Cauchy problem for derivors in finite dimension (Q2718933)

The following Cauchy problem is studied: NEWLINE\[NEWLINEu'(t)+ Au(t)= f(t),\quad t\in [0, T],\quad u(0)= u_0,\tag{1}NEWLINE\]NEWLINE where \(A\) is a continuous operator on \(\mathbb{R}^n\) and \(f\in L^1([0, T],\mathbb{R}^n)\). Moreover, it is assumed that \(A\) is the so-called derivor on \(\mathbb{R}^n\) (i.e. \(-A\) is quasi-monotone with respect to the cone \((\mathbb{R}^+)^n\)) and has some additional properties. It is shown that problem (1) has a unique solution although classical uniqueness conditions of Kamke-type are not satisfied. A lot of interesting applications are pointed out.