Existence and sharp asymptotic behavior of positive decreasing solutions of a class [4pt] of differential systems with power-type nonlinearities

Jaroslav Jaroš; Kusano Takaŝi

Archivum Mathematicum (2014)

  • Volume: 050, Issue: 3, page 131-150
  • ISSN: 0044-8753

Abstract

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The system of nonlinear differential equations x ' + p 1 ( t ) x α 1 + q 1 ( t ) y β 1 = 0 , y ' + p 2 ( t ) x α 2 + q 2 ( t ) y β 2 = 0 , A is under consideration, where α i and β i are positive constants and p i ( t ) and q i ( t ) are positive continuous functions on [ a , ∞ ) . There are three types of different asymptotic behavior at infinity of positive solutions ( x ( t ) , y ( t ) ) of (). The aim of this paper is to establish criteria for the existence of solutions of these three types by means of fixed point techniques. Special emphasis is placed on those solutions with both components decreasing to zero as t → ∞ , which can be analyzed in detail in the framework of regular variation.

How to cite

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Jaroš, Jaroslav, and Takaŝi, Kusano. "Existence and sharp asymptotic behavior of positive decreasing solutions of a class [4pt] of differential systems with power-type nonlinearities." Archivum Mathematicum 050.3 (2014): 131-150. <http://eudml.org/doc/261951>.

@article{Jaroš2014,
abstract = {The system of nonlinear differential equations \begin\{equation*\} x^\{\prime \} + p\_1(t)x^\{\alpha \_1\} + q\_1(t)y^\{\beta \_1\} = 0\,, \qquad y^\{\prime \} + p\_2(t)x^\{\alpha \_2\} + q\_2(t)y^\{\beta \_2\} = 0\,, A \end\{equation*\} is under consideration, where $\alpha _i$ and $\beta _i$ are positive constants and $p_i(t)$ and $q_i(t)$ are positive continuous functions on $[a,\infty )$. There are three types of different asymptotic behavior at infinity of positive solutions $(x(t),y(t))$ of (). The aim of this paper is to establish criteria for the existence of solutions of these three types by means of fixed point techniques. Special emphasis is placed on those solutions with both components decreasing to zero as $t \rightarrow \infty $, which can be analyzed in detail in the framework of regular variation.},
author = {Jaroš, Jaroslav, Takaŝi, Kusano},
journal = {Archivum Mathematicum},
keywords = {systems of nonlinear differential equations; positive solutions; asymptotic behavior; regularly varying functions; systems of nonlinear differential equations; positive solutions; asymptotic behavior; regularly varying functions},
language = {eng},
number = {3},
pages = {131-150},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Existence and sharp asymptotic behavior of positive decreasing solutions of a class [4pt] of differential systems with power-type nonlinearities},
url = {http://eudml.org/doc/261951},
volume = {050},
year = {2014},
}

TY - JOUR
AU - Jaroš, Jaroslav
AU - Takaŝi, Kusano
TI - Existence and sharp asymptotic behavior of positive decreasing solutions of a class [4pt] of differential systems with power-type nonlinearities
JO - Archivum Mathematicum
PY - 2014
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 050
IS - 3
SP - 131
EP - 150
AB - The system of nonlinear differential equations \begin{equation*} x^{\prime } + p_1(t)x^{\alpha _1} + q_1(t)y^{\beta _1} = 0\,, \qquad y^{\prime } + p_2(t)x^{\alpha _2} + q_2(t)y^{\beta _2} = 0\,, A \end{equation*} is under consideration, where $\alpha _i$ and $\beta _i$ are positive constants and $p_i(t)$ and $q_i(t)$ are positive continuous functions on $[a,\infty )$. There are three types of different asymptotic behavior at infinity of positive solutions $(x(t),y(t))$ of (). The aim of this paper is to establish criteria for the existence of solutions of these three types by means of fixed point techniques. Special emphasis is placed on those solutions with both components decreasing to zero as $t \rightarrow \infty $, which can be analyzed in detail in the framework of regular variation.
LA - eng
KW - systems of nonlinear differential equations; positive solutions; asymptotic behavior; regularly varying functions; systems of nonlinear differential equations; positive solutions; asymptotic behavior; regularly varying functions
UR - http://eudml.org/doc/261951
ER -

References

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