## Global invertibility of expanding maps

HTML articles powered by AMS MathViewer

- 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

## Additional Information

- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**116**(1992), 285-291 - MSC: Primary 58C15; Secondary 47H15
- DOI: https://doi.org/10.1090/S0002-9939-1992-1110543-9
- MathSciNet review: 1110543