Advances in Fixed Point Theory Using Control Function Contractions in Pure Mathematics

Abstract

The fixed point theory has a key role in pure mathematics, which gives existence and uniqueness results of an invariant point of a mapping of different structure under different structural assumptions. The aim of this research is to create more unified and general framework of fixed points that surpasses the restrictive conditions usually set in classical principles of contraction. In order to do so, a theoretical and descriptive approach is chosen that is taken in generalized metric spaces, such as b-metric and controlled metric spaces. A control function is an introduction of new-type of contractive mappings into which Banach, Kannan, Chatterjeea, and rational-type contractive mappings are spawned. This approach relies on the building of Picard-type iterative sequences and the study of their convergence using weaker completeness and continuity conditions. Strong mathematical arguments have been presented to prove the existence and uniqueness of fixed points and supported through comparative analysis and examples. The findings reveal that the specified contraction condition guarantees the excellent convergence and uniqueness in addition to greatly weakening such conventional conditions as strict completeness and continuity. Among the contributions of this work to the fixed point theory is that it brings together a number of classical results into one flexible theory that is more general, has better convergence properties and further increased applicability in abstract mathematical spaces, which provides increased strength to the theoretical foundations of fixed point theory in pure mathematics.