Skip to main content
SpringerLink
Log in

Topological degree and boundary value problems for nonlinear differential equations

  • Chapter
  • First Online:
Topological Methods for Ordinary Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1537))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V.V. Amel'kin, I.V. Gaishun and N.N. Ladis, Periodic solutions in the case of a constantly acting perturbation, Differential equations, 11 (1975), 1569–1573

    MATH  Google Scholar 

  2. A. Bahri and H. Berestycki, Existence of forced oscillations for some nonlinear differential equations, Comm. Pure Appl. Math. 37 (1984), 403–442

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Bahri and H. Berestycki, Forced vibrations of superquadratic Hamiltonian systems, Acta Math. 152 (1984), 143–197.

    Article  MathSciNet  MATH  Google Scholar 

  4. Th. Bartsch and J. Mawhin, The Leray-Schauder degree of S 1-equivariant operators associated to autonomous neutral equations in spaces of periodic functions, J. Differential Equations, 92 (1991), 90–99

    Article  MathSciNet  MATH  Google Scholar 

  5. J.W. Bebernes and K. Schmitt, Periodic boundary value problems for systems of second order differential equations, J. Differential Equations, 13 (1973), 32–47

    Article  MathSciNet  MATH  Google Scholar 

  6. I. Berstein and A. Halanay, The index of a critical point and the existence of periodic solutions to a system with small parameter, Doklady Ak. Nauk. 111 (1956), 923–925 (translation and remarks by G. B. Gustafson)

    MATH  Google Scholar 

  7. N.A. Bobylev, The construction of regular guiding functions, Soviet Math. Dokl., 9 (1968), 1353–1355

    MATH  Google Scholar 

  8. L.E.J. Brouwer, Ueber Abbildungen von Mannigfaltigkeiten, Math. Ann. 71 (1912), 97–115

    Article  MATH  Google Scholar 

  9. A. Capietto. Continuation theorems for periodic boundary value problems, Ph.D. Thesis, SISSA, Trieste, 1990

    MATH  Google Scholar 

  10. A. Capietto, J. Mawhin, and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., to appear.

    Google Scholar 

  11. A. Capietto, J. Mawhin and F. Zanolin, The coincidence degree of some functional differential operators in spaces of periodic functions and related continuation theorems, in Delay Differential Equations and Dynamical Systems (Claremont 1990) Busenberg, Martelli eds, Springer, Berlin, 1991, 76–87.

    Chapter  Google Scholar 

  12. A. Capietto, J. Mawhin and F. Zanolin, A continuation approach to superlinear periodic boundary value problems, J. Differential Equations 88 (1990), 347–395

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Capietto, J. Mawhin and F. Zanolin, Periodic solutions of some superlinear functional differential equations, in Ordinary and functional Differential Equations, Kyoto 1990, World Scientific, Singapore, 1991, 19–31

    Google Scholar 

  14. A. Capietto, J. Mawhin and F. Zanolin, A continuation approach to superlinear two point boundary value problems, in preparation

    Google Scholar 

  15. A. Capietto and F. Zanolin, An existence theorem for periodic solutions in convex sets with applications, Results in Mathematics, 14 (1988), 10–29

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Capietto and F. Zanolin, A continuation theorem for the periodic BVP in flow-invariant ENRs with applications, J. Differential Equations, 83 (1990), 244–276

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Cronin, The number of periodic solutions of nonautonomous systems, Duke Math. J., 27 (1960), 183–194

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Cronin, Lyapunov stability and periodic solutions, Bol. Soc. Mat. Mexicana, (1965), 22–27

    Google Scholar 

  19. J. Cronin, The point at infinity and periodic solutions, J. Differential Equations, 1 (1965), 156–170

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Cronin, Periodic solutions of some nonlinear differential equations, J. Differential Equations, 3 (1967), 31–46

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Cronin, Periodic solutions of nonautonomous equations, Boll. Un. Mat. Ital., (4) 6 (1972), 45–54

    MathSciNet  MATH  Google Scholar 

  22. E.N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc., 15 (1976), 321–328

    Article  MathSciNet  MATH  Google Scholar 

  23. E.N. Dancer, Symmetries, degree, homotopy indices and asymptotically homogeneous problems, Nonlinear Analysis, TMA, 6 (1982), 667–686

    Article  MathSciNet  MATH  Google Scholar 

  24. K. Deimling, Nonlinear functional analysis, Springer, Berlin, 1985

    Book  MATH  Google Scholar 

  25. L. Derwidué, Systèmes différentiels non linéaires ayant des solutions périodiques, Acad. Royale de Belgique, Bull. Cl. des Sciences (5) 49 (1963), 11–32, 82–90; (5) 50 (1964), 928–942, 1130–1142

    MATH  Google Scholar 

  26. L. Derwidué, Etude géométrique d'une équation différentielle non linéaire, Bull. Soc. Royale Sciences Liège 34 (1965), 180–187

    MATH  Google Scholar 

  27. L. Derwidué, Systèmes différentiels non linéaires ayant des solutions uniformément bornées dans le futur, Bull. Soc. Royale Sciences Liège 34 (1965), 555–572

    MATH  Google Scholar 

  28. L. Derwidué, L'équation de Forbat, in Colloque du CBRM sur les équations différentielles non linéaires, leur stabilité et leur périodicité, vander, Louvain, 1970, 7–27

    Google Scholar 

  29. T. Ding, R. Iannacci and F. Zanolin, On periodic solutions of sublinear Duffing equation, J. Math. Anal. Appl. 158 (1991), 316–332

    Article  MathSciNet  MATH  Google Scholar 

  30. W.Y. Ding, Generalizations of the Borsuk theorem, J. Math. Anal. Appl. 110 (1985) 553–567

    Article  MathSciNet  MATH  Google Scholar 

  31. C.L. Dolph, Nonlinear integral equations of the Hammerstein type, Trans. Amer. Math. Soc. 66 (1949), 289–307

    Article  MathSciNet  MATH  Google Scholar 

  32. H. Ehrmann. Ueber die Existenz der Lösungen von Randwertaufgaben bei gewöhnlichen nichtlinearen Differentialgleichungen zweiter Ordnung, Math. Ann. 134 (1957), 167–194

    Article  MathSciNet  MATH  Google Scholar 

  33. C. Fabry, J. Mawhin and M. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order differential equations, Bull. London Math. Soc. 18 (1986), 173–180

    Article  MathSciNet  MATH  Google Scholar 

  34. R. Faure, Solutions périodiques d'équations différentielles et méthode de Leray-Schauder (Cas des vibrations forcées), Ann. Inst. Fourier (Grenoble) 14 (1964), 195–204

    Article  MathSciNet  MATH  Google Scholar 

  35. R. Faure, Sur l'application d'un théorème de point fixe à l'existence de solutions périodiques, C.R. Acad. Sci. Paris, 282 (1976), A, 1295–1298

    MathSciNet  MATH  Google Scholar 

  36. R. Faure, Solutions périodiques d'équations admettant des pôles: étude par la méthode de Leray Schauder et par un théorème de point fixe, C.R. Acad. Sci. Paris, 283 (1976), A. 481–484

    MathSciNet  MATH  Google Scholar 

  37. L. Fernandes and F. Zanolin, Periodic solutions of a second order differential equation with one-sided growth restrictions on the restoring term, Arch. Math. (Basel), 51 (1988), 151–163

    Article  MathSciNet  MATH  Google Scholar 

  38. A. Fonda and P. Habets, Periodic solutions of asymptotically positively homogeneous differential equations, J. Differential Equations, 81 (1989), 68–97

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Fonda and F. Zanolin, Periodic solutions to second order differential equations of Liénard type with jumping nonlinearities, Comment. Math. Univ. Carolin., 28 (1987), 33–41

    MathSciNet  MATH  Google Scholar 

  40. N. Forbat and A. Huaux, Détermination approchée et stabilité locale de la solution périodique d'une équation différentielle non linéaire, Mém. et Public. Soc. Sciences, Artts Lettres du Hainaut 76 (1962), 3–13

    MathSciNet  MATH  Google Scholar 

  41. S. Fučik, Solvability of nonlinear equations and boundary value problems, D. Reidel Publishing Company, Dordrecht, 1980.

    MATH  Google Scholar 

  42. S. Fučik and V. Lovicar, Periodic solutions of the equation x″+g(x)=p, Casopis Pest. Mat., 100 (1975), 160–175

    MathSciNet  MATH  Google Scholar 

  43. M. Furi, M. Martelli and A. Vignoli, On the solvability of nonlinear operator equations in normed spaces, Ann. Mat. Pura Appl. (4) 124 (1980), 321–343

    Article  MathSciNet  MATH  Google Scholar 

  44. M. Furi and M.P. Pera, An elementary approach to boundary value problems at resonance, J. Nonlinear Anal. 4 (1980) 1081–1089

    Article  MathSciNet  MATH  Google Scholar 

  45. M. Furi and M.P. Pera, Co-bifurcating branches of solutions for nonlinear eigenvalue problems in Banach spaces, Annali Mat. Pura Appl. 135 (1983), 122

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Furi and M. P. Pera, A continuation principle for forced oscillations on differentiable manifolds, Pacific J. Math., 121 (1986), 321–338.

    Article  MathSciNet  MATH  Google Scholar 

  47. M. Furi and M.P. Pera, The forced spherical pendulum does have forced oscillations, in Delay Differential Equations and Dynamical Systems (Claremont 1990), Busenberg and Martelli eds, Springer, Berlin, 1991, 176–182

    Chapter  Google Scholar 

  48. R.E. Gaines and J. Mawhin, Coincidence degree and nonlinear differential equations, Lecture Notes in Math., 586, Springer-Verlag, Berlin, 1977

    Book  MATH  Google Scholar 

  49. K. Geba, A. Granas, T. Kaczynski and W. Krawcewicz, Homotopie et équations non linéaires dans les espaces de Banach, C.R. Acad. Sci. Paris 300, ser. I (1985), 303–306

    MathSciNet  MATH  Google Scholar 

  50. R.E. Gomory, Critical points at infinity and forced oscillations, in Contributions to the theory of nonlinear oscillations, 3, Ann. of Math. Studies, 36, Princeton Univ. press, N. J. 1956, pp. 85–126

    Google Scholar 

  51. W.B. Gordon, Conservative dynamical systems involving strong forces, Trans. Amer, Math. Soc. 204 (1975), 113–135

    Article  MathSciNet  MATH  Google Scholar 

  52. W.B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math. 99 (1977), 961–971

    Article  MathSciNet  MATH  Google Scholar 

  53. A. Granas, The Leray-Schauder index and the fixed point theory for arbitrary ANRs, Bull. Soc. Math. France, 100 (1972), 209–228

    MathSciNet  MATH  Google Scholar 

  54. P. Habets and L. Sanchez, Periodic solutions of dissipative dynamical systems with singular potentials, J. Differential and Integral Equations, 3 (1990), 1139–1149

    MathSciNet  MATH  Google Scholar 

  55. P. Habets and L. Sanchez, Periodic solutions of some Liénard equations with singularities, Proc. Amer. Math. Soc. 109 (1990), 1035–1044

    MathSciNet  MATH  Google Scholar 

  56. A. Halanay, În legatura cu metoda parametrolui mic (Relativement à la méthode du petit paramètre), Acad. R. P. R., Bul. St., Sect. Mat. fiz., 6 (1954), 483–488

    MathSciNet  Google Scholar 

  57. A. Halanay, Solutions périodiques des systèmes non-linéaires à petit paramètre, Rend. Accad. Naz. Lincei (Cl. Sci. Fis. Mat. Natur.), 22 (Ser. 8) (1957), 30–32

    MathSciNet  MATH  Google Scholar 

  58. A. Halanay, Differential Equations, stability, oscillations, time lags, Academic Press, New-York, London, 1966

    MATH  Google Scholar 

  59. J.K. Hale and A.S. Somolinos, Competition for fluctuating nutrient, J. Math. Biology, 18 (1983), 255–280

    Article  MathSciNet  MATH  Google Scholar 

  60. A. Huaux, Sur l'existence d'une solution périodique de l'équation différentielle non linéaire x″+0.2x′+x/(1−x)=(0,5) coswt, Bull. Cl. Sciences Acad. R. Belgique (5) 48 (1962), 494–504

    MathSciNet  MATH  Google Scholar 

  61. R. Iannacci and M. Martelli, Branches of solutions of nonlinear operator equations in the atypical bifurcation case, preprint.

    Google Scholar 

  62. Yu.S. Kolesov, Study of stability of solutions of second-order parabolic equations in the critical case, Math. USSR-Izvestija 3 (1969), 1277–1291

    Article  MATH  Google Scholar 

  63. M.A. Krasnosel'skii, The operator of translation along the trajectories of differential equations, Amer. Math. Soc., Providence, R.I., 1968

    Google Scholar 

  64. M.A. Krasnosel'skii and P.P. Zabreiko, Geometrical methods of nonlinear Analysis, Springer-Verlag, Berlin, 1984

    Book  Google Scholar 

  65. L. Kronecker. Ueber Systeme von Funktionen mehrerer Variabeln, Monatsber. Berlin Akad. (1869), 159–193

    Google Scholar 

  66. B. Laloux and J. Mawhin, Coincidence index and multiplicity, Trans. Amer. Math. Soc. 217 (1976), 143–162

    Article  MathSciNet  MATH  Google Scholar 

  67. A. Lando, Periodic solutions of nonlinear systems with forcing term, J. Differential Equations, 3 (1971), 262–279

    Article  MathSciNet  MATH  Google Scholar 

  68. A. Lando, Forced oscillations of two-dimensional nonlinear systems, Applicable Analysis, 28 (1988), 285–295

    Article  MathSciNet  MATH  Google Scholar 

  69. A. Lasota, Une généralisation du premier théorème de Fredholm et ses applications à la théorie des équations différentielles ordinaires, Ann. Polon. Math., 18 (1966), 65–77

    MathSciNet  MATH  Google Scholar 

  70. A. Lasota and Z. Opial, Sur les solutions périodiques des équations différentielles ordinaires, Ann. Polon. Math., 16 (1964), 69–94

    MathSciNet  MATH  Google Scholar 

  71. A.C. Lazer and P.J. McKenna, A semi-Fredholm principle for periodically forced systems with homogeneous nonlinearities, Proc. Amer. Math. Soc., 106 (1989), 119–125

    Article  MathSciNet  MATH  Google Scholar 

  72. A.C. Lazer and P.J. McKenna, On the existence of stable periodic solutions of differential equations of Duffing type, Proc. Amer. Math. Soc. 110 (1990), 125–133

    Article  MathSciNet  MATH  Google Scholar 

  73. A.C. Lazer and P.J. McKenna, Nonresonance conditions for the existence, uniqueness and stability of periodic solutions of differential equations with a symmetric nonlinearity, Differential and Integral Equations 4 (1991), 719–730

    MathSciNet  MATH  Google Scholar 

  74. A.C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc. 99 (1987), 109–114

    Article  MathSciNet  MATH  Google Scholar 

  75. J. Leray, Les problèmes non linéaires, L'enseignement math. 35 (1936), 139–151

    MATH  Google Scholar 

  76. J. Leray and J. Schauder, Topologie et equations fonctionelles, Ann. Sci. Ecole Norm. Sup. (3) 51 (1934), 45–78

    MathSciNet  MATH  Google Scholar 

  77. N.G. Lloyd, Degree theory, Cambridge Univ. Press, Cambridge, 1978

    MATH  Google Scholar 

  78. Y. Long, Multiple solutions of perturbed superquadratic second order Hamiltonian systems, Trans. Amer. Math. Soc. 311 (1989), 749–780

    Article  MathSciNet  MATH  Google Scholar 

  79. W.S. Loud, Periodic solutions of x″+cx′+g(x)=ωf(t), Mem. Amer. Math. Soc., 31 (1959)

    Google Scholar 

  80. J.L. Massera. The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950), 457–475

    Article  MathSciNet  MATH  Google Scholar 

  81. J. Mawhin, Equations intégrales et solutions périodiques des systèmes différentiels non linéaires. Acad. Roy. Belg. Bull. Cl. Sci., 55 (1969), 934–947

    MathSciNet  MATH  Google Scholar 

  82. J. Mawhin, Existence of periodic solutions for higher order differential systems that are not of class D, J. Differential Equations, 8 (1970), 523–530

    Article  MathSciNet  MATH  Google Scholar 

  83. J. Mawhin, Equations fonctionnelles non linéaires et solutions périodiques, in Equadiff 70, Centre de Recherches Physiques, Marseille, 1970

    Google Scholar 

  84. J. Mawhin, Periodic solutions of nonlinear functional differential equations, J. Differential Equations, 10 (1971), 240–261

    Article  MathSciNet  MATH  Google Scholar 

  85. 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

    Article  MathSciNet  MATH  Google Scholar 

  86. J. Mawhin, Nonlinear perturbations of Fredholm mappings in normed spaces and applications to differential equations, Trabalho de Mat. 61, Univ. de Brasilia, 1974

    Google Scholar 

  87. J. Mawhin, Periodic solutions of some perturbed differential systems, Boll. Un. Mat. Ital., 11 (4) (1975), 299–305

    MathSciNet  MATH  Google Scholar 

  88. J. Mawhin, Periodic solutions, Workshop on nonlinear boundary value problems for ordinary differential equations and applications, Trieste, 1977, ICTP SMR/42/1

    Google Scholar 

  89. J. Mawhin, Topological degree methods in nonlinear boundary value problems, CBMS 40, Amer. Math. Soc., Providence, R.I., 1979

    Book  MATH  Google Scholar 

  90. J. Mawhin, Compacité, monotonie et convexité dans l'étude des problèmes aux limites semi linéaires, Sémin. Anal. Moderne, 19, Univ. Sherbrooke, 1981

    Google Scholar 

  91. J. Mawhin, Point fixes, point critiques et problems aux limites, Séminaire de Mathématiques Supérieures, vol. 92, Les Presses de l'Université de Montréal, 1985

    Google Scholar 

  92. J. Mawhin, Qualitative behavior of the solutions of periodic first order scalar differential equations with strictly convex coercive nonlinearity, in Dynamics of infinite dimensional systems, Chow and Hale eds., Springer, Berlin, 1987, 151–159

    Chapter  Google Scholar 

  93. J. Mawhin, A simple proof of a semi-Fredholm principle for periodically forced systems with homogeneous nonlinearities, Arch. Math. (Brno), 25 (1989), 235–238

    MathSciNet  MATH  Google Scholar 

  94. J. Mawhin, Bifurcation from infinity and nonlinear boundary value problems, in Ordinary and Partial Differential equations, (Dundee 1989), Pitman, 1990, 119–129

    Google Scholar 

  95. J. Mawhin and C. Munoz, Application du degré topologique à l'estimation du nombre des solutions périodiques d'équations différentielles, Annali Mat. Pura Appl. (4) 96 (1973), 1–19

    Article  MathSciNet  MATH  Google Scholar 

  96. J. Mawhin and C. Rybakowski, Continuation theorems for semi-linear equations in Banach spaces: a survey, in Nonlinear Analysis, World Scientific, Singapore, 1987, 367–405

    Google Scholar 

  97. J. Mawhin and K. Schmitt, Landesman-Lazer type problems at an eigenvalue of odd multiplicity. Results in Math. 14 (1988), 138–146

    Article  MathSciNet  MATH  Google Scholar 

  98. J. Mawhin and K. Schmitt, Nonlinear eigenvalue problems with the parameter near resonance. Ann. Polon. Math. 60 (1990), 241–248

    MathSciNet  MATH  Google Scholar 

  99. J. Mawhin and J.R. Ward, Periodic solutions of some forced Liénard differential equations at resonance, Arch. Math. (Basel), 41 (1983), 337–351

    Article  MathSciNet  MATH  Google Scholar 

  100. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Appl. Math. Sci., vol. 74, Springer, New York, 1989

    MATH  Google Scholar 

  101. G.R. Morris, An infinite class of periodic solutions of x″+2x 3=p(t), Proc. Cambridge Phil. Soc. 61 (1965), 157–164

    Article  MathSciNet  MATH  Google Scholar 

  102. E. Muhamadiev, Construction of a correct guiding function for a system of differential equations, Soviet Math. Dokl., 11 (1970), 202–205

    Google Scholar 

  103. E. Muhamadiev, On the theory of periodic solutions of systems of ordinary differential equations, Soviet Math. Dokl., 11 (1970), 1236–1239

    MathSciNet  Google Scholar 

  104. W. Müller, Ueber die Beschränkheit der Lösungen der Cartwright-Littlewoddschen Gleichung. Math. Nachr. 29 (1965), 25–40

    Article  MathSciNet  Google Scholar 

  105. W. Müller, Qualitative Untersuchungen der Lösungen nichtlinearer Differentialgleichungen zweiter Ordnung nach der direkten Methode von Liapounov, Abhandlungen Deutsche Akad. Wiss., Kl. Math. Phys. Techn., Heft 4, Berlin, 1965

    Google Scholar 

  106. L. Nirenberg, Comments on nonlinear problems, Le Matematiche (Catania) 36 (1981) 109–119

    MathSciNet  MATH  Google Scholar 

  107. R. D. Nussbaum, The fixed point index and some applications, Séminaire de Mathématiques Supérieures, vol. 94, Les Presses de l'Université de Montréal, 1987

    Google Scholar 

  108. P. Omari, Gab. Villari and F. Zanolin, Periodic solutions of the Liénard equation with one-sided growth restrictions, J. Differential Equations, 67 (1987), 278–293

    Article  MathSciNet  MATH  Google Scholar 

  109. P. Omari and F. Zanolin, On forced nonlinear oscillations in n-th order differential systems with geometric conditions, Nonlinear Analysis, TMA, 8 (1984), 723–784

    Article  MathSciNet  MATH  Google Scholar 

  110. Z. Opial. Sur les solutions périodiques de l'équation différentielle x″+g(x)=p(t), Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys., 8 (1960), 151–156

    MathSciNet  MATH  Google Scholar 

  111. R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Ital. (7) 3-B (1989), 533–546

    MathSciNet  MATH  Google Scholar 

  112. R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type, Differential and Integral Equations, 3 (1990), 275–284

    MathSciNet  MATH  Google Scholar 

  113. R. Ortega. Topological degree and stability of periodic solutions for certain differential equations, J. London Math. Soc. (2) 42 (1990), 505–516

    Article  MathSciNet  MATH  Google Scholar 

  114. R. Ortega. A criterion for asymptotic stability based on topological degree, to appear

    Google Scholar 

  115. R. Ortega. The first interval of stability of a periodic equation of Duffing type, Proc. Amer. Math. Soc., to appear

    Google Scholar 

  116. J. Pejsachowicz and A. Vignoli, On the topological coincidence degree for perturbations of Fredholm operators, Boll. Un. Mat. Ital. (5) 17-B (1980), 1457–1466

    MathSciNet  MATH  Google Scholar 

  117. V.A. Pliss, Nonlocal problems of the theory of oscillations, Academic Press, New York, 1966

    MATH  Google Scholar 

  118. P.J. Rabier, Topological degree and the theorem of Borsuk for general covariant mappings with applications, Nonlinear Analysis 16 (1991) 399–420

    Article  MathSciNet  MATH  Google Scholar 

  119. P. Rabinowitz. On bifurcation from infinity, J. Differential Equations 14 (1973), 462–475

    Article  MathSciNet  MATH  Google Scholar 

  120. P. Rabinowitz. Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc. 272 (1982), 753–769

    Article  MathSciNet  MATH  Google Scholar 

  121. R. Reissig, Periodic solutions of a nonlinear n-th order vector differential equation, Ann. Mat. Pura Appl., (4) 87 (1970), 111–123

    Article  MathSciNet  MATH  Google Scholar 

  122. R. Reissig, G. Sansone and R. Conti, Qualitative Theorie nichtlinearer Differentialgleichungen, Cremonese, Roma, 1963

    MATH  Google Scholar 

  123. R. Reissig, G. Sansone and R. Conti, Non-linear Differential Equations of Higher Order, Noordhoff, Leyden, 1974

    MATH  Google Scholar 

  124. N. Rouche and J. Mawhin, Ordinary differential equations, Pitman, London, 1980

    MATH  Google Scholar 

  125. J. Schauder, Der Fixpunktsatz in Funktionalraümen Studia Math. 2 (1930), 171–180

    Google Scholar 

  126. K. Schmitt, Periodic solutions of small period of systems of n-th order differential equations, Proc. Amer. Math. Soc., 36 (1972), 459–463

    MathSciNet  Google Scholar 

  127. R.A. Smith, Massera's convergence theorem for periodic nonlinear differential equations, J. Math. Anal. Appl. 120 (1986), 679–708

    Article  MathSciNet  MATH  Google Scholar 

  128. S. Solimini, On forced dynamical systems with a singularity of repulsive type, J. Nonlinear Analysis 14 (1990), 489–500

    Article  MathSciNet  MATH  Google Scholar 

  129. R. Srzednicki, On rest points of dynamical systems, Fund. Math., 126 (1985), 69–81

    MathSciNet  MATH  Google Scholar 

  130. R. Srzednicki, Periodic and constant solutions via topological principle of Wažewski, Acta Math. Univ. Iag., 26 (1987), 183–190

    MathSciNet  MATH  Google Scholar 

  131. H. Steinlein, Borsuk's antipodal theorem and its generalizations and applications: a survey, in Méth. Topologiques en Analyse Non-Linéaire, Granas ed., Sémin. Math. Sup. n 95, Montréal, 1985, 166–235

    Google Scholar 

  132. F. Stoppelli, Su un'equazione differenziale della meccanica dei fili, Rend. Accad. Sci. Fis. Mat. Napoli, 19 (1952), 109–114

    MathSciNet  MATH  Google Scholar 

  133. M. Struwe, Multiple solutions of anticoercive boundary value problems for a class of ordinary differential equations of the second order, J. Differential Equations 37 (1980), 285–295

    Article  MathSciNet  MATH  Google Scholar 

  134. M. Urabe, Nonlinear autonomous oscillations. Analytical theory, Academic Press, New York, 1967

    MATH  Google Scholar 

  135. G. Villari, Soluzioni periodiche di una classe di equazioni differenziali, Ann. Mat. Pura Appl. (4) 73 (1966), 103–110

    Article  MathSciNet  MATH  Google Scholar 

  136. P. Volkmann, Zur Definition des Koinzidenzgrades, preprint, 1981

    Google Scholar 

  137. P. Volkmann, Démonstration d'un théorème de coincidence par la méthode de Granas, Bull. Soc. Math. Belgique, B 36 (1984), 235–242

    MATH  Google Scholar 

  138. Z.Q. Wang: Symmetries and the calculations of degree, Chinese Ann. of Math. 108 (1989) 520–536

    MathSciNet  MATH  Google Scholar 

  139. J.R. Ward, Asymptotic conditions for periodic solutions of ordinary differential equations, Proc. Amer. Math. Soc. 81 (1981), 415–420

    Article  MathSciNet  MATH  Google Scholar 

  140. J.R. Ward, Conley index and non-autonomous ordinary differential equations, Results in Mathematics, 14 (1988), 191–209

    Article  MathSciNet  MATH  Google Scholar 

  141. E. Zeidler, Nonlinear functional analysis and its applications, vol. I, Springer, New York, 1986

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut mathématique, Université de Louvain, B-1348, Louvain-la-Neuve, Belgium

    J. Mawhin

Authors
  1. J. Mawhin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Massimo Furi Pietro Zecca

Additional information

Dedicated to Jean Leray on the occasion of his 85th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag

About this chapter

Cite this chapter

Mawhin, J. (1993). Topological degree and boundary value problems for nonlinear differential equations. In: Furi, M., Zecca, P. (eds) Topological Methods for Ordinary Differential Equations. Lecture Notes in Mathematics, vol 1537. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085076

Download citation

  • DOI: https://doi.org/10.1007/BFb0085076

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56461-4

  • Online ISBN: 978-3-540-47563-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation