数学季刊

苑天雨, 肖现涛

2026, 41(3): 221-243. doi:10.13371/j.cnki.chin.q.j.m.2026.03.001

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This paper investigates the distributed consensus optimization problem with coupled constraints, where the objective function is the sum of the local objective functions of each agent, and the decision variables are subject to coupled nonlinear constraints. All agents exchange information with one another to ensure consensus on the decision variables and achieve optimality. To address this problem, a distributed proximal method of multipliers is proposed by combining the centralized proximal method of multipliers and the distributed gradient-tracking algorithm. It is shown that this algorithm converges to the optimal solutions of both primal and dual problems for any constant step size. Notably, the algorithm can handle general coupled constraints, requiring only closedness and convexity assumptions.

非局部反应-扩散-平流模型中两种竞争性浮游植物种群的动力学研究

郑露, 彭亚红

2026, 41(3): 244-262. doi:10.13371/j.cnki.chin.q.j.m.2026.03.002

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In a vertical water column, the movements of obligately photosynthetic phytoplankton are not only affected by turbulence in the vertical direction, but also by light exposure. Considering the influence of light on phytoplankton growth, we propose a nonlocal reaction-diffusion-advection model for the competition between mixotrophic
and obligately photosynthetic phytoplankton in this paper. We first prove the stability of semitrivial steady state of system by using eigenvalue theory. Then we prove the existence and boundedness of the solution of system and establish a priori estimates of the solution for the steady state system. Our research indicates successful invasion is determined by the interplay of physiological traits and movement strategies: species with extreme growth
advantages can invade unconditionally, while those with intermediate competitiveness must adopt a vertical migration strategy that either contrasts with the dominant species or enables precise positioning at an optimal depth.

关于数值有效切丛凡诺流形的一个注记

杨奇林

2026, 41(3): 263-268. doi:10.13371/j.cnki.chin.q.j.m.2026.03.003

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In this note we prove that any toric Fano manifold with nef tangent bundle is a product of projective spaces. In particular, it implies that Campana-Peternell conjecture holds for toric manifolds.

J统计de Rham Hodge 算子及相关的 Kastler-Kalau-Walze 类型定理

包开花, 程玉梅, 孙艳红

2026, 41(3): 269-298. doi:10.13371/j.cnki.chin.q.j.m.2026.03.004

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In this paper, we define the basic notions of the J-statistical de Rham Hodge operator and derive its Lichnerowicz-type formula on compact Riemannian manifolds without boundary, and establish the Kastler-Kalau-Walze type theorems related to this operator on four-dimensional and six-dimensional almost product Riemannian manifolds with boundary.

一种求解具有退化核的Hammerstein积分方程精确解的新方法

胡金豆, 闫丽娜, 黄琼敖

2026, 41(3): 299-306. doi:10.13371/j.cnki.chin.q.j.m.2026.03.005

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This paper presents a novel analytical method for obtaining exact solutions of Hammerstein integral equations with degenerate kernels. Subsequently, this approach is systematically extended to equations with general kernels, yielding a convergent numerical approximation whose theoretical analysis establishes sufficient conditions for convergence and provides explicit error bounds. Numerical experiments are conducted to demonstrate the efficiency and accuracy of the proposed method, showing favorable performance compared to existing techniques.

一类完全非线性中立型差分系统周期解的存在唯一性

肖雨茹, 潘晋原, 陈桂玲

2026, 41(3): 307-322. doi:10.13371/j.cnki.chin.q.j.m.2026.03.006

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In this paper, we study the existence and uniqueness of periodic solutions of a totally nonlinear neutral difference system expressed as △x(n)=A(n)h (x(n−τ(n)))+△Q(n,x(n−g(n)))+G(n,x(n), x (n−g(n))) . We first utilize the fundamental matrix solution of △x(n)=A(n)x (n) to transform the aforementioned system. Subsequently, we construct suitable mappings and apply Krasnoselskii’s fixed point theorem to demonstrate the existence of periodic solutions for this nonlinear neutral difference system. Moreover, by means of the contraction mapping
principle, we study the uniqueness of periodic solutions. Unlike linear neutral systems, our model introduces complete nonlinearity, presenting new analytical challenges.

分数阶微分方程边值问题正解存在性

沈春芳

2026, 41(3): 323-330. doi:10.13371/j.cnki.chin.q.j.m.2026.03.007

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In this paper, we investigate positive solutions for a class of boundary value problem of nonlinear differential equation with fractional order derivative. We give sufficient conditions on nonlinear term and the parameter such that the boundary value problem has no positive solutions. Some examples are presented to illustrate the main results.

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