On uniformly convex real Banach spaces, a fixed point theorem in weak topology for successively recurrent system of fuzzy-set-valued nonlinear mapping equations and its application to ring nonlinear network systems are theoretically discussed in detail. An arbitrarily-level likelihood signal estimation is then established.

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 Xi-1(0) ⊂ Xi-1. Moreover, let us introduce n set-valued mappings Fi : Xi-1 Yi → Fc(Xi) (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.

In this paper, we shall construct mathematical theory based on the concept of set-valued mappings, suitable for available operation of extraordinarily complicated large-scale network systems by introducing some connected-block structures. A fine estimation technique for availability of system behaviors of such network systems are obtained finally in the form of fixed point theorem for a special system of fuzzy-set-valued mappings.

Let us introduce n ( 2) nonlinear mappings fi (i = 1,2,,n) defined on complete linear metric spaces (Xi-1,ρ) (i = 1,2,,n), respectively, and let fi:Xi-1 Xi be completely continuous on bounded convex closed subsets Xi-1, (i = 1,2,,n 0), such that fi() . Moreover, let us introduce n fuzzy-set-valued nonlinear mappings Fi:Xi-1Xi {a family of all non-empty closed compact fuzzy subsets of Xi}. Here, by introducing arbitrary constant βi (0,1], for every integer i (i = 1,2,,n 0), separately, we have a fixed point theorem on the recurrent system of βi -level fuzzy-set-valued mapping equations: xi Fiβi(xi-1, fi(xi-1)), (i = 1,2,,n 0), where the fuzzy set Fi is characterized by a membership function µFi(xi):Xi [0,1], and the βi -level set Fiβi of the fuzzy set Fi is defined as Fiβi {ξi Xi |µFi (ξi) βi}, for any constant βi (0,1]. This theorem can be applied immediately to discussion for characteristics of ring nonlinear network systems disturbed by undesirable uncertain fluctuations and to extremely fine estimation of available behaviors of those disturbed systems. In this paper, its mathematical situation and proof are discussed, in detail.

Let us introduce n ( 2) nonlinear mappings fi (i = 1,2,,n) defined on complete linear metric spaces (Xi-1,ρ) (i = 1,2,,n), respectively, and let fi: Xi-1 Xi be completely continuous on bounded convex closed subsets Xi-1,(i = 1,2,,n 0), such that fi() . Moreover, let us introduce n fuzzy-set-valued nonlinear mappings Fi: Xi-1 Xi {a family of all non-empty closed compact fuzzy subsets of Xi}. Here, we have a fixed point theorem on the recurrent system of β-level fuzzy-set-valued mapping equations: xi Fiβ(xi-1,fi(xi-1)), (i = 1,2,,n 0), where the fuzzy set Fi is characterized by a membership function µFi(xi): Xi [0,1], and the β-level set Fiβ of the fuzzy set Fi is defined as Fiβ {ξi Xi | µFi(ξi) β}, for any constant β (0,1]. This theorem can be applied immediately to discussion for characteristics of ring nonlinear network systems disturbed by undesirable uncertain fluctuations and to fine estimation of available behaviors of those disturbed systems. In this paper, its mathematical situation and proof are discussed, in detail.

Let us introduce n ( 2) mappings fi (i=1,2,,n) defined on complete linear metric spaces (Xi-1, ρ) (i=1,2,,n), respectively, and let fi:Xi-1 Xi be completely continuous on bounded convex closed subsets Xi-1(0) Xi-1, (i=1,2,,n 0), such that fi(Xi-1(0)) Xi(0). Moreover, let us introduce n set-valued mappings Fi : Xi-1 Xi (Xi)(the family of all non-empty closed compact subsets of Xi), (i=1,2,,n 0). Here, we have a fixed point theorem on the successively recurrent system of set-valued mapping equations: xi Fi(xi-1, fi(xi-1)), (i=1,2,,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. In this paper, mathematical situation and detailed proof are discussed, about this theorem.

In this paper, we shall construct mathematical theory based on the concept of set-valued mappings, suitable for available operation of network systems extraordinarily complicated and diversified on large scales. Fundamental conditions for availability of system behaviors of such network systems are clarified in a form of fixed point theorem for system of set-valued mappings.

In any ill-conditioned information-transfer system, as in long-distance communication, we often must construct feedback confirmation channels, in order to confirm that informations received at destinations are correct. Unfortunately, for such systems, undesirable uncertain fluctuations may be induced not only into forward communication channels but also into feedback confirmation channels, and it is such difficult that transmitters always confirm correct communications. In this paper, two fuzzy-set-valued mappings are introduced into both the forward communication channel and the feedback confirmation channel, separately, and overall system-behaviors are discussed from the standpoint of functional analysis, by means of fixed point theorem for a system of generalized equations on fuzzy-set-valued mappings. As a result, a good mathematical condition is successfully obtained, for such information-transfer systems, and fine-textured estimations of solutions are obtained, at arbitrary levels of values of membership functions.

In multi-media systems, the type of interactive communication channels is found almost everywhere and plays an important role, as well as the type of unilateral communication channels. In this report, we shall construct a fluctuation theory based on the concept of set-valued mappings, suitable for evaluation, control and operation of interactive communication channels in multi-media systems, complicated and diversified on large scales. Fundamental conditions for availability of such channels are clarified in a form of fixed point theorem for system of set-valued mappings.