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Existence of Solutions to the System of Generalized Implicit Vector Quasivariational Inequality Problems
Lin Zhi
Journal of Inequalities and Applications , 2009,
Abstract: We study the system of generalized implicit vector quasivariational inequality problems and prove a new existence result of its solutions by Kakutani-Fan-Glicksberg's fixed points theorem. As a special case, we also derive a new existence result of solutions to the generalized implicit vector quasivariational inequality problems.
Existence and Uniqueness of Solutions for a Type of Generalized Zakharov System  [PDF]
Rui Li,Xing Lin,Zongwei Ma,Jingjun Zhang
Journal of Applied Mathematics , 2013, DOI: 10.1155/2013/193589
Abstract: We study the Cauchy problem for a type of generalized Zakharov system. With the help of energy conservation and approximate argument, we obtain global existence and uniqueness in Sobolev spaces for this system. Particularly, this result implies the existence of classical solution for this generalized Zakharov system. 1. Introduction In this paper, we study a type of generalized Zakharov system which is given by with initial data where , , and is a fixed constant. In the above system, is a fractional differential operator. With this definition, maps to with the Fourier transform of with respect to the variable . In particular, . When , system (1) and (2) reduces to the usual Zakharov system which was first obtained by Zakharov [1]; here, is the slowly varying amplitude of high-frequency electric field and ?? ?? is the disturbing quantity of ion from its equilibrium. This model turned out to be very useful in laser plasmas, and many contributions have been made both in the physical and mathematical literature. For the local or global existence and uniqueness of smooth solutions for system (4), we refer to [2–6]. Well-posedness of (4) in lower regularity spaces was obtained in [7]. Existence of global attractors for dissipative Zakharov system was studied in [8–11]. For related Zakharov system including magnetic effects, one can see [12–15]. On the other hand, Laskin [16, 17] discovered that the path integral over the Lévy-like quantum mechanical paths allows developing the generalization of the quantum mechanics. That is, if the path integral over Brownian trajectories leads to the well-known Schr?dinger equation, then the path integral over Lévy trajectories leads to the fractional Schr?dinger equation. So fractional Schr?dinger equation is fundamental in the fractional quantum mechanics, and its global well-posedness is studied in [18, 19]. Inspired by this, we then replace the Laplacian in the Schr?dinger equation of (4) by the fractional differential operator , and this is the main motivation of the paper. In this work, we study global existence and uniqueness of smooth solutions for system (1) and (2). The main result is stated in the following theorem. Theorem 1. Let , let be an integer, , and . Then system (1) (3) has a unique solution satisfying Theorem 1 will be proved by using energy conservation and approximate argument. To this end, in the next section, we present some notations and useful lemmas which will be used throughout the paper. In Section 3, we study a regularized system of (1) and (2). Finally, the proof of Theorem 1 is given in
Existence and Iterative Algorithm of Solutions for a System of Generalized Nonlinear Mixed Variational-Like Inequalities
Yan-Mei Du
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/176283
Abstract: We introduce and study a system of generalized nonlinear mixed variational-like inequality problems (SGNMVLIPs) in Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for the SGNMVLIP. First, the existence of solutions of the auxiliary problems for the SGNMVLIP is shown. Second, an iterative algorithm for solving the SGNMVLIP is constructed by using this existence result. Finally, not only the existence of solutions of the SGNMVLIP is shown but also the convergence of iterative sequences generated by the algorithm is also proven. The technique and results presented in this paper generalize and unify the corresponding techniques and results given in the literature.
Existence of Positive Periodic Solutions for a Nonautonomous Generalized Predator-Prey System with Time Delay in Two-Patch Environment
Huilan Wang,Zhengqiu Zhang,Weiping Zhou
Discrete Dynamics in Nature and Society , 2012, DOI: 10.1155/2012/423193
Abstract: By using continuation theorem of coincidence degree theory, sufficient conditions of the existence of positive periodic solutions are obtained for a generalized predator-prey system with diffusion and delays. In this paper, we construct a V-function to make the prior estimation for periodic solutions, which makes the discussion more concise. Moreover, to compute the mapping's topological degree, a polynomial function matrix is constructed straightforwardly as a homotopic mapping for the generalized one, which improves the methods of computation on topological degree for a generalized mapping.
An Existence Theorem of Solutions for the System of Generalized Vector Quasi-Variational-Like Inequalities  [PDF]
Shamshad Husain, Sanjeev Gupta, Vishnu Narayan Mishra
American Journal of Operations Research (AJOR) , 2013, DOI: 10.4236/ajor.2013.33029

In this paper, we introduce and study the system of generalized vector quasi-variational-like inequalities in Hausdorff topological vector spaces, which include the system of vector quasi-variational-like inequalities, the system of vector variational-like inequalities, the system of vector quasi-variational inequalities, and several other systems as special cases. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the g-diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for the system of generalized vector quasi-variational-like inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.

Existence of Global Solutions Via Invariant Regions for a Generalized Reaction-Diffusion System with a Tri-diagonal Toeplitz Matrix of Diffusion Coefficients  [PDF]
Salem Abdelmalek
Mathematics , 2014,
Abstract: The aim of this paper is to construct invariant regions of a generalized m-component reaction-diffusion system with a tri-diagonal Toeplitz matrix of diffusion coefficients and prove the global existence of solutions using Lyapunov functional. The paper assumes nonhomogeneous boundary conditions and polynomial growth for the non-linear reaction term.
Existence and Stability of the Solutions for the System of Generalized Implicit Vector Variational Inequality Problems with Set-Valued Map

LIN Zhi,

系统科学与数学 , 2008,
Abstract: In this paper, the system of generalized implicit vector variational inequality problems with set-valued map is introduced. By Kakutani-Fan-Glicksberg fixed points theorem, the existence result of its solutions is established. Further, the stability and the essential component result of the solution set for the system are given.

- , 2017,
Abstract: 本文研究了一类集值广义强向量拟均衡问题组解的存在性问题.利用集值映射的自然拟C-凸性和集值映射的下(-C)-连续性的定义和Kakutani-Fan-Glicksberg不动点定理,在不要求锥C的对偶锥C*具有弱*紧基的情况下,建立了该类集值广义强向量拟均衡问题组解的存在性定理.所得结果推广了该领域的相关结果.
In this paper, we study existence of solutions to a system of generalized strong vector quasi-equilibrium problems with set-valued mappings. By making use of definitions of natural quasi C-convexity and lower (-C)-continuity of a set-valued mapping and Kakutani-Fan-Glicksberg fixed point theorem, an existence theorem for solutions to the systems of generalized strong vector quasi-equilibrium problems with set-valued mappings (for short, SSGSVQEP) was established without the assumption that the dual of the ordering cone has a weak* compact base, which extends and improves the corresponding results in this area
Existence, regularity and uniqueness of weak solutions for a class of incompressible generalized Navier-Stokes system with slip boundary conditions in $\mathbb{R}^3_+$  [PDF]
Aibin Zang
Mathematics , 2013,
Abstract: We obtain the existence, regularity, uniqueness of the non-stationary problems of a class of non-Newtonian fluid is a power law fluid with $p>9/5$ in the half-space under slip boundary conditions.
Global existence for the generalized two-component Hunter-Saxton system  [PDF]
Hao Wu,Marcus Wunsch
Mathematics , 2010, DOI: 10.1007/s00021-011-0075-9
Abstract: We study the global existence of solutions to a two-component generalized Hunter-Saxton system in the periodic setting. We first prove a persistence result of the solutions. Then for some particular choices of parameters $(\alpha, \kappa)$, we show the precise blow-up scenarios and the existence of global solutions to the generalized Hunter-Saxton system under proper assumptions on the initial data. This significantly improves recent results obtained in [M. Wunsch, DCDS Ser. B 12 (2009), 647-656] and [M. Wunsch, SIAM J. Math. Anal. 42 (2010), 1286-1304].
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