#### Table of Content

26 December 2022, Volume 42 Issue 6
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 The Bonnesen-type Inequalities for Plane Closed Curves Rui Bin,Xingxing Wang,Chunna Zeng Acta mathematica scientia,Series A. 2022, 42 (6):  1601-1610.  Abstract ( 41 )   RICH HTML PDF (354KB) ( 59 )   The isoperimetric inequality is one of the most classical geometric inequalities in differential geometry. The stability of isoperimetric genus can be characterized by Bonnesentype inequality and Bottema-type inequality. In this paper, via the method of differential geometry, Wirtinger inequality, Sachs inequality and divergence theorem and so on, we investigate the Bonnesen-type inequalities and Bottema-type inequalities for plane closed curves, and obtain a series of new Bonnesen-type inequalities and Bottema-type inequalities for curvature integration.
 The Essential Spectrum of a Class of Anti-Triangular Operator Matrices Rui Hua,Yaru Qi Acta mathematica scientia,Series A. 2022, 42 (6):  1611-1618.  Abstract ( 35 )   RICH HTML PDF (335KB) ( 62 )   In this paper, the essential spectrum of a class of unbounded unself-adjoint anti-triangular operator matrices is studied. Firstly, we describe the essential spectrum of operator matrices by using the quadratic operator pencil and the properties of its operator entries, and estimate the essential spectrum of the whole operator matrix. On this basis, the accumulation point of the non-real spectrum of the operator matrix is analyzed.
 The Properties and Applications of the MP Weak Core Inverse Xiaoji Liu,Mengyue Liao,Hongwei Jin Acta mathematica scientia,Series A. 2022, 42 (6):  1619-1632.  Abstract ( 15 )   RICH HTML PDF (338KB) ( 13 )   In this paper, the concept of the Moore-Penrose weak Core inverse (MPWC inverse) is proposed based on the Moore-Penrose inverse and the weak Core inverse. It is described from algebraic and geometric perspectives respectively. The relationship between the MP weak Core inverse and the nonsingular bordered matrix is given. The expression of the MP weak Core inverse is given by using the Hartwig-Spindelböck decomposition and the Core-EP decomposition. The equivalence between the MP weak Core inverse of a matrix and EP matrix, the characterization and the perturbation analysis are given.
 Boundary Schwarz Lemma for Solutions to a Class of Inhomogeneous Biharmonic Equations Xiaojin Bai,Jianfeng Zhu Acta mathematica scientia,Series A. 2022, 42 (6):  1633-1639.  Abstract ( 27 )   RICH HTML PDF (293KB) ( 52 )   Let $\mathbb{D}$ be the unit disk, ${\mathbb T}$ the unit circle. Assume that $f$ is a solution to inhomogeneous biharmonic equation: $\Delta f=g$, satisfying the boundary conditions: $(\Delta f)_{{\mathbb T}}=\psi$ and $f|_{{\mathbb T}}=f^*$, where $g\in {\cal C}(\overline{\mathbb{D}})$, and $\psi, f^*\in {\cal C}({\mathbb T})$ are continuous functions. In this paper, we establish the boundary Schwarz lemma for solutions $f$, this result enriches the related results of boundary Schwarz lemma on the plane.
 Solvability of Several Kinds of Ggeneralized Sylvester Operator Equations Xiaolin Sun,Hua Wang Acta mathematica scientia,Series A. 2022, 42 (6):  1640-1652.  Abstract ( 19 )   RICH HTML PDF (316KB) ( 17 )   In this paper, we discuss the solvability of some kinds of generalized Sylvester operator equations on Hilbert Spaces. Firstly, the necessary and sufficient conditions for the existence of solutions of AXB-XD=EB and AXB-CX = AE are given in the case of normal operators. Secondly, the solvability of three kinds of operator equations is discussed by using the uniform equivalence of operator pairs. Finally, when trigonometric operator matrix is similar to diagonal operator matrix, the corresponding operator equations are solvable.
 On the Bang-Bang Principle for Differential Variational Inequalities in Banach Spaces Cuiyun Shi,Maojun Bin Acta mathematica scientia,Series A. 2022, 42 (6):  1653-1670.  Abstract ( 23 )   RICH HTML PDF (454KB) ( 23 )   In this paper, we discuss a class of differential variational inequalities systems, which are obtained by semilinear evolution equations and generalized variational inequalities. At first, we consider the properties of solution set for generalized variational inequalities. Secondly, the existence results are shown by fixed point method for semilinear differential variational inequality. Our approaches are based on semigroup theory and fixed point theorem. Moreover, by using the density results, the nonlinear and infinite dimensional versions of the "bang-bang" principle for differential variational inequalities systems is derived. Also, an obstacle parabolic-elliptic system is given to illustrate the application of the obtained theory.
 Existence Results of Mild Solutions for Impulsive Neutral Measure Differential Equations with Infinite Delay Wenjie Liu,Shengli Xie Acta mathematica scientia,Series A. 2022, 42 (6):  1671-1681.  Abstract ( 22 )   RICH HTML PDF (373KB) ( 18 )   In this paper, we mainly examine the existence of mild solutions for impulsive neutral measure differential equations with infinite delay. Under the condition that semigroups are non-compact, we obtain sufficient conditions for the existence of mild solutions by using operator semigroup theory, Kuratowski measure of noncompactness, Mönch fixed point theorem and piecewise estimation. Without utilizing a priori estimation and non-compact constraints, we generalize many existing results. Finally, an example is delivered to illustrate the feasibility of the result.
 Multiplicity of Solutions to Fractional Critical Choquard Equation Lin Chen,Fanqin Liu Acta mathematica scientia,Series A. 2022, 42 (6):  1682-1704.  Abstract ( 24 )   RICH HTML PDF (411KB) ( 45 )   In this paper, we are concerned with the multiplicity of solutions for the following fractional Laplacian problem $$$\left\{\begin{array}{ll} { } (-{\Delta})^s{u}=\lambda|u|^{q-2}{u}+ \bigg(\int_\Omega\frac{|u(y)|^{2^\ast_{\mu, s}}}{|x-y|^\mu}{\rm d}y\bigg)|u|^{2^\ast_{\mu, s}-2}u, & x\in\Omega, \\ {u=0}, &x\in{{{\Bbb R}} ^N}\setminus\Omega \end{array}\right.$$$ where $\Omega\subset\mathbb{R} ^N$ is an open bounded set with continuous boundary, $N>2s$ with $s\in(0, 1)$, $\lambda$ is a real parameter, $\mu\in(0, N)$ and $q\in[2, 2^\ast_s)$, where $^\ast_{s}=\frac{2N}{N-2s}$, $^\ast_{\mu, s}=\frac{2N-\mu}{N-2s}$. Using Lusternik-Schnirelman theory, there exists $\bar{\lambda}>0$ such that for any $\lambda\in(0, \bar{\lambda})$, the problem has at least $cat_\Omega(\Omega)$ nontrivial solutions provided that $q=2$ and $N\geq4s$ or $q\in(2, 2^\ast_s)$ and $N>\frac{2s(q+2)}{q}$.
 Existence and Uniqueness of Global Solutions for a Class of Semilinear Time Fractional Diffusion-Wave Equations Xinhai He,Mei Liu,Han Yang Acta mathematica scientia,Series A. 2022, 42 (6):  1705-1718.  Abstract ( 19 )   RICH HTML PDF (368KB) ( 25 )   The purpose of this paper is to study the Cauchy problem of a class of semilinear time fractional diffusion-wave equations. Based on the Lr-Lq estimates obtained from the corresponding linear problem, and combined with the global iteration method, the influence of the exponential of the nonlinear term on the global existence of the solutions is studied with small data, the existence and uniqueness of global solutions are proved under certain conditions of exponential.
 Existence of Schwarz Symmetric Minimizers for Fractional Kirchhoff Constrained Variational Problem Junping Wei,Xiaomeng Huang,Yimin Zhang Acta mathematica scientia,Series A. 2022, 42 (6):  1719-1728.  Abstract ( 20 )   RICH HTML PDF (373KB) ( 23 )   In this paper, the existence and Schwarz symmetry of minimizers for a fractional Kirchhoff constrained variational problem with general nonlinear term were studied in space $H^s(\mathbb{R} ^N)$. Using symmetric decreasing rearrangement inequality and scaling technique, the relation between existence of minimizers for a fractional Kirchhoff constrained variational problem with the exponent of nonlinear term and parameter $c$ in $\int_{\mathbb{R} ^N}|u|^2{\rm d}x=c^2$ was discussed.
 Existence and Asymptotic Behaviour of Solutions for Kirchhoff Type Equations with Zero Mass and Critical Anran Li,Dandan Fan,Chongqing Wei Acta mathematica scientia,Series A. 2022, 42 (6):  1729-1743.  Abstract ( 16 )   RICH HTML PDF (402KB) ( 13 )   In this paper, the existence and asymptotic behaviour of solutions for Kirchhoff type equations with zero mass and critical term are studied. First, under some appropriate assumptions of the nonlinearity, it is proved that the functional associated to the equation enjoys mountain pass structure and the estimate of mountain pass energy level is also given. Then the second concentration compactness lemma is used to verify that the corresponding functional satisfies Palais-Smale local compactness condition, thus the existence of nontrivial solutions is obtained by mountain pass theorem, furthermore, the existence of ground state solutions is obtained by using the definition of the ground state solution. Finally, the asymptotic behaviour of these mountain pass type solutions is studied whenever the parameter tends to zero. It can be proved that they converge to a mountain pass type solution of our problem when the parameter equals to 0.
 The Cauchy Problem for the Forth Order p-Generalized Benney-Luke Equation Xiao Su,Shubin Wang Acta mathematica scientia,Series A. 2022, 42 (6):  1744-1753.  Abstract ( 12 )   RICH HTML PDF (329KB) ( 18 )   We consider the global existence and uniqueness of the Cauchy problem for the forth order $p$-generalized Benney Luke equatoion in the natural energy space $\dot{H}^{1}(\mathbb{R}^n)\cap\dot{H}^{2}(\mathbb{R}^n) \times H^1(\mathbb{R}^n)$. First of all, the local existence and uniqueness of solutions are investigated in the energy space by means of the contraction mapping principle. Secondly, in the case of source nonlinearity, we provide the sufficient conditions of the existence of global solutions.
 Global Regularity for the Incompressible 3D Hall-Magnetohydrodynamics with Partial Dissipation Baoying Du,Jinxing Liu Acta mathematica scientia,Series A. 2022, 42 (6):  1754-1767.  Abstract ( 12 )   RICH HTML PDF (347KB) ( 24 )   In this paper, we study the incompressible 3D Hall-magnetohydrodynamic equations with partial dissipation, by the special property of the energy estimate in term of Sobolev norm, we obtain the local existence of classical solutions for large initial data. We also prove the global existence of the classical solutions, the smallness conditions of which are given by the suitable Sobolev norms.
 The Ill-Posedness of the Solution for the General Power Derivative Schrödinger Equation in Hs Chen Xingfa, Zhong Penghong Acta mathematica scientia,Series A. 2022, 42 (6):  1768-1781.  Abstract ( 10 )   RICH HTML PDF (396KB) ( 38 )   The nonlinear defocusing Schrödinger equations with general power nonlinearity are proved to be ill-posed in the Sobolev space Hs whenever the exponent $s$ is lower than $1/k$ ($2 \leq k \leq 4$) or $1/2-1/k$ ($k>4$).
 Probability Estimation of the Weighted Sum of Independent Identically Distributed Random Variables Li Ma,Liu Ye Acta mathematica scientia,Series A. 2022, 42 (6):  1782-1789.  Abstract ( 10 )   RICH HTML PDF (345KB) ( 7 )   Let $\xi_{i}(1\leq{i}\leq{n})$ be independent identically distributed random variables satisfying $P(\xi_i=1)=P(\xi_i=-1)=\frac{1}{2}$. Let $\overrightarrow{a}=(a_{1}, \cdots, a_{n})$ be random variables uniformly distributed on $S^{n-1}=\{(a_{1}, \cdots, a_{n})\in\mathbb{R} ^n|\sum\limits^n_{i=1}a_i^2=1\}$ which are independent of $\xi_{i}(1\leq{i}\leq{n})$. In this paper, we get the expression of $P (|\sum\limits_{i=1}^n{a_i}{\xi_{i}|\leq1})$ by polar coordination transformation. For $n\leq7$, we give the value of $P (|\sum\limits_{i=1}^n{a_i}{\xi_{i}|\leq1})$ directly which is no less than one half. For $n\geq8$, we can use R software to calculate the value which is also no less than one half. Moreover, for $n=3, 4$, by Beta function, we show that the probability value is still no less than one half.
 E-Bayesian Estimation and E-MSE of Failure Probability and Its Applications Ming Han Acta mathematica scientia,Series A. 2022, 42 (6):  1790-1801.  Abstract ( 18 )   RICH HTML PDF (361KB) ( 28 )   In order to measure the estimated error, this paper based on the E-Bayesian estimation (expected Bayesian estimation) introduced the definition of E-MSE (expected mean square error), and derive the expressions of E-Bayesian estimation of failure probability and their the E-MSE under different loss functions (including: squared error loss function and LINEX loss function). By Monte Carlo simulations compared with the performances of the proposed the estimation method (the comparison of the results is based on the E-MSE). Finally, combined with the engine reliability problem, used respectively E-Bayesian estimation method and the MCMC method the calculation and analysis are performed. When considering evaluating the E-Bayesian estimations under different loss functions, this paper proposed the E-posterior risk as an evaluation standard.
 Lower Bounds for the Symmetric L2-Discrepancy of U-type Designs Yiju Lei,Zujun Ou Acta mathematica scientia,Series A. 2022, 42 (6):  1802-1811.  Abstract ( 10 )   RICH HTML PDF (379KB) ( 13 )   Uniform design is one of the main methods of fractional factorials, which has been widely used in industrial production, systems engineering, pharmacy and other natural sciences. Various discrepancies are used to measure the uniformity of fractional factorials, the key is to find an accurate lower bound of the discrepancy, because it can be used as a benchmark which measures uniformity of design. In this paper, the lower bounds for the symmetric L2-discrepancy on symmetrical U-type designs with four-level and asymmetrical U-type designs with two and three mixed levels and two and four mixed levels are abtained.
 Stability of Weak NS Equilibria for Population Games with Uncertain Parameters Under Bounded Rationality Mingting Wang,Guanghui Yang,Hui Yang Acta mathematica scientia,Series A. 2022, 42 (6):  1812-1825.  Abstract ( 10 )   RICH HTML PDF (484KB) ( 17 )   For population games with uncertain parameters, a weak NS equilibrium is firstly proposed based on the fact that switching strategies cause corresponding costs. The underlying idea of a weak NS equilibrium is that the agents' new gained payoffs from strategy switch are less than or equal to the increased cost for a given uncertainty parameter; simultaneously, each population can not obtain strictly poor net profits under uncertain parameters, thus each agent in every population has no motivation to unilaterally change the current strategy and then they achieve a weak NS equilibrium. Secondly, the existence of weak NS equilibria is proven by Kakutani's fixed point theorem. Thirdly, by constructing an abstract rational function, a corresponding bounded rational model is established, and it is shown structural stability implying robustness. Therefore, the generic stability of weak NS equilibria for population games with uncertain parameters under bounded rationality is also obtained when the net profit function is perturbed. Finally, an example is illustrated the correctness of the above results.
 Stability of Stage-Structured Predator-Prey Models with Beddington-DeAngelis Functional Response Bo Li,Ziwei Liang Acta mathematica scientia,Series A. 2022, 42 (6):  1826-1835.  Abstract ( 12 )   RICH HTML PDF (323KB) ( 21 )   This paper deals with stage-structured predator-prey systems with Beddington-DeAngelis functional response. The local asymptotical stability is given by Routh-Hurwitz criterion, and global asymptotic stability are established from Lyapunov functions.
 Dynamics of an HTLV-I Infection Model with Delayed and Saturated CTL Immune Response and Immune Impairment Rui Xu,Yan Yang Acta mathematica scientia,Series A. 2022, 42 (6):  1836-1848.  Abstract ( 11 )   RICH HTML PDF (712KB) ( 13 )   In this paper, an HTLV-I infection model with delayed and saturated CTL immune response and immune impairment is developed. By calculations, the existences of feasible equilibria are established, immunity-inactivated and immunity-activated reproduction ratios are also derived. Under the assistance of proper Lyapunov functionals and LaSalle's invariance principle, it is shown that the infection-free equilibrium is globally asymptotically stable if the immunity-inactivated reproduction ratio is less than unity; the immunity-inactivated equilibrium is globally asymptotically stable if the immunity-activated reproduction ratio is less than unity, while the immunity-inactivated reproduction ratio is greater than unity; the immunity-activated equilibrium is globally asymptotically stable (when the time delay equals to zero) if the immunity-activated reproduction ratio is greater than unity. A Hopf bifurcation at the immunity-activated equilibrium occurs as the time delay crosses a critical value. Finally, numerical simulations are performed to illustrate the theoretical results.
 Analysis of a Reaction-Diffusion Epidemic Model with Horizontal Transmission and Environmental Transmission Zhenxiang Hu,Linfei Nie Acta mathematica scientia,Series A. 2022, 42 (6):  1849-1860.  Abstract ( 15 )   RICH HTML PDF (647KB) ( 24 )   In this paper, a reaction-diffusion epidemic model with horizontal transmission and environmental transmission is proposed. The well posedness of this model is investigated, including the global existence and uniform boundedness of the solution. Furthermore, the basic reproduction number ${\mathcal R}_{0}$ is defined by the spectral radius of the next generation operator. The threshold dynamics of the model is studied by using monotone dynamical systems theory and uniform persistence theory.
 Qualitative Analysis of Stochastic SIRS Epidemic Model with Logistic Growth and Beddington-DeAngelis Incidence Rate Yanjun Zhao,Xiaohui Sun,Li Su,Wenxuan Li Acta mathematica scientia,Series A. 2022, 42 (6):  1861-1872.  Abstract ( 19 )   RICH HTML PDF (564KB) ( 21 )   In this paper, we considered a class of stochastic SIRS epidemic model with logistic growth and Beddington-DeAngelis incidence rate by white noise in the environment. By constructing Lyapunov function and applying Itô formula, the global existence and uniqueness of positive solution are proved, the sufficient conditions which determines disease extinction and permanence are obtained. Finally, matlab is used for numerical simulation to illustrate the correctness of the theoretical results.
 Soliton Solutions and Its Nonlinear Dynamics Behavior Research of the Three-Component Four-Order Nonlinear Schrödinger System in Alpha Helical Protein Jiayu Liu,Hanyu Wei,Yan Zhang,Tiecheng Xia,Hui Wang Acta mathematica scientia,Series A. 2022, 42 (6):  1873-1885.  Abstract ( 11 )   RICH HTML PDF (1296KB) ( 3 )   Complexes of proteins are central to certain cellular processes, investigated in this paper is the three-component fourth-order nonlinear Schrödinger system, which is used for describing the alpha helical proteins with interspine coupling. First, the matrix Rieman-Hilbert problem for the system is derived by scattering and inverse-scattering transformations through the Rieman-Hilbert method. Then, a unique solution is constructed by using discrete scattering data from the Rieman-Hilbert problem under the reflectionless. Furthermore, the $N$-soliton solution formula of three-component fourth-order nonlinear Schrödinger system is obtained with the help of potential reconstruction. In the case of $N= 1, 2, 3$, the explicit expressions of soliton solutions, breather solutions and interaction solutions are formulated by means of Maple symbolic computation. Finally, the propagation and collision dynamic behaviors as well as localized wave characteristics of these solutions are further analyzed by selecting appropriate parameters with some graphics. The results show that the higher-order linear and nonlinear term coefficient $\gamma$ has important impact on the velocity, phase, period, and wavewidth of wave dynamics. Meanwhile, collisions for the high-order breathers and muli-soliton solutions are elastic interaction which imply they remain bounded all the time.
 Spectral LS-type Projection Algorithm for Solving Nonlinear Pseudo-Monotone Equations Ning Zhang,Jinkui Liu Acta mathematica scientia,Series A. 2022, 42 (6):  1886-1897.  Abstract ( 9 )   RICH HTML PDF (431KB) ( 44 )   Based on the structures of the spectral gradient method and the famous LS conjugate gradient method, in this paper we establish an spectral LS-type derivative-free projection algorithm to solve nonlinear pseudo-monotone equations with convex constraints. By using the spectral parameter, the proposed method can generate the descent direction in each iterate, which is independent of any line search. Under some usual assumptions, the global convergence of the proposed method is proved by using the classical derivative-free line search condition. Numerical experiments show that the proposed method inherits the excellent computational performance of the LS conjugate gradient method and improves its stability.
 A Hybrid Algorithm for Solving Truncated Complex Singular Value Decomposition Yuxin Zhang,Wenting Hou,Xuelin Zhou,Jiaofen Li Acta mathematica scientia,Series A. 2022, 42 (6):  1898-1921.  Abstract ( 14 )   RICH HTML PDF (1197KB) ( 24 )   This paper develops an efficient approach for solving the truncated complex singular value decomposition, which is widely applied in ill-posed model problems. The original problem can be formulated as an optimization problem on a corresponding complex product Stiefel manifold. A hybrid Riemannian Newton-type algorithm with globally and quadratically convergent is proposed to solve the underlying problem, in which the involved Newton's equation is transformed into a standard symmetric linear system with a dimension reduction. Numerical experiments and detailed comparisons are provided to illustrate the efficiency of the proposed method.