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Existence of solutions for infinite nonlinear (p, q)-integral equations 无穷非线性(p, q)-积分方程解的存在性
IF 2.6 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2026-10-01 Epub Date: 2026-02-22 DOI: 10.1016/j.cam.2026.117489
Hamid Reza Sahebi , Manochehr Kazemi , Bipan Hazarika
In this work, we investigate the existence of solutions for an infinite system of nonlinear (p, q)-integral equations within the framework of Banach spaces. Utilizing the concept of measure of noncompactness and Petryshyn’s fixed point theorem, we derive a set of sufficient conditions under which the system admits at least one solution. The methodology integrates the structure of generalized (p, q)-calculus with operator-theoretic techniques to handle infinite-dimensional behavior effectively. The analytical framework is complemented by illustrative examples that demonstrate the validity and applicability of the main results.
在这项工作中,我们研究了Banach空间框架内非线性(p, q)-积分方程无穷系统解的存在性。利用非紧测度的概念和Petryshyn不动点定理,导出了系统至少存在一个解的充分条件。该方法将广义(p, q)-微积分的结构与算子理论技术相结合,有效地处理了无限维行为。分析框架辅以举例说明,证明了主要结果的有效性和适用性。
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引用次数: 0
Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by W1,p energy analysis 用W1,p能量分析描述拟线性双曲抛物型系统的光滑小数据解
IF 1.8 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2026-10-01 Epub Date: 2025-12-24 DOI: 10.1016/j.nonrwa.2025.104580
Leander Claes , Michael Winkler
<div><div>In bounded <em>n</em>-dimensional domains with <em>n</em> ≥ 1, this manuscript examines an initial-boundary value problem for the system<span><span><span><math><mtable><mtr><mtd><mrow><mo>{</mo><mtable><mtr><mtd><mrow><msub><mi>u</mi><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>=</mo><mi>∇</mi><mo>·</mo><mrow><mo>(</mo><mi>γ</mi><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><mi>∇</mi><msub><mi>u</mi><mi>t</mi></msub><mo>)</mo></mrow><mo>+</mo><mi>a</mi><mi>∇</mi><mo>·</mo><mrow><mo>(</mo><mi>γ</mi><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><mi>∇</mi><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>∇</mi><mo>·</mo><mi>f</mi><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mstyle><mi>Θ</mi></mstyle><mi>t</mi></msub><mo>=</mo><mi>D</mi><mstyle><mi>Δ</mi></mstyle><mstyle><mi>Θ</mi></mstyle><mo>+</mo><mstyle><mi>Γ</mi></mstyle><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><msup><mrow><mo>|</mo><mi>∇</mi><msub><mi>u</mi><mi>t</mi></msub><mo>|</mo></mrow><mn>2</mn></msup><mo>+</mo><mi>F</mi><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><mo>·</mo><mi>∇</mi><msub><mi>u</mi><mi>t</mi></msub><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></math></span></span></span>which in the case <span><math><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow></math></span> and with <em>γ</em> ≡ Γ as well as <em>f</em> ≡ <em>F</em> reduces to the classical model for the evolution of displacement and temperatures in thermoviscoelasticity. Unlike in previous related studies, the focus here is on situations in which besides <em>f</em> and <em>F</em>, also the core ingredients <em>γ</em> and Γ may depend on the temperature variable Θ. Firstly, a statement on local existence of classical solutions is derived for arbitrary <em>a</em> > 0, <em>D</em> > 0 as well as 0 < <em>γ</em> ∈ <em>C</em><sup>2</sup>([0, ∞)) and 0 ≤ Γ ∈ <em>C</em><sup>1</sup>([0, ∞)), for functions <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>;</mo><msup><mi>R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>F</mi><mo>∈</mo><msup><mi>C</mi><mn>1</mn></msup><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>;</mo><msup><mi>R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>F</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></mrow></math></span>, and for suitably regular initial data of arbitrary size. Secondly, it is seen that under an additional assumption on smallness of <em>a, f</em>′ and <em>F</em>, as well as on the deviation of the initial data from the constant state given by <span><math><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mstyle><mi>Θ</mi></mstyle><mo>=</mo><msub><mstyle><mi>Θ</mi></mstyle><mi>★</mi>
在有界与n n维域 ≥ 1,这手稿检查系统的初边值问题{utt =∇·(γ(Θ)∇ut) +一个∇·(γ(Θ)∇u) +∇·f(Θ)Θt = DΔΘ+Γ(Θ)|∇ut | 2 + f(Θ)·∇ut,对于n = 1和γ ≡ Γ以及f ≡ f减少的经典模型的进化在thermoviscoelasticity位移和温度。与以往的相关研究不同,这里的重点是除了f和f之外,核心成分γ和Γ也可能取决于温度变量Θ的情况。首先,声明对当地古典解的存在性推导出任意一个 祝辞 0 D 祝辞 0 0 & lt; γ ∈ C2([0,∞))和0 ≤ Γ ∈ C1([0,∞)),函数f∈C2([0,∞);Rn), F∈C1([0,∞);Rn), F(0)=0,对于任意大小的适当规则初始数据。其次,我们可以看到,在附加假设a, f '和f的小性,以及初始数据与任意固定的Θ - ≥ 0给出的u=0和Θ=Θ★的恒定状态的偏差下,在凸域上,这些解在时间上实际上是全局的,并且具有∇ut,∇u和∇Θ在Lp中呈指数级快速衰减的性质。这是通过在Lp空间中检测涉及这些梯度范数的泛函的合适耗散性质来实现的。
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Unlike in previous related studies, the focus here is on situations in which besides &lt;em&gt;f&lt;/em&gt; and &lt;em&gt;F&lt;/em&gt;, also the core ingredients &lt;em&gt;γ&lt;/em&gt; and Γ may depend on the temperature variable Θ. Firstly, a statement on local existence of classical solutions is derived for arbitrary &lt;em&gt;a&lt;/em&gt; &gt; 0, &lt;em&gt;D&lt;/em&gt; &gt; 0 as well as 0 &lt; &lt;em&gt;γ&lt;/em&gt; ∈ &lt;em&gt;C&lt;/em&gt;&lt;sup&gt;2&lt;/sup&gt;([0, ∞)) and 0 ≤ Γ ∈ &lt;em&gt;C&lt;/em&gt;&lt;sup&gt;1&lt;/sup&gt;([0, ∞)), for functions &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and for suitably regular initial data of arbitrary size. Secondly, it is seen that under an additional assumption on smallness of &lt;em&gt;a, f&lt;/em&gt;′ and &lt;em&gt;F&lt;/em&gt;, as well as on the deviation of the initial data from the constant state given by &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mstyle&gt;&lt;mi&gt;Θ&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mstyle&gt;&lt;mi&gt;Θ&lt;/mi&gt;&lt;/mstyle&gt;&lt;mi&gt;★&lt;/mi&gt;","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104580"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Mathematical analysis of a levitation model 悬浮模型的数学分析
IF 1.8 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2026-10-01 Epub Date: 2025-12-23 DOI: 10.1016/j.nonrwa.2025.104573
Rafael Muñoz-Sola
The aim of this paper is to study a model of electromagnetic levitation for a metallic rigid body. The model is constituted by the transient linear model of eddy currents under the hypothesis of axisymmetry, written in terms of a magnetic potential vector, coupled with an ODE which governs the vertical motion of the body. The electromagnetic model is a parabolic-elliptic PDE which parabolicity region is the position occupied by the body, which changes with time. Besides, Lorentz force appears in the RHS of the ODE. Thus, the model exhibits a coupling of geometrical nature. We establish the existence and uniqueness of solution of the coupled problem and we study its maximally defined solution. In particular, we prove that a blow-up of the velocity of the body cannot happen. Our techniques involve: a reformulation of the coupled problem as a causal differential equation, an adaptation of the theory about this kind of equations and a result of locally Lipschitz dependence of the magnetic potential vector with respect to the velocity of the body.
本文的目的是研究金属刚体的电磁悬浮模型。该模型是在轴对称假设下涡流的瞬态线性模型,用磁势向量表示,加上控制物体垂直运动的ODE。电磁模型为抛物-椭圆偏微分方程,抛物区域为物体所占位置,抛物区域随时间变化。此外,洛伦兹力出现在ODE的RHS中。因此,该模型表现出几何性质的耦合。建立了该耦合问题解的存在唯一性,并研究了其最大定义解。特别地,我们证明了物体的速度不可能突然增大。我们的技术包括:耦合问题作为因果微分方程的重新表述,关于这类方程的理论的改编,以及磁势矢量相对于物体速度的局部利普希茨依赖的结果。
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引用次数: 0
An epidemic model for bovine rabies transmission by bats with spatial diffusion 具有空间扩散的蝙蝠传播牛狂犬病的流行模型
IF 1.8 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2026-10-01 Epub Date: 2026-01-19 DOI: 10.1016/j.nonrwa.2026.104603
José Paulo Carvalho dos Santos , Evandro Monteiro , Nelson Henrique Teixeira Lemes , Ana Claudia Pereira
The focus of this research is an epidemic model that examines the spread of rabies in the bovine population, with the spatial diffusion in the bat population, which serves as the vector population. The study investigates both the well-posedness and qualitative behavior of equilibrium points. The paper establishes the well-posedness of the model through Semigroup theory of sectorial operators and existence results for abstract parabolic differential equations. The research also addresses the definition of the basic reproduction number, R0, which acts as a threshold index point using linearization theory for reaction-diffusion equations in the disease-free equilibrium point. Additionally, the global asymptotic stability is established through the use of a Lyapunov function and energy estimates.
本研究的重点是研究狂犬病在牛种群中的传播,以及作为媒介种群的蝙蝠种群的空间扩散的流行病模型。研究了平衡点的适定性和定性行为。本文利用扇形算子的半群理论和抽象抛物型微分方程的存在性结果,建立了该模型的适定性。研究还讨论了基本繁殖数R0的定义,R0作为无病平衡点反应扩散方程的线性化理论的阈值指标点。此外,利用Lyapunov函数和能量估计建立了系统的全局渐近稳定性。
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引用次数: 0
Effect of diffusion rates on a nonlocal SIS model with distinct dispersal kernels and logistic source 扩散速率对具有不同扩散核和逻辑源的非局部SIS模型的影响
IF 1.8 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2026-10-01 Epub Date: 2026-01-31 DOI: 10.1016/j.nonrwa.2026.104613
Boubakr Lamouri , Ahmed Boudaoui , Salih Djilali
We investigate a nonlocal SIS epidemic model that incorporates distinct mobility patterns for susceptible and infected individuals, together with a logistic growth. The model includes distinct nonlocal diffusion kernels, denoted by J1(x) and J2(x), which represent different mobility strategies of the susceptible and infected populations, respectively. This formulation enhances the biological realism of the model by allowing greater flexibility in the representation of individual movement behaviors. Consequently, it introduces additional mathematical challenges in the analysis while providing a more accurate modelling for studying the spatial spread of infectious diseases. We establish the well-posedness, positivity, and uniform boundedness of solutions, and prove the existence of a global attractor. The basic reproduction number R0 is derived, and persistence theory is used to show the existence of an endemic steady state when R0>1. We further analyze the asymptotic profiles of the endemic steady states under extreme diffusion limits, highlighting the impact of mobility on disease persistence.
我们研究了一个非本地SIS流行病模型,该模型结合了易感和感染个体的不同流动模式,以及逻辑增长。模型包含不同的非局部扩散核,分别用J1(x)和J2(x)表示,分别代表易感种群和感染种群的不同迁移策略。这个公式通过允许更大的灵活性来表示个体运动行为,从而增强了模型的生物真实感。因此,它在为研究传染病的空间传播提供更准确的模型的同时,在分析中引入了额外的数学挑战。我们建立了解的适定性、正性和一致有界性,并证明了全局吸引子的存在性。导出了基本繁殖数R0,并利用持续理论证明了在R0>;1时存在地方性稳态。我们进一步分析了极端扩散极限下地方性稳态的渐近分布,强调了流动性对疾病持久性的影响。
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引用次数: 0
Relaxation Crank-Nicolson compact finite difference schemes for Allen-Cahn equation Allen-Cahn方程的松弛Crank-Nicolson紧致有限差分格式
IF 2.6 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2026-10-01 Epub Date: 2026-02-10 DOI: 10.1016/j.cam.2026.117409
Mingrong Cui
Relaxation Crank-Nicolson compact finite difference schemes for solving both one dimensional and two dimensional Allen-Cahn equation are given and analyzed in this paper. Using the idea of relaxation scheme, that is, after introducing a new auxiliary variable, we get a newly added equation to separate the nonlinear term in the original equation. After we discretize the time derivative by Crank-Nicolson scheme with the newly introduced variable approximated on the staggered time mesh points, and approximate the second order spatial derivatives by the compact finite difference method, we obtain the fully discrete relaxation compact finite difference schemes. The linear relaxation schemes have the properties of discrete mass conservation and discrete energy dissipation. Some numerical results are provided, showing that the schemes are second order accurate in time and fourth order accurate in space, verifying the accuracy and efficiency of the proposed algorithm.
本文给出了求解一维和二维Allen-Cahn方程的松弛性Crank-Nicolson紧致有限差分格式,并对其进行了分析。利用松弛方案的思想,即在引入一个新的辅助变量后,得到一个新的附加方程来分离原方程中的非线性项。将新引入的变量近似于交错时间网格点,用Crank-Nicolson格式离散时间导数,用紧致有限差分法近似二阶空间导数,得到了完全离散松弛紧致有限差分格式。线性松弛方案具有离散质量守恒和离散能量耗散的性质。数值结果表明,该算法在时间上具有二阶精度,在空间上具有四阶精度,验证了算法的准确性和高效性。
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引用次数: 0
Optimal error bound and regularization methods for identifying an unknown source in an advection-dispersion equation 平流色散方程中未知源识别的最优误差界和正则化方法
IF 2.6 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2026-10-01 Epub Date: 2026-02-22 DOI: 10.1016/j.cam.2026.117461
Hanghang Wu , Hongqi Yang
We consider the problem of identifying the source term in an advection-dispersion equation based on given terminal data. It is shown that this is an ill-posed problem. The optimal error bound of the problem under certain source conditions is given. Then, the mollification regularization method and the Fourier regularization method are used to solve the problem respectively. Under the selection rules of a-priori and a-posteriori regularization parameters, we derive the a-priori and a-posteriori error estimates. From the theoretical derivation, it can be seen that the error estimates obtained by both regularization methods do not exhibit saturation effects, and the a-posteriori error estimate obtained by using the Fourier regularization is optimal. Finally, numerical experiments are conducted to demonstrate the effectiveness and stability of the proposed regularization methods. Additionally, comparisons between the two regularization methods are presented, along with the conclusions drawn from these comparisons.
我们考虑了基于给定终端数据的平流-色散方程中源项的识别问题。证明了这是一个不适定问题。在一定的源条件下,给出了问题的最优误差界。然后分别采用柔化正则化方法和傅立叶正则化方法求解该问题。在先验和后验正则化参数的选择规则下,推导了先验和后验误差估计。从理论推导可以看出,两种正则化方法得到的误差估计都不存在饱和效应,采用傅里叶正则化方法得到的后验误差估计是最优的。最后,通过数值实验验证了所提正则化方法的有效性和稳定性。此外,提出了两种正则化方法之间的比较,以及从这些比较中得出的结论。
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引用次数: 0
An ensemble-based model reduction method for random diffusion problems in porous media 多孔介质中随机扩散问题的基于集合的模型简化方法
IF 2.6 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2026-10-01 Epub Date: 2026-02-22 DOI: 10.1016/j.cam.2026.117411
Mengnan Li , Shan Zhang , Xiaofei Guan
In this paper, we present an ensemble-based model reduction method to solve random diffusion problems in porous media. The randomness and multiscale pose significant challenges for simulating these problems. To overcome this difficulty, we develop an ensemble-based iterative algorithm by integrating ensemble Monte Carlo (EMC) method with constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). For random diffusion problems, traditional methods strongly rely on sampling the random space. This necessitates solving the corresponding PDEs for a large number of samples, resulting in high computational costs. To this end, we propose an ensemble method to ensure that all samples share a common stiffness matrix. This significantly improves the computational efficiency. Furthermore, this ensemble method can be combined with LU decomposition or block Krylov subspace iterative methods to further improve computational efficiency. Although the ensemble method avoids solving PDEs multiple times, the high-dimensional challenges posed by the multiscale characteristics remain. Therefore, we employ CEM-GMsFEM to construct an effective reduced-order ensemble model, reducing computational costs. For the proposed ensemble method, we provide a rigorous convergence analysis. A few numerical examples are presented to show the effectiveness of the proposed method.
本文提出了一种基于集成的模型简化方法来求解多孔介质中的随机扩散问题。随机性和多尺度对模拟这些问题提出了重大挑战。为了克服这一困难,我们将集成蒙特卡罗(EMC)方法与约束能量最小化广义多尺度有限元法(CEM-GMsFEM)相结合,开发了一种基于集成的迭代算法。对于随机扩散问题,传统的方法强烈依赖于随机空间的采样。这就需要对大量的样本求解相应的偏微分方程,计算成本很高。为此,我们提出了一种集成方法来确保所有样本共享一个共同的刚度矩阵。这大大提高了计算效率。此外,该集成方法可与LU分解或块Krylov子空间迭代方法相结合,进一步提高计算效率。尽管集成方法避免了多次求解偏微分方程,但多尺度特性带来的高维挑战仍然存在。因此,我们采用CEM-GMsFEM来构建有效的降阶集成模型,降低了计算成本。对于所提出的集成方法,我们提供了严格的收敛分析。算例表明了该方法的有效性。
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引用次数: 0
Radau-type inequalities from multiplicative Katugampola fractional integrals theoretical point of view 从乘法Katugampola分数积分的理论观点看radau型不等式
IF 2.6 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2026-10-01 Epub Date: 2026-02-21 DOI: 10.1016/j.cam.2026.117472
Dingyi Ai , Tingsong Du
Herein, we are particularly intrigued by exploring Radau-type inequalities that emerge from multiplicative Katugampola fractional integrals. To that end, we commence by introducing the concept of such integrals. And several analytical properties, such as boundedness, continuity, commutativity, semigroup property, and others, are examined for the freshly introduced operators. Following this, we derive an identity pertinent to multiplicative Katugampola fractional integrals, which forms the basis for establishing a series of Radau-type inequalities in our investigation. These inequalities are derived under the condition that either the function f* is multiplicatively convex or the function (ln ∘f*)q is convex for q > 1, with a specific focus on the scenario where 0 < q ≤ 1. To deepen the readers’ profound comprehension of the acquired results, we present illustrative examples along with accompanying graphs, which confirm the correctness of the derived inequalities. Finally, we showcase the applicability of these inequalities presented here in various contexts, including multiplicative differential equations, quadrature formulas, and special means.
在这里,我们特别感兴趣的是探索从乘法卡图甘波拉分数积分中出现的radau型不等式。为此,我们首先引入此类积分的概念。并对新引入算子的有界性、连续性、交换性、半群性质等进行了分析。在此之后,我们推导了一个与乘法Katugampola分数积分相关的恒等式,它构成了我们研究中建立一系列radau型不等式的基础。这些不等式是在函数f*为乘法凸或函数(ln°f*)q对q >; 1为凸的条件下推导出来的,特别关注0 <; q ≤ 1的情形。为了加深读者对所获得的结果的深刻理解,我们提供了说明示例以及附带的图表,这些示例证实了推导出的不等式的正确性。最后,我们展示了这些不等式在各种情况下的适用性,包括乘法微分方程、正交公式和特殊方法。
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引用次数: 0
Stochastic limited memory BFGS algorithms for solving nonconvex support vector machine problems and nonconvex risk minimization program 随机有限记忆BFGS算法求解非凸支持向量机问题及非凸风险最小化方案
IF 2.6 2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2026-10-01 Epub Date: 2026-02-24 DOI: 10.1016/j.cam.2026.117487
Gonglin Yuan , Yingjie Zhou , Fanhua Shang , Zhijin Ge , Zhongzhou Jin
It is known that the limited memory BFGS (L-BFGS) method is an important method for solving nonlinear optimization. For stochastic optimization problems, especially for nonconvex functions, it is difficult to preserve Bk≻0 (the inverse of the Hessian matrix) for all k due to the noise incurred by estimating the gradient and the nonconvexity of the functions. In this paper, we propose a novel modified stochastic L-BFGS algorithm (MSLBFGS) for solving nonconvex optimization, where the positive definiteness of the designed matrix Bk is automatically maintained. We establish the convergence property of the proposed MSLBFGS algorithm. In the worst-case scenario, we output 1Nk=1NE[f(xk)2]<ε, and prove that the number of the stochastic first-order oracle calls of the proposed algorithm with a diminishing step length is O(ε11β) due to β ∈ (0.5, 1). Moreover, we also present a new modified stochastic L-BFGS algorithm that incorporates the variance reduction technique. Numerical experiments conducted on nonconvex support vector machine problems and nonconvex empirical risk minimization problems validate the effectiveness of the proposed algorithms, showing their advantages over existing methods.
已知有限记忆BFGS (L-BFGS)方法是求解非线性优化问题的重要方法。对于随机优化问题,特别是对于非凸函数,由于估计梯度和函数的非凸性所产生的噪声,很难对所有k保持Bk: 0 (Hessian矩阵的逆)。本文提出了一种新的改进的随机L-BFGS算法(MSLBFGS)来求解非凸优化问题,该算法自动保持设计矩阵Bk的正确定性。建立了所提出的MSLBFGS算法的收敛性。在最坏情况下,我们输出1N∑k=1NE[∥∇f(xk)∥2]<ε,并证明了由于β ∈ (0.5,1),该算法的随机一阶oracle调用次数随步长递减为O(ε−11−β)。此外,我们还提出了一种新的改进的随机L-BFGS算法,该算法结合了方差缩减技术。对非凸支持向量机问题和非凸经验风险最小化问题进行了数值实验,验证了所提算法的有效性,显示了其相对于现有方法的优势。
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