A Common Fixed Point Result in Complex Valued Metric Spaces under Contractive Condition

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A Common Fixed Point Result in Complex Valued Metric Spaces under Contractive Condition

Sultan Ali D

Deeppaarrttmmeenntt ooff MMaatthheemmaattiiccss,, KKaallnnaa CCoolllleeggee,, KKaallnnaa,, BBuurrddwwaann--771133440099,, WWeesstt BBeennggaall,, IInnddiiaa

E-E-mmaaiill:: mmaaiillttoossuullttaannaallii@@gmgmaaiill..ccoomm

ABSTRACT

In [7], S.K. Datta and S. Ali proved a common fixed point theorem for two self- mappings under contractive condition in complex valued metric spaces. Here we prove a different type of common fixed point theorem for two self-mappings in complex valued metric spaces.

Key words: Complex Valued Metric Space, Common Fixed Point, Contractive Type Mapping.

AMS Subject Classification (2010) : 47H10, 54H25

1 INTRODUCTION, DEFINITIONS & NOTATIONS

In 2011, A. Azam, B. Fisher and M. Khan [2] introduced the notion of the complex valued metric space and established sufficient conditions for the existence of common fixed points of a pair of mappings satisfying a contractive condition. W. Sintunavarat and P. Kumam [6], S.K.Datta and S.Ali [8] generalized the results obtained by A. Azam, B. Fisher and M. Khan [2]. There are many common fixed point theorems in complex valued metric spaces see [3], [7], [9]. In this paper we would like to obtain different type of sufficient conditions for existence of common fixed point of two self mappings in complex valued metric spaces.

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𝑧₁ ≾ 𝑧₂ if and only if 𝑅𝑒 𝑧₁ ≤ 𝑅𝑒 𝑧₂ and 𝐼𝑚 𝑧₁ ≤ 𝐼𝑚 𝑧₂ . Thus 𝑧₁ ≾ 𝑧₂ if one of the followings holds:

(1) 𝑅𝑒 𝑧₁ = 𝑅𝑒 𝑧₂ and 𝐼𝑚 𝑧₁ = 𝐼𝑚 𝑧₂ , (2) 𝑅𝑒 𝑧₁ < 𝑅𝑒 𝑧₂ and 𝐼𝑚 𝑧₁ = 𝐼𝑚 𝑧₂ , (3) 𝑅𝑒 𝑧₁ = 𝑅𝑒 𝑧₂ and 𝐼𝑚 𝑧₁ < 𝐼𝑚 𝑧₂ and (4) 𝑅𝑒 𝑧₁ < 𝑅𝑒 𝑧₂ and 𝐼𝑚 𝑧₁ < 𝐼𝑚 𝑧₂ .

We write 𝑧₁ ⋨ 𝑧₂ if 𝑧₁ ≾ 𝑧₂ and 𝑧₁ ≠ 𝑧₂ i.e., one of (2), (3) and (4) is satisfied and we will write 𝑧₁ ≺ 𝑧₂ if only (4) is satisfied.

Remark 1: We can easily check the followings:

(i) 𝑎, 𝑏 ∈ ℝ, 𝑎 ≤ 𝑏 ⇒ 𝑎𝑧 ≾ 𝑏𝑧 ∀ 𝑧 ∈ ℂ.

(ii) 0 ≾ 𝑧₁ ⋨ 𝑧₂ ⇒ 𝑧₁ < 𝑧₂ . (iii) 𝑧₁ ≾ 𝑧₂ and 𝑧₂ ≺ 𝑧₃ ⇒ 𝑧₁ ≺ 𝑧₃.

Azam et al. [2] defined the complex valued metric space in the following way:

Definition 1 ([2]): Let 𝑋 be a nonempty set. Suppose that the mapping 𝑑 ∶ 𝑋 × 𝑋 → ℂ satisfies the following conditions:

(C1) 0 ≾ 𝑑 𝑥, 𝑦 , for all 𝑥, 𝑦 ∈ 𝑋 and 𝑑 𝑥, 𝑦 = 0 if and only if 𝑥 = 𝑦;

(C2) 𝑑 𝑥, 𝑦 = 𝑑 𝑦, 𝑥 for all 𝑥, 𝑦 ∈ 𝑋;

(C3) 𝑑 𝑥, 𝑦 ≾ 𝑑 𝑥, 𝑧 + 𝑑 𝑧, 𝑦 for all 𝑥, 𝑦, 𝑧 ∈ 𝑋.

Then 𝑑 is called a complex valued metric on X and (𝑋, 𝑑) is called a complex valued metric space.

Example 1 ([7]): Let = ℂ . Define the mapping 𝑑 ∶ 𝑋 × 𝑋 → ℂ by 𝑑 𝑧₁, 𝑧₂ = 𝑖 𝑧₁− 𝑧₂ ∀ 𝑧₁, 𝑧₂ ∈ ℂ.

One can easily verify that (𝑋, 𝑑) is a complex valued metric space.

Definition 2 ([2]): Let (𝑋, 𝑑) be a complex valued metric space. Then

(i) A point 𝑥 ∈ 𝑋 is called an interior point of a set 𝐴 ⊆ 𝑋 if there exists 0 ≺ 𝑟 ∈ ℂ such that

𝐵 𝑥, 𝑟 = {𝑦 ∈ 𝑋: 𝑑(𝑥, 𝑦) ≺ 𝑟} ⊆ 𝐴.

A subset 𝐴 ⊆ 𝑋 is called open if each element of A is an interior point of 𝐴.

(ii) A point 𝑥 ∈ 𝑋 is called a limit point of 𝐴 ⊆ 𝑋 if for every 0 ≺ 𝑟 ∈ ℂ, 𝐵(𝑥, 𝑟)⋂(𝐴 − {𝑥}) ≠ 𝜙.

A subset 𝐴 ⊆ 𝑋 is called closed if each element of 𝑋 − 𝐴 is not a limit point of 𝐴.

(iii) The family

𝐹 = {𝐵 𝑥, 𝑟 : 𝑥 ∈ 𝑋, 0 ≺ 𝑟}

is a sub-basis for a Hausdorff topology 𝜏 on 𝑋.

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(i) A sequence {𝑥_𝑛} in 𝑋 is said to converge to 𝑥 ∈ 𝑋 if for every 0 ≺ 𝑟 ∈ ℂ there exists 𝑁 ∈ ℕ such that 𝑑 𝑥_𝑛, 𝑥 ≺ 𝑟, ∀ 𝑛 > 𝑁. We denote this by lim

𝑛→∞𝑥_𝑛 = 𝑥 or 𝑥_𝑛 → 𝑥 as 𝑛 → ∞.

(ii) If for every 0 ≺ 𝑟 ∈ ℂ there exists 𝑁 ∈ ℕ such that 𝑑(𝑥_𝑛 , 𝑥_𝑛+𝑚) ≺ 𝑟 for all 𝑛 > 𝑁, 𝑚 ∈ ℕ, then {𝑥_𝑛} is called a Cauchy sequence in (𝑋, 𝑑).

(iii) If every Cauchy sequence in 𝑋 is convergent in 𝑋 then (𝑋, 𝑑) is called a complete complex valued metric space.

Definition 4 ([9]): The ‘max’ function for the partial order ≾ is defined as follows:

(1) max 𝑧₁, 𝑧₂ = 𝑧₂ ⟺ 𝑧₁ ≾ 𝑧₂.

(2) 𝑧₁ ≾ max 𝑧₂, 𝑧₃ ⇒ 𝑧₁ ≾ 𝑧₂ or 𝑧₁ ≾ 𝑧₃ . (3) max 𝑧₁, 𝑧₂ = 𝑧₂ ⟺ 𝑧₁ ≾ 𝑧₂or 𝑧₁ ≤ 𝑧₂ .

2 RESULTS:

In this section we present the main result of the paper.

Lemma 1 ([2]): Let (𝑋, 𝑑) be a complex valued metric space and {𝑥_𝑛} be a sequence in 𝑋.

Then {𝑥_𝑛} converges to 𝑥 ∈ 𝑋 if and only if 𝑑(𝑥_𝑛, 𝑥) → 0 as 𝑛 → ∞.

Lemma 2 ([2]): Let (𝑋, 𝑑) be a complex valued metric space and {𝑥_𝑛} be a sequence in 𝑋.

Then 𝑥_𝑛 is a Cauchy sequence if and only if 𝑑 𝑥_𝑛, 𝑥_𝑛+𝑚 → 0 as 𝑛 → ∞ where 𝑚 ∈ ℕ.

Lemma 3 ([7]): Let (𝑋, 𝑑) be a complex valued metric space and {𝑥_𝑛} be a sequence in 𝑋 such that lim

𝑛→∞𝑥_𝑛 = 𝑥. Then for any 𝑎 ∈ 𝑋, lim

𝑛→∞𝑑 𝑥_𝑛, 𝑎 = 𝑑 𝑥, 𝑎 .

Theorem 1: Let (𝑋, 𝑑) be a complete complex valued metric space and 𝑆, 𝑇 ∶ 𝑋 → 𝑋 be self- maps satisfying the following condition:

𝑑 𝑆𝑥, 𝑇𝑦 ≾ 𝛼. max 𝑑 𝑥, 𝑦 ,𝑑 𝑥,𝑆𝑥 𝑑(𝑦,𝑇𝑦 ) 1+𝑑(𝑆𝑥,𝑇𝑦 )

for all 𝑥, 𝑦 ∈ 𝑋, where 𝛼 is a real with 0 < 𝛼 < 1. Then 𝑆 and 𝑇 have a unique common fixed point.

Proof : Let 𝑥₀ ∈ 𝑋 be arbitrary.

We define a sequence {𝑥_𝑛} in 𝑋 as 𝑥_2𝑘+1 = 𝑆𝑥_2𝑘

𝑥_2𝑘+2 = 𝑇𝑥_2𝑘+1 , 𝑘 = 0,1,2, … Then

𝑑 𝑥_2𝑘+1, 𝑥_2𝑘+2 = 𝑑 𝑆𝑥_2𝑘 , 𝑇𝑥_2𝑘+1

≾ 𝛼. max 𝑑 𝑥_2𝑘,𝑥_2𝑘+1 ,^𝑑^𝑥^2𝑘^{, 𝑆}^𝑥^2𝑘^𝑑(^𝑥^2𝑘+1^,𝑇^𝑥^2𝑘+1⁾

1+𝑑 𝑆𝑥2𝑘 ,𝑇𝑥2𝑘+1

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1+𝑑 𝑥_2𝑘+1 , 𝑥_2𝑘+2

≾ 𝛼. 𝑑 𝑥_2𝑘,𝑥_2𝑘+1 . Thus

𝑑 𝑥_2𝑘+1, 𝑥_2𝑘+2 ≾ 𝛼. 𝑑 𝑥_2𝑘,𝑥_2𝑘+1 . (1) Similarly

𝑑 𝑥_2𝑘+2, 𝑥_2𝑘+3 = 𝑑 𝑆𝑥_2𝑘+2 , 𝑇𝑥_2𝑘+1

≾ 𝛼. max 𝑑 𝑥_2𝑘+2,𝑥_2𝑘+1 ,^𝑑^𝑥^2𝑘+2^{, 𝑆}^𝑥^2𝑘+2^𝑑(^𝑥^2𝑘+1^,𝑇^𝑥^2𝑘+1⁾

1+𝑑 𝑆𝑥_2𝑘+2 ,𝑇𝑥_2𝑘+1 ≾ 𝛼. max 𝑑 𝑥_2𝑘+2,𝑥_2𝑘+1 ,^𝑑^𝑥^2𝑘+2^,^𝑥^2𝑘+3^𝑑(^𝑥^2𝑘+1^,^𝑥^2𝑘+2⁾

1+𝑑 𝑥2𝑘+3 , 𝑥2𝑘+2

= 𝛼. 𝑑 𝑥_2𝑘+1,𝑥_2𝑘+2 . Hence

𝑑 𝑥_2𝑘+2, 𝑥_2𝑘+3 ≾ 𝛼. 𝑑 𝑥_2𝑘+1,𝑥_2𝑘+2 . (2) Therefore from (1) and (2) for 𝑛 ∈ ℕ we have

𝑑 𝑥_𝑛+1, 𝑥_𝑛+2 ≾ 𝛼𝑑 𝑥_𝑛,𝑥_𝑛+1 ≾ 𝛼²𝑑 𝑥_𝑛−1,𝑥_𝑛 ≾ ⋯ ≾ 𝛼^𝑛+1𝑑 𝑥₀,𝑥₁ . So for 𝑚, 𝑛 ∈ ℕ,

𝑑 𝑥_𝑛, 𝑥_{𝑛 +𝑚} ≾ 𝑑 𝑥_𝑛, 𝑥_{𝑛 +1} +𝑑 𝑥_𝑛+1, 𝑥_{𝑛 +2} +⋯ +𝑑 𝑥_{𝑛+𝑚 −1}, 𝑥_𝑛+𝑚 ≾ 𝛼^𝑛 + 𝛼^𝑛+1 + ⋯ + 𝛼^{𝑛 +𝑚 −1} 𝑑 𝑥₀,𝑥₁

≾ ^𝛼^𝑛

1−𝛼𝑑 𝑥₀,𝑥₁ . Thus

𝑑 𝑥_𝑛, 𝑥_𝑛+𝑚 ≤ ^𝛼^𝑛

1−𝛼 𝑑 𝑥₀,𝑥₁ → 0 as 𝑛 → ∞ where 𝑚 ∈ ℕ.

Therefore from Lemm 2, we see that {𝑥_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete ∃ 𝑢 ∈ 𝑋 such that 𝑥_𝑛 → 𝑢 as 𝑛 → ∞ .

Thus lim

𝑛 →∞𝑆𝑥_2𝑛 = lim

𝑛→∞𝑇𝑥_2𝑛+1 = 𝑢. (3) Now from the given condition we have

𝑑 𝑆𝑢, 𝑢 ≾ 𝑑 𝑆𝑢, 𝑇𝑥_2𝑛+1 + 𝑑(𝑇𝑥_2𝑛+1, 𝑢)

≾ 𝛼. max 𝑑 𝑢,𝑥_2𝑛+1 ,^{𝑑 𝑢,𝑆𝑢 𝑑(}^𝑥^2𝑛+1^,𝑇^𝑥^2𝑛+1⁾

1+𝑑(𝑆𝑢,𝑇𝑥_2𝑛+1) + 𝑑(𝑇𝑥_2𝑛+1, 𝑢) ≾ 𝛼. max 𝑑 𝑢,𝑥_2𝑛+1 ,^{𝑑 𝑢,𝑆𝑢 𝑑(}^𝑥^2𝑛+1^,^𝑥^2𝑛+2⁾

1+𝑑(𝑆𝑢, 𝑥_2𝑛+2) + 𝑑( 𝑥_2𝑛+2, 𝑢)

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𝑑 𝑆𝑢, 𝑢 ≾ 0.

Thus 𝑑 𝑆𝑢, 𝑢 = 0 and hence 𝑆𝑢 = 𝑢.

Again

𝑑 𝑢, 𝑇𝑢 ≾ 𝑑 𝑆𝑢, 𝑇𝑢

≾ 𝛼. max 𝑑 𝑢, 𝑢 ,𝑑 𝑢,𝑆𝑢 𝑑(𝑢,𝑇𝑢 ) 1+𝑑(𝑆𝑢,𝑇𝑢 ) = 0.

Hence 𝑇𝑢 = 𝑢.

Therefore 𝑢 is a common fixed point of 𝑆 and 𝑇.

Now for the uniqueness part, let us suppose that 𝑆𝑢^∗ = 𝑇𝑢^∗ = 𝑢^∗ for some 𝑢^∗ ∈ 𝑋.

Then

𝑑 𝑢, 𝑢^∗ = 𝑑 𝑆𝑢, 𝑇𝑢^∗

≾ 𝛼. max 𝑑 𝑢,𝑢^∗ ,^{𝑑 𝑢,𝑆𝑢 𝑑(}^𝑢^∗^,𝑇^𝑢^∗⁾

1+𝑑(𝑆𝑢,𝑇𝑢^∗) = 𝛼𝑑(𝑢, 𝑢^∗).

This implies (1 − 𝛼) 𝑑 𝑢, 𝑢^∗ ≤ 0.

Since 0 < 𝛼 < 1, we must have 𝑢 =𝑢^∗ and this completes the proof. ∎

By setting 𝑆 = 𝑇 we get the following corollary.

Corollary 1: Let (𝑋, 𝑑) be a complete complex valued metric space and 𝑇 ∶ 𝑋 → 𝑋 be a self-map satisfying the following condition:

𝑑 𝑇𝑥, 𝑇𝑦 ≾ 𝛼. max 𝑑 𝑥, 𝑦 ,𝑑 𝑥,𝑇𝑥 𝑑(𝑦,𝑇𝑦 ) 1+𝑑(𝑇𝑥,𝑇𝑦 )

for all 𝑥, 𝑦 ∈ 𝑋, where 𝛼 is a real with 0 < 𝛼 < 1. Then 𝑆 and 𝑇 has a unique fixed point.

REFERENCE

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3(September 2011), pp. 1385-1389.

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