Vol. 39, No. 1, 1971

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Semi-groups of local Lipschitzians in a Banach space

J. T. Chambers and Shinnosuke Oharu

Vol. 39 (1971), No. 1, 89–112

The purpose of this paper is to construct a nonlinear semi-group determined by a given (multi-valued) nonlinear operator A in a Banach space X, and to investigate the differentiability of this semi-group. The semi-group treated in this paper is the semigroup {T(t);t 0} of nonlinear operators in X such that for each τ > 0,{T(t);0 t τ} is equi-Lipschitz continuous on bounded sets. In order that an operator A in X determine such a semi-group {T(t);t 0} on D(A) with (d∕dt)T(t)x AT(t)x for almost all t 0 and x D(A), it is required that X have a uniformly convex dual, A be dissipative in a local sense, I λA,λ positive and small, satisfy a range condition and an injectiveness condition, and finally the family of operators (I λA)n,n = 1,2,3, be locally equi-bounded.

Mathematical Subject Classification 2000
Primary: 47D05
Secondary: 47H99
Received: 25 May 1970
Revised: 19 April 1971
Published: 1 October 1971
J. T. Chambers
Shinnosuke Oharu