#### Table of Content

26 August 2021, Volume 41 Issue 4
 Meromorphic Solutions of Finite Order to the Equation $f^{n}+f^{m}(z+c)=e^{Az+B}$ Minfeng Chen,Zongsheng Gao,Zhibo Huang Acta mathematica scientia,Series A. 2021, 41 (4):  913-920.  Abstract ( 65 )   RICH HTML PDF (303KB) ( 63 )   In this paper, we study the meromorphic solutions of finite order to the difference equations $f^{n}(z)+f^{m}(z+c)=e^{Az+B}$ $(c\neq 0)$ over the complex plane ${\mathbb C}$ for integers $n, m$, and $A, B, c\in {\mathbb C}$.
 A Generalised Decic Freud-Type Weight Dan Wang,Mengkun Zhu,Chen Yang,Xiaoli Wang Acta mathematica scientia,Series A. 2021, 41 (4):  921-935.  Abstract ( 35 )   RICH HTML PDF (559KB) ( 23 )   In this paper, the authors focus on a generalised decic Freud-type weight function $w(x;t, \alpha)=|x|^{\alpha}{\rm e}^{-x^{10}+tx^2}, t, x\in\mathbb{R}, \alpha>-1,$ and study the properties of the orthogonal polynomials with respect to this weight. The difference-differential equations of their associated recurrence coefficients are derived; meanwhile, the authors also find the asymptotic behavior of recurrence coefficients via above mentioned equations. What's more, the authors discuss the Hankel determinant in regard to this weight as $n\rightarrow\infty$, and calculate the smallest eigenvalues of large Hankel matrices generated by this weight when $\alpha=t=0$.
 Bifurcation of Limit Cycles from a Liénard System of Degree 4 Hongying Zhu,Minzhi Wei,Sumin Yang,Caoqing Jiang Acta mathematica scientia,Series A. 2021, 41 (4):  936-953.  Abstract ( 30 )   RICH HTML PDF (613KB) ( 48 )   In this paper, we study the number of limit cycles by Poincaré bifurcation for some Liénard system of degree 4. We prove that the system can bifurcate at most 6 limit cycles from the periodic annulus, by the tools of regular chain theory in polynomial algebra and Chebyshev criteria, at least 3 limit cycles by asymptotic expansions of the related Abelian integral (first order Melnikov functions).
 Weighted Estimates for Some Multilinear Square Operator and Commutator with Generalized Kernel Xiaoli Chen,Dongxiang Chen,Hongyan Zhu Acta mathematica scientia,Series A. 2021, 41 (4):  954-967.  Abstract ( 21 )   RICH HTML PDF (354KB) ( 36 )   In this paper, the authors investigate some multilinear square operator with generalized kernel. They prove that the multilinear square operator $T$ is bounded from $(L^{p_1}(\omega_1)\times\cdots\times L^{p_m}(\omega_m))$ into $L^{p}(\nu_{\omega})$, where $\frac{1}{p_1}+ \cdots+\frac{1}{p_m}=\frac{1}{p}, \nu_{\omega}=\prod\limits_{i=1}^m\omega_i^{\frac{p_i}{p}}$, the authors proved the commutator $T_{\sum b}$, generalized by multilinear square operator $T$ and BMO function, is also bounded from$(L^{p_1}(\omega_1)\times\cdots\times L^{p_m}(\omega_m))$ into $L^{p}(\nu_{\omega})$上. Finally, the authors also prove the multilinear square operator $T$ is bounded from $L^\infty\times\cdots\times L^{\infty}$ into $BMO$. Some known results are improved.
 On a $p$-Adic Integral Operator Induced by a Homogeneous Kernel and Its Applications Jianjun Jin,Shuan Tang,Xiaogao Feng Acta mathematica scientia,Series A. 2021, 41 (4):  968-977.  Abstract ( 32 )   RICH HTML PDF (343KB) ( 24 )   In this paper, we introduce and study a $p$-adic integral operator induced by a homogeneous kernel of degree $-λ$ and obtain its sharp norm estimates. As applications, we establish some new $p$-adic inequalities with the best constant factors and their equivalent forms, which extend some known results in the literature.
 Nonlinear $\xi$-Jordan *-Triple Derivable Mappings on Factor von Neumann Algebras Fangjuan Zhang,Xinhong Zhu Acta mathematica scientia,Series A. 2021, 41 (4):  978-988.  Abstract ( 15 )   RICH HTML PDF (267KB) ( 11 )   Let ${\cal A}$ be a factor von Neumann algebra and $\xi$ be a non-zero complex number. A nonlinear map $\phi:\mathcal A\rightarrow\mathcal A$ has been demonstrated to satisfy $\phi(A\diamond_{\xi}B\diamond_{\xi}C)=\phi(A)\diamond_{\xi}B\diamond_{\xi}C+A\diamond_{\xi}\phi(B)\diamond_{\xi}C+A\diamond_{\xi}B\diamond_{\xi}\phi(C)$ for all $A, B, C\in\mathcal A$ if and only if $\phi$ is an additive *-derivation and $\phi(\xi A)=\xi\phi(A)$ for all $A\in\mathcal A.$
 Some New Bonnesen-Type Inequalities of the Tetrahedron in $\mathbb{R}^3$ Yan Zhang,Chunna Zeng,Xingxing Wang Acta mathematica scientia,Series A. 2021, 41 (4):  989-996.  Abstract ( 18 )   RICH HTML PDF (350KB) ( 25 )   Discrete isoperimetric problems play an important role in integral geometry and convex geometry. The stability of isoperimetric deficit can be characterized by Bonnesen-type inequality and inverse Bonnesen-type inequality. In this paper, we study the Bonnesen-type inequality and the inverse Bonnesen-type inequality for Tetrahedra in $\mathbb{R}^3$. And we obtain several new Bonnesen-type inequalities for Tetrahedra. It provides a simplified proof which is different from the isoperimetric inequality for Tetrahedra in Sturm [15]; finally, four inverse Bonnesen-type inequalities in terms of the radius of the circumscribed sphere and the radius of the circumscribed sphere are obtained.
 Existence and Asymptotic Behavior of Solution for a Degenerate Elliptic Equation Involving Grushin-Type Operator and Critical Sobolev-Hardy Exponents Jinguo Zhang,Dengyun Yang Acta mathematica scientia,Series A. 2021, 41 (4):  997-1012.  Abstract ( 19 )   RICH HTML PDF (386KB) ( 12 )   In this paper, we study the existence and asymptotic behavior of solutions for a class of degenerate elliptic equation involving Grushin-type operator and Hardy potentials $-(\Delta_{x}+|x|^{2\alpha}\Delta_{y}) u-\mu\frac{\psi^{2}u}{d (z)^{2}}=\frac{\psi^{s}|u|^{2^*(s)-2}u}{d (z)^{s}},\, \, z=(x,y)\in\mathbb{R}^{m}\times\mathbb{R}^{n},$ where $-(\Delta_{x}+|x|^{2\alpha}\Delta_{y})$ is the Grushin-type operator, $\alpha>0, 2^*(s)=\frac{2(Q-s)}{Q-2}$ is the critical Sobolev-Hardy exponent and $Q=m+(\alpha+1)n$ is the homogenous dimension for Grushin operator. If $0 \leq \mu<(\frac{Q-2}{2})^{2}, 0 < s <2$, we will prove the existence of nontrivial, nonnegative solutions for this degenerate problem, and give the asymptotic behavior of solutions, at the singularity and at infinity.
 Multiple Periodic Solutions for a Class of Stationary Dirac Equations with Local Nonlinearity Qingye Zhang,Bin Xu Acta mathematica scientia,Series A. 2021, 41 (4):  1013-1023.  Abstract ( 11 )   RICH HTML PDF (389KB) ( 19 )   In this paper, we mainly study multiple periodic solutions for a class of stationary Dirac equations with local nonlinearity via variational methods. Under some fairly mild conditions on the nonlinearity, we show that the existence of a sequence of nontrivial periodic solutions is rather a local phenomenon, which is only forced by the sublinearity of the nonlinearity near the origin. Meanwhile, some recent results from the literature are generalized and essentially improved.
 Existence of Solutions for Anti-Periodic Boundary Value Problems of Fractional Langevin Equation with p(t)-Laplacian Operator Jifeng Zhang,Wei Zhang,Jinbo Ni,Dandan Ren Acta mathematica scientia,Series A. 2021, 41 (4):  1024-1032.  Abstract ( 27 )   RICH HTML PDF (348KB) ( 39 )   This paper studies the anti-periodic boundary value problems of fractional Langevin equation with p (t)-Laplacian operator. The sufficient conditions for the existence of solutions are obtained by using Schaefer fixed point theorem, and the main result is well illustrated with the aid of an example. The results obtained in this paper extend and enrich the existing related works.
 Lifespan Estimate of Damped Semilinear Wave Equation in Exterior Domain with Neumann Boundary Condition Jinglei Zhao,Jiacheng Lan,Shanshan Yang Acta mathematica scientia,Series A. 2021, 41 (4):  1033-1041.  Abstract ( 13 )   RICH HTML PDF (339KB) ( 20 )   This paper concerns about the upper bound of lifespan estimate to damped semilinear wave equations in exterior domain with vanishing Neumann boundary condition. We find that the initial boundary value problem with Neumann boundary condition admits the same upper bound of lifespan as that of the Cauchy problem in $\mathbb{R}^n (n\ge 1)$. This fact is different from the zero Dirichlet boundary value problem in 2-D exterior domain for lifespan estimate, compared to the corresponding result in [6], and is also different from the zero Dirichlet boundary value problem on half line for critical power, compared to the result in [16].
 Fast Growth or Decay Estimates of Thermoelastic Equations in an External Domain Yuanfei Li,Jincheng Shi,Huishan Zhu,Shiqi Huang Acta mathematica scientia,Series A. 2021, 41 (4):  1042-1052.  Abstract ( 11 )   RICH HTML PDF (401KB) ( 16 )   The Thermoelastic equation of type Ⅲ defined on the outer region of a three-dimensional sphere is considered. It is assumed that the solution of the equation satisfies certain boundary conditions at the boundary of the sphere. By setting an arbitrary parameter greater than zero in the energy expression after making certain constraints on the boundary conditions, the fast growth rate or decay rate of the solution with the distance from the origin is obtained by using energy analysis method and differential inequality technology. In the case of decay, the continuous dependence of solutions on coefficients is proved.
 Study on Periodic Solutions of a Class of Continuous and Discontinuous Piece-Wise Linear Systems Jing Yang,Changcheng Ke,Zhouchao Wei Acta mathematica scientia,Series A. 2021, 41 (4):  1053-1065.  Abstract ( 25 )   RICH HTML PDF (385KB) ( 15 )   In recent years, the study of non-smooth systems has become a hot spot, and the qualitative analysis of piece-wise linear systems has become an indispensable research problem. In this paper, a transformed Michelson differential system is studied, and the existence of periodic solutions for continuous and discontinuous piece-wise linear systems is proved by means of average theory.
 A Class of Efficient Modified Algorithms Based onHalley-Newton Methods Yajun Xie Acta mathematica scientia,Series A. 2021, 41 (4):  1066-1078.  Abstract ( 14 )   RICH HTML PDF (387KB) ( 7 )   In this paper, based on the Halley method and the classical Newton method, a class of new modified Halley-Newton method is presented for solving the systems of nonlinear equations with two concrete modified iteration schemes. The convergence performances of the two new variants of Newton iteration method are analyzed in details under appropriate assumptions. Some numerical experiments are given to illustrate the efficiency of the proposed methods.
 Optimality Conditions for DC Composite Optimization Problems with Conical Constraints Lingli Hu,Liping Tian,Donghui Fang Acta mathematica scientia,Series A. 2021, 41 (4):  1079-1087.  Abstract ( 9 )   RICH HTML PDF (313KB) ( 7 )   In the case when the functions are not necessarily lower semicontinuous and the sets are not necessarily closed, we first define the dual problem for DC composite optimization problems with conical constraints by using convexification technique, then some optimality conditions and saddle point theorems are obtained, which extend the corresponding results in the previous papers.
 The PDP Feedback Control and Stability Analysis of a Diffusive Wave Equation Dongxia Fan,Dongxia Zhao,Na Shi,Tingting Wang Acta mathematica scientia,Series A. 2021, 41 (4):  1088-1096.  Abstract ( 11 )   RICH HTML PDF (339KB) ( 10 )   In this paper, a position and delayed position (PDP) feedback controller is established for the Hayami diffusive wave equation. The well-posedness of the closed system is studied firstly, and then, by the method of Lyapunov function, the exponential stability is obtained. Finally, compared with the previous results, the range of the control parameters is expanded in this paper, which shows the efficiency and feasibility of PDP feedback.
 The Analysis of Evolution Process in a Time-Inhomogeneous Two-State Quantum Walk Yunguo Lin Acta mathematica scientia,Series A. 2021, 41 (4):  1097-1110.  Abstract ( 17 )   RICH HTML PDF (413KB) ( 22 )   In this paper, we establish a mathematical model for a time-inhomogeneous two-state quantum walk and give a calculation of the position probability distribution. By calculating spectral values and spectral vectors, we analyze the evolution of quantum states. Furthermore, we derive an Itô formula and get a matrix decomposition and its interpretation.
 Containment Control for Partial Differential Multi-Agent Systems via Iterative Learning Algorithm Dan Zhang,Qin Fu,Zhenjie Chen Acta mathematica scientia,Series A. 2021, 41 (4):  1111-1123.  Abstract ( 13 )   RICH HTML PDF (557KB) ( 13 )   In this paper, we deals with the containment control problem for a class of partial differential multi-agent systems, which are composed of the second-order parabolic equations or the second-order hyperbolic equations. Based on the framework of network topologies, the P-type iterative learning law is designed depending on the output form of the follower system, and the convergence condition of the system in the sense of iterative learning stability is obtained. By using the contraction mapping method, it is proved that the containment errors of two kinds of systems on the finite time interval converge to zero on $L^2$ space with the increasing of iterations. Finally, simulation examples demonstrate the validity of the theoretical analysis.
 Research on Change-Point Detection for Parameters in Regression Model Jinhui Zheng,Jinghu Yu,Yiming Ding,Zeyu Bao Acta mathematica scientia,Series A. 2021, 41 (4):  1124-1134.  Abstract ( 17 )   RICH HTML PDF (963KB) ( 10 )   This paper constructs a method to detect the change point of regression model parameters based on the non-stationary measurement index (NS). Under the premise of selecting the appropriate parameter estimation method and window size, the residual sequence of the sample in the window and the corresponding NS value are calculated by judging the stationarity of the residual sequence within the window to achieve the purpose of change point detection. A series of two-segment regression models are constructed for experimental verification. The results show that this method can effectively detect the position of the change point of the two-segment regression model. The experimental results of comparison with other methods also show that the method is more accurate in the detection of regression model parameter change points.
 Pricing Asian Options Under Time-Changed Mixed Fractional Brownian Motion with Transactions Costs Yi Ding,Jingjun Guo Acta mathematica scientia,Series A. 2021, 41 (4):  1135-1146.  Abstract ( 14 )   RICH HTML PDF (503KB) ( 13 )   Considering that the classical Black-Scholes(B-S) option pricing model can not describe the characteristics of constant value periodicity and long-term dependence of financial asset prices, the time-changed mixed fractional Brownian motion is used to describe the changes of financial asset prices. By using self-financing delta hedging strategy, the partial differential equation of geometric average Asian call option price in discrete case and the pricing formula of geometric average Asian call and put option are obtained, and the influence of parameters in pricing model on option price is analyzed. Based on the daily closing price of Vanke stock, this article makes an empirical analysis on the established pricing model, and verifies the effectiveness of the pricing model.
 Optimal Investment and Proportional Reinsurance Strategies to Minimize the Probability of Drawdown Under Ambiguity Aversion Yuying Zhao,Yuzhen Wen Acta mathematica scientia,Series A. 2021, 41 (4):  1147-1165.  Abstract ( 12 )   RICH HTML PDF (696KB) ( 10 )   In this paper, we consider the optimal investment and reinsurance control problem for insurers with ambiguity, and we obtain the minimum drawdown probability, optimal robust investment-reinsurance strategies and the associated drift distortion. Moreover, some numerical examples are presented to show the impact of model parameters on the optimal results.
 The Performance Analysis of the $M/G/1$ Queue with Multiple Adaptive Vacations under the Modified Dyadic Min($N, D$)-Policy Min Wang,Yinghui Tang,Shaojun Lan Acta mathematica scientia,Series A. 2021, 41 (4):  1166-1180.  Abstract ( 10 )   RICH HTML PDF (582KB) ( 8 )   This paper considers an $M/G/1$ queueing system with multiple adaptive vacations for exhausted services under the modified dyadic Min($N, D$) in which the server who is on vacation resumes its service if either $N$ customers accumulate in the system or the total workload of the server for all the waiting customers is not less than a given threshold $D$. The essential meaning of the workload of the server for every customer refers to the quantity of events included in the completed service items required by the customer. The unit of measurement for the workload may be a counting unit, a weight unit, etc. According to the well-known stochastic decomposition property of the steady-state queue size, both the probability generating function of the steady-state queue length distribution and the expression of the expected queue length are obtained. Additionally, the mean server busy period and busy cycle period are discussed. Based on the analytical results, the explicit expressions of the expected queue length and the expected length of server busy cycle period for some special cases (e.g., the number of vacations is a fixed positive integer $J$) are derived. Finally, through the renewal theory, the explicit expression of the long-run expected cost per unit time is derived. Meanwhile, numerical examples are provided to determine the optimal joint control policy for economizing the system cost.
 Optimal Control of Initial Distributions in a Hierarchical Size-Structured Population System with Delay Zerong He,Mengjie Han Acta mathematica scientia,Series A. 2021, 41 (4):  1181-1191.  Abstract ( 8 )   RICH HTML PDF (403KB) ( 8 )   This article is concerned with an optimal control problem for a hierarchical size-dependent population model with delay, the control function is the initial distribution. It is expected that the difference between the terminal state and the given target can be minimized in a least costs. The uniform continuity of states in controls is established by the method of characteristic lines and priori estimates, the minimal principle is derived by the construction of a normal cone and an adjoint system, and the existence of unique optimal strategy is proved by means of the Ekeland variational theorem and fixed-point approach. These results pave the way to applications.
 Complex Dynamics and Stochastic Sensitivity Analysis of a Predator-Prey Model with Crowley-Martin Type Functional Response Tengfei Wang,Tao Feng,Xinzhu Meng Acta mathematica scientia,Series A. 2021, 41 (4):  1192-1203.  Abstract ( 11 )   RICH HTML PDF (1439KB) ( 17 )   Predator-prey interactions serve a pivotal role in protecting species diversity. In this study, the parameter $λ$ is presented to show the stochastic dynamics of a predator-prey model. The results show that the predator population tends to become extinct if $λ$ < 0. Furthermore, the solution of the prey population converges weakly to an invariant probability distribution. If $λ$>0, we find that the stochastic system admits a unique ergodic stationary distribution. In addition, stochastic sensitivity analysis is proposed to study the effects of noise on the dynamics of predator-prey populations. Our findings suggest that small-intensity noise can suppress population size enlargement, while large-intensity noise can lead to the extinction of populations.
 Stability and Bifurcation of a Pathogen-Immune Model with Delay and Diffusion Effects Jingnan Wang,Dezhong Yang Acta mathematica scientia,Series A. 2021, 41 (4):  1204-1217.  Abstract ( 15 )   RICH HTML PDF (903KB) ( 37 )   In order to understand the effects of diffusion and time-delay factors on the dynamics between pathogens and immune cells, a delayed pathogen-immune reaction diffusion model with homogeneous Neumann boundary condition is established. By using the diffusion ratio of pathogen-immune cells and immune delay as two parameters, the characteristic root distribution of the linearized system at the positive steady state is analyzed and the necessary and sufficient conditions for the positive steady state to undergo Turing instability and Hopf bifurcation are obtained by using the bifurcation theory of functional differential equations. In addition, the dynamic behavior close to the critical value of Turing instability and Hopf bifurcation is intuitively shown by Matlab numerical simulation. The biological and medicinal significance of corresponding dynamic behaviors are discussed. Furthermore, the obtained results provide certain theoretical support for controlling the growth of pathogen.
 Qualitative Analysis of a Stochastic SIVS Epidemic Model with Nonlinear Perturbations Under Regime Switching Zhonghua Zhang,Qian Zhang Acta mathematica scientia,Series A. 2021, 41 (4):  1218-1234.  Abstract ( 10 )   RICH HTML PDF (696KB) ( 12 )   In this paper, we present a stochastic SIVS epidemic model with nonlinear perturbations under regime switching. For the non-autonomous stochastic SIVS epidemic system with white noise, we provide results regarding the stochastic boundedness, stochastic permanence in mean, and we prove that the system has at least one nontrivial positive T-periodic solution by using Lyapunov function and Hasminskii's theory. For the system with Markov conversion, we establish sufficient conditions for existence of ergodic stationary distribution, and the thresholds for persistence in mean and the extinction of infected persons was obtained, respectively. Finally, some numerical simulations are carried out to support the theoretical results.
 Global Stability of an Epidemic Model with Quarantine and Incomplete Treatment Lixiang Feng,Defen Wang Acta mathematica scientia,Series A. 2021, 41 (4):  1235-1248.  Abstract ( 29 )   RICH HTML PDF (855KB) ( 29 )   An epidemic model with quarantine and incomplete treatment is constructed. The model allows for the susceptibles to the unconscious and conscious susceptible compartment. We establish that the global dynamics are completely detremined by the basic reproduction number $R_{0}$. If $R_{0}≤1$, then the disease free equilibrium is globally asymptotically stable. If $R_{0}>1$, the endemic equilibrium is globally asymptotically stable. Some numerical simulations are also carried out to confirm the analytical results.