An Optimal Lower Eigenvalue System

and Applied Analysis 3 by Fujimoto 3 and Liu and Chen 4, 5 with the functional analysis approach. As for b , in case X is convex compact, and S is convex compact-valued with and without the upper hemicontinuous condition, it has also been studied by Liu and Zhang 6, 7 with the nonlinear analysis methods attributed to 8–10 , in particular, using the classical RogalskiCornet Theorem see 8, Theorem 15.1.4 and some Rogalski-Cornet type Theorems see 6, Theorems 2.8, 2.9 and 2.12 . However, since the methods to tackle 1.3 are quite different from those to study 1.1 , we do not consider it here. Another is the von-Neumann type and Leontief type inequality problems which can be viewed as some special examples of 1.1 or 1.2 . i Assume that F ⊆ R or c ∈ R is an expected demand set or an expected demand of the market, and X ⊆ R some enterprise’s raw material bundle set. Then the von-Neumann type inequality problems including input-output inequalities, alongwith growth and optimal growth factors can be stated, respectively, as follows. 1 If T, S : X → intR are supposed to be the enterprise’s output or producing and consuming maps, respectively, by taking λ 1, then from both a of 1.1 and 1.2 , we obtain the von-Neumann type input-output inequalities: