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

Research paper by **Xiao Ma, Daxiong Piao, Yiqian Wang**

Indexed on: **22 Jan '13**Published on: **22 Jan '13**Published in: **Mathematics - Dynamical Systems**

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#### 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 $(\phi_p(x'))'+a\phi_p(x^+)-b\phi_p(x^-)=G_x(x,t)+f(t)$,
where$x^+=\max (x,0)$,$x^- =\max(-x,0)$,$\phi_p(s)=|s|^{p-2}s$,$p\geq2$, $a $
and $b$ are positive constants $(a\not=b)$, the perturbation $f(t)\in {\cal
C}^{23}(\RR/2\pi_p \ZZ)$, the oscillating term $G\in {\cal
C}^{21}(\RR\times\RR/2\pi_p \ZZ)$,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 $\frac{\pa \hat{G}}{\pa x}=G$.