Table of Content

30 September 2026, Volume 41 Issue 3

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A Gradient-Tracking Based Proximal Method of Multipliers for Distributed Optimization with Coupled Constraints and Consensus Scheme

YUAN Tian-yu, XIAO Xian-tao

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.


Dynamics of Two Competing Phytoplankton Species in a Nonlocal Reaction-Diffusion-Advection Mode

ZHENG Lu, PENG Ya-hong

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.

A Note on Toric Fano Manifolds with Nef Tangent Bundles

YANG Qi-lin

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-Statistical de Rham Hodge Operator and Related Kastler-Kalau-Walze Type Theorems

BAO Kai-Hua, CHENG Yu-mei, SUN Yan-hong

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.

A Novel Method for Exact Solutions of Hammerstein Integral Equations with Degenerate Kernels 

HU Jin-dou1, YAN Li-na1, HUANG Qiong-ao 1, 2

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.

Existence and Uniqueness of Periodic Solutions of a Totally Nonlinear Neutral Difference System

XIAO Yu-ru, PAN Jin-yuan, CHEN Gui-ling

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.

Existence Results of Positive Solutions of a Boundary Value Problem for Fractional Differential Equation

SHEN Chun-fang

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.