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Research Article

Open Access Peer-reviewed

Mumtaz Ali^{ }, Muhammad Arshad

Received April 11, 2017; Revised May 18, 2017; Accepted July 13, 2017

In this paper we study some fixed point results in pseudocompact Tichnovo space using Edelstein type contractive conditions. The results presented in this paper include the generalization of some fixed point theorems established by Fisher and Pathak.

Fixed point theory is a fascinating topic for research in modern mathematics. In this direction the Banach contraction mapping theorem of 1922 popularly known as Banach contraction mapping principle is a rewarding result in analysis and fixed point theory. It has widespread applications in both pure and applied mathematics. The well known Banach ^{ 2} contraction mapping principle states that if "*X* is a complete metric space and is a contraction mapping of *X* into itself then T has unique fixed point in *X*". This celebrated principle has been generalized by several authors. In 1961, Edelstein ^{ 6} introduced the concept of contractive mapping defined on compact metric spaces which is generalizetion of Banach contraction mapping principle. According to Edelstein "if *T** *is a continuous mapping of a compact metric space *X* into itself satisfying for all then *T* has unique fixed point in X". Edelstein’s contractive mapping theorem has been extensively generalized and improved by several mathematicians for fixed points in several different ways viz, Bailey ^{ 1}, Chatterjee ^{ 4}, Ciric ^{ 5}, Iseki ^{ 9}, Kannan and Sharma ^{ 12}, Pachpatte ^{ 15}, Popa ^{ 17}, Sahu ^{ 18}, Sharma and Sahu ^{ 19} and Soni ^{ 20} on complete and compact metric spaces. The concept of fixed point results for contractive mappings in pseudocompact Tichonov spaces was introduced by Harinath ^{ 10}. Later on, Jain and Dixit ^{ 11} and Liu ^{ 13} also established fixed point results for several classes of contractive type mappings in pseudocompact Tichonov spaces. Inspired by the ideas of Fisher ^{ 7} and Pathak ^{ 16} the aim of the present paper is to prove the existence and uniqueness of fixed point results for self mapping in the setting of pseudocompact Tichonov spaces satisfying contractive type conditions.

The following fixed point theorems were proved in ^{ 7} and ^{ 16}.

**Theorem**** ****1.1. **^{ 7} If *T* is a mapping of the complete metric space *X* into itself satisfying the inequality

for all in where and then has a fixed point.

**Theorem 1.2.** ^{ 16} Let *P* be a Pseudocompact Tichonov space and be a non-negative real valued continuous function over ( is Tichnovo but need not be pseudocompact). Suppose μ also satisfies

(*i*)

If *S* and *T* are two continuos self maps of *P* satisfying

(*ii*) *ST* = *TS* and

(*ii**i*)

for all distinct with where Then *S* and *T* have a unique common fixed point in *P* which is unique whenever

**Theorem 2.1.** Let *P* be a pseudocompact Tichonov space and *d* be a nonnegative real valued continuous function over satisfying the conditions:

(*i*)

(*ii*)

If *S* and *T* are continuous self maps of *P* satisfying

(2.1) |

(*iii*)

(2.2) |

for all distinct with and then *S* and *T* have a common fixed point in *X*, which is unique whenever

**Proof. **Define a function by for all . Clearly is continuous being the composite of three continuous functions and* ** *Since is compact, every real valued continuous function over is bounded and attain its bounds. Thus there exists a point such that We now affirm that is a fixed point for If not, let us suppose that then using (2.2) we have

or

or

or

or

which is contradiction because Hence is a fixed point for that is Using (2.1), we have

(2.3) |

Now we shall prove that If possible, let then by using (2.2) and (2.3), we have

which is a contradiction because Hence is a fixed point of i.e.

**Uniqueness:** To prove the uniqueness of *v*, if possible, let *w* be another fixed point for *S* and *T* i.e. and Then, using (2.2), we have

leading to a contradiction because which proves that is unique and this establishes the theorem.

Theorem (2.1) yields the following corollary.

**Corollary 2.2** Let *P* be a pseudocompact Tichonov space and *d* be a non-negative real valued continuous function over satisfying:

(*i*)

(*ii*)

and is a continuous map satisfying the inequality

for all distinct with and then *S* has a fixed point in *P* which is unique.

**Proof** If we take *T* = *S*, then theorem (2.1) shows that S has a unique fixed point in

**Theorem 2.3.** Let *P* be a pseudocompact Tichonov space and d be a non-negative real valued continuous function over satisfy the conditions:

(*i*)

(*ii*)

If *S *and *T* are continuous self maps of *P *satisfying

(2.4) |

(2.5) |

for all and are non negative real numbers such that then *S* and *T* have a common fixed point in *X*, which is unique whenever,

**Proof**. Define a function by for all Clearly is continuous being the composite of three continuous functions *S*, *T* and* **d*. Since *X* is compact, every real valued continuous function over *X* is bounded and attain its bounds. Thus there exists a point* ** *such that We now affirm that *v* is a fixed point for *S*. If not, let us suppose that then using (2.5) we have

Case(*i*)-If

Then

or

or

or

or

where

But which is a contradiction.

Case-(*ii*) If

Then

or

or

or

where

But which is a contradiction.

Case-(*iii*) If

Then

or

or

where

But which is a contradiction.

Here

Hence, is a fixed point of and so Using (2.4), we have

(2.6) |

Now we shall prove that If possible, let then by using (2.5) and (2.6) we have

which is contradiction because Hence is a fixed point of that is

**Uniqueness:** Let, if possible be another fixed pioint of *S* and *T* that is and then using (2.5), we get

This is a contradiction because Hence is unique fixed point of and

Now we give an example to support our results.

**Remark:** If we in Theorem (2.3) then we get Theorem (2.1).

**Example**** ****2.1.** Let and let is the discrete topology on Define as

and let *d* be a non-negative real valued continuous function over such that Then, it is clear that is a pseudocompact Tichonov space and and are continuous self maps of which satisfy all the conditions of Theorem (2.1) and Theorem (2.3) with 1 as the only common fixed point by setting for

[1] | Bailey D.F. some theorems on contractive mappings, J. London Math. Soc.41, 101-106 (1966). | ||

In article | View Article | ||

[2] | Banach S. Sur les operation dans les ensembles abstraits etleur application aux equations integrals Fund. Math. 3: 133-181 (1922). | ||

In article | View Article | ||

[3] | Bhardwaj et al:some fixed point theorems in compact metric spaces Int. Journal of math.analysis, Vol. 2.2. 2008, no. 11, 543-550 | ||

In article | |||

[4] | Chatterjee H. Remarks on some theorems of K. Iseki Indian J. pure appl. Math., 10(2): 158-160(1979). | ||

In article | |||

[5] | Ljubomir Ciric Fixed-point mappings on compact metric spaces Publications De l.institut mathematique Nouvelle series. tome 30(44): 29-31(1981). | ||

In article | View Article | ||

[6] | Edelstein M. An extension of Banach’s contraction principle Pro. Amer. Math. Soc.12: 7-10 (1961). | ||

In article | View Article | ||

[7] | Fisher B. Fixed point and constant mappings on metric spaces, Atti Accad, Naz., Lincci Rend. Ci. Sci. Mat. Natur. 61, 329-332 (1976). | ||

In article | |||

[8] | Fisher B. On three fixed point mappings for compact metric spaces Indian J. Pure and Appl. Math. 8: 479-481 (1977). | ||

In article | |||

[9] | Iseki K. A simple application of Reich’s fixed point theorem Maths. Sem. Notes Kobe University. 5(1), 75-80. | ||

In article | View Article | ||

[10] | Harinath K.S. A chain of results on fixed points Indian J. Pure and Appl. Math., 184-190, 10(1979). | ||

In article | |||

[11] | Jain R.K. and Dixit S.P. Some results on fixed points in pseudocompact Tichonov spaces Indian J. Pure and Appl. Math., 445-458, 1591984). | ||

In article | |||

[12] | Kannan S. and Sharma P.L. Fixed point mappings for compact metric spaces. The Math. Educ. Vol. (XXIV), No.1 (1990). | ||

In article | |||

[13] | Ze-Qing Liu on pseudocompact tichonov spaces Soochow J. Math. Vol. 20, 393-399 (1994). | ||

In article | |||

[14] | Namdeo et al. Related fixed point theorems on two complete and compact metric spaces Internat.J.Math.Sci.Vol.21 No.3, 559-564 (1998). | ||

In article | View Article | ||

[15] | Pachpatte B.G. On certain Fixed point mapping in metric space, Journal of M.A.C.T Vol. 13, 59-63 (1980). | ||

In article | |||

[16] | Pathak H.K. Some theorems on fixed points in pseudocompact Tichonov spaces Indian J. Pure and Appl. Math., 180-186, 17(2), 1986. | ||

In article | View Article | ||

[17] | Popa V.A general fixed point theorem for weakly compatible mappings in compact metric spaces Turk. J. Math. 25, 465-474 (2001). | ||

In article | View Article | ||

[18] | Sahu M.K. Some fixed point theorems on compact metric space International review of pure and applied mathematics, Vol.2, No.2, 151-154 (2006). | ||

In article | |||

[19] | Sharma P.L. and Sahu M.K A unique fixed point theorem in complete metric space Acta Ciencia Indica Vol. XVII, M, 4,685 (1991). | ||

In article | |||

[20] | Soni G. K. Fixed point theorem in compact metric space the mathematical education Vol.(XXVII), No.4(1993).ration dans les ensembles abstraits etleur application aux equations integrals Fund. Math. 3: 133-18 (1922). | ||

In article | |||

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Mumtaz Ali, Muhammad Arshad. Generalization of Fixed Point Theorems in Pseudocompact Tichonov Space. *Turkish Journal of Analysis and Number Theory*. Vol. 5, No. 5, 2017, pp 159-164. http://pubs.sciepub.com/tjant/5/5/3

Ali, Mumtaz, and Muhammad Arshad. "Generalization of Fixed Point Theorems in Pseudocompact Tichonov Space." *Turkish Journal of Analysis and Number Theory* 5.5 (2017): 159-164.

Ali, M. , & Arshad, M. (2017). Generalization of Fixed Point Theorems in Pseudocompact Tichonov Space. *Turkish Journal of Analysis and Number Theory*, *5*(5), 159-164.

Ali, Mumtaz, and Muhammad Arshad. "Generalization of Fixed Point Theorems in Pseudocompact Tichonov Space." *Turkish Journal of Analysis and Number Theory* 5, no. 5 (2017): 159-164.

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[1] | Bailey D.F. some theorems on contractive mappings, J. London Math. Soc.41, 101-106 (1966). | ||

In article | View Article | ||

[2] | Banach S. Sur les operation dans les ensembles abstraits etleur application aux equations integrals Fund. Math. 3: 133-181 (1922). | ||

In article | View Article | ||

[3] | Bhardwaj et al:some fixed point theorems in compact metric spaces Int. Journal of math.analysis, Vol. 2.2. 2008, no. 11, 543-550 | ||

In article | |||

[4] | Chatterjee H. Remarks on some theorems of K. Iseki Indian J. pure appl. Math., 10(2): 158-160(1979). | ||

In article | |||

[5] | Ljubomir Ciric Fixed-point mappings on compact metric spaces Publications De l.institut mathematique Nouvelle series. tome 30(44): 29-31(1981). | ||

In article | View Article | ||

[6] | Edelstein M. An extension of Banach’s contraction principle Pro. Amer. Math. Soc.12: 7-10 (1961). | ||

In article | View Article | ||

[7] | Fisher B. Fixed point and constant mappings on metric spaces, Atti Accad, Naz., Lincci Rend. Ci. Sci. Mat. Natur. 61, 329-332 (1976). | ||

In article | |||

[8] | Fisher B. On three fixed point mappings for compact metric spaces Indian J. Pure and Appl. Math. 8: 479-481 (1977). | ||

In article | |||

[9] | Iseki K. A simple application of Reich’s fixed point theorem Maths. Sem. Notes Kobe University. 5(1), 75-80. | ||

In article | View Article | ||

[10] | Harinath K.S. A chain of results on fixed points Indian J. Pure and Appl. Math., 184-190, 10(1979). | ||

In article | |||

[11] | Jain R.K. and Dixit S.P. Some results on fixed points in pseudocompact Tichonov spaces Indian J. Pure and Appl. Math., 445-458, 1591984). | ||

In article | |||

[12] | Kannan S. and Sharma P.L. Fixed point mappings for compact metric spaces. The Math. Educ. Vol. (XXIV), No.1 (1990). | ||

In article | |||

[13] | Ze-Qing Liu on pseudocompact tichonov spaces Soochow J. Math. Vol. 20, 393-399 (1994). | ||

In article | |||

[14] | Namdeo et al. Related fixed point theorems on two complete and compact metric spaces Internat.J.Math.Sci.Vol.21 No.3, 559-564 (1998). | ||

In article | View Article | ||

[15] | Pachpatte B.G. On certain Fixed point mapping in metric space, Journal of M.A.C.T Vol. 13, 59-63 (1980). | ||

In article | |||

[16] | Pathak H.K. Some theorems on fixed points in pseudocompact Tichonov spaces Indian J. Pure and Appl. Math., 180-186, 17(2), 1986. | ||

In article | View Article | ||

[17] | Popa V.A general fixed point theorem for weakly compatible mappings in compact metric spaces Turk. J. Math. 25, 465-474 (2001). | ||

In article | View Article | ||

[18] | Sahu M.K. Some fixed point theorems on compact metric space International review of pure and applied mathematics, Vol.2, No.2, 151-154 (2006). | ||

In article | |||

[19] | Sharma P.L. and Sahu M.K A unique fixed point theorem in complete metric space Acta Ciencia Indica Vol. XVII, M, 4,685 (1991). | ||

In article | |||

[20] | Soni G. K. Fixed point theorem in compact metric space the mathematical education Vol.(XXVII), No.4(1993).ration dans les ensembles abstraits etleur application aux equations integrals Fund. Math. 3: 133-18 (1922). | ||

In article | |||