• No results found

Some Common Fixed Point Theorems for Rational Type Contractive Mappings in Complex Valued Metric Spaces

N/A
N/A
Info
Download
Protected

Academic year: 2021

Share "Some Common Fixed Point Theorems for Rational Type Contractive Mappings in Complex Valued Metric Spaces"

Copied!
7
0
0

Loading.... (view fulltext now)

Download now ( 7 Page )

Full text

(1)

Some Common Fixed Point Theorems for Rational Type

Contractive Mappings in Complex Valued Metric Spaces

Krati Shukla

Institute for Excellence in Higher Education, Bhopal, (M.P.), India

Sandhya Shukla

Sagar Institute of Science and Technology, Bhopal (M.P.), India

Abstract

We prove some common fixed point theorems for two pairs of weakly compatible mappings satisfying a rational type contractive condition using (E.A.) and (CLR)-property in Complex Valued metric spaces. The proved results generalize and extend some of the results in the literature.

Keywords: Complex Valued metric space, weakly compatible mappings, (E.A.) - property, (CLR)-property.

Mathematics subject classification: 47H10, 54H25.

1. Introduction

Azam et al. [2] introduced the concept of complex valued metric spaces and obtained sufficient conditions for the existence of common fixed point of a pair of contractive type mappings involving rational expressions. Subsequently several authors have studied the existence and uniqueness of the fixed points and common fixed points of self mapping in view of contrasting contractive conditions. Some of these investigations are noted in [3-7].

Aamri and Moutawakil [1] introduced the notion of (E.A.) - property.

Sintunavrat and P. Kumam [8] introduced the notion of CLR-property. Then many authors proved several fixed point theorems using the concept of weakly compatible maps with (E.A.) and (CLR)-property. The main purpose of this paper is to present fixed points results for two pair of maps satisfying a generalize contractive condition by using the concept of weakly compatible maps with (E.A.) and (CLR)-property in complex valued metric space. The proved results generalize and extend some of the results in the literature.

2. Preliminaries

Let 𝐢 be the set of complex numbers and let z1 , z2 ∈ 𝐢. Define a partial order ≀ on 𝐢 as follows:

z1≀z2 if and only if Re(z1)≀Re(z2), Im(z1)≀Im(z2). It follows that z1≀z2 if one of the following conditions is satisfied:

(i) Re(z1 )=Re(z2), Im(z1)<Im(z2), (ii) Re(z1)<Re(z2), Im(z1 )=Im(z2), (iii) Re(z1)<Re(z2), Im(z1)<Im(z2), (iv) Re(z1)=Re(z2), Im(z1 )=Im(z2).

In particular, we will write 𝑧1≀ 𝑧2 if one of (i), (ii) and (iii) is satisfied and we will write 𝑧1< 𝑧2 if only (iii) is satisfied.

Definition2.1. Let 𝑋 be a non-empty set. Suppose that the mapping 𝑑: 𝑋 Γ— 𝑋 β†’ 𝐢 satisfies: (i) 0 ≀ 𝑑(π‘₯, 𝑦) for all π‘₯, 𝑦 ∈ 𝑋 and 𝑑(π‘₯, 𝑦) = 0 if and only if π‘₯ = 𝑦;

(2)

(iii)𝑑(π‘₯, 𝑦) ≀ 𝑑(π‘₯, 𝑧) + 𝑑(𝑧, 𝑦) for all π‘₯, 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. Let 𝑋 = 𝐢. Define a mapping 𝑑: 𝑋 Γ— 𝑋 β†’ 𝐢 by 𝑑(𝑧1, 𝑧2) = π‘’π‘–π‘˜ |𝑧1βˆ’ 𝑧2|, Where π‘˜ ∈ [0, πœ‹ 2⁄ ]. Then , (𝑋, 𝑑) is called a complex valued metric space.

A point π‘₯ ∈ 𝑋 is called an interior point of a set 𝐴 βŠ† 𝑋 whenever there exists 0 < π‘Ÿ ∈ 𝐢 such that 𝐡(π‘₯, π‘Ÿ) =

{𝑦 ∈ 𝑋: 𝑑(π‘₯, 𝑦) < π‘Ÿ} βŠ† 𝐴. A subset 𝐴 in 𝑋 is called open whenever each point of A is an interior point of A. The family 𝐹 = {𝐡(π‘₯, π‘Ÿ): π‘₯ ∈ 𝑋, 0 < π‘Ÿ} is a sub-basis for a Hausdorff topology 𝜏 on 𝑋. A point π‘₯ ∈ 𝑋 is called a limit point of 𝐴 whenever for every 0 < π‘Ÿ ∈ 𝐢, 𝐡(π‘₯, π‘Ÿ) ∩ (𝐴\𝑋) β‰  πœ™.

A subset 𝐡 βŠ† 𝑋 is called closed whenever each limit point of 𝐡 belongs to 𝐡.

Let {π‘₯𝑛} be a sequence in 𝑋 and π‘₯ ∈ 𝑋. If for every 𝑐 ∈ 𝐢, with 0 < 𝑐 there is 𝑛0∈ 𝑁 such that for all 𝑛 > 𝑛0,

𝑑(π‘₯𝑛, π‘₯) < 𝑐, then π‘₯ is called the limit point of {π‘₯𝑛} and we write limπ‘›β†’βˆžπ‘₯𝑛= π‘₯ or π‘₯𝑛→ π‘₯ as 𝑛 β†’βˆž.

If for every 𝑐 ∈ 𝐢, with 0 < 𝑐 there is 𝑛0∈ 𝑁 such that for all 𝑛 > 𝑛0, 𝑑(π‘₯𝑛, π‘₯𝑛+π‘š) < 𝑐, then {π‘₯𝑛} is called a Cauchy sequence in (𝑋, 𝑑) is called a complete complex valued metric space.

Lemma2.2. Let (𝑋, 𝑑) be a complex valued metric space and {π‘₯𝑛} is a sequence in 𝑋. Then {π‘₯𝑛} converges to

π‘₯ if and only if |𝑑(π‘₯𝑛, π‘₯)| β†’ 0 as 𝑛 β†’βˆž.

Lemma2.3. Let (𝑋, 𝑑) be a complex valued metric space and {π‘₯𝑛} is a sequence in 𝑋. Then {π‘₯𝑛} is a Cauchy sequence if and only if |𝑑(π‘₯𝑛, π‘₯𝑛+π‘š)| β†’ 0 as 𝑛 β†’βˆž.

Definition2.4. Let 𝑓 and 𝑔 be self-maps on a set 𝑋, if 𝑀 = 𝑓π‘₯ = 𝑔π‘₯ for some π‘₯ in 𝑋, then π‘₯ is called coincidence point of 𝑓 and 𝑔, 𝑀 is called a point of coincidence of 𝑓 and 𝑔, 𝑀 is called a point of coincidence of

𝑓 and g.

Definition2.5. Let 𝑓 and 𝑔 be two self-maps defined on a set 𝑋, then 𝑓 and 𝑔 are said to be weakly compatible if they commute at coincidence points.

Definition 2.6. Let f and g be two self-mappings of a complex valued metric space (𝑋, 𝑑). We say that f and g satisfy the (E.A.)-property if there exist a sequence {π‘₯𝑛} in X such that limπ‘›β†’βˆžπ‘“π‘₯𝑛= limπ‘›β†’βˆžπ‘”π‘₯𝑛=

𝑑 , for some 𝑑 ∈ 𝑋.

Definition 2.7. Let f and g be two self-mappings of a complex valued metric space (𝑋, 𝑑). We say that f and g satisfy the (𝐢𝐿𝑅𝑓) property if there exist a sequence {π‘₯𝑛} in X such thatlimπ‘›β†’βˆžπ‘“π‘₯𝑛= limπ‘›β†’βˆžπ‘”π‘₯𝑛= 𝑓π‘₯.

3. Main Results

Theorem: 3.1 Let (𝑋, 𝑑) be a Complex valued metric space and 𝑓, 𝑔, β„Ž, π‘˜: 𝑋 β†’ 𝑋 are four self-maps of 𝑋 satisfying the following conditions:

(i) 𝑓(𝑋) βŠ† β„Ž(𝑋), 𝑔(𝑋) βŠ† π‘˜(𝑋),

(ii) 𝑑(𝑓π‘₯, 𝑔𝑦) ≀ π‘Ž1[𝑑(𝑓π‘₯,β„Žπ‘¦)+𝑑(𝑔𝑦,π‘˜π‘₯)𝑑(β„Žπ‘¦,π‘˜π‘₯)𝑑(𝑓π‘₯,π‘˜π‘₯)] + π‘Ž2[𝑑(𝑓π‘₯, π‘˜π‘₯) + 𝑑(β„Žπ‘¦, 𝑔𝑦)]

+π‘Ž3[𝑑(𝑓π‘₯, β„Žπ‘¦) + 𝑑(π‘˜π‘₯, 𝑔𝑦)] + π‘Ž4𝑑(π‘˜π‘₯, β„Žπ‘¦) (3.1) if 𝑑(𝑓π‘₯, β„Žπ‘¦) + 𝑑(𝑔𝑦, π‘˜π‘₯) β‰  0 and if 𝑑(𝑓π‘₯, β„Žπ‘¦) + 𝑑(𝑔𝑦, π‘˜π‘₯) = 0 then 𝑑(𝑓π‘₯, 𝑔𝑦) = 0

for all π‘₯, 𝑦 ∈ 𝑋 and π‘Ž2+ 2π‘Ž3+ π‘Ž4< 1, π‘Ž1, π‘Ž2, π‘Ž3, π‘Ž4> 0; (iii) the pairs (𝑓, π‘˜) and (𝑔, β„Ž)are weakly compatible;

(iv) One of the pairs (𝑓, π‘˜) or (𝑔, β„Ž)satisfy (E.A.)-property.

If the range of one of the mapping π‘˜(𝑋)or β„Ž(𝑋) is closed subspace of 𝑋. Then the mappings 𝑓, 𝑔, β„Ž and π‘˜ have a unique common fixed point in 𝑋.

(3)

lim

π‘›β†’βˆžπ‘”π‘₯𝑛= limπ‘›β†’βˆžβ„Žπ‘₯𝑛= 𝑑 , for some 𝑑 ∈ 𝑋.

Further, since 𝑔(𝑋) βŠ† π‘˜(𝑋), there exists a sequence {𝑦𝑛} in 𝑋, such that 𝑔π‘₯𝑛= π‘˜π‘¦π‘›. Hence limπ‘›β†’βˆžπ‘˜π‘¦π‘›= 𝑑. Now, we claim that limπ‘›β†’βˆžπ‘“π‘¦π‘›= 𝑑. Let limπ‘›β†’βˆžπ‘“π‘¦π‘›= 𝑑1β‰  𝑑 then putting π‘₯ = 𝑦𝑛, 𝑦 = π‘₯𝑛 in (3.1), and we have 𝑑(𝑓𝑦𝑛, 𝑔π‘₯𝑛) ≀ π‘Ž1[ 𝑑(β„Žπ‘₯𝑛, π‘˜π‘¦π‘›)𝑑(𝑓𝑦𝑛, π‘˜π‘¦π‘›) 𝑑(𝑓𝑦𝑛, β„Žπ‘₯𝑛) + 𝑑(𝑔π‘₯𝑛, π‘˜π‘¦π‘›)] + π‘Ž2[𝑑(𝑓𝑦𝑛, π‘˜π‘¦π‘›) + 𝑑(β„Žπ‘₯𝑛, 𝑔π‘₯𝑛)] +π‘Ž3[𝑑(𝑓𝑦𝑛, β„Žπ‘₯𝑛) + 𝑑(π‘˜π‘¦π‘›, 𝑔π‘₯𝑛)] + π‘Ž4𝑑(π‘˜π‘¦π‘›, β„Žπ‘₯𝑛) Letting 𝑛 β†’βˆž, we have 𝑑(𝑑1, 𝑑) ≀ π‘Ž1[𝑑(𝑑𝑑(𝑑, 𝑑)𝑑(𝑑1, 𝑑) 1, 𝑑) + 𝑑(𝑑, 𝑑)] + π‘Ž2[𝑑(𝑑1, 𝑑) + 𝑑(𝑑, 𝑑)] +π‘Ž3[𝑑(𝑑1, 𝑑) + 𝑑(𝑑, 𝑑)] + π‘Ž4𝑑(𝑑, 𝑑) β‡’ [1 βˆ’ (π‘Ž2+ π‘Ž3)] 𝑑(𝑑, 𝑑1) ≀ 0 as π‘Ž2+ 2π‘Ž3+ π‘Ž4< 1.

β‡’ |𝑑(𝑑, 𝑑1)| ≀ 0. Hence 𝑑1= 𝑑 and that is, lim𝑛→

βˆžπ‘“π‘¦π‘›= limπ‘›β†’βˆžπ‘”π‘₯𝑛= 𝑑.

Now suppose that π‘˜(𝑋) is a closed subspace of 𝑋, then 𝑑 = π‘˜π‘’ for some 𝑒 ∈ 𝑋, subsequently we have

lim

π‘›β†’βˆžπ‘“π‘¦π‘›= limπ‘›β†’βˆžπ‘”π‘₯𝑛= limπ‘›β†’βˆžβ„Žπ‘₯𝑛= limπ‘›β†’βˆžπ‘˜π‘¦π‘›= 𝑑 = π‘˜π‘’.

We claim that 𝑓𝑒 = π‘˜π‘’. For this putting π‘₯ = 𝑒, 𝑦 = π‘₯𝑛 in (3.1), and we have

𝑑(𝑓𝑒, 𝑔π‘₯𝑛) ≀ π‘Ž1[ 𝑑(β„Žπ‘₯𝑛, π‘˜π‘’)𝑑(𝑓𝑒, π‘˜π‘’) 𝑑(𝑓𝑒, β„Žπ‘₯𝑛) + 𝑑(𝑔π‘₯𝑛, π‘˜π‘’)] + π‘Ž2[𝑑(𝑓𝑒, π‘˜π‘’) + 𝑑(β„Žπ‘₯𝑛, 𝑔π‘₯𝑛)] +π‘Ž3[𝑑(𝑓𝑒, β„Žπ‘₯𝑛) + 𝑑(π‘˜π‘’, 𝑔π‘₯𝑛)] + π‘Ž4𝑑(π‘˜π‘’, β„Žπ‘₯𝑛) Letting 𝑛 β†’βˆž, we have 𝑑(𝑓𝑒, 𝑑) ≀ π‘Ž1[𝑑(𝑓𝑒,𝑑)+𝑑(𝑑,𝑑)𝑑(𝑑,𝑑)𝑑(𝑓𝑒,𝑑)] + π‘Ž2[𝑑(𝑓𝑒, 𝑑) + 𝑑(𝑑, 𝑑)] + π‘Ž3[𝑑(𝑓𝑒, 𝑑) + 𝑑(𝑑, 𝑑)] + π‘Ž4𝑑(𝑑, 𝑑) β‡’ [1 βˆ’ (π‘Ž2+ π‘Ž3)] 𝑑(𝑓𝑒, 𝑑) ≀ 0 as π‘Ž2+ 2π‘Ž3+ π‘Ž4< 1.

β‡’|𝑑(𝑓𝑒, 𝑑)| ≀ 0, which is a contradiction. Hence 𝑒 is a coincidence point of (𝑓, π‘˜). Now the weak compatibility of pair (𝑓, π‘˜) implies that π‘“π‘˜π‘’ = π‘˜π‘“π‘’ or 𝑓𝑑 = π‘˜π‘‘. On the other hand, Since 𝑓(𝑋) βŠ† β„Ž(𝑋), there exists 𝑣 in 𝑋 such that 𝑓𝑒 = β„Žπ‘£.

Thus, 𝑓𝑒 = π‘˜π‘’ = β„Žπ‘£ = 𝑑. Now, we show that 𝑣 is a coincidence point of (𝑔, β„Ž); that is 𝑔𝑣 = β„Žπ‘£ = 𝑑. Putting π‘₯ = 𝑒, 𝑦 = 𝑣 in (3.1), and we have

𝑑(𝑓𝑒, 𝑔𝑣) ≀ π‘Ž1[

𝑑(β„Žπ‘£, π‘˜π‘’)𝑑(𝑓𝑒, π‘˜π‘’)

𝑑(𝑓𝑒, β„Žπ‘£) + 𝑑(𝑔𝑣, π‘˜π‘’)] + π‘Ž2[𝑑(𝑓𝑒, π‘˜π‘’) + 𝑑(β„Žπ‘£, 𝑔𝑣)]

+π‘Ž3[𝑑(𝑓𝑒, β„Žπ‘£) + 𝑑(π‘˜π‘’, 𝑔𝑣)] + π‘Ž4𝑑(π‘˜π‘’, β„Žπ‘£) or

(4)

𝑑(𝑑, 𝑔𝑣) ≀ π‘Ž1[ 𝑑(𝑑, 𝑑)𝑑(𝑑, 𝑑) 𝑑(𝑑, 𝑑) + 𝑑(𝑔𝑣, 𝑑)] + π‘Ž2[𝑑(𝑑, 𝑑) + 𝑑(𝑑, 𝑔𝑣)] +π‘Ž3[𝑑(𝑑, 𝑑) + 𝑑(𝑑, 𝑔𝑣)] + π‘Ž4𝑑(𝑑, 𝑑) β‡’ [1 βˆ’ (π‘Ž2+ π‘Ž4)] 𝑑(𝑑, 𝑔𝑣 ) ≀ 0, as π‘Ž2+ 2π‘Ž3+ π‘Ž4< 1.

β‡’|𝑑( 𝑑, 𝑔𝑣)| ≀ 0, which is a contradiction. Thus 𝑔𝑣 = 𝑑.

Hence, 𝑔𝑣 = β„Žπ‘£ = 𝑑, and 𝑣 is the coincidence point of 𝑔 and β„Ž. Further, the weak compatibility of pair (𝑔, β„Ž) implies that π‘”β„Žπ‘£ = β„Žπ‘”π‘£, or 𝑔𝑑 = β„Žπ‘‘. Therefore, 𝑑 is a common coincidence point of 𝑓, 𝑔, β„Ž and π‘˜.

Now, we show that t is a common fixed point. Putting π‘₯ = 𝑒, 𝑦 = 𝑑 in (3.1), and we have

𝑑(𝑑, 𝑔𝑑) = 𝑑(𝑓𝑒, 𝑔𝑑) ≀ π‘Ž1[𝑑(𝑓𝑒, 𝑔𝑑) + 𝑑(𝑔𝑑, π‘˜π‘’)𝑑(β„Žπ‘‘, π‘˜π‘’)𝑑(𝑓𝑒, π‘˜π‘’) ] + π‘Ž2[𝑑(𝑓𝑒, π‘˜π‘’) + 𝑑(β„Žπ‘‘, 𝑔𝑑)] +π‘Ž3[𝑑(𝑓𝑒, β„Žπ‘‘) + 𝑑(π‘˜π‘’, 𝑔𝑑)] + π‘Ž4𝑑(π‘˜π‘’, β„Žπ‘‘) or 𝑑(𝑑, 𝑔𝑑) ≀ π‘Ž1[ 𝑑(𝑔𝑑,𝑑)𝑑(𝑑,𝑑) 𝑑(𝑑,𝑔𝑑)+𝑑(𝑔𝑑,𝑑)] + π‘Ž2[𝑑(𝑑, 𝑑) + 𝑑(𝑔𝑑, 𝑔𝑑)] + π‘Ž3[𝑑(𝑑, 𝑔𝑑) + 𝑑(𝑑, 𝑔𝑑)] + π‘Ž4𝑑(𝑑, 𝑔𝑑) β‡’ [1 βˆ’ (2π‘Ž3+ π‘Ž4)] 𝑑(𝑑, 𝐡𝑑 ) ≀ 0, as π‘Ž2+ 2π‘Ž3+ π‘Ž4< 1.

β‡’|𝑑( 𝑑, 𝑔𝑑)| ≀ 0, which is a contradiction. Thus 𝑔𝑑 = 𝑑. Hence, 𝑓𝑑 = 𝑔𝑑 = β„Žπ‘‘ = π‘˜π‘‘ = 𝑑.

Similar arguments arises if we assume that β„Ž(𝑋) is closed subspace of 𝑋. Similarly, the (E.A.)- property of the pair (𝑓, π‘˜) will give a similar result.

For uniqueness of the common fixed point, let us assume that 𝑀 is another common fixed point of 𝑓, 𝑔, β„Ž and π‘˜. Therefore 𝑓𝑀 = 𝑔𝑀 = β„Žπ‘€ = π‘˜π‘€ = 𝑀. Then, Putting π‘₯ = 𝑀 and 𝑦 = 𝑑 in (3.1), and we have

𝑑(𝑀, 𝑑) = 𝑑(𝑓𝑀, 𝑔𝑑) ≀ π‘Ž1[ 𝑑(β„Žπ‘‘, π‘˜π‘€)𝑑(𝑓𝑀, π‘˜π‘€) 𝑑(𝑓𝑀, β„Žπ‘‘) + 𝑑(𝑔𝑑, π‘˜π‘€)] + π‘Ž2[𝑑(𝑓𝑀, π‘˜π‘€) + 𝑑(β„Žπ‘‘, 𝑔𝑑)] +π‘Ž3[𝑑(𝑓𝑀, β„Žπ‘‘) + 𝑑(π‘˜π‘€, 𝑔𝑑)] + π‘Ž4𝑑(π‘˜π‘€, β„Žπ‘‘) or 𝑑(𝑀, 𝑑) ≀ π‘Ž1[𝑑(𝑀,𝑑)+𝑑(𝑑,𝑀)𝑑(𝑑,𝑀)𝑑(𝑀,𝑀)] + π‘Ž2[𝑑(𝑀, 𝑀) + 𝑑(𝑑, 𝑑)]+π‘Ž3[𝑑(𝑀, 𝑑) + 𝑑(𝑀, 𝑑)] + π‘Ž4𝑑(𝑀, 𝑑) β‡’ [1 βˆ’ (2π‘Ž3+ π‘Ž4)] 𝑑(𝑀, 𝑑 ) ≀ 0, as π‘Ž2+ 2π‘Ž3+ π‘Ž4< 1.

β‡’ |𝑑( 𝑀, 𝑑)| ≀ 0, which is a contradiction. Thus, 𝑀 = 𝑑. Hence 𝑓𝑑 = 𝑔𝑑 = β„Žπ‘‘ = π‘˜π‘‘ = 𝑑, and 𝑑 is the unique common fixed point of 𝑓, 𝑔, β„Ž and π‘˜.

Corollary 3.2. Let (𝑋, 𝑑) be a Complex valued metric space and 𝑓, β„Ž: 𝑋 β†’ 𝑋 are two self-maps of 𝑋 satisfying the following conditions:

(i) 𝑓(𝑋) βŠ† β„Ž(𝑋),

(5)

if 𝑑(𝑓π‘₯, β„Žπ‘¦) + 𝑑(𝑓𝑦, β„Žπ‘₯) β‰  0 and if 𝑑(𝑓π‘₯, β„Žπ‘¦) + 𝑑(𝑓𝑦, β„Žπ‘₯) = 0 then 𝑑(𝑓π‘₯, 𝑓𝑦) = 0 for all π‘₯, 𝑦 ∈ 𝑋 and π‘Ž2+ 2π‘Ž3+ π‘Ž4< 1, π‘Ž1, π‘Ž2, π‘Ž3, π‘Ž4> 0;

(iii) the pairs (𝑓, β„Ž) is weakly compatible; (iv) the pair (𝑓, β„Ž) satisfies (E.A.)-property.

If the range of one of the mapping β„Ž(𝑋) is closed subspace of 𝑋. Then the mappings 𝑓 and β„Ž have a unique common fixed.

Theorem: 3.3 Let (𝑋, 𝑑) be a Complex valued metric space and 𝑓, 𝑔, β„Ž, π‘˜: 𝑋 β†’ 𝑋 are four self-maps of 𝑋 satisfying the following conditions:

(i) 𝑓(𝑋) βŠ† β„Ž(𝑋), 𝑔(𝑋) βŠ† π‘˜(𝑋),

(ii) 𝑑(𝑓π‘₯, 𝑔𝑦) ≀ π‘Ž1[𝑑(𝑓π‘₯,β„Žπ‘¦)+𝑑(𝑔𝑦,π‘˜π‘₯)𝑑(β„Žπ‘¦,π‘˜π‘₯)𝑑(𝑓π‘₯,π‘˜π‘₯)] + π‘Ž2[𝑑(𝑓π‘₯, π‘˜π‘₯) + 𝑑(β„Žπ‘¦, 𝑔𝑦)]

+π‘Ž3[𝑑(𝑓π‘₯, β„Žπ‘¦) + 𝑑(π‘˜π‘₯, 𝑔𝑦)] + π‘Ž4𝑑(π‘˜π‘₯, β„Žπ‘¦) (3.3) if 𝑑(𝑓π‘₯, β„Žπ‘¦) + 𝑑(𝑔𝑦, π‘˜π‘₯) β‰  0 and if 𝑑(𝑓π‘₯, β„Žπ‘¦) + 𝑑(𝑔𝑦, π‘˜π‘₯) = 0 then 𝑑(𝑓π‘₯, 𝑔𝑦) = 0

for all π‘₯, 𝑦 ∈ 𝑋 and π‘Ž2+ 2π‘Ž3+ π‘Ž4< 1, π‘Ž1, π‘Ž2, π‘Ž3, π‘Ž4> 0; (iii) the pairs (𝑓, π‘˜) and (𝑔, β„Ž)are weakly compatible;

If the pair (𝑓, π‘˜) satisfies (CL𝑅𝑓) property or (𝑔, β„Ž) satisfies (CL𝑅𝑔)property, then 𝑓, 𝑔, β„Ž and π‘˜ have a unique common fixed point in 𝑋.

Proof: First, we suppose that the pair (𝑔, β„Ž) satisfies (CL𝑅𝑔) property then there exists a sequence {π‘₯𝑛} in 𝑋 such that

lim

π‘›β†’βˆžπ‘”π‘₯𝑛= limπ‘›β†’βˆžβ„Žπ‘₯𝑛= 𝑔π‘₯, for some π‘₯ ∈ 𝑋.

Further, since 𝑔(𝑋) βŠ† π‘˜(𝑋),we have 𝑔π‘₯ = π‘˜π‘’, for some 𝑒 ∈ 𝑋.

we claim that 𝑓𝑒 = π‘˜π‘’ = 𝑑 (say). put π‘₯ = 𝑒, 𝑦 = π‘₯𝑛 in (3.3), and we have

𝑑(𝑓𝑒, 𝑔π‘₯𝑛) ≀ π‘Ž1[ 𝑑(β„Žπ‘₯𝑛, π‘˜π‘’)𝑑(𝑓𝑒, π‘˜π‘’) 𝑑(𝑓𝑒, β„Žπ‘₯𝑛) + 𝑑(𝑔π‘₯𝑛, π‘˜π‘’)] + π‘Ž2[𝑑(𝑓𝑒, π‘˜π‘’) + 𝑑(β„Žπ‘₯𝑛, 𝑔π‘₯𝑛)] +π‘Ž3[𝑑(𝑓𝑒, β„Žπ‘₯𝑛) + 𝑑(π‘˜π‘’, 𝑔π‘₯𝑛)] + π‘Ž4𝑑(π‘˜π‘’, β„Žπ‘₯𝑛) Letting 𝑛 β†’βˆž we have, 𝑑(𝑓𝑒, 𝑔π‘₯) ≀ π‘Ž1[ 𝑑(𝑔π‘₯, π‘˜π‘’)𝑑(𝑓𝑒, π‘˜π‘’) 𝑑(𝑓𝑒, 𝑔π‘₯) + 𝑑(𝑔π‘₯, π‘˜π‘’)] + π‘Ž2[𝑑(𝑓𝑒, π‘˜π‘’) + 𝑑(𝑔π‘₯, 𝑔π‘₯)] +π‘Ž3[𝑑(𝑓𝑒, 𝑔π‘₯) + 𝑑(π‘˜π‘’, 𝑔π‘₯)] + π‘Ž4𝑑(π‘˜π‘’, 𝑔π‘₯) β‡’ [1 βˆ’ (π‘Ž2+ π‘Ž3)] 𝑑(𝑓𝑒, 𝑔π‘₯ ) ≀ 0, as π‘Ž2+ 2π‘Ž3+ π‘Ž4< 1

β‡’ |𝑑( 𝑓𝑒, 𝑔π‘₯)| ≀ 0, which is a contradiction Thus, 𝑓𝑒 = 𝑔π‘₯. Hence, 𝑓𝑒 = π‘˜π‘’ = 𝑔π‘₯ = 𝑑.

Now the weak compatibility of pair (𝑓, π‘˜) implies that π‘“π‘˜π‘’ = π‘˜π‘“π‘’ or 𝑓𝑑 = π‘˜π‘‘.

Further, since 𝑓(𝑋) βŠ† β„Ž(𝑋), there exist 𝑣 in 𝑋 such that 𝑓𝑒 = β„Žπ‘£. Thus, 𝑓𝑒 = π‘˜π‘’ = β„Žπ‘£ = 𝑑. Now, we show that

𝑣 is a coincidence point of (𝑔, β„Ž) that is, 𝑔𝑣 = β„Žπ‘£ = 𝑑.Put π‘₯ = 𝑒, 𝑦 = 𝑣 in (3.3) and we have

𝑑(𝑓𝑒, 𝑔𝑣) ≀ π‘Ž1[

𝑑(β„Žπ‘£, π‘˜π‘’)𝑑(𝑓𝑒, π‘˜π‘’)

𝑑(𝑓𝑒, β„Žπ‘£) + 𝑑(𝑔𝑣, π‘˜π‘’)] + π‘Ž2[𝑑(𝑓𝑒, π‘˜π‘’) + 𝑑(β„Žπ‘£, 𝑔𝑣)]

+π‘Ž3[𝑑(𝑓𝑒, β„Žπ‘£) + 𝑑(π‘˜π‘’, 𝑔𝑣)] + π‘Ž4𝑑(π‘˜π‘’, β„Žπ‘£) or

(6)

𝑑(𝑑, 𝑔𝑣) ≀ π‘Ž1[ 𝑑(𝑑, 𝑑)𝑑(𝑑, 𝑑) 𝑑(𝑑, 𝑑) + 𝑑(𝑔𝑣, 𝑑)] + π‘Ž2[𝑑(𝑑, 𝑑) + 𝑑(𝑑, 𝑔𝑣)] +π‘Ž3[𝑑(𝑑, 𝑑) + 𝑑(𝑑, 𝑔𝑣)] + π‘Ž4𝑑(𝑑, 𝑑) β‡’ [1 βˆ’ (π‘Ž2+ π‘Ž3)] 𝑑(𝑑, 𝑔𝑣 ) ≀ 0, as π‘Ž2+ 2π‘Ž3+ π‘Ž4< 1

β‡’ |𝑑( 𝑑, 𝑔𝑣)| ≀ 0, which is a contradiction. Thus 𝑔𝑣 = 𝑑.

Hence, 𝑔𝑣 = β„Žπ‘£ = 𝑑 and 𝑣 is coincidence point of 𝑔 and β„Ž. Further, the weak compatibility of pair (𝑔, β„Ž) implies that π‘”β„Žπ‘£ = β„Žπ‘”π‘£, or 𝑔𝑑 = β„Žπ‘‘. Therefore, 𝑑 is a common coincidence point of 𝑓, 𝑔, β„Ž and π‘˜.

Now, we show that t is a common fixed point. Put π‘₯ = 𝑒 and 𝑦 = 𝑑 in (3.3), and we have

𝑑(𝑑, 𝑔𝑑) = 𝑑(𝑓𝑒, 𝑔𝑑) ≀ π‘Ž1[ 𝑑(β„Žπ‘‘, π‘˜π‘’)𝑑(𝑓𝑒, π‘˜π‘’) 𝑑(𝑓𝑒, β„Žπ‘‘) + 𝑑(𝑔𝑑, π‘˜π‘’)] + π‘Ž2[𝑑(𝑓𝑒, π‘˜π‘’) + 𝑑(β„Žπ‘‘, 𝑔𝑑)] +π‘Ž3[𝑑(𝑓𝑒, β„Žπ‘‘) + 𝑑(π‘˜π‘’, 𝑔𝑑)] + π‘Ž4𝑑(π‘˜π‘’, β„Žπ‘‘) or 𝑑(𝑑, 𝑔𝑑) ≀ π‘Ž1[ 𝑑(𝑑,𝑑)𝑑(𝑑,𝑑) 𝑑(𝑑,𝑑)+𝑑(𝑔𝑑,𝑑)] + π‘Ž2[𝑑(𝑑, 𝑑) + 𝑑(𝑑, 𝑔𝑑)] + π‘Ž3[𝑑(𝑑, 𝑑) + 𝑑(𝑑, 𝑔𝑑)] + π‘Ž4𝑑(𝑑, 𝑑) β‡’ [1 βˆ’ (π‘Ž2+ 2π‘Ž3)] 𝑑(𝑑, 𝑔𝑑 ) ≀ 0, as π‘Ž2+ 2π‘Ž3+ π‘Ž4< 1

β‡’ |𝑑( 𝑑, 𝑔𝑑)| ≀ 0, which is a contradiction. Thus 𝑔𝑑 = 𝑑.

Therefore 𝑓𝑑 = 𝑔𝑑 = β„Žπ‘‘ = π‘˜π‘‘ = 𝑑. The uniqueness of the common fixed point follows easily.

In a similar way, the argument that the pair (𝑓, π‘˜) satisfies property (CL𝑅𝑓) will also give the unique common fixed point of 𝑓, 𝑔, β„Ž and π‘˜. Hence the result follows.

REFERENCES

[1] M. Aamri and D.El Moutawakil, "Some new common fxed point theorems under strict contractive conditions", J. Math. Anal. Appl. 270 (2002), 181-188.

[2] A. Azam, B. Fisher, and M. Khan, β€œCommon fixed point theorems in complex valued metric spaces,”Numerical Functional Analysis and Optimization, vol. 32, no. 3, pp. 243–253, 2011.

[3] S. Bhatt, S. Chaukiyal, and R. C. Dimri, β€œA common fixed point theorem for weakly compatible maps in complex valued metric spaces,” International Journal of Mathematical Sciences & Applications, vol. 1, no. 3, pp. 1385–1389, 2011.

[4] S. Chandok and D. Kumar, β€œSome common Fixed point results for rational type contraction mappings in complex valued metric spaces,” Journal of operators, vol.2013 , Article ID813707,6 pages, 2013.

[5] S. Chauhan, W. Sintunavarat, and P. Kumam, β€œCommon fixed point theorems for weakly compatible mappings in fuzzy metric spaces using (JCLR)-property,” Applied Mathematics, vol. 3, no. 9, pp. 976–982, 2012.

[6] G. Jungck and B. E. Rhoades, β€œFixed point theorems for occasionally weakly compatible mappings,”Fixed Point Theory, vol. 7, no. 2, pp. 287–296, 2006.

[7] P. kumar, M. kumar and S. kumar, " Common fixed point theorems for a rational inequality in complex valued metric spaces, ” Journal of complex systems, vol.2013 , Article ID942058,7 pages, 2013.

[8] W. Sintunavarat and P. Kumam, β€œCommon fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces,” Journal of Applied Mathematics, vol. 2011, Article ID 637958, 14 pages, 2011.

[9] W. Sintunavarat and P. Kumam, β€œGeneralized common fixed point theorems in complex valued metric spaces and applications,” Journal of Inequalities and Applications, vol. 2012, article 84, 2012.

[10] R. K. Verma and H. K. Pathak, β€œCommon fixed point theorems using property (E.A) in complex-vauled metric spaces,” Thai Journal of Mathematics.

(7)

More information about the firm can be found on the homepage:

http://www.iiste.org

CALL FOR JOURNAL PAPERS

There are more than 30 peer-reviewed academic journals hosted under the hosting platform.

Prospective authors of journals can find the submission instruction on the following

page:

http://www.iiste.org/journals/

All the journals articles are available online to the

readers all over the world without financial, legal, or technical barriers other than those

inseparable from gaining access to the internet itself. Paper version of the journals is also

available upon request of readers and authors.

MORE RESOURCES

Book publication information:

http://www.iiste.org/book/

Academic conference:

http://www.iiste.org/conference/upcoming-conferences-call-for-paper/

IISTE Knowledge Sharing Partners

EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open

Archives Harvester, Bielefeld Academic Search Engine, Elektronische Zeitschriftenbibliothek

EZB, Open J-Gate, OCLC WorldCat, Universe Digtial Library , NewJour, Google Scholar

References

Download now ( PDF - 7 Page - 399.02 KB )

Related documents