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26 February 2023, Volume 43 Issue 1 Previous Issue    Next Issue

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Composition Operators on $ \boldsymbol{H^{p,q,s}}({\Bbb D})$ Chen Hongxin,Zhang Xuejun,Zhou Min Acta mathematica scientia,Series A. 2023, 43 (1):  1-13.  Abstract ( 137 )   RICH HTML PDF (330KB) ( 256 )   Save Let $\varphi$ be an analytic self-map of the unit disc ${\Bbb D}$ in the complex plane ${\Bbb C}$. In this paper, the authors characterize those symbols $\varphi$ such that composition operators $C_{\varphi}$ are bounded or compact on the general Hardy type space $H^{p,q,s}({\Bbb D})$. References | Related Articles | Metrics

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Refined Lower Bound for Sums of Eigenvalues of the Laplace Operator He Yue, Ruan Qihua Acta mathematica scientia,Series A. 2023, 43 (1):  14-26.  Abstract ( 147 )   RICH HTML PDF (407KB) ( 220 )   Save In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain $\Omega$ in $\Bbb R ^n$. It is well known that the $k$-th Dirichlet eigenvalue $\lambda_k(\Omega)$ obeys the Weyl asymptotic formula, that is, $ \lambda_k(\Omega)\sim\frac{4\pi^2}{[\omega_nV(\Omega)]^\frac{2}{n}}k^\frac{2}{n} \qquad\hbox{as}\quad k\rightarrow\infty, $ where $\omega_n$ and $V(\Omega)$ are the volume of $n$-dimensional unit ball in $\Bbb R ^n$ and the volume of $\Omega$ respectively. In view of the above formula, Pólya conjectured that $ \lambda_k(\Omega)\geq\frac{4\pi^2}{[\omega_nV(\Omega)]^\frac{2}{n}}k^\frac{2}{n} \qquad\hbox{for}\quad k\in{\Bbb N}. $ This is the well-known conjecture of Pólya. Studies on this topic have a long history with much work. In particular, one of the more remarkable achievements in recent tens years has been achieved independently by Berezin[2] and Li and Yau[4], respectively. They solved partially the conjecture of Pólya with a slight difference by a factor $n/(n+2)$. Later, Melas[7] improved Berezin-Li-Yau's estimate by adding an additional positive term of the order of $k$ to the right side. Here, following almost the same argument as Melas, we further refine Melas's estimate. References | Related Articles | Metrics

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The Unitary Equivalence of the Toeplitz Operators on the Harmonic Hardy Space Ding Xuanhao, Huang Yuhao, Li Yongning Acta mathematica scientia,Series A. 2023, 43 (1):  27-34.  Abstract ( 106 )   RICH HTML PDF (348KB) ( 222 )   Save Let $H^{2}$ be the Hardy space on the unit disk ${\Bbb D}=\{\xi\in{\Bbb C}:|\xi|<1\}$. Suppose $u$ and $v$ are inner functions and at least one of them is nonconstant, the harmonic Hardy space $H_{u,v}^{2}$ is defined by $H_{u,v}^{2}=uH^{2}\oplus\overline{v}(H^{2})^{\bot}=uH^{2}\oplus\overline{vzH^{2}}.$ For any $x\in H_{u,v}^{2},$ define the Toeplitz operator on the $H_{u,v}^{2}$ by $\widehat{T}_{\varphi}x=Q_{u,v}(\varphi x),$ where $Q_{u,v}$ is the orthogonal projection from $L^{2}\rightarrow H_{u,v}^{2}.$ In this paper, the unitary equivalence of the harmonic Toeplitz operator and the dual truncated Toeplitz operator is obtained, moreover, the sufficient and necessary conditions for when two Toeplitz operators commute is given, and the properties of the harmonic Toeplitz algebra and the commutant of $\widehat{T}_{z}$ are described. Finally, the essential spectrum for the product of finitely many harmonic Toeplitz operators with continuous symbols is obtained in this paper. References | Related Articles | Metrics

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Spectral Geometry of Willmore and Extremal Hypersurfaces Yang Dengyun, Zhang Jinguo, Tao Yongqian Acta mathematica scientia,Series A. 2023, 43 (1):  35-42.  Abstract ( 76 )   RICH HTML PDF (282KB) ( 108 )   Save Let $M$ be a Willmore (or extremal) hypersurface in $S^{n+1}$ with the same squared length of the second fundamental form of Willmore torus $W_{m,n-m}$ (or Clifford torus $C_{m,n-m}$). In this article the authors proved that if $Spec^p(M)=Spec^p(W_{m,n-m})$ (or $Spec^p(M)=Spec^p(C_{m,n-m})$) for $p=0,1,2$, then $M$ is $W_{m,n-m}$ (or $C_{m,m}$). References | Related Articles | Metrics

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A Note on Generalized Douglas-Weyl Spray Zheng Daxiao Acta mathematica scientia,Series A. 2023, 43 (1):  43-52.  Abstract ( 79 )   RICH HTML PDF (251KB) ( 90 )   Save In this paper, we study Generalized Douglas-Weyl spray. We show that a spray $G$ is a Generalized Douglas-Weyl spray if and only if it is $W$-quadratic. As a corollary, we show that a Finsler metric $F$ is a Generalized Douglas-Weyl metric if and only if it is $W$-quadratic. Furthermore, we consider $R$-quadratic spray and prove that a spray $G$ is $R$-quadratic if and only if $\dot{B}^{~i}_{j~kl}=0$. References | Related Articles | Metrics

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Positive Solutions for a High Order Riemann-Liouville Type Fractional Impulsive Differential Equation Integral Boundary Value Problem Xu Jiafa, Yang Zhichun Acta mathematica scientia,Series A. 2023, 43 (1):  53-68.  Abstract ( 77 )   RICH HTML PDF (398KB) ( 125 )   Save In this paper, we study a high order Riemann-Liouville type fractional impulsive differential equation integral boundary value problem involving semipositone the nonlinear and impulsive terms. By virtue of the fixed point index, we obtain the positive solutions theorems under some appropriate superlinear and sublinear growth conditions. The results here extend the existing study. References | Related Articles | Metrics

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Relative Morse Index and Multiple Solutions for Asymptotically Linear Dirac Equation Shan Yuan Acta mathematica scientia,Series A. 2023, 43 (1):  69-81.  Abstract ( 60 )   RICH HTML PDF (352KB) ( 293 )   Save This paper is concerned with the existence and multiplicity of periodic solutions for Dirac equation. We will establish a relative Morse index theory to classify the associated linear Dirac equation. Under a general twist condition for the nonlinear part via the relative Morse index, existence of multiple solutions are obtained. References | Related Articles | Metrics

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Global Attractors for a Class of Reaction-Diffusion Equations with Distribution Derivatives in $\Bbb R ^{n}$ Zhu Kaixuan, Sun Tao, Xie Yongqin Acta mathematica scientia,Series A. 2023, 43 (1):  82-92.  Abstract ( 74 )   RICH HTML PDF (354KB) ( 126 )   Save In this paper, we consider the long-time behavior of solutions for a class of reaction-diffusion equations in $\Bbb R ^{n}$ with weighted terms $V(x)$ and some distribution derivatives in inhomogeneous terms. We prove that the $\big(L^{2}(\Bbb R ^{n}), L^{2}(\Bbb R ^{n})\cap L^{p}(\Bbb R ^{n})\big)$-global attractors indeed can attract every bounded subset of $L^{2}(\Bbb R ^{n})$ with the $L^{2}\cap L^{p+\delta_{1}}$-norm for any $\delta_{1}\in [0,+\infty)$. References | Related Articles | Metrics

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Existence of Multiple Solutions for a Class of Quasilinear Schrödinger Equations Xue Yanfang, Zhu Xincai Acta mathematica scientia,Series A. 2023, 43 (1):  93-100.  Abstract ( 71 )   RICH HTML PDF (394KB) ( 149 )   Save The multiple solutions are studied for a class of Quasilinear Schr?dinger equation under the coercive potential. By using the method of variable substitution, the quasilinear problem is transformed into a semilinear one, then infinitely many high energy solutions of the equation are obtained with the help of the fountain theorem. References | Related Articles | Metrics

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Simple-Pole and Double-Pole Solutions for the Mixed Chen-Lee-Liu Derivative Nonlinear Schrödinger Equation with Nonzero Boundary Conditions Wang Chunjiang, Zhang Jian Acta mathematica scientia,Series A. 2023, 43 (1):  101-122.  Abstract ( 60 )   RICH HTML PDF (693KB) ( 110 )   Save This paper is concerned with simple-pole and double-pole solutions for the mixed Chen-Lee-Liu derivative nonlinear Schr?dinger equation with non-zero boundary conditions at infinity. By solving a direct scattering problem, the Jost eigenfunctions and scattering matrix are given, their symmetries and asymptotic behaviors are also presented. Then the inverse scattering problems are solved in terms of the matrix Riemann-Hilbert method. In addition, the trace formulae for analytic scattering coefficients and theta conditions are derived. Finally, the explicit formulae of double-pole solutions for the equation are obtained. Figures and Tables | References | Related Articles | Metrics

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Limit Behavior of Mass Critical Inhomogeneous Schrödinger Equation at the Threshold Li Deke, Wang Qingxuan Acta mathematica scientia,Series A. 2023, 43 (1):  123-131.  Abstract ( 76 )   RICH HTML PDF (333KB) ( 101 )   Save This paper is concerned with the following time-independent critical inhomogeneous Schr?dinger equation with attractive interactions: $\begin{eqnarray*} -\triangle u+ |x|^2u-\, a m(x)|u|^\frac{4}{N} u= \mu u, \ \ \mbox{in $\Bbb R^N$, $N\geq 1$.} \end{eqnarray*}$ where $a>0$, $0< m(x)\leq 1$. We will show the existence of ground states at the threshold $a=a^*$ for $m(x)=1-\lambda g(x)$ with $\lambda>0$ and suitable $0\leq g(x)<1$, and then investigate the limit behavior of those threshold ground states as $\lambda\rightarrow 0^+$. These conclusions extend the results of Deng-Guo-Lu[10, 11]. In particular, compared to the arguments of [10, 11], we use a direct and simpler method to obtain the lower bound of energy. References | Related Articles | Metrics

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The Rogue Wave Solution of MNLS Equation Based on Hirota's Bi-linear Derivative Transformation Tang Yuxuan, Zhou Guoquan Acta mathematica scientia,Series A. 2023, 43 (1):  132-142.  Abstract ( 123 )   RICH HTML PDF (519KB) ( 178 )   Save The modified nonlinear Schrodinger (MNLS for brevity) equation and the Derivative nonlinear Schrodinger (DNLS for brevity) equation are two nonlinear differential equations that are closely correlated and fully integrable. Firstly, the spatially periodic breather solution of the MNLS equation has been obtained by method of Hirota's bilinear derivative transform, and then its rogue wave solution is also obtained by a long-wave limit of the Akhmediev-type breather, which can be naturally reduced to a rogue wave solution of the DNLS equation by a simple operation of parameters. The existence of global soliton/rogue wave solutions for the MNLS/DNLS equations is briefly discussed. Figures and Tables | References | Related Articles | Metrics

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Nonlocal KP-type Equations with Generalized Bilinear Derivative Zhao Qian, Yan Lu Acta mathematica scientia,Series A. 2023, 43 (1):  143-158.  Abstract ( 57 )   RICH HTML PDF (413KB) ( 72 )   Save A generalized bilinear transformation method, as an extension of bilinear transformation method due to Hirota, is applied to study the nonlinear dispersive equations with nonlocal terms. With this method, the KP-type equations including the KP-II, BKP and (3+1)-dimensional generalized KP equations are studied and their nonlocal forms are derived. As the conclusions, the solitary wave solutions and approximate solutions of those KP-type equations are constructed by developing the bilinear transformation method. References | Related Articles | Metrics

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Global BMO Estimation for the Gradient of Weak Solutions to a Class of Elliptic Obstacle Problems Tong Yuxia, Guo Yanmin, Gu Jiantao Acta mathematica scientia,Series A. 2023, 43 (1):  159-168.  Abstract ( 53 )   RICH HTML PDF (336KB) ( 100 )   Save In this paper, the global BMO estimates for the gradient of weak solutions to a class of elliptic equation obstacle problems is considered by using the Hardy-Littlewood maximal functions, the Jensen inequality for Young function, the perturbation method and other techniques. References | Related Articles | Metrics

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Blow-up of Solutions to the Euler-Poisson-Darboux-Tricomi Equation with a Nonlinear Memory Term Ouyang Baiping Acta mathematica scientia,Series A. 2023, 43 (1):  169-180.  Abstract ( 56 )   RICH HTML PDF (338KB) ( 97 )   Save Blow-up phenomenon of solutions to the Euler-Poisson-Darboux-Tricomi equation with a nonlinear memory term in the subcritical case is studied. By using functional methods associated with a modified Bessel equation, an iteration frame and the first lower bound are derived. Then, nonexistence of global solutions to the Cauchy problem for the Euler-Poisson-Darboux-Tricomi equation and an upper bound estimate of solutions for the lifespan are obtained via the iteration technique. References | Related Articles | Metrics

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On the Optimal Global Estimates of Boundary Blow-up Solutions to the Monge-Ampère Equation Feng Meiqiang, Zhang Xuemei Acta mathematica scientia,Series A. 2023, 43 (1):  181-202.  Abstract ( 68 )   RICH HTML PDF (488KB) ( 87 )   Save This paper is dedicated to studying the optimal global estimates and nonexistence of strictly convex solutions to the boundary blow-up Monge-Ampère problem $ M[u](x)=K(x)f(u) \mbox{ for } x \in \Omega,\; u(x)\rightarrow +\infty \mbox{ as } {\rm dist}(x,\partial \Omega)\rightarrow 0. $ Here $M[u]=\det\, (u_{x_{i}x_{j}})$ is the Monge-Ampère operator, and $\Omega$ denotes a smooth, bounded, strictly convex domain in $ \Bbb R^N (N\geq 2)$. The interesting features in our proof are that we not only obtain the relations among various conditions imposed on $K(x)$ and $f(u)$, but make comparison of some results of global estimates in previous literatures and make clear what conditions lead to what estimations. Moreover, when $\Omega$ is a general region, we give some nonexistence results which is rarely discussed in previous literatures. Figures and Tables | References | Related Articles | Metrics

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Quasi-Neutral Limit of Compressible Electro-Diffusion System with the Different Mobilities Jiang Limin, He Jinman Acta mathematica scientia,Series A. 2023, 43 (1):  203-218.  Abstract ( 46 )   RICH HTML PDF (333KB) ( 61 )   Save In this paper, by using the Sobolev inequality, the Green formula coupling and the elaborate energy method, we study the quasi-neutral limit of compressible Planck-Nernest-Poisson-Navier-Stokes(PNPNS) system with the general mobilities of two kinds of charges, which arises in the electro-hydrodynamics. References | Related Articles | Metrics

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Well-Posedness and Convergence Rates of Three-Dimensional Incompressible Euler Flows in Axisymmetric Nozzles with Symmetric Body Lin Jie, Wang Tianyi Acta mathematica scientia,Series A. 2023, 43 (1):  219-237.  Abstract ( 89 )   RICH HTML PDF (487KB) ( 104 )   Save This paper studies the three-dimensional incompressible Euler flows in axisymmetric nozzles with a symmetric body. The well-posedness is established by stream function method and barrier function. Base on the above well-posedness, the far field convergence rates of the solutions are studied: if the infinite nozzles are the flat boundary outside the finite length, the solution of the equation converges to an asymptotic state at the exponential rate; if the infinite nozzles converge to the flat boundary with the polynomial rates, the solutions converge to the asymptotic states at the same polynomial rates. Figures and Tables | References | Related Articles | Metrics

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Existence and Regularity of Solutions for a Class of Neutral Stochastic Evolution Equations Song Yuying, Fan Hongxia Acta mathematica scientia,Series A. 2023, 43 (1):  238-248.  Abstract ( 59 )   RICH HTML PDF (340KB) ( 84 )   Save In this paper, we study the existence and regularity of solutions for a class of neutral stochastic partial functional integro-differential equations in Hilbert space. By using resolvent operator theory and fixed point theorem, the existence results of mild solutions on Hilbert space $X$ and $X_{\alpha}$ are obtained. It is verified that the mild solution of the equation is its classical solution under some conditions, which generalizes the relevant results. References | Related Articles | Metrics

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Parameter Estimation for Nonlinear Stochastic Differential Equations Driven by $\boldsymbol\alpha$-Stable Processes: Non-ergodic Case Zhang Xuekang, Wan Shanlin, Shu Huisheng Acta mathematica scientia,Series A. 2023, 43 (1):  249-260.  Abstract ( 44 )   RICH HTML PDF (372KB) ( 76 )   Save The present paper deals with the parameter estimation problem for nonlinear stochastic differential equations driven by $\alpha$-stable processes based on continuous-time observation. We first discuss the consistency and the rate of convergence of the weighted trajectory fitting estimator. Then, we have established the asymptotic distribution of the estimator. References | Related Articles | Metrics

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Asymptotic Analysis of a Tumor Model with Angiogenesis and a Periodic Supply of External Nutrients Song Huijuan, Huang Qian, Wang Zejia Acta mathematica scientia,Series A. 2023, 43 (1):  261-273.  Abstract ( 60 )   RICH HTML PDF (414KB) ( 84 )   Save In this paper, we consider a free boundary problem modeling the growth of tumors with angiogenesis and a $\omega$-periodic supply of external nutrients $\phi(t)$. Denote by $S(\sigma)$ the proliferation rate of tumor cells. We first establish the well-posedness and then give a complete classification of asymptotic behavior of solutions according to the sign of $\frac1{\omega}\int_0^\omega S(\phi(t)){\rm d}t$. It is shown that if $\frac1{\omega}\int_0^\omega S(\phi(t)){\rm d}t\le0$, then all evolutionary tumors will finally vanish; the converse is also true. If instead $\frac1{\omega}\int_0^\omega S(\phi(t)){\rm d}t>0$, then there exists a unique and stable positive periodic solution. References | Related Articles | Metrics

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A New Tseng-like Extragradient Algorithm for Common Solutions of Variational Inequalities and Fixed Point Problems Duan Jie, Xia Fuquan Acta mathematica scientia,Series A. 2023, 43 (1):  274-290.  Abstract ( 61 )   RICH HTML PDF (380KB) ( 94 )   Save In this paper, we introduce a new inertial Tseng-like extragradient algorithm for finding a common element of the set of solutions of a variational inequalitiy problem with a pseudomonotone and non-Lipschitz continuous mapping and the set of a fixed point problem with a quasi-nonexpansive mapping in Hilbert spaces. The strong convergence of the sequences generated by the algorithm is proved under some suitable assumptions imposed on the parameters. Finally, numercial experiments are carried out on the algorithm to verify the effectiveness of the algorithm. Figures and Tables | References | Related Articles | Metrics

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Convergence Analysis of Bregman ADMM for Three-Block Nonconvex Indivisible Optimization Problems with Linearization Technique Liu Fuqin, Peng Jianwen, Luo Honglin Acta mathematica scientia,Series A. 2023, 43 (1):  291-304.  Abstract ( 91 )   RICH HTML PDF (362KB) ( 74 )   Save Alternating direction multiplier method is an effective method to solve two separable convex optimization problems, but the convergence of alternating direction multiplier method may not be guaranteed for three nonseparable nonconvex optimization problems. This paper mainly studies the convergence analysis of the linearized generalized Bregman alternating direction multiplier method (L-G-BADMM) for solving the nonconvex minimization problem whose objective function is three indivisible blocks. Under appropriate assumptions, we solve the subproblem of the algorithm and construct a benefit function satisfying Kurdyka-Lojasiewicz property. The convergence of the algorithm can be obtained through theoretical proof. References | Related Articles | Metrics

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Approximate Farkas Lemma and Approximate Duality for Fractional Optimization Problems Xie Feifei, Fang Donghui Acta mathematica scientia,Series A. 2023, 43 (1):  305-320.  Abstract ( 66 )   RICH HTML PDF (347KB) ( 108 )   Save By using the infimal convolution of conjugate functions and the epigraph technique, we introduce some new constraint qualifications. Under those new constraint qualifications, approximate Farkas lemmas and approximate duality results of the fractional optimization problem with conic constraint are established, which extend the corresponding results in the previous papers. References | Related Articles | Metrics