Product measurability with applications to a stochastic contact problem with friction

A new product measurability result for evolution equations with random inputs, when there is no uniqueness of the ω-wise problem, is established using results on measurable selection theorems for measurable multifunctions. 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.

[1]  M. Shillor,et al.  Regularity of solutions to a dynamic frictionless contact problem with normal compliance , 2004 .

[2]  W. Han,et al.  Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity , 2002 .

[3]  Shouchuan Hu,et al.  Handbook of multivalued analysis , 1997 .

[4]  C. Fabre,et al.  A quasistatic viscoplastic contact problem with normal compliance and friction , 2004 .

[5]  M. Fowler,et al.  Function Spaces , 2022 .

[6]  J. T. Oden,et al.  Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws , 1987 .

[7]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[8]  J. J. Telega,et al.  Models and analysis of quasistatic contact , 2004 .

[9]  Mircea Sofonea,et al.  Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems , 2012 .

[10]  A. Bensoussan,et al.  Equations stochastiques du type Navier-Stokes , 1973 .

[11]  Meir Shillor,et al.  Dynamic Contact with Normal Compliance Wear and Discontinuous Friction Coefficient , 2002, SIAM J. Math. Anal..

[12]  Kenneth Kuttler.  Dynamic friction contact problems for general normal and friction laws , 1997 .

[13]  A. Klarbring,et al.  FRICTIONAL CONTACT PROBLEMS WITH NORMAL COMPLIANCE , 1988 .

[14]  J. Lions,et al.  Inequalities in mechanics and physics , 1976 .

[15]  Jacques Simeon,et al.  Compact Sets in the Space L~(O, , 2005 .

[16]  Meir Shillor,et al.  On the dynamic behaviour of a thermoviscoelastic body in frictional contact with a rigid obstacle , 1997, European Journal of Applied Mathematics.

[17]  Meir Shillor,et al.  Dynamic bilateral contact with discontinuous friction coefficient , 2001 .

[18]  Isabel N. Figueiredo,et al.  A class of contact and friction dynamic problems in thermoelasticity and in thermoviscoelasticity , 1995 .

[19]  Meir Shillor,et al.  DYNAMIC CONTACT WITH SIGNORINI'S CONDITION AND SLIP RATE DEPENDENT FRICTION , 2004 .

[20]  Meir Shillor,et al.  Existence for dynamic contact of a stochastic viscoelastic Gao Beam , 2015 .

[21]  Meir Shillor,et al.  On friction problems with normal compliance , 1989 .

[22]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .