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

摘要

This paper is devoted to a characterization of semigroups of Lipschitz operators on a closed subset D of a Banach space X and the abstract Cauchy problem for an operator A in X satisfying the following condition: There exists a proper lower semicontinuous functional phi from X into [0, oaf such that the effective domain of phi is D(A) and such that lim(n ->infinity) Ax(n) = Ax in X for any x is an element of D(A) and any sequence {x(n)} in D(A) satisfying two conditions x = x in X and lim sup(n ->infinity) phi(x(n)) <= phi(x). The main result asserts that a semigroup of Lipschitz operators on D can be generated by an operator A satisfying the above-mentioned condition, a dissipative condition with respect to a metric-like functional and a subtangential condition. The Kirchhoff equations with acoustic boundary conditions are solved by the method based on the abstract result with the construction of suitable Liapunov functionals and the use of a metric-like functional on a suitable set. 2014 Elsevier Inc. All rights reserved.