Generalization of Fixed Point Results via Iterative Process of F-Contraction

Aftab Hussain^1, and Arshad Muhammad²

¹Department of Basic Sciences, Khwaja Fareed University of Engineering & Information Technology, Rahim Yar Khan Pakistan

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

Cite this paper:
Aftab Hussain and Arshad Muhammad. Generalization of Fixed Point Results via Iterative Process of F-Contraction. Turkish Journal of Analysis and Number Theory. 2017; 5(4):132-138. doi: 10.12691/tjant-5-4-3

Abstract

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.

Keywords:
metric space fixed point F contraction iterative process

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References:

[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.
[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.
[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.
[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.
[5]	S.Banach, Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales, Fund. Math., 3 (1922) 133-181.
[6]	R. Baskaran, P.V. Subrahmanyam, A note on the solution of a class of functional equations. Appl. Anal. 22(3-4), 235-241.
[7]	R. Bellman, Methods of Nonlinear Analysis. Vol. II. Mathematics in Science and Engineering, vol. 61. Academic Press, New York (1973).
[8]	R. Bellman, E.S. Lee, Functional equations in dynamic programming. Aequ. Math. 17, 1-18 (1978).
[9]	P.C. Bhakta, S. Mitra, Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl. 98, 348-362 (1984).
[10]	LB. Ćirić, 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.
[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]	N. Hussain and P. Salimi, suzuki-wardowski type fixed point theorems for α-GF-contractions, Taiwanese J. Math., 20 (20) (2014).
[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.
[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.
[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.
[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.
[19]	MA. Kutbi, M. Arshad and A. Hussain, On Modified α-η-Contractive mappings, Abstr. Appl. Anal., Volume 2014, Article ID 657858, 7 pages.
[20]	SB. Nadler, Multivalued contraction mappings, Pac. J. Math., 30 (1969), 475-488.
[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.
[22]	H. Piri and P. Kumam, Some fixed point theorems concerning F- contraction in complete metric spaces, Fixed Poin Theory Appl. 2014, 2014: 210.
[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).
[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.
[25]	P. Salimi, A. Latif and N. Hussain, Modified (α-ψ)-Contractive mappings with applications, Fixed Point Theory Appl. (2013) 2013: 151.
[26]	M. Sgroi, C. Vetro, Multi-valued F-contractions and the solution of certain functional and integral equations, Filomat 27: 7(2013), 1259-1268.
[27]	B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal. 75 (2012) 2154-2165.
[28]	NA. Secelean, Iterated function systems consisting of F-contractions, Fixed Point Theory Appl. 2013, Article ID 277 (2013).
[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).