A nonautonomous Olech type result (Q1677179)

Let \(F,G : D \times I \multimap \mathbb{R}^n\) be uniformly bounded and measurable multimaps, where \(D \subset \mathbb{R}^n\) and \(I \subset \mathbb{R}\) is a compact interval. Under the assumptions that \(G(\cdot,t)\) is upper semicontinuous and convex-valued, \(F(\cdot,t)\) satisfies some mixed semicontinuity type condition, and \(F\) and \(G\) obey a certain joint tangential condition, the author proves the existence of a local \(D\)-viable solution of the problem \[ \dot{x}(t) \in F(x(t),t) + G(x(t),t),\quad x(t_0) = x_0 \in D,\quad t_0 \in I. \]