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Research Article

Open Access Peer-reviewed

Aftab Hussain^{ }, Arshad Muhammad

Received January 27, 2017; Revised February 27, 2017; Accepted June 14, 2017

The aim of this paper to discuss generalized iterative process of F-contraction and establish new fixed point theorems in complete metric spaces. As an application of our results, we prove existence and uniqueness of functional equations and system of differential equations. Our results provide extension as well as substantial generalizations and improvements of several well known results in the existing comparable literature.

In 2012, Wardowski ^{ 29} introduce a new type of contractions called F-contraction and proved new fixed point theorems concerning F-contraction. Afterwards Se-celean ^{ 28}, proved fixed point theorems consisting of F-contractions by Iterated function systems. He generalized the Banach contraction principle in a different way than as it was done by different investigators.

Cosentino et al. ^{ 11} established some fixed point results of Hardy-Rogers-type for self-mappings on complete metric spaces or complete ordered metric spaces. Lately, Acar et al. ^{ 1} introduced the concept of generalized multivalued *F*-contraction mappings further Altun et al. ^{ 2} extended multivalued mappings with δ-Distance and established fixed point results in complete metric space. Sgroi et al. ^{ 24} established fixed point theorems for multivalued *F*-contractions and obtained the solution of certain functional and integral equations, which was a proper generalization of some multivalued fixed point theorems including Nadler’s. Thereafter, many papers have published on *F*-contractive mappings in various spaces. For more detail see ^{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} and references therein.

**Definition 1** ^{ 16} *Let** ** **be a metric space. A mapping** ** **is said to be an F contraction if there exists** ** **such that*

(1.1) |

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

*(F1) **F **is strictly increasing, i.e. for all** ** **such that **x < y**,** *

*(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).

**Example 2** ^{ 29} *Let** ** **be given by the formula** ** It is clear that F **satis**fi**ed **(F1)-(F2)-(F3) for any** ** Each mapping** ** satisfying (1:1) is an F-contraction such that*

*It is clear that for** ** **such that** ** **the inequality** ** **also holds, i.e. **T **is a Banach contraction*.

**Example 3** ^{ 29} *If** ** **then F satis.es (F1)-(F3) and the condition (1**.**1) is of the form*

**Remark 4** *From (F1) and (1**.**1) it is easy to conclude that every F-contraction is necessarily continuous*.

Wardowski ^{ 29} stated a modified version of the Banach contraction principle as follows.

**Theorem 5** ^{ 29} *Let** ** **be a complete metric space and let** ** **be an F contraction. Then** ** has a unique **fi**xed point** ** **and for every** ** **the sequence** ** **converges to** *

We recollect some essential notations, required definitions, and primary results coherent with the literature.

Let be a metric space. We denote by the class of all nonempty closed subsets of For a nonempty set we denote by the class of all nonempty subsets of For define a set

The Hausdorff metric *H* on induced by metric *d* is given as:

Let and A point in is called a fixed point of if The set of all fixed points of is denoted by Furthermore, a point in is called a coincidence point of and if The set of all such points is denoted by If for some point in we have then a point is called a common fixed point of and We denote set of all common fixed points of and by

**Definition 6** *Let** ** **and** ** **Let** ** **be an arbitrary but **fi**xed element in** ** **A sequence** ** **in** ** is called** ** **iterative sequence of** ** **starting with** ** **if** ** Then** ** ** **is called a generalized iterative process of f and T starting at** ** **Note that** ** **reduces to dynamic process of** ** **starting at** ** **if** ** **(an identity map on** **) *^{ 16}*. The generalized iterative process** ** **will simply be written as** *

**Example 7** ^{ 16} *Consider a Banach space X = C*(*I*)* with a norm** ** ** **where** ** **Let** ** **be such that for any** ** ** **is a family of the functions** ** **where ** **i.e.*,

and let Then the sequence is a dynamic process of the operator *T** *starting at

**Definition 8** *Let** ** **be a self map on a metric space** ** **A **multivalued mapping** ** **is called generalize**d multivalued Ciric type **F**-**contraction with** **respect to a iterative process** ** **if there exist** ** **and** ** **such that*

*where*

*and*

**Example 9** *Let** ** ** **and** ** **be de**fi**ned as** ** ** **for all** ** **we obtain a sequence** ** ** **these can be many **f **iterative sequence of **T **starting at** *

Let be an arbitrary point in Then

*Then clearly** ** **We **obtain a generalized iterative process*

*is called a generalized dynamic process of f and T starting at** ** **So you can construct many f iterative sequences of T starting at x**0 **for different values*.

Throughout this section, we assume that the mapping *F* is right continuous. In the following we will consider only the dynamic processes satisfying the following condition:

If dynamic processes does not satisfy property then there exists such that and which implies that that is, the set of coincidence point of hybrid pair is nonempty. Under suitable conditions on hybrid pair one obtaines the existence of common fixed point of

Our main result is the following.

**Theorem 10** *Let** ** ** **and** ** **a generalized multivalued Ciric type F- contraction with respect to generalized iterative process** ** **Then** ** **provided that** ** **is complete and F is continuous or T is closed multivalued mapping. Moreover** ** **if one of the following conditions holds*:

*(a) for some** ** **f **is** **- weakly commuting at **x**,** ** **(b)** ** **is a singleton subset of** *

**Proof**. Let be a generalized iterative process of the mapping and starting at We observe that if there exists such that then the existence of a fixed point is obvious. Hence we can assume that for all Since is a generalized multivalued Ciric type *F*- contraction with respect to a generalized iterative process, it follows that

(2.1) |

implies

Implies that

(2.2) |

Since *F* is strictly increasing, therefore

If

for some *n*, then,

gives a contradiction. So we have

Consequently

(2.3) |

for all By definition 8, there exists and such that for all Thus, we obtain

On taking limit as we have By (F1), we get By (F3) there exists an such that

(2.4) |

Hence it follows that

On taking limit as tends to we obtain that is, This implies that is convergent and hence the sequence is a Cauchy sequence in so there is such that Suppose that is in such that Next we prove that Indeed, assume the contrary, then because is closed: Since is strictly increasing, we deduce that

for all Therefore

Since from definition 8, we have

(2.5) |

for all

Next suppose that *F* is continuous. Since

we deduce that

so, by continuity of *F*,

which provides a contradiction. We conclude that and thus Now let (a) holds, that is for is -weakly commuting at So we get By the given hypothesis and hence Consequently (b) Since (say) and this implies that Thus

**Example 11** *Let** ** ** **and** ** **be defined as** ** ** **and d be the usual metric on X. De**fi**ne** ** **and** ** **by** ** **and ** **for all** ** **Then for all** ** ** **we obtain*

Thus all conditions of above Theorem 10 is satisfied and 0 is a fixed point of *T*.

**Example 12*** **Let** ** **be the usual metric space. **De**fi**ne** ** ** **and** ** **by** ** **and** ** **for all** ** **and** ** **and** ** **for all** ** **Note that so** ** **is complete. It is easy to check that for all** ** **with** ** **(equivalently with** **)*, *one has*

*So we can apply Theorem 10*.

**Corollary 13** *Let** ** **be a complete metric space,** ** **be an arbitrary point in** ** **and** ** **a **multivalued Ciric type **F**-**contraction with respect** **to dynami**c process** ** either **F **is continuous or **T **is closed multivalued** **mapping. Then there exists a **fi**xed point of **T*.

**(1) Applications**

Decision space and a state space are two basic components of dynamic programming problem. State space is a set of states including initial states, action states and transitional states. So a state space is set of parameters representing different states. A decision space is the set of possible actions that can be taken to solve the problem. These general settings allow us to formulate many problems in mathematical optimization and computer programming. In particular the problem of dynamic programming related to multistage process reduces to the problem of solving functional equations

(2.6) |

(2.7) |

where and are Banach spaces, and and

for more details on dynamic programming we refer to ^{ 6, 7, 8, 9, 23}. Suppose that and are the state and decision spaces respectively. We aim to give the existence and uniqueness of common and bounded solution of functional equations given in (2.6) and (2.7). Let denotes the set of all bounded real valued functions on For an arbitrary define Then is a Banach space endowed with the metric defined as

(2.8) |

Suppose that the following conditions hold:

(*C*1): and are bounded.

(*C*2): For and define

(2.9) |

(2.10) |

Moreover assume that and such that for every and implies

(2.11) |

where

(*C*3): For any there exists such that for

(*C*4): There exists such that

**Theorem 14** *Assume that the conditions* (*C*1)-(*C*4*) are satisfied. If** ** is a closed convex subspace of **, then the functional equations *(2.6)* and *(2.7)* have a unique, common and bounded solution*.

**Proof.** Note that is a complete metric space. By (C1), *J*, *K* are self-maps of The condition (C3) implies that It follows from (C4) that *J* and *K* commute at their coincidence points. Let be an arbitrary positive number and Choose and such that

(2.12) |

where Further from (2.9) and (2.10), we have

(2.13) |

(2.14) |

Then (2.12) and (2.14) together with (2.11) imply

(2.15) |

Then (2.12) and (2.13) together with (2.11) imply

(2.16) |

From (2.15) and (2.16), we have

(2.17) |

The inequality (2.17) implies

(2.18) |

(2.19) |

Therefore by Theorem 10, the pair (*K*, *J*) has a common fixed point , that is, is unique, bounded and common solution of (2.6) and (2.7).

**(1) Application of system of integral equations:**

Now we discuss an application of fixed point theorem we proved in the previous section in solving the system of Volterra type integral equations. Such system is given by the following equations:

(2.20) |

(2.21) |

for where We find the solution of the system (2.20) and (2.21). Let be the space of all continuous functions defined on For define supremum norm as: where is taken as a function. Let be endowed with the metric

(2.22) |

for all With these setting becomes Banach space.

Now we prove the following theorem to ensure the existence of solution of system of integral equations. For more details on such applications we refer the reader to ^{ 3, 21}.

**Theorem 15** *Assume the following conditions are satis**fi**ed*:

*(i) ** **and** ** **are continuous;*

*(ii) De**fi**ne*

*Suppose there exist** ** **and** ** **such that*

for all and where

(*iii*) *there exists** ** **such that ** **implies** ** **Then the system of integral equations given in (2:20) and (2:21) has a solution*.

**Proof. **By assumption (iii)

This implies

That is

which further implies

So all the conditions of Theorem 10 are satisfied. Hence the system of integral equations given in (2.20) and (2.21) has a unique common solution.

This paper presents fixed point theorems for generalized iterative process under the improved notion of dynamic process. The presented theorem provide extension as well as substantial generalizations and improvements of several well known results. The present version of these results make significant and useful contribution in the existing literature.

The authors declare that they have no competing interests.

[1] | Ö. Acar, G. Durmaz and G. Minak, Generalized multivalued F-contractions on complete metric spaces, Bulletin of the Iranian Mathematical Society. 40(2014), 1469-1478. | ||

In article | View Article | ||

[2] | 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 | View Article | ||

[3] | A. Augustynowicz, Existence and uniqueness of solutions for partial differential-functional equations of the first order with deviating argument of the derivative of unknown function, Serdica Mathematical Journal 23 (1997) 203-210. | ||

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

In article | View Article | ||

[6] | R. Baskaran, P.V. Subrahmanyam, A note on the solution of a class of functional equations. Appl. Anal. 22(3-4), 235-241. | ||

In article | View Article | ||

[7] | R. Bellman, Methods of Nonlinear Analysis. Vol. II. Mathematics in Science and Engineering, vol. 61. Academic Press, New York (1973). | ||

In article | |||

[8] | R. Bellman, E.S. Lee, Functional equations in dynamic programming. Aequ. Math. 17, 1-18 (1978). | ||

In article | View Article | ||

[9] | P.C. Bhakta, S. Mitra, Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl. 98, 348-362 (1984). | ||

In article | View 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 | View Article | ||

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

In article | View Article | ||

[15] | 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 | View Article | ||

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

In article | View Article | ||

[17] | 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 | ||

[18] | MA. Kutbi, W. Sintunavarat, On new fixed point results for (α,ψ,ξ)-contractive multi-valued mappings on α-complete metric spaces their consequences, Fixed Point Theory Appl., (2015) 2015: 2. | ||

In article | View Article | ||

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

In article | |||

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

In article | View Article | ||

[21] | D. ÓRegan, A. Petruşel, Fixed point theorems for generalized contractions in ordered metric spaces, Journal of Mathematical Analysis and Applications 341 (2008) 1241-1252. | ||

In article | View Article | ||

[22] | 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 | ||

[23] | H.K. Pathak, Y.J. Cho, S.M. Kang, B.S. Lee, Fixed point theorems for compatible mappings of type P and applications to dynamic programming. Matematiche 50, 15-33 (1995). | ||

In article | View Article | ||

[24] | 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 | ||

[25] | 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 | ||

[26] | 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 | View Article | ||

[27] | 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 | ||

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

In article | View Article | ||

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

In article | |||

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/

Aftab Hussain, Arshad Muhammad. Generalization of Fixed Point Results via Iterative Process of *F*-Contraction. *Turkish Journal of Analysis and Number Theory*. Vol. 5, No. 4, 2017, pp 132-138. https://pubs.sciepub.com/tjant/5/4/3

Hussain, Aftab, and Arshad Muhammad. "Generalization of Fixed Point Results via Iterative Process of *F*-Contraction." *Turkish Journal of Analysis and Number Theory* 5.4 (2017): 132-138.

Hussain, A. , & Muhammad, A. (2017). Generalization of Fixed Point Results via Iterative Process of *F*-Contraction. *Turkish Journal of Analysis and Number Theory*, *5*(4), 132-138.

Hussain, Aftab, and Arshad Muhammad. "Generalization of Fixed Point Results via Iterative Process of *F*-Contraction." *Turkish Journal of Analysis and Number Theory* 5, no. 4 (2017): 132-138.

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[1] | Ö. Acar, G. Durmaz and G. Minak, Generalized multivalued F-contractions on complete metric spaces, Bulletin of the Iranian Mathematical Society. 40(2014), 1469-1478. | ||

In article | View Article | ||

[2] | 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 | View Article | ||

[3] | A. Augustynowicz, Existence and uniqueness of solutions for partial differential-functional equations of the first order with deviating argument of the derivative of unknown function, Serdica Mathematical Journal 23 (1997) 203-210. | ||

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

In article | View Article | ||

[6] | R. Baskaran, P.V. Subrahmanyam, A note on the solution of a class of functional equations. Appl. Anal. 22(3-4), 235-241. | ||

In article | View Article | ||

[7] | R. Bellman, Methods of Nonlinear Analysis. Vol. II. Mathematics in Science and Engineering, vol. 61. Academic Press, New York (1973). | ||

In article | |||

[8] | R. Bellman, E.S. Lee, Functional equations in dynamic programming. Aequ. Math. 17, 1-18 (1978). | ||

In article | View Article | ||

[9] | P.C. Bhakta, S. Mitra, Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl. 98, 348-362 (1984). | ||

In article | View 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 | View Article | ||

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

In article | View Article | ||

[15] | 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 | View Article | ||

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

In article | View Article | ||

[17] | 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 | ||

[18] | MA. Kutbi, W. Sintunavarat, On new fixed point results for (α,ψ,ξ)-contractive multi-valued mappings on α-complete metric spaces their consequences, Fixed Point Theory Appl., (2015) 2015: 2. | ||

In article | View Article | ||

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

In article | |||

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

In article | View Article | ||

[21] | D. ÓRegan, A. Petruşel, Fixed point theorems for generalized contractions in ordered metric spaces, Journal of Mathematical Analysis and Applications 341 (2008) 1241-1252. | ||

In article | View Article | ||

[22] | 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 | ||

[23] | H.K. Pathak, Y.J. Cho, S.M. Kang, B.S. Lee, Fixed point theorems for compatible mappings of type P and applications to dynamic programming. Matematiche 50, 15-33 (1995). | ||

In article | View Article | ||

[24] | 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 | ||

[25] | 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 | ||

[26] | 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 | View Article | ||

[27] | 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 | ||

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

In article | View Article | ||

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

In article | |||