Evolution inclusions with Clarke subdifferential type in Hilbert space
In this paper we mainly consider the following evolution inclusion with Clarke subdifferential: {(x(0) = x0) (- t) is an element of partial derivative c phi(x(t)) + G(t, x(t))) where partial derivative(c) is the Clarke subdifferential and t is an element of I = [0, T] is a nonempty, bounded closed interval in R(+) = [0, + infinity); H is a separable Hilbert space; x(0) is an element of D(partial derivative(c)phi) = {x is an element of H : partial derivative(c)phi(x) not equal phi}. First we introduce some hypotheses on phi, under which we prove that there is a unique solution of (*) when G is a single function. Based on this result,we obtain an existence theorem for G being a u.s.c. or l.s.c multifunction. Then the existence of an extremal solution of (*) and the relaxation theorem are studied. In the end,we present two examples of PDE to illustrate the application of our results. (C) 2009 Elsevier Ltd. All rights reserved.
evolution inclusions clarke subdifferential hilbert space extremal solution relaxation theorem