In the setting of 2-uniformly smooth and $q$-uniformly convex Banach spaces, we prove the existence of solutions of the following multivalued differential equation: $$-\frac{d}{dt} J(u(t)) \in N^C(C(t,u(t));u(t)) \mbox{ a.e. in } [0,T]. \:\:\: \mathrm{(SDNSP)}$$ This inclusion is called State Dependent Nonconvex Sweeping Process (SDNSP). Here $N^C(C(t,u(t)); u(t))$ stands for the Clarke normal cone. The perturbed (SDNSPP) is also considered. Our results extend recent existing results from the setting of Hilbert spaces to the setting of Banach spaces. In our proofs we use some new results on $V$-uniformly generalized prox-regular sets in Banach spaces.

中文翻译：

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光滑Banach空间中依赖状态的非凸扫描过程

在2-一致光滑和$ q $-一致凸Banach空间的设置中，我们证明了以下多值微分方程解的存在：$$-\ frac {d} {dt} J（u（t））\在N ^ C（C（t，u（t））; u（t））\ mbox {ae in} [0，T]中。\：\：\：\ mathrm {（SDNSP）} $$此包含项称为状态相关的非凸扫描过程（SDNSP）。这里$ N ^ C（C（t，u（t））; u（t））$代表Clarke法锥。还考虑了受干扰（SDNSPP）。我们的结果将最近的现有结果从Hilbert空间的设置扩展到Banach空间的设置。在我们的证明中，我们对Banach空间中的$ V $-一致广义正则-正则集使用了一些新结果。