Contraction principle

The contraction mapping principle proved independently by Banach [1] and Cacciopoli [7] is a fundamental fixed point theorem, with an elementary proof. This theorem has a wide spectrum of applications and is a natural choice in approximating solutions to nonlinear problems. According to Rall [18], the applications of the contraction principle would fill volumes and Bollabos [4] calls it a doyen of fixed point theorems. Charmed by both the simplicity and utility of this theorem, many authors have generalized it in diverse directions. This chapter samples a few of these.