We define in this work a notion of Young differential inclusion $$ dz_t \in F(z_t)\,dx_t, $$ for an $\alpha$-Hölder control $x$, with $\alpha > 1/2$, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, $\gamma$-Hölder continuous set-valued map on the interval $[0,1]$ has a selection with finite $p$-variation, for $p > 1/\gamma$. We also give a notion of solution to the rough differential inclusion $$ dz_t \in F(z_t)\,dt + G(z_t)\,d{\mathbf X}_t, $$ for an $\alpha$-Hölder rough path $\mathbf X$ with $\alpha\in(1/3,1/2]$, a set-valued map $F$ and a single-valued one form $G$. Then, we prove the existence of a solution to the inclusion when $F$ is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values.

中文翻译：

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年轻和粗糙的微分夹杂物

我们在这项工作中为 $\alpha$-Hölder 控件 $x$ 定义了 Young 微分包含 $$ dz_t \in F(z_t)\,dx_t, $$ 的概念，其中 $\alpha > 1/2$，以及给出这样一个微分系统的存在结果。作为我们证明的副产品，我们证明了区间 $[0,1]$ 上的有界、紧值、$\gamma$-Hölder 连续集值映射具有有限 $p$-variation 的选择, 对于 $p > 1/\gamma$。我们还给出了粗微分包含 $$ dz_t \in F(z_t)\,dt + G(z_t)\,d{\mathbf X}_t, $$ 对于 $\alpha$-Hölder 粗糙的解的概念路径 $\mathbf X$ 与 $\alpha\in(1/3,1/2]$、一个集合值映射 $F$ 和一个单值形式 $G$。然后，我们证明存在一个当 $F$ 有界且下半连续与紧凑值或上半连续与紧凑和凸值时包含的解决方案。