Asymptotic behaviour of oscillatory solutions of n -th order differential equations with quasiderivatives

Miroslav Bartušek

Czechoslovak Mathematical Journal (1997)

  • Volume: 47, Issue: 2, page 245-259
  • ISSN: 0011-4642

Abstract

top
Sufficient conditions are given under which the sequence of the absolute values of all local extremes of y [ i ] , i ∈ { 0 , 1 , ⋯ , n - 2 } of solutions of a differential equation with quasiderivatives y [ n ] = f ( t , y [ 0 ] , ⋯ , y [ n - 1 ] ) is increasing and tends to ∞ . The existence of proper, oscillatory and unbounded solutions is proved.

How to cite

top

Bartušek, Miroslav. "Asymptotic behaviour of oscillatory solutions of $n$-th order differential equations with quasiderivatives." Czechoslovak Mathematical Journal 47.2 (1997): 245-259. <http://eudml.org/doc/30362>.

@article{Bartušek1997,
abstract = {Sufficient conditions are given under which the sequence of the absolute values of all local extremes of $y^\{[i]\}$, $i\in \lbrace 0,1,\dots , n-2\rbrace $ of solutions of a differential equation with quasiderivatives $y^\{[n]\}=f(t,y^\{[0]\},\dots , y^\{[n-1]\})$ is increasing and tends to $\infty $. The existence of proper, oscillatory and unbounded solutions is proved.},
author = {Bartušek, Miroslav},
journal = {Czechoslovak Mathematical Journal},
keywords = {differential equation with quasiderivatives; proper solution; oscillatory solution; unbounded solution},
language = {eng},
number = {2},
pages = {245-259},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic behaviour of oscillatory solutions of $n$-th order differential equations with quasiderivatives},
url = {http://eudml.org/doc/30362},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Bartušek, Miroslav
TI - Asymptotic behaviour of oscillatory solutions of $n$-th order differential equations with quasiderivatives
JO - Czechoslovak Mathematical Journal
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 2
SP - 245
EP - 259
AB - Sufficient conditions are given under which the sequence of the absolute values of all local extremes of $y^{[i]}$, $i\in \lbrace 0,1,\dots , n-2\rbrace $ of solutions of a differential equation with quasiderivatives $y^{[n]}=f(t,y^{[0]},\dots , y^{[n-1]})$ is increasing and tends to $\infty $. The existence of proper, oscillatory and unbounded solutions is proved.
LA - eng
KW - differential equation with quasiderivatives; proper solution; oscillatory solution; unbounded solution
UR - http://eudml.org/doc/30362
ER -

References

top
  1. Monotonicity theorem for second-order non-linear differential equations, Arch. Math. XVI (1980), no. 3, 127–136. (1980) 
  2. The asymptotic behaviour of solutions of the differential equation of the third order, Arch. Math. XX (1984), no. 3, 101–112. (1984) MR0784861
  3. The asymptotic behaviour of oscillatory solutions of the equation of the fourth Order, Arch. Math. 21 (1985), no. 2, 93–104. (1985) MR0817551
  4. On oscillatory solution of the differential equation of the n -th order, Arch. Math. 22 (1986), no. 3, 145–156. (1986) MR0868130
  5. Asymptotic Properties of Oscillatory Solutions of Differential Equations of the n -th Order, FOLIA FSN Univ. Masaryk. Brunensis, Math. 3, Masaryk Univ. Brno, 1992. (1992) MR1271586
  6. 10.1007/BF02256721, Georgian Math. J. 3 (1996), no. 4, 301–314. (1996) MR1397813DOI10.1007/BF02256721
  7. Monotone and oscillatory solutions of a class of nonlinear differential equations, Math. Čas. 19 (1969), no. 3, 169–187. (1969) Zbl0271.34045MR0304773
  8. 10.1007/BF02023866, Acta Math. Acad. Sci. Hung. IX (1958), no. 1–2, 83–104. (1958) MR0095321DOI10.1007/BF02023866
  9. Some properties of oscillatory solutions of certain differential equations of second order, Ann. Soc. Math. Polonae XI (1967), 39–48. (1967) Zbl0166.35002MR0218668
  10. 10.1007/BF01897153, Acta Math. Acad. Sci. Hung. 15 (1964), no. 3–4, 449–456. (1964) MR0176155DOI10.1007/BF01897153
  11. On certain properties of oscillatory solutions of the second order nonlinear differential equation (Polish), Fasc. Math. 4 (1969), 57–64. (1969) 
  12. Some Singular Boundary-Value Problems for Ordinary Differential Equations, Izd. Tbiliss. Univ., Tbilisi, 1975. (Russian) (1975) MR0499402
  13. Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Nauka, Moscow, 1990. (Russian) (1990) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.