Korean J. Math. Vol. 32 No. 3 (2024) pp.561-591
DOI: https://doi.org/10.11568/kjm.2024.32.3.561

Cone $\mathfrak{C}$-class functions using $(CLR_{\Gamma \mathfrak{L} })$-property on cone $b$-normed spaces with application

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Published: Sep 30, 2024
Keywords:
cone b-metric space,, Banach space, common fixed point, CLR-property,, normed space
Mathematics Subject Classification:
47H10, 54H25

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K. Maheshwaran
Department of mathematics, School of Engineering and Technology, Dhanalakshmi Srinivasan University, Perambalur-621212,Tamil Nadu, India.
Arslan Hojat Ansari
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa-0204, South Africa.
Stojan N Radenovic
Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Belgrade 11120, Serbia.
M.S. Khan
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa-0204, South Africa.
Yumnam Mahendra Singh
Department of Basic Science and Humanities, Manipur Institute of Technology (A Constituent College of Manipur University), Takyelpat - 795004, India.

Abstract

In this article, we demonstrate the conditions for the existence of common fixed points $(CFP)$ theorems for four self-maps satisfying the common limit range $(CLR)$-property on cone $b$-normed spaces $(CbNS)$ via $\mathfrak{C}$-class functions. Furthermore, we have a unique common fixed point for two weakly compatible $ (WC)$ pairings. Towards the end, the existence and uniqueness of common solutions for systems of functional equations arising in dynamic programming are discussed as an application of our main result.



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References

[1] M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Appl. 341 (2008), 1.3, 1.8. https://dx.doi.org/10.1016/j.jmaa.2007.09.070 Google Scholar

[2] S. Aleksic, Z. Kadelburg, Z.D. Mitrovic and S. Radenovic, A new survey: Cone metric spaces, J. Inter. Math. Virtual Institute 9 (2019), 93–121. Google Scholar

[3] A.H. Ansari, Note on φ−ψ-contractive type mappings and related fixed points, The 2nd Regional Conference on Mathematics and Applications PNU (2014), 377–380. Google Scholar

[4] A.H. Ansari, S. Chandok, N. Hussin and L. Paunovic, Fixed points of (ψ, φ)-weak contractions in regular cone metric spaces via new function, J. Adv. Math. Stud. 9 (1) (2016), 72–82. https://www.isroset.org/pub_paper/IJSRMSS/32-IJSRMSS-01061.pdf Google Scholar

[5] A. Aghajani, M. Abbas and J.R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca 64 (4) (2014), 941–960. https://dx.doi.org/10.2478/s12175-014-0250-6 Google Scholar

[6] I.A. Bakhtin, The contraction mapping principle in almost metric space, Funct. Anal. Gos. Ped. Inst. Unianowsk 30 (1989), 26–37. https://www.scirp.org/reference/referencespapers?referenceid=2207100 Google Scholar

[7] R. Bellman and E.S. Lee, Functional equations in dynamic programming, Aequationes Mathematicae 17 (1) (1978), 1–18. https://dx.doi.org/10.1007/BF01818535 Google Scholar

[8] M. Boriceanu, M. Bota and A. Petrusel, Mutivalued fractals in b-metric spaces, Cent. Eur. J. Math. 8 (2) (2010), 367–377. https://dx.doi.org/10.2478/s11533-010-0009-4 Google Scholar

[9] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math Inform Univ Ostrav 5 (1993), 5–11. http://dml.cz/dmlcz/120469 Google Scholar

[10] E. Hewitt and K. Stromberg, Real and Abstract Analysis, New York: Springer-Verlag, (1965). https://link.springer.com/book/10.1007/978-3-642-88044-5 Google Scholar

[11] L.G. Huang and X. Zhang, Cone metric space and fixed point theorems of contractive mappings, J. Mat. Anal. App. 332 (2) (2007), 1468–1476. https://dx.doi.org/10.1016/j.jmaa.2005.03.087 Google Scholar

[12] H. Huang and S. Xu, Fixed point theorems of contractive mappings in cone b-metric spaces and applications, Fixed Point Theory Appl. 112 (2013), 1–10. https://dx.doi.org/10.1186/1687-1812-2014-55 Google Scholar

[13] N. Hussain and M.H. Shah, KKM mappings in cone b-metric space, Comp. Math. Appl. 62 (2011), 1677–1684. https://dx.doi.org/10.1016/j.camwa.2011.06.004 Google Scholar

[14] S. Jankovic, Z. Kadelburg and S. Radenovic, On cone metric spaces: a survey, Nonlinear Analysis: Theory, Methods & Applications 74 (7) (2011), 2591–2601. https://dx.doi.org/10.1016/j.na.2010.12.014 Google Scholar

[15] Jitender Kumar, Common Fixed Point Theorems Of Weakly Compatible Maps Satisfying (E.A.) and (CLR) Property, International Journal of Pure and Applied Mathematics 88 (3) (2013), 363–376. https://dx.doi.org/10.12732/ijpam.v88i3.4 Google Scholar

[16] G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Sci. 9 (4) (1986), 771–777. https://dx.doi.org/10.1155/S0161171286000935 Google Scholar

[17] E. Karapinar, Fixed Point Theorems in cone Banach spaces, Fixed Point Theory and Applications (2009), 1087–1093. https://dx.doi.org/10.1155/2009/609281 Google Scholar

[18] M.S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc. 30 (1984), 1–9. https://dx.doi.org/10.1017/S0004972700001659 Google Scholar

[19] Z. Liu, X. Li, S.M. Kang and S.Y. Cho, Fixed point theorems for mappings satisfying contractive conditions of integral type and applications, Fixed Point Theory and Applications 64 (2011), 1–18. https://dx.doi.org/10.1186/1687-1812-2011-64 Google Scholar

[20] Z. Liu, X. Zou, S.M. Kang and J.S. Ume, Common fixed points for a pair of mappings satisfying contractive conditions of integral type, Journal of Inequalities and Applications 394 (2014), 1–19. https://dx.doi.org/10.1186/1029-242X-2014-394 Google Scholar

[21] H.K. Nashine, Common fixed point theorems satisfying integral type rational contractive conditions and applications, Miskolc Mathematical Notes 16 (1) (2015), 321–351. https://dx.doi.org/10.18514/MMN.2015.1133 Google Scholar

[22] H.K. Pathak, Common Fixed Point Theorems Using Property (E.A) In Complex-Valued Metric Spaces, Thai Journal of Mathematics 11 (2) (2013), 347–355. https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/382/378 Google Scholar

[23] K.P.R. Rao, Common Fixed Point Theorems For Mappings Under (Clrs)-Property In Partial Metric Spaces, Demonstratio Mathematica 49 (3) (2016), 357–371. https://dx.doi.org/10.1515/dema-2016-0030 Google Scholar

[24] Sh. Rezapour and R. Hamlbarani, Some notes on the paper: Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications 345 (2) (2008), 719–724. https://dx.doi.org/10.1016/j.jmaa.2008.04.049 Google Scholar

[25] W. Sintunavarat and P. Kumam, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, Journal of Applied Mathematics (2011), 1–14. https://dx.doi.org/10.1155/2011/637958 Google Scholar

[26] D. Turkoglu, M. Abuloha and T. Abdeljawad, Some theorems and examples of cone metric spaces, J. Comput. Anal. 12 (4) (2010), 739–753. Google Scholar

[27] C. Yang and X. Zhu, The concept of cone b-metric space and fixed point theorems, Open Mathematics 19 (2021), 1187–1196. https://dx.doi.org/10.1515/math-2021-0068 Google Scholar