View of COMMON FIXED - POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS IN COMPLEX VALUED B-METRIC SPACES

(1)

VOLUME: 08, Special Issue 08, (IC-RAAPAMAA-2021) Paper id-IJIERM-VIII-

VIII, October 2021

21 COMMON FIXED - POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS

IN COMPLEX VALUED B-METRIC SPACES Rajesh Pandya

School of studies in Mathematics, Vikram University (M.P.), India Аklesh Раriyа

Deраrtment оf Mаthemаtiсs, S.V.Р. Gоvt. Соllege, Kukshi, (M.Р.), Indiа Sandeep Kumar Tiwari

School of studies in Mathematics, Vikram University (M.P.), India

Abstract- Azam et al. [1] introduced the definition of complex-valued metric spaces and created several typical fixed point results in the sense of complex-valued metric spaces.

This paper proves the existence of common fixed points for weakly compatible mappings on complex-valued b-metric spaces. Related observations in the literature are unified, generalized and supplemented by our findings.

Keywords: complex-valued metric space; compatible mappings; common fixed point;

rational contractions, compatible mappings.

2010 MSC: 47H10; 54H25 1. INTRODUCTION AND PRELIMINARIES

Bakhtin[2] proposed the notion of b- metric space. Rouzkard and Imdad[9]

established certain common fixed point theorems fulfilling certain rational expressions in complex valued metric spaces in which is an extension and improvement of Azam et al[1] finding. By substituting the constant of the contractive condition to control function, Sintunavarat and Kumam[10] produced similar fixed point results. Rao et al [7]

proposed the notion of complex valued b- metric spaces in which was more broad than the well-known complex valued metric spaces[1].

Definition1.1 [2]. Let 𝑋 be a nonempty set and let 𝑠≥1 be a given real number. A function 𝑑:𝑋×𝑋→ [0, ∞) is called a b-metric if for all 𝑥, ∈𝑋 the following conditions are satisfied:

(i) (𝑥,) = 0 if and only if 𝑥=𝑦;

(ii) (𝑥,) =𝑑(𝑦,𝑥) for all 𝑥,𝑦 ∈𝑋;

(iii) 𝑑(𝑥,𝑦) ≤𝑠[𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦)] for all 𝑥,𝑦,𝑧∈ 𝑋.

The pair (𝑋, 𝑑) is called a b-metric space.

The number 𝑠≥1 is called the coefficient of (𝑋,).

Example 1.1[8]- Let (𝑋, 𝑑) be a metric space and 𝜌(𝑥,𝑦) = (𝑑(𝑥,𝑦))𝑝, where 𝑝>1 is a real number. Then (𝑋, 𝜌) is b-metric space with 𝑠 = 2𝑝−1.

Definition 1.2 [7]- Let 𝑋 be a nonempty set and let 𝑠≥1 be a given real number. A function 𝑑:𝑋×𝑋→ℂ is called a complex

valued b-metric on 𝑋 if for all 𝑥, 𝑦, 𝑧∈𝑋 the following conditions are satisfied:

(i) 0 ≾𝑑(𝑥,𝑦) and 𝑑(𝑥,𝑦) = 0 if and only if 𝑥=𝑦;

(ii) d(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋;

(iii) 𝑑(𝑥,𝑦) ≾𝑠[𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦)] for all 𝑥,𝑦,𝑥∈𝑋 .

The pair (𝑋, 𝑑) is called a complex valued b-metric space.

Example 1.2 [7]- Let 𝑋= [0, 1]. Define the mapping 𝑑:𝑋×𝑋→ℂ by (𝑥,) = |𝑥−𝑦|2 + i|𝑥−𝑦|2, for all 𝑥, 𝑦 ∈𝑋.

Then (𝑋, 𝑑) is a complex valued b-metric space with 𝑠 = 2.

Definition1.3 [7]- Let (𝑋, 𝑑) be a complex valued b-metric space and 𝐵⊆𝑋.

(i) 𝑏∈𝐵 is called an interior point of a set 𝐵 whenever there is

0 ≺𝑟∈ℂ such that (𝑏,) ⊆𝐵, Where 𝑁(𝑏,𝑟) ={ 𝑦∈𝑋: 𝑑(𝑏,𝑦)≺𝑟 }.

(ii) A point 𝑥∈𝑋 is called a limit point of 𝐵 whenever for every

0 ≺𝑟∈ℂ, (𝑥,)∩ (𝐵∖{𝑋})≠𝜙.

(iii) A subset 𝐴⊆ 𝑋 is called open whenever each element of 𝐴 is an interior point of 𝐴 whereas a subset 𝐵 ⊆𝑋 is called closed whenever each limit point of 𝐵 belongs to 𝐵. The family 𝐹={𝑁(𝑥,𝑟):𝑥∈𝑋,0 ≺ 𝑟}

is a sub-basis for a topology on 𝑋. We denote this complex topology by 𝜏c.

Indeed, the topology 𝜏c is Hausdorff.

Definition 1.4[7]- Let (𝑋, 𝑑) be a complex valued b-metric space and {𝑥𝑛} be a sequence in X and 𝑥∈𝑋. We say that

(2)

VIII, October 2021

22 (i) The sequence {𝑥𝑛} converges to 𝑥 if

for every 𝑐∈ ℂ with 0 ≺ 𝑐 there is 𝑛0 ∈ ℕ such that for all 𝑛>𝑛0, 𝑑(𝑥𝑛,𝑥) ≺𝑐 . We denote this by lim 𝑛𝑥𝑛 or 𝑥𝑛→𝑥 as 𝑛→ ∞,

(ii) the sequence {𝑥𝑛} is Cauchy sequence if for every 𝑐∈ℂ with 0 ≺𝑐 there is 𝑛0∈ ℕ such that for all 𝑛>𝑛0 , d(𝑥𝑛, 𝑥𝑛+𝑚) ≺𝑐,

(iii) the metric space (𝑋,𝑑) is a complete complex valued metric space

(iv) If every Cauchy sequence is convergent.

Lemma 1.1 [7]- Let (𝑋, 𝑑) be a complex valued b-metric space and {𝑥𝑛} be a sequence in 𝑋. Then {𝑥𝑛} converges to 𝑥 if and only if |(𝑥𝑛,𝑥)| → 0 as 𝑛→ ∞.

Lemma 1.2 [7]- Let (𝑋, 𝑑) be a complex valued b-metric space and {𝑥𝑛} be a sequence in 𝑋. Then {𝑥𝑛} is a Cauchy sequence if and only if |(𝑥𝑛 ,𝑥𝑛+𝑚)|→ 0 as 𝑛,𝑚→ ∞.

Theorem 1.1[6]- Let (𝑋, 𝑑) be a complex valued b-metric space with the coefficient 𝑠≥1 and let 𝜆 be nonnegative real number such that 0≤𝜆<1𝑠2+𝑠. Suppose that 𝑆, 𝑇:𝑋→𝑋 are a pair of self mappings satisfying:

𝑑(𝑆𝑥,𝑇𝑦)≾ 𝜆

𝑚𝑎𝑥{𝑑(𝑥,𝑦),𝑑(𝑥,𝑆𝑥),𝑑(𝑦,𝑇𝑦),𝑑(𝑥,𝑇𝑦),𝑑(𝑦,𝑆𝑥)}, for all 𝑥,𝑦∈𝑋. Then 𝑆, 𝑇 have unique common fixed point in 𝑋.

We recall the following definitions:

Definition 1.5[3]- If 𝑓 and 𝑔 are mappings from a metric space (𝑋,𝑑) into itself, are called compatible on 𝑋, if lim𝑛→∞𝑑(𝑓𝑔𝑥𝑛, 𝑔𝑓𝑥𝑛)=0, whenever {𝑥𝑛} is a

sequence in 𝑋 such that

lim𝑛→∞𝑓𝑥𝑛=lim𝑛→∞𝑔𝑥𝑛=𝑥 , for some point 𝑥∈𝑋.

Definition 1.6 [4]- Let 𝑓 and 𝑔 be two self-maps defined on a set 𝑋, and then 𝑓 and 𝑔 are said to be weakly compatible if they commute at coincidence point.

2. MAIN RESULTS

In this section, we use weak compatibility in complex valued b-metric spaces to establish several common fixed point theorems for four mappings in complex valued b-metric spaces. Our finding generalizes recent results in the literature

of complex valued metric spaces and complex valued b-metric spaces by Azam et al.[1], Rao et al.[7], Mukheimer[5,6], and others.

Theorem 2.1 Let (𝑋,)be a complete complex valued b-metric space with the coefficient 𝑠≥1 and let 𝛼 be nonnegative real number such that 0≤𝛼<1𝑠2+𝑠 . Suppose that 𝐴,,, 𝑇:𝑋→𝑋 are self- mappings satisfying:

(2.1) 𝐴⊂, 𝐵⊂𝑆

(2.2) 𝑑(𝐴𝑥,𝐵𝑦)≾𝛼(𝑚𝑎𝑥(𝑑(𝐴𝑥,𝑆𝑥),𝑑(𝐵𝑦,𝑇𝑦), 𝑑(𝐵𝑦,𝑆𝑥),𝑑(𝐴𝑥,𝑇𝑦),𝑑(𝑆𝑥,𝑇𝑦))) for all 𝑥,𝑦∈𝑋.

(2.3) one of pair (𝐴,) or (𝐵,) is weakly compatible. Then, 𝐴,, and 𝑇 have a unique common fixed point in 𝑋.

Proof. Let 𝑥0 be an arbitrary point in 𝑋.

We define a sequence {𝑦2𝑛} in 𝑋 such that 𝑦2𝑛 = 𝐴𝑥2𝑛 = 𝑇𝑥2𝑛+1

𝑦2𝑛+1 = 𝐵𝑥2𝑛+1 = 𝑆𝑥2𝑛+2; 𝑛= 0, 1, 2,… . (2.1)

Then

(𝑦2𝑛, 𝑦2𝑛+1) =(𝐴𝑥2𝑛 ,𝐵𝑥2𝑛+1)

≾

2.2 (𝑦2𝑛, 𝑦2𝑛+1) ≾ 𝛼(𝑑(𝑦2𝑛, 𝑦2𝑛+1)

Which implies that

|(𝑦2𝑛,𝑦2𝑛+1)| ≤𝛼𝑠|𝑑(𝑦2𝑛,𝑦2𝑛+1)| (2.3)

By using the fact that 0≤𝛼<1𝑠2+𝑠, it is easy to see that 0<𝛼𝑠<12.Thus The inequality (2.3) is true only if

|(𝑦2𝑛,𝑦2𝑛+1)|=0 which implies that 𝑑(𝑦2𝑛, 𝑦2𝑛+1)=0. Hence 𝑦2𝑛= 𝑦2𝑛+1. Continuing this process one easily can show that 𝑦2𝑛=

𝑦2𝑛+1= 𝑦2𝑛+2= 𝑦2𝑛+3=⋯.Hence {𝑦2𝑛} is a Cauchy sequence. Assume that 𝑦2𝑛≠ 𝑦2𝑛−1, for all n in (2.2) and if

Since , 𝑛𝑒𝑞𝑢𝑎𝑙𝑡𝑦 (2.4) is true only if

| (𝑦2𝑛+1, 𝑦2𝑛)|=0, which implies that 𝑑(𝑦2𝑛, 𝑦2𝑛+1)=0. Hence 𝑦2𝑛= 𝑦2𝑛+1. Which

(3)

VIII, October 2021

23 contradicts the fact 𝑦2𝑛≠ 𝑦2𝑛−1, for all n?

Therefore (2.2) becomes

So, we have two cases Case 1: If 𝛽 = 𝛼 then 𝑠𝛽= ¹

𝑎+𝑠 ≤ ¹

2 < 1

From the two cases, we conclude that 𝑠𝛽<1. Taking the modulus of (2.5), we get

Similarly, we obtain

Thus, for any 𝑚>𝑛, 𝑚, ∈𝑁 we get

Using (2.7), we get

Since 𝑠𝛽, 𝛽< 1, we have

Hence {𝑦2𝑛} is a Cauchy sequence in 𝑋.

Since 𝑋 is complete, there exists a point 𝑢∈𝑋 such that 𝑦2𝑛 →𝑢 as 𝑛→∞ and its subsequences 𝐴𝑥2𝑛, 𝑇𝑥2𝑛+1, 𝐵𝑥2𝑛+1 and 𝑆𝑥2𝑛+2 of sequence {𝑦2𝑛} also converges to 𝑢 in 𝑋.

Now we assert that 𝑢=𝑆𝑢. Suppose if not, then there exist 𝑧∈𝑋 such that

Consider

𝑧 =𝑑(𝑢,𝑆𝑢)

Taking modulus of above inequality, we get

|𝑧|= |𝑑(𝑢,𝑆𝑢)| ≾𝑠

|𝑧|= | (𝑢, 𝑆𝑢)| ≾𝑠2 |𝑧| < |𝑧|, where 0 ≤𝛼

< 1𝑠2.

Which is a contradiction? So |𝑧| = 0.

Hence 𝑆𝑢=𝑢.

Now consider

Now since pair (𝐵,) is weak compatible on 𝑋 and 𝐵𝑣=𝑇𝑣 and 𝐵𝑇𝑣=𝑇𝐵𝑣.

Implies that 𝐵𝑢=𝐵𝑇𝑣=𝑇𝐵𝑣=𝑇𝑢.

Consider

(4)

VIII, October 2021

24 𝑑(𝑢,𝐵𝑢)=𝑑(𝐴𝑢,𝐵𝑢)

≾𝛼(𝑚𝑎𝑥(𝑑(𝐴𝑢,𝑆𝑢),𝑑(𝐵𝑢,𝑇𝑢),𝑑(𝐵𝑢,𝑆𝑢),𝑑(𝐴𝑢,𝑇𝑢), 𝑑(𝑆𝑢,𝑇𝑢))).

Implies that (𝑢, 𝑢)≾ 0 so that 𝐵𝑢=𝑢.

Hence 𝐴𝑢=𝐵𝑢=𝑆𝑢=𝑇𝑢=𝑢. Therefore 𝑢 is a common fixed point of 𝐴,, and 𝑇.

In a similar manner, we can prove that 𝑢 is a common fixed point of 𝐴,, and 𝑇 when we take pair (A, S) is weakly compatible.

Now, we prove the uniqueness of 𝑢, suppose that 𝑢′ be another common fixed point of 𝐴,, and 𝑇.

Then

𝑑(𝑢,𝑢′)=𝑑(𝐴𝑢,𝐵𝑢′)

≾𝛼(max

(𝑑(𝐴𝑢,𝑆𝑢),𝑑(𝐵𝑢′,𝑇𝑢′),𝑑(𝐵𝑢′,𝑆𝑢),𝑑(𝐴𝑢,𝑇𝑢′),𝑑(𝑆𝑢, 𝑇𝑢′)))

≾𝛼(max (𝑑(𝑢,𝑢),𝑑(𝑢′,𝑢′),𝑑(𝑢′,𝑢),𝑑(𝑢,𝑢′),𝑑(𝑢,𝑢′)))

≾𝛼 𝑑(𝑢,𝑢′).

This implies that |(𝑢,𝑢′)|<𝛼|𝑑(𝑢,𝑢′)|, which leads us to a contradiction. Hence

| (𝑢,′)| = 0 and that 𝑢=𝑢′. Therefore 𝑢 is unique common fixed point of 𝐴, 𝐵, 𝑆 and 𝑇.

By putting A =B =I (Identity map) in theorem 2.1, we get following corollary:

Corollary 2.1. Let (𝑋, 𝑑) be a complete complex valued b-metric space with the coefficient 𝑠≥1 and let 𝛼 be nonnegative

real number such that .

Suppose that 𝑆, 𝑇:𝑋→𝑋 are self-mappings satisfying:

If the pair (S, T) is weakly compatible, then 𝑆 and 𝑇 have a unique common fixed point in 𝑋.

Corollary 2.2 Let (𝑋,𝑑)be a complete complex valued b-metric space with the coefficient 𝑠≥1 and let 𝛼 be nonnegative

real number such that .

Suppose that 𝑆, 𝑇:𝑋→𝑋 are self-mappings satisfying:

𝑑(𝑆𝑥,𝑇𝑦)≾𝛼(𝑚𝑎𝑥(𝑑(𝑥,𝑆𝑥),𝑑(𝑦,𝑇𝑦),𝑑(𝑦,𝑆𝑥),𝑑(𝑥,𝑇 𝑦),𝑑(𝑥,𝑦))) for all 𝑥,𝑦∈𝑋.

Then 𝑆 and 𝑇 have a unique common fixed point in 𝑋.

3. CONCLUSION

Some common fixed point theorems for weakly compatible mappings in complete complex valued b-metric spaces are proved. Our result generalizes, extends

and improve the results of Mukheimer[5,6] (Corollary 2.2 is the main result of Mukheimer [5]). Also in result of Rao et al [7] (Theoem 3.1 in [7]) both (A, S) and (B, T) are weakly compatible and closedness is required for T(X), but here we take only one of the pair is weakly compatible and condition of closedness of the mapping T(X) is relaxed, still the results follows.

REFERENCES

1. Azam, A., Fisher, B. and Khan, M., Common fixed point theorems in complex valued metric spaces, Num. Func. Anal. Opt. 32 (2011), 243- 253.

2. Bakhtin, I.A., The contraction principle in quasimetric spaces, Funct. Anal. 30 (1989), 26-37.

3. Jungck, G., Compatible mappings and common fixed points, Int. J. Math. Sci. 9(4), (1986), 771-779.

4. Jungck, G., and Rhoades, B.E., Fixed point for set valued functions without continuity, I.

J. Pure Appl. Math.,29(3)(1998), 227-238.

5. Mukheimer, A.A., Common fixed point Theorems for a pair of mappings in complex valued b-metric spaces, Adv. Fixed Point Theory, Vol. 4, No.3 (2014), 344-354.

6. Mukheimer, A.A. Some common fixed point theorems in complex valued b- metric spaces, Sci. World J. 2014 (2014), Article ID 587825.

7. Rao, K., Swamy, P. and Prasad, J., A Common fixed point theorem in complex valued b-metric spaces, Bull. Math. Stat. Res.

1 (2013), 1-8.

8. Roshan, J., Parvaneh, V., Sedghi, S., Shobkolaei, N. and Shatanawi, W., Common fixed points of almost Generalized theorems (𝜓,𝜑)𝑠-contractive mappings in ordered b- metric spacces, Fixed Point Theory Appl.

2013 (2013), Article ID 159.

9. Rouzkard, F. and Imdad, M., Some common fixed point on complex valued metric spaces, Comp. Math. Appls. 64(2012), 1866-1874.

10. Sintunavarat, W. and Kumam, P., Generalized common fixed point theorems in complex valued metric spaces and applications, J. Inequalities Appl. 2012(2012), 11.