COMMON FIXED POINT THEOREMS IN COMPLEX PARTIAL b-metric SPACE

2010 Mathematics Subject Classification: admitting: Primary 47H10; Secondary 54H25. Keywords: Complex valued b-metric space, complex partial b-metric space, fixed point.

COMMON FIXED POINT THEOREMS IN COMPLEX

PARTIAL b-METRIC SPACE

A. LEEMA MARIA PRAKASAM and M. GUNASEELAN

PG and Research Department of Mathematics Holy Cross College (Autonomous)

Trichy-620002, India

E-mail: [email protected] Department of Mathematics

Sri Sankara Arts and Science College (Autonomous) Enathur, Kanchipuram-631561, India

E-mail: [email protected]

Abstract

In this paper, we obtain existence and unique common fixed point theorem for four self-maps in complex partial b-metric space.

1. Introduction

The fixed point theory is one of the most important tools in many branches of science, computer science, engineering and the development of non-linear analysis. Backhtin [1] introduced the concept of b-metric spaces in 1989. Azam et al. [2] introduced the concept of complex valued metric spaces. Rao et.al [3] introduced complex valued b-metric space. P. Dhivya and M. Marudai [4] introduced the concept of complex valued partial metric space and extended the common fixed point theorems under the contraction condition of rational expression. In 2019, M. Gunaseelan [6] introduced the notion of complex valued partial b-metric space and proved existence and uniqueness of fixed point theorem.

In this paper, we prove an existence and uniqueness of common fixed point theorems using weakly compatible in complex partial b-metric space.

2. Preliminaries

Let  be the set of complex numbers and e₁,e₂ . Define a partial order   as follows:

2 1 e

e  iff Re

 

e₁  Re

 

e₂ ,Im

 

e₁  Im

 

e₂.

It follows that e₁  e₂ if one of the following conditions is satisfied. i. Re

 

e₁  Re

 

e₂, Im

 

e₁  Im

 

e₂.

ii. Re

 

e₁  Re

 

e₂ , Im

 

e₁  Im

 

e₂ . iii. Re

 

e₁  Re

 

e₂ ,Im

 

e₁  Im

 

e₂. iv. Re

 

e₁  Re

 

e₂ ,Im

 

e₁  Im

 

e₂.

In particular, we write e₁e₂ if e₁  e₂ and one of i, ii, iii is satisfied and we write e₁  e₂ if only iii is satisfied Notice that

(a) If 0  e₁e₂, then e₁  e₂ . (b) If e₁  e₂, and e₂  e₃ then e₁  e₃.

(c) If f , g  and f  g then fh  gh, for all h.

Definition 2.1 [5]. A complex partial metric on a non-void set R is a

mapping c : RR   satisfying the following conditions: (i) 0  _c



u,u



 _c



u,v



(small self distance)

(ii) _c



u,v



 _c



v,u



(symmetry)

(iii) _c



u,u



 _c



u,v



 _c



v,v



iff uv (equality) (iv) _c



u,v



 _c



u,w



_c



w,v



_c



w,w



(triangularity)

for all u,v,wR. A complex partial metric spaces is a pair



R, _c



such that R is a non-void set and _c is a complex partial metric on R.

Definition 2.2 [6]. A complex partial b-metric on a non-void set R is a

(i) 0  _cb



x, x



 _cb



x, y



(small self distance) (ii) _cb



x, y



 _cb



y, x



(symmetry)

(iii) _cb



x,x



 _cb



x, y



 _cb



y, z



iff x  y (equality) (iv) _cb



x, y



 s



_cb



x, z



_c



z, y



_cb



z, z



(triangularity)

for all x, y, zR. A complex partial b-metric space is a pair



R,_cb



such that R is a non-void set and _cb is a complex partial b-metric on. The number s is called the coefficient of



R, _cb



.

Remark 2.1 [6]. In a complex partial b-metric space



R, _cb



if x, yR and _cb



x, y



 0, then x  y. But the converse may not be true.

Remark 2.2 [6]. It is clear that every complex partial metric is a complex

partial b-metric space with coefficient s 1 and every complex valued b-metric is a complex partial b-b-metric space with the same coefficient and zero self-distance. However, the converse of the fact need not hold.

Definition 2.3 [6]. Let



R, cb



be a complex partial b-metric space with coefficient s. Let

 

xn be any sequence in R and x  R. Then

1. The sequence

 

xn is said to be convergent w.r.to cb and converges to x, if cb



xn x



cb



x x



nlim ,   ,

2. The sequence

 

xn is said to be Cauchy sequence in



R, cb



if



n m



cb

nlim x ,x exists and is finite.

3. Every Cauchy sequence

 

xn in R there exists x  R such that



,



lim



,





,



. lim x x cb xn x cb x x n m n cb n     Then



R,cb



is said to be a complete complex partial b-metric space.

4. A mapping p: R  R is said to be continuous at x₀ R if for every ,

0



 there exists   0, such that



B_cb



₀, 







B_cb



₀, 



. P

Definition 2.4 [3]. Let R be a non-void set and let s 1 be a given real number. A mapping  : RR   is said to be a complex valued b-metric if the following conditions are satisfied:

1. 0  



x, y



and 



x, y



 0  x  y,x, yR. 2. 



x, y



 



y, x



, x, yR.

3. 



x, y



 s







x, z







z, y





,x, y, zR.

The pair



x, 



is called a complex valued b-metric space.

Definition 2.5 [6]. Let V and W be two self maps defined on a set R, then

V and W are said to be weakly compatible if they commute at coincide points.

3. Main Results 3.1. Common Fixed Point Theorem

We prove fixed point theorem for four self-maps in complex partial b-metric space.

Theorem 3.1. Let



R, cb



be a complete complex partial b-metric space

with coefficient s 1 and let C, D, I and J are four self maps of R such that

 

R C

 

R J  and I

 

R  D

 

R satisfying (i)





p



z w



s Jw Iz cb ,  ₂ ,  if s 1 and  



0,1



,z,wR. where p



z w



 

_cbCz Dw



_cb



Cz Iz



_cb



Dw Jw





_cb



Dw, Iz



2 1 , , , , , , max ,     











_



_





. , 1 , , , , Dw Cz Jw Dw Iz Cz Jw Cz cb cb cb cb  _   

(ii) Suppose that the pairs



C, I



and



D, J



are weakly compatible, then I

D

C, , and J have a unique common fixed point.

Proof. Let z₀ R be arbitrary from the condition J

 

R  C

 

R and

 

R D

 

R,

I  there exists z₁, z₂ such that w₀  Dz₁  Iz₀ and .

1 2 1 Dz Jz

We can construct successively the sequences

 

w_n and

 

z_n in R as follows: n n n DZ Iz w₂  ₂ ₁  ₂ and w2n1  Cz2n2  Jz2n1 (3.1)

using equation (3.1) in (i), we get



₂ , _{2 1}



 



₂ , _{2 1}



 ₂



₂ , _{2 1}



_{cb n n cb n n} pz _n z _n s Jz Iz w w (3.2) where



z_2n,z_2n1



 max



_cb



Cz_2n, Dz_2n1



,_cb



Cz_2n, Iz_2n



, _cb



Dz_2n1,Jz_2n1



P















_





_





1 2 2 1 2 1 2 2 2 1 2 2 2 1 2 , , , ₁, _, , 2 1        _   n n cb n n cb n n cb n n cb n n cb Dz Iz Cz Jz Cz Jz _{Cz Dz}Dz Jz





,



,



,



,



,



, max  _{2 1 2}  _{2 1 2}  _{2 2 1}  _cb w _n w _{n cb} w _n w _{n cb} w _n w _n















_





_





n n cb n n cb n n cb n n cb n n cb w w w w w w _w w_w w 2 1 2 1 2 2 2 1 2 1 2 1 2 2 2 , , , ₁ , _, , 2 1       _    



z_2n, z_2n1



 max



_cb



w_2n1,w_2n



, _cb



w_2n,w_2n1



. P (3.3) Substitute (3.3) in equation (3.2)



₂ , _{2 1}



 ₂







₂ , _{2 1}





_{cb n n cb} w _n w _n s w w



,



0 1 ₂ _{2 2 1}       _   n n cb w w s

which is a contradiction. Since  



0,1



and s 1.

We conclude that _cb



_2n, _{2n 1}



₂



_cb



w_{2n 1},w_2n





. s w w _    _  Similarly we get



_{2 1}, _{2 2}



 ₂







₂ , _{2 1}





. _{cb n n cb} w _n w _n s w w It follows that



, ₁



₂





₁,



₂



w₀,w₁



. s w w s w w _cb n n n cb n n cb               _{ } 

Which implies



,



₂







, 1







1,







1, 1





       _{cb n m} s _cb w_n w_{n cb} w_n w_{m cb} w_n w_n s w w



cb



wn wn



cb



wn wm





s s ₂  , _1  _1,       



cb



n n







cb



wn wn



cb



wn wm





s s w w s s ₂ , 1 2 ₂  1, 2  2,                





2 2





2_   ,        _cb w_n w_n s s



cb



n n







cb



n n





cb



wn wm





s s w w s s w w s s ₂ , 1 2 ₂ 1, 2 2 ₂  2,                         



,





0, 1



. 1 2 2 1 0 2 w w s _s w w s s cb n cb n                   













     _{ }                  n m i cb n i i cb n m n m _{w w} s s w w s s 1 1 0 1 2 1 0 2 , , 













     _{ }          n m i cb n i n i m n cb w w s s w w 1 1 0 1 2 1 _, ,







           n m i cb t t _{w w} s s 1 1 0 2 ,







           n m i cb t w w s 1 1 0 2 ,











,



. 1 s w0 w1 s cb n      Hence



₁





_



_



₀, ₁



1 , w w s s w w_{n n} n _cb cb  _ _   _ as n.

Thus

 

w_n is a Cauchy sequence.

Since R is complete, so there exist some u R such that w_n u, as

  n and



,



 lim 



,



 lim 



,



 0.      cb n n cb n n n cb u u u w w w

For its sequence we have Dz2_n1  u, Iz2_n u,Cz2_n1 u and .

2 u

Jz _n 

Since, J

 

R  C

 

R, there exist a point v R such that u  Cv. Suppose that _cb



Iv,u



 0. Then





cb



n



cb



n



cb



n n



cb Iv,u  s Iv, Jz2 s Jz2 , z  Jz2 , Jz2 



v z n



P s s ₂ , 2 



v, z_2n



max



_cb



Cv, Dz_2n



, _cb



Cv, Iv



, _cb



DZ_2n, Jz_2n



, P    















_





_





n cb cb n n cb n cb n cb Dz Iv Cv Jz Dz Jz_{Cv Dz} Cv Iv 2 2 2 2 2 , , , ₁ , _, , 2 1         As



















,





,





, 2 1 , , , , , , max u u u Iv u u u Iv u u n    cb cb cb cb cb











u u





u u u u cb cb cb , 1 , ,     



u Iv



cb ,  



,



₂ s



u, Iv



. s u Iv cb cb     Since, s 1 and 



0,1





,

 

1



 0 _cb Iv u s which is a contradiction.

Hence Cv  Iv  u.

Since I

 

R  D



 

R



, there exists a point y R such that u  Dy. Suppose that _cb



u, Jy



 0. Then









P



v y



s Jy Iv Jy u _cb cb ,   ,  ₂ , 



v, y



max





Cv, Dy



,



Cv, Iv



,



Dy,Jy



, P  _cb _cb _cb

















_



_





Dy v C Jy Dy Iv Cv Jy Cv Iv Dy cb cb cb cb cb , , , ₁ , _, , 2 1         As n 



u,v



max





u,u



,



u,u



,



u,Jy



, P  _cb _cb _cb

















_



_





u u Jy u u u Jy u u y cb cb cb cb cb , , , ₁, _, , 2 1        



u Jy



s2 cb ,  







u jy



s Jy u _cb cb ,  ₂  , 



,

 

1 2



 0 _cbu Jy s which is a contradiction. Therefore Jy  Dy  u. Hence Cv  Iv  Jy  Dy  u.

Since C and I are weakly compatible maps then ICv  CIv. Therefore Iu  Cu.

Now we claim that u is a fixed point of I if u  u we have









P



u y



s Jy Iu u Iu _cb cb ,   ,  ₂ , 



u, y



max





Cu, Dy



,



Cu, Iu



,



Dy, Jy



, P  _cb _cb _cb

















_



_





Dy u C Jy Dy Iu Cu Jy Cu Iu Dy cb cb cb cb cb , , , ₁ , _, , 2 1        





,



,



,



,



,



, max _cb Iu u _cb Iu Iu _cb u u 

















_



_





u Iu u u Iu Iu u Iu Iu u cb cb cb cb cb , , , ₁ , _, , 2 1        



Iu u



cb ,  







Iu u



s u Iu cb cb ,  ₂  , 



1 s2



_cb



Iu,u



 0 which is a contradiction. . u Iu  Hence Iu  cu  u.

Similarly, D and J weakly compatible maps, we have Du  Ju. Now, we claim that u is a fixed point of J. suppose that Ju  u.

Then we have









P



u u



s Ju Iu Ju u _cb cb ,   ,  ₂ , 



u,u



max





Cu, Du



,



Cu, Iu



,



Du,Ju



, P  _cb _cb _cb

















_



_





Du u C Ju Du Iu Cu Ju Cu Iu Du cb cb cb cb cb , , , ₁ , _, , 2 1        





,



,



,



,



,



, max _cbu Ju _cb u u _cb Ju Ju 

















_



_





Ju u Ju Ju u u Ju u u Ju cb cb cb cb cb , , , ₁, _, , 2 1        



u Ju



cb ,  







u Ju



s Ju u cb cb ,  ₂  , 



1 s2



cb



u,Ju



 0 which is contradiction.

Hence Du  Ju  u.

Hence Cu  Iu  Du  Ju  u. And it follows that u is a common fixed point of C, D, I and J. Next we claim that the uniqueness of u.

Let u and v are distinct common fixed point of C, D, J and I. Suppose not









P



u v



s Ju Iu v u cb cb ,   ,  ₂ , 



u,v



max





Cu, Dv



,



Cu, Iv



,



DvS, Jv



, P  _cb _cb _cb

















_



_





Dv u C Jv Dv Du Cu Jv Cu Iu Dv cb cb cb cb cb , , , ₁ , _, , 2 1        





,



,



,



,



,



, max _cb u v _cb u u _cb v v 

















_



_





v u v v u u v u u v cb cb cb cb cb , , , ₁ , _, , 2 1        



u v



cb ,  







u v



s v u _cb cb ,  ₂  , 



1 s2



_cb



u,v



 0.

Therefore u v. Hence u is the unique common fixed point of C, D, I and J.

Corollary 3.2. Let



R,cb



be a complete complex partial b-matric space with coefficient s 1 and let C, D, I and J are four self maps of R such that

 

R D

 

R J  and I

 

R  D

 

R satisfying (i)





P



z w



s Jw Iz cb ,  ₂ ,  if s 1 and  



0,1



, z,wR



z,w



max





Cz, Dw



,



Cz, Iz



,



Dw, Jw



.

P  _cb _cb _cb

(ii) Suppose that the pairs



C, I



and



D, I



are weakly compatible. Then I

D

C, , and J have a unique common fixed point.

Corollary 3.3. Let



R,_cb



be a complete complex partial b-matric space with coefficient s 1 and let C, D, I and J are four self maps of R such that

 

R D

 

R J  and I

 

R  D

 

R satisfying (i)



Iz, Jw



a_{1 cb}



Cz, Dw



a_{2 cb}



Cz, Iz



a_{3 cb}



Dw, Iw



a_{4 cb}



Cz, Iw



. cb          Where z,wR and a₁, a₂, a₃,a₄  0, a₁ a₂ a₃ 2a₄ 1

(ii) The pairs



C, I



and



D,J



are weakly compatible. Then C, D, I and J have a unique common points.

Example 3.4. Let



0,2



2 1 _        R and _cb



u,v

 

 max

  

u,v 2 1iv



where u,vR then



R, cb



be a complete complex partial b-matric space.

Let C, D, I and J : R  R be defined by

 

 





            1 , 0 if 2 1 2 , 1 2 1 if 1 r r r C 

 

 





            1 , 0 if 4 1 2 , 1 2 1 if 1 r r r D 

 









                  1 , 0 2 1 if 1 1 , 0 if 3 4if 1 1  r r r r r I

 









. 2 , 1 2 1 if 2 2 1 , 0 if 4 5if 1 1 2                     r r r r r r M Then

 

 

 





 



1, 2



4 5 1 , 0 3 4 , 2 3 , 4 1 , 1 , 2 1 , 1                               D R I R M R R C and (i) J

 

R  D

 

R and I

 

R  D

 

R

(ii) For all z,wR and s  0, 



0,1



are can verify that





P



z w



s Iw Iz cb ,  ₂ , 



u,v



max





Cz, Dw



,



Cz, Iz



,



Dw, Jw



, P  _cb _cb _cb

















_



_





Dw Cz Jw Dw Iz Cz w J Cz Iz Dw cb cb cb cb cb , , , ₁ , _, , 2 1        

(iii) The pairs



C, I



and



D,J



are weakly compatible. Hence by theorem 3.1, 1 is a unique common fixed point of C, D, I and J.

References

[1] I. A. Bakthin, The Contraction mappings principle in quasi-metric spaces, Functional Analysis 30 (1989), 26-37.

[2] A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optim. 32(3) (2011), 243-253.

[3] K. P. R. Rao, P. R. Swamy and J. R. Prasad, A common fixed point theorem in complex valued b-metric spaces, Bulletin of Mathematics and Statistics Research 1(1), 2013. [4] P. Dhivya and M. Marudai, Common fixed point theorems for mappings satisfying a

contractive condition of rational expression on an ordered complex partial metric space, Cogent Mathematics, 4, 2017:1389622.

[5] E. Karapinar and B. Samet, Generalised  contractive type mappings and related fixed point theorems with applications, Abstr. Appl. Anal. (2012), 17 pages.

[6] M. Gunaseelan, Generalized fixed point theorems on complex partial b-Metric spaces, International Journal of Research and Analytical Reviews 6(2) (2019).

[7] Anil Kumar Dubey, Shweta Bibay and R. P. Dubey, Fixed point theorems for Rational contractions in complex valued b-metric spaces, International Journal of Advances in Mathematics 3 (2018), 25-53.