<abstract xmlns="http://www.w3.org/1999/xhtml"> This paper is concerned with asymptotic analysis of moderately growing solutions of the third-order differential equation with singular nonlinerity((((x′)α₁*)′)α₂*)′ +q(t)x-β= 0; (A) where α1, α2 and β are positive constants and σ : [α;∞)→(0;∞) is a continuous regularly varying function of index σ, α > 0 and uγ* = |u|γ sgnu. An application of the theory of regular variation allows us to establish necessary and sufficient conditions for the existence of regularly varying solutions of (A) which are moderately growing and to acquire precise information about the asymptotic behavior at infinity of these solutions. The Schauder-Tychonoff fixed point technique is used. </abstract>

This paper is concerned with asymptotic analysis of moderately growing solutions of the third-order differential equation with singular nonlinerity((((x′)α₁*)′)α₂*)′ +q(t)x-β= 0; (A) where α1, α2 and β are positive constants and σ : [α;∞)→(0;∞) is a continuous regularly varying function of index σ, α > 0 and uγ* = |u|γ sgnu. An application of the theory of regular variation allows us to establish necessary and sufficient conditions for the existence of regularly varying solutions of (A) which are moderately growing and to acquire precise information about the asymptotic behavior at infinity of these solutions. The Schauder-Tychonoff fixed point technique is used.

Moderately growing solutions of third-order differential equations with a singular nonlinearity and regularly varying coefficients

Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava

Journal of Applied Mathematics, Statistics and Informatics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

This paper is concerned with asymptotic analysis of moderately growing solutions of the third-order differential equation with singular...

Moderately growing solutions of third-order differential equations with a singular nonlinearity and regularly varying coefficients

Published Online: Jun 27, 2015

Page range: 33 - 61

DOI: https://doi.org/10.1515/jamsi-2015-0003

Keywords
Third order nonlinear differential equation, singular nonlinearity, positive solution, moderately growing solution, asymptotic behavior, regularly varying functions

© Faculty of Natural Sciences, University of Saint Cyril and Metodius

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Moderately growing solutions of third-order differential equations with a singular nonlinearity and regularly varying coefficients

Published Online: Jun 27, 2015

Page range: 33 - 61

DOI: https://doi.org/10.1515/jamsi-2015-0003

KeywordsThird order nonlinear differential equation, singular nonlinearity, positive solution, moderately growing solution, asymptotic behavior, regularly varying functions

© Faculty of Natural Sciences, University of Saint Cyril and Metodius

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Third order nonlinear differential equation, singular nonlinearity, positive solution, moderately growing solution, asymptotic behavior, regularly varying functions