### Comparison results and existence of bounded solutions to strongly nonlinear second order differential equations

DOI: http://dx.doi.org/10.12775/TMNA.2009.031

#### Abstract

We investigate the existence of bounded solutions on the whole real

line of the following strongly non-linear non-autonomous

differential equation

$$

(a(x(t))x'(t))'= f(t,x(t),x'(t)) \quad \text{a.e } t\in \mathbb R

\tag \text{\rm E}

$$

where $a(x)$ is a

generic continuous positive function, $f$ is a Carathéodory

right-hand side.

We get existence results by combining the upper and lower-solutions

method to fixed-point techniques. We also provide operative

comparison criteria ensuring the well-ordering of pairs of upper and

lower-solutions.

line of the following strongly non-linear non-autonomous

differential equation

$$

(a(x(t))x'(t))'= f(t,x(t),x'(t)) \quad \text{a.e } t\in \mathbb R

\tag \text{\rm E}

$$

where $a(x)$ is a

generic continuous positive function, $f$ is a Carathéodory

right-hand side.

We get existence results by combining the upper and lower-solutions

method to fixed-point techniques. We also provide operative

comparison criteria ensuring the well-ordering of pairs of upper and

lower-solutions.

#### Keywords

Nonlinear ordinary differential operator; bounded solutions; non-compact interval; upper and lower solutions; comparison result

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