Existence Of Solutions For Abstract Economic Equilibrium Problems And Arithmetic

The general equilibrium theory was proposed by Walras at the end of 19 centuries. It regard economic analysis in complete competition as the main contents. It include the economic system which was composed of the consumer and producer and large quantity of the money and ware in his model. His theory is, if this time exist appropriate price system, under this price system each main body act as acceptors, then the consumer can get the biggest benefits, producer can acquire the biggest profits, and make the money and ware attain the complete competition equilibrium state. This conclusion was called the Walras compete competition equilibrium exist theorem. It was proposed by Walras, but the strict proof was gave by the mathematician G. Debreu after half a century in 1952. Therefore Debreu won the Nobel economics prize in 1983, Debreu proved the Walras compete competition equilibrium exist theorem by fixed-point theorem of set-valued mapping. Henceforth, many economists start study different ware spaces the price system influenced by different factor, discuss the existence of equilibrium point, because of the factor which influence the price more and more, the ware space also extend to infinite dimension from finite dimension. Many mathematicians start prove under what conditions the equilibrium point exist, and discuss the algorithm of the equilibrium point, studying the convergence of the algorithm.At the foundation of predecessors, this paper studied a new kind of vector equilibrium problem, we proved the existence of solutions for this kind of equilibrium problem by using the section theorem and KKM theorem these two tools, then we generalized this kind of vector equilibrium problem to a more general case. Equilibrium problems with lower and upper bounds is a open problem proposed by Isac, Sehgal and Singh in 1999. J.Li, O.Chadli, Y.Chiang and J.C.Yao, Zhang Cong-jun drived some results of the equilibrium problems with lower and upper bounds under certain conditions. In this paper we derived some more results of the open problem on certain conditions, constructed an iteration algorithm, and discussed the convergence of the algorithm.