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2025, Vol. 45 No. 2   Published date: 28 June 2025
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• On Hilbert's Sixteenth Problem

  LI Chengzhi

  2025, 45(2): 1-21. doi: 10.3969/j.issn.1006-8074.2025.02.001

  Abstract ( 739 )   PDF (1563KB) ( 274 )   　　

  This paper aims to introduce the second part of Hilbert's 16th problem, including its formulation, research history, current developments and prospects, as well as the challenges involved.

• The Tamed Euler Method for Random Periodic Solution of Semilinear SDEs with One-sided Lipschitz Coefficient

  GUO Yujia , NIU Yuanling

  2025, 45(2): 22-39. doi: 10.3969/j.issn.1006-8074.2025.02.002

  Abstract ( 726 )   PDF (288KB) ( 120 )   　　

   This paper aims to investigate the tamed Euler method for the random periodic solution of semilinear SDEs with one-sided Lipschitz coefficient. We introduce a novel approach to analyze mean-square error bounds of the novel schemes, without relying on a priori high-order moment bound of the numerical approximation. The expected order-one mean square convergence is attained for the proposed scheme.Moreover, a numerical example is presented to verify our theoretical analysis.

• Property  (ω)  for Bounded Linear Operators and Its Stability

  DAI Lei, YI Jialu, CAO Xiaohong

  2025, 45(2): 40-52. doi: 10.3969/j.issn.1006-8074.2025.02.003

  Abstract ( 716 )   PDF (177KB) ( 105 )   　　

  In this paper, using the property of uniform Fredholm non-positive index of bounded linear operators, we give criteria for operators and their functions to possess property $(\omega)$, and several equivalent conditions for the stability of property $(\omega)$, and investigate the relationship between the stability of property $(\omega)$ and the $(\omega)$-property of operator functions.

• Unimodality of Independence Polynomials of Rooted Products of a Kind of Tree

  XIE Yuan, ZHANG Xiaoqian

  2025, 45(2): 53-75. doi: 10.3969/j.issn.1006-8074.2025.02.004

  Abstract ( 803 )   PDF (1830KB) ( 119 )   　　

  In 1987, Alavi, Malde, Schwenk and Erd\H{o}s conjectured that the independence polynomial of any tree or forest is unimodal. Although many researchers have been attracted by it, it is still open. Inspired by this conjecture, in this paper, we prove that rooted products of some trees preserve real-rootedness of independence polynomials. In particular, we can obtain that their independence polynomials are unimodal and log-concave.

• Blow-up Phenomenon for a Class of Logarithmic Viscoelastic Equations with Delay and Nonlocal Term under Acoustic Boundary Conditions

  DUAN Jisong, XIANG Changyong

  2025, 45(2): 76. doi: 10.3969/j.issn.1006-8074.2025.02.005

  Abstract ( 854 )   PDF (205KB) ( 260 )   　　

  In this paper, we investigate the blow-up phenomenon for a class of logarithmic viscoelastic equations with delay and nonlocal terms under acoustic boundary conditions. Using the energy method, we prove that nontrivial solutions with negative initial energy will blow up in finite time, and provide an upper bound estimate for the blow-up time. Additionally, we also derive a lower bound estimate for the blow-up time. 

• Study on a Respiratory Syncytial Virus SIRS Model with Age Structure

  LIN Caihong, GAO Shukun, WANG Wencong, ZHANG Long

  2025, 45(2): 93-109. doi: 10.3969/j.issn.1006-8074.2025.02.006

  Abstract ( 803 )   PDF (659KB) ( 106 )   　　

  In this paper, we study the epidemic model of respiratory syncytial virus SIRS with age structure. Firstly, the basic reproduction number $R_{0}$ of the model is calculated and the positivity and ultimate boundedness of the solution to the model under initial conditions are proven. Secondly, it is proven that when $R_{0}<1$, the disease-free equilibrium is locally and globally asymptotically stable;~and when $R_{0}>1$, the disease is uniformly persistent and there is at least a positive equilibrium. Finally, the effectiveness of the theoretical results is demonstrated by numerical simulation, and the impact of vaccination on disease transmission is predicted.

• Extinction and Optimal Control of Stochastic Epidemic Model with Multiple Vaccinations and Time Delay

  YANG Rujie, QIU Hong, JU Xuewei

  2025, 45(2): 110-121. doi: 10.3969/j.issn.1006-8074.2025.02.007

  Abstract ( 785 )   PDF (319KB) ( 112 )   　　

  In this paper, based on the SVIQR model we develop a stochastic epidemic model with multiple vaccinations and time delay. Firstly, we prove the existence and uniqueness of the global positive solution of the model, and construct suitable functions to obtain sufficient conditions for disease extinction. Secondly, in order to effectively control the spread of the disease, appropriate control strategies are formulated by using optimal control theory. Finally, the results are verified by numerical simulation.