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Existence of Global Solutions to the Nonlocal mKdV Equation on the Line

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Chinese Annals of Mathematics,Series B 2024年第4期45卷 497-528页

作者： Anran LIU Engui FAN School of Mathematical Sciences Fudan UniversityShanghai 200433China

In this paper,the authors address the existence of global solutions to the Cauchy problem for the integrable nonlocal modified Korteweg-de Vries(nonlocal mKdV for short)equation under the initial data u0∈H^(3)(R)∩H^(1,1)(R)with the L^(1)(R)small-norm assumption.A lipschitz L2-bijection map between potential and reflection coefficient is established by using inverse scattering method based on a Riemann-Hilbert problem associated with the Cauchy *** map from initial potential to reflection coefficient is obtained in direct scattering *** inverse scattering transform goes back to the map from scattering coefficient to potential by applying the reconstruction formula and Cauchy integral *** bijective relation naturally yields the existence of global solutions in a Sobolev space H^(3)(R)∩H^(1,1)(R)to the Cauchy problem.

关键词： Nonlocal mKdV equation Riemann-Hilbert problem Plemelj projection operator lipschitz continuous Global solutions

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EXISTENCE AND ITERATIVE ALGORITHMS OF SOLUTIONS FOR GENERALIZED MIXED QUASI_VARIATIONAL INEQUALITIES

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Applied Mathematics and Mechanics(English Edition) 1999年第9期20卷 1038-1045页

作者：罗春林 Department of Mathematics Kangding Teacher's College Kangding P R China

In this paper, by using the auxiliary technique of variational inequalities, the existence and iterative algorithm; of solutions for a class of generalized mixed quasi-variational inequalities are studied. Our results... 详细信息

关键词： generalized mixed quasi-variational inequalities strongly monotone lipschitz continuous Hausdorff metric

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A New Modified Two-Subgradient Extragradient Algorithm for Solving Variational Inequality Problems

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Journal of Mathematical Research with Applications 2022年第4期42卷 402-412页

作者： Jiaxin CHEN Minglu YE School of Mathematical Sciences Sichuan Normal University School of Mathematics and Information China West Normal University

In this paper, we propose a modified two-subgradient extragradient algorithm(MTSEGA) for solving monotone and lipschitz continuous variational inequalities with the feasible set being a level set of a smooth convex function in Hilbert space. The advantage of MTSEGA is that all the projections are computed onto a half-space per iteration. Moreover, MTSEGA only needs one computation of the underlying mapping per iteration. Under the same assumptions with the known algorithm, we show that the sequence generated by this algorithm is weakly convergent to a solution of the concerned problem.

关键词： two-subgradient extragradient algorithm monotone lipschitz continuous variational inequality Hilbert space

Modification of the Convergence of GG-PPA for Solving Generalized Equations

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Journal of Applied Mathematics and Physics 2023年第1期11卷 260-275页

作者： Asraful Alom Zaidur Rahman Bayezid Gazi Imran Hossain Department of Mathematics Khulna University of Engineering & Technology Khulna Bangladesh

A modified Gauss-type Proximal Point Algorithm (modified GG-PPA) is presented in this paper for solving the generalized equations like 0 ∈T(x), where T is a set-valued mapping acts between two different Banach spaces X and Y. By considering some necessary assumptions, we show the existence of any sequence generated by the modified GG-PPA and prove the semi-local and local convergence results by using metrically regular mapping. In addition, we give a numerical example to justify the result of semi-local convergence.

关键词： Set-Valued Mappings Metrically Regular Mappings Semi-Local Convergence lipschitz continuous Fixed Point Lemma

The Family of Exponential Attractors and Random Attractors for a Class of Kirchhoff Equations

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Journal of Applied Mathematics and Physics 2021年第12期9卷 3143-3154页

作者： Guoguang Lin Chunmeng Zhou Department of Mathematics Yunnan University Kunming China

To prove the existence of the family of exponential attractors, we first define a family of compact, invariant absorbing sets Bk. Then we prove that the solution semigroup has lipschitz property and discrete squeezing property. Finally, we obtain a family of exponential attractors and its estimation of dimension by combining them with previous theories. Next, we obtain Kirchhoff-type random equation by adding product white noise to the right-hand side of the equation. To study the existence of random attractors, firstly we transform the equation by using Ornstein-Uhlenbeck process. Then we obtain a family of bounded random absorbing sets via estimating the solution of the random differential equation. Finally, we prove the asymptotic compactness of semigroup of the stochastic dynamic system;thereby we obtain a family of random attractors.

关键词： Family of Exponential Attractors lipschitz continuous Squeezing Property Stochastic Dynamic System Family Random Attractors

A Family of Inertial Manifolds for a Class of Asymmetrically Coupled Generalized Higher-Order Kirchhoff Equations

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Open Journal of Applied Sciences 2022年第7期12卷 1174-1183页

作者： Guoguang Lin Min Shao Department of Mathematics Yunnan University Kunming China

In this paper, we study the inertial manifolds for a class of asymmetrically coupled generalized Higher-order Kirchhoff equations. Under appropriate assumptions, we firstly exist Hadamard’s graph transformation method to structure a graph norm of a lipschitz continuous function, then we prove the existence of a family of inertial manifolds by showing that the spectral gap condition is true.

关键词： Inertial Manifold Hadamard’s Graph Transformation Method lipschitz continuous Spectral Gap Condition