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Oscillation criteria for a class of nonlinear vector delay-differential equations

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Summary

Oscillation criteria are obtained for a class of vector delay differential equation of the form

$$u''\left( t \right) + F_1 \left( {t,u\left( t \right),u\left( {g\left( t \right)} \right)} \right)u\left( t \right) + F_2 \left( {t,u\left( t \right),u\left( {g\left( t \right)} \right)} \right)u\left( {g\left( t \right)} \right) = 0$$

t≥0, u ε C2 ([0, ∞),R n), where F 1 and F 2 are continuous nonnegative valued functionals in [0, ∞)×R n×R n, and g is a continuous real valued function with g(t)≤t, g(t)→∞ as t→ ∞. A solution u: [0, ∞)→R n is called h-oscillatory in [0, ∞) whenever the scalar product [u(t), h] (│h│=1) has zeros in [a, ∞) with a arbitrary large. The method involves oscillatory behaviour of solutions of a nonlinear scalar delay differential inequality satisfied by h-nonoscillatory solutions of the above equation.

References

  1. [1]

    F. V. Atkinson,On second order nonlinear oscillation, Pacific J. Math.,5 (1955), pp. 643–647.

  2. [2]

    N. P. Bhatia,Some oscillation for second order differential equations, J. Math. Anal. Appl.,15 (1966), pp. 442–446.

  3. [3]

    L. E. Bobisud,Oscillation of solution of nonlinear equations, Proc. Amer. Math. Soc.,23 (1969), pp. 501–505.

  4. [4]

    K. L. Chiou,Oscillation and nonoscillation theorems for second order functional differential equations, J. Math. Anal. Appl.,45 (1974), pp. 382–403.

  5. [5]

    K. M. Das,Properties of solutions of certain non-linear differential equations, J. Math. Anal. Appl.,8 (1964), pp. 445–451.

  6. [6]

    J. I. Domslak,On the oscillation of solutions of vector differential equations, Dokl Akad. Nauk SSSR,193 (1970), pp. 21–23; Soviet Math. Dokl,11 (1970), pp. 839–841.

  7. [7]

    L. Erbe,Oscillation criteria for second order nonlinear delay equations, Can. Math. Bull.,16 (1973), pp. 49–56.

  8. [8]

    L. Erbe,Oscillation in functional differential equations, Lecture notes in Mathematics, Springer-Verlag, Vol.243, pp. 289–291. Japan-United States seminar on ordinary differential and functional equations.

  9. [9]

    H. E. Gollwitzer,On nonlinear oscillations for a second order delay equation, J. Math. Anal. Appl.,26 (1969), pp. 385–389.

  10. [10]

    J. Jones jr.,On the extension of a theorem of Atkinson, Quart. J. Math.,7 (1956), pp. 306–309.

  11. [11]

    G. C. T. Kung,Oscillation and nonoscillation of differential equations with a time lag, SIAM J. Appl. Math.,21 (1971), pp. 207–213.

  12. [12]

    J. W. Macki -J. S. Wong,Oscillation to second order nonlinear differential equations, Pacific J. Math.,24 (1968), pp. 111–117.

  13. [13]

    R. A. Moore,The behaviour of solutions of a linear differential equation of second order, Pacific J. Math.,5 (1955), pp. 125–145.

  14. [14]

    E. S. Noussair -C. A. Swanson,Oscillation theorems for vector differential equations, Utilitas Mathematica,1 (1972), pp. 97–109.

  15. [15]

    E. S. Noussair -C. A. Swanson,Oscillation theory for semilinear Schrödinger equations and inequalities, Proc. Roy. Soc. Edinburgh, Sect. A,75 (1975/76), pp. 67–86.

  16. [16]

    E. S. Noussair -C. A. Swanson,Oscillation of nonlinear vector differential equations, Annali di Matematica Pura ed Applicata,109 (1976), pp. 305–315.

  17. [17]

    H. Teufel jr.,A note on second order differential ineuqalities and functional differential equations, Pacific J. Math.,41 (1972), pp. 537–541.

  18. [18]

    P. Waltman,Oscillations of solutions of non-linear equations, SIAM Rev.,5 (1963), pp. 128–130.

  19. [19]

    J. S. Wong,Second order oscillation with retarded arguments, Ordinary differential equation (1971), NRL-MRC-Conference.

  20. [20]

    J. S. Wong,A note on second order nonlinear oscillation, SIAM Rev.,10 (1968), pp. 89–91.

  21. [21]

    J. S. Wong,On the second order non-linear oscillation, FUNCKCIAL EKAV,11 (1968), pp. 207–234; MR 39, No. 722.

  22. [22]

    M. Zlamal,Oscillation criterions, Cas pêst Mat. a Fis.,75 (1950), pp. 213–217.

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Entrata in Redazione il 24 novembre 1976.

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Nababan, S. Oscillation criteria for a class of nonlinear vector delay-differential equations. Annali di Matematica 117, 55–66 (1978) doi:10.1007/BF02417884

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Keywords

  • Differential Equation
  • Scalar Product
  • Oscillatory Behaviour
  • Delay Differential Equation
  • Differential Inequality