|
Global invertibility of expanding maps
Authors:
Jorge E. Hernández and M. Zuhair Nashed
Journal:
Proc. Amer. Math. Soc. 116 (1992), 285-291
MSC:
Primary 58C15; Secondary 47H15
MathSciNet review:
1110543
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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  -expanding Fréchet differentiable map in Banach space under the assumption that the logarithmic norm of the derivative is negative.
- [1]
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
(53 #8982)
- [2]
Felix
E. Browder, On the Fredholm alternative for
nonlinear operators, Bull. Amer. Math. Soc.
76 (1970),
993–998. MR 0265999
(42 #908), http://dx.doi.org/10.1090/S0002-9904-1970-12527-7
- [3]
Kung
Ching Chang and Shu
Jie Li, A remark on expanding maps,
Proc. Amer. Math. Soc. 85 (1982),
no. 4, 583–586. MR 660608
(83j:47045), http://dx.doi.org/10.1090/S0002-9939-1982-0660608-4
- [4]
Mihai
Cristea, A note on global inversion theorems and applications to
differential equations, Nonlinear Anal. 5 (1981),
no. 11, 1155–1161. MR 636727
(84d:58006), http://dx.doi.org/10.1016/0362-546X(81)90009-2
- [5]
Klaus
Deimling, Nonlinear functional analysis, Springer-Verlag,
Berlin, 1985. MR
787404 (86j:47001)
- [6]
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
(88j:58009)
- [7]
T.
M. Flett, Differential analysis, Cambridge University Press,
Cambridge, 1980. Differentiation, differential equations and differential
inequalities. MR
561908 (82e:26021)
- [8]
Fritz
John, On quasi-isometric mappings. I, Comm. Pure Appl. Math.
21 (1968), 77–110. MR 0222666
(36 #5716)
- [9]
Erwin
Kreyszig, Introductory functional analysis with applications,
John Wiley & Sons, New York-London-Sydney, 1978. MR 0467220
(57 #7084)
- [10]
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 (86b:47116), http://dx.doi.org/10.1016/0022-1236(84)90057-0
- [11]
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
(43 #2580)
- [12]
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
(58 #7672)
- [13]
Roy
Plastock, Homeomorphisms between Banach
spaces, Trans. Amer. Math. Soc. 200 (1974), 169–183. MR 0356122
(50 #8593), http://dx.doi.org/10.1090/S0002-9947-1974-0356122-6
- [14]
Marius
Rădulescu and Sorin
Radulescu, Global inversion theorems and applications to
differential equations, Nonlinear Anal. 4 (1980),
no. 5, 951–965. MR 586858
(81m:58018), http://dx.doi.org/10.1016/0362-546X(80)90007-3
- [15]
Sorin
Rădulescu and Marius
Rădulescu, Global univalence and global inversion theorems
in Banach spaces, Nonlinear Anal. 13 (1989),
no. 5, 539–553. MR 993257
(90d:58017), http://dx.doi.org/10.1016/0362-546X(89)90063-1
- [16]
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
(87f:47083)
- [1]
- F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Pure Math., vol. 18, Amer. Math. Soc., Providence, RI, 1976. MR 0405188 (53:8982)
- [2]
- -, On the Fredholm alternative for nonlinear operators, Bull. Amer. Math. Soc. 76 (1970), 993-998. MR 0265999 (42:908)
- [3]
- K.-C. Chang and L. Shujie, A remark on expanding maps, Proc. Amer. Math. Soc. 85 (1982), 583-586. MR 660608 (83j:47045)
- [4]
- M. Cristea, A note on global inversion theorems and applications to differential equations, Nonlinear Anal. 5 (1981), 1155-1161. MR 636727 (84d:58006)
- [5]
- K. Deimling, Nonlinear functional analysis, Springer-Verlag, New York, 1985. MR 787404 (86j:47001)
- [6]
- M. Fabian and D. Preiss, A generalization of the interior mapping theorem of Clarke and Pourciau, Comment. Math. Univ. Carolinae 28 (1987), 311-324. MR 904756 (88j:58009)
- [7]
- T. M. Flett, Differential analysis, Cambridge Univ. Press, Cambridge, UK, 1980. MR 561908 (82e:26021)
- [8]
- F. John, On quasi-isometric mappings. I, Comm. Pure Appl. Math. 21 (1968), 77-110. MR 0222666 (36:5716)
- [9]
- E. Kreyszig, Introductory functional analysis with applications, Wiley, New York, 1978. MR 0467220 (57:7084)
- [10]
- J. M. Morel and H. Steinlein, On a problem of Nirenberg concerning expanding maps, J. Funct. Anal. 59 (1984), 145-150. MR 763781 (86b:47116)
- [11]
- M. Z. Nashed, Differentiability and related properties of nonlinear operators: some aspects of the role of differentials in nonlinear functional analysis, Nonlinear Functional Analysis and Applications (L. B. Rall, ed.), Academic Press, New York, 1971, pp. 103-309. MR 0276840 (43:2580)
- [12]
- L. Nirenberg, Topics in nonlinear functional analysis, Lecture notes, Courant Institute, New York, 1974. MR 0488102 (58:7672)
- [13]
- R. Plastock, Homeomorphisms between Banach spaces, Trans. Amer. Math. Soc. 200 (1974), 169-183. MR 0356122 (50:8593)
- [14]
- M. Radulescu and S. Radulescu, Global inversion theorems and applications to differential equations, Nonlinear Anal. 4 (1980), 951-965. MR 586858 (81m:58018)
- [15]
- -, Global univalence and global inversion theorems in Banach spaces, Nonlinear Anal. 13 (1989), 539-553. MR 993257 (90d:58017)
- [16]
- E. Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York, 1986. MR 816732 (87f:47083)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
58C15,
47H15
Retrieve articles in all journals
with MSC:
58C15,
47H15
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1992-1110543-9
PII:
S 0002-9939(1992)1110543-9
Keywords:
Global inverse mapping theorems,
 -expanding maps,
logarithmic norm,
interior mapping theorem,
Fréchet derivative
Article copyright:
© Copyright 1992 American Mathematical Society
|
