#### Table of Content

26 April 2022, Volume 42 Issue 2
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 Classification of Calabi Hypersurfaces in ${\mathbb R}^5$ with Parallel Fubini-Pick Form Ruiwei Xu,Miaoxin Lei Acta mathematica scientia,Series A. 2022, 42 (2):  321-337.  Abstract ( 53 )   RICH HTML PDF (391KB) ( 68 )   The classifications of locally strongly convex equiaffine hypersurfaces (centroaffine hypersurfaces) with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Blaschke-Berwald metric (centroaffine metric) have been completed in the last decades. In [20], the authors studied Calabi hypersurfaces with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Calabi metric and classified 2 and 3-dimensional cases. In this paper, we extend such calssification results to 4-dimensional Calabi hypersurfaces in the affine space ${\mathbb R}^5$.
 On the Optimal Voronoi Partitions for Moran Measures on ${\Bbb R} ^{1}$ with Respect to the Geometric Mean Error Yi Cao Acta mathematica scientia,Series A. 2022, 42 (2):  338-352.  Abstract ( 25 )   RICH HTML PDF (395KB) ( 22 )   Let $E$ be a Moran set on ${{\Bbb R}} ^{1}$ associated with a bounded closed interval $J$ and two sequences $(n_{k})^{\propto}_{k=1}$ and ${\cal C}_{k}=((c_{k, j})_{j=1}^{n_{k}})_{k\geq 1}$ of numbers. Let $\mu$ be the Moran measure on $E$ determined by a sequence $({\cal P}_{k})_{k\geq1}$ of positive probability vectors. For every $n\geq 1$, let $C_{n}(\mu)$ denote the collection of all the $n$-optimal sets for $\mu$ with respect to the geometric mean error; let $\alpha_n\in C_n(\mu)$ and $\{P_{a}(\alpha_{n})\}_{a\in\alpha_{n}}$ be an arbitrary Voronoi partition with respect to $\alpha_n$. We prove that $\min\limits_{a\in\alpha_n}\mu (P_{a}(\alpha_{n})), \max\limits_{a\in\alpha_n}\mu (P_{a}(\alpha_{n}))\asymp n^{-1}.$ For each $a\in\alpha_{n}$, we show that the set $P_{a}(\alpha_{n})$ contains a closed interval of radius $d_{2}|P_{a}(\alpha_{n})\cap E|$ which is centered at $a$, where $d_{2}$ is a constant and $|B|$ denotes the diameter of a set $B\subset{{\Bbb R}} ^1$. Let $e_n(\mu)$ denote the $n$th geometric mean error for $\mu$ and $\hat{e}_n(\mu):=\log e_n(\mu)$. We show that $\hat{e}_n(\mu)-\hat{e}_{n+1}(\mu)\asymp n^{-1}$.
 Limit Circle Bifurcations of Polynomial Differential System with Two Parallel Switch Straight Lines Xueli Jiang,Xuan Deng,Qiuhao Wen,Yanqin Xiong Acta mathematica scientia,Series A. 2022, 42 (2):  353-364.  Abstract ( 19 )   RICH HTML PDF (378KB) ( 34 )   In this paper, the limit cycle bifurcation problem is investigated for a class of polynomial differential system with two parallel switch straight lines. We use the generalized first order Melnikov function and related qualitative theory of knowledge to export the algebraic and corresponding coefficients of expression, then, use coefficient of change to research generalized double homoclinic bifurcation, and get a lower bound of its ring number.
 Characterization of Optimality to Constrained Vector Equilibrium Problems via Approximate Subdifferential Shengxin Hua,Guolin Yu,Wenyan Han,Xiangyu Kong Acta mathematica scientia,Series A. 2022, 42 (2):  365-378.  Abstract ( 11 )   RICH HTML PDF (380KB) ( 16 )   In this paper, we study the optimality conditions and duality theorems for a class of Constrained Vector Equilibrium Problem (CVEP) with respect to approximate quasi weak efficient solutions. Firstly, a necessary optimality condition related to approximate subdifferential of approximation quasi weak efficient solution to problem (CVEP) is established. Secondly, a kind of generalized convexity, named pseudo quasi type-Ⅰ function, is introduced, and under its assumption, a sufficient optimality condition is also obtained. Finally, the generalized approximate Mond-Weir dual model of problem (CVEP) is presented, and the dual theorems between with the primal problem are established.
 Existence of Convex Solutions for a Discrete Mixed Boundary Value Problem with the Mean Curvature Operator Lei Duan,Tianlan Chen Acta mathematica scientia,Series A. 2022, 42 (2):  379-386.  Abstract ( 15 )   RICH HTML PDF (296KB) ( 21 )   In this paper, by using the fixed point theorem in cones, we discuss the existence of $\Delta[\phi(\Delta v(t-1))]=f(t, -v(t)), {\quad} t\in[2, T-1]_{{\Bbb Z}},$ $\Delta v(1)=0, {\quad} v(T)=0$ nontrivial convex solutions for a discrete mixed boundary value problem of mean curvature operator in Minkowski space, where $\phi(s)=\frac{s}{\sqrt{1-s^{2}}}, s\in(-1, 1),$ $[2, T-1]_{{\Bbb Z}}:=\{2, 3, \cdots, T-2,$ $T-1\},$ $T\geqslant4$ and $T\in{\Bbb N}^{\ast}$, the nonlinear term $f(t, u)$ is nonnegative and continuous, and singularity is allowed at $u=1$.
 Global Existence of Weak Solutions to the Quantum Navier-Stokes Equations Tong Tang,Cong Niu Acta mathematica scientia,Series A. 2022, 42 (2):  387-400.  Abstract ( 20 )   RICH HTML PDF (405KB) ( 20 )   In this paper, we proved the global existence of weak solutions to the quantum Navier-Stokes equations with non-monotone pressure. Motivated by the work of Antonelli-Spirito(Arch Ration Mech Anal, 2017, 255: 1161–1199) and Ducomet-Ne?asová-Vasseur (Z Angew Math Phys, 2010, 61: 479–491), we construct the suitable approximate system and obtain the corresponding compactness by B-D entropy estimate and Mellet-Vasseur inequality.
 Multiple Solutions to Logarithmic Kirchhoff Equations Die Hu,Qi Gao Acta mathematica scientia,Series A. 2022, 42 (2):  401-417.  Abstract ( 29 )   RICH HTML PDF (383KB) ( 27 )   We study a class of logarithmic Kirchhoff equations with two types of potentials, and obtain the existence of positive solutions and sign-changing solutions.
 Existence of Positive Ground State Solutions for a Class of Kirchhoff Type Problems with Critical Exponent Lei Ji,Jiafeng Liao Acta mathematica scientia,Series A. 2022, 42 (2):  418-426.  Abstract ( 16 )   RICH HTML PDF (346KB) ( 24 )   The following Kirchhoff type equations with critical exponent $\begin{eqnarray*}\left\{\begin{array}{ll}\displaystyle-\Big[a+b\Big(\int_{\Omega}|\nabla u|^2{\rm d}x\Big)^{m}\Big]\Delta u= |u|^{2^{\ast}-2}u+\lambda|u|^{q-2}u,\ & x \in \Omega, \\u=0, & x\in\partial\Omega,\end{array}\right.\end{eqnarray*}$ where $\Omega\subset\mathbb{R} ^{N}(N\geq3)$ is a smooth bounded domain with smooth boundary $\partial\Omega,$ $a, \lambda>0,$$b\geq0,$ $0 < m < \frac{2}{N-2}, 2 < q < 2m+2, 2^{\ast}=\frac{2N}{N-2}$ are parameters, are considered. By using the Moutain-Pass Theorem, the existence positive solutions is obtained. Moreover, the existence positive ground state solutions is obtained by the Nehari method. Our result completes and improves the recent corresponding results in the literature.
 Global Weak Solutions to a High-Order Camass-Holm Type Equation Kexin Luo,Shaoyong Lai Acta mathematica scientia,Series A. 2022, 42 (2):  427-441.  Abstract ( 12 )   RICH HTML PDF (331KB) ( 21 )   The viscous approximation technique is employed to investigate the existence of global weak solutions for a high-order Camassa-Holm type equation. A higher integrability estimate of the viscous solutions for the equation and the upper bound estimate on the space derivative of its viscous solutions are derived to prove the existence.
 Multiplicity of Normalized Solutions for Nonlinear Schrödinger-Poisson Equation with Hardy Potential Mengxue Du,Fanghui Li,Zhengping Wang Acta mathematica scientia,Series A. 2022, 42 (2):  442-453.  Abstract ( 20 )   RICH HTML PDF (337KB) ( 17 )   In this paper, we concern the multiplicity of normalized solutions for a class of nonlinear Schrödinger-Poisson equation with Hardy potential. By using some ideas of the fountain theorem, we define a sequence of minimax values and prove that these minimax values are critical values of the energy functional limited to a constraint set. Then we get the multiplicity of normalized solutions, which extends some related results in the literature.
 Nondegeneracy and Uniqueness of Periodic Solution for Third-Order Nonlinear Differential Equations Shaowen Yao,Wenjie Li,Zhibo Cheng Acta mathematica scientia,Series A. 2022, 42 (2):  454-462.  Abstract ( 7 )   RICH HTML PDF (302KB) ( 15 )   In this paper, we study the nondegeneracy of periodic solution for a third-order linear differential equation $x'''(t)+a_{2}x''(t)+a_{1}x'(t)=a_{0}(t)x(t).$ By the Writinger inequality, we give a nondegeneracy condition of the above equation. By the nondegeneracy of third-order linear differential equation, we consider the existence and uniqueness of periodic solution third-order nonlinear differential equation with semilinear and superlinear terms.
 Existence and Multiplicity of Anti-Periodic Solutions for a Class of Second Order Duffing Equation Jun Lan Acta mathematica scientia,Series A. 2022, 42 (2):  463-469.  Abstract ( 8 )   RICH HTML PDF (279KB) ( 13 )   In this paper, we establish the existence and multiplicity of solutions for second order Duffing equations with anti-periodic boundary conditions through using variational approach.
 Existence, Multiplicity and Concentration of Positive Solutions for a Fractional Choquard Equation Weiqiang Zhang,Peihao Zhao Acta mathematica scientia,Series A. 2022, 42 (2):  470-490.  Abstract ( 11 )   RICH HTML PDF (451KB) ( 24 )   We are concerned with the existence, multiplicity and concentration of positive solutions for the following fractional Choquard equation with subcritical nonlinearity $\begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u+V(x)u=\varepsilon^{\mu-N}(|x|^{-\mu}\ast F(u))f(u), \quad x\in \mathbb{R} ^{N}, \end{eqnarray*}$ where $\varepsilon>0$ is a parameter, $s\in(0, 1)$, $(-\Delta)^{s}$ is the fractional Laplace operator, $V:\mathbb{R} ^{N}\rightarrow\mathbb{R}$ is a positive potential having global minimum, $0<\mu<\min\{4s, N\}$, and $F$ is the primitive of $f\in C^{1}(\mathbb{R} , \mathbb{R} )$ which is subcritical growth. The main research methods of this article are variational method and the Ljusternik-Schnirelmann theory.
 Forced Waves of a Delayed Reaction-Diffusion Equation with Nonlocal Diffusion Under Shifting Environment Hongmei Cheng,Rong Yuan Acta mathematica scientia,Series A. 2022, 42 (2):  491-501.  Abstract ( 8 )   RICH HTML PDF (380KB) ( 15 )   This paper is devoted to establish the existence and uniqueness of the forced waves for a general reaction-diffusion equation with time delay and nonlocal diffusion term in a shifting environment. We first obtain the existence of the forced waves with the speed at which the habitat is shifting by using super- and sub-solutions method and monotone iteration method. Then we will give the uniqueness by applying the sliding method with the strong maximum principle. Finally, these analytical conclusions are applied to the nonlocal delayed Logistic model and the nonlocal delayed quasi-Nicholson's Blowfiles population model.
 Boundedness and Stabilization of a Chemotaxis Model Describing Tumor Invasion with Signal-Dependent Motility Shijie Shi,Zhengrong Liu,Hui Zhao Acta mathematica scientia,Series A. 2022, 42 (2):  502-519.  Abstract ( 13 )   RICH HTML PDF (419KB) ( 15 )   In this paper, we study the following problem $\left\{\begin{array}{l} u_{t}=\Delta{(\gamma(v)u)}, & x\in\Omega, t>0, \\ v_{t}=\Delta v+wz, & x\in\Omega, t>0, \\ w_{t}=-wz, & x\in\Omega, t>0, \\ z_{t}=\Delta z+ u- z, & x\in\Omega, t>0, \\ \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=\frac{\partial z}{\partial \nu}=0, & x\in\partial\Omega, t>0, \\ (u, v, w, z)(x, 0)=(u_0, v_0, w_0, z_0)(x), &x\in\Omega \end{array}\right.$ in a bounded domain $\Omega\subset\mathbb{R} ^n(1\leqq n\leqq 5)$ with smooth boundary and $\nu$ denotes the outward normal vector of $\partial \Omega$, where $0 <\gamma(v)\in C^3[0, \infty)$. Under suitably regular initial data, we show the existence of global classical solution with uniform-in-time bound under one of the following conditions$\bullet\; 1\leq n\leq 3$,$\bullet\; 4\leq n\leq 5$ and $\gamma_2\geq \gamma(v)\geq \gamma_1>0$,$\left|\gamma'(v)\right|\leq \gamma_3,$ $v\in [0, \infty)$ with some constants $\gamma_i>0\ (i=1, 2, 3)$.Moreover, we confirm that the solution $(u, v, w, z)$ will exponentially converge to the homogeneous equilibrium $(\bar{u}_0, \bar{v}_0+\bar{w}_0, 0, \bar{u}_0)$ as $t\rightarrow\infty$, where $\bar{u} _0: =\frac{1}{\left|\Omega\right|}\int_{\Omega}u_0{\rm d}x$, $\bar{v}_0: =\frac{1}{\left|\Omega\right|}\int_{\Omega}v_0{\rm d}x$ and $\bar{w}_0: =\frac{1}{\left|\Omega\right|}\int_{\Omega}w_0{\rm d}x$.
 Wong-Zakai Approximations of Anticipated Backward Doubly Stochastic Differential Equations with Jumps in Non-Lipschitz Conditions Jie Xu,Yanhua Sun Acta mathematica scientia,Series A. 2022, 42 (2):  520-556.  Abstract ( 6 )   RICH HTML PDF (419KB) ( 9 )   In this paper we will prove the Wong-Zakai approximation of anticipated backward doubly stochastic differential equations with Poisson jumps under the non-Lipschitz conditions.
 Characteristic Endpoints Question for Piecewise Monotone Functions Xiao Tang,Lin Li Acta mathematica scientia,Series A. 2022, 42 (2):  557-569.  Abstract ( 6 )   RICH HTML PDF (469KB) ( 9 )   For PM functions of height 1, the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition. This raises an open problem about iterative roots without this condition, called characteristic endpoints problem. This problem is solved almost completely when the number of forts is equal to or less than the order. In this paper, we study the case that the number of forts is greater than the order and give a sufficient condition for existence of continuous iterative roots of order 2 with height 2, answering the characteristic endpoints problem partially.
 Hopf Bifurcation for a Fractional Differential-Algebraic Predator-Prey System with Time Delay and Economic Profit Daoxiang Zhang,Ben Li,Dandan Chen,Yating Lin,Xinmei Wang Acta mathematica scientia,Series A. 2022, 42 (2):  570-582.  Abstract ( 14 )   RICH HTML PDF (959KB) ( 17 )   A fractional differential-algebraic biological economic system with harvest and time delay is firstly proposed and investigated in this paper. By using the Hopf bifurcation theory, some sufficient conditions for the existence of Hopf bifurcation induced by delay are obtained. The results show that in the case of zero economic profit, the biological equilibrium point of the system is asymptotically stable. Under positive economic profit condition, the system produces limit cycles at the positive equilibrium point as the delay increases through a certain threshold. It is found that fractional exponent, economic interest and delay can affect the other dynamic behavior of the system through some numerical simulations.
 The Improved Convergence Theorems of Modulus-Based Matrix Splitting Iteration Methods for a Class of Nonlinear Complementarity Problems with H-Matrices Changfeng Ma,Feiyang Ma Acta mathematica scientia,Series A. 2022, 42 (2):  583-593.  Abstract ( 5 )   RICH HTML PDF (376KB) ( 9 )   In this paper, we proved the convergence theories of the modulus-based matrix splitting iteration methods and the corresponding acceleration method for nonlinear complementarity problems of $H$-matrices in a weaker condition. It implies that we have more choices for the splitting $A=M-N$ which makes the modulus-based matrix splitting iteration methods converge. The improved convergence theories extend the scope of modulus-based matrix splitting iteration methods in applications.
 Asymptotic Optimality of Quantized Stationary Policies in Continuous-Time Markov Decision Processes with Polish Spaces Xiao Wu,Yinying Kong,Zhenbin Guo Acta mathematica scientia,Series A. 2022, 42 (2):  594-604.  Abstract ( 4 )   RICH HTML PDF (384KB) ( 12 )   In this paper, we study the asymptotic optimality of the quantized stationary policies for continuous-time Markov decision processes (CTMDPs) with Polish space and state-dependent discount factors. Firstly, the existence and uniqueness of the discounted optimal equation (DOE) and its solution are established. Secondly, the existence of the optimal deterministic stationary policies is proved under appropriate conditions. In addition, in order to discretize the action space, a series of quantization policies are constructed to approximate the optimal stationary policies of the discounted CTMDPs in general state (Polish) space by using the policies in finite action space. Finally, an example is given to illustrate the asymptotic approximation results of this paper.
 A Modified HS-DY-Type Method with Nonmonotone Line Search for Image Restoration and Unconstrained Optimization Problems Gonglin Yuan,Yulun Wu,Hongtruong Pham Acta mathematica scientia,Series A. 2022, 42 (2):  605-620.  Abstract ( 9 )   RICH HTML PDF (2098KB) ( 17 )   A modified conjugate gradient algorithm for solving image restoration problems and unconstrained optimization problems is proposed, where the conjugate gradient (CG) parameter is the convex combination of the improved HS and DY methods, and the CG parameter contains function information. In addition, the method does not require any line searches, and it can generate sufficient descent directions. Moreover, under certain conditions, the new method is globally convergent with nonmonotone line search. Finally, experiments on unconstrained optimization and image restoration problems show that the new method has good application prospects and advantages when compared with other conjugate gradient algorithms.
 Stochastic Ordering of Sample Minima from Scale Proportional Hazards Random Variables Under Archimedean Copula Dependence Longxiang Fang,Balakrishnan Narayanaswamy,Wenyu Huang Acta mathematica scientia,Series A. 2022, 42 (2):  621-630.  Abstract ( 3 )   RICH HTML PDF (378KB) ( 9 )   In this paper, we discuss stochastic comparisons of the smallest order statistics arising from two sets of dependent scale proportional hazards random variables with respect to multivariate chain majorization. When a scale proportional hazards model with possibly different frailty and scale parameters has its matrix of parameters changing to another matrix of parameters in a certain mathematical sense, we study the first sample minimum is less than the second minimum with respect to the usual stochastic order, based on certain conditions. Some examples are also presented for illustrating the results established here.
 Randomized Dividends in a Discrete Risk Model with Time-Correlated Claims Mi Chen,Changwei Nie,Haiyan Liu Acta mathematica scientia,Series A. 2022, 42 (2):  631-640.  Abstract ( 4 )   RICH HTML PDF (387KB) ( 14 )   In this paper, the compound binomial risk model is extended by involving the random premium income with time-correlated claims and random dividend strategy. By the method of generating function, the difference equation and its solution for the expected cumulated discounted dividends until ruin are obtained. Finally, the effect of related parameters on the total expected discounted dividends are shown in several numerical examples.