This work establishes the existence of measurable solutions to evolution inclusions involving set-valued pseudomonotone operators that depend on a random variable $ \omega\in \Omega $ that is an element of a measurable space $ (\Omega, \mathcal{F}) $. This result considerably extends the current existence results for such evolution inclusions since there are no assumptions made on the uniqueness of the solution, even in the cases where the parameter $ \omega $ 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 under 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 $ \omega = t $ is made, where $ t $ is the time and $ \Omega = [0, T] $, the results apply to a wide range of quasistatic inclusions, many of which arise naturally in contact mechanics, among many other applications.

中文翻译：

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通用演化包含物的可测解决方案

这项工作建立了包含定值伪单调算子的演化包含物的可测解的存在，该值依赖于随机变量$ \ omega \ in \ Omega $，它是可测量空间$（\ Omega，\ mathcal {F}）的元素$。由于没有对解的唯一性进行假设，因此即使在参数$ \ omega $保持恒定的情况下，也导致了通常的演化包含，因此该结果大大扩展了此类演化包含的当前存在结果。此外，当假设解决方案的唯一性时，就可以在合理且温和的假设下，对集值算子，初始数据和强迫函数建立渐进可测解的存在。此处开发的理论允许包含记忆或历史相关项以及混合类型的退化方程。该证明是基于一个新的结果，该结果是可测解的一个依赖参数的椭圆方程族。最后，当做出选择$ \ omega = t $时，其中$ t $是时间，而$ \ Omega = [0，T] $，结果适用于各种准静态夹杂物，其中许多是自然产生的。联系技术，以及许多其他应用程序。