Advertisement

Introduction

  • Seshadev Padhi
  • Smita Pati

Abstract

In this chapter, we consider the third-order linear differential equation with constant coefficients of the form
$$ x^{\prime\prime\prime}+ax^{\prime\prime}+bx^{\prime}+cx=0,\quad t \geq \sigma, $$
(1.1)
where a, b, and c are real constants. We have considered eight different cases for the constants a, b, and c while studying the structures of spaces of oscillatory and nonoscillatory solutions of (7.5). Further, it is observed that different structures of solution spaces of (7.5) appear for the eight different cases on a, b, and c. Two comparison theorems are also given in this chapter on the oscillation theory of the nonhomogeneous equation
$$ x^{\prime\prime\prime}+ax^{\prime\prime}+bx^{\prime}+cx=f, $$
(1.2)
where a, b, c, and f are constants. Introductions to third-order delay differential equations and third-order canonical differential equations are also given. The results obtained in this chapter provide some basic ideas on the oscillation and nonoscillation of third-order linear and nonlinear differential equations with variable coefficients.

Keywords

Cauchy Sequence Delay Differential Equation Oscillatory Solution Oscillation Theory Nonoscillatory Solution 
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.

References

  1. 1.
    M. R. Abdollahpour and A. Najati; Hyers-Ulam stability of a differential equation of third order, International Journal of Mathematical Analysis, 6(59) (2012), 2943–2948. MathSciNetMATHGoogle Scholar
  2. 2.
    R. P. Agarwal, M. F. Aktas and A. Tiryaki; On oscillation criteria for third order nonlinear delay differential equations, Archivum Mathematicum, 45(1) (2009), 1–18. MathSciNetMATHGoogle Scholar
  3. 3.
    S. Ahmad and A. C. Lazer; On the oscillatory behavior of a class of linear third order differential equations, Journal of Mathematical Analysis and Applications, 28(3) (1970), 681–689. MathSciNetCrossRefGoogle Scholar
  4. 4.
    J. H. Barrett; Third order differential equations with nonnegative coefficients, Journal of Mathematical Analysis and Applications, 24(1) (1968), 212–224. MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    J. H. Barrett; Oscillation theory of ordinary linear differential equations, Advances in Mathematics, 3(4) (1969), 415–509. (Reprinted in Lectures in Ordinary Differential Equations, Edited by R. McKelvev, Academic Press, New York, 1970). MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    G. D. Birkhoff; On the solutions of ordinary linear homogeneous differential equations of the third order, Annals of Mathematics, 12(3) (1911), 103–127. MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    W. S. Burnside and A. W. Panton; The Theory of Equations, Third Edition, S. Chand and Company Ltd., New Delhi, 1979. Google Scholar
  8. 8.
    E. A. Cox and M. P. Mortell; The evolution of resonant water-wave oscillations, Journal of Fluid Mechanics, 162 (1986), 99–116. MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    P. Das; Contributions to the Study of Qualitative Behaviour of Solutions of Third-Order Differential Equations, Ph.D. Thesis, Berhampur University, India, 1991. Google Scholar
  10. 10.
    P. Das; On oscillation of third order forced equations, Journal of Mathematical Analysis and Applications, 196(2) (1995), 502–513. MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Z. Došlá; On oscillatory solutions of third-order linear differential equations, Časopis pro Pěstování Matematiky, 114(1) (1989), 28–34. MATHGoogle Scholar
  12. 12.
    R. D. Driver; Ordinary and Delay Differential Equations, Springer-Verlag, New York, 1977. CrossRefMATHGoogle Scholar
  13. 13.
    L. Erbe; Disconjugacy conditions for the third order linear differential equations, Canadian Mathematical Bulletin, 12 (1969), 603–613. MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    L. Erbe; Existence of oscillatory solutions and asymptotic behaviour for a class of third order linear differential equations, Pacific Journal of Mathematics, 64(2) (1976), 369–385. MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    L. Erbe; Oscillation and asymptotic behaviour of solutions of third order differential delay equations, SIAM Journal of Mathematical Analysis, 7(4) (1976), 491–500. MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    L. Erbe, Q. Kong and B. G. Zhang; Oscillation Theory for Functional Differential Equations, Marcel Dekker Inc., New York, 1995. Google Scholar
  17. 17.
    G. J. Etgen and C. D. Shih; Conditions for the nonoscillation of third order differential equations with nonnegative coefficients, SIAM Journal of Mathematical Analysis, 6 (1975), 1–8. MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    N. Finizio and G. Ladas; Ordinary Differential Equations with Modern Applications, Third Edition, Wadsworth Pub. Co., Belmont, 1988. Google Scholar
  19. 19.
    M. Greguš; On certain new properties of solutions of the differential equation y′′′+Qy′+Qy=0, Publ. Fac. Sci. Univ. Masaryk Brno Czech., 362 (1955), 237–251 (in Czech). Google Scholar
  20. 20.
    M. Greguš; Oscillatory properties of solutions of a third order differential equation of the type y′′′+2A(x)y′+[A′(x)+b(x)]y=0, Acta Facultatis Rerum Naturalium Universitatis Comenianae, Mathematica, 6 (1961), 275–300. MATHGoogle Scholar
  21. 21.
    M. Greguš; Third Order Linear Differential Equations, D. Reidel Publishing Company, Boston, 1987. CrossRefMATHGoogle Scholar
  22. 22.
    M. Greguš and M. Gera; Some results in the theory of a third order linear differential equation, Annales Polonici Mathematici, 42 (1983), 93–101. MathSciNetMATHGoogle Scholar
  23. 23.
    M. Greguš, J. R. Graef and M. Gera; Oscillating nonlinear third order differential equations, Nonlinear Analysis; Theory Methods and Applications, 28(10) (1997), 1611–1622. MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    I. Gyori and G. Ladas; Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991. Google Scholar
  25. 25.
    M. Hanan; Oscillation criteria for third order linear differential equations, Pacific Journal of Mathematics, 11 (1961), 919–944. MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    P. Hartman; Ordinary Differential Equations, Wiley, New York, 1964, and Birkhäuser, Boston, 1982. MATHGoogle Scholar
  27. 27.
    E. Hille; Non-oscillation theorems, Transactions of the American Mathematical Society, 64 (1948), 234–252. MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Z. Jackiewicz, M. Klaus and C. O’Cinneide; Asymptotic behaviour of solutions to Volterra integro-differential equations, Journal of Integral Equations and Applications, 1(4) (1988), 501–516. MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    G. Jayaraman, N. Padmanabhan and R. Mehrotra; Entry flow into a circular tube of slowly varying cross section, Fluid Dynamics Research, 1(2) (1986), 131–144. CrossRefGoogle Scholar
  30. 30.
    G. D. Jones; A property of \(y^{\prime\prime\prime} + p(x) y^{\prime} + \frac{1}{2}p^{\prime}(x)y = 0\), Proceedings of American Mathematical Society, 33 (1972), 420–422. Google Scholar
  31. 31.
    G. D. Jones; An asymptotic property of solutions of y′′′+py′+qy=0, Pacific Journal of Mathematics, 48(1) (1973), 135–138. CrossRefGoogle Scholar
  32. 32.
    G. D. Jones; Oscillation properties of third order differential equations, Rocky Mountain Journal of Mathematics, 3 (1973), 507–513. MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    G. D. Jones; Properties of solutions of a class of third order differential equations, Journal of Mathematical Analysis and Applications, 48(1) (1974), 165–169. MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    M. S. Keener; On the solutions of certain linear nonhomogeneous second order differential equations, Applicable Analysis, 1 (1971), 57–63. MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    I. T. Kiguradze and T. A. Chanturia; Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Academic Publishers, Dordrecht, 1993. CrossRefMATHGoogle Scholar
  36. 36.
    W. J. Kim; Oscillatory properties of linear third order differential equations, Proceedings of American Mathematical Society, 26 (1970), 286–293. MATHGoogle Scholar
  37. 37.
    Y. Kuramoto and T. Yamada; Turbulent state in chemical reaction, Progress of Theoretical Physics, 56 (1976), 679. MathSciNetCrossRefGoogle Scholar
  38. 38.
    T. Kusano and M. Naito; Comparison theorems for functional differential equations with deviating arguments, Journal of the Mathematical Society of Japan, 3 (1981), 509–532. MathSciNetCrossRefGoogle Scholar
  39. 39.
    T. Kusano, M. Naito and K. Tanaka; Oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations, Proceedings of the Royal Society of Edinburgh, 90A (1981), 25–40. MathSciNetCrossRefGoogle Scholar
  40. 40.
    G. Ladas, Y. G. Sficas and I. P. Stavroulakis; Necessary and sufficient conditions for oscillations of higher order delay differential equations, Transactions of the American Mathematical Society, 285(1) (1984), 81–90. MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    G. S. Ladde, V. Laxmikantham and B. G. Zhang; Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker Inc., New York, 1987. Google Scholar
  42. 42.
    A. C. Lazer; The behaviour of solutions of the differential equation y′′′+p(x)y′+q(x)y=0, Pacific Journal of Mathematics, 17 (1966), 435–466. MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    W. Leighton and Z. Nehari; On the oscillation of solutions of self adjoint linear differential equations of the fourth order, Transactions of the American Mathematical Society, 89 (1958), 325–377. MathSciNetCrossRefGoogle Scholar
  44. 44.
    H.P. McKean; Nagumo’s equation, Advances in Mathematics, 4 (1970), 209–223. MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    D. Michelson; Steady solutions of the Kuramoto-Sivashinsky equation, Physica D, 19 (1986), 89–111. MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    S. Padhi; Contributions to the Oscillation Theory of Ordinary and Delay Differential Equations of Third Order, Ph.D. Thesis, Berhampur University, India, 1998. Google Scholar
  47. 47.
    S. Padhi; On oscillatory linear third order forced differential equations, Differential Equations and Dynamical Systems, 13 (2005), 343–358. MathSciNetMATHGoogle Scholar
  48. 48.
    N. Parhi and P. Das; On the zeros of solutions of nonhomogeneous third order differential equations, Czechoslovak Mathematical Journal, 41(116) (1991), 641–652. MathSciNetGoogle Scholar
  49. 49.
    N. Parhi and P. Das; Oscillation and nonoscillation of nonhomogeneous third order differential equations, Czechoslovak Mathematical Journal, 44(119) (1994), 443–459. MathSciNetMATHGoogle Scholar
  50. 50.
    N. Parhi and S. K. Nayak; Nonoscillation of second order nonhomogeneous differential equations, Journal of Mathematical Analysis and Applications, 102(1) (1984), 62–74. MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    D. W. Reynolds; Bifurcation of harmonic solutions of an integro differential equation modelling resonant sloshing, SIAM Journal of Applied Mathematics, 49(2) (1989), 362–372. CrossRefMATHGoogle Scholar
  52. 52.
    J. Rovder; Oscillation criteria for third-order linear differential equations, Matematický Časopis, 25(3) (1975), 231–244. MathSciNetMATHGoogle Scholar
  53. 53.
    H. L. Royden; Real Analysis, Second Edition, McMillan Publishing Co. Inc., New York, 1968. Google Scholar
  54. 54.
    W. Rudin; Principles of Mathematical Analysis, Third Edition, McGraw-Hill Kogakusha Ltd., Tokyo, 1976. MATHGoogle Scholar
  55. 55.
    Y. P. Singh; The asymptotic behaviour of solutions of linear third order differential equations, Proceedings of American Mathematical Society, 20(2) (1969), 309–314. CrossRefMATHGoogle Scholar
  56. 56.
    A. Škerlik; Integral criteria of oscillation for a third order linear differential equation, Mathematica Slovaca, 45(4) (1995), 403–412. MathSciNetMATHGoogle Scholar
  57. 57.
    A. Škerlik; An integral condition of oscillation for equation y′′′+p(t)y′+q(t)y=0 with nonnegative coefficients, Archivum Mathematicum, 31(2) (1995), 155–161. MathSciNetMATHGoogle Scholar
  58. 58.
    J. C. F. Sturm; Memoire Sur les equations differentielles linearies du second order, Journal de Mathématiques Pures et Appliquées, 11 (1836), 106–186. Google Scholar
  59. 59.
    M. Svec; Some remarks on a third order linear differential equation, Czechoslovak Mathematical Journal, 15 (1965), 42–49. MathSciNetGoogle Scholar
  60. 60.
    C. A. Swanson; Comparison and Oscillation Theory of Linear Differential equations, Academic Press, New York, 1968. MATHGoogle Scholar
  61. 61.
    W. F. Trench; Canonical forms and principal systems for general disconjugate equations, Transactions of the American Mathematical Society, 189 (1974), 319–327. MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    D. D. Vreeke and G. M. Sandquist; Phase plane analysis of reactor kinetics, Nuclear Science and Engineering 42 (1970), 295–305. Google Scholar

Copyright information

© Springer India 2014

Authors and Affiliations

  • Seshadev Padhi
    • 1
  • Smita Pati
    • 1
  1. 1.Department of Applied MathematicsBirla Institute of Technology, MesraRanchiIndia

Personalised recommendations