#### Table of Content

26 April 2019, Volume 39 Issue 2
 The Invariance of Two Subclasses of Biholomorphic Mappings Under the Roper-Suffridge Extension Operators Chaojun Wang,Yanyan Cui,Hao Liu Acta mathematica scientia,Series A. 2019, 39 (2):  209-219.  Abstract ( 46 )   RICH HTML PDF (360KB) ( 74 )   In this paper, we generalize the Roper-Suffridge operator on Bergman-Hartogs domains. Applying the geometric properties and the growth theorems of spirallike mappings of type β and order α as well as almost starlike mappings of complex order λ, we obtain that the generalized Roper-Suffridge operators preserve spirallikeness of type β and order α as well as almost starlikeness of complex order λ on Bergman-Hartogs domains which lead to some special cases. The conclusions provide new approaches to construct spirallike mappings of type β and order α and almost starlike mappings of complex order λ in several complex variables.
 Milin Coefficient Estimation and Adjacent Coefficient Problem for Bazilevič Functions of Type α and Order β Xiaomeng Niu,Shuhai Li Acta mathematica scientia,Series A. 2019, 39 (2):  220-234.  Abstract ( 28 )   RICH HTML PDF (418KB) ( 60 )   In this paper, we study the Milin coefficients of Bazilevič functions of type α and order β and obtain estimate of the difference of moduli of adjacent coefficients for functions in this class. The accurate results are obtained. The results present here generalize some known results. In addition, as a special case, the geometric features of Milin function are given.
 Coincidence Points and Common Fixed Points for Mappings with ϕ-Contractive Conditions on Metric Spaces Yongjie Piao Acta mathematica scientia,Series A. 2019, 39 (2):  235-243.  Abstract ( 41 )   RICH HTML PDF (273KB) ( 75 )   We obtain the existence theorems of common fixed points and coincidence points for multi-valued mappings and single-valued mappings satisfying ϕ-contractive type conditions on metric spaces, also give several fixed point theorems.
 A Trace Formula for Integro-Differential Operators Shirong Chen Acta mathematica scientia,Series A. 2019, 39 (2):  244-252.  Abstract ( 26 )   RICH HTML PDF (269KB) ( 29 )   The trace formulae for the integro-differential operator are studied, which have many applications in the inverse problem, the numerical computation of eigenvalues and the theory of integrable system, etc. The trace formula for integro-differential operators with Dirichlet-Robin boundary conditions or Dirichlet boundary conditions are obtained.
 A Quantitative Weighted Estimate for Bilinear Fourier Multiplier Operators Aiwen Sun,Meng Qu,Min Wang Acta mathematica scientia,Series A. 2019, 39 (2):  253-263.  Abstract ( 26 )   RICH HTML PDF (342KB) ( 26 )   In this paper, we obtain the quantitative estimate with multiple-weight for bilinear Fourier multiplier operator, by dominating the bilinear Fourier multiplier operator by sparse operator, and establishing the weighted estimate for sparse operator. It improves the result in[1].
 Least Energy Solution for Nonlinear Kirchhoff Type Elliptic Equation Zhide Liu,Zhengping Wang Acta mathematica scientia,Series A. 2019, 39 (2):  264-276.  Abstract ( 28 )   RICH HTML PDF (350KB) ( 43 )   In this paper, we study the existence of nontrivial solution and nonnegative least energy solution for the following nonlinear Kirchhoff type elliptic equation \left\{ \begin{align} & -(a+b\int_{{{\mathbb{R}}^{3}}}{|}\nabla u{{|}^{2}}\text{d}x)\Delta u+V(x)u=\mu u+|u{{|}^{p-1}}u,\ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{3}}, \\ & u\in {{H}^{1}}({{\mathbb{R}}^{3}}),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{3}}, \\ \end{align} \right. where $p\in (3, 5)$, $a, b>0$, $V\in C(\mathbb{R} ^3, \mathbb{R} ^+)$ and $\lim\limits_{|x|\to +\infty}V(x)=\infty$. By using variational methods, firstly we prove that for any $b>0$, there exists $\delta(b)>0$ such that problem (0.1) (0.1) with $\mu_1\leq\mu <\mu_1+\delta(b)$ has a nontrivial solution, where $\mu_1$ denotes the first eigenvalue of the Schrödinger operator $-\triangle+V$. Secondly, we show that there exists $\delta_1(b)\in(0, \delta(b))$ such that problem (0.1) (0.1) with $\mu_1 <\mu <\mu_1+\delta_1(b)$ has a nonnegative least energy solution. Finally, by using the symmetric Mountain Pass lemma we prove that problem (0.1) (0.1) has infinitely many nontrivial solutions for any $\mu\in \mathbb{R}$.
 Periodic Solution for p-Laplacian Liénard Equation with Attractive Singularity and Time-Dependent Deviating Argument Zhibo Cheng,Zhonghua Bi,Shaowen Yao Acta mathematica scientia,Series A. 2019, 39 (2):  277-285.  Abstract ( 28 )   RICH HTML PDF (298KB) ( 27 )   In this paper, we consider a kind of $p$-Laplacian singular Liénard equation with time-dependent deviating argument $(\varphi_p(x'(t)))'+f(x(t))x'(t)+g(t, x(t-\sigma(t)))=e(t),$ where $g$ has a attractive singularity at $x=0$. By applications of Manásevich-Mawhin continuation theorem and some analysis skills, sufficient conditions for the existence of periodic solution is established.
 Multiple Solutions for Quasilinear Nonhomogeneous Elliptic Equations with a Parameter Hongxue Song,Yunfeng Wei Acta mathematica scientia,Series A. 2019, 39 (2):  286-296.  Abstract ( 27 )   RICH HTML PDF (387KB) ( 18 )   In this paper, we study the following quasilinear nonhomogeneous elliptic equations of the form $-\triangle_{p}u-\triangle_{p}(|u|^{2\alpha})|u|^{2\alpha -2}u + V(x)|u|^{p-2}u=h(u)+g(x), ~~~x\in \mathbb{R}^{N},$ where $1  Generalized Solution to the Singular Perturbation Problem for a Class of Nonlinear Differential-Integral Time Delay Reaction Diffusion System Xianglin Han,Weigang Wang,Jiaqi Mo Acta mathematica scientia,Series A. 2019, 39 (2): 297-306. Abstract ( 31 ) RICH HTML PDF (361KB) ( 18 ) A class of nonlinear differential-integral system for the singular perturbation generalized reaction diffusion equations with time delay is considered. Under suitable conditions, the asymptotic expansions of generalized solution to the initial boundary problem is obtained by using the singular perturbation method. And the theory of differential inequality for generalized solution is constructed. Corresponding existence and the uniformly validity of the asymptotic expansion for the solution are proved.  The Non-Existence of Non-Radial Blow-Up Solutions for the Quasilinear Elliptic System Ting Ji,Lianggen Hu,Jing Zeng Acta mathematica scientia,Series A. 2019, 39 (2): 307-315. Abstract ( 17 ) RICH HTML PDF (336KB) ( 29 ) In this paper, we consider the following quasilinear elliptic system$\begin{eqnarray*}\Delta_{p_i}u_i+\zeta_i (|x|)|\nabla u_i|^{p_i-1}=\eta_i(|x|)f_i (u_1, \cdots, u_m), \;\\mbox{in}\;\ \mathbb{R} ^N, \end{eqnarray*}$where$i=1, \cdots, m$,$p_i\ge 2$,$\zeta_i$and$\eta_i$are positive continuous functions, and$f_i$is a non-negative continuous function and nondecreasing in each component for every$i\in \{1, 2, \cdots, m\}\$. After using some new comparison principle, we are able to show that the system does not admit any nonradial blow-up solutions.
 Remarks on Regularity Criteria for 3D Generalized MHD Equations and Boussinesq Equations Hua Qiu,Changping Xie,Shaomei Fang Acta mathematica scientia,Series A. 2019, 39 (2):  316-328.  Abstract ( 27 )   RICH HTML PDF (357KB) ( 22 )   In this paper, we study the 3D generalized MHD system with dissipation and diffusion in terms of fractional Laplacian. We obtain a regularity criterion of solution for the generalized MHD equations in terms of the summation of velocity field u and magnetic field b under the framework of homogeneous Besov space with negative indices, which extends the previous results. We also present a regularity criterion of smooth solution to the 3D generalized Boussinesq equations with fractional dissipation in terms of the gradient of velocity only.
 A Nonconforming Locking-Free Triangular Prism Element Analysis for Linear Elasticity Problem Yanping Sun,Shaochun Chen Acta mathematica scientia,Series A. 2019, 39 (2):  329-338.  Abstract ( 21 )   RICH HTML PDF (393KB) ( 16 )   This paper discuss the linear elasticity problem and constructs a nonconforming triangular prism element with 18 degrees of freedom. The shape functions of this element satisfy that the divergence of displacement is zeroth polynomial. We can deduce that the energy norm has the first order convergence rate and the L2 norm has the second order convergence rate.
 Geometric Analysis of a Class of the New Chaotic System Xiezhen Huang,Yongjian Liu,Qiujian Huang Acta mathematica scientia,Series A. 2019, 39 (2):  339-347.  Abstract ( 23 )   RICH HTML PDF (1574KB) ( 24 )   Based on Poincaré compactification technology, the global dynamics behavior of three dimensional chaotic system is studied. The results show that the equilibria at infinity are unstable and highly degraded. The controlled system with a linear controller which does not change the singularity structure has a bunch of degenerate singular orbits. The chaotic attractors for the system in the case of small parameters b and c are found numerically, and thus the nearby singularly degenerate heteroclinic cycles. It is hoped that the investigation of this paper will be quite beneficial for further studies of the geometrical structure for the chaotic attractor.
 Jeep Problems with Container Restriction Xu Cheng,Yiming Ding,Xiangying Hua Acta mathematica scientia,Series A. 2019, 39 (2):  348-357.  Abstract ( 26 )   RICH HTML PDF (391KB) ( 19 )   Jeep problems with and without container restriction are studied. Optimal strategies of problems without container restriction are obtained for one-way and round trip. Strategies of jeep problems with container restriction are also given for one-way trip and round trip respectively, from which we prove that jeep with container restriction can travel any distance if enough fuel is available. To get optimal strategy of jeep problems with container restriction is a challenging task.
 Hopf Bifurcation of Delayed Density-Dependent Predator-Prey Model Haiyin Li Acta mathematica scientia,Series A. 2019, 39 (2):  358-371.  Abstract ( 46 )   RICH HTML PDF (460KB) ( 38 )   In this paper, we investigate stability and Hopf bifurcation of a delayed density-dependent predator-prey system with Beddington-DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered such that the studied predator-prey system conforms to the realistically biological environment. Firstly, stability transformation of the system was given to prepare for discussion of bifurcating periodic solution. Secondly, we discussed properties of Hopf bifurcation about bifurcating direction, stability and period by center manifold theorem and normal form theory. Finally, an example with numerical simulations is given to illustrate stability transformation and Hopf bifurcation of the system.
 Research on the Application of Central Scheme in Saint-Venant System Jian Dong Acta mathematica scientia,Series A. 2019, 39 (2):  372-385.  Abstract ( 30 )   RICH HTML PDF (1085KB) ( 27 )   Shallow water wave equations have high requirements for their numerical schemes. In practical applications, we are more concerned with the behavior in the vicinity of the steady-state solution, especially when the dry and wet fronts occur in the computation area. In this case, not only the scheme is required to be well-balanced, but also the water depth needs to be kept non-negative, at the same time, the scheme is required with high resolution. Therefore, it is difficult to design a numerical scheme that is both well-balanced and positivity preserving. The core of the dissertation is to summarize and study the central schemes of the concerned shallow water wave equations:the KP scheme, the BCKN scheme and the T scheme. We investigate their advantages and disadvantages. We use the one-dimensional shallow water wave equations to show the applications of the each scheme to some examples.
 The Flow Equations of the Modified Discrete KP Hierarchy Xiaoyi Wang,Jipeng Cheng,Hongyun Li Acta mathematica scientia,Series A. 2019, 39 (2):  386-392.  Abstract ( 25 )   RICH HTML PDF (216KB) ( 26 )   In this paper, we mainly study the flow equations of the modified discrete KP hierarchy, and derive a general formula of the flow equation.
 Static Multidimensional Risk Measures Research Hongwei Liu,Caibo Xiao,Yijun Hu Acta mathematica scientia,Series A. 2019, 39 (2):  393-401.  Abstract ( 27 )   RICH HTML PDF (354KB) ( 25 )   In this paper, static risk measures is established in the multidimensional framework, the concepts of multidimensional monetary risk measures and acceptable set are introduced, and the relationships between multidimensional risk measures and acceptable set are investigated. Finally, the representation theorem of multidimensional risk measures is provided, some properties of multidimensional risk measures are given.
 Propagation of Computer Virus on Generalized Networks Chunming Zhang,Dongyao Chen Acta mathematica scientia,Series A. 2019, 39 (2):  402-416.  Abstract ( 30 )   RICH HTML PDF (1319KB) ( 27 )   To explore the mechanism that computer viruses spread on generalized networks, this paper proposes a novel nonlinear and a novel linear computer viruses propagation models. Theoretical analysis demonstrates that the maximum eigenvalue of the network is a vital parameter determining the computer viruses propagation. Then, the global stability of virus-free equilibriums in both nonlinear and linear models has been proved. The global attractivity of the viral equilibrium of the linear model has also been proved. Eventually, some numerical simulation results verify the main conclusions of the theoretical analysis.