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Open mappings and solvability of nonlinear equations in Banach space

Published online by Cambridge University Press:  14 November 2011

Stewart C. Welsh
Stewart C. Welsh
Affiliation:
Department of Mathematics, Southwest Texas State University, San Marcos, TX 78666, U.S.A.
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Extract

We give sufficient conditions under which a funclion f: X → Y is an open mapping, where X and y are Banach spaces. This function is not necessarily continuous, but is assumed to have closed graph. We prove our results without requiring that f be Gateaux differentiable; instead, f is assumed to possess a weak type of Gateaux inverse derivative.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 2 , 1996 , pp. 239 - 246
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Altman, M.. Contractors and Contractor Directions. Theory and Applications (New York; Dekker, 1977).Google Scholar
2Brézis, H. and Browder, F. E.. A general principle on ordered sets in nonlinear functional analysis. Adv. Math. 21 (1976), 567–890.CrossRefGoogle Scholar
3Kacurowskii, R. I., Generalization of the Fredholm theorems and of theorems on linear operators withclosed range to some classes of nonlinear operators. Soviet Math. Dokl. 12 (1971), 168–72.Google Scholar
4Krasnosel'skii, M. A.. On several new fixed point principles. Dokl. Akad. Nauk. SSSR 14 (1973), 259–61.Google Scholar
5Ray, W. O. and Cramer, W. J.. Solvability of nonlinear operator equations. Pacific J. Math. 95 (1981), 37–50.Google Scholar
6Ray, W. O. and Walker, A. M.. Mapping theorems for Gateaux differentiable and accretive operators. Nonlinear Anad. 6 (1982), 423–33.CrossRefGoogle Scholar
7Rosenholtz, I. and Ray, W. O.. Mapping theorems for differentiable operators. Bull. polon. Sci. Math. 29(1981), 265–73.Google Scholar