Some Generalized Contraction Classes and Common Fixed Points in b-Metric Space Endowed with a Graph †
Abstract
:1. Introduction
2. Preliminaries
- (1)
- F is continuous,
- (2)
- ,
- (3)
- or
- for each , for each there exists such that .
3. Main Results
3.1. R-Weakly Graph Preserving and R-Weakly -Admissible Mappings
- (9.1)
- For , there exists such that
- (9.2)
- For , there exists such that
- (13.1a)
- For , we can find such that .
- (13.1b)
- For , we can find such that .
3.2. Common Fixed Point Theorems in b-Metric Space Endowed with a Graph
- (14.1)
- there exists , and such that
- and
- (1.1)
- For some arbitrary there exists such that ,
- (1.2)
- S and T are pairwise R-weakly graph preserving,
- (1.3)
- for some and .
- (15.1)
- there exists , and such that
- and
- (2.1)
- for some , .
- (16.1)
- there exists , such that
- and
- (3.1)
- G satisfies transitivity property,
- (3.2)
- for some , .
- (1.1)
- For all with
- and
- (2.1)
- For all with
- and
- For , we have and
- For , we have and
- In addition, for , we have , , , . Simple calculations shows that , , , , , and
- (17.1)
- there exists , and such that
- and
- (4.1)
- rational contractions.
- (18.1)
- there exists , and such that
- and
- (5.1)
- rational contractions class.
3.3. Common Fixed Point Theorems for R-Weakly -Admissible Mappings in a b-Metric Space
- (6.1)
- There exists such that and ,
- (6.2)
- The pair is R-weakly α-admissible of type S,
- (6.3)
- for some , and for all with
- and
- (7.1)
- for some , and for all with
- and
- (8.1)
- α is a triangular function, that is if and then ,
- (8.2)
- for some , and for all with
- and
- (9.1)
- for some , and for all with
- and
- (10.1)
- for some , and for all with
- and
- (3.1)
- For all with
- and
4. Discussions
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Khan, M.S.; Swaleh, M.; Sessa, S. Fixed Point Theorems by Altering Distances Between the Points. Bull. Aust. Math. Soc. 1984, 30, 1–9. [Google Scholar] [CrossRef]
- Haitham, Q.; Noorani, M.S.N.; Shatanawi, W.; Aydi, H.; Alsamir, H. Fixed point results for multi-valued contractions in b-metric spaces and an application. Mathematics 2019, 7, 132. [Google Scholar]
- George, R.; Tamrakar, E.; Vujaković, J.; Pathak, H.K.; Velusamy, S. (C,ψ*,G) class of contractions and fixed points in a metric space endowed with a graph. Mathematics 2019, 7, 481. [Google Scholar] [CrossRef]
- Doric, D. Common Fixed Point for Generalized (ψ − φ) Contractions. Appl. Math. Lett. 2009, 22, 1896–1900. [Google Scholar] [CrossRef]
- Rhoades, B.E. Some theorems on weakly contractive maps. Nonlinear Anal. 2001, 47, 2683–2693. [Google Scholar] [CrossRef]
- Dutta, P.N.; Choudhury, B.S. A generalization of contraction principle in metric spaces. Fixed Point Theory Appl. 2008, 8, 406368. [Google Scholar] [CrossRef]
- Zhang, Q.; Song, Y. Fixed point theory for generalized ϕ-weak contractions. Appl. Math. Lett. 2009, 22, 75–78. [Google Scholar] [CrossRef]
- Jachymski, J. The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 2008, 136, 1359–1373. [Google Scholar] [CrossRef]
- Phon-on, A.; Sama, A.; Makaje, N.; Riyapan, P.; Busaman, S. Coincidence point theorems for weak graph preserving multi-valued mapping. Fixed Point Theory Appl. 2014, 2014, 248. [Google Scholar] [CrossRef] [Green Version]
- Bojor, F. Fixed point theorems for Reich type contractios on metric spaces with graph. Nonlinear Anal. 2012, 75, 3895–3901. [Google Scholar] [CrossRef]
- Mohanta, S.; Patra, S. Coincidence Points and Common Fixed points for Hybrid Pair of Mappings in b-metric spaces Endowed with a Graph. J. Linear Topol. Algebra 2017, 6, 301–321. [Google Scholar]
- Cholamjiak, W.; Suantai, S.; Suparatulatorn, R.; Kesornprom, S.; Cholamjiak, P. Viscosity approximation methods for fixed point problems in Hilbert spaces endowed with graphs. J. Appl. Numer. Optim. 2019, 1, 25–38. [Google Scholar]
- Sauntai, S.; Charoensawan, P.; Lampert, T.A. Common Coupled Fixed Point Theorems for θ − ψ-Contraction mappings Endowed with a Directed Graph. Fixed Point Theory Appl. 2015, 2015, 224. [Google Scholar] [CrossRef]
- Sultana, A.; Vetrivel, V. Fixed points of Mizoguchi-Takahashi contraction on a metric space with a graph and applications. J. Math. Anal. Appl. 2014, 417, 336–344. [Google Scholar] [CrossRef]
- Tiammee, J.; Suantai, S. Coincidence point theorems for graph-preserving multi-valued mappings. Fixed Point Theory Appl. 2014, 2014, 70. [Google Scholar] [CrossRef] [Green Version]
- Beg, I.; Butt, A.R.; Radojevic, S. The contraction principle for set valued mappings on a metric space with a graph. Comput. Math. Appl. 2010, 60, 1214–1219. [Google Scholar] [CrossRef] [Green Version]
- Alfuraidan, M.R. Remarks on monotone multivalued mappings on a metric spaces with a graph. J. Inequal. Appl. 2015, 2015, 202. [Google Scholar] [CrossRef]
- Hanjing, A.; Suantai, S. Coincidence point and fixed point theorems for a new type of G-contraction multivalued mappings on a metric space endowedwith a graph. Fixed Point Theory Appl. 2015, 2015, 171. [Google Scholar] [CrossRef]
- Nicolae, A.; O’Regan, D.; Petrusel, A. Fixed point theorems for single-valued and multi-valued generalized contractions in metric spaces endowed with a graph. Georgian Math. J. 2011, 18, 307–327. [Google Scholar]
- Samet, B.; Vetro, C.; Vetro, P. Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Anal. 2012, 75, 2154–2165. [Google Scholar] [CrossRef]
- Sintunavarat, W. Nonlinear integral equations with new admissibility types in b- metric spaces. J. Fixed Point Theory Appl. 2016, 18, 397–416. [Google Scholar] [CrossRef]
- Arshad, M.; Ameer, E.; Karapinar, E. Generalized contractions with triangular α-orbital admissible mapping on Branciari metric spaces. J. Inequal. Appl. 2016, 2016, 63. [Google Scholar] [CrossRef]
- Ameer, E.; Arshad, M.; Shatanawi, W. Common fixed point results for generalized α* − ψ−contraction multivalued mappings in b-metric spaces. J. Fixed Point Theory Appl. 2017, 19, 3069–3086. [Google Scholar] [CrossRef]
- Ansari, A.H. Note on φ–ψ-contractive type mappings and related fixed point. In The 2nd Regional Conference on Mathematics And Applications; Payame Noor University: Tehran, Iran, 2014; pp. 377–380. [Google Scholar]
- Miculescu, R.; Mihail, A. New fixed point theorems for set-valued contractions in b-metric spaces. J. Fixed Point Theory Appl. 2017, 19, 2153–2163. [Google Scholar] [CrossRef]
- Aleksić, S.; Dosenovic, T.; Mitrović, Z.D.; Radenović, S. Remarks on common fixed point results for generalized α* − ψ-contraction multivalued mappings in b-metric spaces. Adv. Fixed Point Theory 2019, 9, 1–16. [Google Scholar]
- Huang, H.; Deng, G.; Chen, Z.; Radenović, S. On some recent fixed point results for α-admissible mappings in b-metric spaces. J. Comp. Anal. Appl. 2018, 25, 255–269. [Google Scholar]
- Zhao, X.Y.; He, F.; Sun, Y.Q. Common fixed point results for quasi-contractions of Ciric type in b-metric spaces with Qt-functions. J. Nonlinear Funct. Anal. 2017, 2017, 1–18. [Google Scholar] [CrossRef]
- Simkhah, M.; Turkoglu, D.; Sedghi, S.; Shobe, N. Suzuki type fixed point results in p-metric spaces. Commun. Optim. Theory 2019, 2019, 1–12. [Google Scholar] [CrossRef]
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George, R.; Nabwey, H.A.; Ramaswamy, R.; Radenović, S. Some Generalized Contraction Classes and Common Fixed Points in b-Metric Space Endowed with a Graph. Mathematics 2019, 7, 754. https://doi.org/10.3390/math7080754
George R, Nabwey HA, Ramaswamy R, Radenović S. Some Generalized Contraction Classes and Common Fixed Points in b-Metric Space Endowed with a Graph. Mathematics. 2019; 7(8):754. https://doi.org/10.3390/math7080754
Chicago/Turabian StyleGeorge, Reny, Hossam A. Nabwey, Rajagopalan Ramaswamy, and Stojan Radenović. 2019. "Some Generalized Contraction Classes and Common Fixed Points in b-Metric Space Endowed with a Graph" Mathematics 7, no. 8: 754. https://doi.org/10.3390/math7080754