Upper and Lower Solutions in the Theory of Ode Boundary Value Problems: Classical and Recent Results

  • C. De Coster
  • P. Habets
Part of the International Centre for Mechanical Sciences book series (CISM, volume 371)


The method of upper and lower solutions for ordinary differential equation was introduced in 1931 by G. Scorza Dragoni for a Dirichlet problem. Since then a large number of contributions enriched the theory. Among others, one has to point out the pioneer work of M. Nagumo who associated his name with derivative dependent right hand side.


Periodic Solution Dirichlet Problem Lower Solution Periodic Problem Multiplicity Result 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Adje, Sur et sous solutions dans les équations différentielles discontinues avec conditions aux limites non linéaires, Thèse de Doctorat, U.C.L., Louvainla-Neuve (1987).Google Scholar
  2. [2]
    A. Adje, Sur et sous-solutions généralisées et problèmes aux limites du second ordre, Bull. Soc. Math. Belg. 42 sen. B (1990), 347–368.Google Scholar
  3. [3]
    K. Ako, On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan 13 (1961), 45–62.CrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    K. Ako, Subfunctions for ordinary differential equations II, Funkcialaj Ekvacioj 10 (1967), 145–162.MATHMathSciNetGoogle Scholar
  5. [5]
    K. Ako, Subfunctions for ordinary differential equations VI, J. Fac. Sci. Univ. Tokyo 16 (1969), 149–156.MATHMathSciNetGoogle Scholar
  6. [6]
    H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971), 125–146.CrossRefMathSciNetGoogle Scholar
  7. [7]
    H. Amann, Existence of multiple solutions for nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1972), 925–935.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    H. Amann, On the number of solutions of nonlinear equations in ordered Ba-nach spaces, J. Funct. Anal. 11 (1972), 346–384.CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review 18 (1976), 620–709.CrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    H. Amann, A. Ambrosetti and G. Mancini, Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities, Math. Z. 158 (1978), 179–194.CrossRefMATHMathSciNetGoogle Scholar
  11. [11]
    H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Royal Soc. Edinburgh 84A (1979), 145–151.CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pure Appl. 93 (1972), 231–247.CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    P.B. Bailey, L.F. Shampine and P.E. Waltman, Nonlinear two point boundary value problems, Academic Press, New York (1968).MATHGoogle Scholar
  14. [14]
    J.V. Baxley, A singular nonlinear boundary value problem: membrane response of a spherical cap, SIAM J. Appl. Math. 48 (1988), 497–505.CrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    A.K. Ben-Naoum and C. De Coster, On the existence and multiplicity of positive solutions of the p-Laplacian separated boundary value problem, Recherches de mathématique 46, preprint UCL 1995.Google Scholar
  16. [16]
    H. Berestycki, Le nombre de solutions de certains problèmes semilinéaires elliptiques, J. Funct. Anal. 40 (1981), 1–29.CrossRefMATHMathSciNetGoogle Scholar
  17. [17]
    M.S. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J. 24 (1975), 837–846.CrossRefMATHMathSciNetGoogle Scholar
  18. [18]
    L.E. Bobisud, D. O’Regan and W.D. Royalty, Solvability of some nonlinear boundary value problem, Nonlinear Anal. T.M.A. 12 (1988), 855–869.CrossRefMATHMathSciNetGoogle Scholar
  19. [19]
    Bongsoo Ko, The third solution of semilinear elliptic boundary value problems and applications to singular perturbation problems, J. Diff. Equ. 101 (1993), 1–14.CrossRefMATHGoogle Scholar
  20. [20]
    H. Brezis, Analyse fonctionnelle: Théorie et applications, Masson, Paris (1983).MATHGoogle Scholar
  21. [21]
    H. Brezis and L. Nirenberg, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris 317 (1993), 465–472.MATHMathSciNetGoogle Scholar
  22. [22]
    R.F. Brown, A topological introduction to nonlinear analysis, Birkhäuser, Boston 1993.CrossRefMATHGoogle Scholar
  23. [23]
    K.J. Brown and H. Budin, Multiple positive solutions for a class of nonlinear boundary value problem, J. Math. Anal. Appl. 60 (1977), 329–338.CrossRefMATHMathSciNetGoogle Scholar
  24. [24]
    A. Cabada and L. Sanchez, A positive operator approach to the Neumann problem for a second order ordinary differential equation, preprint.Google Scholar
  25. [25]
    K.C. Chang, A variant mountain pass lemma, Scientia Sinica (Series A) 26 (1983), 1241–1255.MATHMathSciNetGoogle Scholar
  26. [26]
    K.C. Chang, Variational methods and sub-and super-solutions, Scientia Sinica (Series A) 26 (1983), 1256–1265.MATHMathSciNetGoogle Scholar
  27. [27]
    R. Chiappinelli, J. Mawhin and R. Nugari, Generalized Ambrosetti-Prodi conditions for nonlinear two-points boundary value problems, J. Diff. Equ. 69 (1987), 422–434.CrossRefMATHMathSciNetGoogle Scholar
  28. [28]
    P. Clement and L.A. Peletier, An anti-maximum principle for second order elliptic operators, J. Diff. Equ. 34 (1979), 218–229.CrossRefMATHMathSciNetGoogle Scholar
  29. [29]
    F.J.S.A. Corrêa, On pairs of positive solutions for a class of sub-superlinear elliptic problems, Diff. Int. Equ. 5 (1992), 387–392.MATHGoogle Scholar
  30. [30]
    D.G. Costa and J.V.A. Goncalves, On the existence of positive solutions for a class of non-selfadjoint elliptic boundary value problems, Applicable Analysis 31 (1989), 309–320.CrossRefMATHMathSciNetGoogle Scholar
  31. [31]
    R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. II, Inter-science, New York, 1962.MATHGoogle Scholar
  32. [32]
    E.N. Dancer, On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. Pures Appl. 57 (1978), 351–366.MATHMathSciNetGoogle Scholar
  33. [33]
    C. De Coster, La méthode des sur et sous solutions dans l’étude de problèmes aux limites, Thèse de Doctorat, U.C.L., Louvain-la-Neuve (1994).Google Scholar
  34. [34]
    C. De Coster, Pairs of positive solutions for the one-dimensional p-laplacian, Nonlinear Anal. T.M.A. 23 (1994), 669–681.CrossRefMATHGoogle Scholar
  35. [35]
    C. De Coster, M.R. Grossinho and P. Habets, On pairs of positive solutions for a singular boundary value problem,to appear in Applicable Analysis.Google Scholar
  36. [36]
    C. De Coster and P. Habets, A two parameters Ambrosetti-Prodi problem, to appear in Portugaliæ Mathematica.Google Scholar
  37. [37]
    D.G. de Figueiredo, Lectures on boundary value problems of the AmbrosettiProdi type, Atas 12e Semin. Brasileiro Analise, Sao Paulo, (1980), 230–291.Google Scholar
  38. [38]
    D.G. de Figueiredo, Positive solutions of semilinear elliptic problems, Lecture Notes in Math., vol. 957, Springer-Verlag, Berlin, 1982, 34–87.Google Scholar
  39. [39]
    D.G. de Figueiredo, Positive solutions for some classes of semilinear elliptic problems, Proceedings of Symposia in Pure Mathematics 45 (1986), 371–379.CrossRefGoogle Scholar
  40. [40]
    D.G. de Figueiredo and P.L. Lions, On pairs of positive solutions for a class of semilinear elliptic problems, Indiana University Math. J. 34 (1985), 591–606.CrossRefMATHGoogle Scholar
  41. [41]
    D.G. de Figueiredo and W.N. Ni, Perturbations of second order linear elliptic problems by nonlinearities without Landesman-Lazer condition, Nonlinear Anal., T.M.A. 5 (1979), 629–634.CrossRefGoogle Scholar
  42. [42]
    D.G. de Figueiredo and S. Solimini, A variational approach to superlinear elliptic problems, Comm. P.D.E. 9 (1984), 699–717.CrossRefMATHGoogle Scholar
  43. [43]
    J. Deuel and P. Hess, A criterion for the existence of solutions of non-linear elliptic boundary value problems,Proc. Royal Soc. Edinburgh 74A (1974/75), 49–54.Google Scholar
  44. [44]
    J.I. Diaz, Nonlinear differential equations and free boundaries. Vol 1. Elliptic equations, Pitman Research Notes in Math. 106 (1985).Google Scholar
  45. [45]
    H. Epheser, Ober die existenz der lösungen von randwertaufgaben mit gewöhnlichen, nichtlinearen differentialgleichungen zweiter ordnung, Math. Zeitschr. 61 (1955), 435–454.CrossRefMATHMathSciNetGoogle Scholar
  46. [46]
    C. Fabry, personal communication.Google Scholar
  47. [47]
    C. Fabry and P. Habets, The Picard boundary value problem for nonlinear second order vector differential equations, J. Diff. Equ. 42 (1981), 186–198.CrossRefMATHMathSciNetGoogle Scholar
  48. [48]
    C. Fabry and P. Habets, Upper and lower solutions for second-order boundary value problems with nonlinear boundary conditions, Nonlinear Anal. T.M.A. 10 (1986), 985–1007.CrossRefMATHMathSciNetGoogle Scholar
  49. [49]
    A. Fonda, On the existence of periodic solutions for scalar second order differential equations when only the asymptotic behaviour of the potential is known, Proc. A.M.S 119 (1993), 439–445.CrossRefMATHMathSciNetGoogle Scholar
  50. [50]
    A. Fonda and J. Mawhin, Quadratic forms, weighted eigenfunctions and boundary value problems for nonlinear second order ordinary differential equations, Proc. R. Soc. Edinburgh 112A (1989), 145–153.CrossRefMATHMathSciNetGoogle Scholar
  51. [51]
    G. Fournier and J. Mawhin, On periodic solutions of forced pendulum-like equations, J. Diff. Equ. 60 (1985), 381–395.CrossRefMATHMathSciNetGoogle Scholar
  52. [52]
    S. Fucík, Solvability of nonlinear equations and boundary value problems, Reidel, Dordrecht, 1980.MATHGoogle Scholar
  53. [53]
    J.A. Gatica, V. Oliker and P. Waltman, Singular nonlinear boundary value problems for second order ordinary differential equations, J. Diff. Equ. 79 (1989), 62–78.CrossRefMATHMathSciNetGoogle Scholar
  54. [54]
    J.P. Gossez and P. Omani, Periodic solutions of a second order ordinary differential equation: a necessary and sufficient condition for nonresonance, J. Diff. Equ. 94 (1991), 67–82.CrossRefMATHGoogle Scholar
  55. [55]
    J.P. Gossez and P. Omani, A necessary and sufficient condition of nonresonance for a semilinear Neumann problem, Proc. A.M.S. 114 (1992), 433–442.CrossRefMATHGoogle Scholar
  56. [56]
    J.P. Gossez and P. Omani, Non-ordered lower and upper solutions in semi-linear elliptic problems, Comm. P. D. E. 19 (1994), 1163–1184.CrossRefMATHGoogle Scholar
  57. [57]
    J.P. Gossez and P. Omani, On a semilinear elliptic Neumann problem with asymmetric nonlinearities, Trans. A.M.S. 347 (1995), 2553–2562.Google Scholar
  58. [58]
    V.V. Gudkov and A.J. Lepin, On necessary and sufficient conditions for the solvability of certain boundary-value problems for a second-order ordinary differential equation, Dokl. Akad. Nauk SSSR 210 (1973), 800–803.MathSciNetGoogle Scholar
  59. [59]
    Z. Guo, Solvability of some singular nonlinear boundary value problems and existence of positive radial solutions of some nonlinear elliptic problems, Nonlinear Anal. T.M.A. 16 (1991), 781–790.CrossRefMATHGoogle Scholar
  60. [60]
    P. Habets and M. Laloy, Perturbations singulières de problèmes aux limites: Les sur-et sous-solutions, Séminaire de Mathématique Appliquée et Mécanique 76, U.C.L. (1974).Google Scholar
  61. [61]
    P. Habets and P. Omani, Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order, to appear in Topological Methods in Nonlinear Analysis.Google Scholar
  62. [62]
    P. Habets and L. Sanchez, Periodic,solutions of some Liénard equations with singularities, Proc. A.M.S. 109 (1990), 1035–1044.MATHMathSciNetGoogle Scholar
  63. [63]
    P. Habets and F. Zanolin, Upper and lower solutions for a generalized Emden-Fowler equation, J. Math. Anal. Appl. 181 (1994), 684–700.CrossRefMATHMathSciNetGoogle Scholar
  64. [64]
    P. Habets and F. Zanolin, Positive solutions for a class of singular boundary value problem, Boll. U.M.I. (7) 9-A (1995), 273–286.Google Scholar
  65. [65]
    G. Harris, A nonlinear Dirichlet problem with nonhomogeneous boundary data, Applicable Anal. 33 (1989), 169–182.CrossRefMATHGoogle Scholar
  66. [66]
    P. Hess, An antimaximum principle for linear elliptic equations with an indefinite weight function, J. Diff. Equ. 41 (1981), 369–374.CrossRefMathSciNetGoogle Scholar
  67. [67]
    S.I. Hudjaev, Boundary problems for certain quasi-linear elliptic equations, Soviet Math. Dokl. 5 (1964), 188–192.Google Scholar
  68. [68]
    R. Iannacci, M.N. Nkashama and J.R. Ward, Jr., Nonlinear second order elliptic partial differential equations at resonance, Trans. A.M.S. 311 (1989), 711–726.CrossRefMATHMathSciNetGoogle Scholar
  69. [69]
    F. Inkmann, Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions, Indiana Univ. Math. J. 31 (1982), 213–221.CrossRefMATHMathSciNetGoogle Scholar
  70. [70]
    L.K. Jackson, Subfunctions and second-order ordinary differential inequalities, Advances in Math. 2 (1967), 307–363.CrossRefGoogle Scholar
  71. [71]
    J. Janus and J. Myjak, A generalized Emden-Fowler equation with a negative exponent, Nonlinear Analysis T.M.A. 23 (1994), 953–970.CrossRefMATHMathSciNetGoogle Scholar
  72. [72]
    R. Kannan and K. Nagle, Forced oscillations with rapidly vanishing nonlinearites, Proc. A.M.S. 111 (1991), 385–393.CrossRefMATHMathSciNetGoogle Scholar
  73. [73]
    R. Kannan and R. Ortega, An asymptotic result in forced oscillations of pendulum-type equations, Applicable Analysis 22 (1986), 45–53.CrossRefMATHMathSciNetGoogle Scholar
  74. [74]
    J. Kazdan and F. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567–597.CrossRefMATHMathSciNetGoogle Scholar
  75. [75]
    H.B. Keller and D.S. Cohen, Some positone problems suggested by non-linear heat generation, J. Math. Mech. 16 (1967), 1361–1376.MATHMathSciNetGoogle Scholar
  76. [76]
    H.W. Knobloch, Eine neue methode zur approximation periodischer lösungen nicht-linearer differentialgleichungen zweiter ordnung, Math. Zeitschr. 82 (1963), 177–197.CrossRefMATHMathSciNetGoogle Scholar
  77. [77]
    Y.S. Kolesov, Periodic solutions of quasilinear parabolic equations of second order, Trans. Moscow Math. Soc. 21 (1970), 114–146.MathSciNetGoogle Scholar
  78. [78]
    E.M. Landesman and A.C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609–623.MATHMathSciNetGoogle Scholar
  79. [79]
    A.C. Lazer and P.J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl. 84 (1981), 282–294.CrossRefMATHMathSciNetGoogle Scholar
  80. [80]
    A.C. Lazer and P.J. McKenna, On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc. Royal Soc. Edinburgh 95A (1983), 275–283.CrossRefMATHMathSciNetGoogle Scholar
  81. [81]
    A.C. Lazer and P.J. McKenna, On a conjecture on the number of solutions of a nonlinear Dirichlet problem with jumping nonlinearity, in “Trends in theory of nonlinear differential equations (Arlington Texas 1982) L. N. Pure and Appl. Math. 90, Dekker, New-York, (1984), 301–313.Google Scholar
  82. [82]
    A.C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. A.M.S. 99 (1987), 109–114.CrossRefMATHMathSciNetGoogle Scholar
  83. [83]
    S. Leela, Monotone method for second order periodic boundary value problems, Nonlinear Analysis, T.M.A. 7 (1983), 349–355.CrossRefMATHMathSciNetGoogle Scholar
  84. [84]
    P.L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Review 24 (1982), 441–467.CrossRefMATHMathSciNetGoogle Scholar
  85. [85]
    A.G. Lomtatidze, Positive solutions of boundary value problems for second order ordinary differential equations with singular points, Differentsial’nye Uravneniya 23 (1987), 1685–1692.MathSciNetGoogle Scholar
  86. [86]
    J. Mawhin, Compacité, monotonie et convexité dans l’étude de problèmes aux limites semi-linéaires, Séminaire d’analyse moderne 19, Université de Sherbrooke (1981).Google Scholar
  87. [87]
    J. Mawhin, Periodic oscillations of forced pendulum-like equations, in “Ordinary and Partial Diff. Equ., Proceed. Dundee 1982”, Lectures Notes in Math. 964, Springer, Berlin (ed: Everitt, Sleeman ) (1982), 458–476.Google Scholar
  88. [88]
    J. Mawhin, Boundary value problems with nonlinearities having infinite jumps, Comment. Math. Univ. Carolin. 25 (1984), 401–414.MATHMathSciNetGoogle Scholar
  89. [89]
    J. Mawhin, Points fixes, points critiques et problèmes aux limites, Sém. de Math. Supérieures, Univ. Montréal, (1985).MATHGoogle Scholar
  90. [90]
    J. Mawhin, Problèmes de Dirichlet variationnels non linéaires, Sém. de Math. Supérieures, Univ. Montréal, (1987).MATHGoogle Scholar
  91. [91]
    J. Mawhin, Recent results on periodic solutions of the forced pendulum equation, Rend. Ist. Matem. Univ. Trieste 19 (1987), 119–129.MATHMathSciNetGoogle Scholar
  92. [92]
    J. Mawhin, The forced pendulum: A paradigm for nonlinear analysis and dynamical systems, Expo. Math. 6 (1988), 271–287.MATHMathSciNetGoogle Scholar
  93. [93]
    J. Mawhin, Boundary value problems for nonlinear ordinary differential equations: from successive approximations to topology, in “Development of Mathematics, 1900–1950” (ed: J.P. Pier ), Birkhauser, Basel (1994), 445–478.Google Scholar
  94. [94]
    J. Mawhin and J.R. Ward, Nonresonance and existence for nonlinear elliptic boundary value problems, Nonlinear Anal. T.M.A. 5 (1981), 677–684.CrossRefMATHMathSciNetGoogle Scholar
  95. [95]
    J. Mawhin and J.R. Ward, Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Lienard and Duffing equations, Rocky Mountain J. Math. 12 (1982), 643–654.CrossRefMATHMathSciNetGoogle Scholar
  96. [96]
    J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Diff. Equ. 52 (1984), 264–287.CrossRefMATHMathSciNetGoogle Scholar
  97. [97]
    E.J. McShane, Integration, Princeton Univ. Press., Princeton (1944).MATHGoogle Scholar
  98. [98]
    M. Nagumo, Ober die differentialgleichung y″ = f (t, y, y′), Proc. Phys-Math. Soc. Japan 19 (1937), 861–866.Google Scholar
  99. [99]
    M. Nagumo, On principally linear elliptic differential equations of the second order, Osaka Math. J. 6 (1954), 207–229.MATHMathSciNetGoogle Scholar
  100. [100]
    J.J. Nieto, Nonlinear second order boundary value problems with Carathéodory function, Applicable Analysis 34 (1989), 111–128.CrossRefMATHGoogle Scholar
  101. [101]
    P. Omani, Non-ordered lower and upper solutions and solvability of the periodic problem for the Liénard and the Rayleigh equations, Rend. Ist. Mat. Univ. Trieste 20 (1988), 54–64.Google Scholar
  102. [102]
    P. Omani and M. Trombetta, Remarks on the lower and upper solution method for second and third-order periodic boundary value problem, Applied Math. Comput. 50 (1992), 71–82.Google Scholar
  103. [103]
    P. Omani and W. Ye, Necessary and sufficient conditions for the existence of periodic solutions of second order ODE with singular nonlinearities, Diff. Int. Equ. 8 (1995), 1843–1858.Google Scholar
  104. [104]
    P. Omani and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, preprint SISSA 1995.Google Scholar
  105. [105]
    L.C. Piccinini, G. Stampacchia and G. Vidossich, Ordinary differential equations inn: Problems and methods, Applied Math. Sciences 39, Springer-Verlag (1984).Google Scholar
  106. [106]
    N. Rouche and J. Mawhin, Equations différentielles ordinaires, Masson, Paris, (1973).MATHGoogle Scholar
  107. [107]
    D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972), 979–1000.CrossRefMATHMathSciNetGoogle Scholar
  108. [108]
    D. Sattinger, Topics in stability and bifurcation theory, Lecture Notes vol. 309, Springer, Berlin, 1973.MATHGoogle Scholar
  109. [109]
    K. Schmitt, Bounded solutions of nonlinear second order differential equations, Duke Math. J. 36 (1969), 237–244.CrossRefMATHMathSciNetGoogle Scholar
  110. [110]
    K. Schmitt, Boundary value problems for quasilinear second order elliptic equations, Nonlinear Anal. T.M.A. 2 (1978), 263–309.CrossRefMATHGoogle Scholar
  111. [111]
    K. Schrader, Differential inequalities for second and third order equations, J. Differential Equations 23 (1977), 203–215.CrossRefMathSciNetGoogle Scholar
  112. [112]
    G. Scorza Dragoni, Il problema dei valori ai limiti studiato in grande per gli integrali di una equazione differenziale del secondo ordine, Giornale di Mat (Battaglini) 69 (1931), 77–112.Google Scholar
  113. [113]
    G. Scorza Dragoni, Il problema dei valori ai limiti studiato in grande per le equazioni differenziali del secondo ordine, Math. Ann. 105 (1931), 133–143.CrossRefMathSciNetGoogle Scholar
  114. [114]
    G. Scorza Dragoni, Su un problema di valori ai limite per le equazioni differenziali ordinarie del secondo ordine, Rend. Semin. Mat. R. Univ. Roma 2 (1938), 177–215. Aggiunta, ibid, 253–254.Google Scholar
  115. [115]
    G. Scorza Dragoni, Elementi uniti di transformazioni funzionali e problemi di valori ai limiti, Rend. Semin. Mat. R. Univ. Roma 2 (1938), 255–275.Google Scholar
  116. [116]
    G. Scorza Dragoni, Intorno a un criterio di esistenza per un problema di valori ai limiti, Rend. Semin. R. Accad. Naz. Lincei 28 (1938), 317–325.Google Scholar
  117. [117]
    L.F. Shampine and G.M. Wing, Existence and Uniqueness of Solutions of a Class of Nonlinear Elliptic Boundary Value Problems, J. Math. and Mech. 19 (1970), 971–979.MATHMathSciNetGoogle Scholar
  118. [118]
    R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear Analysis and Applications (Ed: V. Lakshmikantham ), Dekker Inc, New York and Basel, (1987), 561–566.Google Scholar
  119. [119]
    S.D. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Analysis T.M.A. 3 (1979), 897–904.CrossRefMATHMathSciNetGoogle Scholar
  120. [120]
    A. Tineo, Existence of two periodic solutions for the periodic equation x = g(t, x), J. Math. Anal. Appl. 156 (1991), 588–596.CrossRefMATHMathSciNetGoogle Scholar
  121. [121]
    A. Tineo, Existence theorems for a singular two-point Dirichlet problem, Nonlinear Analysis T.M.A. 19 (1992), 323–333.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 1996

Authors and Affiliations

  • C. De Coster
    • 1
  • P. Habets
    • 1
  1. 1.Catholic University of LouvainLouvain-la-NeuveBelgium

Personalised recommendations