Skip to main content
Figure 2 | Fixed Point Theory and Applications

Figure 2

From: The existence of equilibrium in a simple exchange model

Figure 2

This figure explains the idea of the proof of theorem forn=3. Values of l assigned to the vertices in V( S 2 ,m) are independent of z ˜ if the considered vertex is above or on the line p 3 1 ε 1 /2 - here the second and third row of the formula (3) are used to define vales of l. If a vertex is below the line p 3 1 ε 1 /2 - but not at the bottom of S 2 - then z ˜ is used to compute the value of l. The thick curve presents a hypothetical sequence of simplices σ 1 ,, σ J . For vertices of simplices above ( p 3 1 ε 1 )-line (below ( p 3 ε 1 )-line) values of z n are negative (positive). If σ j is below ( p 3 1 ε 1 /2)-line then each coordinate of z ˜ admits a non-positive value at a vertex of σ j . Somewhere between ( p 3 1 ε 1 ) and ( p 3 ε 1 )-lines there is a simplex σ j such that z n (p) z n ( p )0 for a pair of vertices p, p of σ j - that simplex is what we are looking for.

Back to article page