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.

In a file transmission net N with vertex set V and arc set B, copies of a file J are distributed from a vertex to every vertex, subject to certain rules on file transmission. A cost of making one copy of J at each vertex µ is called a copying cost at µ, a cost of transmitting one copy of J through each arc (x, y) is called a transmission cost (x, y), and the number of copies of J demanded at each vertex u in N is called a copy demand at u. A scheduling of distributing copies of J from a vertex, say s, to every vertex on N is called a file transfer from s. The vertex s is called the source of the file transfer. A cost of a file transfer is defined, a file transfer from s is said to be optimal if its cost is not larger than the cost of any other file transfer from s, and an optimal file transfer from s is said to be optimum on N if its cost is not larger than that of an optimal file transfer from any other vertex. In this note, it is proved that an optimal file transfer from a vertex with a minimum copying cost is optimum on N, if there holds M U where M and U are the mother vertex set and the positive demand vertex set of N, respectively. Also it is shown by using an example that an optimal file transfer from a vertex with a minimum copying cost is not always optimum on N when M ⊃ U holds.

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 problem of synthesizing an optimal file transfer on a file transmission net N is to consider how to distribute, with a minimum total cost, copies of a file J with some information from source vertex set S to all vertices of N by the respective vertices' copy demand numbers. The case of |S| =1 has been studied so far. This paper deals with N such that |S|1, where a forest-type file transfer is defined. This paper proposes a polynomial time algorithm to synthesize an optimal forest-type file transfer on such N satisfying SM U, where M and U are mother vertex set and positive demand vertex set of N, respectively.

A problem of obtaining an optimal file transfer on a file transmission net N is to consider how to distribute, with a minimum total cost, copies of a file with some information from a vertex of N to all vertices of N by the respective vertices' copy demand numbers (i.i., needed numbers of copies). The maximum number of copies of file which can be made at a vertex is called the copying number of the vertex. In this paper, we consider as N an arborescence-net with constraints on copying numbers, and give a necessary and sufficient condition for a file transfer to be optimal on N, and furthermore propose an O(n2) algorithm for obtaining an optimal file transfer on N, where n is the number of vertices of N.

A vertex-capacitated network is a graph whose edges and vertices have infinite positive capacities and finite positive capacities, respectively. Such a network is a model of a communication system in which capacities of links are much larger than those of stations. This paper considers a problem of realizing a flow-saturation in an undirected vertex-capacitated network by adding the least number of edges. By defining a set of influenced vertex pairs by adding edges, we show the follwing results.(1) It suffices to add the least number of edges to unsaturated vertex pairs for realizing flow-saturation.(2) An associated graph of a flow-unsaturated network defined in this paper gives us a sufficient condition that flow-saturation is realized by adding a single edge.

This paper presents and effective acceleration technique for the homotopy method using a rectangular subdivision (which will be called the rectangular algorithm" in this paper). The rectangular algorithm is an efficient solution algorithm that is globally convergent for a large class of nonlinear resistive networks. In the rectangular algorithm, the restart technique is usually used in order to improve the accuracy of the solution. In this paper, we show that certain realization of the restart technique achieves local quadratic convergence. In other words, if we determine the grid size of the rectangular subdivision pertinently in each restart, the sequence of the approximate solutions generated by the algorithm converges to the exact solution quadratically. And in this case, the computational work involved in each restart is almost identical to that of Newton's method. Therefore, our algorithm becomes as efficient as Newton's method when it reaches sufficiently close to the solution.

Recently, a fail-safe principle in wide sense was proposed by one of the authors, in order to accomplish the flexible security against accidents or failures. This new principle is established by the following idea: i.e., if the tolerable range of the deviated output response as a behavior of system is definitely estimated, in advance, and if the response can be found in this range, we can always use this response as available. Even if the deviation range of the output response extends far from the tolerable one, it is sufficient that the system itself has a methodology to find the available response in the tolerable range, and the system behaves by this response. This new concept of an extended type of the fail-safe principle is a significant application of the mathematical theory of nondeterministic fluctuations of systems which was presented by the authors. In this paper, the foundation is given to this principle by developing its mathematical theory, in detail.

In this letter, a constructive simplified Newton method is presented for calculating solutions of infinite dimensional nonlinear equations, which uses a projection scheme from an infinite dimensional space to finite dimensional subspaces. A convergence theorem of the method is shown based on Urabe's theorem.

We analyze the nonlinear dynamics of PLL from the "complex" singularity structure by introducing the complex time. The most important results which we have obtained in this work are as follow: (1) From the psi-series expansion of the solution, the local behavior in the neighbourhood of a movable singularity is mapped onto an integrable differential equation: the Ricatti equation. (2) From the movable pole of the Ricatti equation, a set of infinitly clustered singularities about a movable singularity is shown to exist for the equation of PLL by the multivalued mapping. The above results are interesting because the clustering and/or the fractal distribution of singularities is known to be a characteristic feature of the non-integrability or chaos. By using the method in this letter, we can present a circumstantial evidence for chaotic dynamics without assuming any small parameters in the equation of PLL.

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.

A 2-switch node network is one of the most fundamental structure among communication nets such as telephone networks and local area networks etc. In this letter, we prove that a problem of designing a 2-switch node network satisfying capacity conditions of switch nodes and their link, which we call 2-switch node network problem, is NP-complete.

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.

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.