Comparison and nonoscillation results for perturbed nonlinear differential equations


Nonoscillation theorems for perturbed second order nonlinear differential equations are obtained. A nonlinear Picone type identity is introduced to obtain some Sturm-Picone type comparison theorems for nonlinear equations.


  1. [1]

    M. S. P. Eastham,The Picone identity for self-adjoint differential equations of even order, Mathematika,20 (1973), pp. 197–200.

    MathSciNet  Article  Google Scholar 

  2. [2]

    J. R. Graef,A comparison and oscillation result for second order nonlinear differential equations, to appear.

  3. [3]

    J. R. Graef -P. W. Spikes,A nonoscillation result for second order ordinary differential equations, Rend. Accad. Sci. Fis. Mat. Napoli, (4)41 (1974), pp. 92–101, (1975).

    MathSciNet  MATH  Google Scholar 

  4. [4]

    J. R. Graef -P. W. Spikes,Sufficient conditions for nonoscillation of a second order nonlinear differential equation, Proc. Amer. Math. Soc.,50 (1975), pp. 289–292.

    MathSciNet  Article  Google Scholar 

  5. [5]

    J. R. Graef -P. W. Spikes,Sufficient conditions for the equation (a(t)x′)′ + h(t,x,x′) + + q(t)f(x,x′) = e(t,x,x′) to be nonoscillatory, Funkcial. Ekvac.,18 (1975), pp. 35–40.

    MathSciNet  MATH  Google Scholar 

  6. [6]

    J. R. Graef -P. W. Spikes,Nonoscillation theorems for forced second order nonlinear differential equations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (8)59 (1975), pp. 694–701, (1976).

    MathSciNet  MATH  Google Scholar 

  7. [7]

    M. E. Hammett,Oscillation and nonoscillation theorems for nonhomogeneous linear differential equations of second order, Ph. D. Dissertation, Auburn University (1967).

  8. [8]

    M. S. Keener,On the solutions of certain linear nonhomogeneous second-order differential equations, Applicable Analysis,1 (1971), pp. 57–63.

    MathSciNet  Article  Google Scholar 

  9. [9]

    K. Kreith,Oscillation Theory, Lecture Notes in Mathematics, no. 324, Springer-Verlag, Ne York (1973).

    Google Scholar 

  10. [10]

    K. Kreith,A Picone identity for fourth order differential equations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (8)52 (1972), pp. 455–456.

    MathSciNet  MATH  Google Scholar 

  11. [11]

    V. N. Ševelo -V. G. Štelik,Certain problems concerning the oscillation of solutions of nonlinear nonautonomous second-order equations, Soviet Math. Dokl.,4 (1963), pp. 383–387.

    Google Scholar 

  12. [12]

    M. Švec,On various properties of the solutions of third- and fourth-order linear differential equations, in « Differential Equations and Their Applications, Proceedings of a Conference held in Prague, September, 1962 », Academic Press, Ne York (1963), pp. 187–198.

    Google Scholar 

  13. [13]

    C. A. Swanson,Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York (1968).

    Google Scholar 

  14. [14]

    C. A. Swanson,Picone's identity, Rend. Mat., (6)8 (1975), pp. 373–397.

    MathSciNet  MATH  Google Scholar 

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Entrata in Redazione il 19 gennaio 1977.

Research supported by the Mississippi State University Biological and Physical Sciences Research Institute.

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Graef, J.R., Spikes, P.W. Comparison and nonoscillation results for perturbed nonlinear differential equations. Annali di Matematica 116, 135–142 (1978).

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  • Differential Equation
  • Nonlinear Equation
  • Nonlinear Differential Equation
  • Type Identity
  • Comparison Theorem