Multivalued F-Contraction Involving Fixed Point in Closed Ball

Authors

1 Department of Mathematics and Statistics, International Islamic university, H-10 Islamabad Pakistan Department of Mathematical Sciences, Lead University Lahore-54000, Pakistan

2 Department of Mathematics, International Islamic University, H-10, Islamabad - 44000, Pakistan.

3 Department of Mathematics and Statistics, International Islamic university, H-10 Islamabad Pakistan

Abstract

This paper is a continuation of the investigations of F-contraction. The aim of this article is to extend
the concept of F-contraction on a closed ball. We introduce the notion of multivalued F-contraction on
a closed ball and establish new fixed point theorems in a complete metric space. Our results are very
useful for the contraction of the mapping only on closed ball instead on the whole space. Some comparative
examples are constructed which illustrate the superiority of our results. Our results provide extension as well
as substantial generalizations and improvements of several well known results in the existing comparable
literature.

Keywords


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

[2] O. Acar, G. Durmaz, G. Minak, Generalized multivalued F-contractions on complete metric spaces , Bull. Iranian Math.
Soc, 40 (2014), 1469-1478. 

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

[4] M. Arshad, A. Hussain, M. Nazam, S. U. Khan, Some xed point results for multivalued Fcontraction on closed ball, Func.
Anal.-TMA 2 (2016), 69-80. 

[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. 

[6] M. Arshad, A. Shoaib, 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. 

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

[8] A. Azam, M. Waseem, M. Rashid, Fixed point theorems for fuzzy contractive mappings in quasi-pseudo-metric spaces,
Fixed Point Theory Appl., 2013 (2013):27. 

[9] S.Banach, Sur les op´ erations dans les ensembles abstraits et leur application aux equations itegrales, Fund. Math., 3 (1922) 133–181. 

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

[11] M. Cosentino, P. Vetro, Fixed point results for F-contractive mappings of Hardy-Rogers-Type, Filomat 28:4(2014), 715-722. doi:10.2298/FIL1404715C. 

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

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

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

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

[16] N. Hussain, P. Salimi, suzuki-wardowski type fixed point theorems for α-GF-contractions, Taiwanese J. Math., 20 (20)
(2014), doi: 10.11650/tjm.18.2014.4462.

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

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

[19] 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. 

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

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

[22] 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 and Appl., (2015) 2015:2.

[23] MA. Kutbi, M. Arshad, A.Hussain, Multivalued Ciric type α-η-GF-Contractions, J. Comput. Anal. Appl. Accepted. 

[24] MA. Kutbi, M. Arshad, A.Hussain, Fixed Point Results for Ciric type α-η-GF-Contractions, J. Comput. Anal. Appl. 21
(3) 2016, 466-481. 

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

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

[27] M. Nazam, M. Arshad, A. Hussain, Fixed Point Theorems For Chatterjea’s type Contraction on Closed ball, J. Ana. Num. Theor., 5(1) 2017 1-8. 

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

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

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

[31] NA. Secelean, Iterated function systems consisting of F -contractions, Fixed Point Theory Appl. 2013, Article ID 277
(2013). doi:10.1186/1687-1812-2013-277. 

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

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

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

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