Some fixed point theorem for successively recurrent system of set-valued mapping equations

Let us introduce n (greater than or equal to 2) mappings f(i) (i = 1, 2,..., n) defined on complete linear metric spaces (Xi-1, p) (i = 1, 2,..., n), respectively, and let f(i) : Xi-1 --> X-i be completely continuous on bounded convex closed subsets X-i-1((0)) subset of Xi-1, (i = 1, 2,..., n equivalent to 0), such that f(i)(X-i-1((0))) subset of X-i((0)). Moreover, let us introduce n set-valued mappings F-i : Xi-1 x X-i --> F(X-i)(the family of all non-empty closed compact subsets of X-i), (i = 1, 2,..., n equivalent to 0). Here; we have a fixed point theorem on the successively recurrent system of set-valued mapping equations: x(i) is an element of F-i(x(i-1), f(i)(x(i-1))), (i = 1, 2,...,n equivalent to 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. In this paper, mathematical situation and detailed proof are discussed, about this theorem.