Existence and iterative algorithm of solutions for a class of bilevel generalized mixed equilibrium problems in Banach spaces

A new class of bilevel generalized mixed equilibrium problems involving set-valued mappings is introduced and studied in Banach spaces. First, an auxiliary generalized mixed equilibrium problem (AGMEP) to compute the approximate solutions of the generalized mixed equilibrium problems (GMEP) and bilevel generalized mixed equilibrium problems (BGMEP) involving set-valued mappings is introduced. By using a minimax inequality, the existence and uniqueness of solutions of the AGMEP is proved under quite mild conditions. By using auxiliary principle technique, new iterative algorithm to compute the approximate solutions of the GMEP and the BGMEP are suggested and analyzed. The strong convergence of the iterative sequences generated by the algorithms are proved under quite mild assumptions. These results are new and generalize some recent results in this field.     

Existence and algorithms for bilevel generalized mixed equilibrium problems in Banach spaces

A new class of bilevel generalized mixed equilibrium problems involving set-valued mappings is introduced and studied in Banach spaces. First, an auxiliary generalized mixed equilibrium problem (AGMEP) to compute the approximate solutions of the generalized mixed equilibrium problems (GMEP) and bilevel generalized mixed equilibrium problems (BGMEP) involving set-valued mappings is introduced. By using a minimax inequality, the existence and uniqueness of solutions of the AGMEP is proved under quite mild conditions. By using auxiliary principle technique, new iterative algorithm to compute the approximate solutions of the GMEP and the BGMEP is suggested and analyzed. Strong convergence of the iterative sequences generated by the proposed algorithms is proved under quite mild assumptions. These results are new and generalize some recent results in this field.     

Iterative algorithm of solutions for a system of generalized mixed implicity equilibrium problems in reflexive Banach spaces

In this paper, a new system of generalized mixed implicity equilibrium problems involving non-monotone set-valued mappings is introduced and studied in real reflexive Banach spaces. Following the idea of Moudafi, we consider a system of generalized equations problems and show its equivalence with the system of generalized mixed implicity equilibrium problems. By using a fixed point formulation of the system of generalized equation problems, a new iterative algorithm for solving the system of generalized mixed implicity equilibrium problems is suggested and analyzed. The strong convergence of the iterative sequences generated by the algorithm is proved under suitable conditions. These results are new and unify and generalize some recent results in this field.     

反Banach空间内涉及广义混合似臣分不等式问题的双水平广义混合平衡问题

在自反Banach空间内,引入和研究了一类新的涉及广义混合似变分不等式问题的双水平广义混合平衡问题(BGMEP).首先,为了计算BGMEP的近似解,引入了一类辅助广义混合平衡问题(AGMEP).由使用一极小极大不等式,在没有任何强制条件的相当温和假设下,证明了AGMEP解的存在性和唯一性.利用辅助原理技巧,建议和分析了一类计算BGMEP的近似解的新迭代算法.在没有任何强制条件的相当温和假设下,证明了由算法生成的迭代序列的强收敛性.这些结果是新的并且推广了这一领域内某些最近结果.     

Existence and algorithm of solutions for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces

Some classes of mixed equilibrium problems and bilevel mixed equilibrium problems are introduced and studied in reflexive Banach spaces. First, by using a minimax inequality, some new existence results of solutions and the behavior of solution set for the mixed equilibrium problems and the bilevel mixed equilibrium problems are proved under suitable assumptions without the coercive conditions. Next, by using auxiliary principle technique, some new iterative algorithms for solving the mixed equilibrium problems and the bilevel mixed equilibrium problems are suggested and analyzed. The strong convergence of the iterative sequences generated by the proposed algorithms is proved under suitable assumptions without the coercive conditions. These results are new and generalize some recent results in this field.     

Sensitivity analysis for a system of generalized mixed implicit equilibrium problems in uniformly smooth Banach spaces

In this paper, a new system of parametric generalized mixed implicit equilibrium problems involving non-monotone set-valued mappings in real Banach spaces is introduced and studied. We first generalize the notion of the Yosida approximation in Hilbert spaces introduced by Moudafi to reflexive Banach spaces. Further, by using the notion of the Yosida approximation, we consider a system of parametric generalized Wiener-Hopf equation problems and show its equivalence to the system of parametric generalized mixed implicit equilibrium problems. By using a fixed point formulation of the system of parametric generalized Wiener-Hopf equation problems, we study the behavior and sensitivity analysis of a solution set of the system of parametric generalized mixed implicit equilibrium problems. We prove that, under suitable assumptions, the solution set of the system of parametric generalized mixed implicit equilibrium problems is nonempty, closed and Lipschitz continuous with respect to the parameters. Our results are new, and improve and generalize some known results in this field.     

求解广义混合隐拟平衡问题的预测修正算法

引入和研究了一类新的具有四元函数的广义混合隐拟平衡问题．这类平衡问题包含了很多已知的广义平衡问题和广义混合隐拟变分不等式问题作为很特殊的情形．利用辅助原理技巧建议和分析了求解广义混合隐拟平衡问题的预测修正迭代算法．所建议算法的收敛性仅需要映象的连续性和部分松弛强单调性．     

Auxiliary Principle and Algorithm of Solutions for a New System of Generalized Mixed Equilibrium Problems in Banach Spaces

In this paper, we study a new system of generalized mixed equilibrium problems involving nonmonotone set-valued mappings and nondifferential bifunctions in Banach spaces. First, a system of auxiliary generalized mixed equilibrium problems are introduced. By using a minimax inequality, the existence and uniqueness of solutions of the system of auxiliary generalized mixed equilibrium problems is proved. Next, by using the auxiliary principle technique, an iterative algorithm for solving the system of generalized mixed equilibrium problems is constructed. Finally, the existence of solutions and the convergence of iterative sequences generated by the algorithm are proved under suitable conditions. These results improve, unify, and generalize many corresponding known results given in literature.     

扰动广义向量平衡问题解集的下半连续性

In this paper,by a scalarization method,the lower semicontinuity of the solution mappings to two kinds of parametric generalized vector equilibrium problems involving set-valued mappings is established under new assumptions which are weaker than the C-strict monotonicity.These results extend the corresponding ones.Some examples are given to illustrate our results.     

局部FC-一致空间内凝聚映象的极大元和广义对策及应用(Ⅰ)

首先在没有凸性结构的局部FC-一致空间内引入了非紧性测度和凝聚集值映象概念．在局部FC-一致空间内对涉及凝聚集值映象的集值映象族证明了新的极大元存在性定理．作为应用,在局部FC-一致空间内对涉及凝聚集值映象的广义对策建立了某些新的平衡存在性定理．这些结果改进和推广了文献中的某些已知结果到局部FC-一致空间．对广义矢量拟平衡组的进一步应用,我们将在文(Ⅱ)中给出．     

On Generalized Vector Equilibrium Problems

A new generalized vector equilibrium problem involving set-valued mappings and the proper quasi concavity of set-valued mappings in topological vector spaces are introduced; its existence theorems and the convexity of the solution sets are established.     

Iterative algorithms for generalized mixed equilibrium problems

In this paper, we consider a generalized mixed equilibrium problem in real Hilbert space. Using the auxiliary principle, we define a class of resolvent mappings. Further, using fixed point and resolvent methods, we give some iterative algorithms for solving generalized mixed equilibrium problem. Furthermore, we prove that the sequences generated by iterative algorithms converge weakly to the solution of generalized mixed equilibrium problem. These results require monotonicity (θ-pseudo monotonicity) and continuity (Lipschitz continuity) for mappings.     

含极大η-单调算子的广义非线性混合似变分包含组的扰动算法和稳定性

研究了一类含极大η-单调算子的广义非线性混合似变分包含组.依据不动点理论和极大η-单调算子的预解算子技巧,在Hilbert空间中提出了一种求这类变分不等式组的逼近解的扰动迭代算法,并证明了这类算法的收敛性和稳定性.所得结果是新的,并推广和统一了近期文献中的一些相关结论.     

Banach空间中集值混和变分不等式问题解的迭代算法

介绍了集值映象的伪单调定义，并在Banach空间中构造了集值混和变分不等式问题近似解的迭代算法．应用伪单调映象定义，证明了该迭代算法收敛于集值混和变分不等式问题的近似解．特别值得注意的是：在文章中对集值映象没有Lipschitz连续性假设．     

含参广义集值强向量平衡问题的稳定性

本文借助集合极限的性质和弱f-性假设证明了含参广义集值强向量平衡问题解集映射的下半连续性,其方法不同于最近文献(Zhao,2016和Meng,2018).此外,建立了含参广义集值强向量平衡问题解集连通性的充分条件,并举例验证了所得结果的正确性.本文得结果推广和改进了已有文献(Gong,2008,Xu,2009,Chen,2010,Xu,2013和Zhao,2013)中相应结果.     

Auxiliary Principle and Algorithm for Mixed Equilibrium Problems and Bilevel Mixed Equilibrium Problems in Banach Spaces

A new class of bilevel mixed equilibrium problems is introduced and studied in real Banach spaces. By using the auxiliary principle technique, new iterative algorithms for solving the mixed equilibrium problems and bilevel mixed equilibrium problems are suggested and analyzed. Strong convergence of the iterative sequences generated by the algorithms is proved under suitable conditions. The behavior of the solution set of the bilevel mixed equilibrium problem is also discussed.     

Vector Equilibrium Problems. Existence Theorems and Convexity of Solution Set

The natural quasi-concavity of set-valued mappings in an ordered vector space is introduced. Existence theorems for vector equilibrium problems involving set-valued monotone mappings are obtained and the convexity of the solution set is shown.     

关于集值目标映射的向量平衡问题的广义Tykhonov适定性

引入集值目标映射的向量平衡问题的两类广义Tykhonov适定性,利用非紧性Kuratowski测度给出它们的度量刻划和讨论这两类适定性的充分性条件.最后证明向量平衡问题的广义Tykhonov适定性与约束极小化问题的广义Tykhonov适定性之间的等价关系.     

Auxiliary principle and iterative algorithm for a new system of generalized mixed equilibrium problems in Banach spaces

In this paper, we study a new system of generalized mixed equilibrium problems involving skew-symmetric bifunctions (SGMEP) in reflexive Banach spaces. A system of auxiliary mixed equilibrium problems (SAMEP) for solving the SGMEP is introduced and the existence and uniqueness of the solutions of the SAMEP is first proved. Next, by using the auxiliary principle technique, a new iterative algorithm to compute the approximate solutions of the SGMEP is suggested and analyzed. Finally, the strong convergence of the iterative sequences generated by the algorithm is also proved under quite mild conditions. These results improve, unify and generalize some known results in recent literature.     

广义平衡问题与非扩张映象不动点问题公解的一个新迭代方法

In this paper,we introduce a new iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems and the set of fixed points for nonexpansive mappings in Hilbert space.Under suitable conditions,some strong convergence theorems are proved.Our results extend and improve some recent results.