Existence of solutions to differential inclusions in Banach spaces (Q1292912)

Let \(R_{0}=\{(t,u)\in \mathbb{R}^{2}\mid t_{0}\leq t_{0}+a\), \(|u-u_{0}|\leq b\}\), with \(a>0\), \(b>0 \). One of the two results of the paper is the following theorem: Let \(E\) be a real Banach space, and the set-valued map \(F:R_{0}\to D\subset E\) bounded upper semicontinuous with convex compact values. \(D\) is a convex subset of \(E\), \(0\in D\) and \( 0<T\leq a\), \(x_{n}\in C^{1}[[t_{0},t_{0}+T]\), \(\overline{B}(x_{0},b)]\) satisfies \( x_{n+1}(t)\in F(t,x_{n}(t))+y_{n}(t)\), \(x_{n}(t_{0})=x_{0}\), \(\|y_{n}(t)\|\leq\varepsilon_{n}\) \((n=1,2,\cdots,\forall t \in [t_{0},t_{0}+T]), \) with \(\varepsilon_{n}>0, \varepsilon_{n}\to 0, y_{n}\in C[[t_{0},t_{0}+T],D]\). If \(\|x_{n}(t)-x(t)\|\to 0\) uniformly \(\forall t\in [t_{0},t_{0}+T]\), then \(x\in C[[t_{0},t_{0}+T],\overline{B}(x_{0},b)]\) and \(x'(t)\in F(t,x(t)), x(t_{0})=x_{0}, \forall t\in [t_{0},t_{0}+T]\). Then the existence theorem on solutions to differential inclusions is obtained.