Abstract
In this paper, some fixed point results for relation theoretic weak \(\varphi \)-contractions on cone metric spaces which is defined over a Banach algebra and equipped with a binary relation are proved. These results extend and generalize several results of metric and cone metric spaces. Some examples are provided which demonstrate the results proved herein. An application of our main result to a system of Volterra type equations is given.
Similar content being viewed by others
References
Alam, A., Imdad, M.: Relation-theoretic contraction principle. J. Fixed Point Theory Appl. 17(4), 693–702 (2015)
Alam, A., Imdad, M.: Nonlinear contractions in metric spaces under locally \(T\)-transitive binary relations. Fixed Point Theory 19(1), 13–24 (2018)
Aleksic, S., Kadelburg, Z., Mitrovic, Z., Radenović, S.: A new survey: cone metric spaces. J. Int. Math. Virtual Inst. 9, 93–121 (2019)
Altun, I., Rakočević, V.: Ordered cone metric spaces and fixed point results. Comput. Math. Appl. 60, 1145–1151 (2010)
Azam, A., Arshad, M., Beg, I.: Common fixed points of two maps in cone metric spaces. Rend. Cir. Mat. Palermo 57, 433–441 (2008)
Azam, A., Beg, I., Arshad, M.: Fixed point in topological vector space-valued cone metric spaces. Fixed Point Theory Appl. 2010(1), 604084 (2010)
Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations integrals. Fundam. Math. 3, 133–181 (1922)
Ben-El-Mechaiekh, H.: The Ran-Reurings fixed point theorem without partial order: a simple proof. J. Fixed Point Theory Appl. 16(01), 337–383 (2014)
Dordević, M., Dorić, D., Kadelburg, Z., Radenović, S., Spasić, D.: Fixed point results under c-distance in tvs-cone metric spaces. Fixed Point Theory Appl. 2011, 29 (2011)
Huang, H., Deng, G.-T., Radenović, S.: Some topological properties and fixed point results in cone metric spaces over Banach algebras. Positivity 23(01), 21–34 (2019)
Huang, H., Radenović, S.: Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras and applications. J. Nonlinear Sci. Appl. 8, 787–799 (2015)
Huang, H., Radenović, S.: Some fixed point results of generalized Lipschitz mappings on cone \(b\)-metric spaces over Banach algebras. J. Comput. Anal. Appl. 20(3), 566–583 (2016)
Huang, L.G., Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332(2), 1468–1476 (2007)
Janković, S., Kadelburg, Z., Radenović, S.: On cone metric spaces: a survey. Nonlinear Anal. Theory Methods Appl. Int. Multidiscip. J. 74(7), 2591–2601 (2011)
Jungck, G., Radenović, S., Radojević, S., Rakočević, V.: Common fixed point theorems for weakly compatible pairs on cone metric spaces. Fixed Point Theory Appl. 2009(1), 13 (2009). https://doi.org/10.1155/2009/643840. Article ID 643840
Kadelburg, Z., Radenović, S., Rakočević, V.: Remarks on Quasi-contraction on a cone metric space. Appl. Math. Lett. 22(11), 1674–1679 (2009)
Li, B., Huang, H.: Fixed point results for weak \(\varphi \)-contractions in cone metric spaces over Banach algebras and applications. J. Funct. Spaces 2017, 6 (2017)
Liu, H., Xu, S.-Y.: Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings. Fixed Point Theory Appl. 2013, 320 (2013). https://doi.org/10.1186/1687-1812-2013-320
Malhotra, S.K., Sharma, J.B., Shukla, S.: Relation-theoretic contraction principle in cone metric spaces with Banach algebra. Ser. A: Appl. Math. Inf. Mech. 8(1), 87–100 (2016)
Malhotra, S.K., Sharma, J.B., Shukla, S.: Fixed points of generalized Kannan type \(\alpha \)-admissible mappings in cone metric spaces with Banach algebra. Theory Appl. Math. Comput. Sci. 7(1), 1–13 (2017)
Mehmood, N., Rawashdeh, A.-A., Radenović, S.: New fixed point results for E-metric spaces. Positivity 23(5), 1101–1111 (2019)
Nieto, J.J., López, R.R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equation. Order 2005(2005), 223–239 (2005)
Radenović, S., Vetro, F.: Some remarks on Perov type mappings in cone metric spaces. Mediterr. J. Math. 14(6), 240 (2017)
Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some application to matrix equations. Proc. Am. Math. Sco. 132, 1435–1443 (2004)
Roldán-López-De-Hierro, A., Karapinar, E., De-La-Sen, M.: Coincidence point theorems in quasi-metric spaces without assuming the mixed monotone property and consequences in \(G\)-metric spaces. Fixed Point Theory Appl. 2014, 184 (2014)
Turinici, M.: Ran-Reurings fixed point results in ordered metric spaces. Lib. Math. 31, 49–55 (2011)
Turinici, M.: Nieto–Lopez theorems in ordered metric spaces. Math. Stud. 81, 219–229 (2012)
Turinici, M.: Linear contractions in product ordered metric spaces. Ann. Univ. Ferrara Sez. VII Sci. Mat. 59, 187–198 (2013)
Vetro, F., Radenović, S.: Some results of Perov type on rectangular cone metric spaces. J. Fixed Point Theory Appl. 20(1), 41 (2018)
Vetro, P.: Common fixed points in cone metric spaces. Rend. Cir. Mat. Palermo, Serie II 56, 464–468 (2007)
Xu, S., Radenović, S.: Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality. Fixed Point Theory Appl. 2014, 1–12 (2014)
Acknowledgements
Authors are thankful to the Editor and Reviewers of this paper for their critical and important suggestions.
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
Shukla, S., Dubey, N. Some fixed point results for relation theoretic weak \(\varphi \)-contractions in cone metric spaces equipped with a binary relation and application to the system of Volterra type equations. Positivity 24, 1041–1059 (2020). https://doi.org/10.1007/s11117-019-00719-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-019-00719-8
Keywords
- Fixed point
- Cone metric space
- Relation theoretic weak \(\varphi \)-contraction
- Transitive relation
- Integral equation