Semigroups of Lipschitz operators. (English) Zbl 1045.47051

Let \(D\) be a closed subset of a real Banach space \(X\). The semigroup \(\{ T(t)| t\geq 0\}\) of operators from \(D\) into itself, related to the abstract Cauchy problem \(u'(t)=A u(t), t\geq 0, u(0)=x, A:D\rightarrow X\), is a semigroup of Lipschitz operators if for each \(T>0\) there exists \(M_{\tau}\geq 1\) such that \(|| T(t) x-T(t)y|| \leq M_{\tau} || x-y||\) for \(x,y\in D\) and \(t\in [0,T]\). Two theorems are proved. (1) A semigroup of Lipschitz operators is a quasi-contractive semigroup with respect to a certain metric-like functional. (2) A continuous operator \(A\) from \(D\) into \(X\) is the infinitesimal generator of a semigroup of Lipschitz operators on \(D\) iff it satisfies the subtangential condition and a general type of dissipative condition that there is a metric-like functional with respect to which \(A\) is dissipative. Some results concerning the uniqueness, continuous dependence on initial data and continuation of solutions to the abstract Cauchy problem are obtained under such a dissipative condition. For the construction of approximate solutions of the abstract Cauchy problem, the Euler difference scheme is used instead of the Cauchy polygons method. As an application of the obtained abstract theory, the Cauchy problem for the quasilinear wave equation with damping is investigated.


47H20 Semigroups of nonlinear operators
34G20 Nonlinear differential equations in abstract spaces