EVOLUTION EQUATIONS AND doi:10.3934/eect.2020055 CONTROL THEORY Volume 9, Number 4, December 2020 pp. 935–960 MEASURABLE SOLUTIONS TO GENERAL EVOLUTION INCLUSIONS Kevin T. Andrews ∗ Department of Mathematics and Statistics Oakland University Rochester MI 48309 USA Kenneth L. Kuttler Retired Ji Li School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan, Hubei 430074, China Meir Shillor Department of Mathematics and Statistics Oakland University Rochester MI 48309 USA Abstract. This work establishes the existence of measurable solutions to evo- lution inclusions involving set-valued pseudomonotone operators that depend on a random variable ω ∈ Ω that is an element of a measurable space (Ω, F ). This result considerably extends the current existence results for such evolu- tion inclusions since there are no assumptions made on the uniqueness of the solution, even in the cases where the parameter ω is held constant, which leads to the usual evolution inclusion. Moreover, when one assumes the uniqueness of the solution, then the existence of progressively measurable solutions un- der reasonable and mild assumptions on the set-valued operators, initial data and forcing functions is established. The theory developed here allows for the inclusion of memory or history dependent terms and degenerate equations of mixed type. The proof is based on a new result for measurable solutions to a parameter dependent family of elliptic equations. Finally, when the choice ω = t is made, where t is the time and Ω = [0,T ], the results apply to a wide range of quasistatic inclusions, many of which arise naturally in contact mechanics, among many other applications. 1. Introduction. In this paper we establish the existence of product measurable solutions (u,u ∗ ) to a general class of evolution inclusions that depend on a random 2000 Mathematics Subject Classification. Primary: 35R60 ; Secondary: 35Q74, 60H25. Key words and phrases. Set-valued differential inclusions, measurable solutions, random coef- ficients, quasistatic inclusions, product measurability. * Corresponding author: andrews@oakland.edu. 935