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October  2018, 23(8): 3297-3308. doi: 10.3934/dcdsb.2018252

## 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

Received  March 2018 Published  October 2018 Early access  August 2018

Fund Project: 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).

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.
Citation: Zuzana Došlá, Mauro Marini, Serena Matucci. Global Kneser solutions to nonlinear equations with indefinite weight. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3297-3308. doi: 10.3934/dcdsb.2018252
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