Kenneth L. Kuttler, Meir Shillor
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.
Submitted August 27, 2014. Published December 11, 2014.
Math Subject Classifications: 60H15, 34F05, 35R60, 60H10, 74M10.
Key Words: Stochastic differential equations; product measurability;
dynamic contact.
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Kenneth L. Kuttler Department of Mathematics Brigham Young University Provo, UT 84602, USA email: klkuttle@math.byu.edu | |
Meir Shillor Department of Mathematics and Statistics Oakland University Rochester, MI 48309, USA email: shillor@oakland.edu |
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