Abstract
In the present paper, we introduce the concept of F-contraction on vector-valued metric space. Then, we achieved a fixed point result that includes the famous Perov fixed point theorem as properly. We provided a nontrivial and illustrative example showing this fact. Finally, we give an existence result for semilinear operator system in a Banach space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abbas, M., Rakočević, V., Iqbal, A.: Fixed points of Perov type contractive mappings on the set endowed with a graphic structure. RACSAM 112, 209–228 (2018)
Cvetković, M.: On the equivalence between Perov fixed point theorem and Banach contraction principle. Filomat 31(11), 3137–3146 (2017)
Cvetković, M., Rakočević, V.: Common fixed point results for mappings of Perov type. Math. Nachr. 288(16), 1873–1890 (2015). https://doi.org/10.1002/mana.201400098
Cvetković, M., Rakočević, V.: Extensions of Perov theorem. Carpath. J. Math. 31(2), 181–188 (2015)
Filip, A.D., Petruşel, A.: Fixed point theorems on space endowed with vector valued metrics. Fixed Point Theory Appl. (2010). https://doi.org/10.1155/2010/281381
Ilić, D., Cvetković, M., Gajić, L., Rakoč ević, V.: Fixed points of sequence of Ćirić generalized contractions of Perov type. Mediterr. J. Math. 13, 3921–3937 (2016)
Mınak, G., Helvacı, A., Altun, I.: Ćirić type generalized \(F\)-contractions on complete metric spaces and fixed point results. Filomat 28(6), 1143–1151 (2014)
Perov, A.I.: On the Cauchy problem for a system of ordinary differential equations. Pviblizhen. Met. Reshen. Differ. Uvavn. 2, 115–134 (1964)
Petruşel, A., Mlesnite, O., Urs, C.: Vector-valued metrics in fixed point theory. In: Infinite Products of Operators and their Applications. In: Reich, S., Zaslavski A. J. (eds.) American Mathematical Society, Israel Mathematical Conference Proceedings, Contemporary Mathematics, vol. 636, pp. 149-165, (2015) https://doi.org/10.1090/conm/636/12734
Petruşel, A., Petruşel, G.: A study of a general system of operator equations in \(b\)-metric spaces via the vector approach in fixed point theory. J. Fixed Point Theory Appl. 19(3), 1793–1814 (2017)
Petruşel, A., Rus, I.A.: Fixed point theory in terms of a metric and of an order relation. Fixed Point Theory 20(2), 601–622 (2019)
Precup, R.: The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comput. Model. 49, 703–708 (2009)
Varga, R.S.: Matrix Iterative Analysis, Springer Series in Computational Mathematics, vol. 27. Springer, Berlin (2000)
Vetro, F., Radenović, S.: Some results of Perov type in rectangular cone metric spaces. J. Fixed Point Theory Appl. 20, 41 (2018). https://doi.org/10.1007/s11784-018-0520-y
Wardowski, D.: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 94 (2012). 6pp
Acknowledgements
The authors are thankful to the referee for making valuable suggestions leading to the better presentations of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Altun, I., Olgun, M. Fixed point results for Perov type F-contractions and an application. J. Fixed Point Theory Appl. 22, 46 (2020). https://doi.org/10.1007/s11784-020-00779-4
Published:
DOI: https://doi.org/10.1007/s11784-020-00779-4