On the asymptotic behavior of solutions of \(v''(x)+q(x)\sin v(x)=0\) (Q1362811)

The authors study the asymptotic behavior of the solution \(v(x)\) of the initial value problem \[ v''(x)+ q(x)\sin v(x)=0,\quad v'(0)=0,\quad v(0)=a,\quad a\in(0,\pi),\tag{1} \] where \(q(0)\geq 0\) and \(q'(x)>0\) for all \(x\in(0,+\infty)\); if \(q\in C^1[0,+\infty)\cap C^2(0,+\infty)\), \(\lim_{x\to+\infty} q(x)\) \(f+ \infty\), there exists \(x_0\geq 0\) such that \[ y''(x)q(x)-{5\over 4}(q'(x))^2\leq 0 \] for all \(x\in[x_0,+ \infty)\) and \[ \lim _{x\to+\infty} (q(x))^{-{1\over 2}}(q'(x))^{- 1}q''(x)= 0. \] Then the solution \(v\) of (1) satisfies \[ \lim_{x\to+\infty} v(x,a)=0 \] for all \(a\in(0,+\infty)\); moreover, the zeros of \(v\), denoted by \(x_1<x_2<\cdots< x_k<\cdots\), satisfy \[ \lim_{k\to+\infty} |x_k- x_{k-1}|= 0. \] The problem (1) arises from a mathematical model describing large deformations of a nonuniform cantilever.