Annali di Matematica Pura ed Applicata (1923 -)

, Volume 195, Issue 4, pp 1347–1371 | Cite as

Generalizing the Poincaré–Miranda theorem: the avoiding cones condition

  • Alessandro FondaEmail author
  • Paolo Gidoni


After proposing a variant of the Poincaré–Bohl theorem, we extend the Poincaré–Miranda theorem in several directions, by introducing an avoiding cones condition. We are thus able to deal with functions defined on various types of convex domains, and situations where the topological degree may be different from \(\pm \)1. An illustrative application is provided for the study of functionals having degenerate multi-saddle points.


Poincaré–Miranda Fixed point Topological degree 

Mathematics Subject Classification

47H10 54H25 37C25 



We thank the referee for suggesting us simpler proofs of Theorem 3 and Proposition 7. We also acknowledge the support of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).


  1. 1.
    Alonso, J.M., Ortega, R.: Roots of unity and unbounded motions of an asymmetric oscillator. J. Differ. Equ. 143, 201–220 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amann, H.: A note on degree theory for gradient mappings. Proc. Am. Math. Soc. 85, 591–595 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  4. 4.
    Ben-El-Mechaiekh, H., Kryszewski, W.: Equilibria of set-valued maps on nonconvex domains. Trans. Am. Math. Soc. 349, 4159–4179 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Breuer, F.: Uneven splitting of ham sandwiches. Discrete Comput. Geom. 43, 876–892 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cinquini, S.: Problemi di valori al contorno per equazioni differenziali di ordine \(n\). Ann. R. Sc. Norm. Sup. Pisa (2) 9, 61–77 (1940)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  8. 8.
    Ćwiszewski, A., Kryszewski, W.: The constrained degree and fixed-point index theory for set-valued maps. Nonlinear Anal. 64, 2643–2664 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dancer, E.N.: Degenerate critical points, homotopy indices and Morse inequalities. J. Reine Angew. Math. 350, 1–22 (1984)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Derivière, S., Kaczynski, T., Vallerand, P.O.: On the decomposition and local degree of multiple saddles. Ann. Sci. Math. Québec 33, 45–62 (2009)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fabry, C., Fonda, A.: Unbounded motions of perturbed isochronous Hamiltonian systems at resonance. Adv. Nonlinear Stud. 5, 351–373 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fonda, A., Mawhin, J.: Planar differential systems at resonance. Adv. Differ. Equ. 11, 1111–1133 (2006)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fonda, A., Ureña, A.J.: On the higher dimensional Poincaré–Birkhoff theorem for Hamiltonian flows. 2. The avoiding rays condition. Preprint (2014).
  14. 14.
    Giaquinta, M., Hildebrandt, S.: Calculus of Variations II. Springer, Berlin (1996)zbMATHGoogle Scholar
  15. 15.
    Haddad, G.: Topological properties of the sets of solutions for functional differential inclusions. Nonlinear Anal. 5, 1349–1366 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Idzik, A., Junosza-Szaniawski, K.: Combinatorial lemmas for polyhedrons. Discuss. Math. Graph Theory 25, 439–448 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Krasnosel’skii, M.A.: The operator of translation along the trajectories of differential equations. Transl. Math. Monogr. vol. 19, Am. Math. Soc. Providence, R.I. (1968)Google Scholar
  18. 18.
    Kryszewski, W.: On the existence of equilibria and fixed points of maps under constraints. In: Brown, R.F. , Furi, M., Górniewicz, L., Jiang B. (eds.) Handbook of Topological Fixed Point Theory. Springer, Berlin, pp. 783–866 (2005)Google Scholar
  19. 19.
    Kulpa, W.: The Poincaré–Miranda theorem. Am. Math. Mon. 104, 545–550 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mawhin, J.: Le théorème du point fixe de Brouwer: un siècle de métamorphoses. Sci. Tech. Perspect. (2) 10, fasc. 1–2, Blanchard, Paris, pp. 175–220 (2006)Google Scholar
  21. 21.
    Mawhin, J.: Variations on some finite-dimensional fixed-point theorems. Ukr. Math. J. 65, 294–301 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Miranda, C.: Un’osservazione su un teorema di Brouwer. Boll. Un. Mat. Ital. (2) 3 (1940–41), pp. 5–7Google Scholar
  23. 23.
    Pireddu, M., Zanolin, F.: Fixed points for dissipative-repulsive systems and topological dynamics of mappings defined on \(N\)-dimensional cells. Adv. Nonlinear Stud. 5, 411–440 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pireddu, M., Zanolin, F.: Cutting surfaces and applications to periodic points and chaotic-like dynamics. Topol. Methods Nonlinear Anal. 30, 279–319 (2007)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Poincaré, H.: Sur certaines solutions particulières du problème des trois corps. C. R. Acad. Sci. Paris 97, 251–252 (1883)zbMATHGoogle Scholar
  26. 26.
    Rockafellar, R.T., Wets, R.J.: Variational Analysis. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  27. 27.
    Rybakowski, K.P.: The Homotopy Index and Partial Differential Equations. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  28. 28.
    Scorza, G.: Dragoni, Un’osservazione sulle radici di un sistema di equazioni non lineari. Rend. Sem. Mat. Univ. Padova 15, 135–138 (1946)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sovrano, E., Zanolin, F.: Dolcher fixed point theorem and its connections with recent developments on compressive/expansive maps. Rend. Istit. Mat. Univ. Trieste 46, 101–122 (2014)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Srzednicki, R.: On rest points of dynamical systems. Fundam. Math. 126, 69–81 (1985)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Vidossich, G.: A correction and an extension of Stampacchia’s work on the geometric BVP. Adv. Nonlinear Stud. 14, 813–837 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Vrahatis, M.N.: A short proof and a generalization of Miranda’s existence theorem. Proc. Am. Math. Soc. 107, 701–703 (1989)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Zgliczyn’ski, P.: On periodic points for systems of weakly coupled 1-dim maps. Nonlinear Anal. 46, 1039–1062 (2001)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zwirner, G.: Sulle radici dei sistemi di equazioni non lineari. Rend. Sem. Mat. Univ. Padova 15, 132–134 (1946)MathSciNetzbMATHGoogle Scholar

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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Dipartimento di Matematica e GeoscienzeUniversità di TriesteTriesteItaly
  2. 2.SISSA - International School for Advanced StudiesTriesteItaly

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