Solution of an Algebraic Linear System of Equations Using Fixed Point Results in C∗-Algebra Valued Extended Branciari Sb-Metric Spaces

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DOI: https://doi.org/10.28924/2291-8639-22-2024-139
Published: Aug 15, 2024
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Khairul Habib Alam, Yumnam Rohen, Imen Ali Kallel, Junaid Ahmad, Solution of an Algebraic Linear System of Equations Using Fixed Point Results in C∗-Algebra Valued Extended Branciari Sb-Metric Spaces, Int. J. Anal. Appl., 22 (2024), 139.

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Khairul Habib Alam, Yumnam Rohen, Imen Ali Kallel, Junaid Ahmad

Abstract

This study explores the realm of metric spaces, advancing beyond conventional boundaries by introducing two innovative types of metrics known as generalized Branciari-type metrics. Through exacting examination and exemplification, we shed light on the intricacies of these newly defined metric spaces and their extended versions. By drawing parallels with established theorems such as Banach and Kannan, we unveil corollaries that establish necessary symmetric conditions for the existence and uniqueness of fixed points concerning self-operators within these spaces. The inclusion of illustrative examples not only bolsters our theoretical framework but also underscores the practical relevance of our findings. Furthermore, we utilize our research to address real-world applications, showcasing how our results can be employed to determine the existence of unique solutions for algebraic systems of linear equations, thereby bridging the theoretical and applied aspects of mathematical exploration. Through these interventions, our study significantly contributes to the comprehensive understanding and utilization of all the properties in metric spaces within diverse mathematical contexts.

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References

  1. G.J. Murphy, C∗−Algebras and Operator Theory, Academic Press, 1990.
  2. Q.H. Xu, T.E.D. Bieke, Z.Q. Chen, Introduction to Operator Algebras and Noncommutative Lp Spaces, Science Press, Beijing, 2010.
  3. Z. Ma, L. Jiang, H. Sun, C∗−Algebra Valued Metric Spaces and Related Fixed Point Theorems, Fixed Point Theory Appl. 2014 (2014), 206. https://doi.org/10.1186/1687-1812-2014-206.
  4. H. Liu, S. Xu, Cone Metric Spaces With Banach Algebras and Fixed Point Theorems of Generalized Lipschitz Mappings, Fixed Point Theory Appl. 2013 (2013), 320. https://doi.org/10.1186/1687-1812-2013-320.
  5. P. Kumam, H. Rahimi, G.S. Rad, The Existence of Fixed and Periodic Point Theorems in Cone Metric Type Spaces, J. Nonlinear Sci. Appl. 7 (2014), 255–263.
  6. L. Aryanpour, H. Rahimi, G.S. Rad, Common Fixed Points of Two Mappings Regarding a Generalized c-Distance over a Banach Algebra, J. Funct. Spaces 2021 (2021), 6623606. https://doi.org/10.1155/2021/6623606.
  7. Z. Kadelburg, Lj. Paunovic, S. Radenovic, S. Rad, Non-Normal Cone Metric and Cone b-Metric Spaces and Fixed Point Results, Sci. Publ. State Univ. Novi Pazar Ser. A. 8 (2016), 177–186. https://doi.org/10.5937/SPSUNP1602177K.
  8. H. Huang, G. Deng, S. Radenovi´c, Some Topological Properties and Fixed Point Results in Cone Metric Spaces Over Banach Algebras, Positivity 23 (2019), 21–34. https://doi.org/10.1007/s11117-018-0590-5.
  9. A. Tomar, M. Joshi, On Existence of Fixed Points and Applications to a Boundary Value Problem and a Matrix Equation in C∗ -Algebra Valued Partial Metric Spaces, Acta Univ. Sapientiae Math. 14 (2022), 341–355. https://doi.org/10.2478/ausm-2022-0023.
  10. S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inf. Univ. Ostrav. 1 (1993), 5–11.
  11. K.H. Alam, Y. Rohen, N. Saleem, Fixed Points of (α, β, F∗) and (α, β, F∗∗)−Weak Geraghty Contractions With an Application, Symmetry, 15 (2023), 243. https://doi.org/10.3390/sym15010243.
  12. K.H. Alam, Y. Rohen, A. Tomar, M. Sajid, On Fixed Points in Mbv−Metric Space and Solutions to Nonlinear Matrix Equations and Differential Equations Related to Beam Theory, J. Nonlinear Convex Anal. In Press.
  13. K.H. Alam, Y. Rohen, A. Tomar, (α, F)−Geraghty Type Generalized F−Contractions on Non-Archimedean Fuzzy Metric-Unlike Spaces, Demonstr. Math. In Press.
  14. A.Z. Rezazgui, A.A. Tallafha, W. Shatanawi, Common Fixed Point Results via Av-α-Contractions With a Pair and Two Pairs of Self-Mappings in the Frame of an Extended Quasi b-Metric Space, AIMS Math. 8 (2023), 7225–7241. https://doi.org/10.3934/math.2023363.
  15. M. Joshi, A. Tomar, T. Abdeljawad, On Fixed Points, Their Geometry and Application to Satellite Web Coupling Problem in S−Metric Spaces, AIMS Math. 8 (2023), 4407–4441. https://doi.org/10.3934/math.2023220.
  16. Z. Ma, L. Jiang, C∗-Algebra-Valued b-Metric Spaces and Related Fixed Point Theorems, Fixed Point Theory Appl 2015 (2015), 222. https://doi.org/10.1186/s13663-015-0471-6.
  17. D. Ali, A. Hussain, E. Karapinar, P. Cholamjiak, Efficient Fixed-Point Iteration for Generalized Nonexpansive Mappings and Its Stability in Banach Spaces, Open Math. 20 (2022), 1753–1769. https://doi.org/10.1515/math-2022-0461.
  18. S. Sedghi, N. Shobe, A. Aliouche, A Generalization of Fixed Point Theorems in S-Metric Spaces, Mat. Vesnik, 64 (2012), 258–266.
  19. M.P. Singh, Y. Rohen, N. Saleem, K.H. Alam, K.A. Singh, A. Razzaque, On Fixed-Point Equations Involving Geraghty-Type Contractions with Solution to Integral Equation, Mathematics 11 (2023), 4882. https://doi.org/10.3390/math11244882.
  20. M.E. Ege, C.I. Alaca, C∗−Algebra Valued S−Metric Spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 67 (2018), 165–177. https://doi.org/10.1501/commua1_0000000871.
  21. N. Souayah, N. Mlaiki, A Fixed Point Theorem in Sb−Metric Spaces, J. Math. Comput. Sci. 16 (2016), 131–139.
  22. S. Sedghi, A. Gholidahneh, T. Došenovi´c, J. Esfahani, S. Radenovi´c, Common Fixed Point of Four Maps in Sb−Metric Spaces, J. Linear Topol. Algebra. 5 (2016), 93–104.
  23. Y. Rohen, T. Dosenovic, S. Radenovic, A Note on the Paper "A Fixed point Theorems in Sb−Metric Space", Filomat, 31 (2017), 3335–3346. https://www.jstor.org/stable/26195059.
  24. C. Kalaivani, G. Kalpana, Fixed Point Theorems in C∗−Algebra Valued S-Metric Spaces with Some Applications, U.P.B. Sci. Bull. Ser. A, 80 (2018), 93–102.
  25. K. Roy, M. Saha, Branciari Sb−Metric Space and Related Fixed Point Theorems with an Application, Appl. Math. E-Notes, 22 (2022), 8–17.
  26. T. Kamran, M. Samreen, Q. UL Ain, A Generalization of b-Metric Space and Some Fixed Point Theorems, Mathematics. 5 (2017), 19. https://doi.org/10.3390/math5020019.
  27. M. Asim, M. Imdad, C∗−Algebra Valued Extended b−Metric Spaces and Fixed Point Results with an Application, U.P.B. Sci. Bull. Ser. A, 82 (2020), 207–218.
  28. N. Mlaiki, Extended Sb−Metric Spaces, J. Math. Anal. 9 (2018), 124–135.
  29. A. Buyukkaya, A. Fulga, M. Ozturk, On Generalized Suzuki-Proinov Type (α,Z∗E )−Contractions in Modular b-Metric Spaces, Filomat 37 (2023), 1207–1222. https://doi.org/10.2298/fil2304207b.
  30. K.H. Alam, Y. Rohen, A. Tomar, On Fixed Point and Its Application to the Spread of Infectious Diseases Model in Mbv−Metric Space, Math Meth. Appl. Sci. 47 (2024), 6489–6503. https://doi.org/10.1002/mma.9933.
  31. M.P. Singh, Y. Rohen, K.H. Alam, J. Ahmad, W. Emam, On Fixed Point and an Application of C∗−Algebra Valued (α, β)−Bianchini-Grandolfi Gauge Contractions, AIMS Math. 9 (2024), 15172–15189. https://doi.org/10.3934/math.2024736.
  32. K.H. Alam, Y. Rohen, An Efficient Iterative Procedure in Hyperbolic Space and Application to Non-Linear Delay Integral Equation, J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02134-z.
  33. K.H. Alam, Y. Rohen, N. Saleem, M. Aphane, A. Rzzaque, Convergence of Fibonacci-Ishikawa Iteration Procedure for Monotone Asymptotically Nonexpansive Mappings, J Inequal Appl 2024 (2024), 81. https://doi.org/10.1186/s13660-024-03156-8.