Article Contents
Article Contents

# Global Kneser solutions to nonlinear equations with indefinite weight

• * Corresponding author: Serena Matucci

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.

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

 Citation:

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