‎Nonlinear Contraction Mappings in b-metric Space and Related Fixed Point Results with Application

Subject Areas : Transactions on Fuzzy Sets and Systems

Muhammed Raji ¹ , Arvind Kumar Rajpoot ² , Laxmi Rathour ³ , Lakshmi Narayan Mishra ⁴ , Vishnu Narayan Mishra ⁵

1 - Department of Mathematics, Confluence University of Science and Technology, Osara, Kogi State, Nigeria.
2 - Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India.
3 - Department of Mathematics, National Institute of Technology, Chaltlang, Aizawl 796 012, Mizoram, India.
4 - Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, Tamil Nadu, India.
5 - Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484 887, India.

Received: 2024-05-27 Accepted : 2024-08-19 Published : 2024-11-05

Keywords: Fixed point‎, Coincidence point‎, ‎Eldeisten-Suzuki-type contraction‎, ‎b-metric space‎,

Abstract :

The paper aims to introduce some fixed point results in the setting of sequential compact b-metric spaces to prove Eldeisten-Suzuki-type contraction for self-mappings‎. ‎These contributions extend the existing literature on fixed point for ordered metric spaces and fixed point theory‎. ‎Through illustrative examples‎, ‎we showcase the practical applicability of our proposed notions and results‎, ‎demonstrating their effectiveness in real-world scenarios‎.

References:

[1] Banach S. Sur les op´erations dans les ensembles abstraits et leur application aux equations itegrales. Fundamenta mathematicae. 1922; 3(1): 133-181. DOI: https://doi.org/10.4064/fm-3-1-133-181
[2] Ran ACM, Reurings MCB. A fixed point theorem in partially ordered sets and some applications to matrix equations. proceedings of the American Mathematical Society. 2004; 132(5): 1435-1443. https://www.ams.org/journals/proc/2004-132-05/S0002-9939-03-07220-4/S0002-9939-03-07220-4.pdf
[3] Nieto JJ, Rodrıguez-Lopez R. Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Mathematica Sinica, English Series. 2007; 23(12): 2205-2212. DOI: https://doi.org/10.1007/s10114-005-0769-0
[4] Rus I A, Petru¸sel A, Petru¸sel G. Fixed Point Theory. Cluj-Napoca: Cluj University Press; 2(2), 2008.
[5] Raji M, Ibrahim MA. Fixed point theorems for fuzzy contractions mappings in a dislocated bmetric spaces with applications. Annals of Mathematics and Computer Science. 2024; 21: 1-13. DOI: https://doi.org/10.56947/amcs.v21.233
[6] Shah MH, Hussain N. Nonlinear contractions in partially ordered quasi b-metric spaces. Communications of the Korean Mathematical Society. 2012; 27(1): 117-128. DOI: http://dx.doi.org/10.4134/CKMS.2012.27.1.117
[7] Kannan R. Some results on fixed points. Bull Calcutta Math Soc. 1968; 60: 71-76. DOI: https://doi.org/10.2307/2316437
[8] Raji M, Ibrahim MA, Rauf K, Kehinde R. Common fixed point results for fuzzy Fcontractive mappings in a dislocated metric spaces with application. Qeios. 2024; 7; 1-21. DOI: https://doi.org/10.32388/SV98CN
[9] Dewangan K, Rathour L, Mishra VN, Raiz M. On multi-valued nonexpansive mappings in UCBS, Journal of Computational Analysis and Applications. 2024; 33(1): 396-406. https://eudoxuspress.com/index.php/pub/article/view/31
[10] Mishra LN, Raiz M, Rathour L, Mishra VN. Tauberian Theorems for Weighted Means of Double Sequences in Intuitionistic Fuzzy Normed Spaces. Yugoslav Journal of Operations Research. 2022; 377-388. DOI: https://doi.org/10.2298/YJOR210915005M
[11] Czerwik S. Contraction mappings in b-metric spaces. Acta mathematica et informatica universitatis ostraviensis. 1993; 1(1): 5-11.
[12] Boriceanu M. Strict fixed point theorems for multivalued operators in b-metric Spaces. International Journal of Modern Mathematical Sciences. 2009; 4(3): 285-301.
[13] Hussain N, Doric D, Kadelburg Z, Radenovic S. Suzuki-type fixed point results in metric type spaces. Fixed point theory and applications. 2012; 2012: 1-12. DOI: https://doi.org/10.1186/1687-1812-2012-126
[14] Edelstein M. On fixed and periodic points under contractive mappings. Journal of the London Mathematical Society. 1962; 1(1): 74-79. DOI: https://doi.org/10.1112/jlms/s1-37.1.74
[15] Suzuki T. A new type of fixed point theorem in metric spaces. Nonlinear Analysis: Theory, Methods & Applications. 2009; 71(11): 5313-5317. DOI: https://doi.org/10.1016/j.na.2009.04.017
[16] Doric D, Kadelburg Z, Radenovic S. Edelstein-Suzuki-type fixed point results in metric and abstract metric spaces.Nonlinear Analysis: Theory, Methods & Applications. 2012; 75(4): 1927-1932. DOI: https://doi.org/10.1016/j.na.2011.09.046
[17] Paesano D, Vetro P. Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topology and its Applications. 2012; 159(3): 911-920. DOI:
https://doi.org/10.1016/j.topol.2011.12.008
[18] Salimi P, Karapinar E. Suzuki-Edelstein type contractions via auxiliary functions. Mathematical Problems in Engineering. 2013; 2013(1): 648528. DOI: https://doi.org/10.1155/2013/648528
[19] Radenovic S, Salimi P, Vetro C, Dosenovic T. Edelstein-Suzuki-type results for self-mappings in various abstract spaces with application to functional equations. Acta Mathematica Scientia. 36(1): 94-110. DOI:
https://doi.org/10.1016/S0252-9602(15)30081-3
[20] Cosentino M, Salimi P, Vetro P. Fixed point results on metric-type spaces. Acta Mathematica Scientia. 2014; 34(4):1237-1253. DOI: https://doi.org/10.1016/S0252-9602(14)60082-5
[21] Czerwik S. Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem. Mat. Univ. Modena. 1998; 46: 263-276.
[22] Nieto JJ, Rodrıguez-Lopez R. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order. 2005; 22: 223-239. DOI: https://doi.org/10.1007/s11083-005- 9018-5
[23] Raji M, Rathour L, Mishra LN, Mishra VN. Generalized twisted (α, β)-ψ contractive type mappings and related fixed point results with applications. Int. J. Adv. Sci. Eng. 2024; 10(4): 3639-3654. DOI:
https://doi.org/10.29294/IJASE.10.4.2024.3639-3654
[24] Raji M. Generalized α-ψ contractive type mappings and related coincidence fixed point theorems with applications.The Journal of Analysis. 2023; 31(2): 1241-1256. DOI: https://doi.org/10.1007/s41478-022- 00498-8
[25] Raji M, Ibrahim MA. Fixed point theorems for modified F-weak contractions via α- admissible mapping with application to periodic points. Annals of Mathematics and Computer Science. 2024; 20: 82-97. DOI:
https://doi.org/10.56947/amcs.v20.232
[26] Nieto JJ, Pouso RL, Rodrıguez-Lopez R. Fixed point theorems in ordered abstract spaces. Proceedings of the American Mathematical Society. 2007; 135(8): 2505-2517. DOI: https://doi.org/10.1090/S0002- 9939-07-08729-1