Evolution governed by ''generalized'' dissipative operators (Q1803579)

The authors consider the Cauchy problem (P) \(u'(t)\in Au(t)+f(t)\), \(t\in[0,T]\); \(u(0)=x\) is a Banach space \((X,|\cdot|)\) under the assumption that \(f\in L^ 1(0,T;X)\) and the multivalued operator \(A: D(A)\subset X\to X\) satisfy the condition: there exists \(\omega: \mathbb{R}_ +\to\mathbb{R}\) such that \(\omega(0)=0\) and problem \(r'(t)=\omega(r(t))\), \(t\geq 0\), \(r(0)=0\) has only the zero-solution and \([x_ 1-x_ 2,y_ 1,y_ 2]_ -\leq \omega(| x_ 1-x_ 2|)\) for \(x_ 1,x_ 2\in D(A)\), \(y_ 1\in Ax_ 1\) and \(y_ 2\in Ax_ 2\). Here we have set \([x,y]_ -=\lim_{s\to 0^ -}\bigl(| x+sy|-| x|\bigr)s^{-1}\). The following result is stated (the detailed proofs will appear elsewhere): if \(\limsup_{r\to\infty}{\omega(r)\over r}<\infty\), \(x\in\overline{D(A)}\) and for a.e. \(t\in ]0,T[\) we have \(f(t)\in S=\bigl\{z\in X;\;\liminf_{s\to 0^ +} s^{-1} d(R(I-sA),y+sz)=0\) for any \(y\in\overline{D(A)}\bigr\}\) then there exists a unique mild solution of problem (P) in \([0,T]\).