Fixed point theorems for weakC-contractions in partially ordered 2-metric spaces

In [], Choudhury proved that if X is a complete metric space, then every weak C-contraction has a unique fixed point; see [, Theorem .]. This result was generalized to a complete, partially ordered metric space in []; see [, Theorems ., . and .]. There were some generalizations of a metric such as a -metric, aD-metric, aG-metric, a conemetric, and a complex-valuedmetric. The notion of a -metric has been introduced by Gähler in []. Note that a -metric is not a continuous function of its variables, whereas an ordinary metric is. This led Dhage to introduce the notion of a D-metric in []. But in [] Mustafa and Sims showed that most of topological properties of D-metric were not correct. In [] Mustafa and Sims introduced the notion of a G-metric to overcome flaws of a D-metric. After that, many fixed point theorems on G-metric spaces have been stated. However, it was shown in [] and [] that in several situations fixed point results in G-metric spaces can be in fact deduced from fixed point theorems in metric or quasimetric spaces.