The origin of metric fixed point theory goes back to the remarkable work of Polish mathematician Stefan Banach in his Ph.D. dissertation in 1920. The scholarly outcome of the dissertation is now known as the Banach contraction principle. The beauty of the Banach contraction principle is that it requires only completeness and contraction condition on the underlying metric space and mapping, respectively. With these conditions it provides the following assertions:

• • The existence and uniqueness of a fixed point.

• • The method to compute the approximate fixed points.

• • The error estimates for approximate fixed points.

A large number of research papers have already appeared in the literature on extensions and generalizations of the Banach contraction principle. One of the most important generalizations is due to Boyd and Wong [43] for 𝜓-contraction mappings. Another important generalization of the Banach contraction principle is given by Rhoades [148] for weakly contraction mappings introduced by Alber and Guerre-Delabriere [2]. However, a remarkable generalization of the Banach contraction principle is the Caristi’s fixed point theorem [52].

In this chapter, the Banach contraction principle and some of its extensions, namely, Boyd–Wong fixed point theorem for 𝜓-contraction mappings, a fixed point theorem for weakly contraction mappings, and Caristi’s fixed point theorem, are presented. The converse of the Banach contraction principle that provides the characterization of completeness of the metric space is also discussed.

Fixed Points

Definition 2.1 Let X be a nonempty set and TX → X be a mapping.

• A point x ∈ X is said be a fixed point of T if T(x) = x.

• The problem of finding a point x ∈ X such that T(x) = x is called a fixed point problem.

• We denote by Fix(T) = {x ∈ XT(x) = x} the set of all fixed points of T.

Example 2.1 (a) Let T →be a mapping defined by T(x) = x + a for some fixed number a ≠ 0. Then, T has no fixed point.