Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Coincidence index and multiplicity

Authors: B. Laloux and J. Mawhin
Journal: Trans. Amer. Math. Soc. 217 (1976), 143-162
MSC: Primary 47H10
MathSciNet review: 0423138
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is devoted to the extension, in the frame of coincidence degree theory in normed spaces, of the concept of Leray-Schauder index of an isolated fixed point. The generalization includes basic properties of the coincidence index, Krasnosel'skiĭ type theorems for the case of noninvertible linear part and a Leray-Schauder's type formula relating the index and spectral theory in the linear case. This last problem needs the introduction of the concept of characteristic value for some couples of linear mappings and of its multiplicity.

References [Enhancements On Off] (What's this?)

  • [1] Robert A. Bonic, Linear functional analysis, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. MR 0257686
  • [2] Felix E. Browder, Non-linear functional equations in locally convex spaces, Duke Math. J. 24 (1957), 579–589. MR 0092940
  • [3] J. Dieudonné, Eléments d'analyse. Tome I: Fondements de l'analyse moderne, 2nd rev. ed., Cahiers Scientifiques, fasc. 28, Gauthiers-Villars, Paris, 1968. MR 38 #4246.
  • [4] Jack K. Hale and J. Mawhin, Coincidence degree and periodic solutions of neutral equations, J. Differential Equations 15 (1974), 295–307. MR 0336004,
  • [5] M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR 0159197
  • [6] B. Laloux, Indice de coincidence et bifurcations, Équations différentielles et fonctionnelles non linéaires (Actes Conf. Internat. “Equa-Diff 73”, Brussels/Louvain-la-Neuve, 1973) Hermann, Paris, 1973, pp. 109–121 (French). MR 0470782
  • [7] Jean Leray and Jules Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. (3) 51 (1934), 45–78 (French). MR 1509338
  • [8] J. Leray, Topologie des espaces abstraits de M. Banach, C. R. Acad. Sci. Paris 200 (1935), 1082-1084.
  • [9] J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations 12 (1972), 610–636. MR 0328703,
  • [10] J. Mawhin, A further invariance property of coincidence degree in convex spaces, Ann. Soc. Sci. Bruxelles Sér. I 87 (1973), 51–57. MR 0320839
  • [11] J. Mawhin, The solvability of some operator equations with a quasibounded nonlinearity in normed spaces, J. Math. Anal. Appl. 45 (1974), 455–467. MR 0333865,
  • [12] J. Mawhin, Periodic solutions of some vector retarded functional differential equations, J. Math. Anal. Appl. 45 (1974), 588–603. MR 0333400,
  • [13] -, Problémes aux limites du type de Neumann pour certaines équations différentielles ou aux dérivées partielles non linéaires, Equations Différentielles et Fonctionnelles non Linéaires, P. Janssens, J. Mawhin and N. Rouche, Editors, Hermann, Paris, 1973, pp. 123-134.
  • [14] V. B. Melamed, Calculation of the index of a fixed point of a completely continuous vector field, Dokl. Akad. Nauk SSSR 126 (1959), 501–504 (Russian). MR 0121681
  • [15] V. B. Melamed, Computing the rotation of a completely continuous vector field in the critical case, Sibirsk. Mat. Ž. 2 (1961), 414–427 (Russian). MR 0132382
  • [16] A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 53, Cambridge University Press, New York, 1964. MR 0162118
  • [17] N. Rouche and J. Mawhin, Équations différentielles ordinaires, Masson et Cie, Éditeurs, Paris, 1973 (French). Tome II: Stabilité et solutions périodiques. MR 0481182
  • [18] J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher; Notes on Mathematics and its Applications. MR 0433481
  • [19] P. P. Zabreĭko and M. A. Krasnosel′skiĭ, Calculation of the index of a fixed point of a vector field, Sibirsk. Mat. Ž. 5 (1964), 509–531 (Russian). MR 0166620
  • [20] Felix E. Browder, Topology and non-linear functional equations, Studia Math. 31 (1968), 189–204. MR 0238134,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47H10

Retrieve articles in all journals with MSC: 47H10

Additional Information

Keywords: Coincidence degree, coincidence index, characteristic value, multiplicity, Leray-Schauder theory, bifurcation theory
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society