In this note, we develop a framework which allows to prove an existence result for nonlinear evolution problems involving time-dependent, pseudo-monotone operators. This abstract existence result is applicable to a large class of concrete problems where the standard theorem on evolutionary pseudo-monotone operators (cf. Theorem 1.3) is not applicable. To this end we introduce the notion of Bochner pseudo-monotonicity, and Bochner coercivity, which are appropriate extensions of the concepts of pseudo-monotonicity and coercivity to the evolutionary setting. Moreover, we give sufficient conditions for these new notions, which are easily and widely applicable.

在本说明中，我们开发了一个框架，该框架可以证明涉及时间相关的伪单调算子的非线性演化问题的存在结果。这个抽象的存在结果适用于一类具体的问题，其中关于演化伪单调算子的标准定理（参见定理1.3）不适用。为此，我们介绍了Bochner伪单调性和Bochner矫顽力的概念，它们是伪单调性和矫顽力概念到演化环境的适当扩展。此外，我们为这些新概念提供了足够的条件，这些条件易于广泛应用。