A topological property of solution sets of semilinear differential inclusions (Q2926561)

Consider the differential inclusion NEWLINE\[NEWLINE \dot{x}\in Ax+ F(t,x),\quad x(0) =x_{0}, \tag{1} NEWLINE\]NEWLINE where \(X\) is a real separable Banach space, \(P(X)\) is the family of all subsets of \(X\), \(F:[0,\infty)\times X\to P(X)\) and \(A\) is the infinitesimal generator of a strongly continuous semigroup \(\{G(t): t\geq 0\}\) on \(X\). It is proved that the set of selections that correspond to solutions of (1) is a retract of the space of integrable functions on an unbounded interval.