Equilibrium Problems And Their Applications To Economics

Equilibrium problems have extensive applications on optimization, control theo-ry and mathematical economics. Equilibrium problems have close relationship with fixed point problems, variational inequality problems and complementarity problems etc.. They have provided an unified framework for us to study the financial prob-lems, economical problems and network analysis. Equilibrium problems have become a powerful tool to solve these problems. In this paper, we focus on the existence of equilibrium problems, behaviors of the solution correspondence to equilibrium prob-lems, iterative schemes of solutions to equilibrium problems and the applications of equilibrium problems.Firstly, we prove an order-theoretic fixed point theorem in Banach lattice, and apply this fixed point theorem to obtain an existence result of solutions to equilibrium problems. As extensions, we also explore the quasi-equilibrium problems in Hilbert lattices, chain complete lattices and chain complete posets, respectively. In contrast to many traditional studies, our approach is order-theoretic, which implies that all the results do not require the considered mappings to be topological continuous and topological semi-continuous.Secondly, we study the behaviors of solutions correspondence to parametric gen-eralized variational inequality problems. Unlike the traditional work that focus on the continuities of solutions correspondence, we mainly examine the order-preservation of solutions correspondence. We use order-theoretic fixed point theorems to study the upper order-preservation and lower order-preservation of solution correspondence in Hilbert lattices. Furthermore, based on the order-preservation of generalized metric projection operators, the order-preservation of solutions correspondence is considered in Banach lattices.Thirdly, we use Wiener-Hopf equation techniques and axillary principle to con-struct some iterative schemes for finding the common element of equilibrium problems, variationsl inequality problems and fixed points of a non-expansive mapping. Strong convergence theorems are proved in Hilbert spaces. Moreover, applying generalized Wiener-Hopf techniques, two classes of generalized variational inequality problems and a finite family of non-expansive mappings are also considered. Our algorithms have less iterative steps, which implies that the Wiener-Hopf techniques are more flexible than projection techniques.Fourthly, we utilize the Ekeland’s principle to study the existence of solutions to equilibrium problems with upper and lower bounds, and answer the open problems introduced by Isac et al. in 1999. Furthermore, the Holder continuity of solution correspondence to this problem is also examined based on the generalized monotonicity.Finally, as the applications of equilibrium problems to economics, we consider the personal income tax which involves the trade-off between equality and efficiency, and mainly study the existence of equally personal income tax. To this end, we choose Gini coefficient to measure the inequality of income distribution and introduce the concept of (α,β)-equal income tax. Then we derive the functional form between Gini coefficient and personal income tax. By using this function, the existence of ((α,β)-equal income tax is converted to a convex feasible problem. Applying the FKKM theorem, an existence theorem of ((α,β)-equal income tax is obtained. Furthermore, we examine the existence of progressive (0.3001,0.3156)-equality tax rate for the urban of China based on our models and the grouped data published on China Statistical Yearbook 2012. Finally, we construct the mathematical model for the personal income tax based on the equality and efficiency.