Fixed point results and application for cyclic contractive maps in b-metric spaces

Authors

Abbas Karim Nahi Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad, Baghdad, Iraq https://orcid.org/0009-0006-3158-1348
Salwa Salman Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad, Baghdad, Iraq. https://orcid.org/0000-0002-0581-253X

DOI:

https://doi.org/10.30526/38.2.3828

Keywords:

Altering distance functions, Complete b-metric spaces, Contractive conditions, Cyclic representation, Fixed points

Abstract

One of the generalizations of the usual metric function is b-metric, which allows the researchers a wider aria to derive many results and applications regarding the fixed point theory. This aim of the article is to advance new three fixed point principles in complete b-metric space instance is continuous function in two variables. Here, there are three directions to prove existence and uniqueness of fixed point. First, deriving a result in sense of Branciari’s theorem by combining integral contractive conditions with the idea of cyclic map. Second, applying the notion of cyclic representation regarding maps satisfying general weak conditions including an altering distance function to simulate the content of Boyd and Wong theorem. By this result, an application is given about the existence and uniqueness solution of an integral equation. Lastly, using an implicit relation with an altering distance function to construct cyclic contractive map. Furthermore, some examples are presented to analyze and illustrate the main results.

Author Biographies

Abbas Karim Nahi, Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad, Baghdad, Iraq

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Salwa Salman , Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad, Baghdad, Iraq.

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References

1. Kirk WA, Srinivasan PS, Veeramani P. Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory. 2003;4(1):79-89.

2. Asem V, Singh YM, Khan MS, Sessa S. On (α, p)-Cyclic Contractions and Related Fixed Point Theorems. Symmetry. 2023;15(10):1826. http://doi.org/10.3390/sym15101826

3. Konar P, Bhandari SK, Chandok S, Mukheimer A. Multivalued weak cyclic δ-contraction mappings. Journal of Inequalities and Applications. 2020;2020(1):1-11. http://doi.org/10.1186/s13660-020-02373-1

4. Karapinar E, Sadarangani K. Fixed point theory for cyclic (ϕ-ψ)-contractions. Fixed Point Theory and Applications. 2011;2011(1):1-8. http://doi.org/10.1186/1687-1812-2011-1

5. Hussain N, Parvaneh V, Roshan JR, Kadelburg Z. Fixed points of cyclic weakly (ψ, φ, L, A, B)-contractive mappings in ordered b-metric spaces with applications. Fixed Point Theory and Applications. 2013;2013(1):1-18. http://doi.org/10.1186/1687-1812-2013-1

6. Hiranmoy G, Karapınar E, Kanta Dey L. Best proximity point results for contractive and cyclic contractive type mappings. Numerical Functional Analysis and Optimization. 2021;42(7):849-864. http://doi.org/10.1080/01630563.2021.1929578

7. Chaipunya P, Cho YJ, Sintunavarat W, Kumam P. Fixed point and common fixed point theorems for cyclic quasi-contractions in metric and ultrametric spaces. Advances in Pure Mathematics. 2012;2(6):401-407. http://doi.org/10.4236/apm.2012.26059

8. Abou Bakr, Sahar MA. Cyclic G‐Ω-Weak Contraction-Weak Nonexpansive Mappings and Some Fixed Point Theorems in Metric Spaces. Abstract and Applied Analysis. 2021;2021:1-9.

9. Karapınar E, Romaguera S, Taş K. Fixed points for cyclic orbital generalized contractions on complete metric spaces. Open Mathematics. 2013;11(3):552-560. http://doi.org/10.2478/s11533-013-0245-5

10. Bakhtin I. The contraction mapping principle in quasimetric spaces. Functional Analysis. 1989;30:26-37.

11. Al-Bundi SS. Iterated function system in metric spaces. Bol Soc Paran Mat. 2022;40:1-10.

12. Abed SS. Fixed point principles in general b-metric Spaces and b-Menger probabilistic spaces. Journal of AL-Qadisiyah for Computer Science and Mathematics. 2018;10(2):42.

13. Karapınar E. Fixed point theory for cyclic weak ϕ-contraction. Applied Mathematics Letters. 2011;24(6):822-825. http://doi.org/10.1016/j.aml.2011.01.021

14. Hammad HA, De la Sen M. Solution of nonlinear integral equation via fixed point of cyclic α L ψ-rational contraction mappings in metric-like spaces. Bulletin of the Brazilian Mathematical Society. 2020;51(1):81-105. http://doi.org/10.1007/s00574-019-00162-1

15. Jabbar HA, Kadhim SN, Abed SS. Best approximation in b-modular spaces. Baghdad Science Journal. 2023. http://doi.org/10.21123/bsj.2023.8230

16. Qawaqneh H, Noorani M, Shatanawi W, Alsamir H. Common fixed point theorems for generalized geraghty (α, ψ, ϕ)-quasi contraction type mapping in partially ordered metric-like spaces. Axioms. 2018;7(4):74. http://doi.org/10.3390/axioms7040074

17. Mustafa Z, Mustafa Z, Khan SU, Jaradat MMM, Arshad M, Jaradat HM. Fixed point results of F-rational cyclic contractive mappings on 0-complete partial metric spaces. Forum. 2018;40:394-409.

18. Păcurar M, Rus IA. Fixed point theory of cyclic operators. Journal of Fixed Point Theory and Applications. 2022;24(4):79.

19. Theivaraman R, Srinivasan P, Radenovic S, Park C. New approximate fixed point results for various cyclic contraction operators on E-metric spaces. Journal of the Korean Society for Industrial and Applied Mathematics. 2023;27(3):160-179.

20. Kumari PS, Panthi D. Cyclic contractions and fixed point theorems on various generating spaces. Fixed Point Theory and Applications. 2015;2015(1):1-17. http://doi.org/10.1186/s13663-015-0365-7

21. Hussein Luaibi H, Abed SS. Fixed point theorems in general metric space with an application. Baghdad Science Journal. 2021;18(1 Suppl.):0812.

22. Branciari A. A fixed point theorem for mappings satisfying a general contractive condition of integral type. International Journal of Mathematics and Mathematical Sciences. 2002;29:531-536. http://doi.org/10.1155/S0161171202007524

23. Khan MS, Swaleh M, Sessa S. Fixed point theorems by altering distances between the points. Bulletin of the Australian Mathematical Society. 1984;30(1):1-9. http://doi.org/10.1017/S0004972700001659

24. Nieto JJ, Rodríguez-López R. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order. 2005;22(3):223-239. http://doi.org/10.1007/s11083-005-9018-5

25. Rabaiah A, Tallafha A, Shatanawi W. Common fixed point results for mappings under nonlinear contraction of cyclic form in b-metric spaces. Nonlinear Functional Analysis and Applications. 2021;26(2):289-301.

26. Nashine HK, Kadelburg Z, Kumam P. Implicit-relation-type cyclic contractive mappings and applications to integral equations. Abstract and Applied Analysis. 2012;2012:1-15.

27. Matkowski J. Fixed point theorems for mappings with a contractive iterate at a point. Proceedings of the American Mathematical Society. 1977;62(2):344-348. http://doi.org/10.2307/2041931

28. Abed SS, Abed AN. Convergence and Stability of Iterative Scheme for a Monotone Total Asymptotically Non-expansive Mapping. Iraqi Journal of Science. 2022;63(1):241-250.

29. Abed SS, Taresh NS. On stability of iterative sequences with error. Mathematics. 2019;7(8):765. http://doi.org/10.3390/math7080765

30. Monje ZAM, Ahmed BA. A study of stability of first-order delay differential equations using fixed point theorem Banach. Iraqi Journal of Science. 2019;60(12):2719-2724