Rectifiable oscillations in second-order half-linear differential equations
Rectifiable oscillations in second-order half-linear differential equations
复制标题
二阶半线性微分方程中的可整流振荡
DOI：
10.1007/s10231-008-0087-0
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发表时间：
2009
影响因子：
1
通讯作者：
J. Wong
中科院分区：
文献类型：
--
作者：
Mervan Pašić;J. Wong
摘要
将研究有限区间 I = (0,1] 上的二阶半线性微分方程 (H): $${(\Phi (y'))'+f(x)\Phi (y)=0}$$，其中 $${\Phi (u)=|u|^{p-2}u}$$ , p > 1 且 I, $${f\in 上的系数 f(x) > 0 C^{2}((0,1])}$$ 和 $${\lim_{x\rightarrow0}f(x)=\infty }$$ 。当 p = 2 时，方程 (H) 简化为谐振子方程 (P)：y′′ + f(x)y = 0。在本文中，我们研究 (H) 解的振荡，特别关注图的一些几何和分形性质$${G(y)=\{(x,y(x)):0\leq x\leq 1\}\subseteq {\bf {R}}^{2}}$$ 。我们为具有有限和无限弧长的图的振荡解建立必要且充分的积分标准。在 $${f(x)\sim \lambda x^{-\alpha}}$$、λ > 0、α > p 时，我们还确定分形。解 y(x) 的图 G(y) 的维数 最后，我们研究方程 (H) 所有解的导数的 Lp 不可积性。
Abstract
Second-order half-linear differential equation (H): $${(\Phi (y'))'+f(x)\Phi (y)=0}$$ on the finite interval I = (0,1] will be studied, where $${\Phi (u)=|u|^{p-2}u}$$ , p > 1 and the coefficient f(x) > 0 on I, $${f\in C^{2}((0,1])}$$ , and $${\lim_{x\rightarrow0}f(x)=\infty }$$ . In case when p = 2, the equation (H) reduces to the harmonic oscillator equation (P): y′′ + f(x)y = 0. In this paper, we study the oscillations of solutions of (H) with special attention to some geometric and fractal properties of the graph $${G(y)=\{(x,y(x)):0\leq x\leq 1\}\subseteq {\bf {R}}^{2}}$$ . We establish integral criteria necessary and sufficient for oscillatory solutions with graphs having finite and infinite arclength. In case when $${f(x)\sim \lambda x^{-\alpha}}$$, λ > 0, α > p, we also determine the fractal dimension of the graph G(y) of the solution y(x). Finally, we study the Lp nonintegrability of the derivative of all solutions of the equation (H).