of the foundation are random processes. The existence and uniqueness of a measurable solution for the problem with Lipschitz friction coefficient, and just existence for a discontinuous one, is obtained by using a sequence of approximate problems and then passing to the limit. The new result shows that the limit exists and is measurable. This new result opens the way to establish the existence of measurable solutions for various problems with random inputs in which the uniqueness of the solution is not known, which is the case in many problems involving frictional contact.

Keywords

• Stochastic differential equations
• product measurability
• dynamic contact