JOURNAL OF MATHEMATICAL AF;ALYSIS AND APPLICATIONS 131. 17&17Y (198X) Global lnvertibility of Nonsmooth Mappings BRUCE POURCIAU * Department q/ Marhematic.s, Lrr~wnce Universif~, Appleron, Wisronsin 54912 Submilted by R. P. Boas Received September 13, 198.5 In 1906 Hadamard established the following criterion for the global invertihility of continuously differentiable maps ,f from R” into R”: HA~IAMAR~I’S THEOKEM. Suppose the derivative f’(x) is invertible for every x in R” and suppose jg inf,,, <, (l/l/f’(x)-‘11 ) dt = io. Thenfmaps R” diffeomorphically onto R”. If./ is only locally Lipschitz continuous, then the derivative f’(x) may not exist, but we can replace it with a generalized derivative JJ(x) which is a certain compact, convex collection of linear transformations of R”. Using df(x) we extend Hadamard’s theorem to the class of locally Lipschitz continuous maps. The question of global invertibihty for nonsmooth maps arises naturally in several applied areas, including electrical network theory and nonlinear elasticity. ti” ,9** Academic Press. Inc. 1. INTRODUCTION The theorem below was proved by Hadamard [6]. In the statement of this theorem, we use the notation //A//, where A is a linear transformation of R”, to mean the co-norm inf,,, =, IAu(. HADAMARD’S THEOREM. Suppose f is u continuously d&ferentiable map from R” into R” and suppose the derivative f’(x) is invertible for every x. Let m(l) = infIx, <I /lf’(xY/. If I m m(t) dt = co, 0 then f maps R” diffeomorphically onto R”. *This paper was written while the author was visiting the University of California, San Diego. 170 0022-247X/88 $3.00 Copyright (~1 1988 by Academc Press. Inc. All rights of reproduction in any form rcscrvcd.