Existence and Algorithm of Solutions for Generalized Mixed Implicit Quasi-Variational-Like Inequalities

By applying the auxiliary variational principle technique, the existence of solutions for a new class of generalized mixed implicit quasi-variational-like inequalities and the convergence criteria of a new iterative algorithm to compute approximate solutions are proved in Hilbert spaces. The obtained result is a improvement over and generalization of the main theorem proposed by Ding.

A PROJECTION-TYPE ALGORITHM FOR SOLVING GENERALIZED MIXED VARIATIONAL INEQUALITIES

We propose a projection-type algorithm for generalized mixed variational in- equality problem in Euclidean space Rn. We establish the convergence theorem for the pro- posed algorithm, provided the multi-valued mapping is continuous and f-pseudomonotone with nonempty compact convex values on dom（f）, where f ： Rn --RU{＋∞} is a proper func- tion. The algorithm presented in this paper generalize and improve some known algorithms in literatures. Preliminary computational experience is also reported.

EXISTENCE AND ITERATIVE ALGORITHMS OF SOLUTIONS FOR GENERALIZED MIXED QUASI_VARIATIONAL INEQUALITIES

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 answer the open problems mentioned by Noor, improve and generalize some recent known results.

EXISTENCE OF SOLUTIONS FOR GENERALIZEDQUASI-VARIATIONAL-LIKE INEQUALITIES

In this paper, some existence theorems of solutions for a class of generalized quasi-variational-like inequalities with discontinuous mappings ape proved under paracompact setting in topological vector spaces. These theorems unify, improve and generalize many recent results.

Existence of Solutions to Generalized Vector Quasi-variational-like Inequalities with Set-valued Mappings

In this paper, we introduce and study a class of generalized vector quasivariational-like inequality problems, which includes generalized nonlinear vector variational inequality problems, generalized vector variational inequality problems and generalized vector variational-like inequality problems as special cases. We use the maximal element theorem with an escaping sequence to prove the existence results of a solution for generalized vector quasi-variational-like inequalities without any monotonicity conditions in the setting of locally convex topological vector space.

An Existence Theorem of Solutions for the System of Generalized Vector Quasi-Variational-Like Inequalities

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.

Bilevel generalized mixed equilibrium problems involving generalized mixed variational-like inequality problems in reflexive Banach spaces

A new bilevel generalized mixed equilibrium problem （BGMEP） involving generalized mixed variational-like inequality problems （GMVLIPs） is introduced and studied in the reflexive Banach spaces. First, an auxiliary generalized mixed equilibrium problem （AGMEP） is introduced to compute the approximate solutions of the BGMEP involving the GMVLIPs. By using a minimax inequality, the existence and the unique- ness of solutions of the AGMEP are proved under mild conditions without any coercive assumptions. By using an auxiliary principle technique, the new iterative algorithms are proposed and analyzed, with which the approximate solutions of the BGMEP are computed. The strong convergence of the iterative sequence generated by the algorithms is shown under mild conditions without any coercive assumptions. These new results can generalize some recent results in this field.

A NEW SELF-ADAPTIVE ITERATIVE METHOD FOR GENERAL MIXED QUASI VARIATIONAL INEQUALITIES

The general mixed quasi variational inequality containing a nonlinear term φ is a useful and an important generalization of variational inequalities. The projection method can not be applied to solve this problem due to the presence of nonlinear term. It is well known that the variational inequalities involving the nonlinear term φ are equivalent to the fixed point problems and resolvent equations. In this article, the authors use these alternative equivalent formulations to suggest and analyze a new self-adaptive iterative method for solving general mixed quasi variational inequalities. Global convergence of the new method is proved. An example is given to illustrate the efficiency of the proposed method.

EXISTENCE AND ALGORITHM OF SOLUTIONS FOR GENERAL MULTIVALUED MIXED IMPLICIT QUASI-VARIATIONAL INEQUALITIES

A new class of general multivalued mixed implicit quasi-variational inequalities in a real Hilbert space was introduced, which includes the known class of generalized mixed implicit quasi-variational inequalities as a special case , introduced and studied by Ding Xie-ping . The auxiliary variational principle technique was applied to solve this class of general multivalued mixed implicit quasi-variational inequalities. Firstly, a new auxiliary variational inequality with a proper convex , lower semicontinuous , binary functional was defined and a suitable functional was chosen so that its unique minimum point is equivalent to the solution of such an auxiliary variational inequality . Secondly , this auxiliary variational inequality was utilized to construct a new iterative algorithm for computing approximate solutions to general multivalued mixed implicit quasi-variational inequalities . Here , the equivalence guarantees that the algorithm can generate a sequence of approximate solutions. Finally, the existence of solutions and convergence of approximate solutions for general multivalued mixed implicit quasi-variational inequalities are proved. Moreover, the new convergerce criteria for the algorithm were provided. Therefore, the results give an affirmative answer to the open question raised by M. A . Noor, and extend and improve the earlier and recent results for various variational inequalities and complementarity problems including the corresponding results for mixed variational inequalities, mixed quasi-variational inequalities and quasi-complementarity problems involving the single-valued and set- valued mappings in the recent literature .

Generalized mixed equilibrium problem in Banach spaces

This paper uses a hybrid algorithm to find a common element of the set of solutions to a generalized mixed equilibrium problem, the set of solutions to variational inequality problems, and the set of common fixed points for a finite family of quasi-C- nonexpansive mappings in a uniformly smooth and strictly convex Banach space. As applications, we utilize our results to study the optimization problem. It shows that our results improve and extend the corresponding results announced by many others recently.

An Iterative Algorithm for Generalized Mixed Equilibrium Problems and Fixed Points of Nonexpansive Semigroups

In this works, by using the modified viscosity approximation method associated with Meir-Keeler contractions, we proved the convergence theorem for solving the fixed point problem of a nonexpansive semigroup and generalized mixed equilibrium problems in Hilbert spaces.

Behavior of solution set for bilevel generalized mixed equilibrium problems in topological vector spaces

A new bilevel generalized mixed equilibrium problem （BGMEF） is introduced and studied in topological vector spaces. By using a minimax inequality, the existence of solutions and the behavior of solution set for the BGMEP are studied under quite mild conditions. These results are new and generalize some recent results in this field.

PREDICTOR-CORRECTOR ALGORITHMS FOR SOLVING GENERALIZED MIXED IMPLICIT QUASI-EQUILIBRIUM PROBLEMS

A new class of generalized mixed implicit quasi-equilibrium problems （GMIQEP） with four-functions is introduced and studied. The new class of equilibrium problems includes many known generalized equilibrium problems and generalized mixed implicit quasi-variational inequality problems as many special cases. By employing the auxiliary principle technique, some predictor-corrector iterative algorithms for solving the GMIQEP are suggested and analyzed. The convergence of the suggested algorithm only requires the continuity and the partially relaxed implicit strong monotonicity of the mappings

ALGORITHM OF SOLUTIONS FOR MIXEDNONLINEAR VARIATIONAL-LIKE INEQUALITIES IN REFLEXIVE BANACH SPACE

In this paper, the author studies a class of mixed nonlinear variational-like inequalities in reflexive Banach space. By applying a minimax inequality obtained by the author, some existence uniqueness theorems of solutions for the mixed nonlinear variational-like inequalities are proved. Next, by applying the auxiliary problem technique, rite author suggests an innovative iterative algorithm to compute the approximate solutions of the mixed nonlinear variational-like inequalities. Finally, the convergence criteria is also discussed.

Generalized Vector Quasi-Variational-Like Inequalities without Monotonity and Compactness

In this paper, some existence theorems of a solution for generalized vector quasivariational-like inequalities without any monotonity conditions in a noncompact topological space setting are proven by the maximal element theorem.

EXISTENCE AND ALGORITHM OF SOLUTIONS FOR GENERALIZED STRONGLY MIXED IMPLICIT QUASI-VARIATIONAL INEQUALITIES

The purpose of this paper is to introduce and study a new class of generalized strongly mixed implicit quasi-variational inequalities in Hilbert spaces, which includes the known class of generalized mixed implicit quasi-variational inequalities as a special case.By applying the auxiliary variational principle technique, the existence of solutions for this class of quasi-variational inequalities is proved. Moreover, a new iterative algorithm for computing approximate solutions is constructed and the convergence criteria for this iterative algorithm are also established.

Petty Projection Inequalities for the General L_p-Mixed Projection Bodies

In this paper,the definition of the general L p-mixed projection bodies is introduced,and the general L p-projection bodies given by Ludwig is a special case for the general L p-mixed projection bodies.Then the Petty projection inequality for the general L p-mixed projection bodies is shown.Moreover,the monotonicity for the general L p-mixed projection bodies is obtained.

求解逆混合变分不等式问题的动力系统方法

本文主要研究求解逆混合变分不等式问题的动力系统方法。首先,在强单调和Lipschitz连续性的假设下,通过借助广义f-投影算子的性质,得到逆混合变分不等式问题有唯一解的充分条件。进一步考虑一个动力系统方法来求解逆混合变分不等式问题,而且该动力系统具有唯一均衡点。最后,证明在强单调和Lipschitz连续性的条件下该动力系统的唯一均衡点是全局指数稳定的。

广义L_(p)-混合弦长积分的Brunn-Minkowski型不等式与逆循环不等式

研究凸几何分析中星体的广义L_(p)-混合弦长积分,建立关于广义L_(p)-混合弦长积分的两类Brunn-Minkowski型不等式和一类逆循环不等式.

Existence Theorem of Solutions for Mixed Variational Inequality in Banach Spaces

By using the property of generalized f-projection operator and FKKM theorem, the existence theorem of solutions for the mixed variational inequality is proved under weaker assumption in reflexive and smooth Banach space. The results improve and extend the corresponding results shown recentlv.