Fixed Point Theory of Weak Contractions in Partially Ordered Metric Spaces

The well-known Banach’s fixed point theorem asserts that if D = X, f is contractive and (X, d) is complete, then f has a unique fixed point in X. In nonlinear analysis, the study of fixed points of given mappings satisfying certain contractive conditions in various abstract spaces has been investigated deeply. The Banach contraction principle [1] is one of the initial and crucial results in this direction. Also, this principle has many generalizations. For instance, Alber and Guerre-Delabriere in [2] suggested a generalization of the Banach contractionmapping principle by introducing the concept of weak contraction in Hilbert spaces. In [2], the authors also proved that the result of Eslamian and Abkar [3] is equivalent to the result of Dutta and Choudhury [4]. Later, weakly contractive mappings and mappings satisfying other weak contractive inequalities have been discussed in several works, some of which are noted in [4–16] In 2008, Dutta and Choudhury proved the following theorem. Theorem 1 (see [4]). Let (X, d) be a complete metric space, and let f : X → X be such that