Table of Content

26 June 2022, Volume 42 Issue 3
 The General Inverse Bonnesen-Style Inequalities in $\mathbb{R}^n$ Xu Dong,Yan Zhang,Chunna Zeng,Xingxing Wang Acta mathematica scientia,Series A. 2022, 42 (3):  641-650.  Abstract ( 124 )   RICH HTML PDF (337KB) ( 122 )   The isoperimetric problem plays an important role in integral geometry. In this paper we mainly investigate the inverse form of the isoperimetric inequality, i.e. the general inverse Bonnesen-type inequalities. The upper bounds of several new general isoperimetric genus are obtained. Futhermore, as corollaries, we get a series of classical inverse Bonnesen-type inequalities in the plane. Finally, the best estimate between the results of three upper bounds is given.
 Multiple Spectra of Self-Similar Measures with Three Digits Haixiong Li,Xinlin Wu,Daoxin Ding Acta mathematica scientia,Series A. 2022, 42 (3):  705-715.  Abstract ( 18 )   RICH HTML PDF (390KB) ( 18 )   It is well known that the self-similar measure $\mu_{k, a, b}$ defined by $\mu_{k, a, b}(\cdot)=\frac{1}{3}\sum\limits_{i=0}^{2}\mu_{k, a, b}(3k(\cdot)-ki)$ is a spectral measure with a spectrum $\Lambda(3k, C)=\left\{\sum\limits_{j=0}^{{\rm finite}}(3k)^jc_j:c_j\in C=\{0, 1, 2\}\right\}.$ In this paper, by applying the properties of congruences and the order of elements in the finite group, we obtain some conditions on the integer $p$ such that the set $p\Lambda(3k, C)$ is also a spectrum for $\mu_{k, a, b}$. Moreover, an example is given to explain our theory.
 Research on the Lowest Energy Solution ofChern-Simons-Schrödinger Equation with Trapping Potential Ying Yang,Liejun Shen Acta mathematica scientia,Series A. 2022, 42 (3):  716-729.  Abstract ( 20 )   RICH HTML PDF (394KB) ( 25 )   In this paper, we mainly study the existence of solutions with prescribed $L^{2}$-norm to the Chern-Simons-Schrödinger (CSS) equation. This type problem can be transformed into look for the minimizer of the corresponding energy functional $E^\beta_{p} (u)$ under the constraint $\|u\|_{L^{ 2}(\mathbb{R}^2)}=1$. Concerning the subcritical mass case, that is, $p\in(0,2)$, no matter whether the potential function $V(x)$ equals to $0$, we prove that the constraint minimization can be achieved by some simple methods. We are also concerned with the critical mass case of $p=2$:if $V(x)\equiv0$, there exist two constants $\beta^*>\beta_*>0$ which can be explicitly determined such that the constraint minimization cannot achieved for any $\beta\in(0,\beta_{*}]\cup(\beta^{*},+\infty)$; if $V(x)\not\equiv0$, the constraint minimization cannot be achieved for $\beta>\beta^{*}$, but can be achieved for $\beta\in(0,\beta_{*}]$. In addition, we discuss the limit behavior of the mass subcritical constrained minimum energy when $p\nearrow2$.
 Ground State Solutions for Quasilinear Schrödinger Equation of Choquard Type Yanan Wang,Kaimin Teng Acta mathematica scientia,Series A. 2022, 42 (3):  730-748.  Abstract ( 25 )   RICH HTML PDF (399KB) ( 40 )   In this paper, we consider the following quasilinear Schrödinger equations of Choquard type $\begin{eqnarray*} -\triangle u+\frac{k}{2}u\triangle u^2+V(x)u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u, \, \, x\in\mathbb{R}^N, \end{eqnarray*}$ where $N\geq3$, 0 < $\alpha$ < $N$, $ Existence of Positive Ground State Solutions for the Choquard Equation Xudong Shang,Jihui Zhang Acta mathematica scientia,Series A. 2022, 42 (3): 749-759. Abstract ( 30 ) RICH HTML PDF (355KB) ( 41 ) In this paper we study the following nonlinear Choquard equation$ $$-\Delta u + V(x)u= (I_{\alpha}* F(u))f(u), \hskip0.5cm x\in{{\Bbb R}} ^{N} ,$$ $where$N \geq 3$,$\alpha \in (0, N)$,$I_{\alpha}$is the Riesz potential,$V(x):\mathbb{R} ^{N} \rightarrow \mathbb{R} $is a given potential function, and$F\in {\cal C}^{1}(\mathbb{R}, \mathbb{R})$with$F'(s)=f(s)$. Under assumptions on$V$and$f$, we do not require the$(AR)$condition of$f$, the existence of ground state solutions are obtained via variational methods.  Existence and Multiplicity of Solutions for a 2$n$th-Order Discrete Boundary Value Problems with a Parameter Zhenguo Wang Acta mathematica scientia,Series A. 2022, 42 (3): 760-766. Abstract ( 22 ) RICH HTML PDF (313KB) ( 24 ) In this paper, we consider the existence and multiplicity of solutions for a$2n$th-order discrete boundary value problems depending on a parameter$\lambda$. When$\lambda\in\left(\frac{p(T)}{2B}, \frac{1}{2A}\right)$, we obtain a sufficient condition for the existence of solutions of a discrete boundary value problems by means of critical point theory. Finally, one example is given to illustrate our main result.  Infinitely Solutions for a Class of Nonlocal Quasilinear Elliptic Equations Qian Wang,Lin Chen,Nan Tang Acta mathematica scientia,Series A. 2022, 42 (3): 767-774. Abstract ( 19 ) RICH HTML PDF (281KB) ( 24 ) In this paper, we study the existence of multiple solutions for a class of nonlocal quasilinear elliptic problem$\left\{\begin{array}{ll} M\Big(\int_{\mathbb{R} ^{N}}(|\nabla u|^{p}+V(x)|u|^{p}){\rm d}x\Big)(-\Delta_{p}u+V(x)|u|^{p-2}u)=\sigma d^{-1}F_{u}(x, u, v)+\lambda|u|^{q-2}u, \nonumber\\ M\Big(\int_{\mathbb{R} ^{N}}(|\nabla v|^{p}+V(x)|v|^{p}){\rm d}x\Big)(-\Delta_{p}v+V(x)|v|^{p-2}v)=\sigma d^{-1}F_{v}(x, u, v)+\mu|v|^{q-2}v, \nonumber\\ u, v\in W^{1, p}(\mathbb{R} ^{N}), x\in\mathbb{R} ^{N}\nonumber \end{array}\right. $where$M(s)=s^{k}, k>0, N\geq3, 10, \sigma\in\mathbb{R} ^{N}$, and in which$p^{\ast}=\frac{Np}{N-p}, $and$p^{\ast}=\infty$if$p=N.$The weight function$V(x)\in C(\mathbb{R} ^{N})$satisfy some conditions.  Breather Wave Solutions, Lump Solutions and Semi-Rational Solutions of a Reduced (3+1)Dimensional Hirota Equation Chunmei Fang,Shoufu Tian Acta mathematica scientia,Series A. 2022, 42 (3): 775-783. Abstract ( 21 ) RICH HTML PDF (1085KB) ( 20 ) In this paper, the long wave limit method is used to study the exact solutions of the (3+1)dimensional Hirota equation under dimensional reduction$z$=$x$. First, the bilinear form is constructed by using Bell polynomials. Based on the bilinear form, the$n$-order breather wave solutions are obtained under some parameter constraints on the$N$-order soliton solution. Secondly, by using the long wave limit method, high order lump wave solutions are obtained. Finally, the combined solutions of the first-order, second-order lump wave solutions and single solitary wave solutions are derived, i.e. semi-rational solutions. All the obtained solutions were analyzed with Maple software for physical characteristics.  Statistical Solutions and Its Limiting Behavior for the Impulsive Discrete Ginzburg-Landau Equations Caidi Zhao,Huite Jiang,Chunqiu Li,Caraballo Tomás Acta mathematica scientia,Series A. 2022, 42 (3): 784-806. Abstract ( 11 ) RICH HTML PDF (497KB) ( 16 ) In this article we first prove the global well-posedness of the impulsive discrete Ginzburg-Landau equations. Then we establish that the generated process by the solution operators possesses a pullback attractor and a family of invariant Borel probability measures. Further, we formulate the definition of statistical solution for the addressed impulsive system and prove the existence. Our results reveal that the statistical solution of the impulsive system satisfies merely the Liouville type theorem piecewise, which implies that the Liouville type equation for impulsive system will not always hold true on the interval containing any impulsive point. Finally, we prove that the statistical solution of the impulsive discrete Ginzburg-Landau equations converges to that of the impulsive discrete Schrödinger equations.  Asymptotic Stability Analysis of Solutions to Transport Equations in Structured Bacterial Population Growth Hongxing Wu,Dengbin Yuan,Shenghua Wang Acta mathematica scientia,Series A. 2022, 42 (3): 807-817. Abstract ( 19 ) RICH HTML PDF (377KB) ( 17 ) With the help of linear operator theory, the transport equation with more general boundary condition for the structured equation with bacterial population as background is discussed. By means of resolving operator and comparison operator, it is proved that the corresponding transfer operator spectrum of the transfer equation consists of only a finite number of discrete eigenvalues with finite algebraic multiplicity in band domain$\Gamma_{\alpha, \beta}$. It is proved that the solution of the transfer equation is asymptotically stable when$\psi_0 \in D (A_{H_{\alpha, \beta}})$.  A Dynamic Model for a Class of New Generalized Absolute Value Equations Wenli Zheng,Jia Tang,Cairong Chen Acta mathematica scientia,Series A. 2022, 42 (3): 818-825. Abstract ( 25 ) RICH HTML PDF (352KB) ( 30 ) In this paper, a dynamic model for solving a class of new generalized absolute value equations (GAVE) is proposed. Under suitable conditions, it can be proved that the solution of the GAVE is equivalent to the equilibrium point of the dynamic model and that the equilibrium point of the dynamic model is asymptotically stable. Numerical experiments show that the proposed dynamic model is feasible and effective. Articles  BOUNDEDNESS AND EXPONENTIAL STABILIZATION IN A PARABOLIC-ELLIPTIC KELLER–SEGEL MODEL WITH SIGNAL-DEPENDENT MOTILITIES FOR LOCAL SENSING CHEMOTAXIS Jie JIANG Acta mathematica scientia,Series A. 2022, 42 (3): 825-846. DOI: 10.1007/s10473-022-0301-y In this paper we consider the initial Neumann boundary value problem for a degenerate Keller—Segel model which features a signal-dependent non-increasing motility function. The main obstacle of analysis comes from the possible degeneracy when the signal concentration becomes unbounded. In the current work, we are interested in the boundedness and exponential stability of the classical solution in higher dimensions. With the aid of a Lyapunov functional and a delicate Alikakos—Moser type iteration, we are able to establish a time-independent upper bound of the concentration provided that the motility function decreases algebraically. Then we further prove the uniform-in-time boundedness of the solution by constructing an estimation involving a weighted energy. Finally, thanks to the Lyapunov functional again, we prove the exponential stabilization toward the spatially homogeneous steady states. Our boundedness result improves those in [1] and the exponential stabilization is obtained for the first time.  Oscillation Analysis of a Kind of Systems with Piecewise Continuous Arguments Ying Liu,Jianfang Gao Acta mathematica scientia,Series A. 2022, 42 (3): 826-838. Abstract ( 13 ) RICH HTML PDF (378KB) ( 20 ) In this paper, we mainly use$\theta$-method to analyze the oscillation of differential equations with piecewise continuous arguments of retarded type, and discuss the oscillation and non-oscillation of analytic solution and numerical solution. The sufficient conditions for the numerical methods to preserve the oscillation of the equation under the condition of the analytic solution oscillation are obtained. Meanwhile, some numerical experiments are given.  Oscillation of Second-Order Nonlinear Nonautonomous Delay Dynamic Equations on Time Scales Ping Zhang,Jiashan Yang,Guijiang Qin Acta mathematica scientia,Series A. 2022, 42 (3): 839-850. Abstract ( 17 ) RICH HTML PDF (418KB) ( 13 ) The oscillatory behavior of a class of second-order nonlinear nonautonomous variable delay damped dynamic equations are studied on a time scale T, where the equations are noncanonical form. By using the generalized Riccati transformation, and the time scales theory and the classical inequality, we establish some new oscillation criteria for the equation. The results fully reflect the influential actions of delay functions and damping terms in system oscillation. Finally, some examples are given to show that our results extend, improve and enrich those reported in the literature, and that they have good effectiveness and practicability. Articles  ABSOLUTE MONOTONICITY INVOLVING THE COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND WITH APPLICATIONS Zhenhang YANG, Jingfeng TIAN Acta mathematica scientia,Series A. 2022, 42 (3): 847-864. DOI: 10.1007/s10473-022-0302-x Let$\mathcal{K}\left( r\right) $be the complete elliptic integrals of the first kind for$r\in \left( 0,1\right) $and$f_{p}\left( x\right) =\left[ \left( 1-x\right) ^{p}\mathcal{K}\left( \sqrt{x}\right) \right] $. Using the recurrence method, we find the necessary and sufficient conditions for the functions$-f_{p}^{\prime }$,$\ln f_{p}$,$-\left( \ln f_{p}\right) ^{\left( i\right) }$($i=1,2,3$) to be absolutely monotonic on$\left( 0,1\right) $. As applications, we establish some new bounds for the ratios and the product of two complete integrals of the first kind, including the double inequalities \begin{eqnarray*} \frac{\exp \left[ r^{2}\left( 1-r^{2}\right) /64\right] }{\left( 1+r\right) ^{1/4}} &<&\frac{\mathcal{K}\left( r\right) }{\mathcal{K}\left( \sqrt{r}% \right) }<\exp \left[ -\frac{r\left( 1-r\right) }{4}\right] , \\ \frac{\pi }{2}\exp \left[ \theta _{0}\left( 1-2r^{2}\right) \right] &<&\frac{% \pi }{2}\frac{\mathcal{K}\left( r^{\prime }\right) }{\mathcal{K}\left( r\right) }<\frac{\pi }{2}\left( \frac{r^{\prime }}{r}\right) ^{p}\exp \left[ \theta _{p}\left( 1-2r^{2}\right) \right] , \\ \mathcal{K}^{2}\left( \frac{1}{\sqrt{2}}\right) &\leq &\mathcal{K}\left( r\right) \mathcal{K}\left( r^{\prime }\right) \leq \frac{1}{\sqrt{% 2rr^{\prime }}}\mathcal{K}^{2}\left( \frac{1}{\sqrt{2}}\right) \end{eqnarray*}% for$r\in \left( 0,1\right) $and$p\geq 13/32$, where$r^{\prime }=% \sqrt{1-r^{2}}$and$\theta _{p}=2\Gamma \left( 3/4\right) ^{4}/\pi ^{2}-p$.  Dynamics of an Anthrax Epidemiological Model with Time Delay and Seasonality Tailei Zhang,Junli Liu,Mengjie Han Acta mathematica scientia,Series A. 2022, 42 (3): 851-866. Abstract ( 26 ) RICH HTML PDF (480KB) ( 38 ) In this paper, we developed a time-delayed epidemiological model to describe the anthrax transmission, which incorporates seasonality and the incubation period of the animal population. The basic reproduction number$R_{0}$can be obtained. It is shown that the threshold dynamics is completely determined by the basic reproduction number. If$R_{0}<1$, the disease-free periodic solution is globally attractive and the disease will die out; if$R_{0} >1$, then there exists at least one positive periodic solution and the disease persists. We further investigate the corresponding autonomous system, the global stability of the disease-free equilibrium and the positive equilibrium is established in terms of$[R_0]$. Numerical simulations are carried out to investigate the sensitivity of$R_0$about the parameters, the effects of vaccination and carcass disposal on controlling the spread of anthrax is also analyzed. Articles  THE$\partial\bar{\partial}$-LEMMA UNDER SURJECTIVE MAPS Lingxu MENG Acta mathematica scientia,Series A. 2022, 42 (3): 865-875. DOI: 10.1007/s10473-022-0303-9 We consider the$\partial\bar{\partial}$-lemma for complex manifolds under surjective holomorphic maps. Furthermore, using Deligne-Griffiths-Morgan-Sullivan's theorem, we prove that a product compact complex manifold satisfies the$\partial\bar{\partial}$-lemma if and only if all of its components do as well.  Optimal Boundary Control for a Hierarchical Size-Structured Population Model with Delay Zerong He,Yimeng Dou,Mengjie Han Acta mathematica scientia,Series A. 2022, 42 (3): 867-880. Abstract ( 13 ) RICH HTML PDF (438KB) ( 26 ) In this article, we formulate a population control model, which is based upon the hierarchical size-structure and the incubation delay. For a given ideal population distribution, we investigate the optimal input problem: How to choose a inflow way such that the sum of the deviation between the terminal state and the given one and the total costs is minimal. The well-posedness is established by the method of characteristics, the existence of unique optimal policy is shown by the Ekeland variational principle, and the optimal policy is exactly described by a normal cone and an adjoint system. These results set a foundational framework for practical applications. Articles  PARAMETER ESTIMATION OF PATH-DEPENDENT MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS Meiqi LIU, Huijie QIAO Acta mathematica scientia,Series A. 2022, 42 (3): 876-886. DOI: 10.1007/s10473-022-0304-8 This work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters. First, we prove the existence and uniqueness of these equations under non-Lipschitz conditions. Second, we construct maximum likelihood estimators of these parameters and then discuss their strong consistency. Third, a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered. Finally, we estimate the errors between solutions of these equations and that of their numerical equations.  Regular Kernel Method for State Space Model Chao Wang,Bo Li,Lei Wang Acta mathematica scientia,Series A. 2022, 42 (3): 881-890. Abstract ( 19 ) RICH HTML PDF (465KB) ( 13 ) State space models(SSMs) provide a general framework for studying stochastic processes, which has been applied in revealing the true underlying economic processes of an economy, recognizing cellphone signals, detecting the loaction of an airplane on a radar screen, et al. In this paper, we study the Markov state space models by modeling the space transformation with reproducing kernel Hilbert space. Not only the existence and uniqueness of solutions are given, but also the error is estimate in L2 spaces. We applied our method in Visibility prediction at airport in National Post-Graduate Mathematical Contest in Modeling supported by China Academic Degrees & Graduate Education Development Center. Articles  GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW Ning-An LAI, Wei XIANG, Yi ZHOU Acta mathematica scientia,Series A. 2022, 42 (3): 887-902. DOI: 10.1007/s10473-022-0305-7 In this paper, we are concerned with the long time behavior of the piecewise smooth solutions to the generalized Riemann problem governed by the compressible isothermal Euler equations in two and three dimensions. A non-existence result is established for the fan-shaped wave structure solution, including two shocks and one contact discontinuity which is a perturbation of plane waves. Therefore, unlike in the one-dimensional case, the multi-dimensional plane shocks are not stable globally. Moreover, a sharp lifespan estimate is established which is the same as the lifespan estimate for the nonlinear wave equations in both two and three space dimensions.  Moving-Water Equilibria Preserving Central Scheme for the Saint-Venant System Yiming Luo,Dingfang Li,Min Liu,Jian Dong Acta mathematica scientia,Series A. 2022, 42 (3): 891-903. Abstract ( 21 ) RICH HTML PDF (5011KB) ( 18 ) In this paper, we propose a second-order unstaggered central finite volume scheme for the Saint-Venant system. Classical central scheme can preserve still-water steady state solution by reconstructing conservative variables and the water level, but generates enormous numerical oscillation when considering moving-water steady state. We choose to reconstruct conservative variables and the energy, and design a new discretization of the source term to preserve moving-water equilibria and capture its small perturbations. In the end, several classical numerical experiments are performed to verify the proposed scheme which is convergent, well-balanced and robust. Articles  ESTIMATES FOR EXTREMAL VALUES FOR A CRITICAL FRACTIONAL EQUATION WITH CONCAVE-CONVEX NONLINEARITIES Jianghao HAO, Yajing ZHANG Acta mathematica scientia,Series A. 2022, 42 (3): 903-918. DOI: 10.1007/s10473-022-0306-6 In this paper we study the critical fractional equation with a parameter λ and establish uniform lower bounds for Λ, which is the supremum of the set of λ, related to the existence of positive solutions of the critical fractional equation.  A New Projection Algorithm for Solving Pseudo-Monotone Variational Inequality and Fixed Point Problems Jing Yang,Xianjun Long Acta mathematica scientia,Series A. 2022, 42 (3): 904-919. Abstract ( 30 ) RICH HTML PDF (694KB) ( 28 ) In this paper, we propose a new projection algorithm for finding a common element of psedomonotone variational inequality problems and fixed point set of demicontractive mappings in Hilbert spaces. We prove that this new algorithm converges strongly to the common element for a psedomonotone and uniformly continuous mapping. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the new projection algorithm. Articles  THE SYSTEMS WITH ALMOST BANACH-MEAN EQUICONTINUITY FOR ABELIAN GROUP ACTIONS Bin ZHU, Xiaojun HUANG, Yuan LIAN Acta mathematica scientia,Series A. 2022, 42 (3): 919-940. DOI: 10.1007/s10473-022-0307-5 In this paper, we present the concept of Banach-mean equicontinuity and prove that the Banach-, Weyl- and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent. Furthermore, we obtain that the topological entropy of a transitive, almost Banach-mean equicontinuous dynamical system of Abelian group action is zero. As an application of our main result, we show that the topological entropy of the Banach-mean equicontinuous system under the action of an Abelian groups is zero.  The Greedy Simplex Algorithm for Double Sparsity Constrained Optimization Problems Tingwei Pan,Suxiang He Acta mathematica scientia,Series A. 2022, 42 (3): 920-933. Abstract ( 16 ) RICH HTML PDF (400KB) ( 27 ) In view of the shortcomings of the alternating minimization method in which it needs to calculate the Lipschitz constant of the objective function gradient and the Lipschitz condition is needed to construct the L-stable point of the problem, this paper proposes a greedy simplex algorithm to solve the optimization problems with double sparse constraints. The CW optimality condition for double sparse constrained optimization problems is described. Based on the CW optimality condition, the iterative steps of the algorithm are designed, and the global convergence of the sequence of iterative points generated by the algorithm to the optimal solution of the problem is proved under the weaker assumptions.  Precise Large Deviations for a Bidimensional Risk Model with the Regression Dependent Structure Zhenlong Chen,Yang Liu,Ke-ang Fu Acta mathematica scientia,Series A. 2022, 42 (3): 934-942. Abstract ( 13 ) RICH HTML PDF (374KB) ( 28 ) In this paper, we consider a non-standard bidimensional risk model, in which the claim sizes$ \{\vec{X}_k=(X_{1k}, X_{2k})^T, k\ge 1\}$form a sequence of independent and identically distributed random vectors with nonnegative components that are allowed to be dependent on each other, and there exists a regression dependent structure between these vectors and the inter-arrival times. By assuming that the univariate marginal distributions of claim vectors have consistently varying tails, we obtain the precise large deviation formulas for the bidimensional risk model with the regression dependent structure. Articles  A NEW SUFFICIENT CONDITION FOR SPARSE RECOVERY WITH MULTIPLE ORTHOGONAL LEAST SQUARES Haifeng LI, Jing ZHANG Acta mathematica scientia,Series A. 2022, 42 (3): 941-956. DOI: 10.1007/s10473-022-0308-4 A greedy algorithm used for the recovery of sparse signals, multiple orthogonal least squares (MOLS) have recently attracted quite a big of attention. In this paper, we consider the number of iterations required for the MOLS algorithm for recovery of a$K$-sparse signal$\mathbf{x}\in\mathbb{R}^n$. We show that MOLS provides stable reconstruction of all$K$-sparse signals$\mathbf{x}$from$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{w}$in$\lceil\frac{6K}{M}\rceil$iterations when the matrix$\mathbf{A}$satisfies the restricted isometry property (RIP) with isometry constant$\delta_{7K}\leq0.094$. Compared with the existing results, our sufficient condition is not related to the sparsity level$K$.  α-Robust Optimal Investment Strategy for Target Benefit Pension Plans Under Default Risk Yuan Shi,Yongxia Zhao Acta mathematica scientia,Series A. 2022, 42 (3): 943-960. Abstract ( 25 ) RICH HTML PDF (560KB) ( 28 ) This paper considers the optimal investment and benefit payment problem for target benefit pension plan with default risk and model uncertainty. We assume that pension funds are invested in a risk-free asset, a defaultable bond and a stock satisfied a constant elasticity of variance(CEV) model. The payment of pensions depends on the financial status of the plan, with risk sharing between different generations. At the same time, in order to protect the rights of pension holders who dies before retirement, the return of premiums clauses is added to the model. In addition, our model allows the pension manager to have different levels of ambiguity aversion, instead of only considering extremely ambiguity-averse attitude. Using the stochastic optimal control approach, we establish the Hamilton-Jacobi-Bellman equations for both the post-default case and the pre-default case, respectively. We derive the closed-form solutions for α-robust optimal investment strategies and optimal benefit payment adjustment strategies. Finally, numerical analyses illustrate the influence of financial market parameters on optimal control problems. Articles  CENTRAL LIMIT THEOREM AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT Yingqiu LI, Xulan HUANG, Zhaohui PENG Acta mathematica scientia,Series A. 2022, 42 (3): 957-974. DOI: 10.1007/s10473-022-0309-3 We are interested in the convergence rates of the submartingale${W}_{n} =\frac{Z_{n}}{\Pi_{n}}$to its limit${W},$where$(\Pi_{n})$is the usually used norming sequence and$(Z_{n})$is a supercritical branching process with immigration$(Y_{n})$in a stationary and ergodic environment$\xi$. Under suitable conditions, we establish the following central limit theorems and results about the rates of convergence in probability or in law: (i)$W-W_{n}$with suitable normalization converges to the normal law$N(0,1)$, and similar results also hold for$W_{n+k}-W_{n}$for each fixed$k\in \mathbb{N}^{\ast};$(ii) for a branching process with immigration in a finite state random environment, if$W_{1}$has a finite exponential moment, then so does$W,$and the decay rate of$\mathbb{P}(|W-W_{n}|>\varepsilon)$is supergeometric; (iii) there are normalizing constants$a_{n}(\xi)$(that we calculate explicitly) such that$a_{n}(\xi)(W-W_{n})$converges in law to a mixture of the Gaussian law.  QUASILINEAR EQUATIONS USING A LINKING STRUCTURE WITH CRITICAL NONLINEARITIES Edcarlos D. SILVA, Jefferson S. SILVA Acta mathematica scientia,Series A. 2022, 42 (3): 975-1002. DOI: 10.1007/s10473-022-0310-x It is to establish existence of a weak solution for quasilinear elliptic problems assuming that the nonlinear term is critical. The potential V is bounded from below and above by positive constants. Because we are considering a critical term which interacts with higher eigenvalues for the linear problem, we need to apply a linking theorem. Notice that the lack of compactness, which comes from critical problems and the fact that we are working in the whole space, are some obstacles for us to ensure existence of solutions for quasilinear elliptic problems. The main feature in this article is to restore some compact results which are essential in variational methods. Recall that compactness conditions such as the Palais-Smale condition for the associated energy functional is not available in our setting. This difficulty is overcame by taking into account some fine estimates on the critical level for an auxiliary energy functional.  SINGULAR CONTROL OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS Nacira AGRAM, Saloua LABED, Bernt ØKSENDAL, Samia YAKHLEF Acta mathematica scientia,Series A. 2022, 42 (3): 1003-1017. DOI: 10.1007/s10473-022-0311-9 This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations, where the solution$X^{u,\xi}(t)=X(t)$is given by $$X(t) =\phi(t)+{ \int_{0}^{t}}b\left( t,s,X(s),u(s)\right){\rm d}s+% { \int_{0}^{t}} \sigma\left( t,s,X(s),u(s)\right) {\rm d}B(s) + { \int_{0}^{t}} h\left( t,s\right) {\rm d}\xi(s).$$ Here${\rm d}B(s)$denotes the Brownian motion Itô type differential,$\xi$denotes the singular control (singular in time$t$with respect to Lebesgue measure) and$u$denotes the regular control (absolutely continuous with respect to Lebesgue measure).Such systems may for example be used to model harvesting of populations with memory, where$X(t)$represents the population density at time$t$, and the singular control process$\xi$represents the harvesting effort rate. The total income from the harvesting is represented by $$J(u,\xi) =\mathbb{E}\bigg[ \int_{0}^{T} f_{0}(t,X(t),u(t)){\rm d}t+ \int_{0}^{T} f_{1}(t,X(t)){\rm d}\xi(t)+g(X(T))\bigg]$$ for the given functions$f_{0},f_{1}$and$g$, where$T>0$is a constant denoting the terminal time of the harvesting. Note that it is important to allow the controls to be singular, because in some cases the optimal controls are of this type.Using Hida-Malliavin calculus, we prove sufficient conditions and necessary conditions of optimality of controls. As a consequence, we obtain a new type of backward stochastic Volterra integral equations with singular drift.Finally, to illustrate our results, we apply them to discuss optimal harvesting problems with possibly density dependent prices.  HE EXPONENTIAL OF QUASI BLOCK-TOEPLITZ MATRICES Elahe BOLOURCHIAN, Bijan Ahmadi KAKAVANDI Acta mathematica scientia,Series A. 2022, 42 (3): 1018-1034. DOI: 10.1007/s10473-022-0312-8 The matrix Wiener algebra,$\mathcal{W}_N:=\mathrm{M}_{N} (\mathcal{W})$of order$N>0$, is the matrix algebra formed by$N \times N$matrices whose entries belong to the classical Wiener algebra$\mathcal{W}$of functions with absolutely convergent Fourier series. A block-Toeplitz matrix$T(a)=[A_{i,j}]_{i,j \geq 0}$is a block semi-infinite matrix such that its blocks$A_{i,j}$are finite matrices of order$N$,$A_{i,j}=A_{r,s}$whenever$i-j=r-s$and its entries are the coefficients of the Fourier expansion of the generator$a:\mathbb{T} \rightarrow \mathrm{M}_{N} (\mathbb{C})$. Such a matrix can be regarded as a bounded linear operator acting on the direct sum of$N$copies of$L^2(\mathbb{T})$. We show that$\exp(T(a))$differes from$T(\exp(a))$only in a compact operator with a known bound on its norm. In fact, we prove a slightly more general result: for every entire function$f$and for every compact operator$E$, there exists a compact operator$F$such that$f(T(a)+E)=T(f(a))+F$. We call these$T(a)+E's$matrices, the quasi block-Toeplitz matrices, and we show that via a computation-friendly norm, they form a Banach algebra. Our results generalize and are motivated by some recent results of Dario Andrea Bini, Stefano Massei and Beatrice Meini.  BOUNDEDNESS AND ASYMPTOTIC STABILITY IN A PREDATOR-PREY CHEMOTAXIS SYSTEM WITH INDIRECT PURSUIT-EVASION DYNAMICS Shuyan QIU, Chunlai MU, Hong YI Acta mathematica scientia,Series A. 2022, 42 (3): 1035-1057. DOI: 10.1007/s10473-022-0313-7 This work explores the predator-prey chemotaxis system with two chemicals \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=\Delta u+\chi\nabla\cdot(u\nabla v)+\mu_1u(1-u-a_1w),\quad &x\in \Omega, t>0,\\ v_t=\Delta v-\alpha_1 v+\beta_1w,\quad &x\in \Omega, t>0,\\ w_t=\Delta w-\xi\nabla \cdot(w\nabla z)+\mu_2 w(1+a_2u-w),\quad &x\in\Omega, t>0,\\ z_t=\Delta z-\alpha_2 z+\beta_2u,\quad &x\in \Omega, t>0,\\ \end{array} \right. \end{eqnarray*} in an arbitrary smooth bounded domain$\Omega\subset \mathbb{R}^n$under homogeneous Neumann boundary conditions. The parameters in the system are positive.We first prove that if$n\leq3$, the corresponding initial-boundary value problem admits a unique global bounded classical solution, under the assumption that$\chi, \xi$,$\mu_i, a_i, \alpha_i$and$\beta_i(i=1,2)$satisfy some suitable conditions. Subsequently, we also analyse the asymptotic behavior of solutions to the above system and show that$\bullet$when$a_1<1$and both$\frac{\mu_1}{\chi^2}$and$\frac{\mu_2}{\xi^2}$are sufficiently large, the global solution$(u, v, w, z)$of this system exponentially converges to$(\frac{1-a_1}{1+a_1a_2}, \frac{\beta_1(1+a_2)}{\alpha_1(1+a_1a_2)}, \frac{1+a_2}{1+a_1a_2}, \frac{\beta_2(1-a_1)}{\alpha_2(1+a_1a_2)})$as$t\rightarrow \infty$;$\bullet$when$a_1>1$and$\frac{\mu_2}{\xi^2}$is sufficiently large, the global bounded classical solution$(u, v, w, z)$of this system exponentially converges to$(0, \frac{\alpha_1}{\beta_1}, 1, 0)$as$t\rightarrow \infty$;$\bullet$when$a_1=1$and$\frac{\mu_2}{\xi^2}$is sufficiently large, the global bounded classical solution$(u, v, w, z)$of this system polynomially converges to$(0, \frac{\alpha_1}{\beta_1}, 1, 0)$as$t\rightarrow \infty$.  THE GLOBAL EXISTENCE AND A DECAY ESTIMATE OF SOLUTIONS TO THE PHAN-THEIN-TANNER MODEL Ruiying WEI, Yin LI, Zheng-an YAO Acta mathematica scientia,Series A. 2022, 42 (3): 1058-1080. DOI: 10.1007/s10473-022-0314-6 In this paper, we study the global existence and decay rates of strong solutions to the three dimensional compressible Phan-Thein-Tanner model. By a refined energy method, we prove the global existence under the assumption that the$H^3$norm of the initial data is small, but that the higher order derivatives can be large. If the initial data belong to homogeneous Sobolev spaces or homogeneous Besov spaces, we obtain the time decay rates of the solution and its higher order spatial derivatives. Moreover, we also obtain the usual$L^p-L^2(1\leq p\leq2)$type of the decay rate without requiring that the$L^p$norm of initial data is small.  THE TIME DECAY RATES OF THE CLASSICAL SOLUTION TO THE POISSON-NERNST-PLANCK-FOURIER EQUATIONS IN$\mathbb{R}^3$Leilei TONG, Zhong TAN, Xu ZHANG Acta mathematica scientia,Series A. 2022, 42 (3): 1081-1102. DOI: 10.1007/s10473-022-0315-5 In this work, the Poisson-Nernst-Planck-Fourier system in three dimensions is considered. For when the initial data regards a small perturbation around the constant equilibrium state in a$H^3\cap\dot{H}^{-s} (0\leq s\leq 1/2)$norm, we obtain the time convergence rate of the global solution by a regularity interpolation trick and an energy method.  DK SPACES AND CARLESON MEASURES Dongxing LI, Hasi WULAN, Ruhan ZHAO Acta mathematica scientia,Series A. 2022, 42 (3): 1103-1112. DOI: 10.1007/s10473-022-0316-4 We give some characterizations of Carleson measures for Dirichlet type spaces by using Hadamard products. We also give a one-box condition for such Carleson measures.  A FRACTIONAL CRITICAL PROBLEM WITH SHIFTING SUBCRITICAL PERTURBATION Qi LI, Chang-Lin XIANG Acta mathematica scientia,Series A. 2022, 42 (3): 1113-1124. DOI: 10.1007/s10473-022-0317-3 In this paper, we consider a class of fractional problem with subcritical perturbation on a bounded domain as follows: \begin{equation*} (P_{k})\quad \left\{ \begin{array}{ll} \displaystyle (-\Delta)^s u=g(x)[(u-k)^+]^{q-1}+u^{2^{*}_{s}-1},\ \ &x\in \Omega,\\ \displaystyle u>0,\ \ &x\in \Omega,\\ \displaystyle u=0,\ \ &x\in \mathbb{R}^N\backslash \Omega. \end{array} \right. \end{equation*} We prove the existence of nontrivial solutions$u_{k}$of$(P_{k})$for each$k\in (0,\infty)$. We also investigate the concentration behavior of the solutions$u_{k}$as$k\to \infty$.  THE EXISTENCE AND CONCENTRATION OF GROUND STATE SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH A STEEP WELL POTENTIAL Jinlan TAN, Yongyong LI, Chunlei TANG Acta mathematica scientia,Series A. 2022, 42 (3): 1125-1140. DOI: 10.1007/s10473-022-0318-2 In this paper, we investigate a class of nonlinear Chern-Simons-Schrödinger systems with a steep well potential. By using variational methods, the mountain pass theorem and Nehari manifold methods, we prove the existence of a ground state solution for λ > 0 large enough. Furthermore, we verify the asymptotic behavior of ground state solutions as λ → +∞.  CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT Mingzhan HUANG, Shouzong LIU, Xinyu SONG, Xiufen ZOU Acta mathematica scientia,Series A. 2022, 42 (3): 1141-1159. DOI: 10.1007/s10473-022-0319-1 This paper mainly studies the stochastic character of tumor growth in the presence of immune response and periodically pulsed chemotherapy. First, a stochastic impulsive model describing the interaction and competition among normal cells, tumor cells and immune cells under periodically pulsed chemotherapy is established. Then, sufficient conditions for the extinction, non-persistence in the mean, weak and strong persistence in the mean of tumor cells are obtained. Finally, numerical simulations are performed which not only verify the theoretical results derived but also reveal some specific features. The results show that the growth trend of tumor cells is significantly affected by the intensity of noise and the frequency and dose of drug deliveries. In clinical practice, doctors can reduce the randomness of the environment and increase the intensity of drug input to inhibit the proliferation and growth of tumor cells.  A COMPACTNESS THEOREM FOR STABLE FLAT$SL(2,\mathbb{C})$CONNECTIONS ON 3-FOLDS Teng HUANG Acta mathematica scientia,Series A. 2022, 42 (3): 1160-1172. DOI: 10.1007/s10473-022-0320-8 Let$Y$be a closed$3$-manifold such that all flat$SU(2)$-connections on$Y$are non-degenerate. In this article, we prove a Uhlenbeck-type compactness theorem on$Y$for stable flat$SL(2,\mathbb{C})$connections satisfying an$L^{2}$-bound for the real curvature. Combining the compactness theorem and a result from [7], we prove that the moduli space of the stable flat$SL(2,\mathbb{C})$connections is disconnected under certain technical assumptions.  LEARNING RATES OF KERNEL-BASED ROBUST CLASSIFICATION Shuhua WANG, Baohuai SHENG Acta mathematica scientia,Series A. 2022, 42 (3): 1173-1190. DOI: 10.1007/s10473-022-0321-7 This paper considers a robust kernel regularized classification algorithm with a non-convex loss function which is proposed to alleviate the performance deterioration caused by the outliers. A comparison relationship between the excess misclassification error and the excess generalization error is provided; from this, along with the convex analysis theory, a kind of learning rate is derived. The results show that the performance of the classifier is effected by the outliers, and the extent of impact can be controlled by choosing the homotopy parameters properly.  BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES Qianjun HE, Xinfeng WU, Dunyan YAN Acta mathematica scientia,Series A. 2022, 42 (3): 1191-1208. DOI: 10.1007/s10473-022-0322-6 In this paper, we study a boundedness property of the Adams type for multilinear fractional integral operators with the multilinear$L^{r^{\prime},\alpha}$-Hörmander condition and their commutators with vector valued BMO functions on a Morrey space and a predual Morrey space. Moreover, we give an endpoint estimate for multilinear fractional integral operators. As an application, we obtain the boundedness of multilinear Fourier multipliers with limited Sobolev regularity on a Morrey space.  EXISTENCE RESULTS FOR SINGULAR FRACTIONAL p-KIRCHHOFF PROBLEMS Mingqi XIANG, Vicenţiu D. RǍDULESCU, Binlin ZHANG Acta mathematica scientia,Series A. 2022, 42 (3): 1209-1224. DOI: 10.1007/s10473-022-0323-5 This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional$p-Laplacian operator. More precisely, we study the following nonlocal problem: \begin{align*} \begin{cases} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|x|^{\alpha_1p}|y|^{\alpha_2p}|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\right) \mathcal{L}^{s}_pu= |x|^{\beta} f(u)\,\, \ &{\rm in}\ \Omega,\\ u=0\ \ \ \ &{\rm in}\ \mathbb{R}^N\setminus \Omega, \end{cases} \end{align*} where\mathcal{L}^{s}_p$is the generalized fractional$p$-Laplacian operator,$N\geq1$,$s\in(0,1)$,$\alpha_1,\alpha_2,\beta\in\mathbb{R}$,$\Omega\subset \mathbb{R}^N$is a bounded domain with Lipschitz boundary, and$M:\mathbb{R}^+_0\rightarrow \mathbb{R}^+_0$,$f:\Omega\rightarrow\mathbb{R}$are continuous functions. Firstly, we introduce a variational framework for the above problem. Then, the existence of least energy solutions is obtained by using variational methods, provided that the nonlinear term$f$has$(\theta p-1)$-sublinear growth at infinity. Moreover, the existence of infinitely many solutions is obtained by using Krasnoselskii's genus theory. Finally, we obtain the existence and multiplicity of solutions if$f$has$(\theta p-1)$-superlinear growth at infinity. The main features of our paper are that the Kirchhoff function may vanish at zero and the nonlinearity may be singular.  CONTINUOUS SELECTIONS OF THE SET-VALUED METRIC GENERALIZED INVERSE IN 2-STRICTLY CONVEX BANACH SPACES Shaoqiang SHANG Acta mathematica scientia,Series A. 2022, 42 (3): 1225-1237. DOI: 10.1007/s10473-022-0324-4 In this paper, we prove that if$X$is an almost convex and 2-strictly convex space, linear operator$T: X \to Y$is bounded,$N(T)$is an approximative compact Chebyshev subspace of$X$and$R(T)$is a 3-Chebyshev hyperplane, then there exists a homogeneous selection${T^\sigma }$of${T^\partial }$such that continuous points of${T^\sigma }$and${T^\partial }$are dense on$Y\$.