Asymptotic behavior of solutions of a nonlinear evolution equation

Abstract

The purpose of this paper is to study the asymptotic behavior of the solutions of a certain nonlinear evolution equation in Banach spaces (see (1) below). This time-dependent initial value problem involves an accretive operator A and a function g: R + --f R +. It was first considered by Browder in [ 11, and was the basis of the iterative procedures for constructing zeros of accretive operators presented in [2] and [5]. These papers treat only the case when 0 E R(A). We begin with several results (e.g., Theorems 3 and 6) that are of interest when 0 & R(A). These results are, in some sense, analogs of known results for nonlinear semigroups and resolvents. (There is, however, a difference-see, for example, Corollary 4). Then we show that under certain mild restrictions, 0 E R(A) if and only if the solutions to (1) are bounded (Theorem 7). When an additional condition is imposed on g, we can shown that A is zero free if and only if lim,,, ] u(t)] = co for each solution of (1) (Theorem 8 and Proposition 9). In the last part of the paper we assume that 0 E R(A) and obtain several strong convergence results for solutions of (1) (e.g., Theorem 12). These results show that the asymptotic behavior of the solutions to (1) resembles the behavior of resolvents more than that of nonlinear semigroups. Theorem 12 yields improvements of the convergence results for the iterative procedure of [S], and leads to a geometric result concerning the fixed point set of a nonexpansive mapping (Corollary 13). We also mention that the sequence constructed from the iterative procedure of [5] imitates the behavior of the solutions of (1) not only when 0 E R(A), but also when A is zero free. Finally, we remark that many of our results are new even in Hilbert space. Let E be a real Banach space, and let I denote the identity operator. Recall that a subset A of E x E with domain D(A) and range R(A) is said to be accretive if lx, -x2) 0. We denote the closure of a subset D of E by cl(D), its closed convex hull by clco(D), and its distance from a point x in E by d(x, 0). We 43 0022-247X/81/090043-11$02.00/0