标题： 振荡链序列的构建与表征 显示英文标题

标题： Construction and Characterization of Oscillatory Chain Sequences

摘要： 本文开始对$\frac{1}{4}$-振荡链序列$\{a_n\}$进行理论研究，将 Szwarc 对非振荡链\cite{Sz94, Sz98, Sz02, Sz03}的经典框架推广到围绕$\frac{1}{4}$波动的序列。 我们证明了临界映射$f(x)=1-\frac{1}{4x}$的不动点存在性，并建立了将振荡行为与参数序列$\{g_n\}$联系起来的收敛性质。 通过一个必要充分条件提供了完整的表征，并以显式解$a_n=\frac{1}{4}\left(1+(-1)^{n}\varepsilon_{n}\right)$作为例子。 关键的是，我们构造了振荡链序列，使得级数$\sum_{n=1}^{\infty} \left(a_n - \frac{1}{4}\right)$发散，从而违反了Chihara的猜测界限。 显示英文摘要

摘要： This paper initiates a theoretical investigation of $\frac{1}{4}$-oscillatory chain sequences $\{a_n\}$, generalizing Szwarc's classical framework for non-oscillatory chains \cite{Sz94, Sz98, Sz02, Sz03} to sequences fluctuating around $\frac{1}{4}$. We prove the existence of a fixed point for the critical map $f(x)=1-\frac{1}{4x}$ and establish convergence properties linking oscillatory behavior to parameter sequences $\{g_n\}$. A complete characterization is provided via a necessary and sufficient condition, exemplified by explicit solutions $a_n=\frac{1}{4}\left(1+(-1)^{n}\varepsilon_{n}\right)$. Crucially, we construct oscillatory chain sequences for which the series $\sum_{n=1}^{\infty} \left(a_n - \frac{1}{4}\right)$ diverges, thus violating Chihara's conjectured bound.