Fixed point and periodic point theorems

Abstract

We introduce a weaker form of continuity which is a necessary and sufficient condition for the existence of fixed points. The obtained theorems exhibit interesting fixed point – eventual fixed point patterns. If we slightly weaken the conditions then the mappings admit periodic points besides fixed points and such mappings possess interesting combinations of fixed and periodic points. Our results are applicable to contractive type as well as non-expansive type mappings. Our theorems are independent of almost all the existing results for contractive type mappings. The last theorem of Sect. 2 is applicable to mappings having various geometric patterns as their domain and is perhaps the first result of its type that also opens up scope for the study of periodic points and periodic point structures. We also give an application of our theorem to obtain the solutions of a nonlinear Diophantine equation; and also show that various well-known fixed point theorems are not applicable in solving this equation.