Abstract
In a metric space with a directed graph G, Jachymski (Proc Am Math Soc 1(136):1359–1373, 2008) introduced the concept of Banach G-contraction and proved two fixed point theorems for such mappings. Bojor (Nonlinear Anal 75:3895–3901, 2012) generalized this concept to Reich G-contraction and obtain a fixed point theorem. Note that Bojor’s theorem is established under the additional type of connectedness of G and it does not include Jachymski’s results as a special case. Moreover, there are some mistakes in several corollaries. Some examples and counterexamples are illustrated. It is our purpose to improve Bojor’s theorem and to present two fixed point theorems for Reich G-contractions. Our results are extensions of the two Jachymski’s theorems. Finally, we also discuss some priori error estimates.
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References
Alfuraidan, M.R.: On Reich fixed point theorem of \(G\)-contraction mappings on modular function spaces. J. Nonlinear Sci. Appl. 9, 4600–4606 (2016)
Alfuraidan, M.R., Bachar, M., Khamsi, M.A.: A graphical version of Reich’s fixed point theorem. J. Nonlinear Sci. Appl. 9, 3931–3938 (2016)
Alfuraidan, M.R., Khamsi, M.A.: On multivalued \(G\)-monotone Ćirić and Reich contraction mappings. Filomat 31(11), 3285–3290 (2017)
Bojor, F.: Fixed point theorems for Reich type contractions on metric spaces with a graph. Nonlinear Anal. 75, 3895–3901 (2012)
Boonsri, N., Saejung, S.: A fixed point theorem for Caristi-type cyclic mappings. Fixed Point Theory 18, 481–492 (2017)
Hicks, T.L., Rhoades, B.E.: A Banach type fixed-point theorem. Math. Jpn. 24(3), 327–330 (1979/1980)
Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 1(136), 1359–1373 (2008)
Nieto, J.J., Rodríguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2006)
Nieto, J.J., Rodríguez-López, R.: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. (Engl. Ser.) 23, 2205–2212 (2007)
Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2004)
Reich, S.: Some remarks concerning contraction mappings. Can. Math. Bull. 14, 121–124 (1971)
Reich, S.: Fixed points of contractive functions. Boll. Unione Mat. Ital. 5, 26–42 (1972)
Turinici, M.: Ran–Reurings fixed point results in ordered metric spaces. Lib. Math. 31, 49–55 (2011)
Turinici, M.: Ran–Reurings theorems in ordered metric spaces. J. Indian Math. Soc. (N.S.) 78, 207–214 (2011)
Acknowledgements
The first author is thankful to the Development and Promotion of Science and Technology Talents Project (DPST) for the financial support. The second author is supported by the Thailand Research Fund and Khon Kaen University under Grant RSA5980006.
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Boonsri, N., Saejung, S. Fixed point theorems for contractions of Reich type on a metric space with a graph. J. Fixed Point Theory Appl. 20, 84 (2018). https://doi.org/10.1007/s11784-018-0565-y
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DOI: https://doi.org/10.1007/s11784-018-0565-y