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.
Mitsunori MAKINO Shin'ichi OISHI Masahide KASHIWAGI Kazuo HORIUCHI
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.
Yoshihiro KANEKO Shoji SHINODA Kazuo HORIUCHI
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.