Unbounded Solutions of Asymmetric Oscillator

Abstract

We obtain sufficient conditions for the existence of unbounded solutions of the following nonlinear differential equation ${({\phi }_p(x^{\prime }))}^{\prime }+(p-1)[\alpha {\phi }_p(x^{+})-\beta {\phi }_p(x^{-})]=(p-1)f(t,x,x^{\prime })$, where ${\phi }_p(u)={|u|}^{p-2}u,p>1,x^{+}=\text{max}\{x,0\},x^{-}=\text{max}\{-x,0\},\alpha ,\beta$ are positive constants, and $f$ is continuous, bounded, and $T$-periodic in $t$ for some $T>0$.