On the solution set of some classes of nonconvex nonclosed differential inclusions
The author studies the following semilinear differential inclusion \[ x'\in Ax+F(t,x,H(t,x)), \quad x(0)=x_0, \] where \(X\) is a real separable Banach space, \(F:[0,T]\times X^2 \to 2^X\) and \(H:[0,T]\times X \to 2^X\) are two multifunctions with nonempty values, \(A\) is the infinitesimal generator of strongly continuous semigroups \(\{G(t): t\geq 0\}\) on \(X\). In the main result, the arcwise connectedness of the solution set of the above problem is proved in the case that \(F\) and \(H\) are Lipschitzian with respect to the second variable and \(F\) is contractive in the third variable. A similar result is stated also for a class of second order differential inclusions.
scientific article; zbMATH DE number 5381762