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

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 Caratheodory 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.