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Abstract
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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) [7,10,5,6]). 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 T which guarantees the
existence of its fixed point. Our results generalize some old results.
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