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 � 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 � 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 � 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 � [0, 1), f : X�X a mapping such that, for each x, y �X,

 

 

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 z�X such that, for each x�X, fⁿx = 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 �[0, 1), T : X�X a mapping such that, for each x, y, z�X,

                                                  �(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 �X such that, for each x�X,  = u.

Proof :- Let x � X, and for brevity, define x_n = Tⁿx

For each integer m � n � 1, from (1)

���� kⁿ                �(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)) � 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 �[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.