Artículos de revistas
Global Injectivity Of C1 Maps Of The Real Plane, Inseparable Leaves And The Palais-smale Condition
Registro en:
Canadian Mathematical Bulletin. , v. 50, n. 3, p. 377 - 389, 2007.
84395
2-s2.0-34548598149
Autor
Gutierrez C.
Jarque X.
Llibre J.
Teixeira M.A.
Institución
Resumen
We study two sufficient conditions that imply global injectivity for a C1 map X: ℝ2 → ℝ2 such that its Jacobian at any point of ℝ2 is not zero. One is based on the notion of half-Reeb component and the other on the Palais-Smale condition. We improve the first condition using the notion of inseparable leaves. We provide a new proof of the sufficiency of the second condition. We prove that both conditions are not equivalent, more precisely we show that the Palais-Smale condition implies the nonexistence of inseparable leaves, but the converse is not true. Finally, we show that the Palais-Smale condition it is not a necessary condition for the global injectivity of the map X. © Canadian Mathematical Society 2007. 50 3 377 389 Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maier, A.L., (1973) Qualitative theory of second-order dynamic systems, , Wiley, New York Ambrosetti, A., Rabinowitz, P.H., Dual variational methods in critical point theory and applications (1973) J. Functional Analysis, 14, pp. 349-381 Cobos, M., Gutierrez, C., Llibre, J., On the injectivity of ρ{variant}1 maps on the real plane (2001) Canad. J. Math, 54 (6), pp. 1187-1201 van den Essen, A., Polynomial Automorphisms and the Jacobian Conjecture (2000) Progress in Mathematics, 190. , Birkhauser Verlag, Basel Fernandes, A., Gutierrez, C., Rabanal, R., Global asymptotic stability for differentiate vector fields of ℝ2 (2004) J. Differential Equations, 206 (2), pp. 470-482 Gonzales Velasco, E.A., Generic properties of polynomial vector fields at infinity (1969) Trans. Amer. Math. Soc, 143, pp. 201-222 Gutierrez, C., Nguyen, N.C., A remark on an eigenvalue condition for theghbal injectivity of differentiable maps of ℝ2 (2007) Disc. Cont. Dyn. Syst, 17 (2), pp. 397-402 Jarque, X., Llibre, J., Polynomial foliations of ℝ 2 (2001) Pacific J. Math, 197 (1), pp. 53-72 Jarque, X., Nitecki, Z., Hamiltonian stability in the plane (2000) Ergodic Theory Dynam. Systems, 20 (3), pp. 775-799 Kaplan, W., Regular curve-families filling the plane. II (1941) Duke Math. J, 8, pp. 11-46 Kotus, J., Krych, M., Nitecki, Z., Global structural stability of flows on open surfaces (1982) Mem. Amer. Math. Soc, 37 (261) Neumann, D., Classification of continuous flows on 2-manifolds (1975) Proc. Amer. Math. Soc, 48, pp. 73-81 Pinchuck, S., A counterexample to the strong Jacobian conjecture (1994) Math. Z, 217 (1), pp. 1-4 P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Eqautions, C.B.M.S. Regional Conference Series in Mathematics 65, American Mathematical Society, Providence, RI, 1986E. A. de B. e Silva and M. A. Teixeira, A version of Rolle's Theorem and applications. Bol. Soc. Brasil. Mat. 29(1998), no. 2, 301-328Pinchuck, S., Global injectivity and asymptotic stability via minimax method (2000) Progressin Nonlinear Analysis Nankai Ser. Pure Appl. Math. Theoret. Phys, 6, pp. 339-358. , World Sci. Publishing, River Edge, NJ