On the asymptotic behavior at infinity of solutions to quasi-linear differential equations

Irina Astashova

Mathematica Bohemica (2010)

  • Volume: 135, Issue: 4, page 373-382
  • ISSN: 0862-7959

Abstract

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Sufficient conditions are formulated for existence of non-oscillatory solutions to the equation y ( n ) + ∑ j = 0 n - 1 a j ( x ) y ( j ) + p ( x ) | y | k sgn y = 0 with n ≥ 1 , real (not necessarily natural) k > 1 , and continuous functions p ( x ) and a j ( x ) defined in a neighborhood of + ∞ . For this equation with positive potential p ( x ) a criterion is formulated for existence of non-oscillatory solutions with non-zero limit at infinity. In the case of even order, a criterion is obtained for all solutions of this equation at infinity to be oscillatory. Sufficient conditions are obtained for existence of solution to this equation which is equivalent to a polynomial.

How to cite

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Astashova, Irina. "On the asymptotic behavior at infinity of solutions to quasi-linear differential equations." Mathematica Bohemica 135.4 (2010): 373-382. <http://eudml.org/doc/197233>.

@article{Astashova2010,
abstract = {Sufficient conditions are formulated for existence of non-oscillatory solutions to the equation \[y^\{(n)\}+\sum \_\{j=0\}^\{n-1\}a\_j(x)y^\{(j)\}+p(x)|y|^k \mathop \{\rm sgn\} y =0\] with $ n\ge 1$, real (not necessarily natural) $k>1$, and continuous functions $p(x)$ and $a_j(x)$ defined in a neighborhood of $+\infty $. For this equation with positive potential $p(x)$ a criterion is formulated for existence of non-oscillatory solutions with non-zero limit at infinity. In the case of even order, a criterion is obtained for all solutions of this equation at infinity to be oscillatory. Sufficient conditions are obtained for existence of solution to this equation which is equivalent to a polynomial.},
author = {Astashova, Irina},
journal = {Mathematica Bohemica},
keywords = {quasi-linear ordinary differential equation of higher order; existence of non-oscillatory solution; oscillatory solution; quasi-linear ordinary differential equation; higher order; existence; non-oscillatory solution; oscillatory solution},
language = {eng},
number = {4},
pages = {373-382},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the asymptotic behavior at infinity of solutions to quasi-linear differential equations},
url = {http://eudml.org/doc/197233},
volume = {135},
year = {2010},
}

TY - JOUR
AU - Astashova, Irina
TI - On the asymptotic behavior at infinity of solutions to quasi-linear differential equations
JO - Mathematica Bohemica
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 135
IS - 4
SP - 373
EP - 382
AB - Sufficient conditions are formulated for existence of non-oscillatory solutions to the equation \[y^{(n)}+\sum _{j=0}^{n-1}a_j(x)y^{(j)}+p(x)|y|^k \mathop {\rm sgn} y =0\] with $ n\ge 1$, real (not necessarily natural) $k>1$, and continuous functions $p(x)$ and $a_j(x)$ defined in a neighborhood of $+\infty $. For this equation with positive potential $p(x)$ a criterion is formulated for existence of non-oscillatory solutions with non-zero limit at infinity. In the case of even order, a criterion is obtained for all solutions of this equation at infinity to be oscillatory. Sufficient conditions are obtained for existence of solution to this equation which is equivalent to a polynomial.
LA - eng
KW - quasi-linear ordinary differential equation of higher order; existence of non-oscillatory solution; oscillatory solution; quasi-linear ordinary differential equation; higher order; existence; non-oscillatory solution; oscillatory solution
UR - http://eudml.org/doc/197233
ER -

References

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