We obtain two generalizations of a
known theorem of A. Alam and M. Imdad (J. Fixed Point Theory Appl. 17 (2015)
693-702) showing that some standard proofs can be obtained involving only
Cauchy sequences of the successive approximations. Suitable examples prove the
effective generalization of our results in metric spaces not necessarily
complete.
References
[1]
Alam, A., Khan, A.R. and Imdad, M. (2014) Some Coincidence Theorems for Generalized Nonlinear Contractions in Ordered Metric Spaces with Applications. Fixed Point Theory and Applications, 2014, 216. https://doi.org/10.1186/1687-1812-2014-216
[2]
Alam, A. and Imdad, M. (2015) Relation-Theoretic Contraction Principle. Fixed Point Theory and Applications, 17, 693-702. https://doi.org/10.1007/s11784-015-0247-y
[3]
Alam, A. and Imdad, M. (2017) Relation-Theoretic Metrical Coincidence Theorems. Filomat, 31, 4421-4439. https://doi.org/10.2298/FIL1714421A
[4]
Alam, A., Khan, Q.H. and Imdad, M. (2017) Comparable Nonlinear Contractions in Ordered Metric Spaces. Journal of Nonlinear Sciences and Applications, 10, 1652-1674. https://doi.org/10.22436/jnsa.010.04.29
[5]
Baha, S.S. (2018) Best Proximity Point Theorems for Some Classes of Contractions. Acta Mathematica Hungarica, 156, 336-360. https://doi.org/10.1007/s10474-018-0882-z
[6]
Berzig, M. (2012) Coincidence and Common Fixed Point Results on Metric Spaces Endowed with an Arbitrary Binary Relation and Applications. Journal of Fixed Point Theory and Applications, 12, 221-238. https://doi.org/10.1007/s11784-013-0094-7
[7]
Nieto, J.J. and Rodrguez-Lopez, R. (2005) Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations. Order, 22, 223-239. https://doi.org/10.1007/s11083-005-9018-5
[8]
Popa, V. and Patriciu, A.-M. (2018) Other Fixed Point Theorems for Two Pairs of Absorbing Mappings Satisfying an Implicit Relation in Partial Metric Spaces. Acta Universitatis Apulensis, 55, 35-47.
[9]
Romaguera Bonilla, S. (2013) A Common Fixed Point Theorem for Weakly Compatible Self-Maps on Complete Preordered Quasi-Metric Spaces. In: Sessa, S., Ed., Weakly Commuting Mappings after Thirty Years in Metric Fixed Point Theory, Aracne Editrice int.le S.r.l., Rome, 55-70. https://doi.org/10.1186/1687-1812-2013-120
[10]
Sintunavarat, W., Plubtien, S. and Katchang, P. (2013) Fixed Point Result and Applications on a b-Metric Space Endowed with an Arbitrary Binary Relation. Fixed Point Theory and Applications, 2013, 296. https://doi.org/10.1186/1687-1812-2013-296
[11]
Turinici, M. (2013) Linear Contractions in Product Ordered Metric Spaces. Annali dell’ Università di Ferrara, 59, 187-198. https://doi.org/10.1007/s11565-012-0164-6
[12]
Alam, A. and Imdad, M. (2016) Monotone Generalized Contractions in Ordered Metric Spaces. Bulletin of the Korean Mathematical Society, 53, 61-81. https://doi.org/10.4134/BKMS.2016.53.1.061
[13]
Aleksic, S., Mitrovic, Z.D. and Radenovic, S. (2018) A Fixed Point Theorem of Jungck in bv(s)-Metric Spaces. Periodica Mathematica Hungarica, 77, 224-231. https://doi.org/10.1007/s10998-018-0236-1
[14]
Karapinar, E. and Agarwal, R.P. (2017) A Note on Ciric Type Nonunique Fixed Point Theorems. Fixed Point Theory and Applications, 2017, 20.
[15]
Suzuki, T. (2017) Basic Inequality on a b-Metric Space and Its Applications. Journal of Inequalities and Applications, 2017, 256. https://doi.org/10.1186/s13660-017-1528-3
[16]
Bonsangue, M.M., van Breugel, F. and Rutten, J.J.M.M. (1998) Generalized Metric Spaces: Completion, Topology, and Power Domains via the Yoneda Embedding. Theoretical Computer Science, 193, 1-51. https://doi.org/10.1016/S0304-3975(97)00042-X
[17]
Turkoglu, D. and Manav, N. (2018) Fixed Point Theorems in a New Type of Modular Metric Spaces. Fixed Point Theory and Applications, 2018, 25.
[18]
Cho, Y.Y. (2017) Bartlett Adjustments to the Likelihood Ratio Statistic and the Distribution of the Maximum Likelihood Estimator. In: Ruzhansky, M., et al., Eds., Advances in Real and Complex Analysis with Applications, Trends in Mathematics, Springer Nature, Singapore, 183-241.
[19]
Diviccaro, M.L. (2009) Fixed Point Theorems for Contractions in Posets. International Mathematical Forum, 4, 1709-1713.
[20]
Kolman, B., Busby, R.C. and Ross, S. (2009) Discrete Mathematical Structures. 6th Edition, Pearson, London.
[21]
Som, T., Kundu, A. and Choudhury, B.S. (2016) Some Results on Fixed Points of Weak Contractions for Non Compatible Mappings via (E.A)-Like Property. In: Cushing, J.M., et al., Eds., Applied Analysis in Biological and Physical Sciences, Springer Proceedings in Mathematics & Statistics 186, Springer India, New Delhi, 427-438. https://doi.org/10.1007/978-81-322-3640-5_27
