Differential inclusions on Banach spaces with nonlocal state constraints (Q2778301)

Here, the author studies a system governed by a differential inclusion on a Banach space \(E\) with nonlocal state constraints. Actually, he considers the problem NEWLINE\[NEWLINE\dot{x}(t) - Ax(t) \in F(t,x(t)),\;t \in I=[0,1] \quad\text{and}\quad x(0)+ \int_I x(t) da(t)=\xi,NEWLINE\]NEWLINE where \(A\) is the infinitesimal generator of a \(C_0\)-semigroup in \(E\), \(F\) is a multifunction, \(a\) is a scalar-valued function of bounded variation on \(I\) and \(\xi\) is an element of \(E\). The author proves sufficient conditions for the existence of functions \(x \in C(I,E)\) that solve the above problem in some general sense called mild solutions.NEWLINENEWLINENEWLINEIn the last part of the paper, some open problems are presented.