Global Kneser solutions to nonlinear equations with indefinite weight

Zuzana Došlá; Mauro Marini; Serena Matucci

doi:10.3934/dcdsb.2018252

Article Contents

2018, Volume 23, Issue 8: 3297-3308. Doi: 10.3934/dcdsb.2018252

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Global Kneser solutions to nonlinear equations with indefinite weight

1.
Department of Mathematics and Statistics, Masaryk University, CZ-61137 Brno, Czech Republic
2.
Department of Mathematics and Computer Sciences "Ulisse Dini", University of Florence, I-50139 Florence, Italy

* Corresponding author: Serena Matucci
* Corresponding author: Serena Matucci

Received: February 28, 2018

Early access: August 2018

Published: August 2018

The first author is supported by the grant GA 17-03224S of the Czech Grant Agency. The third author is partially supported by Gnampa, National Institute for Advanced Mathematics (INdAM)

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Abstract

The paper deals with the nonlinear differential equation

$\bigl(a(t)\Phi(x^{\prime})\bigr)^{\prime}+b(t)F(x)=0,\ \ \ t\in\lbrack1,\infty),$

in the case when the weight $b$ has indefinite sign. In particular, the problem of the existence of the so-called globally positive Kneser solutions, that is solutions $x$ such that $x(t)>0, {{x}'}(t)<0$ on the whole closed interval $[1,\infty )$, is considered. Moreover, conditions assuring that these solutions tend to zero as $t\rightarrow\infty$ are investigated by a Schauder's half-linearization device jointly with some properties of the principal solution of an associated half-linear differential equation. The results cover also the case in which the weight $b$ is a periodic function or it is unbounded from below.

Keywords:

Second order nonlinear differential equations,
boundary value problems on infinite intervals,
global positive solutions,
half-linear equations,
disconjugacy,
principal solution.

Mathematics Subject Classification: Primary: 34B40; Secondary: 34B18.

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