Global invertibility of expanding maps

Hernández, Jorge E.; Nashed, M. Zuhair

doi:10.1090/S0002-9939-1992-1110543-9

Global invertibility of expanding maps
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by Jorge E. Hernández and M. Zuhair Nashed PDF

Proc. Amer. Math. Soc. 116 (1992), 285-291 Request permission

Abstract:

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 $\alpha$-expanding Fréchet differentiable map in Banach space under the assumption that the logarithmic norm of the derivative is negative.

References

Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308. MR 0405188
Felix E. Browder, On the Fredholm alternative for nonlinear operators, Bull. Amer. Math. Soc. 76 (1970), 993–998. MR 265999, DOI 10.1090/S0002-9904-1970-12527-7
Kung Ching Chang and Shu Jie Li, A remark on expanding maps, Proc. Amer. Math. Soc. 85 (1982), no. 4, 583–586. MR 660608, DOI 10.1090/S0002-9939-1982-0660608-4
Mihai Cristea, A note on global inversion theorems and applications to differential equations, Nonlinear Anal. 5 (1981), no. 11, 1155–1161. MR 636727, DOI 10.1016/0362-546X(81)90009-2
Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
M. Fabián and D. Preiss, A generalization of the interior mapping theorem of Clarke and Pourciau, Comment. Math. Univ. Carolin. 28 (1987), no. 2, 311–324. MR 904756
T. M. Flett, Differential analysis, Cambridge University Press, Cambridge-New York, 1980. Differentiation, differential equations and differential inequalities. MR 561908, DOI 10.1017/CBO9780511897191
Fritz John, On quasi-isometric mappings. I, Comm. Pure Appl. Math. 21 (1968), 77–110. MR 222666, DOI 10.1002/cpa.3160210107
Erwin Kreyszig, Introductory functional analysis with applications, John Wiley & Sons, New York-London-Sydney, 1978. MR 0467220
Jean-Michel Morel and Heinrich Steinlein, On a problem of Nirenberg concerning expanding maps, J. Funct. Anal. 59 (1984), no. 1, 145–150. MR 763781, DOI 10.1016/0022-1236(84)90057-0
M. Z. Nashed, Differentiability and related properties of nonlinear operators: Some aspects of the role of differentials in nonlinear functional analysis, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1971, pp. 103–309. MR 0276840
L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. With a chapter by E. Zehnder; Notes by R. A. Artino; Lecture Notes, 1973–1974. MR 0488102
Roy Plastock, Homeomorphisms between Banach spaces, Trans. Amer. Math. Soc. 200 (1974), 169–183. MR 356122, DOI 10.1090/S0002-9947-1974-0356122-6
Marius Rădulescu and Sorin Radulescu, Global inversion theorems and applications to differential equations, Nonlinear Anal. 4 (1980), no. 5, 951–965. MR 586858, DOI 10.1016/0362-546X(80)90007-3
Sorin Rădulescu and Marius Rădulescu, Global univalence and global inversion theorems in Banach spaces, Nonlinear Anal. 13 (1989), no. 5, 539–553. MR 993257, DOI 10.1016/0362-546X(89)90063-1
Eberhard Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. MR 816732, DOI 10.1007/978-1-4612-4838-5