In this paper, we shall describe a basic fuzzy-estimation theory based on the concept of set-valued operators, suitable for available operation of extremely complicated large-scale network systems. Fundamental conditions for availability of system behaviors of such network systems are clarified in a form of β-level fixed point theorem for system of fuzzy-set-valued operators. Here, the proof of this theorem is accomplished by the concept of Hausdorff's ball measure of non-compactness introduced into the Banach space.

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 describe about a fuzzy estimation theory based on the concept of set-valued operators, suitable for available operation of extremely complicated large-scale network systems. Fundamental conditions for availability of system behaviors of such network systems are clarified in a form of β-level fixed point theorem for system of fuzzy-set-valued operators. Here, the proof of this theorem is accomplished in a weak topology introduced into the Banach space.

In this paper, we shall describe about a basic theory based on the concept of set-valued operators, suitable for available operation of extremely complicated large-scale network systems. 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 operators. Here, the proof of this theorem is accomplished by the concept of Hausdorff's ball measure of non-compactness.

In this paper, we shall describe about a refined theory based on the concept of set-valued operators, suitable for available operation of extremely complicated large-scale network systems. The deduction of theory is accomplished in a weak topology introduced into the Banach space. Fundamental conditions for availability of system behaviors of such network systems are clarified, as a result, in a form of fixed point theorem for system of set-valued operators.

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.

In this paper, we introduce the following m-layered hard constrained convex feasibility problem HCF(m): Find a point u m, where 0:=H (a real Hilbert space), i: = arg min gi(i-1) and gi(u):=wi,jd 2(u,Ci,j) are defined for (i) nonempty closed convex sets Ci,jH and (ii) weights wi,j > 0 satisfying wi,j=1 (i {1,,m}, j {1,,Mi}. This problem is regarded as a natural extension of the standard convex feasibility problem: find a point u Ci, where Ci H (i {1,, M}) are closed convex sets. Unlike the standard problem, HCF(m) can handle the inconsistent case; i.e., i,j Ci,j = , which unfortunately arises in many signal processing, estimation and design problems. As an application of the hybrid steepest descent method for the asymptotically shrinking nonexpansive mapping, we present an algorithm, based on the use of the metric projections onto Ci,j, which generates a sequence (un) satisfying limn d(un,3) = 0 (for M1 = 1) when at least one of C1,1 or C2,j's is bounded and H is finite dimensional. An application of the proposed algorithm to the pulse shaping problem is given to demonstrate the great flexibility of the method.

A mathematical theory is proposed based on the concept of functional analysis, suitable for operation of network systems extraordinarily complicated and diversified on large scales, through connected-block structures. Fundamental conditions for existence and evaluation of system behaviors of such network systems are obtained in a form of fixed point theorem for system of nonlinear 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-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.

A mathematical theory is proposed, based on the concept of functional analysis, suitable for operation of network systems extraordinarily complicated and diversified on large scales, through connected-block structures. Fundamental conditions for existence and evaluation of system behaviors of such network systems are obtained in a form of fixed point theorem for system of nonlinear mappings.

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 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.

This paper proposes an associative memory neural network whose limiting state is the nearest point in a polyhedron from a given input. Two implementations of the proposed associative memory network are presented based on Dykstra's algorithm and a fixed point theorem for nonexpansive mappings. By these implementations, the set of all correctable errors by the network is characterized as a dual cone of the polyhedron at each pattern to be memorized, which leads to a simple amplifying technique to improve the error correction capability. It is shown by numerical examples that the proposed associative memory realizes much better error correction performance than the conventional one based on POCS at the expense of the increase of necessary number of iterations in the recalling stage.

We analyze the dynamics of self-organizing cortical maps under the influence of external stimuli. We show that if the map is a contraction, then the system has a unique equilibrium which is globally asymptotically stable; consequently the system acts as a stable encoder of external input stimuli. The system converges to a fixed point representing the steady-state of the neural activity which has as an upper bound the superposition of the spatial integrals of the weight function between neighboring neurons and the stimulus autocorrelation function. The proposed theory also includes nontrivial interesting solutions.

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.

This paper proposes a methodology for fine evaluation of the uncertain behaviors of systems affected by any fluctuation of internal structures and internal parameters, by the use of a new concept on the fuzzy mapping. For a uniformly convex real Banach space X and Y, a fuzzy mapping G is introduced as the operator by which we can define a bounded closed compact fuzzy set G(x,y) for any (x,y)∈X×Y. An original system is represented by a completely continuous operator f defined on X, for instance, in a form xλ(f(x)) by a continuous operator λ: YX. The nondeterministic fluctuations induced into the original system are represented by a generalized form of the fuzzy mapping equation xGβ (x,f(x)) {ζX|µG(x,f(x))(ζ)β}, in order to give a fine evaluation of the solutions with respect to an arbitrarily–specified β–level. By establishing a useful fixed point theorem, the existence and evaluation problems of the "β–level-likely" solutions are discussed for this fuzzy mapping equaion. The theory developed here for the fluctuation problems is applied to the fine estimation of not only the uncertain behaviors of system–fluctuations but also the validity of system–models and -simulations with uncertain properties.

In the direct product space of a complete metric linear space X and its related space Y, a fuzzy mapping G is introduced as an operator by which we can define a projective fuzzy set G(x,y) for any xX and yY. An original system is represented by a completely continuous operator f(x)Y, e.g., in the form x=λ(f(x)), (λ is a linear operator), and a nondeterministic or fuzzy fluctuation induced into the original system is represented by a generalized form of system equation xβG(x,f(x)). By establishing a new fixed point theorem for the fuzzy mapping G, the existence and evaluation problems of solution are discussed for this generalized equation. The analysis developed here for the fluctuation problem goes beyond the scope of the perturbation theory.