On positive solutions of fourth order nonlinear differential inequalities

  • Manabu Naito


This paper is concerned with positive solutions of the fourth order nonlinear differential inequality
$$\left( {r_3 \left( t \right)\left( {r_2 \left( t \right)\left( {r_1 \left( t \right)x'} \right)'} \right)'} \right)' + f\left( {x,t} \right)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } 0, t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } a.$$
Necessary and sufficient conditions are presented for the existence of special types of positive solutions of(A). When r 1 (t) ≡ r 3 (t), necessary and sufficient conditions are established for the existence of positive solutions of(A) which is either strongly superlinear or strongly sublinear.


Fourth Order Differential Inequality Nonlinear Differential Inequality 
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.


  1. [1]
    Ya. V. Bykov -G. D. Merzlyakova,On the oscillation of solutions of nonlinear differential equations with deviating argument, Differencial'nye Uravnenija,10 (1974), pp. 210–220 (Russian).zbMATHGoogle Scholar
  2. [2]
    T. Kusano -M. Naito,Nonlinear oscillation of second order differential equations with retarded argument, Ann. Mat. Pura Appl.,106 (1975), pp. 171–185.CrossRefMathSciNetzbMATHGoogle Scholar
  3. [3]
    T. Kusano -M. Naito,Nonlinear oscillation of fourth order differential equations, Can. J. Math.,28 (1976), pp. 840–852.MathSciNetzbMATHGoogle Scholar
  4. [4]
    T. Kusano -M. Naito,On fourth-order non-linear oscillations, J. London Math. Soc.,14 (1976), pp. 91–105.MathSciNetzbMATHGoogle Scholar
  5. [5]
    W. Leighton -Z. Nehari,On the oscillation of solutions of self-adjoint linear differential equations of the fourth order, Trans. Amer. Math. Soc.,89 (1958), pp. 325–377.CrossRefMathSciNetGoogle Scholar
  6. [6]
    B. M. Levitan,Some problems of the theory of almost periodic functions I, Uspehi Mat. Nauk,2–5 (1947), pp. 133–192 (Russian).MathSciNetGoogle Scholar
  7. [7]
    D. L. Lovelady,Some oscillation criteria for fourth order differential equations, Rocky Mountain J. Math.,5 (1975), pp. 593–600.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    R. D. Terry,Oscillatory properties of a fourth-order delay differential equation, 2,Funkcial. Ekvac.,16 (1973), pp. 213–224.zbMATHMathSciNetGoogle Scholar
  9. [9]
    R. D. Terry,Oscillatory and asymptotic properties of homogeneous and nonhomogeneous delay differential equations of fourth order, Funkcial. Ekvac.,18 (1975), pp. 207–218.zbMATHMathSciNetGoogle Scholar
  10. [10]
    R. D. Terry -P. K. Wong,Oscillatory properties of a fourth-order delay differential equation, Funkcial. Ekvac.,15 (1972), pp. 209–221.MathSciNetzbMATHGoogle Scholar
  11. [11]
    P. K. Wong,On a class of nonlinear fourth order differential equations, Ann. Mat. Pura Appl.,81 (1969), pp. 331–346.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Nicola Zanichelli Editore 1978

Authors and Affiliations

  • Manabu Naito
    • 1
  1. 1.HiroshimaJapan

Personalised recommendations