#### Table of Content

25 October 2021, Volume 41 Issue 5
Articles
 THE UNIQUENESS OF THE Lp MINKOWSKI PROBLEM FOR q-TORSIONAL RIGIDITY Guangling SUN, Lu XU, Ping ZHANG Acta mathematica scientia,Series B. 2021, 41 (5):  1405-1416.  DOI: 10.1007/s10473-021-0501-x In this paper, we prove the uniqueness of the Lp Minkowski problem for q-torsional rigidity with p>1 and q>1 in smooth case. Meanwhile, the Lp Brunn-Minkowski inequality and the Lp Hadamard variational formula for q-torsional rigidity are established.
 RIGIDITY RESULTS FOR SELF-SHRINKING SURFACES IN $\mathbb{R}^4$ Xuyong JIANG, Hejun SUN, Peibiao ZHAO Acta mathematica scientia,Series B. 2021, 41 (5):  1417-1427.  DOI: 10.1007/s10473-021-0502-9 In this paper, we give some rigidity results for complete self-shrinking surfaces properly immersed in $\mathbb{R}^4$ under some assumptions regarding their Gauss images. More precisely, we prove that this has to be a plane, provided that the images of either Gauss map projection lies in an open hemisphere or $\mathbb{S}^2(1/\sqrt{2})\backslash \bar{\mathbb{S}}^1_+(1/\sqrt{2})$. We also give the classification of complete self-shrinking surfaces properly immersed in $\mathbb{R}^4$ provided that the images of Gauss map projection lies in some closed hemispheres. As an application of the above results, we give a new proof for the result of Zhou. Moreover, we establish a Bernstein-type theorem.
 GLOBAL STRONG SOLUTION AND EXPONENTIAL DECAY OF 3D NONHOMOGENEOUS ASYMMETRIC FLUID EQUATIONS WITH VACUUM Guochun WU, Xin ZHONG Acta mathematica scientia,Series B. 2021, 41 (5):  1428-1444.  DOI: 10.1007/s10473-021-0503-8 We prove the global existence and exponential decay of strong solutions to the three-dimensional nonhomogeneous asymmetric fluid equations with nonnegative density provided that the initial total energy is suitably small. Note that although the system degenerates near vacuum, there is no need to require compatibility conditions for the initial data via time-weighted techniques.
 CONTINUOUS TIME MIXED STATE BRANCHING PROCESSES AND STOCHASTIC EQUATIONS Shukai CHEN, Zenghu LI Acta mathematica scientia,Series B. 2021, 41 (5):  1445-1473.  DOI: 10.1007/s10473-021-0504-7 A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes. The process can also be obtained by the pathwise unique solution to a stochastic equation system. From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup. Meanwhile, we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.
 SOME OSCILLATION CRITERIA FOR A CLASS OF HIGHER ORDER NONLINEAR DYNAMIC EQUATIONS WITH A DELAY ARGUMENT ON TIME SCALES Xin WU Acta mathematica scientia,Series B. 2021, 41 (5):  1474-1492.  DOI: 10.1007/s10473-021-0505-6 In this paper, we establish some oscillation criteria for higher order nonlinear delay dynamic equations of the form \begin{align*}[r_n\varphi(\cdots r_2(r_1x^{\Delta})^{\Delta}\cdots)^{\Delta}]^{\Delta}(t)+h(t)f(x(\tau(t)))=0 \end{align*} on an arbitrary time scale $\mathbb{T}$ with $\sup\mathbb{T}=\infty$, where $n\geq 2$, $\varphi(u)=|u|^{\gamma}$sgn$(u)$ for $\gamma>0$, $r_i(1\leq i\leq n)$ are positive rd-continuous functions and $h\in {\mathrm{C}_{\mathrm{rd}}}(\mathbb{T},(0,\infty))$. The function $\tau\in {\mathrm{C}_{\mathrm{rd}}}(\mathbb{T},\mathbb{T})$ satisfies $\tau(t)\leq t$ and $\lim\limits_{t\rightarrow\infty}\tau(t)=\infty$ and $f\in {\mathrm{C}}(\mathbb{R},\mathbb{R})$. By using a generalized Riccati transformation, we give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero. The obtained results are new for the corresponding higher order differential equations and difference equations. In the end, some applications and examples are provided to illustrate the importance of the main results.
 COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES Yuqi SUN, Wen ZHANG Acta mathematica scientia,Series B. 2021, 41 (5):  1493-1502.  DOI: 10.1007/s10473-021-0506-5 Assume that $X$ and $Y$ are real Banach spaces with the same finite dimension. In this paper we show that if a standard coarse isometry $f:X\rightarrow Y$ satisfies an integral convergence condition or weak stability on a basis, then there exists a surjective linear isometry $U:X\rightarrow Y$ such that $\|f(x)-Ux\|=o(\|x\|)$ as $\|x\|\rightarrow\infty$. This is a generalization about the result of Lindenstrauss and Szankowski on the same finite dimensional Banach spaces without the assumption of surjectivity. As a consequence, we also obtain a stability result for $\varepsilon$-isometries which was established by Dilworth.
 ZERO KINEMATIC VISCOSITY-MAGNETIC DIFFUSION LIMIT OF THE INCOMPRESSIBLE VISCOUS MAGNETOHYDRODYNAMIC EQUATIONS WITH NAVIER BOUNDARY CONDITIONS Fucai LI, Zhipeng ZHANG Acta mathematica scientia,Series B. 2021, 41 (5):  1503-1536.  DOI: 10.1007/s10473-021-0507-4 We investigate the uniform regularity and zero kinematic viscosity-magnetic diffusion limit for the incompressible viscous magnetohydrodynamic equations with the Navier boundary conditions on the velocity and perfectly conducting conditions on the magnetic field in a smooth bounded domain $\Omega\subset\mathbb{R}^3$. It is shown that there exists a unique strong solution to the incompressible viscous magnetohydrodynamic equations in a finite time interval which is independent of the viscosity coefficient and the magnetic diffusivity coefficient. The solution is uniformly bounded in a conormal Sobolev space and $W^{1,\infty}(\Omega)$ which allows us to take the zero kinematic viscosity-magnetic diffusion limit. Moreover, we also get the rates of convergence in $L^\infty(0,T; L^2)$, $L^\infty(0,T; W^{1,p})\,(2\leq p<\infty)$, and $L^\infty((0,T)\times \Omega)$ for some $T>0$.
 THE PRECISE NORM OF A CLASS OF FORELLI-RUDIN TYPE OPERATORS ON THE SIEGEL UPPER HALF SPACE Lifang ZHOU, Yun FAN, Jin LU Acta mathematica scientia,Series B. 2021, 41 (5):  1537-1546.  DOI: 10.1007/s10473-021-0508-3 The precise Lp norm of a class of Forelli-Rudin type operators on the Siegel upper half space is given in this paper. The main result not only implies the upper Lp norm estimate of the Bergman projection, but also implies the precise Lp norm of the Berezin transform.
 A PRIORI BOUNDS AND THE EXISTENCE OF POSITIVE SOLUTIONS FOR WEIGHTED FRACTIONAL SYSTEMS Pengyan WANG, Pengcheng NIU Acta mathematica scientia,Series B. 2021, 41 (5):  1547-1568.  DOI: 10.1007/s10473-021-0509-2 In this paper, we prove the existence of positive solutions to the following weighted fractional system involving distinct weighted fractional Laplacians with gradient terms:$$\left\{\begin{array}{lll} (-\Delta)^{\frac{\alpha}{2}}_{a_1} u_1(x)=u_1^{q_{11}}(x)+u_2^{q_{12}}(x)+ h_1(x,u_1(x),u_2(x),\nabla u_1(x),\nabla u_2(x)),~~~x\in \Omega,\\ (-\Delta)^{\frac{\beta}{2}}_{a_2} u_2(x)=u_1^{q_{21}}(x)+u_2^{q_{22} }(x)+h_2(x,u_1(x),u_2(x),\nabla u_1(x),\nabla u_2(x)),~~~x\in \Omega,\\ u_1(x)=0,~u_2(x)=0,~~~x\in \mathbb{R}^n \backslash \Omega. \end{array}\right.$$ Here $(-\Delta)^{\frac{\alpha}{2}}_{a_1}$ and $(-\Delta)^{\frac{\beta}{2}}_{a_2}$ denote weighted fractional Laplacians and $\Omega \subset \mathbb{R}^n$ is a $C^2$ bounded domain. It is shown that under some assumptions on $h_i(i=1,2)$, the problem admits at least one positive solution $(u_1(x),u_2(x))$. We first obtain the {a priori} bounds of solutions to the system by using the direct blow-up method of Chen, Li and Li. Then the proof of existence is based on a topological degree theory.
 THE NONEMPTINESS AND COMPACTNESS OF MILD SOLUTION SETS FOR RIEMANN-LIOUVILLE FRACTIONAL DELAY DIFFERENTIAL VARIATIONAL INEQUALITIES Yirong JIANG, Zhouchao WEI, Jingping LU Acta mathematica scientia,Series B. 2021, 41 (5):  1569-1578.  DOI: 10.1007/s10473-021-0510-9 This paper investigates the nonemptiness and compactness of the mild solution set for a class of Riemann-Liouville fractional delay differential variational inequalities, which are formulated by a Riemann-Liouville fractional delay evolution equation and a variational inequality. Our approach is based on the resolvent technique and a generalization of strongly continuous semigroups combined with Schauder's fixed point theorem.
 A STRONG SOLUTION OF NAVIER-STOKES EQUATIONS WITH A ROTATION EFFECT FOR ISENTROPIC COMPRESSIBLE FLUIDS Tuowei CHEN, Yongqian ZHANG Acta mathematica scientia,Series B. 2021, 41 (5):  1579-1605.  DOI: 10.1007/s10473-021-0511-8 We study the initial boundary value problem for the three-dimensional isentropic compressible Navier-Stokes equations in the exterior domain outside a rotating obstacle, with initial density having a compact support. By the coordinate system attached to the obstacle and an appropriate transformation of unknown functions, we obtain the three-dimensional isentropic compressible Navier-Stokes equations with a rotation effect in a fixed exterior domain. We first construct a sequence of unique local strong solutions for the related approximation problems restricted in a sequence of bounded domains, and derive some uniform bounds of higher order norms, which are independent of the size of the bounded domains. Then we prove the local existence of unique strong solution of the problem in the exterior domain, provided that the initial data satisfy a natural compatibility condition.
 THE PRODUCT OPERATOR BETWEEN BLOCH-TYPE SPACES OF SLICE REGULAR FUNCTIONS Yuxia LIANG Acta mathematica scientia,Series B. 2021, 41 (5):  1606-1618.  DOI: 10.1007/s10473-021-0512-7 There is little work concerning the properties of quaternionic operators acting on slice regular function spaces defined on quaternions. In this paper, we present an equivalent characterization for the boundedness of the product operator $C_\varphi D^m$ acting on Bloch-type spaces of slice regular functions. After that, an equivalent estimation for its essential norm is established, which can imply several existing results on holomorphic spaces.
 AN EXTENSION OF ZOLOTAREV'S PROBLEM AND SOME RELATED RESULTS Tran Loc HUNG, Phan Tri KIEN Acta mathematica scientia,Series B. 2021, 41 (5):  1619-1634.  DOI: 10.1007/s10473-021-0513-6 The main purpose of this paper is to extend the Zolotarev's problem concerning with geometric random sums to negative binomial random sums of independent identically distributed random variables. This extension is equivalent to describing all negative binomial infinitely divisible random variables and related results. Using Trotter-operator technique together with Zolotarev-distance's ideality, some upper bounds of convergence rates of normalized negative binomial random sums (in the sense of convergence in distribution) to Gamma, generalized Laplace and generalized Linnik random variables are established. The obtained results are extension and generalization of several known results related to geometric random sums.
 ZERO DISSIPATION LIMIT TO RAREFACTION WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SELECTED DENSITY-DEPENDENT VISCOSITY Yifan SU, Zhenhua GUO Acta mathematica scientia,Series B. 2021, 41 (5):  1635-1658.  DOI: 10.1007/s10473-021-0514-5 This paper is devoted to studying the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations with selected density-dependent viscosity. In particular, we focus our attention on the viscosity taking the form $\mu(\rho)=\rho^\epsilon (\epsilon > 0).$ For the selected density-dependent viscosity, it is proved that the solutions of the one-dimensional compressible Navier-Stokes equations with centered rarefaction wave initial data exist for all time, and converge to the centered rarefaction waves as the viscosity vanishes, uniformly away from the initial discontinuities. New and subtle analysis is developed to overcome difficulties due to the selected density-dependent viscosity to derive energy estimates, in addition to the scaling argument and elementary energy analysis. Moreover, our results extend the studies in[Xin Z P. Comm Pure Appl Math, 1993, 46(5):621-665].
 THE $\partial\overline{\partial}$-BOCHNER FORMULAS FOR HOLOMORPHIC MAPPINGS BETWEEN HERMITIAN MANIFOLDS AND THEIR APPLICATIONS Kai TANG Acta mathematica scientia,Series B. 2021, 41 (5):  1659-1669.  DOI: 10.1007/s10473-021-0515-4 In this paper, we derive some $\partial\overline{\partial}$-Bochner formulas for holomorphic maps between Hermitian manifolds. As applications, we prove some Schwarz lemma type estimates, and some rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a compact Hermitian manifold with positive (resp. non-negative) $\ell$-second Ricci curvature to a Hermitian manifold with non-positive (resp. negative) real bisectional curvature. These theorems generalize the results[5, 6] proved recently by L. Ni on Kähler manifolds to Hermitian manifolds. We also derive an integral inequality for a holomorphic map between Hermitian manifolds.
 ADDITIVE MAPPINGS PRESERVING FREDHOLM OPERATORS WITH FIXED NULLITY OR DEFECT Ruihan ZHANG, Weijuan SHI, Guoxing JI Acta mathematica scientia,Series B. 2021, 41 (5):  1670-1678.  DOI: 10.1007/s10473-021-0516-3 Let $\mathcal{X}$ be an infinite-dimensional real or complex Banach space, and $\mathcal{B}(\mathcal{X})$ the Banach algebra of all bounded linear operators on $\mathcal{X}$. In this paper, given any non-negative integer $n$, we characterize the surjective additive maps on $\mathcal{B}(\mathcal{X})$ preserving Fredholm operators with fixed nullity or defect equal to $n$ in both directions, and describe completely the structure of these maps.
 APPLICATION OF MULTI-DIMENSIONAL OF CONFORMABLE SUMUDU DECOMPOSITION METHOD FOR SOLVING CONFORMABLE SINGULAR FRACTIONAL COUPLED BURGER'S EQUATION Hassan ELTAYEB, Said MESLOUB Acta mathematica scientia,Series B. 2021, 41 (5):  1679-1698.  DOI: 10.1007/s10473-021-0517-2 In this article, several theorems of fractional conformable derivatives and triple Sumudu transform are given and proved. Based on these theorems, a new conformable triple Sumudu decomposition method (CTSDM) is intrduced for the solution of singular two-dimensional conformable functional Burger's equation. This method is a combination of the decomposition method (DM) and Conformable triple Sumudu transform. The exact and approximation solutions obtained by using the suggested method in the sense of conformable. Particular examples are given to clarify the possible application of the achieved results and the exact and approximate solution are sketched by using Matlab software.
 NON-INSTANTANEOUS IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS WITH STATE DEPENDENT DELAY AND PRACTICAL STABILITY Ravi AGARWAL, Ricardo ALMEIDA, Snezhana HRISTOVA, Donal O'REGAN Acta mathematica scientia,Series B. 2021, 41 (5):  1699-1718.  DOI: 10.1007/s10473-021-0518-1 Nonlinear delay Caputo fractional differential equations with non-instantaneous impulses are studied and we consider the general case of delay, depending on both the time and the state variable. The case when the lower limit of the Caputo fractional derivative is fixed at the initial time, and the case when the lower limit of the fractional derivative is changed at the end of each interval of action of the impulse are studied. Practical stability properties, based on the modified Razumikhin method are investigated. Several examples are given in this paper to illustrate the results.
 ON THE (p,q)-MELLIN TRANSFORM AND ITS APPLICATIONS Pankaj JAIN, Chandrani BASU, Vivek PANWAR Acta mathematica scientia,Series B. 2021, 41 (5):  1719-1732.  DOI: 10.1007/s10473-021-0519-0 In this paper, we introduce and study a (p,q)-Mellin transform and its corresponding convolution and inversion. In terms of applications of the (p,q)-Mellin transform, we solve some integral equations. Moreover, a (p,q)-analogue of the Titchmarsh theorem is also derived.
 THE INITIAL BOUNDARY VALUE PROBLEMS FOR A NONLINEAR INTEGRABLE EQUATION WITH 3×3 LAX PAIR ON THE FINITE INTERVAL Yu XIAO, Jian XU, Engui FAN Acta mathematica scientia,Series B. 2021, 41 (5):  1733-1748.  DOI: 10.1007/s10473-021-0520-7 In this paper, we apply Fokas unified method to study the initial boundary value (IBV) problems for nonlinear integrable equation with $3\times 3$ Lax pair on the finite interval $[0,L]$. The solution can be expressed by the solution of a $3\times 3$ Riemann-Hilbert (RH) problem. The relevant jump matrices are written in terms of matrix-value spectral functions $s(k),S(k),S_{l}(k)$, which are determined by initial data at $t=0$, boundary values at $x=0$ and boundary values at $x=L$, respectively. What's more, since the eigenvalues of $3\times 3$ coefficient matrix of $k$ spectral parameter in Lax pair are three different values, search for the path of analytic functions in RH problem becomes a very interesting thing.
 A PENALTY FUNCTION METHOD FOR THE PRINCIPAL-AGENT PROBLEM WITH AN INFINITE NUMBER OF INCENTIVE-COMPATIBILITY CONSTRAINTS UNDER MORAL HAZARD Jia LIU, Xianjia WANG Acta mathematica scientia,Series B. 2021, 41 (5):  1749-1763.  DOI: 10.1007/s10473-021-0521-6 In this paper, we propose an iterative algorithm to find the optimal incentive mechanism for the principal-agent problem under moral hazard where the number of agent action profiles is infinite, and where there are an infinite number of results that can be observed by the principal. This principal-agent problem has an infinite number of incentive-compatibility constraints, and we transform it into an optimization problem with an infinite number of constraints called a semi-infinite programming problem. We then propose an exterior penalty function method to find the optimal solution to this semi-infinite programming and illustrate the convergence of this algorithm. By analyzing the optimal solution obtained by the proposed penalty function method, we can obtain the optimal incentive mechanism for the principal-agent problem with an infinite number of incentive-compatibility constraints under moral hazard.
 POSITIVE SOLUTIONS OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM Lingjun LIU, Feilin SHI Acta mathematica scientia,Series B. 2021, 41 (5):  1764-1776.  DOI: 10.1007/s10473-021-0522-5 In this paper, we will study the nonlocal and nonvariational elliptic problem $$\left\{\begin{array}{ll}\label{eq0.1} -(1+a||u||_q^{\alpha q})\Delta u=|u|^{p-1}u+h(x,u,\nabla u) & \mbox{in}\ \ \Omega,\\ u=0 & \mbox{on}\ \ \partial\Omega,\\ \end{array} (0.1)\right.$$ where $a>0, \alpha>0, 1< q< 2^*, p\in(0,2^*-1)\setminus\{1\}$ and $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ $(N\geq 2)$. Under suitable assumptions about $h(x,u,\nabla u)$, we obtain \emph{a priori} estimates of positive solutions for the problem (0.1). Furthermore, we establish the existence of positive solutions by making use of these estimates and of the method of continuity.
 THE CONVERGENCE OF NONHOMOGENEOUS MARKOV CHAINS IN GENERAL STATE SPACES BY THE COUPLING METHOD Zhifeng ZHU, Shaoyi ZHANG, Fanji TIAN Acta mathematica scientia,Series B. 2021, 41 (5):  1777-1787.  DOI: 10.1007/s10473-021-0523-4 We investigate the convergence of nonhomogeneous Markov chains in general state space by using the f norm and the coupling method, and thus, a sufficient condition for the convergence of nonhomogeneous Markov chains in general state space is obtained.
 ON NONCOERCIVE (p,q)-EQUATIONS Nikolaos S. PAPAGEORGIOU, Nikolaos S. PAPAGEORGIOU, Calogero VETRO Acta mathematica scientia,Series B. 2021, 41 (5):  1788-1808.  DOI: 10.1007/s10473-021-0524-3 We consider a nonlinear Dirichlet problem driven by a (p,q)-Laplace differential operator (1 < q < p). The reaction is (p-1)-linear near ±∞ and the problem is noncoercive. Using variational tools and truncation and comparison techniques together with critical groups, we produce five nontrivial smooth solutions all with sign information and ordered. In the particular case when q=2, we produce a second nodal solution for a total of six nontrivial smooth solutions all with sign information.
 HYERS-ULAM STABILITY OF SECOND-ORDER LINEAR DYNAMIC EQUATIONS ON TIME SCALES Douglas R. ANDERSON, Masakazu ONITSUKA Acta mathematica scientia,Series B. 2021, 41 (5):  1809-1826.  DOI: 10.1007/s10473-021-0525-2 We investigate the Hyers-Ulam stability (HUS) of certain second-order linear constant coefficient dynamic equations on time scales, building on recent results for first-order constant coefficient time-scale equations. In particular, for the case where the roots of the characteristic equation are non-zero real numbers that are positively regressive on the time scale, we establish that the best HUS constant in this case is the reciprocal of the absolute product of these two roots. Conditions for instability are also given.