On Cyclic Generalized Weakly C-Contractions on Partial Metric Spaces

The notion of partial metric space [1], represented by the abbreviation PMS, departs from the usual metric spaces due to removing the assumption of self-distance. In other words, in PMS self-distance needs not to be zero. This interesting distance function is defined by Matthews [1], as a generalization metric to study in computer science, in particular, to get a more efficient programs in computer science. In the remarkable publication of Matthews [1], a characterization of the Banach Contraction Principle was given in the context of PMS. Due to its wide application potential [2–6], PMS and its topological properties are considered bymany authors [7– 25]. Very recently, Haghi et al. [26] proved that some obtained results in the context of PMS can be deduced from earlier results in the setting of usual metric space. In the sequel, R, N∗ will represent the set of all real nonnegative numbers and the set of all positive natural numbers, respectively. Moreover, we use the abbreviations MS, CMS, PMS, andCMPS formetric space, completemetric space, partialmetric space, and complete partialmetric space, respectively. Let Λ be the collection of function φ : [0, 1) → [0, 1) which is nondecreasing, continuous together with the property φ(t) > 0 for t ∈ (0, 1) and φ(0) = 0. The following definition introduced by Chatterjea [27] to generalize the Banach contraction principle. Definition 1. Suppose that (X, d) is an MS. A mapping T : X → X is said to be a C-contraction if there exists α ∈ (0, 1/2) such that the following inequality holds: