# zbMATH — the first resource for mathematics

Partial linearization for noninvertible mappings. (English) Zbl 0813.47075
The authors prove a Hartman-Grobman result for noninvertible mappings in a rather general situation: Let $${\mathcal U}$$, $$\mathcal V$$, $$\mathcal W$$ be Banach spaces and $$F: {\mathcal U}\to {\mathcal U}$$, $$G: {\mathcal V}\to {\mathcal V}$$, $$H: {\mathcal W}\to {\mathcal W}$$ bounded linear operators such that there is an $$a\in (0,1)$$ with $$\sigma(F)\subset \{\lambda\in \mathbb{C}\mid|\lambda |< a\}$$, $$\sigma(G)\subset \{\lambda\in \mathbb{C}\mid a< | \lambda|< 1\}$$, and $$\sigma(H)\subset \{\lambda\in \mathbb{C}\mid |\lambda|> 1\}$$. The Banach spaces are assumed to possess a smooth norm in the sense that there exists a real-valued $$C^ 1$$- function $$\lambda$$ with bounded and uniformly Lipschitzian derivative which vanishes outside the unit ball and equals 1 on some ball around zero. They consider nonlinear operators on a neighbourhood $$\mathcal N$$ of zero in $${\mathcal E}:= {\mathcal U}\times {\mathcal V}\times {\mathcal W}$$, $$U: {\mathcal N}\to {\mathcal U}$$, $$V: {\mathcal N}\to {\mathcal V}$$, $$W: {\mathcal N}\to {\mathcal W}$$ of the form $$U(u,v,w)= Fu+ f(u,v,w)$$, $$V(u,v,w)= Gv+ g(u,v,w)$$, $$W(u,v,w)= Hw+ h(u,v,w)$$ where $$f$$, $$g$$, $$h$$ are assumed to be partially differentiable with respect to $$u$$, $$v$$, $$w$$ with partial derivatives which are bounded and uniformly Lipschitzian in $$u$$, $$v$$, $$w$$. Moreover, $$f$$, $$g$$, $$h$$ are assumed to be flat at zero (i.e., $$f$$, $$g$$, $$h$$ and their first partial derivatives with respect to $$u$$, $$v$$, $$w$$ take the value zero at zero). It is conjectured that under these conditions there is an open neighbourhood $${\mathcal N}_ 0$$ of zero in $$\mathcal E$$ and a continuous function $$f_ 0: {\mathcal N}_ 0\to {\mathcal U}$$ with $$f_ 0(0,0,0)= 0$$ such that $$(U,V,W)$$ is locally conjugate to $$(F+ f_ 0,G,H)$$ (i.e., there is a homeomorphism of neighbourhoods of zero in $$\mathcal E$$ which carries $$(U,V,W)$$ to $$(F+ f_ 0,G,H)$$). (For example, the classical Hartman-Grobman lemma states that the conjecture is true if $$F$$ is invertible.) Using rather elaborate computations the authors prove that the conjecture is true if one assumes in addition that there are finitely many numbers $$a_ 1,\dots, a_ n$$ in $$(a,1)$$ such that $$|\lambda|= a_ k$$ for some $$k\in \{1,\dots, n\}$$ whenever $$\lambda\in \sigma(G)$$.
Reviewer: C.Fenske (Gießen)

##### MSC:
 47J05 Equations involving nonlinear operators (general) 58C40 Spectral theory; eigenvalue problems on manifolds 46G05 Derivatives of functions in infinite-dimensional spaces 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations
Full Text:
##### References:
 [1] B. Aulbach and B. M. Garay,Linearizing the expanding part of noninvertible mappings. J. Appl. Math. Phys. (ZAMP),44, 469-494 (1993). · Zbl 0803.46045 · doi:10.1007/BF00953663 [2] B. Aulbach and B. M. Garay,Discretization of semilinear differential equations with an exponential dichotomy. Computers & Mathematics with Applications, to appear. · Zbl 0806.65067 [3] B. Aulbach and T. Wanner,Invariant fiber bundles and topological equivalence in dynamic processes. Lecture Notes (in preparation). · Zbl 1049.39020 [4] G. R. Belickii,Equivalence and normal forms of germs of smooth mappings. Russian Math. Surveys,33, 107-177 (1978). · Zbl 0398.58009 · doi:10.1070/RM1978v033n01ABEH002237 [5] G. R. Belickii,Sternberg’s theorem for Banach spaces. Functional Anal. Appl.,108, 238-239 (1984). [6] S. N. Chow and K. Lu,C k center unstable manifolds. Proc. Soc. Edinb. Sect. A,108, 303-320 (1988). · Zbl 0707.34039 [7] Yu. L. Daletskii and G. K. Roginskii,A topological classification of certain types of semilinear evolution equations in a Banach space. Differential Equations,28, 463-472 (1992). [8] B. M. Garay,Parallelizability, mild mixing and topological conjugacy for strongly continuous one-parameter groups of unitary operators. J. London Math. Soc.,47, 533-541 (1993). · Zbl 0791.47034 · doi:10.1112/jlms/s2-47.3.533 [9] J. K. Hale,Asymptotic behaviour of dissipative systems. AMS, Providence, R.I. (1988). · Zbl 0642.58013 [10] P. Hartman,On local homeomorphisms of Euclidean spaces. Bol. Soc. Math. Mexicana,5, 220-241 (1960). · Zbl 0127.30202 [11] P. Hartman,A lemma in the theory of structural stability of differential equations. Proc. Amer. Math. Soc.,11, 610-622 (1960). · Zbl 0132.31904 · doi:10.1090/S0002-9939-1960-0121542-7 [12] D. Henry,Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin 1981. · Zbl 0456.35001 [13] D. Henry,Geometric theory of semilinear parabolic equations. Dep. de Math. Applicada, Univ. de Sao Paulo, Sao Paulo, Brazil 1983. [14] S. Hilger,Ein Ma?kettenkalkül mit Anwendungen auf Zentrumsmannigfaltigkeiten. Dissertation, Universität Würzburg 1988. · Zbl 0695.34001 [15] S. Hilger,Smoothness of invariant manifolds. J. Funct. Analysis,106, 95-129 (1992). · Zbl 0762.93041 · doi:10.1016/0022-1236(92)90065-Q [16] M. Hirsch, C. Pugh and M. Shub,Invariant Manifolds. Springer, Berlin 1977. [17] M. C. Irwin,A new proof of the pseudo-stable manifold theorem. J. London Math. Soc.,21, 557-566 (1980). · Zbl 0436.58021 · doi:10.1112/jlms/s2-21.3.557 [18] M. C. Irwin,Smooth Dynamical Systems. Academic Press, New York 1980. · Zbl 0465.58001 [19] B. Jakubczyk,Equivalence and invariants of nonlinear control systems. In H. Sussman, ed.,Nonlinear Controllability and Optimal Control, pp. 177-218, Dekker, Basel 1990. · Zbl 0712.93027 [20] T. Kato,Perturbation Theory for Linear Operators. Springer, Berlin 1966. · Zbl 0148.12601 [21] U. Kirchgraber and K. Palmer,Geometry in the neighborhood of invariant manifolds of maps and flows and linearization. Pitman, London 1991. · Zbl 0746.58008 [22] K. Lu,A Hartman-Grobman theorem for scalar reaction-diffusion equations. J. Diff. Eqns.,93, 364-394 (1991). · Zbl 0767.35039 · doi:10.1016/0022-0396(91)90017-4 [23] J. Marsden and J. Scheurle,The construction and smoothness of invariant manifolds by the deformation method. SIAM J. Math. Anal.,18, 1261-1274 (1987). · Zbl 0674.58040 · doi:10.1137/0518092 [24] X. Mora and J. Sola-Morales,Existence and nonexistence of finite-dimensional globally attracting invariant manifolds in semilinear damped wave equation. In S. N. Chow and J. K. Hale eds.,Dynamics of Infinite-Dimensional Systems, pp. 187-210, Springer, Berlin 1987. [25] N. V. Nikolaenko,The method of Poincaré normal forms in problems of integrability of equations of evolution type. Russian Math. Surveys,41, 63-114 (1986). · Zbl 0632.35026 · doi:10.1070/RM1986v041n05ABEH003423 [26] C. Pugh,On a theorem of P. Hartman. Amer. J. Math.,91, 363-367 (1969). · Zbl 0197.20701 · doi:10.2307/2373513 [27] J. Quandt,On inverse limit stability for maps. J. Diff. Equ.,79, 316-339 (1989). · Zbl 0677.58028 · doi:10.1016/0022-0396(89)90106-X [28] D. Ruelle,Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press, New York 1989. · Zbl 0684.58001 [29] G. R. Sell,Smooth linearization near a fixed point. J. Amer. J. Math.,107, 1035-1091 a(1985). · Zbl 0574.34025 · doi:10.2307/2374346 [30] M. Shub,Global Stability of Dynamical Systems. Springer, Berlin 1987. · Zbl 0606.58003 [31] N. Sternberg,A Hartman-Grobman theorem for a class of retarded functional differential equations. J. Math. Anal. Appl.,176, 156-165 (1993). · Zbl 0779.34061 · doi:10.1006/jmaa.1993.1206 [32] S. Sternberg,On the structure of local homeomorphisms of Euclidean n-space II. Amer. J. Math.,80, 623-631 (1958). · Zbl 0083.31406 · doi:10.2307/2372774 [33] S. van Strien,Smooth linearization of hyperbolic fixed points without resonance conditions. J. Diff. Eqns.,85, 66-90 (1990). · Zbl 0726.58039 · doi:10.1016/0022-0396(90)90089-8 [34] A. Vanderbauwhede and S. A. van Gils,Center manifolds and contractions on a scale of Banach spaces. J. Fund. Anal.,72, 209-224 (1987). · Zbl 0621.47050 · doi:10.1016/0022-1236(87)90086-3 [35] T. Wanner,Invariante Faserbündel und topologische Äquivalenz bei dynamischen Prozessen. Diplomarbeit, Augsburg 1991. [36] T. Wanner,A Hartman-Grobman theorem for discrete random dynamical systems. Preprint No. 269, University of Augsburg, 1992. [37] E. Zehnder,Siegers linearization theorem in infinite dimensions. Manuscripta Math.,23, 363-371 (1978). · Zbl 0374.47037 · doi:10.1007/BF01167695 [38] W. Zang,Invariant manifolds for differential equations. Acta Math. Sinica,8, 375-398 (1992). · Zbl 0785.58043 · doi:10.1007/BF02583264
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.