The contraction theorem has many fields of application, including linear algebraic equations, differential and integral equations, control systems theory, optimization, etc. The paper aims at showing how contraction mapping can be applied to the computation and the training of adaptive structures with algebraic loops. These structures are used for the approximation of unknown functional relations (mappings) represented by training sets. The technique is extended to multilayer neural networks with algebraic loops. Application of a two-layer neural network to breast cancer diagnosis is described.