**Turkish Journal of Analysis and Number Theory**

##
New Approach of *F*-Contraction Involving Fixed Point on a Closed Ball

^{1}Department of Mathematics, International Islamic University, H-10, Islam-abad, Pakistan

^{2}Department of Mathematical Sciences, Lahore Leads University, Lahore, Pakistan

### Abstract

The article is written with a view to introducing the new idea of *F*-contraction on a closed ball and have new theorems in a complete metric space. That is why this outcome becomes useful for the contraction of the mapping on a closed ball instead of the whole space. At the same time, some comparative examples are constructed which establish the superiority of our results. It can be stated that the results that have come into being give proof of extension as well as substantial generalizations and improvements of several well known results in the existing comparable literature.

**Keywords:** metric space, fixed point, F contraction, closed ball

Received August 24, 2016; Revised November 20, 2016; Accepted November 28, 2016

**Copyright**© 2016 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Aftab Hussain. New Approach of
*F*-Contraction Involving Fixed Point on a Closed Ball.*Turkish Journal of Analysis and Number Theory*. Vol. 4, No. 6, 2016, pp 159-163. http://pubs.sciepub.com/tjant/4/6/2

- Hussain, Aftab. "New Approach of
*F*-Contraction Involving Fixed Point on a Closed Ball."*Turkish Journal of Analysis and Number Theory*4.6 (2016): 159-163.

- Hussain, A. (2016). New Approach of
*F*-Contraction Involving Fixed Point on a Closed Ball.*Turkish Journal of Analysis and Number Theory*,*4*(6), 159-163.

- Hussain, Aftab. "New Approach of
*F*-Contraction Involving Fixed Point on a Closed Ball."*Turkish Journal of Analysis and Number Theory*4, no. 6 (2016): 159-163.

Import into BibTeX | Import into EndNote | Import into RefMan | Import into RefWorks |

### 1. Introduction

Shoaib et al. ^{[40]} proved significant results concerning the existence of fixed points of the dominated self mappings satisfying some contractive conditions on a closed ball in a 0-complete quasi-partial metric space. Other results on closed ball can be seen in ^{[5, 6, 7, 8, 25]}. Over the years, Fixed Point Theory has been generalized in different ways by several mathematicians (see [2,3,4,9-15,17-21,23-27]).

For and is a closed ball in

**Definition 1** ^{[18]} *Let** ** **be a metric space. Let** ** **be self mappings and** ** **be two functions.** ** **is called** **-**continuous if for** **given** ** **and sequence** ** **with*

In 2012, Wardowski ^{[42]} introduce a concept of *F*-contraction as follows:

**Definition 2** ^{[33]}* Let** ** **be a metric space. A self mapping** ** **is said to be an** ** **contraction if there exists** ** **such that*

(1.1) |

*where ** **is a mapping satisfying the following conditions*:

*(F1) ** **is strictly increasing, i.e. for all** ** **such that** ** *

*(F2) For each sequence** ** **of positive numbers,** ** **if and only if** ** *

*(F3) There exists** ** **such that** *

We denote by , the set of all functions satisfying the conditions (F1)-(F3).

Furthermore, it was done by different investigators see [1,3,15,16,23,27,28,29,31,32,33,34,37,38,39].

Hussain et al. ^{[18]} introduced the following family of new functions.

Let denotes the set of all functions satisfying:

(*G*) for all with there exists such that

### 2. Banach Fixed Point Theorem for *F*-Contraction on a Closed Ball

In this section, we introduce Banach fixed point theorem for modified *F*-contraction on a closed ball in complete metric spaces.

Now we state our main result.

**Theorem 3*** Let T be a continuous self mapping in a complete metric space** ** **and** ** **be an arbitrary point in** ** **Assume that** ** **and** ** **for all** ** **with** ** **such that*

(2.1) |

*Morever*

(2.2) |

*Then there exist* *a point** ** **in** ** **such that** *

**Proof**. Choose a point in such that continuing in this way, so we get for all and this implies that is a nonincreasing sequence. First we show that for all by using mathematical induction. Since from (2.2), we have

(2.3) |

thus, Suppose for some Thus from (2.1), we obtain

As *F* is strictly increasing, we have

(2.4) |

Now,

Thus Hence for all Continuing this process, we get

This implies that

(2.5) |

From (2.5), we obtain Since we have

(2.6) |

From (*F*3), there exists such that

(2.7) |

From (2.5), for all we obtain

(2.8) |

By using (2.6), (2.7) and letting in (2.8), we have

(2.9) |

We observe that from (2.9), then there exists such that for all we get

(2.10) |

Now, such that Then, by the triangle inequality and from (2.10) we have

(2.11) |

The series is convergent. By taking the limit as in (2.11), we have Hence is a Cauchy sequence. Since is a complete metric space there exists such that as is a continuous then ** **as That is, Hence is a fixed point of To prove uniqueness, let and be any two fixed point of then from (2.1), we have

we obtain

which is a contradiction. Hence, Therefore, has a unique fixed point in

**Example 4*** **Let** ** **and** ** **Then** **is a complete** **metric space. De**fi**ne the mapping** ** **by*,

* ** If** ** **and** ** **then*

*If ** **then*

*This implies that*

*If ** **then*

*Hence the contraction does not satisfy on** *

### 3. Fixed Point Theoreminvolving *GF*-Contraction on a Closed Ball

In this section, we define a new contraction called -*GF*-contraction on a closed ball and obtained Banach fixed point theorems for such contraction in the setting of complete metric spaces. We define -*GF*-contraction on a closed ball as follows:

**Definition 5*** Let** ** **be a self mapping in a metric space** ** **and** ** **an arbitrary point in** ** **with** ** **Also suppose that** ** ** **two functions. We say that** ** **is called** **-GF-contraction on a closed ball if for all** ** **with** ** **and** ** **we have*

(3.1) |

*and*

(3.2) |

*where ** ** **and** *

**Definition 6** (^{[36]}). *Let** ** **and** ** **be two functions. We say that** ** **is **-admissible mapping with respect to** ** if** ** ** **implies that** *

**Theorem 7** *Let** ** **be a complete metric space. Let** ** **be** **GF-contraction mapping on a closed ball satisfying the following assertions:*

(i) *T* is an -admissible mapping with respect to

(ii) there exists such that

(iii) is an continuous.

Then there exist a point in such that

**Proof**. Let in such that For we construct a sequence such that Continuing this way, for all Now since, is an -admissible mapping with respect to then By continuing in this process we have,

(3.3) |

If there exists such that there is nothing to prove. So, we assume that with

First we show that for all Since be a *GF*-contraction on closed ball, we have

(3.4) |

thus, Suppose for some such that

which implies

(3.5) |

Now by definition of

so there exists such that,

Therefore

(3.6) |

Rest of the proof follows the similar lines of Theorem 3. Since is a complete metric space there exists such that as is an continuous and for all then as That is, Hence is a fixed point of

**Example 8 ***Let** ** **and d be the usual metric on** ** **de**fi**ne** ** ** ** ** **and** ** **by*

for all and with *then*

*If** ** then** ** **On the other hand,** ** **for all** ** **Then** ** with** ** **clearly** ** **Hence we have*

*Consequently*,

*If** ** **or** ** t**hen** ** ** **either*

*On the other hand the contraction does not satisfy*.

### 4. Conclusion

This research focus on introducing new idea of *F*-contraction on a closed ball which is different from *F*-contraction given in ^{[18, 33, 42]}. Therefore a generalization of results is very useful so far as it requires the *F*-contraction mapping only on a closed ball rather than the whole space. This new idea however guides the researcher towards further investigations and applications. At the same time, it will be interesting to apply these concepts in a various spaces.

### Conflict of Interests

The author declare that he has no competing interests.

### References

[1] | M. Abbas, B. Ali and S. Romaguera, Fixed and periodic points of generalized contractions in metric spaces, Fixed Point Theory Appl. 2013, 2013: 243. | ||

In article | |||

[2] | T. Abdeljawad, Meir-Keeler α-contractive fixed and common fixed point theorems, Fixed PoinTheory Appl. 2013. | ||

In article | PubMed | ||

[3] | Ö. Acar and I. Altun, A Fixed Point Theorem for Multivalued Mappings with δ-Distance, Abstr. Appl. Anal., Volume 2014, Article ID 497092, 5 pages. | ||

In article | |||

[4] | M. Arshad, Fahimuddin, A. Shoaib and A. Hussain, Fixed point results for α-ψ-locally graphic contraction in dislocated qusai metric spaces, Math Sci., (2014), 7 pages. | ||

In article | |||

[5] | M. Arshad , A. Shoaib, I. Beg, Fixed point of a pair of contractive dominated mappings on a closed ball in an ordered complete dislocated metric space, Fixed Point Theory and Appl. (2013), 2013: 115, 15 pp. | ||

In article | |||

[6] | M. Arshad, A. Shoaib, and P. Vetro, Common Fixed Points Of A Pair Of Hardy Rogers Type Mappings On A Closed Ball In Ordered Dislocated Metric Spaces, Journal of Function Spaces and Appl. 2013 (2013), article ID 638181, 9 pages. | ||

In article | |||

[7] | A. Azam, S. Hussain and M. Arshad, Common fixed points of Chatterjea type fuzzy mappings on closed balls, Neural Computing & Applications, (2012) 21 (Suppl 1): S313-S317. | ||

In article | View Article | ||

[8] | A. Azam, M. Waseem, M. Rashid, Fixed point theorems for fuzzy contractive mappings in quasi-pseudo-metric spaces, Fixed Point Theory Appl., 20132013:27. | ||

In article | View Article | ||

[9] | S.Banach, Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales, Fund. Math., 3 (1922) 133-181. | ||

In article | |||

[10] | LB. Ćirić, A generalization of Banach.s contraction principle. Proc. Am. Math. Soc., 45, (1974) 267-273. | ||

In article | View Article | ||

[11] | M. Cosentino, P. Vetro, Fixed point results for F-contractive mappings of Hardy-Rogers-Type, Filomat 28:4(2014), 715-722. | ||

In article | View Article | ||

[12] | M. Edelstein, On fixed and periodic points under contractive mappings. J. Lond. Math. Soc., 37, 74-79 (1962). | ||

In article | View Article | ||

[13] | B. Fisher, Set-valued mappings on metric spaces, Fundamenta Mathematicae, 112 (2) (1981) 141-145. | ||

In article | |||

[14] | M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973) 604-608. | ||

In article | View Article | ||

[15] | A. Hussain and M. Arshad, New Type of Multivalued F-Contraction Involving Fixed Point on Closed Ball, J. Math. Comp. Sci. Accepted. | ||

In article | |||

[16] | A. Hussain, M. Arshad and Sami Ullah Khan, τ-Generalization of Fixed Point Results for F-Contractions, Bangmod Int. J. Math & Comp. Sci. 1(1) (2015), 136-146. | ||

In article | |||

[17] | N. Hussain, M. Arshad, A. Shoaib and Fahimuddin, Common fixed point results for α-ψ-contractions on a metric space endowed with graph, J. Inequal. Appl., (2014) 2014: 136. | ||

In article | View Article | ||

[18] | N. Hussain and P. Salimi, suzuki-wardowski type fixed point theorems for α-GF-contractions, Taiwanese J. Math., 20 (20) (2014). | ||

In article | |||

[19] | N. Hussain, E. Karapınar, P. Salimi and F. Akbar, α-admissible mappings and related fixed point theorems, J. Inequal. Appl., 114 (2013) 1-11. | ||

In article | View Article | ||

[20] | N. Hussain, P Salimi and A. Latif, Fixed point results for single and set-valued α-η-ψ-contractive mappings, Fixed Point Theory Appl. 2013, 2013: 212. | ||

In article | |||

[21] | N. Hussain, E. Karapinar, P. Salimi, P. Vetro, Fixed point results for G^{m}-Meir-Keeler contractive and G-(α,ψ)-Meir-Keeler contractive mappings, Fixed Point Theory Appl. 2013, 2013:34. | ||

In article | |||

[22] | N. Hussain, S. Al-Mezel and P. Salimi, Fixed points for α-ψ-graphic con-tractions with application to integral equations, Abstr. Appl. Anal., (2013) Article 575869. | ||

In article | View Article | ||

[23] | D. Jain, A. Padcharoen, P. Kumam and D. Gopal, A new approach to study fixed point of multivalued mappings in modular metric spaces and applications, Mathematics 2016, 4, 51. | ||

In article | View Article | ||

[24] | E. Karapinar and B. Samet, Generalized (α-ψ) contractive type mappings and related fixed point theorems with applications, Abstr. Appl. Anal., (2012) Article id:793486. | ||

In article | View Article | ||

[25] | E. Kryeyszig., Introductory Functional Analysis with Applications, John Wiley & Sons, New York, (Wiley Classics Library Edition) 1989. | ||

In article | |||

[26] | MA. Kutbi, M. Arshad and A. Hussain, On Modified α-η-Contractive mappings, Abstr. Appl. Anal., (2014) Article ID 657858, 7 pages. | ||

In article | |||

[27] | MA. Kutbi, M. Arshad and A.Hussain, Multivalued Ćirić type α-η-GF- Contractions, Journal of Computational Analysis and Applications. Ac-cepted. | ||

In article | |||

[28] | MA. Kutbi, M. Arshad and A.Hussain, Fixed Point Results for ´Ciri´c type α-η-GF-Contractions, journal of Computational Analysis and Applications 21 (3) 2016, 466-481. | ||

In article | |||

[29] | G. Minak, A. Halvaci and I. Altun, Ćirić type generalized F-contractions on complete metric spaces and fixed point results, Filomat, 28 (6) (2014), 1143-1151. | ||

In article | View Article | ||

[30] | SB. Nadler, Multivalued contraction mappings, Pac. J. Math., 30 (1969), 475-488. | ||

In article | View Article | ||

[31] | M. Nazam, M. Arshad and A. Hussain, Fixed Point Theorems For Chatterjea’s type Contraction on Closed ball, Journal of Analysis and Number Theory. 5(1) 2017 1-8. | ||

In article | |||

[32] | A. Padcharoen, D. Gopa, P. Chaipunya and P. Kumam, Fixed point and periodic point results for α- type F-contractions in modular metric spaces, Fixed Point Theory Appl. 2016, 2016: 39. | ||

In article | |||

[33] | H. Piri and P. Kumam, Some fixed point theorems concerning F-contraction in complete metric spaces, Fixed Poin Theory Appl. (2014) 2014:210. | ||

In article | View Article | ||

[34] | H. Piri and P. Kumam, Fixed point theorems for generalized F-Suzukicontraction mappings in complete b- metric spaces, Fixed Point Theory Appl. 2016, 2016: 90. | ||

In article | |||

[35] | M. Sgroi and C. Vetro, Multi-valued F-contractions and the solution of certain functional and integral equations, Filomat 27:7 (2013), 1259-1268. | ||

In article | View Article | ||

[36] | P. Salimi, A. Latif and N. Hussain, Modified α-ψ-Contractive mappings with applications, Fixed Point Theory Appl. (2013) 2013:151. | ||

In article | View Article | ||

[37] | SU. Khan, M. Arshad and A. Hussain, Two new Types of fixed point theorems for F-contraction, Journal of Advanced Studies in Topology, 7(4) 2016, 251-260. | ||

In article | View Article | ||

[38] | NA. Secelean, Iterated function systems consisting of F-contractions, Fixed Point Theory Appl. 2013, Article ID 277 (2013). | ||

In article | View Article | ||

[39] | M. Sgroi, C. Vetro, Multi-valued F-Contractions and the Solution of certain Functional and integral Equations, Filomat 27:7, (2013), 1259-1268. | ||

In article | |||

[40] | A. Shoaib, M. Arshad and J. Ahmad, Fixed point results of locally cotractive mappings in ordered quasi-partial metric spaces, The Scientific World Journal, 2013 (2013), Article ID 194897, 8 pp. | ||

In article | |||

[41] | B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ –contractive type mappings, Nonlinear Anal. 75 (2012) 2154-2165. | ||

In article | View Article | ||

[42] | D. Wardowski, Fixed point theory of a new type of contractive mappings in complete metric spaces. Fixed PoinTheory Appl. (2012) Article ID 94. | ||

In article | |||