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) 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.

A numerical method is proposed for identifying bifurcating solution of infinite dimensional nonlinear equations by making use of the infinite dimensional homotopy method. In this paper in the first place, in order to show the existence of bifurcating solutions for a certain class of the Fredholm operators, a mapping degree is defined which has the similar properties as in a finite dimensional space. Using this, under certain conditions a primary bifurcation point exists for a certain type of infinite dimensional nonlinear equations. Furthermore, in case of the Leray-Schauder operator, it is shown that a certain bifurcating solution of the Leray-Schauder operator equation can be identified by making use of the infinite dimensional homotopy method.

An estimation method of region is presented, in which a solution path of the so-called Newton type homotopy equation in guaranteed to exist, it is applied to a certain class of uniquely solvable nonlinear equations. The region can be estimated a posteriori, and its upper bound also can be estimated a priori.

This paper surveys the research topics and results on nonlinear theory and its applications which have been achieved in Japan or by Japanese researchers during the last decade. The paticular emphasis is placed on chaos, neural networks, nonlinear circuit analysis, nonlinear system theory, and numerical methods for solving nonlinear systems.

This letter considers a problem of realizing an undirected vertex-capacitated network from the view of two types; flow-saturated and flow-unsaturated. As a result, two necessary and sufficient conditions for a given matrix to be realizable as a flow-saturated network and as both a flow-saturated network and a flow-unsaturated one have been shown.

Algorithms for computing channel capacity have been proposed by many researchers. Recently, one of the authors proposed an efficient algorithm using Newton's method. Since this algorithm has local quadratic convergence, it is advantageous when we want to obtain a numerical solution with high accuracy. In this letter, it is shown that this algorithm can be extended to the algorithm for computing the constrained capacity, i.e., the capacity of discrete memoryless channels with linear constraints. The global convergence of the extended algorithm is proved, and its effectiveness is verified by numerical examples.

A priori estimation is presented for a computational complexity of the homotopy method applying to a certain class of strongly monotone nonlinear equations. In the present papers, a condition is presented for a certain class of uniquely solvable equations, under which an upper bound of a computational complexity of the Newton type homotopy method can be a priori estimated. In this paper, a condition is considered in a case of linear homotopy equations including the Newton type homotopy equations. In the first place, the homotopy algorithm based on the simplified Newton method is introduced. Then by using Urabe type theorem, which gives a sufficient condition guaranteeing the convergence of the simplified Newton method, a condition is presented under which an upper bound of a computational complexity of the algorithm can be a priori estimated, when it is applied to a certain class of strongly monotone nonlinear equations. The presented condition is demonstrated by numerical experiments.

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 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.

A priori estimation is presented for a computational complexity of the homotopy method applying to a certain class of uniquely solvable nonlinear equations. In the first place, the reason is explained why a computational complexity of the homotopy method can not be a priori estimated in general. In this paper, the homotopy algorithm is considered in which a numerical path following algorithm is executed based on the simplified Newton method. Then by introducing Urabe's theorem, which gives a sufficient condition guaranteeing the convergence of the simplified Newton method, it is shown that a computational complexity of the algorithm can be a priori estimated, when it is applied to a certain class of uniquely solvable nonlinear equation. In this paper, two types of path following algorithms are considered, one with a numerical error estimation in the domain of a nonlinear operator and another with one in the range of the operator.

The fundamental methodologies derived from the functional analysis theory are discussed for the analysis of nonlinear system fluctuations. Starting from the orthodox methods based on the classical fixed point theorems, up to the newly proposed methods based on the fixed point theorems for set-valued functions, several kinds of methodology have been shown, in detail. In the latter half of this paper, clear descriptions are given for the tolerability and quasitolerability of the system fluctuation, the fail-safe principle in the wide sense, the stability and quasistability of the system, two standpoints in sensitivity analysis, and the tolerability and quasitolerability of the system model and simulation by model. Particularly, the new concepts of quasitolerability and quasistability are proposed for the wide utilization of the notion of system. The fail-safe principle in the wide sense is a new idea for security design of system. The recent rapid development of computers lets us expect that these new concepts may become truly important in the near future.

In this paper, in the first place, a slightly version upped Urabe type theorem of convergence criterion is presented for the modified Newton method. Then, based on this theorem, a posteriori stopping criterion is presented for a class of numerical methods of calculating solutions including the simplicial approximate homotopy method for nonlinear equations. By this criterion it is estimated whether an approximate solution satisfies the conditions of the Urabe theorem or not. Finally, it is shown that under a certain mild condition a class of simplicial approximate homotopy methods such as Merrill's method generate an approximate solution which satisfies our stopping criterion in restarting finite steps.

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 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 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.

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 the first place, a similarity degree is defined, as a membership function being a measure of a relation between two fuzzy points in a fuzzy set. Then, by using the degree, a fuzzy metric is proposed, and some properties are shown. Furthermore, an application of this is considered to a recognition system.

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.

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.