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- W2528565205 abstract "Using Sperner's lemma for modified partition of a simplex we will constructively prove the existence of equilibrium in competitive economy with sequentially locally non-constant excess demand functions. And we will show the converse result that the existence of equilibrium in competitive econ- omy with sequentially locally non-constant excess demand functions implies Sperner's lemma. We follow the Bishop style constructive mathematics. It is well known that Brouwer's fixed point theorem can not be constructively proved in general case. Thus, the existence of competitive equilibrium also can not be constructively proved in general case. Sperner's lemma which is used to prove Brouwer's theorem, however, can be constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer's theorem using Sperner's lemma. See (5) and (7). Also Dalen in (5) states a conjecture that a uniformly continuous function f from a simplex to itself, with property that each open set contains a point x such that x 6 f(x), which means jx − f(x)j> 0, and also for every point x on the faces of the simplex x 6 f(x), has an exact fixed point. We call such a property of functions local non-constancy. Further we define a stronger property sequential local non-constancy. Then, Can we prove the existence of an exact equilibrium in competi- tive economy with sequentially locally non-constant excess demand functions directly by Sperner's lemma? We present such a proof. We consider Sperner's lemma for modified partition of a simplex, and utilizing it prove the existence of such an equilibrium. Also we will show the converse result that the existence of equilibrium in competitive economy with sequentially locally non-constant excess demand functions implies Sperner's lemma. In the next section we prove a modified version of Sperner's lemma. In Section 3 we present a proof of the existence of an equilibrium in competitive economy" @default.
- W2528565205 created "2016-10-14" @default.
- W2528565205 creator A5007696556 @default.
- W2528565205 date "2011-03-10" @default.
- W2528565205 modified "2023-09-27" @default.
- W2528565205 title "CONSTRUCTIVE PROOF OF THE EXISTENCE OF EQUILIBRIUM IN COMPETITIVE ECONOMY WITH SEQUENTIALLY LOCALLY NON-CONSTANT EXCESS DEMAND FUNCTIONS BY SPERNER'S LEMMA" @default.
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