Lyapunov pairs for continuous perturbations of nonlinear evolutions (Q1026062)

For the abstract Cauchy problem \[ y'(t)\in Ay(t) + F(y(t))\;,\;y(0)=\xi \] with \(A:D(A)\subseteq X\mapsto X\) a (possibly) multivalued \(m\)-dissipative operator, \(X\) being a real Banach space, \(\emptyset\neq M\subset \overline{D(A)}\) and \(F:M\mapsto X\) a function. The pair \(V,g:X\mapsto R\) is a Liapunov pair of the above equation if \(\mathrm{dom} (V)\subseteq M\) and the dissipativeness inequality holds \[ V(y(t)) + \int_0^t g(y(\tau))d\tau \leq V(\xi)\;,\;0\leq t\leq T \] The necessary and sufficient condition for \((V,g)\) to be a Liapunov pair is \[ \underline{D}^AV(\xi)F(\xi)+g(\xi)\leq 0\;,\;\xi\in \mathrm{dom}(V) \] where \(\underline{D}^AV(\xi)(v)\) is the \(A\)-contingent derivative of \(V\) at \(\xi\in \mathrm{dom}(V)\) in the direction \(v\in X\).