Generalized Browder-type fixed point theorem with strongly geodesic convexity on Hadamard manifolds with applications

In this paper, a generalized Browder-type fixed point theorem on Hadamard manifolds is introduced, which can be regarded as a generalization of the Browder-type fixed point theorem for the set-valued mapping on an Euclidean space to a Hadamard manifold. As applications, a maximal element theorem, a section theorem, a Ky Fan-type Minimax Inequality and an existence theorem of Nash equilibrium for non-cooperative games on Hadamard manifolds are established.     

Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications

In this paper, maximal element theorem on Hadamard manifolds is established. First, we prove the existence of solutions for maximal element theorem on Hadamard manifolds. Further, we prove that most of problems in maximal element theorem on Hadamard manifolds (in the sense of Baire category) are essential and that, for any problem in maximal element theorem on Hadamard manifolds, there exists at least one essential component of its solution set. As applications, we study existence and stability of solutions for variational relation problems on Hadamard manifolds, and existence and stability of weakly Pareto-Nash equilibrium points for n-person multi-objective games on Hadamard manifolds.     

Split equilibrium problems for related games and applications to economic theory

ABSTRACT

In this paper we introduce the concept of split Nash equilibrium problems associated with two related noncooperative strategic games. Then we apply the Fan-KKM theorem to prove the existence of solutions to split Nash equilibrium problems of related noncooperative strategic games, in which the strategy sets of the players are nonempty closed and convex subsets in Banach spaces. As an application of this existence to economics, an example is provided that studies the existence of split Nash equilibrium of utilities of two related economies. As applications, we study the existence of split Nash equilibrium in the dual (playing twice) extended Bertrand duopoly model of price competition.     

A more refined convexity idea for Nash equilibria

In this paper, we relax the classical quasi-concavity assumption for the existence of pure Nash equilibria in the setting of constrained and unconstrained games in normal form. Multiconnected convexity (H. Ben-El-Mechaiekh et al., 1998) in spaces without any linear structure is a keen point. We present two games in which we show how the generalized continuity and quasi-concavity hypotheses are unrelated to each other as sufficient conditions for existence of Nash equilibria for games in normal form. Then our results are applied to two non-zero-sum games lacking the classical quasi-concavity assumption (Nash, 1950) and the more recent improvements (Ziad, 1999) and (Abalo and Kostreva, 2004). As minor results, we introduce new concept of convexity, named a-convexity, and some counterexamples of the relationships between some continuity conditions on players’ payoffs imposed by Lignola (1997), Reny (1999) and Simon (1987).     

ON WELL-POSEDNESS OF THE MULTIOBJECTIVE GENERALIZED GAME

§1　IntroductionHadamard type well-posedness and Tikhonov type well-posedness are two main typesof concepts of well-posedness. At the beginning of last century,Hadamard firstintroduced the concept of well-posedness in study of optimal problem. Hadamard typewell-posedness of a problem means the continuous dependence of the solution on the dataof such problem. Later,Tikhonov introduced another concept of well-posedness.Tikhonov type well-posedness deals with the behavior ofa prescribed class …     

Nonempty Intersection Theorems and Generalized Multi-objective Games in Product <Emphasis Type="Italic">FC</Emphasis>-Spaces

A new class of generalized multi-objective games is introduced and studied in FC-spaces where the number of players may be finite or infinite, and all payoff are all set-valued mappings and get their values in a topological space. By using an existence theorems of maximal elements for a family of set-valued mappings in product FC-spaces due to author, some new nonempty intersection theorems for a family of set-valued mappings are first proved in FC-spaces. As applications, some existence theorems of weak Pareto equilibria for the generalized multi-objective games are established in noncompact FC-spaces. These theorems improve, unify and generalize the corresponding results in recent literatures.     

Essential Stability of Solutions for Maximal Element Theorem with Applications

In this paper, we prove that most of problems in maximal element theorem (in the sense of Baire category) are essential and that, for any problem in maximal element theorem, there exists at least one essential component of its solution set. As applications, we deduce the existence of essential components of the set of Ky Fan’s points based on Ky Fan Minimax Inequality, the existence of essential components of the set of Nash equilibrium points for general n-person non-cooperative games and the existence of essential components of the set of solutions of vector Ky Fan Minimax Inequality.     

Maximal Element Theorems with Applications to Generalized Games and Systems of Generalized Vector Quasi-Equilibrium Problems in <Emphasis Type="Italic">G</Emphasis>-Convex Spaces

By applying the maximal element theorems on product of G-convex spaces due to the first author, some equilibrium existence theorems for generalized games with fuzzy constraint correspondences are proved in G-convex spaces. As applications, some existence theorems of solutions for the system of generalized vector quasiequilibrium problem are established in noncompact product of G-convex spaces. Our results improve and generalize some recent results in the literature to product of G-convex spaces.The authors thank the referees for valuable comments and suggestionsThe research of this author was supported by the National Science Foundation of China, Sichuan Education Department.The research of this author was supported by the National Science Council of the Republic of China.     

Solutions of system of generalized vector quasi-equilibrium problems in locally G-convex uniform spaces

In this paper we establish a collectively fixed point theorem and an equilibrium existence theorem for generalized games in product locally G-convex uniform spaces. As applications, some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are derived in product locally G-convex uniform spaces. These theorems are new and generalize some known results in the literature.     

Pareto equilibria for generalized constrained multiobjective games in FC-spaces without local convexity structure

In this paper, we further study a class of generalized constrained multiobjective games where the number of players may be finite or infinite, the strategy sets may be general FC-spaces without local convexity structure, and all payoff functions get their values in infinite-dimensional topological vector spaces. By using an existence theorem of maximal elements for a family of set-valued mappings in FC-spaces due to the author, an existence theorem of solutions for a system of generalized vector quasivariational inclusions is first proved in general FC-spaces. By applying the existence result of solutions of the system of generalized vector quasivariational inclusions, some existence theorems of (weak) Pareto equilibria for the generalized constrained multiobjective games are established in noncompact product FC-spaces. Some special cases of our results are also discussed. Our results are new and different from the corresponding known results in the literature.     

非紧集上不连续函数的Ky Fan不等式及其等价形式和应用

证明了非紧集上不具有任何连续性的函数弱Ky Fan点的存在性,给出了在函数只具非常弱的连续性和凸性条件下非紧集上Ky Fan不等式解的存在性,并给出它的两种等价形式.作为应用:(1)得到Ky Fan截口定理和Fan-Browder不动点定理的推广;(2)应用于博弈理论,得到几个新的Nash平衡存在性定理.     

Equilibrium existence theorems for multi-leader-follower generalized multiobjective games in FC-spaces

In this paper, we introduce and study a class of multi-leader-follower generalized multiobjective games in FC-spaces where the number of leaders and followers may be finite or infinite and the objective functions of leaders and followers get their values in infinite-dimensional spaces. By using a Pareto equilibrium existence theorem of generalized constrained multiobjective games in FC-spaces due to author, some equilibrium existence theorems for the multi-leader-follower generalized multiobjective games are established in noncompact FC-spaces. These results improve and generalize some corresponding results in recent literatures.     

Rate of convergence for proximal point algorithms on Hadamard manifolds

In this paper, an estimate of convergence rate concerned with an inexact proximal point algorithm for the singularity of maximal monotone vector fields on Hadamard manifolds is discussed. We introduce a weaker growth condition, which is an extension of that of Luque from Euclidean spaces to Hadamard manifolds. Under the growth condition, we prove that the inexact proximal point algorithm has linear/superlinear convergence rate. The main results presented in this paper generalize and improve some corresponding known results.     

New Characterization of Geodesic Convexity on Hadamard Manifolds with Applications

In this paper, some new results, concerned with the geodesic convex hull and geodesic convex combination, are given on Hadamard manifolds. An S-KKM theorem on a Hadamard manifold is also given in order to generalize the KKM theorem. As applications, a Fan–Browder-type fixed point theorem and a fixed point theorem for the a new mapping class are proved on Hadamard manifolds.     

New systems of generalized quasi-variational inclusions in FC-spaces and applications

In this paper, we study some new systems of generalized quasi-variational inclusion problems in FC-spaces without convexity structure.By applying an existence theorem of maximal elements of set-valued mappings due to the author, some new existence theorems of solutions for the systems of generalized quasi-variational inclusion problems are proved in noncompact FC-spaces. As applications, some existence results of solutions for the system of quasi-optimization problems and mathematical programs with the systems of generalized quasi-variational inclusion constraints are obtained in FC-spaces.     

Halpern Iteration in CAT（k） Spaces

In this paper we show that an iterative sequence generated by the Halpern algorithm converges to a fixed point in the case of complete CAT(κ) spaces. Similar results for Hadamard manifolds were obtained in [Li, C., López, G., Martín-Márquez, V.: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J. Math., 14, 541–559 (2010)], but we study a much more general case. Moreover, we discuss the Halpern iteration procedure for set-valued mappings.     

Existence of nash equilibria for constrained stochastic games

In this paper, we consider constrained noncooperative N-person stochastic games with discounted cost criteria. The state space is assumed to be countable and the action sets are compact metric spaces. We present three main results. The first concerns the sensitivity or approximation of constrained games. The second shows the existence of Nash equilibria for constrained games with a finite state space (and compact actions space), and, finally, in the third one we extend that existence result to a class of constrained games which can be “approximated” by constrained games with finitely many states and compact action spaces. Our results are illustrated with two examples on queueing systems, which clearly show some important differences between constrained and unconstrained games.Mathematics Subject Classification (2000): Primary: 91A15. 91A10; Secondary: 90C40     

Generalized vector quasi-equilibrium problems with applications

In this paper, we consider the generalized vector quasi-equilibrium problem with or without involving Φ-condensing maps and prove the existence of its solution by using known fixed point and maximal element theorems. As applications of our results, we derive some existence results for a solution to the vector quasi-optimization problem for nondifferentiable functions and vector quasi-saddle point problem.     

Network capacity management under competition

We consider capacity management games between airlines who transport passengers over a joint airline network. Passengers are likely to purchase alternative tickets of the same class from competing airlines if they do not get tickets from their preferred airlines. We propose a Nash and a generalized Nash game model to address the competitive network revenue management problem. These two models are based on well-known deterministic linear programming and probabilistic nonlinear programming approximations for the non-competitive network capacity management problem. We prove the existence of a Nash equilibrium for both games and investigate the uniqueness of a Nash equilibrium for the Nash game. We provide some further uniqueness and comparative statics analysis when the network is reduced to a single-leg flight structure with two products. The comparative statics analysis reveals some useful insights on how Nash equilibrium booking limits change monotonically in the prices of products. Our numerical results indicate that airlines can generate higher and more stable revenues from a booking scheme that is based on the combination of the partitioned booking-limit policy and the generalized Nash game model. The results also show that this booking scheme is robust irrespective of which booking scheme the competitor takes.