Impulsive mild solutions for semilinear differential inclusions with nonlocal conditions in Banach spaces (Q651155)

The authors study the existence of mild solutions to an impulsive nonlocal Cauchy problem driven by a semilinear differential inclusion in a real Banach space \(E\) \[ \left\{\begin{aligned} &x'(t)\in A(t)x(t)+F(t,x(t)), \quad t \in [0,b],\;t\neq t_i, \;i=1,\dotsc ,p,\\ &x(t_i^+)=x(t_i)+I_i(x(t_i)), \quad i=1,\dotsc ,p,\\ &x(0)=g(x),\end{aligned}\right. \] where \(\left\{ A(t)\right\} _{t\in[0,b]}\) is a family of linear operators in \(E\); \(F\) is a multifunction; \(g\) is a function; \(0=t_0<t_1<\dotsb <t_p<t_{p+1}=b\); \(I_i\) is an impulsive function for every \(i=1,\dotsc ,p\) and \(x(t_i^+)=\lim_{s\to t_i^+}x(s)\). The authors achieve existence results under a lower semicontinuity-type assumption on the nonlinearity \(F\), thanks to a recently proven selection theorem joined with the classical Schauder fixed point theorem.