#### Table of Content

25 June 2020, Volume 40 Issue 3
Articles
 A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES Qinrui SHEN Acta mathematica scientia,Series B. 2020, 40 (3):  603-613.  DOI: 10.1007/s10473-020-0301-8 This article is committed to deal with measure of non-compactness of operators in Banach spaces. Firstly, the collection C(X) (consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication) is a normed semigroup, and the mapping J from C(X) onto F(Ω) is a fully order-preserving positively linear surjective isometry, where Ω is the closed unit ball of X* and F(Ω) the collection of all continuous and w*-lower semicontinuous sublinear functions on X* but restricted to Ω. Furthermore, both EC=JC-JC and EK=JK-JK are Banach lattices and EK is a lattice ideal of EC. The quotient space EC/EK is an abstract M space, hence, order isometric to a sublattice of C(K) for some compact Haudorspace K, and (FQJ)C which is a closed cone is contained in the positive cone of C(K), where Q:EC → EC/EK is the quotient mapping and F:EC/EK → C(K) is a corresponding order isometry. Finally, the representation of the measure of non-compactness of operators is given:Let BX be the closed unit ball of a Banach space X, thenμ(T)=μ(T(BX))=||(F QJ)T(BX)||C(K), ∀T ∈ B(X).
 MINIMAL PERIOD SYMMETRIC SOLUTIONS FOR SOME HAMILTONIAN SYSTEMS VIA THE NEHARI MANIFOLD METHOD Chouha?d SOUISSI Acta mathematica scientia,Series B. 2020, 40 (3):  614-624.  DOI: 10.1007/s10473-020-0302-7 For a given T > 0, we prove, under the global ARS-condition and using the Nehari manifold method, the existence of a T-periodic solution having the W-symmetry introduced in[21], for the hamiltonian systemz+ V'(z)=0, z ∈ RN, N ∈ N*.Moreover, such a solution is shown to have T as a minimal period without relaying to any index theory. A multiplicity result is also proved under the same condition.
 TOEPLITZ OPERATORS WITH POSITIVE OPERATOR-VALUED SYMBOLS ON VECTOR-VALUED GENERALIZED FOCK SPACES Jianjun CHEN, Xiaofeng WANG, Jin XIA Acta mathematica scientia,Series B. 2020, 40 (3):  625-640.  DOI: 10.1007/s10473-020-0303-6 In this article, we study some characterizations of Toeplitz operators with positive operator-valued function as symbols on the vector-valued generalized Bargmann-Fock spaces Fψ2. Main results including Fock-Carleson condition, bounded Toeplitz operators, compact Toeplitz operators, and Toeplitz operators in the Schatten-p class are all considered.
 THE QUASI-BOUNDARY VALUE METHOD FOR IDENTIFYING THE INITIAL VALUE OF THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION Fan YANG, Yan ZHANG, Xiao LIU, Xiaoxiao LI Acta mathematica scientia,Series B. 2020, 40 (3):  641-658.  DOI: 10.1007/s10473-020-0304-5 In this article, we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation. This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem. We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule. Some numerical results in one-dimensional case and two-dimensional case show that our method is efficient and stable.
 LIE-TROTTER FORMULA FOR THE HADAMARD PRODUCT Jing WANG, Yonggang LI, Huafei SUN Acta mathematica scientia,Series B. 2020, 40 (3):  659-669.  DOI: 10.1007/s10473-020-0305-4 Suppose that A and B are two positive-definite matrices, then, the limit of (Ap/2BpAp/2)1/p as p tends to 0 can be obtained by the well known Lie-Trotter formula. In this article, we generalize the usual product of matrices to the Hadamard product denoted as * which is commutative, and obtain the explicit formula of the limit (Ap * Bp)1/p as p tends to 0. Furthermore, the existence of the limit of (Ap * Bp)1/p as p tends to +∞ is proved.
 AN ABLOWITZ-LADIK INTEGRABLE LATTICE HIERARCHY WITH MULTIPLE POTENTIALS Wen-Xiu MA Acta mathematica scientia,Series B. 2020, 40 (3):  670-678.  DOI: 10.1007/s10473-020-0306-3 Within the zero curvature formulation, a hierarchy of integrable lattice equations is constructed from an arbitrary-order matrix discrete spectral problem of Ablowitz-Ladik type. The existence of infinitely many symmetries and conserved functionals is a consequence of the Lax operator algebra and the trace identity. When the involved two potential vectors are scalar, all the resulting integrable lattice equations are reduced to the standard Ablowitz-Ladik hierarchy.
 MULTIPLICITY OF POSITIVE SOLUTIONS FOR A NONLOCAL ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX NONLINEARITIES Jinguo ZHANG, Tsing-San HSU Acta mathematica scientia,Series B. 2020, 40 (3):  679-699.  DOI: 10.1007/s10473-020-0307-2 In this article, we study the following critical problem involving the fractional Laplacian:$\left\{ \begin{array}{l} { - \Delta)^{\frac{\alpha }{2}}}u - \gamma \frac{u}{{|x{|^\alpha }}} = \lambda \frac{{|u{|^{q - 2}}}}{{|x{|^s}}} + \frac{{|u{|^{2_\alpha ^*(t) - 2}}u}}{{|x{|^t}}}\quad {\rm{in }}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \Omega,\\ u = 0\quad {\rm{ }}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{in }}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {{\rm{\mathbb{R} }}^N}\backslash \Omega , \end{array} \right.$where Ω ? RN (N > α) is a bounded smooth domain containing the origin, α ∈ (0, 2), 0 ≤ s, t < α, 1 ≤ q < 2, λ > 0, 2α*(t)=2(N-t)/N -α is the fractional critical Sobolev-Hardy exponent, 0 ≤ γ < γH, and γH is the sharp constant of the Sobolev-Hardy inequality. We deal with the existence of multiple solutions for the above problem by means of variational methods and analytic techniques.
 ASYMPTOTIC CONVERGENCE OF A GENERALIZED NON-NEWTONIAN FLUID WITH TRESCA BOUNDARY CONDITIONS Adelkader SAADALLAH, Hamid BENSERIDI, Mourad DILMI Acta mathematica scientia,Series B. 2020, 40 (3):  700-712.  DOI: 10.1007/s10473-020-0308-1 The goal of this article is to study the asymptotic analysis of an incompressible Herschel-Bulkley fluid in a thin domain with Tresca boundary conditions. The yield stress and the constant viscosity are assumed to vary with respect to the thin layer parameter ε. Firstly, the problem statement and variational formulation are formulated. We then obtained the existence and the uniqueness result of a weak solution and the estimates for the velocity field and the pressure independently of the parameter ε. Finally, we give a specific Reynolds equation associated with variational inequalities and prove the uniqueness.
 BOUNDEDNESS OF THE HIGHER-DIMENSIONAL QUASILINEAR CHEMOTAXIS SYSTEM WITH GENERALIZED LOGISTIC SOURCE Qingquan TANG, Qiao XIN, Chunlai MU Acta mathematica scientia,Series B. 2020, 40 (3):  713-722.  DOI: 10.1007/s10473-020-0309-0 This article considers the following higher-dimensional quasilinear parabolic-parabolic-ODE chemotaxis system with generalized Logistic source and homogeneous Neumann boundary conditions $\left\{ \begin{array}{lll} u_t=\nabla\cdot(D(u)\nabla u)-\nabla \cdot (S(u)\nabla v)+f(u), &x\in \Omega,t>0\\ v_t = \Delta v+w-v, &x\in \Omega, t>0,\\ w_t=u-w, &x\in \Omega, t>0, \end{array} \right.$ in a bounded domain $\Omega \subset R^{n}(n\geq 2)$ with smooth boundary $\partial\Omega$, where the diffusion coefficient $D(u)$ and the chemotactic sensitivity function $S(u)$ are supposed to satisfy $D(u)\geq M_{1}(u+1)^{-\alpha}$ and $S(u)\leq M_{2}(u+1)^\beta$, respectively, where $M_{1},M_{2}>0$ and $\alpha, \beta\in R$. Moreover, the logistic source $f(u)$ is supposed to satisfy $f(u)\leq a-\mu u^{\gamma}$ with $\mu>0$, $\gamma\geq 1$, and $a\geq 0$. As $\alpha+2\beta<\gamma-1+\frac{2\gamma}{n}$, we show that the solution of the above chemotaxis system with sufficiently smooth nonnegative initial data is uniformly bounded.
 A NOVEL METHOD FOR NONLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS IN BROKEN REPRODUCING KERNEL SPACE Liangcai MEI Acta mathematica scientia,Series B. 2020, 40 (3):  723-733.  DOI: 10.1007/s10473-020-0310-7 In this article, a new algorithm is presented to solve the nonlinear impulsive differential equations. In the first time, this article combines the reproducing kernel method with the least squares method to solve the second-order nonlinear impulsive differential equations. Then, the uniform convergence of the numerical solution is proved, and the time consuming Schmidt orthogonalization process is avoided. The algorithm is employed successfully on some numerical examples.
 A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS Qian YU Acta mathematica scientia,Series B. 2020, 40 (3):  734-754.  DOI: 10.1007/s10473-020-0311-6 Let $B=\{B^H(t)\}_{t\geq0}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in (0,1)$. Consider the functionals of $k$ independent $d$-dimensional fractional Brownian motions $$\frac{1}{\sqrt{n}} \int^{e^{nt_1}}_0\cdots\int^{e^{nt_k}}_0 f(B^{H,1}(s_1)+\cdots +B^{H,k}(s_k)){\rm d}s_1\cdots{\rm d}s_k,$$ where the Hurst index $H=k/d$. Using the method of moments, we prove the limit law and extending a result by Xu \cite{xu} of the case $k=1$. It can also be regarded as a fractional generalization of Biane \cite{biane} in the case of Brownian motion.
 SPECTRAL PROPERTIES OF DISCRETE STURM-LIOUVILLE PROBLEMS WITH TWO SQUARED EIGENPARAMETER-DEPENDENT BOUNDARY CONDITIONS Chenghua GAO, Yali WANG, Li LV Acta mathematica scientia,Series B. 2020, 40 (3):  755-781.  DOI: 10.1007/s10473-020-0312-5 In this article, we consider a discrete right-definite Sturm-Liouville problems with two squared eigenparameter-dependent boundary conditions. By constructing some new Lagrange-type identities and two fundamental functions, we obtain not only the existence, the simplicity, and the interlacing properties of the real eigenvalues, but also the oscillation properties, orthogonality of the eigenfunctions, and the expansion theorem. Finally, we also give a computation scheme for computing eigenvalues and eigenfunctions of specific eigenvalue problems.
 ON BLOW-UP PHENOMENON OF THE SOLUTION TO SOME WAVE-HARTREE EQUATION IN d ≥ 5 Suxia XIA Acta mathematica scientia,Series B. 2020, 40 (3):  782-794.  DOI: 10.1007/s10473-020-0313-4 This article mainly considers the blow up phenomenon of the solution to the wave-hartree equation utt-△u=(|x|-4 *|u|2)u in the energy space for high dimensions d ≥ 5. The main result of this article is that:if the initial data (u0, u1) satisfy the conditions E(u0, u1) < E(W, 0) and||▽u0||22 >||▽W||22 for some ground state W, then the corresponding solution must blows up in finite time.
 A BLOW-UP CRITERION OF STRONG SOLUTIONS TO THE QUANTUM HYDRODYNAMIC MODEL Guangwu WANG, Boling GUO Acta mathematica scientia,Series B. 2020, 40 (3):  795-804.  DOI: 10.1007/s10473-020-0314-3 In this article, we focus on the short time strong solution to a compressible quantum hydrodynamic model. We establish a blow-up criterion about the solutions of the compressible quantum hydrodynamic model in terms of the gradient of the velocity, the second spacial derivative of the square root of the density, and the first order time derivative and first order spacial derivative of the square root of the density.
 ON THE ASYMPTOTIC SPECTRUM OF A TRANSPORT OPERATOR WITH ELASTIC AND INELASTIC COLLISION OPERATORS Abdul-Majeed AL-IZERI, Khalid LATRACH Acta mathematica scientia,Series B. 2020, 40 (3):  805-823.  DOI: 10.1007/s10473-020-0315-2 In this article, we investigate the spectral properties of a class of neutron transport operators involving elastic and inelastic collision operators introduced by Larsen and Zweifel[1]. Our analysis is manly focused on the description of the asymptotic spectrum which is very useful in the study of the properties of the solution to Cauchy problem governed by such operators (when it exists). The last section of this work is devoted to the properties of the leading eigenvalue (when it exists). So, we discuss the irreducibility of the semigroups generated by these operators. We close this section by discussing the strict monotonicity of the leading eigenvalue with respect to the parameters of the operator.
 ON NEW APPROXIMATIONS FOR GENERALIZED CAUCHY FUNCTIONAL EQUATIONS USING BRZDȨK AND CIEPLIŃSKI'S FIXED POINT THEOREMS IN 2-BANACH SPACES Laddawan AIEMSOMBOON, Wutiphol SINTUNAVARAT Acta mathematica scientia,Series B. 2020, 40 (3):  824-834.  DOI: 10.1007/s10473-020-0316-1 In this work, we apply the Brzdȩk and Ciepliński's fixed point theorem to investigate new stability results for the generalized Cauchy functional equation of the formf(ax + by)=af(x) + bf(y),where a,b ∈ N and f is a mapping from a commutative group (G, +) to a 2-Banach space (Y,||·,·||). Our results are generalizations of main results of Brzdȩk and Ciepliński[J Brzdȩk, K Ciepliński. On a fixed point theorem in 2-normed spaces and some of its applications. Acta Mathematica Scientia, 2018, 38B(2):377-390].
 PARTIAL REGULARITY FOR STATIONARY NAVIER-STOKES SYSTEMS BY THE METHOD OF A-HARMONIC APPROXIMATION Yichen DAI, Zhong TAN Acta mathematica scientia,Series B. 2020, 40 (3):  835-854.  DOI: 10.1007/s10473-020-0317-0 In this article, we prove a regularity result for weak solutions away from singular set of stationary Navier-Stokes systems with subquadratic growth under controllable growth condition. The proof is based on the A-harmonic approximation technique. In this article, we extend the result of Shuhong Chen and Zhong Tan[7] and Giaquinta and Modica[18] to the stationary Navier-Stokes system with subquadratic growth.
 ON THE ENTROPY OF FLOWS WITH REPARAMETRIZED GLUING ORBIT PROPERTY Peng SUN Acta mathematica scientia,Series B. 2020, 40 (3):  855-862.  DOI: 10.1007/s10473-020-0318-z We show that a flow or a semiflow with a weak form of reparametrized gluing orbit property has positive topological entropy if it is not minimal.
 SYNCHRONIZATION OF SINGULAR MARKOVIAN JUMPING NEUTRAL COMPLEX DYNAMICAL NETWORKS WITH TIME-VARYING DELAYS VIA PINNING CONTROL K. S. ANAND, J. YOGAMBIGAI, G. A. HARISH BABU, M. SYED ALI, S. PADMANABHAN Acta mathematica scientia,Series B. 2020, 40 (3):  863-886.  DOI: 10.1007/s10473-020-0319-y This article discusses the synchronization problem of singular neutral complex dynamical networks (SNCDN) with distributed delay and Markovian jump parameters via pinning control. Pinning control strategies are designed to make the singular neutral complex networks synchronized. Some delay-dependent synchronization criteria are derived in the form of linear matrix inequalities based on a modified Lyapunov-Krasovskii functional approach. By applying the Lyapunov stability theory, Jensen's inequality, Schur complement, and linear matrix inequality technique, some new delay-dependent conditions are derived to guarantee the stability of the system. Finally, numerical examples are presented to illustrate the effectiveness of the obtained results.
 ON APPROXIMATE EFFICIENCY FOR NONSMOOTH ROBUST VECTOR OPTIMIZATION PROBLEMS Tadeusz ANTCZAK, Yogendra PANDEY, Vinay SINGH, Shashi Kant MISHRA Acta mathematica scientia,Series B. 2020, 40 (3):  887-902.  DOI: 10.1007/s10473-020-0320-5 In this article, we use the robust optimization approach (also called the worst-case approach) for finding ε-efficient solutions of the robust multiobjective optimization problem defined as a robust (worst-case) counterpart for the considered nonsmooth multiobjective programming problem with the uncertainty in both the objective and constraint functions. Namely, we establish both necessary and sufficient optimality conditions for a feasible solution to be an ε-efficient solution (an approximate efficient solution) of the considered robust multiobjective optimization problem. We also use a scalarizing method in proving these optimality conditions.