An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems

By using Gerstewitz functions, we establish a new equilibrium version of Ekeland variational principle, which improves the related results by weakening both the lower boundedness and the lower semi-continuity of the ob jective bimaps. Applying the new version of Ekeland principle, we obtain some existence theorems on solutions for set-valued vector equilibrium problems, where the most used assumption on compactness of domains is weakened. In the setting of complete metric spaces(Z,d), we present an existence result of solutions for set-valued vector equilibrium problems, which only requires that the domain XZ is countably compact in any Hausdorff topology weaker than that induced by d. When(Z, d) is a Féchet space(i.e., a complete metrizable locally convex space), our existence result only requires that the domain XZ is weakly compact. Furthermore, in the setting of non-compact domains, we deduce several existence theorems on solutions for set-valued vector equilibrium problems,which extend and improve the related known results.     

Fixed point theorems for set-valued contractions

In this paper, we obtain some fixed point theorems for new set-valued contractions in complete metric spaces. Then by using these results and the scalarization method, we present some fixed point theorems for set-valued contractions in complete cone metric spaces without the normality assumption. We also present some examples to support our results.     

Error analysis of approximate solutions to parametric vector quasiequilibrium problems

In this paper, we give some results on error estimates of approximate solutions to parametric vector quasiequilibrium problems in metric linear spaces. Under some special cases, the error estimates are equivalent to H?lder stability or Lipschitz stability of the set-valued solution map at a given point. An application to variational inequalities is also presented.     

Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces

Investigations concerning the existence of dynamic processes convergent to fixed points of set-valued nonlinear contractions in cone metric spaces are initiated. The conditions guaranteeing the existence and uniqueness of fixed points of such contractions are established. Our theorems generalize recent results obtained by Huang and Zhang [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive maps, J. Math. Anal. Appl. 332 (2007) 1467–1475] for cone metric spaces and by Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (1) (2007) 132–139] for metric spaces. The examples and remarks provided show an essential difference between our results and those mentioned above.     

Generic Existence of Fixed Points for Set-Valued Mappings

We first consider a complete metric space of nonexpansive set-valued mappings acting on a closed convex subset of a Banach space with a nonempty interior, and show that a generic mapping in this space has a fixed point. We then establish analogous results for two complete metric spaces of set-valued mappings with convex graphs.     

H?lder metric regularity of set-valued maps

This paper is devoted to metric regularity of set-valued maps from a complete metric space to a Banach space. In particular we extend a known characterization of the regularity modulus to maps defined on reflexive spaces. The higher order metric regularity, i.e. an extension of metric regularity to H?lder context, is also investigated using high order variations of set-valued maps and results of similar nature are obtained for conical metric regularity.     

Endpoints of set-valued asymptotic contractions in metric spaces

By introducing a new concept called “set-valued asymptotic contraction” in metric spaces, the existence and uniqueness of endpoints for a set-valued asymptotic contraction which has the approximate endpoint property have been established.     

Endpoints of set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and strict contractions in uniform spaces

In this paper, we introduce the concepts of the set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and set-valued dynamical systems of strict contractions in uniform spaces and we present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions. The result, concerning the investigations of problems of the set-valued asymptotic fixed point theory, include some well-known results of Meir and Keeler, Kirk and Suzuki concerning the asymptotic fixed point theory of single-valued maps in metric spaces. The result, concerning set-valued strict contractions (in which the contractive coefficient is not constant), is different from the result of Yuan concerning the existence of endpoints of Tarafdar–Vyborny generalized contractions (in which the contractive coefficient is constant) in bounded metric spaces and provides some examples of Tarafdar–Yuan topological contractions in compact uniform spaces. Definitions and results presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our results and the well-known ones.     

概率度量空间中的Ekeland变分原理与集值映象的Caristi重合定理

借助偏序方法,本文得到概率度量空间中之一推广形式的Ekeland变分原理及一集值形式的Caristi重合定理,同时证明了这两个定理之间的等价性.本文结果是[1,2,5,6,7,9]中相应结果的改进和推广.     

Ekeland's ε-variational principle for set-valued mappings

In this paper, we introduce the concept of approximate solutions for set-valued mappings and provide a sufficient condition for the existence of approximate solutions of set-valued mappings. We obtain an approximate variational principle for set-valued mappings. Revised version received November 1997     

The Set-Valued Dynamic System and Its Applications to Pareto Optima

In this paper, we first study the existence of endpoints for set-valued dynamic systems which are either upper or lower semicontinuous in metric spaces. Then the existence, uniqueness and algorithms of endpoints for set-valued dynamic systems which are either generalized contractions (defined in metric spaces) or topological contractions (defined in topological spaces which do not necessarily have any metric). These results are then applied to derive the existence of Pareto optima for mappings which take values in ordered Banach spaces. Finally, the stability of (generalized) nucleolus sets is also established.     

完备凸度量空间中(次)相容和集值广义非扩张映象的公共不动点定理

本文在完备凸度量空间中,利用集值和单值映象(次)相容的一些条件,建立了数值广义非扩张映象存在公共不动点的一个充要条件和一个充分条件.我们的结果改进、扩充和发展了文[2～7]中的主要结果.     

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.     

The Generic Stability of Fixed Points for Upper Semicontinuous Mappings in Hyperconvex Metric Spaces

The main goal of this paper is to establish the generic stability of Fan-Glicksberg type fixed points in hyperconvex metric spaces. In order to do so, we first give Fan-Glicksberg type fixed point theorem in hyperconvex metric spaces and then the generic stability of fixed points for upper semicontinuous set-valued mappings is obtained. Our generic stability results show that almost all of fixed points of upper semicontinuous set-valued mappings defined in compact hyperconvex metric spaces are stable in the sense of Baire category theory     

Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems

In this paper, we establish the Hölder continuity of solution mappings to parametric vector quasiequilibrium problems in metric spaces under the case that solution mappings are set-valued. Our main assumptions are weaker than those in the literature, and the results extend and improve the recent ones. Furthermore, as an application of Hölder continuity, we derive upper bounds for the distance between an approximate solution and a solution set of a vector quasiequilibrium problem with fixed parameters.     

An Approximate Exact Penalty in Constrained Vector Optimization on Metric Spaces

In this paper, we use the penalty approach in order to study a class of constrained vector minimization problems on complete metric spaces. A penalty function is said to have the generalized exact penalty property iff there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. For our class of problems, we establish the generalized exact penalty property and obtain an estimation of the exact penalty.     

An exact penalty approach to constrained minimization problems on metric spaces

In this paper we use the penalty approach in order to study a class of constrained minimization problems on complete metric spaces. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. For our class of problems we establish the generalized exact penalty property and obtain an estimation of the exact penalty.     

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.     

Set-valued dynamic systems and random iterations of set-valued weaker contractions in uniform spaces

Set-valued weaker contractions in uniform, locally convex and metric spaces are defined and dynamic systems of such weaker contractions are studied. Conditions guaranteeing the convergence of generalized sequences of random iterations and iterations and the existence and uniqueness of endpoints of set-valued weaker contractions are established. Our definitions and results are new for set-valued maps in uniform, locally convex and metric spaces and even for single-valued maps.     

Calculus of sequential normal compactness in variational analysis

In this paper we study some properties of sets, set-valued mappings, and extended-real-valued functions unified under the name of “sequential normal compactness.” These properties automatically hold in finite-dimensional spaces, while they play a major role in infinite-dimensional variational analysis. In particular, they are essential for calculus rules involving generalized differential constructions, for stability and metric regularity results and their broad applications, for necessary optimality conditions in constrained optimization and optimal control, etc. This paper contains principal results ensuring the preservation of sequential normal compactness properties under various operations over sets, set-valued mappings, and functions.