Abstract: The Banach contraction theorem not only plays a vital role in functional analysis, but also is an important theoretical basis for the algebraic ectuations of numerical analysis, the existence and uniqueness of ordinary differential equations and the integral ectuation of mathematical analysis. It is one of the most common methods in mathematical and engineering calculations. This paper presented adaptive functional networks loop structures which were designed based on the I3anach contraction theorem and the learning algorithm. These structures are used for the approximation of the fixed point of unknown functional relations(mappings) represented by training sets. Finally, the simulation results do monstrate that the structure presented in the paper has high precision and stable. The results obtained in this paper are very important for researching the methods of neural computation.

Key words: Contraction theorem, Functional networks, Loop structures,Learning algorithm