**
**

*Abstract:*
In this paper a fixed point theorem for mappings satisfying a
contractive inequality of integral type in generalized metric space is
established. This result is analogous to the result of Branciari.

** **

**Introduction:**
Dhage [1] introduced the notion of D-metric space (Generalized Metric
Space) as follows.

A non-empty set X together with a
function D : X
´
X ´
X ®
R^{+}, R^{+} denote the set of all non-negative real
numbers, called a D-metric on X, becomes a D-metric space (X, D) if D
satisfies the following properties :

(i) D(x, y, z) = 0 if and only if x = y =
z (coincidence)

(ii) D(x, y, z) = D(x, z, y) = …..
(symmetry)

(iii) D(x, y, z)
£
D(x, y, a) + D(x, a, z) + D(a, y, z) for all x, y, z, a
Î
X (tetrahedral inequality)

**Example:**
Define a function D :: X
´
X ´
X ®R
by D(x, y, z) = max {d(x, y), d(y, z), d(z, x)} for all x, y, z
Î
X and where d is an ordinary metric on X. Then D defines a D-metric on
X.

A sequence {x_{n}} in
a D-metric space (X, D) is said to be D-convergent and converges to a
point x Î
X if D(x_{m}, x_{n}, x) =0

A sequence {x_{n}} in
(X, D) is said to be D-Cauchy if
D(x_{m}, x_{n}, x_{p}) = 0.
A complete D-metric space X is one in which every D-Cauchy sequence
converges to a point in it.

In a recent paper Branciari
[2] established the following theorem.

**Theorem :**
Let (X, d) be a complete metric space, C
Î
[0, 1), f : X®X
a mapping such that, for each x, y
ÎX,

where
f
: R^{+}
®
R^{+} is a Lebesgue-integrable mapping which is summable,
non-negative, and such that, for each
e
> 0, dt > 0. Then f has a unique fixed point zÎX
such that, for each xÎX,
f^{n}x = z.

The aim of this paper is to
prove the above result of Branciari [2] in Generalized Metric Space.

**Theorem:**
Let X be a complete D-metric space, k
Î[0,
1), T : X®X
a mapping such that, for each x, y, zÎX,

…(1)

where
f
: R^{+}
®
R^{+} is a Lebesgue – integrable mapping which is summable,
non-negative and such that for each
e
> 0,

> 0, …(2)

Then T has a unique fixed point u
ÎX
such that, for each xÎX,
= u.

**Proof :-**
Let x Î
X, and for brevity, define x_{n} = T^{n}x

For each integer m
³
n ³
1, from (1)

£……£
k^{n} …(3)

Taking the limit of (3), as n®¥,
yields

= 0 …(4)

which, from (2), implies that

…(5)

We now show that {x_{n}} is
Cauchy. Suppose that it is not. Then there exists an
e
> 0 and subsequences {m(k)}, {n(k)}, {p(k)} such that

k
£
m(k) < p(k) < n(k)

D(x_{m(k)}, x_{n(k)}, x_{p(k)})
³
e,
D(x_{m(k)}, x_{n(k)}_{-1},
x_{p(k)}_{-1})

<
e
…(6)

Using the triangular inequality and (6),

D(x_{m(k)}_{-1},
x_{n(k)}_{-1},
x_{p(k)}_{-1})
£
D(x_{m(k)}_{-1},
x_{m(k)}, x_{p(k)}_{-1})

+ D(x_{m(k)}_{-1},
x_{n(k)}_{-1},
x_{m(k)}) + D(x_{m(k)}, x_{n(k)}_{-1},
x_{p(k)}_{-1})

< D(x_{m(k)}_{-1},
x_{m(k)}, x_{p(k)}_{-1})

+ D(x_{m(k)}_{-1},
x_{n(k)}_{-1},
x_{m(k)}) +
e
…(7)

Using (5) and (7)

…(8)

Using (1), (6) and (8), it then follows
that

£
k

which is a contradiction. Therefore {x_{n}}
is Cauchy, hence convergent. Call the limit u.

From (1)

…(9)

Taking the limit of (9) as n®¥,
we obtain

which implies that

= 0

which from (2), implies that D(Tu, u, u)
= 0 or Tu = u. This implies u is a fixed point of T.

For uniqueness,

Suppose that u and v are fixed point of
T. Then from (1),

£
k

which implies that

which, from (2), implies that D(u, u, v)
= 0 or u = v, and the fixed point is unique.

**Example:**
Let X = [0, 2] and D be a D-metric on X defined by D(x, y, z) =
max{d(x, y), d(y, z), d(z, x)}, where d is a usual metric on X. Define
T : X®X
such that T(x) = . Let
f
: R^{+}®R^{+}
be such that
f(t) = t,
then all the conditions of theorem are satisfied and Pr
Î[1/4,
1) and clearly 1 is the unique fixed point of T.

** **

**References:**

** **

[1]
Dhage, B.C., Generalized metric spaces and mappings with fixed
point, Bull. Cal. Math. Soc. **84**, 329, 1992.

[2]
Branciari, A., A fixed point theorem for mappings satisfying a
general contractive condition of Integral type, Int. J. Math. Math.
Sci.** 29**, no. 9, 531, 2002.