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Green’s Functions in the Theory of Ordinary Differential Equations

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Green’s Functions in the Theory of Ordinary Differential Equations

Part of the book series: SpringerBriefs in Mathematics ((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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Appell, J., Zabrejko, P.P.: Nonlinear superposition operators. Cambridge Tracts in Mathematics, vol. 95. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cabada, A.: The monotone method for first-order problems with linear and nonlinear boundary conditions. Appl. Math. Comput. 63, 163–186 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Cabada, A.: An overview of the lower and upper solutions method with nonlinear boundary value conditions. Bound. Value Prob. 2011, 18 (2011) Article ID 893753

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cabada, A., Enguiça, R.: Positive solutions of fourth order problems with clamped beam boundary conditions. Nonlinear Anal. 74, 3112–3122 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cabada, A., Lois, S.: Maximum principles for fourth and sixth order periodic boundary value problems. Nonlinear Anal. 29(10), 1161–1171 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cabada, A., Lois, S.: Existence results for nonlinear problems with separated boundary conditions. Nonlinear Anal. 35, 449–456 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  20. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New Delhi (1987)

    Google Scholar 

  21. Coppel, W.A.: Disconjugacy. In: Lecture Notes in Mathematics, vol. 220. Springer, Berlin (1971)

    Google Scholar 

  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. 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. 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. Duffy, D.G.: Green’s functions with applications. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton (2001)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Fried, H.M.: Green’s Functions and Ordered Exponentials. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  28. Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)

    MATH  Google Scholar 

  29. Karlin, S.: Positive operators. J. Math. Mech. 8(6), 907–937 (1959)

    MathSciNet  MATH  Google Scholar 

  30. Karlin, S.: The existence of eigenvalues for integral operators. Trans. Am. Math. Soc. 113, 1–17 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  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. Ladde, G.S., Lakshmikantham, V., Vatsala, A.S.: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston (1985)

    MATH  Google Scholar 

  33. Lloyd, N.G.: Degree theory. Cambridge Tracts in Mathematics, vol. 73. Cambridge University Press, Cambridge (1978)

    Google Scholar 

  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. Mawhin, J.: Twenty years of ordinary differential equations through twelve Oberwolfach meetings. Results Math. 21(1–2), 165–189 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Melnikov, Y.A.: Green’s functions and infinite products. Bridging the Divide. Birkhäuser/Springer, New York (2011)

    Book  MATH  Google Scholar 

  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. Müller, M.: Über das Fundamentaltheorem in der theorie der gewöhnlichen differentialgleichungen. Math. Z. 26, 619–649 (1926)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  42. Novo, S., Obaya, R., Rojo, J.: Equations and Differential Systems (in Spanish). McGraw-Hill, New York (1995)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  44. Perron, O.: Ein neuer existenzbeweis für die integrale der differentialgleinchung y′ = f(t, y). Math. Ann. 76, 471–484 (1915)

    Google Scholar 

  45. Picard, E.: Mémoire sur la théorie des équations aux derivés partielles et las méthode des approximations succesives. J. Math. 6, 145–210 (1890)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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. Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New York (1976)

    MATH  Google Scholar 

  49. Schröder, J.: On linear differential inequalities. J. Math. Anal. Appl. 22, 188–216 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  50. Schröder, J.: Operator inequalities. Mathematics in Science and Engineering, vol. 147. Academic [Harcourt Brace Jovanovich, Publishers], New York (1980)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Şeremet, V.D.: Handbook of Green’s Functions and Matrices. With 1 CD-ROM (Windows and Macintosh). WIT Press, Southampton (2003)

    Google Scholar 

  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. Zeidler, E.: Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems. Translated from the German by Peter R. Wadsack. Springer, New York (1986)

    Book  MATH  Google Scholar 

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© 2014 Alberto Cabada

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Cabada, A. (2014). Green’s Functions in the Theory of Ordinary Differential Equations. In: Green’s Functions in the Theory of Ordinary Differential Equations. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9506-2_1

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