Abstract
In this paper, we have some observations on F-weak contractions due to Wardowski and Van Dung (Demonstr Math 47(1):146–155, 2014). Our observations lead us to introduce the notion of \(F^*\)-weak contractions and utilize the same to prove some fixed point results. The proven results give an affirmative answer to certain open questions raised by Kannan (Bull Calcutta Math Soc 60:71–76, 1968) and Rhoades (Contemp. Math. 72:233–245, 1988) on the existence of contractive definitions not forcing the continuity at the fixed point. Some illustrative examples are also given. As an application, we investigate the existence and uniqueness of the solution of an integral equation of Volterra type.
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References
Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math 3(1), 133–181 (1922)
Berzig, M., Karapınar, E., Roldán-López-de Hierro, A.-F.: Discussion on generalized-(\(\alpha \psi \), \(\beta \))-contractive mappings via generalized altering distance function and related fixed point theorems. Abstr. Appl. Anal. 2014, 1–12 (2014)
Durmaz, G., Mınak, G., Altun, I.: Fixed points of ordered F-contractions. Hacet. J. Math. Stat. 45(1), 15–21 (2016)
Harjani, J., Sadarangani, K.: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. Theory Methods Appl. 72(3–4), 1188–1197 (2010)
Hussain, N., Ahmad, J.: New suzuki-berinde type fixed point results. Carpathian J. Math. 33(1), 59–72 (2017)
Imdad, M., Khan, Q., Alfaqih, W.M., Gubran, R.: A relation theoretic \(({F},\cal{R})\)-contraction principle with applications to matrix equations. Bull. Math. Anal. Appl. 10(1), 1–12 (2018)
Kannan, R.: Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71–76 (1968)
Pant, R.: Discontinuity and fixed points. J. Math. Anal. Appl. 240(1), 284–289 (1999)
Piri, H., Kumam, P.: Some fixed point theorems concerning F-contraction in complete metric spaces. Fixed Point Theory Appl. 2014(210), 1–11 (2014)
Rhoades, B.E.: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 226, 257–290 (1977)
Rhoades, B.E.: Contractive definitions and continuity. Contemp. Math. 72, 233–245 (1988)
Sawangsup, K., Sintunavarat, W., de Hierro, A.F.R.L.: Fixed point theorems for \(F_{\cal{R}}\)-contractions with applications to solution of nonlinear matrix equations. J. Fixed Point Theory Appl. 18, 1–15 (2016)
Secelean, N.-A.: Weak F-contractions and some fixed point results. Bull. Iran. Math. Soc. 42(3), 779–798 (2016)
Wardowski, D.: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012(94), 6 (2012)
Wardowski, D., Van Dung, N.: Fixed points of F-weak contractions on complete metric spaces. Demonstr. Math. 47(1), 146–155 (2014)
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Alfaqih, W.M., Imdad, M. & Gubran, R. An observation on F-weak contractions and discontinuity at the fixed point with an application. J. Fixed Point Theory Appl. 22, 66 (2020). https://doi.org/10.1007/s11784-020-00801-9
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DOI: https://doi.org/10.1007/s11784-020-00801-9