A global existence theorem for autonomous differential equations in a Banach space

Let £ be a Banach space and let A be a continuous function from E into jE. Sufficient conditions are given to insure that the differential equation u'(t) =Au(t) has a unique solution on [0, oo ) for each initial value in E. One consequence of this result is that if — A is monotonie, then —A is m-monotonic and A is the generator of a nonexpansive semigroup of operators. Let £ be a Banach space over the real or complex field and let I • | denote the norm on E. HA is a continuous function from E into E, we will give a sufficient condition for the autonomous differential equation (ADE) u'(t) = Au(t) to have a unique solution u( • , z) defined on [0, oo) such that w(0, z)—z for each z in E. One consequence of this result is that if A is continuous and — A is monotonie, then — A is wz-monotonic and A is the generator of a nonexpansive semigroup of operators. The principal tool in this paper is the one-sided derivative of the norm on E which will be used as a Liapunov function. Definition 1. If A is a function from E into E and x and y are in E, define (i) D+[x, y, A ] =limA_>+0 (|x — y+h[Ax — Ay]\ — \x—y\)/h and (ii) D-[x, y, A ] =lim^_o (|x— y-\-h[Ax—Ay\\ — \x—y\)/h. Remark 1. Each of the above limits exists since the function h—+\x— y-\-h[Ax—Ay]\ is convex. Furthermore, £>_[x, y, A] ^D+[x, y, A ] for all x and y in E. Example 1. Let E* denote the dual of E and for each x in E let G(x) denote the set of all g in E* such that (x, g) = |x| and |g| =1. In [8, Corollary 2.2] it is shown that D+[x, y, A] = sup{Re(Ax—Ay, g):g is in G(x — y)}. Consequently, D_[x, y, A] — — D+[x, y, — A ] =inf {Re(^4x — Ay, g):g is in G(x— y)}. For each x in £ let F(x) denote the set of all / in E* such that (x, f) = |x|2 and |/| =|x|. Note that if xf^O then / is in F(x) if and only if //|x| is in G(x). Thus — A is accretive on E (i.e. Re(Ax—Ay, f) ^0 Received by the editors February 3, 1970. A MS 1969 subject classifications. Primary 3495; Secondary 4750.