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Research Article

Open Access Peer-reviewed

Abdullah Shoaib^{ }, Awais Asif, Muhammad Arshad, Eskandar Ameer

Received September 27, 2016; Revised February 07, 2018; Accepted March 28, 2018

The aim of this paper is to establish common fixed point results for multivalued mappings satisfying generalized F-contractive conditions of Hardy Rogers type with respect to generalized dynamic process in *b*-metric space. Our results improve and generalize several well known results in the existing literature.

Let be a non empty set, be a mapping. A point is called a fixed point of if Fixed points results of mappings, which satisfies some specific contractive conditions on some space have been very useful in research activity (see ^{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}).

Recently, Wardowski ^{ 30} introduced a new concept of contraction named F-contraction and proved a fixed point theorem which generalizes Banach contraction principle. Klim et al. ^{ 22} further established fixed point result for F-contractive mapping in dynamic process. Cosentino et al. ^{ 16} further generalized this concept as F-Contractive Mappings of Hardy-Rogers-Type. Arshad et al. ^{ 4} proved fixed point result in contraction of Hardy-Rogers-type. Following this direction of research, in this paper, we will present some fixed point results of Hardy-Rogers-type for multivalued mappings in b-metric space with generalized dynamic process. This paper contain common fixed point results for two mappings. Throughout our paper and represent set of real numbers, set of natural numbers and family of non-empty closed bounded subsets respectively.

**Definition** **1** ^{ 10}* Let ** be a non-empty set and let ** **be a given real number. A function ** **is called a b-metric provided that, for all *

1) iff

2)

3) The pair is called a b-metric space.

**Definition**** 2** ^{ 15} *Let** ** **be a b**-metric space. Then a sequence** ** **in** ** **is called a Cauchy sequence if and only if for all** ** **there exist** ** **such that for each** **, we have** *

**Definition**** 3** ^{ 30} *Let** ** **be a mapping satisfying:*

(F1) F is strictly increasing.

(F2) for each sequence of positive numbers if and only if

(F3) there exists such that

We denote with the family of all functions that satisfy the conditions (F1)-(F3).

Let be a metric space. A self-mapping on is called an F-contraction if there exist such that

**Theorem**** 4** ^{ 30} *Let** ** **be a complete metric space and let** ** **be an F-contraction. Then** ** **has a unique fixed point** ** **and for every** ** **a sequence** ** **is convergent to** *

**Definition**** 5** ^{ 16} *Let** ** **be a metric space. A self-**mapping** ** **on** ** **is called a generalized F-contraction of **Hardy-Rogers-type if there exist** ** **and** ** **such that*

*for all** ** **with** ** where** ** ** **and** *

**Theorem**** 6** ^{ 16} Let *be a complete metric space and let** ** **be a self-mapping on** **. Assume that there exist** ** **and** ** **such that** ** **is an F-contraction of Hardy-Rogers-type, that is*,

*for all** ** ** where** ** **,** ** **and** ** Then** ** **has a fixed point*. *Moreover*,* if** ** , then the fixed point of** ** **is unique*.

**Definition**** 7** ^{ 20} *Let** ** **be a metric space*. *For** ** the Hausdorff metric** ** **on** ** **induced by metric** ** **is given as*:

*where** ** **and** ** ** **is the Hausdorff metric induced by*

**Definition**** 8** ^{ 22} *Let** ** **and let** ** **satisfies (F1)-(F3). The mapping** ** **is called a set-valued F-contraction with respect to a dynamic process** ** **if there exists a function** ** **such that*

In the above discussion denotes the collection of all non empty closed subsets of and is called a dynamic process of starting at The dynamic process will be written simply as where

**Definition**** 9** ^{ 3} *Let** **, ** **and** ** **be an arbitrary but fixed element in** ** The set** ** **for all** ** **is called a generalized dynamic process of** ** **and** ** **starting at** ** The generalized dynamic process** ** **will simply be written as** ** The sequence** ** **for which** ** **is a generalized dynamic process is called** ** **iterative sequence of** ** **starting at** *

**Lemma**** 10** ^{ 23} *Suppose** ** **be a b-metric space and** ** **be a sequence in** ** **such that*

where Then the sequence is a Cauchy sequence in provided that

**Definition** **11** Let be a mapping satisfying:

(F1) F is strictly increasing.

(F2) for each sequence of positive numbers if and only if

We denote with the family of all functions that satisfy the conditions (F1) and (F2).

Let be a b-metric space. A self-mapping on is called an F_{1} -contraction if there exist such that

(1) |

**Definition** **12** *Let** ** **be a b-metric space,** ** **is continuous** ** **and** ** **be **an arbitrary point in** **. A mapping** ** **is called a generalized** multivalued** ** -contraction of Hardy-Rogers-type with respect to a generalized dynamic process** ** **if there exist** ** **and** ** **is non decreasing such that*

*where *

*for all** ** **with** ** where** ** ** ** ** **and*

Now we state and prove our main result.

**Theorem** **13*** Let** ** **be a complete b-metric space and** ** **If** ** **is a generalized multivalued** -contraction of Hardy-Rogers-type with respect to a generalized dynamic process** ** Then** ** **and** ** **have a common fixed point*.

**Proof****:** Let be an arbitrary point, by definition of generalized multivalued -contraction of Hardy-Rogers-type with respect to a generalized dynamic process we have

As is strictly increasing, therefore

As and we deduce that and so

This implies that

Continuing this process, we can easily say that

Now, to show that is a Cauchy sequence in . Let with

Using Lemma 10, and taking limit we get

which proves that is a Cauchy, so there exist some such that Now we prove that For this, we have

By (F2)

Therefore

From above we can write

Again by (F2) Therefore So . Moreover

Hence, , so , that is is the common fixed point of and .

Putting , we obtain a generalize form of Kannan's result in dynamic process.

**Corollary**** 14**** ***Let** ** **be a complete b-metric **space and** ** Assume that there exist** ** **and** ** **such that ** **is continuous satisfying*:

*for all** ** **with** ** , where** ** ** **,** ** **and** ** . Then** ** **and** ** **have a common fixed point**.*

Choosing , we obtain a generalize version of Reich's result.

**Corollary**** 15**** ***Let** ** **be a complete b-metric space and** ** Assume that there exist** ** **and** ** **such that** ** **is continuous satisfying*:

for all with , where , and . Then and have a common fixed point.

**Theorem**** 16**** **Let * **be a complete b-metric space,** ** **is continuous and** ** **Assume that there exist** ** **and** ** **such that ** **is continuous from the right. Now if** ** **is a generalized dynamic process such that*

(2) |

*where *

*for all** ** **with** ** where** ** ** ** ** **and** **. Then** ** **and** ** **have a common fixed point*.

**Proof:** Let be an arbitrary point of By definition of generalized dynamic process . Since is continuous from the right, there exists a real number such that

Now, from we deduce that there exists such that Consequently, we get

which implies

As is strictly increasing we deduce

As and hence we deduce that and so consequently,

Continuing this way we get

and hence

Proceeding this as in Theorem 13, we obtain that is a Cauchy sequence. Since is a complete metric space, there exists some such that . Now we prove that . For this, since

By (F2)

Therefore, Hence

By using (F2) which implies Also

Therefore hence that is

Replacing b-metric space by metric space in the above result we get the following corollary:

**Corollary**** 17**** ***Assume that** ** **is a complete metric space,** ** **is continuous and** ** **Assume that there exist** ** **and ** such that ** is continuous from the right**. Now if** ** **is a generalized dynamic process in such a way that*

*where *

*for each** ** **with** ** where** ** ** **and** ** Then** ** **and** ** **possess a common fixed point*.

**Example**** 18**** ***Assume that** ** **and** ** **is defined by*

*where** * *Define** ** ** **and** ** **by** ** ** **and** ** **Define a sequence** ** **by for all** ** ** **with** ** **and** ** **Then*

and so on. Here

Fix and clearly . We can check that is a b-metric. Now checking for all we can find some that satisfy the inequality (2) in such a way that for each Moreover is found to be the common fixed point of and

Putting and in the above result we get the following corollary:

**Corollary**** 19** *Assume that** ** **is a complete b-metric space,** ** **is continuous*,* ** there exist a function** ** **and let** ** **satisfy (F1)-(F**2**). Now if** ** **is a generalized dynamic process in such a way that*

*then there is a common fixed point of** ** **and** ** **i.e.*

Putting as an identity function in Corollary 19 we get:

**Corollary**** 20*** Assume that** ** **is a complete metric space*,* ** **is continuo**u**s** ** there exist a function** ** **and suppose** ** **satisfy **(F1)-(F2**). Now if** ** **is a dynamic process in such a way that*:

then there exists a fixed point of

The authors declare that they have no competing interests.

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

In article | |||

[2] | I. Altun, G. Durmaz, Some fixed point theorems on ordered cone metric spaces, Rendiconti del Circolo Matematico di Palermo 58 (2009) 319-325. | ||

In article | View Article | ||

[3] | M. Arshad, M. Abbas, A. Hussain and N. Hussain, Generalized Dynamic Process for Generalized (f,L)-almost F-Contraction with Applications, J. Nonlinear Sci. Appl. 9 (2016), 1702-1715. | ||

In article | View Article | ||

[4] | M. Arshad, E. Ameer and A.Hussain, Hardy-Rogers-Type Fixed Point Theorems for α-GF-Contractions, Archivum Mathematicum (BRNO) Tomus 51 (2015), 129-141. | ||

In article | View 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 pages. | ||

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, 2013 (2013), Article ID 63818. | ||

In article | |||

[7] | M. Arshad , A. Shoaib, M. Abbas and A. Azam, Fixed Points of a pair of Kannan Type Mappings on a Closed Ball in Ordered Partial Metric Spaces, Miskolc Mathematical Notes, 14(3), 2013, 769-784. | ||

In article | |||

[8] | M. Arshad, A. Azam, M. Abbas and A. Shoaib, Fixed point results of dominated mappings on a closed ball in ordered partial metric spaces without continuity U.P.B. Sci. Bull., Series A, 76(2), 2014. | ||

In article | |||

[9] | A. Azam, M. Arshad, I. Beg, Common fixed points of two maps in cone metric spaces, Rendiconti del Circolo Matematico di Palermo 57 (2008) 433-441. | ||

In article | View Article | ||

[10] | I.A. Bakhtin, The contraction mapping principle in quasi-metric spaces, Funct. Anal. Unianowsk Gos. Ped. Inst. 30 (1989), 26-37. | ||

In article | |||

[11] | I. Beg, M. Arshad , A. Shoaib, Fixed Point on a Closed Ball in ordered dislocated Metric Space, Fixed Point Theory, 16(2), 2015. | ||

In article | |||

[12] | V. Berinde, F. Vetro, Common fixed points of mappings satisfying implicit contractive conditions, Fixed Point Theory and Applications 2012:105 (2012). | ||

In article | View Article | ||

[13] | V. Berinde, F. Vetro, Fixed point for cyclic weak (Ψ, C)-contractions in 0-complete partial metric spaces, Filomat 27 (2013) 1405-1413. | ||

In article | View Article | ||

[14] | A. Bhatt and H. Chandra, Common fixed points for JH operators and occasionally weakly g-biased pairs under relaxed condition on probabilistic metric space, Journal of Function Spaces and Applications, vol. 2013, Article ID 846315, 6 pages, 2013. | ||

In article | |||

[15] | M. Boriceanu, Fixed Point theory for multivalued generalized contraction on a set with two b-metrics, studia Univ Babes, Bolya: Math. LIV (3) (2009), 1-14. | ||

In article | |||

[16] | 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 | ||

[17] | N. Hussain, J. Ahmad and A. Azam, Generalized fixed point theorems for multi-valued α - ψ -contractive mappings, J. Inequal. Appl., 2014, 2014:348. | ||

In article | |||

[18] | N. Hussain, S. Al-Mezel and P. Salimi, Fixed points for ψ -graphic contractions with application to integral equations, Abstract and Applied Analysis, Volume 2013, Article ID 575869. | ||

In article | |||

[19] | N. Hussain, M. Arshad, A. Shoaib and Fahimuddin, Common Fixed Point results for α - ψ -contractions on a metric space endowed with graph, J. Inequalities and Appl., 2014, 2014:136. | ||

In article | |||

[20] | M. Jleli, H. Kumar, B. Samet and C. Vetro, On multivalued weakly Picard operators in partial Hausdorff metric spaces, Fixed Point Theory and Applications 2015, 2015: 52. | ||

In article | |||

[21] | Z. Kadelburg, L. Paunović, S. Radenović, A note on fixed point theorems for weakly T-Kannan and weakly T-Chatterjea contractions in b-metric spaces, Gulf Journal of Mathematics 3 (2015) 57-67. | ||

In article | |||

[22] | D. Klim and D. Wardowski, Fixed points of dynamic processes of set-valued F-contractions and application to functional equations, Fixed Point Theory and Applications (2015) 2015: 22. | ||

In article | View Article | ||

[23] | P. Kumar, M. S. Sachdeva and S. K. Banerjee, Some Fixed Point Theorems in b-metric Space, Turkish Journal of Analysis and Number Theory, 2014, 2(1), 19-22. | ||

In article | View Article | ||

[24] | 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 pages. | ||

In article | |||

[25] | A. Shoaib, M. Arshad and M. A. Kutbi, Common fixed points of a pair of Hardy Rogers Type Mappings on a Closed Ball in Ordered Partial Metric Spaces, J. Comput. Anal. Appl., 17(2014), 255-264. | ||

In article | |||

[26] | A. Shoaib, α-η Dominated Mappings and Related Common Fixed Point Results in Closed Ball, Journal of Concrete and Applicable Mathematics, 13(1-2), 2015, 152-170. | ||

In article | |||

[27] | N. Shobkolaei, S. Sedghi, J. R. Roshan, andN.Hussain, Suzuki type fixed point results in metric-like spaces, Journal of Function Spaces and Applications, vol. 2013, Article ID 143686, 9 pages, 2013. | ||

In article | |||

[28] | S. Shukla, S. Radenović, C. Vetro, Set-valued Hardy-Rogers type contraction in 0-complete partial metric spaces, International Journal of Mathematics and Mathematical Sciences, Volume 2014, Article ID 652925, 9 pages. | ||

In article | |||

[29] | S. Shukla, S. Radenović, Z. Kadelburg, Some fixed point theorems for F-generalized contractions in 0-orbitally complete partial metric spaces, Theory and Applications of Mathematics and Computer Science 4(1) (2014) 87-98. | ||

In article | |||

[30] | D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory and Appl. 2012:94 (2012). | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2018 Abdullah Shoaib, Awais Asif, Muhammad Arshad and Eskandar Ameer

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

Abdullah Shoaib, Awais Asif, Muhammad Arshad, Eskandar Ameer. Generalized Dynamic Process for Generalized Multivalued F-contraction of Hardy Rogers Type in b-metric Spaces. *Turkish Journal of Analysis and Number Theory*. Vol. 6, No. 2, 2018, pp 43-48. https://pubs.sciepub.com/tjant/6/2/2

Shoaib, Abdullah, et al. "Generalized Dynamic Process for Generalized Multivalued F-contraction of Hardy Rogers Type in b-metric Spaces." *Turkish Journal of Analysis and Number Theory* 6.2 (2018): 43-48.

Shoaib, A. , Asif, A. , Arshad, M. , & Ameer, E. (2018). Generalized Dynamic Process for Generalized Multivalued F-contraction of Hardy Rogers Type in b-metric Spaces. *Turkish Journal of Analysis and Number Theory*, *6*(2), 43-48.

Shoaib, Abdullah, Awais Asif, Muhammad Arshad, and Eskandar Ameer. "Generalized Dynamic Process for Generalized Multivalued F-contraction of Hardy Rogers Type in b-metric Spaces." *Turkish Journal of Analysis and Number Theory* 6, no. 2 (2018): 43-48.

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

In article | |||

[2] | I. Altun, G. Durmaz, Some fixed point theorems on ordered cone metric spaces, Rendiconti del Circolo Matematico di Palermo 58 (2009) 319-325. | ||

In article | View Article | ||

[3] | M. Arshad, M. Abbas, A. Hussain and N. Hussain, Generalized Dynamic Process for Generalized (f,L)-almost F-Contraction with Applications, J. Nonlinear Sci. Appl. 9 (2016), 1702-1715. | ||

In article | View Article | ||

[4] | M. Arshad, E. Ameer and A.Hussain, Hardy-Rogers-Type Fixed Point Theorems for α-GF-Contractions, Archivum Mathematicum (BRNO) Tomus 51 (2015), 129-141. | ||

In article | View 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 pages. | ||

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, 2013 (2013), Article ID 63818. | ||

In article | |||

[7] | M. Arshad , A. Shoaib, M. Abbas and A. Azam, Fixed Points of a pair of Kannan Type Mappings on a Closed Ball in Ordered Partial Metric Spaces, Miskolc Mathematical Notes, 14(3), 2013, 769-784. | ||

In article | |||

[8] | M. Arshad, A. Azam, M. Abbas and A. Shoaib, Fixed point results of dominated mappings on a closed ball in ordered partial metric spaces without continuity U.P.B. Sci. Bull., Series A, 76(2), 2014. | ||

In article | |||

[9] | A. Azam, M. Arshad, I. Beg, Common fixed points of two maps in cone metric spaces, Rendiconti del Circolo Matematico di Palermo 57 (2008) 433-441. | ||

In article | View Article | ||

[10] | I.A. Bakhtin, The contraction mapping principle in quasi-metric spaces, Funct. Anal. Unianowsk Gos. Ped. Inst. 30 (1989), 26-37. | ||

In article | |||

[11] | I. Beg, M. Arshad , A. Shoaib, Fixed Point on a Closed Ball in ordered dislocated Metric Space, Fixed Point Theory, 16(2), 2015. | ||

In article | |||

[12] | V. Berinde, F. Vetro, Common fixed points of mappings satisfying implicit contractive conditions, Fixed Point Theory and Applications 2012:105 (2012). | ||

In article | View Article | ||

[13] | V. Berinde, F. Vetro, Fixed point for cyclic weak (Ψ, C)-contractions in 0-complete partial metric spaces, Filomat 27 (2013) 1405-1413. | ||

In article | View Article | ||

[14] | A. Bhatt and H. Chandra, Common fixed points for JH operators and occasionally weakly g-biased pairs under relaxed condition on probabilistic metric space, Journal of Function Spaces and Applications, vol. 2013, Article ID 846315, 6 pages, 2013. | ||

In article | |||

[15] | M. Boriceanu, Fixed Point theory for multivalued generalized contraction on a set with two b-metrics, studia Univ Babes, Bolya: Math. LIV (3) (2009), 1-14. | ||

In article | |||

[16] | 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 | ||

[17] | N. Hussain, J. Ahmad and A. Azam, Generalized fixed point theorems for multi-valued α - ψ -contractive mappings, J. Inequal. Appl., 2014, 2014:348. | ||

In article | |||

[18] | N. Hussain, S. Al-Mezel and P. Salimi, Fixed points for ψ -graphic contractions with application to integral equations, Abstract and Applied Analysis, Volume 2013, Article ID 575869. | ||

In article | |||

[19] | N. Hussain, M. Arshad, A. Shoaib and Fahimuddin, Common Fixed Point results for α - ψ -contractions on a metric space endowed with graph, J. Inequalities and Appl., 2014, 2014:136. | ||

In article | |||

[20] | M. Jleli, H. Kumar, B. Samet and C. Vetro, On multivalued weakly Picard operators in partial Hausdorff metric spaces, Fixed Point Theory and Applications 2015, 2015: 52. | ||

In article | |||

[21] | Z. Kadelburg, L. Paunović, S. Radenović, A note on fixed point theorems for weakly T-Kannan and weakly T-Chatterjea contractions in b-metric spaces, Gulf Journal of Mathematics 3 (2015) 57-67. | ||

In article | |||

[22] | D. Klim and D. Wardowski, Fixed points of dynamic processes of set-valued F-contractions and application to functional equations, Fixed Point Theory and Applications (2015) 2015: 22. | ||

In article | View Article | ||

[23] | P. Kumar, M. S. Sachdeva and S. K. Banerjee, Some Fixed Point Theorems in b-metric Space, Turkish Journal of Analysis and Number Theory, 2014, 2(1), 19-22. | ||

In article | View Article | ||

[24] | 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 pages. | ||

In article | |||

[25] | A. Shoaib, M. Arshad and M. A. Kutbi, Common fixed points of a pair of Hardy Rogers Type Mappings on a Closed Ball in Ordered Partial Metric Spaces, J. Comput. Anal. Appl., 17(2014), 255-264. | ||

In article | |||

[26] | A. Shoaib, α-η Dominated Mappings and Related Common Fixed Point Results in Closed Ball, Journal of Concrete and Applicable Mathematics, 13(1-2), 2015, 152-170. | ||

In article | |||

[27] | N. Shobkolaei, S. Sedghi, J. R. Roshan, andN.Hussain, Suzuki type fixed point results in metric-like spaces, Journal of Function Spaces and Applications, vol. 2013, Article ID 143686, 9 pages, 2013. | ||

In article | |||

[28] | S. Shukla, S. Radenović, C. Vetro, Set-valued Hardy-Rogers type contraction in 0-complete partial metric spaces, International Journal of Mathematics and Mathematical Sciences, Volume 2014, Article ID 652925, 9 pages. | ||

In article | |||

[29] | S. Shukla, S. Radenović, Z. Kadelburg, Some fixed point theorems for F-generalized contractions in 0-orbitally complete partial metric spaces, Theory and Applications of Mathematics and Computer Science 4(1) (2014) 87-98. | ||

In article | |||

[30] | D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory and Appl. 2012:94 (2012). | ||

In article | View Article | ||