Fixed point theorems in metric spaces

In this paper, we establish two general theorems for equivalence between the Meir–Keeler type contractive conditions and the contractive definitions involving gauge functions. One of these theorems is an extension of a recent result of Lim (On characterization of Meir–Keeler contractive maps, Nonlinear Anal. 46 (2001) 113–120).Also, we establish the following new fixed point theorem. Suppose ϕ:R+→R+ϕ:R+→R+ is a contractive gauge function in the sense that for any ε>0ε>0 there exists δ>εδ>ε such that ε