Vol. 95, No. 1, 1981

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ISSN: 0030-8730
Solvability of nonlinear operator equations

W. J. Cramer and William O. Ray

Vol. 95 (1981), No. 1, 37–50
Abstract

Let X and Y be Banach spaces, P : X Y a Gateaux differentiable operator having closed graph. Suppose

  1. for each R > 0 there is a δ > 0 such that
    dPx(B (0;1)) ⊇ B (0;δ) whenever ∥x∥ ≦ R

  2. P1(K) is bounded whenever cl (K) Y is compact;

then P is an open mapping of X onto Y . Similar results are obtained for compact Gateaux differentiable operators using a local version of (i); the same local version gives a domain invariance theorem for Gateaux differentiable operators having closed graph. Related results deal with M. Altman’s theory of contractor directions and theory of normal solvability as developed by F. E. Browder and others.

Mathematical Subject Classification 2000
Primary: 47H15
Secondary: 58C15
Milestones
Received: 7 April 1980
Published: 1 July 1981
Authors
W. J. Cramer
William O. Ray