A REMARK ON THE CONSTRUCTION OF NONLINEAR EVOLUTION OPERATORS

ABSTRACT: Here we note that the standard ODE trick of converting a nonautonomous into an autonomous equation can be employed (under a usual type assumption on the t-dependence) to construct in a simple way an evolution operator generated by a family A(t) (t •> 0) of accretive operators. Let X be a Banach space with norm I1 II, and suppose that A(t)(t •> 0)isafamily of single-valued, m-accretive nonlinear operators (see Crandall [2] for definitions). We propose here to study the evolution equation (1) 2c•t(t) + A(t)u(t) = 0 t•> s u(s) = x (where s •> 0 and x G D(a(s)) are given, and u: [s,oo)-, X is the unknown), and to discuss the solvability of (1) in light of nonlinear semigroup theory for the autonomous equation (2) d•y-(t) + Aw(t) = 0 t •> 0 w(0) = y (where A is a m-accretive operator and y G D(A)is given). The time-dependent problem (1) has been investigated by Crandall-Liggett [4, Appendix to õ1], Crandall-Pazy [5], Plant [9], Evans [6], and others; and in each of these papers a weak or generalized "evolution operator " solution of (1) is constructed under various additional hypotheses on the t-dependence of the A(t), all of the general type IlJx(t)x- Jx(s)xll •< Xlf(t)- f(s)lL (3) for all t,s •> 0, X> 0, and x G X. Here Jx(t)- = (I+ XA(t))-1 L is a constant (but see the concluding remarks of this, paper), and f: [0,oo)-, (_o%oo) is a given function: f is Lipschitz continuous in [4], continuous in [5], Riemann integrable in [9], Lebesgue integrable in [6]. The mai