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.

The analytic structure of the governing equation for a 2nd order Phase–Locked Loops (PLL) is studied in the complex time plane. By a local reduction of the PLL equation to the Ricatti equation, the PLL equation is analytically shown to have singularities which form a fractal structure in the complex time plane. Such a fractal structure of complex time singularities is known to be characteristic for nonintegrable, especially chaotic systems. On the other hand, a direct numerical detection of the complex time singularities is performed to verify the fractal structure. The numerical results show the reality of complex time singularities and the fractal structure of singularities on a curve.

This letter presents the results of an analysis concerning the global, dynamical structure of a second order phase–locked loop (PLL) in the presence of the continuous wave (CW) interference. The invariant manifolds of the PLL equation are focused and analyzed as to how they are extended from the hyperbolic periodic orbits. Using the Melnikov integral which evaluates the distance between the stable manifolds and the unstable manifolds, the transversal intersection of these manifolds is proven to occur under some conditions on the power of the interference and the angular frequency difference between the signal and the interference. Numerical computations were performed to confirm the transversal intersection of the system–generated invariant manifolds for a practical set of parameters.

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.

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.

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.

A type of infinite dimensional homotopy method is considered for numerically calculating a solution curve of a nonlinear functional equation being a Fredholm operator with index 1 and an A-proper operator. In this method, a property of so-called A-proper homotopy plays an important role.

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.

A file transmission net N is a directed communication net with vertex set V and arc set B such that each arc e has positive cost ca(e) and each vertex u in V has two parameters of positive cost cv(u) and nonnegative integral demand d(u). Some information to be distributed through N is supposed to have been written in a file and the written file is denoted by J, where the file means an abstract concept of information carrier. In this paper, we define concepts of file transfer, positive demand vertex set U and mother vertex set M, and we consider a problem of distributing d(v) copies of J through a file transfer on N from a vertex v1 to every vertex v in V. As a result, for N such that MU, we propose an O(nm+n2 log n) algorithm, where n=|V| and m=|B|, for synthesizing a file transfer whose total cost of transmitting and making copies of J is minimum on N.

This paper is concerned with the validation of simple turning points of two-point boundary value problems of nonlinear ordinary differential equations. Usually it is hard to validate approximate solutions of turning points numerically because of it's singularity. In this paper, it is pointed out that applying the infinite dimensional Krawcyzk-based interval validation method to enlarged system, the existence of simple turning points can be verified. Taking an example, the result of validation is also presented.

An iterative decomposition method with mesh refinement strategies is presented for the numerical solution of nonlinear two-point boundary value problems. It is shown that this method is more efficient than the traditional finite difference methods and shooting methods.

This paper presents an efficient algorithm for solving bipolar transistor networks. In our algorithm, the network equation f (x)=0 is solved by a homotopy method, in which a homotopy h (x, t)=f (x)-(1-t) f (x0) is introduced and the solution curve of h (x, t)=0 is traced from an obvious solution (x0, 0) to the solution (x*, 1) which we seek. It is shown that the convergence of the algorithm is guaranteed by fairly mild conditions. A rectangular subdivision and an upper bounding technique of linear programming are used for tracing the solution curve. Our rectangular algorithm is much more efficient than the conventional simplicial type algorithms. Some numerical examples are given in order to demonstrate the effectiveness of the algorithm. The advantages of the rectangular algorithm are as follows. (1) Convergence is guaranteed by fairly general conditions. (2) There is no need to evaluate Jacobian matrices. (3) There is no need to invert matrices except for the first step; only pivoting operations are necessary. (4) The replacement rule of vertices is very simple. (5) The computational complexity is markedly reduced compared with the simplicial algorithm. (6) The computational efficiency can be greatly improved by choosing the grid sizes of the rectangular subdivision pertinently according to the nonlinearity of the equation.

An arborescence-net N is a directed connected communication network with arborescence structure. Some information to be distributed through N is supposed to have been written in a file and the written file is denoted by J, where the file means an abstract concept of information carrier. In this letter, we consider a problem of distributing copies of J through N from the root vertex to every vertex, where the cost of transmitting a copy of J through each arc, the cost of making a copy of J at each vertex and the number of copies of J needed at each vertex in N are defined. Definig a file transfer on N, we give a method for designing an optimal file transfer by which we mean a file transfer whose total cost of transmitting and making copies of J is minimum on N.

A method for analyzing the input- and parameter-sensitivities of a broad class of nonlinear continuous systems with nonlinear feedback couplings is proposed. This method is carried out first by formulating the problems in the form of nonlinear integral equations, and then evaluating the solutions by applying fixed point theorems in the appropriate Banach spaces. The actual analysis in this paper is accomplished for the entire function type of nonlinear integral equations, making use of Banach's contraction operator principle, Schauder's fixed point theorem for completely continuous operators and the Leray-Schauder rotation concept of completely continuous vector fields. These procedures can be regarded as systematic and simple even for practical analysis of complicated systems.