Jinlu Li

A lower and upper bounds version of a variational inequality

Abstract Introducing the concept of extremal subset to solve one open problem raised in [1]: find the conditions for a lower and upper bounds version of a variational inequality. A few applications are given.

Order-clustered fixed point theorems on chain-complete preordered sets and their applications to extended and generalized Nash equilibria

In this paper, we introduce the concept of order-clustered fixed point of set-valued mappings on preordered sets and give several generalizations of the extension of the Abian-Brown fixed point theorem provided in (Mas-Colell et al. in Microeconomic Theory, 1995), which is from chain-complete posets to chain-complete preordered sets. By using these generalizations and by applying the order-increasing upward property of set-valued mappings, we prove several existence theorems of the extended and generalized Nash equilibria of nonmonetized noncooperative games on chain-complete preordered sets.MSC:46B42, 47H10, 58J20, 91A06, 91A10.

Applications of fixed point theory to extended Nash equilibriums of nonmonetized noncooperative games on posets

We say that a noncooperative game is nonmonetized if the ranges of the utilities of the players are posets. In this paper, we examine some nonmonetized noncooperative games of which both the collection of strategies and the ranges of the utilities for the players are posets. Then we carry the concept of generalized Nash equilibriums of noncooperative games defined in (Li in J. Nonlinear Anal. Forum 18:1-11, 2013; Li and Park in Br. J. Econ. Manag. Trade 4(1), 2014) to extended Nash equilibriums of nonmonetized noncooperative games. By applying some fixed point theorems in posets and by using the order-preserving property of mappings, we prove an existence theorem of extended Nash equilibriums for nonmonetized noncooperative games.MSC:46B42, 47H10, 58J20, 91A06, 91A10.

Fixed point theorems on partially ordered topological vector spaces and their applications to equilibrium problems with incomplete preferences

AbstractIn this paper, we examine some properties of chain-complete posets and introduce the concept of universally inductive posets. By applying these properties, we provide several extensions of Abian-Brown fixed point theorem from single-valued mappings to set-valued mappings on chain-complete posets and on compact subsets of partially ordered topological spaces. As applications of these fixed point theorems, we explore the existence of generalized Nash equilibrium for strategic games with partially ordered preferences. MSC:06F30, 91A06, 91A18.

The Existence of Maximum and Minimum Solutions to General Variational Inequalities in the Hilbert Lattices

We apply the variational characterization of the metric projection to prove some results about the solvability of general variational inequalities and the existence of maximum and minimum solutions to some general variational inequalities in the Hilbert lattices.

Vector and Ordered Variational Inequalities and Applications to Order-Optimization Problems on Banach Lattices

We investigate the connections between vector variational inequalities and ordered variational inequalities in finite dimensional real vector spaces. We also use some fixed point theorems to prove the solvability of ordered variational inequality problems and their application to some order-optimization problems on the Banach lattices.

An approximation of solutions of variational inequalities

We use a Mann-type iteration scheme and the metric projection operator (the nearest-point projection operator) to approximate the solutions of variational inequalities in uniformly convex and uniformly smooth Banach spaces.

Infinitely split Nash equilibrium problems in repeated games

In this paper, we introduce the concept of infinitely split Nash equilibrium in repeated games in which the profile sets are chain-complete posets. Then by using a fixed point theorem on posets in (J. Math. Anal. Appl. 409:1084–1092, 2014), we prove an existence theorem. As an application, we study the repeated extended Bertrant duopoly model of price competition.

Applications of order-clustered fixed point theorems to generalized saddle point problems and preordered variational inequalities

AbstractIn this paper, we introduce the concepts of preordered Banach spaces, generalized saddle points and preordered variational inequalities. Then we apply the order-clustered fixed point theorems to prove the existence of solutions to these problems. MSC:06A06, 47B48, 47B60, 49J40.