Construction and Characterization of Oscillatory Chain Sequences
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918495101386752 |
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| author | Dai, Zejun Piao, Daxiong Qiao, Jinglai |
| author_facet | Dai, Zejun Piao, Daxiong Qiao, Jinglai |
| contents | 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, demonstrating fundamentally different behavior outside the hypothesis $a_n \ge \frac{1}{4}$ required by Chihara's bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07653 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Construction and Characterization of Oscillatory Chain Sequences Dai, Zejun Piao, Daxiong Qiao, Jinglai Dynamical Systems Spectral Theory 42C05, 47B39 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, demonstrating fundamentally different behavior outside the hypothesis $a_n \ge \frac{1}{4}$ required by Chihara's bound. |
| title | Construction and Characterization of Oscillatory Chain Sequences |
| topic | Dynamical Systems Spectral Theory 42C05, 47B39 |
| url | https://arxiv.org/abs/2508.07653 |