arXiv subjects

Yiqian Wang

Publications and source records attributed to Yiqian Wang.

22 records · Page 2Linked to original sources

arXiv2013

Boundedness of semilinear Duffing equations at resonance with oscillating nonlinearities

In this paper, we prove the boundedness of all the solutions for the equation $\ddot{x}+n^2x+g(x)+ψ'(x)=p(t)$ with the Lazer-Leach condition on $g$ and $p$, where $n\in \mathbb{N^+}$, $p(t)$ and $ψ'(x)$ are periodic and $g(x)$ is bounded. For the critical situation that $\big |\int_0^{2π}p(t)e^{int}dt \big|=2\big|g(+\infty)-g(-\infty)\big|$, we also prove a sufficient and necessary condition for the boundedness if $ψ'(x)\equiv0$.

math.DS↗

arXiv2013

Examples of Discontinuity of Lyapunov Exponent in Smooth Quasi-Periodic Cocycles

We study the regularity of the Lyapunov exponent for quasi-periodic cocycles $(T_ω, A)$ where $T_ω$ is an irrational rotation $x\to x+ 2πω$ on $\SS^1$ and $A\in {\cal C}^l(\SS^1, SL(2,\mathbb{R}))$, $0\le l\le \infty$. For any fixed $l=0, 1, 2, \cdots, \infty$ and any fixed $ω$ of bounded-type, we construct $D_{l}\in {\cal C}^l(\SS^1, SL(2,\mathbb{R}))$ such that the Lyapunov exponent is not continuous at $D_{l}$ in ${\cal C}^l$-topology. We also construct such examples in a smaller Schrödinger class.

math.DS↗

arXiv2013

Quasi-periodic solutions for p-Laplacian equations with jumping nonlinearity and unbounded potential terms

In this paper, we are concerned with the boundedness of all the solutions for a kind of second order differential equations with p-Laplacian term $(ϕ_p(x'))'+aϕ_p(x^+)-bϕ_p(x^-)+f(x)=e(t)$, where $x^+=\max (x,0)$, $x^- =\max(-x,0)$, $ϕ_p(s)=|s|^{p-2}s$, $p\geq2$, $a$ and $b$ are positive constants $(a\not=b)$, and satisfy $\frac{1}{a^{\frac{1}{p}}}+\frac{1}{b^{\frac{1}{p}}}=2ω^{-1} $,where $ω\in \RR^+ \backslash \QQ$, the perturbation $f$ is unbounded, $e(t)\in {\cal C}^{6}$ is is a smooth $2π_p$-periodic function on $t$, where $π_p=\frac{2π(p-1)^{\frac{1}{p}}}{p\sin\fracπ{p}}$.

math.DS↗

arXiv2013

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

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 $(ϕ_p(x'))'+aϕ_p(x^+)-bϕ_p(x^-)=G_x(x,t)+f(t)$, where$x^+=\max (x,0)$,$x^- =\max(-x,0)$,$ϕ_p(s)=|s|^{p-2}s$,$p\geq2$, $a $ and $b$ are positive constants $(a\not=b)$, the perturbation $f(t)\in {\cal C}^{23}(\RR/2π_p \ZZ)$, the oscillating term $G\in {\cal C}^{21}(\RR\times\RR/2π_p \ZZ)$,where $π_p=\frac{2π(p-1)^{\frac{1}{p}}}{p\sin\fracπ{p}},$ 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 $\frac{\pa \hat{G}}{\pa x}=G$.

math.DS↗