The Discrete Derivative and Local Convergence

Abstract

In Chap. 2 we introduced a metric tool which was then used to develop the concepts of contraction and monotonicity. With the aid of these concepts we were then able to study certain behaviours of discrete iterations. The incidence matrix B(F) for an operator F was furthermore found to play a basic role within the framework of contractions through the information it carried (f _i depends or does not depend on x _j ). The convergence results obtained were of a global kind (“independent of the initial configuration, the iteration converges to a fixed point …”). These results transpose the classical results from the continuous context into the discrete framework (see for example [32]).