Aplicação do teorema do ponto fixo na análise de convergência de sequências

Abstract

Numerical sequences represent a very important part of study in Mathematics, being studied from elementary school to advanced levels of research. An important application of sequences can be found in biological models of population growth, where it is possible to predict the size of a population from an initial population. In this context, there are several ways to analyze its convergence. When the sequence is defined as recurrence, that is, a term depends on its predecessors (xn + 1 = f (xn)), this analysis may not be as simple. Strategies such as finding the general term of a recurrence are inefficient. In this sense, the Fixed Point Theorem can be an alternative for this analysis to be successful, besides giving conditions for, in case of convergence, to characterize the behavior of the limit point. Thus, this work seeks to apply the fixed-point theory in the context of convergence of defined sequences in the form of recurrence, and to apply it in the study of some models of population dynamics.