On fixed point generalizations of Suzuki’s method

In order to generalize the well-known Banach contraction theorem, many authors have introduced various types of contraction inequalities. In 2008, Suzuki introduced a new method (Suzuki (2008) [4]) and then his method was extended by some authors (see for example, Dhompongsa and Yingtaweesittikul (2009), Kikkawa and Suzuki (2008) and Mot and Petrusel (2009) [7], [10], [5] and [6]). Kikkawa and Suzuki extended the method in (Kikkawa and Suzuki (2008) [5]) and then Mot and Petrusel further generalized it in (Mot and Petrusel (2009) [6]). In this paper, we shall provide a new condition for TT which guarantees the existence of its fixed point. Our results generalize some old results.