Boundedness for Second Order Differential Equations with Jumping p-Laplacian and an Oscillating Term

Abstract

In this paper, we are concerned with the boundedness of all the solutions for a kind of second order differential equations with p-Laplacian and an oscillating term $({\varphi }_p(x^{\prime }){)}^{\prime }+a{\varphi }_p(x^{+})-b{\varphi }_p(x^{-})=G_x(x,t)+f(t)$, where$x^{+}=\max (x,0)$,$x^{-}=\max (-x,0)$,${\varphi }_p(s)=|s{|}^{p-2}s$,$p\ge 2$, $a$ and $b$ are positive constants $(a\ne b)$, the perturbation $f(t)\in {\cal C}^{23}(\RR/2π_p \ZZ)$, the oscillating term $G\in {\cal C}^{21}(\RR\times\RR/2π_p \ZZ)$,where ${\pi }_p=\frac{2\pi (p-1{)}^{\frac{1}{p}}}{p\sin \frac{\pi }{p}},$ and $G(x,t)$ satisfies $\label{G} |D_x^iD_t^jG(x,t)|\le C,\quad 0\le i+j\le 21,$ and $\label{hatG} |D_t^j\hat{G}|\le C,\quad 0\le j\le 21$ for some $C>0$, where $\hat{G}$ is some function satisfying $\frac{\pa \hat{G}}{\pa x}=G$.