Fixed point results concerning α-F-contraction mappings in metric spaces

[EN] In this paper, we introduce the notions of generalized α-F-contraction and modified generalized α-F-contraction. Then, we present sufficient conditions for existence and uniqueness of fixed points for the above kind of contractions. Necessarily, our results generalize and unify several results of the existing literature. Some examples are presented to substantiate the usability of our obtained results.The authors would like to thank the learned referee for his/her insightful comments and suggestions. The Research is funded by the Council of Scientific and Industrial Research (CSIR), Government of India under the Grant Number:25(0285)/18/EMR−II.Dey, LK.; Kumam, P.; Senapati, T. (2019). Fixed point results concerning α-F-contraction mappings in metric spaces. Applied General Topology. 20(1):81-95. https://doi.org/10.4995/agt.2019.99498195201S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181LB. Ciric, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc. 45, no. 2 (1974), 267-273. https://doi.org/10.2307/2040075P. Das and L. K. Dey, A fixed point theorem in a generalized metric space, Soochow J. Math. 33, no. 1 (2007), 33-39.P. Das and L. K. Dey, Fixed point of contractive mappings in generalized metric spaces, Math. Slovaca 59, no. 4 (2009), 499-504. https://doi.org/10.2478/s12175-009-0143-2L. K. Dey and S. Mondal, Best proximity point of F-contraction in complete metric space, Bull. Alahabad Math. Soc. 30, no. 2 (2015), 173-189.N. V. Dung and V. L. Hang, A fixed point theorem for generalized F-contractions on complete metric spaces, Vietnam. J. Math. 43, no. 4 (2015), 743-753.M. Edelstein, An extension of Banach's contraction principle, Proc. Amer. Math. Soc. 12, no. 1 (1961), 7-10.D. Gopal, M. Abbas, D. K. Patel and C. Vetro, Fixed points of α-type F-contractive mappings with an application to nonlinear fractional differential equation, Acta Math. Scientia 36, no. 3 (2016), 957-970. https://doi.org/10.1016/S0252-9602(16)30052-2N. Hussain, M. H. Shah, A. A. Harandi and Z. Akhtar, Common fixed point theorem for generalized contractive mappings with applications, Fixed Point Theory Appl. 2013:169, 2013. https://doi.org/10.1186/1687-1812-2013-169N. Hussain and P. Salimi, Suzuki-Wardowski type fixed point theorems for α-GF-contractions, Taiwanese J. Math. 18, no. 6 (2014), 1879-1895. https://doi.org/10.11650/tjm.18.2014.4462E. Karapinar, P. Kumam and P. Salimi, On α-ψ-Meir-Keeler contractive mappings, Fixed Point Theory Appl. 2013:9, 2013.H. Piri and P. Kumam, Some fixed point theorem concerning F-contractions in complete metric spaces, Fixed Point Theory Appl. 2014:210, 2014. https://doi.org/10.1186/1687-1812-2014-210H. Piri and P. Kumam, Wardowski type fixed point theorems in complete metric spaces, Fixed Point Theory Appl. 2016:45, 2016. https://doi.org/10.1186/s13663-016-0529-0B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal. 75, no. 4 (2012), 2154-2165. https://doi.org/10.1016/j.na.2011.10.014N. Secelean and D. Wardowski, ψF-contractions: not necessarily nonexpansive Picard operators, Results Math. 70, no. 3 (2016), 415-431. https://doi.org/10.1007/s00025-016-0570-7N. Secelean, Weak F-contractions and some fixed point results, Bull. Iranian Math. Soc. 42, no. 3 (2016), 779-798.T. Senapati, L. K. Dey and D.D. Dekic, Extensions of Ciric and Wardowski type fixed point theorems in D-generalized metric spaces, Fixed Point Theory Appl. 2016:33, 2016. https://doi.org/10.1186/s13663-016-0522-7S. Shukla, D. Gopal and J. M. Moreno, Fixed points of set-valued F-contractions and its application to non-linear integral equations, Filomat 31, no. 11 (2017), 3377-3390. https://doi.org/10.2298/FIL1711377SD. Wardowski, Fixed points of new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012:94, 2012. https://doi.org/10.1186/1687-1812-2012-94D. Wardowski and N. V. Dung, Fixed points of F-weak contractions on complete metric spaces, Demonstr. Math. 47, no. 1 (2014), 146-155. https://doi.org/10.2478/dema-2014-001