Abstract

In this paper, we prove a unique fixed point theorem for generalized d-cyclic contraction in dislocated metric spaces (d-metric spaces). Our result generalizes, extends and improves some known results existing in the references.

1. Introduction

In 2000, Hitzler and Seda ² have introduced the notion of dislocated metric space(also called d-metric space) and established some fixed point theorems in complete dislocated metric spaces, dis-located metric space plays an important role in Topology, Logical programming and in electronics engineering. In 2003, Kirk et al. ⁵ have introduced the notion of cyclic contraction and they obtained some fixed point theorems for cyclic contractions in dislocated metric spaces. In 2013, George et al. ¹ have obtained some fixed point results on d-cyclic contractions in dislocated metric spaces. In this paper, we obtain a unique fixed point theorem for a generalized d-cyclic contraction in dislocated metric spaces.

Definition 1.1 ². Let X be a non-empty set and let d:X × X→[0,∞) be a function satisfying the following conditions

(d1)d(x, y) = d(y, x).

(d2)d(x, y) = d(y, x) = 0 ⇒ x = y.

(d3)d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X.

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

Definition 1.2 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 1.3 ². A sequence {x_n} in a d-metric space (X, d) d-converges with respect to d if there exists x∊X such that d(x_n, x) → 0 as n→∞. In this case x is called limit of {x_n}( in d) and we write x_n → x.

Definition 1.4 ². A d-metric space (X, d) is called d-complete if every Cauchy sequence in it is d-convergent.

Definition 1.5 ⁵. Let A and B be non-empty subsets of a metric space (X, d). A cyclic map

T: A ∪ B →A ∪ B is said to be cyclic map if T(A) ⊂ B and T(B) ⊂ A.

Definition 1.6 ⁵. Let A and B be non-empty subsets of a metric space (X, d). A cyclic map T:A ∪ B →A ∪ B is said to be a cyclic contraction if there exists k∈(0,1) such that d(Tx, Ty) ≤ kd(x, y) for all x∈A and y ∈ B.

We define a generalized d-cyclic contraction mapping in the following way.

Definition 1.7. Let A and B be non-empty subsets of a d-metric space (X, d). A cyclic map T:A ∪ B →A ∪ B is said to be a generalized d-cyclic contraction if there exists α, β, γ > 0 satisfying α + 2β + 4γ < 1 such that

for all x ∈A and y ∈ B.

2. Fixed Point Theorem

Theorem 2.1. Let (X, d) be a complete d-metric space, A and B be non-empty subsets of X and T:A ∪ B →A ∪ B be a generalized d- cyclic contraction in X. Then T has a unique fixed point in A ∩ B.

Proof. Fix x ∈ A. By the definition 1.7 there exists α, β, γ > 0, α + 2β + 4γ < 1 such that

Where, b = (α + β + 3 γ) /1- (β+ γ) < 1.

By induction, we have d(Tⁿ⁺¹x, Tⁿx) ≤ bⁿ d(Tx, x), more generally, for m>n, we have

Since, b < 1, so as m, n → ∞ we have

Hence, d(T^mx, Tⁿx) → 0 ,as m, n → ∞.

Therefore, {Tⁿx} is a Cauchy sequence. Since (X, d) is complete so {Tⁿx} converge to some point z ∈ X. Since {T²ⁿx} ⊆ A and {T^2n-1x} ⊆ B and so z ∈ A ∩ B.

We claim that Tz = z.

Taking limit n→∞, we obtain

Thus, z is a fixed point.

To show the uniqueness, let us assume that there exists two fixed points say z₁ and z₂ such that Tz₁ = z₁ and Tz₂ = z₂.

Now,

Since, α + 2β + 2γ < 1.

Therefore, T has a unique fixed point in A ∩ B.

This completes the proof of the theorem.

Remark 2.2. If we choose β = γ = 0 in the above theorem then we get the d-cyclic contraction theorem3.3 in ¹.

Remark 2.3. If we choose α = γ = 0 in the above theorem then we get the Kannan type d-cyclic contraction theorem3.6 in ¹.

Remark 2.4. If we choose α = β = 0 in the above theorem then we get the Chatterjee type d-cyclic contraction theorem3.8 in ¹.

3. Conclusion

The above Theorem 2.1 is a generalization of Theorems 3.3., 3.6., and 3.8., in ¹.

References