Controllability results for nondensely defined evolution differential inclusions with nonlocal conditions (Q1883084)

The author provides sufficient conditions for the controllability of the following semilinear evolution differential inclusion with nonlocal conditions \[ y'(t)\in Ay(t)+F(t,y(t))+(\Theta u)(t), \quad t\in J=[0,b],\quad y(0)+g(y)=y_{0}, \] where \(A:D(A)\subset E\to E\) is a nondensely defined closed linear operator, \(F: J\times E\to {\mathcal P}(E)\backslash\emptyset\) is a multivalued map \(({\mathcal P}\) is the family of all subsets of \(E\)) and \(g:C(J,E)\to E\) is a continuous function. The control function \(u(\cdot)\) is given in \(L^{2}(J,U)\), a Banach space of admissible control functions with \(U\) as a Banach space. Finally, \(\Theta\) is a bounded linear operator from \(U\) to \(E\) and \(E\) is a separable Banach space. The proofs rely on the theory of integrated semigroups and the Bohnenblust-Karlin fixed-point theorem.