Advertisement

Green’s Functions in the Theory of Ordinary Differential Equations

  • Alberto Cabada
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

In this monograph we will present the main topics concerning Green’s functions related to nth-order ordinary differential equations coupled with linear two-point boundary conditions. To show the potential of this theory and importance of obtaining qualitative and quantitative properties of this kind of functions, we will consider in this preliminary section a simple example to illustrate the results we are dealing with.

References

  1. 1.
    Afuwape, A.U., Omari, P., Zanolin, F.: Nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary value problems. J. Math. Anal. Appl. 143, 35–56 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Appell, J., Zabrejko, P.P.: Nonlinear superposition operators. Cambridge Tracts in Mathematics, vol. 95. Cambridge University Press, Cambridge (1990)Google Scholar
  3. 3.
    Bernfeld, S.R., Lakshmikantham, V.: An introduction to nonlinear boundary value problems. Mathematics in Science and Engineering, vol. 109. Academic Press, New York (1974)Google Scholar
  4. 4.
    Cabada, A.: The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems. J. Math. Anal. Appl. 185, 302–320 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cabada, A.: The monotone method for first-order problems with linear and nonlinear boundary conditions. Appl. Math. Comput. 63, 163–186 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cabada, A.: The method of lower and upper solutions for nth-order periodic boundary value problems. J. Appl. Math. Stoch. Anal. 7, 33–47 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cabada, A.: The method of lower and upper solutions for third-order periodic boundary value problems. J. Math. Anal. Appl. 195, 568–589 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cabada, A.: The monotone method for third order boundary value problems. In: Proceedings of the World Congress of Nonlinear Analysts, vol. I, pp. 211–221, Aug 1992. Walter de Gruyter, Tampa (1996)Google Scholar
  9. 9.
    Cabada, A.: Maximum principles for third-order initial and terminal value problems. In: Differential & Difference Equations and Applications, pp. 247–255. Hindawi Publishing Corporation, New York (2006)Google Scholar
  10. 10.
    Cabada, A.: An overview of the lower and upper solutions method with nonlinear boundary value conditions. Bound. Value Prob. 2011, 18 (2011) Article ID 893753Google Scholar
  11. 11.
    Cabada, A., Cid, J.A.: On the sign of the Green’s function associated to Hill’s equation with an indefinite potential. Appl. Math. Comput. 205, 303–308 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cabada, A., Enguiça, R.: Positive solutions of fourth order problems with clamped beam boundary conditions. Nonlinear Anal. 74, 3112–3122 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cabada, A., Lois, S.: Maximum principles for fourth and sixth order periodic boundary value problems. Nonlinear Anal. 29(10), 1161–1171 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cabada, A., Lois, S.: Existence results for nonlinear problems with separated boundary conditions. Nonlinear Anal. 35, 449–456 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cabada, A., Nieto, J.J.: Approximation of solutions for second order boundary value problems. Bull. Classe Sci. Acad. Roy. Bel. \({6}^{\underline{a}}\) Sér. II 10–11, 287–311 (1991)MathSciNetGoogle Scholar
  16. 16.
    Cabada, A., Sanchez, L.: A positive operator approach to the Neumann problem for a second order ordinary differential equation. J. Math. Anal. Appl. 204(3), 774–785 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cabada, A., Tojo, F.A.: Comparison results for first order linear operators with reflection and periodic boundary value conditions. Nonlinear Anal. 78, 32–46 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cabada, A., Cid, J.A., Sanchez, L.: Positivity and lower and upper solutions for fourth order boundary value problems. Nonlinear Anal. 67, 1599–1612 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cabada, A., Cid, J.A., Máquez-Villamarín, B.: Computation of Green’s functions for boundary value problems with Mathematica. Appl. Math. Comput. 219(4), 1919–1936 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New Delhi (1987)Google Scholar
  21. 21.
    Coppel, W.A.: Disconjugacy. In: Lecture Notes in Mathematics, vol. 220. Springer, Berlin (1971)Google Scholar
  22. 22.
    De Coster, C., Habets, P.: An overview of the method of lower and upper solutions for ODEs. Nonlinear analysis and its applications to differential equations (Lisbon, 1998). Progress in Nonlinear Differential Equations and their Applications, vol. 43, pp. 3–22. Birkhäuser, Boston (2001)Google Scholar
  23. 23.
    De Coster, C., Habets, P.: The lower and upper solutions method for boundary value problems. Handbook of Differential Equations, pp. 69–160, Elsevier/North-Holland, Amsterdam (2004)Google Scholar
  24. 24.
    De Coster, C., Habets, P.: Two-point boundary value problems: lower and upper solutions. Mathematics in Science and Engineering, vol. 205. Elsevier B.V., Amsterdam (2006)Google Scholar
  25. 25.
    Duffy, D.G.: Green’s functions with applications. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton (2001)CrossRefzbMATHGoogle Scholar
  26. 26.
    Fabry, C., Habets, P.: Upper and lower solutions for second-order boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 10(10), 985–1007 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Fried, H.M.: Green’s Functions and Ordered Exponentials. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  28. 28.
    Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)zbMATHGoogle Scholar
  29. 29.
    Karlin, S.: Positive operators. J. Math. Mech. 8(6), 907–937 (1959)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Karlin, S.: The existence of eigenvalues for integral operators. Trans. Am. Math. Soc. 113, 1–17 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kythe, P.: Green’s functions and linear differential equations. Theory, applications, and computation. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. CRC Press, Boca Raton (2011)Google Scholar
  32. 32.
    Ladde, G.S., Lakshmikantham, V., Vatsala, A.S.: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston (1985)zbMATHGoogle Scholar
  33. 33.
    Lloyd, N.G.: Degree theory. Cambridge Tracts in Mathematics, vol. 73. Cambridge University Press, Cambridge (1978)Google Scholar
  34. 34.
    Mawhin, J.: Points fixes, points critiques et problèmes aux limites. (French) Séminaire de Mathématiques Supérieures, vol. 92, p. 162. Presses de l’Université de Montréal, Montreal (1985)Google Scholar
  35. 35.
    Mawhin, J.: Twenty years of ordinary differential equations through twelve Oberwolfach meetings. Results Math. 21(1–2), 165–189 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Mawhin, J.: Boundary value problems for nonlinear ordinary differential equations: from successive approximations to topology. Development of Mathematics 1900–1950 (Luxembourg, 1992), pp. 443–477. Birkhäuser, Basel (1994)Google Scholar
  37. 37.
    Mawhin, J.: Bounded solutions of nonlinear ordinary differential equations. Non-linear analysis and boundary value problems for ordinary differential equations (Udine). CISM Courses and Lectures, vol. 371, pp. 121–147. Springer, Vienna (1996)Google Scholar
  38. 38.
    Melnikov, Y.A.: Green’s functions and infinite products. Bridging the Divide. Birkhäuser/Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  39. 39.
    Melnikov, Y.A., Melnikov, M.Y.: Green’s functions. Construction and applications. de Gruyter Studies in Mathematics, vol. 42. Walter de Gruyter & Co., Berlin (2012)Google Scholar
  40. 40.
    Müller, M.: Über das Fundamentaltheorem in der theorie der gewöhnlichen differentialgleichungen. Math. Z. 26, 619–649 (1926)CrossRefGoogle Scholar
  41. 41.
    Nkashama, M.N.: A generalizaded upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations. J. Math. Anal. Appl. 140, 381–395 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Novo, S., Obaya, R., Rojo, J.: Equations and Differential Systems (in Spanish). McGraw-Hill, New York (1995)Google Scholar
  43. 43.
    Omari, P., Trombetta, M.: Remarks on the lower and upper solutions method for second and third-order periodic boundary value problems. Appl. Math. Comput. 50, 1–21 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Perron, O.: Ein neuer existenzbeweis für die integrale der differentialgleinchung y′ = f(t, y). Math. Ann. 76, 471–484 (1915)Google Scholar
  45. 45.
    Picard, E.: Mémoire sur la théorie des équations aux derivés partielles et las méthode des approximations succesives. J. Math. 6, 145–210 (1890)zbMATHGoogle Scholar
  46. 46.
    Picard, E.: Sur l’application des métodes d’approximations succesives à l’étude de certains équations différentielles ordinaires. J. Math. 9, 217–271 (1893)zbMATHGoogle Scholar
  47. 47.
    Renardy, M., Rogers, R.C.: An introduction to partial differential equations. Texts in Applied Mathematics, vol. 13, 2nd edn. Springer, New York (2004)Google Scholar
  48. 48.
    Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New York (1976)zbMATHGoogle Scholar
  49. 49.
    Schröder, J.: On linear differential inequalities. J. Math. Anal. Appl. 22, 188–216 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Schröder, J.: Operator inequalities. Mathematics in Science and Engineering, vol. 147. Academic [Harcourt Brace Jovanovich, Publishers], New York (1980)Google Scholar
  51. 51.
    Scorza Dragoni, S.: Il problema dei valori ai limiti estudiato in grande per le equazione differenziale del secondo ordine. Math. Ann. 105, 133–143 (1931)MathSciNetCrossRefGoogle Scholar
  52. 52.
    S̆eda, V., Nieto, J.J., Gera, M.: Periodic boundary value problems for nonlinear higher order ordinary differential equations. Appl. Math. Comput. 48, 71–82 (1992)Google Scholar
  53. 53.
    Şeremet, V.D.: Handbook of Green’s Functions and Matrices. With 1 CD-ROM (Windows and Macintosh). WIT Press, Southampton (2003)Google Scholar
  54. 54.
    Stakgold, I., Holst, M.: Green’s functions and boundary value problems, 3rd edn. Pure and Applied Mathematics (Hoboken). Wiley, Hoboken (2011)Google Scholar
  55. 55.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems. Translated from the German by Peter R. Wadsack. Springer, New York (1986)CrossRefzbMATHGoogle Scholar

Copyright information

© Alberto Cabada 2014

Authors and Affiliations

  • Alberto Cabada
    • 1
  1. 1.Department of Mathematical AnalysisUniversity of Santiago de CompostelaGaliciaSpain

Personalised recommendations