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1992 | 57 | 3 | 253-263
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Differential conditions to verify the Jacobian Conjecture

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Let F be a polynomial mapping of ℝ², F(O) = 0. In 1987 Meisters and Olech proved that the solution y(·) = 0 of the autonomous system of differential equations ẏ = F(y) is globally asymptotically stable provided that the jacobian of F is everywhere positive and the trace of the matrix of the differential of F is everywhere negative. In particular, the mapping F is then injective. We give an n-dimensional generalization of this result.
Twórcy
  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland,
  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland,
Bibliografia
  • [B] N. E. Barabanov, On Kalman's problem, Sibirsk. Mat. Zh. 29 (3) (1988), 2-11 (in Russian).
  • [BR] A. Białynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200-203.
  • [BCR] J. Bochnak, M. Coste et M.-F. Roy, Géométrie Algébrique Réelle, Springer, Berlin 1987.
  • [D] F. Dillen, Polynomials with constant Hessian determinant, J. Pure Appl. Algebra 71 (1991), 13-18.
  • [E] A. van den Essen, A note on Meisters and Olech's proof of the global asymptotic stability Jacobian conjecture, Pacific J. Math. 151 (1991), 351-356.
  • [H] P. Hartman, Ordinary Differential Equations, Wiley, New York 1964.
  • [HO] P. Hartman and C. Olech, On global asymptotic stability of solutions of differential equations, Trans. Amer. Math. Soc. 104 (1962), 154-178.
  • [KR] K. Kurdyka and K. Rusek, Surjectivity of certain injective semialgebraic transformations of ℝⁿ, Math. Z. 200 (1988), 141-148.
  • [Ł] S. Łojasiewicz, Introduction to Complex Analytic Geometry, Birkhäuser, Basel 1991.
  • [MY] L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. J. 12 (1960), 305-317.
  • [MO] G. H. Meisters and C. Olech, Solution of the global asymptotic stability Jacobian conjecture for the polynomial case, in: Analyse Mathématique et Applications, Gauthier-Villars, Paris 1988, 373-381.
  • [MO1] G. H. Meisters and C. Olech, A Jacobian condition for injectivity of differentiable plane maps, Ann. Polon. Math. 51 (1990), 249-254.
  • [Md] D. Mumford, Algebraic Geometry, I. Complex Projective Varieties, Springer, Berlin 1976.
  • [O] C. Olech, On the global stability of an autonomous system on the plane, Contributions Differential Equations 1 (1963), 389-400.
  • [P] T. Parthasarathy, On Global Univalence, Lecture Notes in Math. 977, Springer, Berlin 1983.
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Bibliografia
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bwmeta1.element.bwnjournal-article-apmv57z3p253bwm
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