Global invertibility of expanding maps

We prove a global inversion theorem in reflexive Banach spaces utilizing a recent generalization of the interior mapping theorem. As a corollary, we provide, under a mild approximation property, a positive answer to an open problem that was stated by Nirenberg. We also establish global invertibility of an a-expanding Frechet differentiable map in Banach space under the assumption that the logarithmic norm of the derivative is negative. The study of nonlinear operator equations of the form F(x) = y, where F is a map from a Banach space X into a Banach space Y, is a central topic in nonlinear functional analysis and its applications. Typically, it is important to find conditions on F that guarantee the existence of a solution for each y E Y. It is also important, particularly when approximate and iterative methods are involved, to determine conditions that ensure uniqueness and continuous dependence of solutions. In other words, a central problem is to find tractable conditions for the map F to be a global homeomorphism. Problems of global inversions have been studied by several authors, e.g., Browder [1], Cristea [4], Nirenberg [12], Plastock [13], and Radulescu [14, 15]; see references cited therein for earlier contributions. See also the recent books by Deimling [5] and Zeidler [16]. Along this line, Nirenberg [12, p. 175] posed the following interesting problem: Let H be a Hilbert space and T: H -* H be a continuous expanding map (i.e., IITx Tyll > lix yII). Let T(O) = 0 and suppose that T maps a neighborhood of the origin onto a neighborhood of the origin. Does T map H onto H? A partial answer to this problem was given by Chang and Shujie [3]. They prove the surjectivity of T: X -* Y in the case when Y is reflexive, under the additional assumptions that T is Frechet differentiable and lim sup II T'(x) T'(xo) II < 1 for all xo E X . x-+xo Received by the editors February 21, 1991. 1991 Mathematics Subject Classification. Primary 58C 15, 47H 15, 58C20.