A Fixed Point Theorem in Weak Topology for Successively Recurrent System of Set-Valued Mapping Equations and Its Applications

Abstract

Let us introduce n (≥ 2) mappings fi(i = 1, …, n ≡ 0) defined on reflexive real Banach spaces Xi-1 and let fi : Xi-1 → Yi be completely continuous on bounded convex closed subsets $X_{i-1}^{(0)} \\subset X_{i-1}$. Moreover, let us introduce n set-valued mappings $F_i : X_{i-1} imes Y_i o {\\cal F}_c(X_i)$ (the family of all non-empty compact subsets of Xi), (i=1, …, n ≡ 0). Here, we have a fixed point theorem in weak topology on the successively recurrent system of set-valued mapping equations: xi ∈ Fi(xi-1, fi(xi-1)), (i=1, …, n ≡ 0). This theorem can be applied immediately to analysis of the availability of system of circular networks of channels undergone by uncertain fluctuations and to evaluation of the tolerability of behaviors of those systems.