Godard [9] has proved that a

metric space M is 0-hyperbolic if and only if F(M), the Lipschitzfree space over M (see the definition in the next section), is isometric to a subspace of some [L.

12] studied complex valued

metric space and proved common fixed point theorems for two self-mappings satisfying a rational type inequality.

There have been a number of generalizations of

metric spaces such as vector valued

metric spaces, G-

metric spaces, pseudometric spaces, fuzzy

metric spaces, D-

metric spaces, cone

metric spaces, and modular

metric spaces.

rho]], we define a borel [sigma]-field in a

metric space ([F.

Note that that each Caristi mapping on a complete

metric space has a fixed point.

Proposition 12 Let (X, d) be a

metric space, [absolute value of X] = n, and let (X, [[delta].

Lakshmikantham, Fixed point theorems in partially ordered

metric spaces and applications, Nonlinear Anal.

Jain, "Coupled coincidence and common fixed point theorems for set-valued and single-valued mappings in fuzzy

metric space," Journal of Fuzzy Set Valued Analysis, vol.

We recall that (C(I),d) is a complete

metric space (cf.

Henceforth, (X, d) shall be an asymmetric

metric space.

Let X be a geodesic

metric space- that is, a

metric space where any two points x and y are the endpoints of a curve of length d(x, y).

1985): Let (X,d) be a complete

metric space and T: X [right arrow] X a contraction mapping with contraction factor c [member of] [0,1).

Likewise because in the Schwarzschild

metric space is curved a spatial velocity velocity [[upsilon].

5] generalized the Banach contraction principle by using two closed subsets of a complete

metric space.

infinity]]) is a

metric space of fuzzy functions from (-[infinity],0] to [E.