The abstract Cauchy problem for dissipative operators with respect to metric-like functionals (Q401382)

A celebrated result of \textit{R. H. Martin jun.} [J. Funct. Anal. 11, 62--76 (1972; Zbl 0249.47065)] asserts that a semigroup of contractions on a closed subset of a Banach space is generated by a nonlinear operator if and only if this operator satisfies a dissipativity condition and a subtangential condition. The present paper generalizes this result in many ways, providing as a main result a characterization theorem for semigroups of Lipschitz operators defined on a closed subset of a Banach space.NEWLINENEWLINEThe disspativity condition is defined in terms of a nonnegative, Lipschitz continuous functional (called in the paper a metric-like functional), being localized with respect to a \(d\)-uple of Lyapunov functionals. The subtangential condition is also localized by means of the same \(d\)-uple of Lyapunov functionals, while also involving growth estimations in terms of a so-called comparison functional.NEWLINENEWLINEThe proof of the characterization theorem involves the construction of families of approximate solutions (called regularly approximate solutions in the paper) satisfying certain quantitative estimations, by means of a rather involved argument relying upon transfinite induction. The convergence of a family of regularly approximate solutions to a so-called regularly mild solution (which is the notion of solution used in this paper) is proved by means of a key lemma providing an estimation of the difference between two approximate solutions.NEWLINENEWLINEFinally, the existence of a solution of the mixed problem for the Kirchhoff equations with acoustic boundary conditions is determined by employing the previously mentioned abstract result and constructing suitable Lyapunov functionals and metric-like functionals.