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    25 April 2021, Volume 41 Issue 2 Previous Issue    Next Issue
    Articles
    MARTINGALE INEQUALITIES UNDER G-EXPECTATION AND THEIR APPLICATIONS
    Hanwu LI
    Acta mathematica scientia,Series B. 2021, 41 (2):  349-360.  DOI: 10.1007/s10473-021-0201-6
    Abstract ( 37 )   RICH HTML PDF   Save
    In this paper, we study the martingale inequalities under $G$-expectation and their applications. To this end, we introduce a new kind of random time, called $G$-stopping time, and then investigate the properties of a $G$-martingale (supermartingale) such as the optional sampling theorem and upcrossing inequalities. With the help of these properties, we can show the martingale convergence property under $G$-expectation.
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    REDUCIBILITY FOR A CLASS OF ANALYTIC MULTIPLIERS ON SOBOLEV DISK ALGEBRA
    Yong CHEN, Ya LIU, Chuntao QIN
    Acta mathematica scientia,Series B. 2021, 41 (2):  361-370.  DOI: 10.1007/s10473-021-0202-5
    Abstract ( 13 )   RICH HTML PDF   Save
    We prove the reducibility of analytic multipliers $M_\phi$ with a class of finite Blaschke products symbol $\phi$ on the Sobolev disk algebra $R(\mathbb{D})$. We also describe their nontrivial minimal reducing subspaces.
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    MULTIPLICITY OF PERIODIC SOLUTIONS FOR SECOND ORDER HAMILTONIAN SYSTEMS WITH MIXED NONLINEARITIES
    Mingwei WANG, Fei GUO
    Acta mathematica scientia,Series B. 2021, 41 (2):  371-380.  DOI: 10.1007/s10473-021-0203-4
    The multiplicity of periodic solutions for a class of second order Hamiltonian system with superquadratic plus subquadratic nonlinearity is studied in this paper. Obtained via the Symmetric Mountain Pass Lemma, two results about infinitely many periodic solutions of the systems extend some previously known results.
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    THE TWO-LEVEL STABILIZED FINITE ELEMENT METHOD BASED ON MULTISCALE ENRICHMENT FOR THE STOKES EIGENVALUE PROBLEM
    Juan WEN, Pengzhan HUANG, Ya-Ling HE
    Acta mathematica scientia,Series B. 2021, 41 (2):  381-396.  DOI: 10.1007/s10473-021-0204-3
    In this paper, we first propose a new stabilized finite element method for the Stokes eigenvalue problem. This new method is based on multiscale enrichment, and is derived from the Stokes eigenvalue problem itself. The convergence of this new stabilized method is proved and the optimal priori error estimates for the eigenfunctions and eigenvalues are also obtained. Moreover, we combine this new stabilized finite element method with the two-level method to give a new two-level stabilized finite element method for the Stokes eigenvalue problem. Furthermore, we have proved a priori error estimates for this new two-level stabilized method. Finally, numerical examples confirm our theoretical analysis and validate the high effectiveness of the new methods.
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    EXISTENCE AND BOUNDEDNESS OF SOLUTIONS FOR SYSTEMS OF QUASILINEAR ELLIPTIC EQUATIONS
    Abdelkrim MOUSSAOUI, Jean VELIN
    Acta mathematica scientia,Series B. 2021, 41 (2):  397-412.  DOI: 10.1007/s10473-021-0205-2
    This article sets forth results on the existence and boundedness of solutions for quasilinear elliptic systems involving p-Laplacian and q-Laplacian operators. The approach combines Schaefer's fixed point as well as Moser's iteration procedure.
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    S-ASYMPTOTICALLY BLOCH TYPE PERIODIC SOLUTIONS TO SOME SEMI-LINEAR EVOLUTION EQUATIONS IN BANACH SPACES
    Yong-Kui CHANG, Yanyan WEI
    Acta mathematica scientia,Series B. 2021, 41 (2):  413-425.  DOI: 10.1007/s10473-021-0206-1
    This paper is mainly concerned with the $S$-asymptotically Bloch type periodicity. Firstly, we introduce a new notion of $S$-asymptotically Bloch type periodic functions, which can be seen as a generalization of concepts of $S$-asymptotically $\omega$-periodic functions and $S$-asymptotically $\omega$-anti-periodic functions. Secondly, we establish some fundamental properties on $S$-asymptotically Bloch type periodic functions. Finally, we apply the results obtained to investigate the existence and uniqueness of $S$-asymptotically Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces.
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    THE ENDPOINT ESTIMATE FOR FOURIER INTEGRAL OPERATORS
    Guangqing WANG, Jie YANG, Wenyi CHEN
    Acta mathematica scientia,Series B. 2021, 41 (2):  426-436.  DOI: 10.1007/s10473-021-0207-0
    Let $T_{a,\varphi}$ be a Fourier integral operator defined by the oscillatory integral \begin{eqnarray*} T_{a,\varphi}u(x) &=&\frac{1}{(2\pi)^n}\int_{\mathbb{R}^n} e^{ {\rm i} \varphi(x,\xi)}a(x,\xi) \hat{u}(\xi){\rm d}\xi, \end{eqnarray*} where $a\in S^{m}_{\varrho,\delta}$ and $\varphi\in \Phi^{2}$, satisfying the strong non-degenerate condition. It is shown that if $0<\varrho\leq1$, $0\leq\delta<1$ and $m\leq \frac{\varrho^{2}-n}{2}$, then $T_{a,\varphi}$ is a bounded operator from $L^{\infty}(\mathbb{R}^n)$ to ${\rm BMO}(\mathbb{R}^n).$
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    MAXIMUM PRINCIPLE FOR STOCHASTIC OPTIMAL CONTROL PROBLEM WITH DISTRIBUTED DELAYS
    Qixia ZHANG
    Acta mathematica scientia,Series B. 2021, 41 (2):  437-449.  DOI: 10.1007/s10473-021-0208-z
    This paper is concerned with a Pontryagin's maximum principle for the stochastic optimal control problem with distributed delays given by integrals of not necessarily linear functions of state or control variables. By virtue of the duality method and the generalized anticipated backward stochastic differential equations, we establish a necessary maximum principle and a sufficient verification theorem. In particular, we deal with the controlled stochastic system where the distributed delays enter both the state and the control. To explain the theoretical results, we apply them to a dynamic advertising problem.
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    TIME GLOBAL MILD SOLUTIONS OF NAVIER-STOKES-OSEEN EQUATIONS
    Viet Duoc TRINH
    Acta mathematica scientia,Series B. 2021, 41 (2):  450-460.  DOI: 10.1007/s10473-021-0209-y
    In this paper we prove the existence and uniqueness of time global mild solutions to the Navier-Stokes-Oseen equations, which describes dynamics of incompressible viscous fluid flows passing a translating and rotating obstacle, in the solenoidal Lorentz space $L_{\sigma, {\rm{w}}}^3$. Besides, boundedness and polynomial stability of these solutions are also shown.
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    THE BALL-COVERING PROPERTY ON DUAL SPACES AND BANACH SEQUENCE SPACES
    Shaoqiang SHANG
    Acta mathematica scientia,Series B. 2021, 41 (2):  461-474.  DOI: 10.1007/s10473-021-0210-5
    In this paper, we prove that $(X,p)$ is separable if and only if there exists a $w^{*}$-lower semicontinuous norm sequence $\{ {p_n}\} _{n = 1}^\infty $ of $(X^{*},p)$ such that (1) there exists a dense subset $G_{n}$ of $X^{*}$ such that $p_{n}$ is G$\mathrm{\hat{a}}$teaux differentiable on $G_{n}$ and $dp_{n}(G_{n})\subset X$ for all $n\in N$; (2) $p_n \leq p$ and $p_n \to p$ uniformly on each bounded subset of $X^{*}$; (3) for any $\alpha\in(0,1)$, there exists a ball-covering $\{ B({x_{i,n}^{*}},{r_{i,n}})\} _{i = 1}^\infty $ of $(X^{*},p_{n})$ such that it is $\alpha$-off the origin and ${x_{i,n}^{*}}\in G_{n}$. Moreover, we also prove that if $ X_{i}$ is a G$\mathrm{\hat{a}}$teaux differentiability space, then there exist a real number $\alpha > 0$ and a ball-covering $\mathfrak{B_{i}}$ of $X_{i}$ such that $\mathfrak{B_{i}}$ is $\alpha $-off the origin if and only if there exist a real number $\alpha > 0$ and a ball-covering $\mathfrak{B}$ of ${l^\infty }({X_i})$ such that $\mathfrak{B}$ is $\alpha$-off the origin.
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    ON GENERALIZED COMPLETE (p,q)-ELLIPTIC INTEGRALS
    Li YIN, Barkat Ali BHAYO, Nihat Gökhan GÖĞüŞ
    Acta mathematica scientia,Series B. 2021, 41 (2):  475-486.  DOI: 10.1007/s10473-021-0211-4
    In this paper, we study the generalized complete (p,q)-elliptic integrals of the first and second kind as an application of generalized trigonometric functions with two parameters, and establish the monotonicity, generalized convexity and concavity of these functions. In particular, some Tur\'an type inequalities are given. Finally, we also show some new series representations of these functions by applying Alzer and Richard's methods.
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    A BRAY-BRENDLE-NEVES TYPE INEQUALITY FOR A RIEMANNIAN MANIFOLD
    Hongcun DENG
    Acta mathematica scientia,Series B. 2021, 41 (2):  487-492.  DOI: 10.1007/s10473-021-0212-3
    In this paper, for any local area-minimizing closed hypersurface $\Sigma$ with $Rc_{\Sigma}=\frac{R_\Sigma}{n}g_{\Sigma}$, immersed in a $(n+1)$-dimension Riemannian manifold $M$ which has positive scalar curvature and nonnegative Ricci curvature, we obtain an upper bound for the area of $\Sigma$. In particular, when $\Sigma$ saturates the corresponding upper bound, $\Sigma$ is isometric to $\mathbb{S}^n$ and $M$ splits in a neighborhood of $\Sigma$. At the end of the paper, we also give the global version of this result.
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    MULTIPLE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SCHRÖDINGER EQUATIONS WITH SATURABLE NONLINEARITY
    Zhongyuan LIU
    Acta mathematica scientia,Series B. 2021, 41 (2):  493-504.  DOI: 10.1007/s10473-021-0213-2
    In this paper, we construct sign-changing radial solutions for a class of Schrödinger equations with saturable nonlinearity which arises from several models in mathematical physics. More precisely, for any given nonnegative integer $k$, by using a minimization argument, we first obtain a sign-changing minimizer with $k$ nodes of a constrained minimization problem, and show, by a deformation lemma and Miranda's theorem, that the minimizer is the desired solution.
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    ON THE DIFFERENTIAL AND DIFFERENCE INDEPENDENCE OF Γ AND ζ
    Wei CHEN, Qiong WANG
    Acta mathematica scientia,Series B. 2021, 41 (2):  505-516.  DOI: 10.1007/s10473-021-0214-1
    In this paper, we study the algebraic differential and the difference independence between the Riemann zeta function and the Euler gamma function. It is proved that the Riemann zeta function and the Euler gamma function cannot satisfy a class of nontrivial algebraic differential equations and algebraic difference equations.
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    THE LEAST SQUARES ESTIMATOR FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A HERMITE PROCESS WITH A PERIODIC MEAN
    Guangjun SHEN, Qian YU, Zheng TANG
    Acta mathematica scientia,Series B. 2021, 41 (2):  517-534.  DOI: 10.1007/s10473-021-0215-0
    We consider the least square estimator for the parameters of Ornstein-Uhlenbeck processes $${\rm d}Y_s=\Big (\sum\limits_{j=1}^{k}\mu_j \phi_j (s)- \beta Y_s\Big){\rm d}s + {\rm d}Z_s^{q,H},$$ driven by the Hermite process $Z_s^{q,H}$ with order $q \geq 1$ and a Hurst index $H \in (\frac12,1)$, where the periodic functions $\phi_j(s), j=1,\ldots,k$ are bounded, and the real numbers $\mu_j, j=1,\ldots, k$ together with $\beta>0$ are unknown parameters. We establish the consistency of a least squares estimation and obtain the asymptotic behavior for the estimator. We also introduce alternative estimators, which can be looked upon as an application of the least squares estimator. In terms of the fractional Ornstein-Uhlenbeck processes with periodic mean, our work can be regarded as its non-Gaussian extension.
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    COMPARISON THEOREMS FOR MULTI-DIMENSIONAL GENERAL MEAN-FIELD BDSDES
    Juan LI, Chuanzhi XING, Ying PENG
    Acta mathematica scientia,Series B. 2021, 41 (2):  535-551.  DOI: 10.1007/s10473-021-0216-z
    In this paper we study multi-dimensional mean-field backward doubly stochastic differential equations (BDSDEs), that is, BDSDEs whose coefficients depend not only on the solution processes but also on their law. The first part of the paper is devoted to the comparison theorem for multi-dimensional mean-field BDSDEs with Lipschitz conditions. With the help of the comparison result for the Lipschitz case we prove the existence of a solution for multi-dimensional mean-field BDSDEs with an only continuous drift coefficient of linear growth, and we also extend the comparison theorem to such BDSDEs with a continuous coefficient.
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    DYNAMICS OF A NONLOCAL DISPERSAL FOOT-AND-MOUTH DISEASE MODEL IN A SPATIALLY HETEROGENEOUS ENVIRONMENT
    Xiaoyan WANG, Junyuan YANG
    Acta mathematica scientia,Series B. 2021, 41 (2):  552-572.  DOI: 10.1007/s10473-021-0217-y
    Foot-and-mouth disease is one of the major contagious zoonotic diseases in the world. It is caused by various species of the genus Aphthovirus of the family Picornavirus, and it always brings a large number of infections and heavy financial losses. The disease has become a major public health concern. In this paper, we propose a nonlocal foot-and-mouth disease model in a spatially heterogeneous environment, which couples virus-to-animals and animals-to-animals transmission pathways, and investigate the dynamics of the disperal. The basic reproduction number $\mathcal R_0$ is defined as the spectral radius of the next generation operator $\mathcal R(x)$ by a renewal equation. The relationship between $\mathcal R_0$ and a principal eigenvalue of an operator $\mathcal L_0$ is built. Moreover, the proposed system exhibits threshold dynamics in terms of $\mathcal R_0,$ in the sense that $\mathcal R_0$ determines whether or not foot-and-mouth disease invades the hosts. Through numerical simulations, we have found that increasing animals' movements is an effective control measure for preventing prevalence of the disease.
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    PARAMETER ESTIMATION FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A GENERAL GAUSSIAN NOISE
    Yong CHEN, Hongjuan ZHOU
    Acta mathematica scientia,Series B. 2021, 41 (2):  573-595.  DOI: 10.1007/s10473-021-0218-x
    In this paper, we consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\, s)=\mathbb{E}[G_t G_s]$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other of which is bounded by $(ts)^{\beta-1}$ up to a constant factor. This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or to have stationary increments; some examples of this include the subfractional Brownian motion and the bi-fractional Brownian motion. Under this assumption, we study the parameter estimation for a drift parameter in the Ornstein-Uhlenbeck process driven by the Gaussian noise $(G_t)_{t\ge 0}$. For the least squares estimator and the second moment estimator constructed from the continuous observations, we prove the strong consistency and the asympotic normality, and obtain the Berry-Esséen bounds. The proof is based on the inner product's representation of the Hilbert space $\mathfrak{H}$ associated with the Gaussian noise $(G_t)_{t\ge 0}$, and the estimation of the inner product based on the results of the Hilbert space associated with the fractional Brownian motion.
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    TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS
    Yali PAN, Qingying XUE
    Acta mathematica scientia,Series B. 2021, 41 (2):  596-608.  DOI: 10.1007/s10473-021-0219-9
    Let $ 0<\beta <1$ and $\Omega$ be a proper open and non-empty subset of $\mathbf{R}^n$. In this paper, the object of our investigation is the multilinear local maximal operator $\mathcal{M}_{\beta}$, defined by $$\mathcal{M}_{\beta}(\vec{f})(x)= \sup_{\substack{Q \ni x \\ Q\in{\mathcal{F}_{\beta}}}} \prod_{i=1}^m \frac{1}{|Q|} \int_Q |f_i(y_i)|{\rm d}y_i,$$ where $\mathcal{F}_{\beta}=\{Q(x,l):x \in \Omega, l< \beta {\rm d}(x, \Omega^c)\}$, $Q=Q(x,l)$ is denoted as a cube with sides parallel to the axes, and $x$ and $l$ denote its center and half its side length. Two-weight characterizations for the multilinear local maximal operator $\mathcal{M}_{\beta}$ are obtained. A formulation of the Carleson embedding theorem in the multilinear setting is proved.
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    LONG-TIME BEHAVIOR FOR A THERMOELASTIC MICROBEAM PROBLEM WITH TIME DELAY AND THE COLEMAN-GURTIN THERMAL LAW
    Wenjun LIU, Dongqin CHEN, Zhijing CHEN
    Acta mathematica scientia,Series B. 2021, 41 (2):  609-632.  DOI: 10.1007/s10473-021-0220-3
    This study addresses long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law, the convolution kernel of which entails an extremely weak dissipation in the thermal law. By using the semigroup theory, we first establish the existence of global weak and strong solutions as well as their continuous dependence on the initial data in appropriate function spaces, under suitable assumptions on the weight of time delay term, the external force term and the nonlinear term. We then prove that the system is quasi-stable and has a gradient on bounded variant sets, and obtain the existence of a global attractor whose fractal dimension is finite. A result on the exponential attractor of the system is also proved.
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    REGULARITY OF P-HARMONIC MAPPINGS INTO NPC SPACES
    Changyu GUO, Changlin XIANG
    Acta mathematica scientia,Series B. 2021, 41 (2):  633-645.  DOI: 10.1007/s10473-021-0221-2
    Let $M$ be a $C^2$-smooth Riemannian manifold with boundary and $X$ be a metric space with non-positive curvature in the sense of Alexandrov. Let $u\colon M\to X$ be a Sobolev mapping in the sense of Korevaar and Schoen. In this short note, we introduce a notion of $p$-energy for $u$ which is slightly different from the original definition of Korevaar and Schoen. We show that each minimizing $p$-harmonic mapping ($p\geq 2$) associated to our notion of $p$-energy is locally H\"older continuous whenever its image lies in a compact subset of $X$.
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    ENTIRE FUNCTIONS REPRESENTED BY LAPLACE-STIELTJES TRANSFORMS CONCERNING THE APPROXIMATION AND GENERALIZED ORDER
    Hongyan XU, Yinying KONG
    Acta mathematica scientia,Series B. 2021, 41 (2):  646-656.  DOI: 10.1007/s10473-021-0222-1
    The first aim of this paper is to investigate the growth of the entire function defined by the Laplace-Stieltjes transform converges on the whole complex plane. By introducing the concept of generalized order, we obtain two equivalence theorems of Laplace-Stieltjes transforms related to the generalized order, $A_n^*$ and $\lambda_n$. The second purpose of this paper is to study the problem on the approximation of this Laplace-Stieltjes transform. We also obtain some theorems about the generalized order, the error, and the coefficients of Laplace-Stieltjes transforms, which are generalization and improvement of the previous results.
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