2023年, 第39卷, 第2期　刊出日期：2023-04-15

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  Global Stability of a Time-delayed Malaria Model with Standard Incidence Rate

  Song-bai GUO, Min HE, Jing-an CUI

  应用数学学报(英文版). 2023, 39(2): 211-221. https://doi.org/10.1007/s10255-023-1042-y

  摘要 ( 90 ) PDF全文 ( 97 )   可视化   收藏

  A four-dimensional delay differential equations (DDEs) model of malaria with standard incidence rate is proposed. By utilizing the limiting system of the model and Lyapunov direct method, the global stability of equilibria of the model is obtained with respect to the basic reproduction number $R_0$. Specifically, it shows that the disease-free equilibrium $E^0$ is globally asymptotically stable (GAS) for $R_0<1$, and globally attractive (GA) for $R_0=1$, while the endemic equilibrium $E^{\ast }$ is GAS and $E^0$ is unstable for $R_0>1$. Especially, to obtain the global stability of the equilibrium $E^{\ast }$ for $R_0>1$, the weak persistence of the model is proved by some analysis techniques.
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  Best Linear Unbiased Estimators of Location and Scale Ranked Set Parameters under Moving Extremes Sampling Design

  Yan-fei DONG, Wang-xue CHEN, Min-yu XIE

  应用数学学报(英文版). 2023, 39(2): 222-231. https://doi.org/10.1007/s10255-023-1043-x

  摘要 ( 63 ) PDF全文 ( 83 )   可视化   收藏

  In the current paper, the best linear unbiased estimators (BLUEs) of location and scale parameters from location-scale family will be respectively proposed in cases when one parameter is known and when both are unknown under moving extremes ranked set sampling (MERSS). Explicit mathematical expressions of these estimators and their variances are derived. Their relative efficiencies with respect to the minimum variance unbiased estimators (MVUEs) under simple random sampling (SRS) are compared for the cases of some usual distributions. The numerical results show that the BLUEs under MERSS are significantly more efficient than the MVUEs under SRS.
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  Path Factors and Neighborhoods of Independent Sets in Graphs

  Si-zhong ZHOU

  应用数学学报(英文版). 2023, 39(2): 232-238. https://doi.org/10.1007/s10255-022-1096-2

  摘要 ( 67 ) PDF全文 ( 57 )   可视化   收藏

  A path-factor is a spanning subgraph $F$ of $G$ such that every component of $F$ is a path with at least two vertices. Let $k\ge 2$ be an integer. A $P_{\ge k}$-factor of $G$ means a path factor in which each component is a path with at least $k$ vertices. A graph $G$ is a $P_{\ge k}$-factor covered graph if for any $e\in E(G)$, $G$ has a $P_{\ge k}$-factor including $e$. Let $\beta$ be a real number with $\frac{1}{3}\le \beta \le 1$ and $k$ be a positive integer. We verify that (\romannumeral1) a $k$-connected graph $G$ of order $n$ with $n\ge 5k+2$ has a $P_{\ge 3}$-factor if $|N_G(I)|>\beta (n-3k-1)+k$ for every independent set $I$ of $G$ with $|I|=\lfloor \beta (2k+1)\rfloor$; (\romannumeral2) a $(k+1)$-connected graph $G$ of order $n$ with $n\ge 5k+2$ is a $P_{\ge 3}$-factor covered graph if $|N_G(I)|>\beta (n-3k-1)+k+1$ for every independent set $I$ of $G$ with $|I|=\lfloor \beta (2k+1)\rfloor$.
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  Star-factorization of the Complete Bipartite Multigraphs

  Jing SHI, Jian WANG, Bei-liang DU

  应用数学学报(英文版). 2023, 39(2): 239-248. https://doi.org/10.1007/s10255-023-1044-9

  摘要 ( 53 ) PDF全文 ( 33 )   可视化   收藏

  Let $\lambda K_{m,n}$ be a complete bipartite multigraph with two partite sets having $m$ and $n$ vertices, respectively. A $K_{p,q}$-factorization of $\lambda K_{m,n}$ is a set of $K_{p,q}$-factors of $\lambda K_{m,n}$ which partition the set of edges of $\lambda K_{m,n}$. When $\lambda =1$, Martin, in [Complete bipartite factorizations by complete bipartite graphs, Discrete Math., 167/168 (1997), 461-480], gave simple necessary conditions for such a factorization to exist, and conjectured those conditions are always sufficient. In this paper, we will study the $K_{p,q}$-factorization of $\lambda K_{m,n}$ for $p=1$, to show that the necessary conditions for such a factorization are always sufficient whenever related parameters are sufficiently large.
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  Mild Solutions of a Class of Conformable Fractional Differential Equations with Nonlocal Conditions

  Mohamed BOUAOUID

  应用数学学报(英文版). 2023, 39(2): 249-261. https://doi.org/10.1007/s10255-022-1015-6

  摘要 ( 90 ) PDF全文 ( 83 )   可视化   收藏

  This paper deals with the existence, uniqueness and continuous dependence of mild solutions for a class of conformable fractional differential equations with nonlocal initial conditions. The results are obtained by means of the classical fixed point theorems combined with the theory of cosine family of linear operators.
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  The Spread Speed of Multiple Catalytic Branching Random Walks

  Rong-li LIU

  应用数学学报(英文版). 2023, 39(2): 262-292. https://doi.org/10.1007/s10255-023-1046-7

  摘要 ( 56 ) PDF全文 ( 43 )   可视化   收藏

  In this paper we study the asymptotic behavior of the maximal position of a supercritical multiple catalytic branching random walk $(X_n)$ on $\mathbb{Z}$. If $M_n$ is its maximal position at time $n$, we prove that there is a constant $\alpha >0$ such that $M_n/n$ converges to $\alpha$ almost surely on the set of infinite number of visits to the set of catalysts. We also derive the asymptotic law of the centered process $M_n-\alpha n$ as $n\to \mathrm{\infty }$. Our results are similar to those in [13]. However, our results are proved under the assumption of finite $L\log L$ moment instead of finite second moment. We also study the limit of $(X_n)$ as a measure-valued Markov process. For any function $f$ with compact support, we prove a strong law of large numbers for the process $X_n(f)$.
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  Controllability of Nonlinear Discrete Systems with Degeneracy

  Yue LYU, Xue-li TAN, Xue YANG, Yong LI

  应用数学学报(英文版). 2023, 39(2): 293-305. https://doi.org/10.1007/s10255-023-1047-6

  摘要 ( 30 ) PDF全文 ( 43 )   可视化   收藏

  This paper concerns the controllability of autonomous and nonautonomous nonlinear discrete systems, in which linear parts might admit certain degeneracy. By introducing Fredholm operators and coincidence degree theory, sufficient conditions for nonlinear discrete systems to be controllable are presented. In addition, applications are given to illustrate main results.
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  Energy Decay for von Kármán-Gurtin-Pipkin System

  Hanni DRIDI

  应用数学学报(英文版). 2023, 39(2): 306-319. https://doi.org/10.1007/s10255-023-1045-8

  摘要 ( 44 ) PDF全文 ( 42 )   可视化   收藏

  This paper aims to prove the asymptotic behavior of the solution for the thermo-elastic von Karman system where the thermal conduction is given by Gurtin-Pipkins law. Existence and uniqueness of the solution are proved within the semigroup framework and stability is achieved thanks to a suitable Lyapunov functional. Therefore, the stability result clarified that the solutions energy functional decays exponentially at infinite time.
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  Estimation of Treatment Effects in Nonlinear Models with Unobserved Confounding

  Yu-ling LI, Jun WANG

  应用数学学报(英文版). 2023, 39(2): 320-336. https://doi.org/10.1007/s10255-023-1051-x

  摘要 ( 30 ) PDF全文 ( 36 )   可视化   收藏

  Estimation of treatment effects is one of the crucial mainstays in economics and sociology studies. The problem will become more serious and complicated if the treatment variable is endogenous for the presence of unobserved confounding. The estimation and conclusion are likely to be biased and misleading if the endogeny of treatment variable is ignored. In this article, we propose the pseudo maximum likelihood method to estimate treatment effects in nonlinear models. The proposed method allows the unobserved confounding and random error terms to exist in an arbitrary relationship (such as, add or multiply), and the unobserved confounding have different influence directions on treatment variables and outcome variables. The proposed estimator is consistent and asymptotically normally distributed. Simulation studies show that the proposed estimator performs better than the special regression estimator, and the proposed method is stable for various distribution of error terms. Finally, the proposed method is applied to the real data that studies the influence of individuals have health insurance on an individual’s decision to visit a doctor.
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  A.S. Convergence Rate and $L^p$-Convergence of Bisexual Branching Processes in a Random Environment and Varying Environment

  Sheng XIAO, Xiang-dong LIU, Ying-qiu LI

  应用数学学报(英文版). 2023, 39(2): 337-353. https://doi.org/10.1007/s10255-023-1052-9

  摘要 ( 33 ) PDF全文 ( 36 )   可视化   收藏

  Let $(Z_n)$ be a supercritical bisexual branching process in a random environment $\xi$. We study the almost sure (a.s.) convergence rate of the submartingale ${\overline{W}}_n=Z_n/I_n$ to its limit $\overline{W}$, where $(I_n)$ is an usually used norming sequence. We prove that under a moment condition of order $p\in (1,2),\overline{W}-{\overline{W}}_n=o(e^{-na})$ a.s. for some $a>0$ that we find explicitly; assuming the logarithmic moment condition holds, we have $\overline{W}-{\overline{W}}_n=o(n^{-\alpha })$ a.s.. In order to obtain these results, we provide the $L^p-$ convergence of $({\overline{W}}_n)$; similar conclusions hold for a bisexual branching process in a varying environment.
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  A Bregman-style Partially Symmetric Alternating Direction Method of Multipliers for Nonconvex Multi-block Optimization

  Peng-jie LIU, Jin-bao JIAN, Guo-dong MA

  应用数学学报(英文版). 2023, 39(2): 354-380. https://doi.org/10.1007/s10255-023-1048-5

  摘要 ( 55 ) PDF全文 ( 37 )   可视化   收藏

  The alternating direction method of multipliers (ADMM) is one of the most successful and powerful methods for separable minimization optimization. Based on the idea of symmetric ADMM in two-block optimization, we add an updating formula for the Lagrange multiplier without restricting its position for multiblock one. Then, combining with the Bregman distance, in this work, a Bregman-style partially symmetric ADMM is presented for nonconvex multi-block optimization with linear constraints, and the Lagrange multiplier is updated twice with different relaxation factors in the iteration scheme. Under the suitable conditions, the global convergence, strong convergence and convergence rate of the presented method are analyzed and obtained. Finally, some preliminary numerical results are reported to support the correctness of the theoretical assertions, and these show that the presented method is numerically effective.
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  On the Existence of Ground State Solutions to a Quasilinear Schrödinger Equation involving p-Laplacian

  Ji-xiu WANG, Qi GAO

  应用数学学报(英文版). 2023, 39(2): 381-395. https://doi.org/10.1007/s10255-023-1053-8

  摘要 ( 38 ) PDF全文 ( 50 )   可视化   收藏

  We consider the following quasilinear Schrödinger equation involving $p$-Laplacian $$\begin{array}{r}-{\mathrm{\Delta }}_{p}u+V(x)|u{|}^{p-2}u-{\mathrm{\Delta }}_p(|u{|}^{2\eta })|u{|}^{2\eta -2}u=\lambda \frac{|u{|}^{q-2}u}{|x{|}^{\mu }}+\frac{|u{|}^{2\eta p^{\ast }(\nu )-2}u}{|x{|}^{\nu }}\text{in}{\mathbb{R}}^N,\end{array}$$ where $N>p>1,\eta \ge \frac{p}{2(p-1)}$, $p<q<2\eta p^{\ast }(\mu )$, $p^{\ast }(s)=\frac{p(N-s)}{N-p}$, and $\lambda ,\mu ,\nu$ are parameters with $\lambda >0$, $\mu ,\nu \in [0,p)$. Via the Mountain Pass Theorem and the Concentration Compactness Principle, we establish the existence of nontrivial ground state solutions for the above problem.
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  Defined Contribution Pension Planning with the Return of Premiums Clauses and HARA Preference in Stochastic Environments

  Hao CHANG, Xing-jiang CHEN

  应用数学学报(英文版). 2023, 39(2): 396-423. https://doi.org/10.1007/s10255-023-1050-y

  摘要 ( 28 ) PDF全文 ( 35 )   可视化   收藏

  This paper studies a defined contribution (DC) pension fund investment problem with return of premiums clauses in a stochastic interest rate and stochastic volatility environment. In practice, most of pension plans were subject to the return of premiums clauses to protect the rights of pension members who died before retirement. In the mathematical modeling, we assume that a part of pension members could withdraw their premiums if they died before retirement and surviving members could equally share the difference between accumulated contributions and returned premiums. We suppose that the financial market consists of a risk-free asset, a stock, and a zero-coupon bond. The interest rate is driven by a stochastic affine interest rate model and the stock price follows the Heston’s stochastic volatility model with stochastic interest rates. Different fund managers have different risk preferences, and the hyperbolic absolute risk aversion (HARA) utility function is a general one including a power utility, an exponential utility, and a logarithm utility as special cases. We are concerned with an optimal portfolio to maximize the expected utility of terminal wealth by choosing the HARA utility function in the analysis. By using the principle of dynamic programming and Legendre transform-dual theory, we obtain explicit solutions of optimal strategies. Some special cases are also derived in detail. Finally, a numerical simulation is provided to illustrate our results.
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  Approximate Analytical Solution of the Generalized Kolmogorov-Petrovsky-Piskunov Equation with Cubic Nonlinearity

  Wei-guo ZHANG, Xie-kui HU, Xing-qian LING, Wen-xia LI

  应用数学学报(英文版). 2023, 39(2): 424-449. https://doi.org/10.1007/s10255-023-1054-7

  摘要 ( 50 ) PDF全文 ( 48 )   可视化   收藏

  In this paper, the approximate analytical oscillatory solutions to the generalized KolmogorovPetrovsky-Piskunov equation (gKPPE for short) are discussed by employing the theory of dynamical system and hypothesis undetermined method. According to the corresponding dynamical system of the bounded traveling wave solutions to the gKPPE, the number and qualitative properties of these bounded solutions are received. Furthermore, pulses (bell-shaped) and waves fronts (kink-shaped) of the gKPPE are given. In particular, two types of approximate analytical oscillatory solutions are constructed. Besides, the error estimations between the approximate analytical oscillatory solutions and the exact solutions of the gKPPE are obtained by the homogeneity principle. Finally, the approximate analytical oscillatory solutions are compared with the numerical solutions, which shows the two types of solutions are similar.
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  Solutions of the Camassa-Holm Equation Near the Soliton

  Dan-ping DING, Wei LU

  应用数学学报(英文版). 2023, 39(2): 450-464. https://doi.org/10.1007/s10255-023-1056-5

  摘要 ( 52 ) PDF全文 ( 56 )   可视化   收藏

  In this paper, solutions of the Camassa-Holm equation near the soliton $Q$ is decomposed by pseudo-conformal transformation as follows: ${\lambda }^{1/2}(t)u(t,\lambda (t)y+x(t))=Q(y)+\epsilon (t,y)$, and the estimation formula with respect to $\epsilon (t,y)$ is obtained: $|\epsilon (t,y)|\le C{a}_{3}T{e}^{-\theta |y|}+|{\lambda }^{1/2}(t){\epsilon }_0|$. For the CH equation, we prove that the solution of the Cauchy problem and the soliton $Q$ is sufficiently close as $y\to \mathrm{\infty }$, and the approximation degree of the solution and $Q$ is the same as that of initial data and $Q$, besides the energy distribution of $\epsilon$ is consistent with the distribution of the soliton $Q$ in $H^2$.