Abstract

In this paper, we study a unique common fixed point theorem for four self mappings in dislocated metric spaces, which generalizes, extends and improves some of the recent results existing in the literature.

1. Introduction

In 2000, Hitzler and Seda ² have introduced the notion of dislocated metric space in which self distance of a point need not be equal to zero and also generalized the Banach contraction principle in this dislocated metric space. Later on some of the authors like Aage, Salunke ¹, sufati ³ and Shrivastava et.al., ⁵ have proved some fixed point theorems in dislocated metric space. In 2012, Jha and Panti ⁴ have proved some fixed point theorems for two pairs of weakly compatible maps in dislocated metric space. In this paper, we study a unique common fixed point theorem for four self mappings in dislocated metric space, which generalizes, extends and improves some known results existing in the references.

2. Preliminaries

The following definitions are due to Hitzler and Seda ² .

Definition 2.1 ². Let X be a non-empty set and let d: X × X→ [0,∞) be a function satisfying the following conditions

(i) d(x, y) = d(y,x).

(ii) d(x, y) = d(y, x) = 0 ⇒ x = y.

(iii) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ϵ X.

Then d is called dislocated metric or d-metric on X.

Definition 2.2 ². A sequence {x_n} in a d-metric space (X, d) is called a Cauchy sequence if for given ∊ > 0 , there exists n₀ ∊ℕ such that for all m, n ≥ 0, we have d(x_m, x_n) < ϵ.

Definition 2.3 ². A sequence {x_n} in a d-metric space (X, d) converges with respect to d if there exists x ∊ X such that d(x_n, x) → 0 as n→∞.

Definition 2.4 ². A d-metric space (X, d) is called complete if every Cauchy sequence is convergent with respect to d.

Definition 2.5 ². Let T and S be mappings from a metric space (X, d) itself. Then T and S are said to be weakly compatible if they commute at their coincidence point, that is, Tx = Sx for some x∊X ⇒ TSx = STx.

3. Main Results

Theorem 3.1. Let (X, d) be a complete d-metric space. Suppose S,T, I and J: X→X are continuous mappings satisfying :

(1)

for all x, y ϵ X, where a_i ≥ 0 (i = 1,2,3,4,5), a₁ + a₂ + a₃ +2a₄ +2a₅ < 1.

If S(X) ⊆ J(X), T(X) ⊆ I(X), and if the pairs (S, I) and (T, J) are weakly compatible then S,T, I and J have unique common fixed point.

Proof. Let x₀ be an arbitrary point in X.

Since S(X)⊆ J(X), T(X)⊆I(X) there exists x₁, x₂ ϵ X Such that Sx₀ = Jx₁, Tx₁ = Ix₂ . Continuing this process, we define {x_n} by Jx_2n+1 = Sx_2n , Ix_2n+2 = Tx_2n+1 , n = 0,1,2….. Denote y_2n = Jx_2n+1 = Sx_2n , y_2n+1 = Ix_2n+2 = Tx_2n+1 , n = 0,1,2…..

If y_2n = y_2n+1 for some n, then Jx_2n+1 = Tx_2n+1. Therefore, x_2n+1 is a coincidence point of J and T. Also if y_2n+1 = y_2n+2 for some n, then Ix_2n+2 = Sx_2n+2. Therefore, x_2n+2 is a coincidence point of I and S. Assume that If y_2n ≠ y_2n+1 for all n. Then we have,

Letting, b = (a₁ + a_{2 +} a₄ )/1-( a₃ + a₄ + 2a₅) < 1.

This shows that

For every integer m > 0, we have

Therefore, d(y_n, y_n+m) → 0.

⇒ {y_n} is a Cauchy sequence in a complete d-metric space. So there exists a point zϵX such that y_n → z. Therefore, the subsequences {Sx_2n}→ z, {Jx_2n+1}→ z, {Tx_2n+1}→ z and {Ix_2n+2}→ z. Since, T(X) ⊆ I(X), there exists a point uϵX such that z = Iu. Then we have by (1)

Letting n→∞ we get that

which is a contradiction. So Su = z = Iu. Since, S(X)⊆J(X), there exists a point vϵX such that z = Jv.

We claim that z = Tv. If z ≠ Tv. Then

which is a contradiction. So we get that z = Tv.

Therefore, Su = Iu = Tv = Jv = z. That is z is a common fixed point of S, T, f and g.

Finally in order to prove that the uniqueness of z. Suppose that z and z₁, z ≠ z₁ , are common fixed points of S, T, f and g respectively. Then by (1), we have

which is a contradiction, since a₁ + a₂ + a₃ +2a₄ +2a₅ < 1. Therefore, z = z₁.

Hence, z is the unique common fixed point of S, T, f and g respectively.

Remark 3.2. If we choose f = g = I is an identity mapping in the above Theorem3.1, then we get the following corollary.

Corollary 3.3. Let (X, d) a complete d-metric space. Let S, T: X→X be continuous mappings satisfying the following

for all x, yϵX, where a_i ≥ 0 (i = 1,2,3,4,5), a₁ + a₂ + a₃ +2a₄ +2a₅ < 1.

Then S, and T have unique common fixed point.

Remark 3.4. If we choose S = T in the above Theorem 3.1, then we get the following corollary

Corollary 3.5. Let (X, d) a complete d-metric space. Let S, T: X→X be continuous mappings satisfying the following

for all x, yϵX, where a_i ≥ 0 (i = 1,2,3,4,5), a₁ + a₂ + a₃ +2a₄ +2a₅ < 1.

Then T has a unique common fixed point.

References