Novel Results in Cone Bipolar Metric Spaces With Application in Initial Value Fractional Caputo Differential Equations

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-294
Published: Nov 13, 2025
Cite as:
Gunaseelan Mani, Balaji Ramalingam, Ola Ashour Abdelnaby, Khizar Hayat Khan, Zoran D. Mitrović, Rajagopalan Ramaswamy, Novel Results in Cone Bipolar Metric Spaces With Application in Initial Value Fractional Caputo Differential Equations, Int. J. Anal. Appl., 23 (2025), 294.

Main Article Content

Gunaseelan Mani, Balaji Ramalingam, Ola Ashour Abdelnaby, Khizar Hayat Khan, Zoran D. Mitrović, Rajagopalan Ramaswamy

Abstract

In this paper, we establish fixed point results in the setting of cone bipolar metric space. Some of the well-known results in the literature are extended and generalized by the demonstrated results. We give some examples based on our outcomes to strengthen our results. An application is presented based on integral equations and fractional differential equations that confirm our findings.

Article Details

References

  1. A. Rezazgui, A.A. Tallafha, W. Shatanawi, Common Fixed Point Results via $mathcal{A}_{vartheta} $-$ alpha$-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.
  2. W. Shatanawi, T.A.M. Shatnawi, New Fixed Point Results in Controlled Metric Type Spaces Based on New Contractive Conditions, AIMS Math. 8 (2023), 9314–9330. https://doi.org/10.3934/math.2023468.
  3. M. Joshi, A. Tomar, T. Abdeljawad, On Fixed Points, Their Geometry and Application to Satellite Web Coupling Problem in $mathcal{S}$-metric Spaces, AIMS Math. 8 (2023), 4407–4441. https://doi.org/10.3934/math.2023220.
  4. A. Mutlu, U. Gürdal, Bipolar Metric Spaces and Some Fixed Point Theorems, J. Nonlinear Sci. Appl. 09 (2016), 5362–5373. https://doi.org/10.22436/jnsa.009.09.05.
  5. C. Milici, G. Drăgănescu, J.T. Machado, Introduction to Fractional Differential Equations, Springer, Cham, 2019. https://doi.org/10.1007/978-3-030-00895-6.
  6. A. Mutlu, K. Ozkan, U. Gurdal, Coupled Fixed Point Theorems on Bipolar Metric Spaces, Eur. J. Pure Appl. Math. 10 (2017), 655–667.
  7. U. Gurdal, A. Mutlu, K. Ozkan, Fixed Point Results for $alpha$-$phi$-Contractive Mappings in Bipolar Metric Spaces, J. Inequal. Spec. Funct. 11 (2020), 64–75.
  8. R. Ramaswamy, G. Mani, A.J. Gnanaprakasam, O.A.A. Abdelnaby, V. Stojiljković, et al., Fixed Points on Covariant and Contravariant Maps with an Application, Mathematics 10 (2022), 4385. https://doi.org/10.3390/math10224385.
  9. P.P. Murthy, C.P. Dhuri, S. Kumar, R. Ramaswamy, M.A.S. Alaskar, et al., Common Fixed Point for Meir–Keeler Type Contraction in Bipolar Metric Space, Fractal Fract. 6 (2022), 649. https://doi.org/10.3390/fractalfract6110649.
  10. M. Kumar, P. Kumar, A. Mutlu, R. Ramaswamy, O.A.A. Abdelnaby, et al., Ulam–Hyers Stability and Well-Posedness of Fixed Point Problems in C*-Algebra Valued Bipolar B-Metric Spaces, Mathematics 11 (2023), 2323. https://doi.org/10.3390/math11102323.
  11. G. Mani, R. Ramaswamy, A.J. Gnanaprakasam, V. Stojiljković, Z.M. Fadail, et al., Application of Fixed Point Results in the Setting of $mathcal{F}$-Contraction and Simulation Function in the Setting of Bipolar Metric Space, AIMS Math. 8 (2023), 3269–3285. https://doi.org/10.3934/math.2023168.
  12. G.N.V. Kishore, K.P.R. Rao, H. IsIk, B. Srinuvasa Rao, A. Sombabu, Covarian Mappings and Coupled Fiexd Point Results in Bipolar Metric Spaces, Int. J. Nonlinear Anal. Appl. 12 (2021), 1–15. https://doi.org/10.22075/ijnaa.2021.4650.
  13. A. Mutlu, K. Ozkan, U. Gurdal, Locally and Weakly Contractive Principle in Bipolar Metric Spaces, TWMS . Appl. Eng. Math. 10 (2020), 379–388.
  14. Y.U. Gaba, M. Aphane, H. Aydi, ($alpha$, BK)-Contractions in Bipolar Metric Spaces, J. Math. 2021 (2021), 5562651. https://doi.org/10.1155/2021/5562651.
  15. G. Mani, B. Ramalingam, S. Etemad, İ. Avcı, S. Rezapour, On the Menger Probabilistic Bipolar Metric Spaces: Fixed Point Theorems and Applications, Qual. Theory Dyn. Syst. 23 (2024), 99. https://doi.org/10.1007/s12346-024-00958-5.
  16. G. Mani, R. Ramaswamy, A.J. Gnanaprakasam, A. Elsonbaty, O.A.A. Abdelnaby, et al., Application of Fixed Points in Bipolar Controlled Metric Space to Solve Fractional Differential Equation, Fractal Fract. 7 (2023), 242. https://doi.org/10.3390/fractalfract7030242.
  17. M.I. Pasha, K.R.K. Rao, G. Mani, A.J. Gnanaprakasam, S. Kumar, Solving a Fractional Differential Equation via the Bipolar Parametric Metric Space, J. Math. 2024 (2024), 5533347. https://doi.org/10.1155/2024/5533347.
  18. G. Mani, A.J. Gnanaprakasam, H. Işık, F. Jarad, Fixed Point Results in $mathcal{C}^star $-Algebra-Valued Bipolar Metric Spaces with an Application, AIMS Math. 8 (2023), 7695–7713. https://doi.org/10.3934/math.2023386.
  19. G. Mani, R. Ramaswamy, A.J. Gnanaprakasam, V. Stojiljković, Z.M. Fadail, et al., Application of Fixed Point Results in the Setting of $mathcal{F}$-Contraction and Simulation Function in the Setting of Bipolar Metric Space, AIMS Math. 8 (2023), 3269–3285. https://doi.org/10.3934/math.2023168.
  20. L. Huang, X. Zhang, Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings, J. Math. Anal. Appl. 332 (2007), 1468–1476. https://doi.org/10.1016/j.jmaa.2005.03.087.
  21. M. Gunaseelan, M. Narayan, M. Narayan, Fixed Point Theorems of Generalized Multi-Valued Mappings in Cone B-Metric Spaces, Math. Moravica 25 (2021), 31–45. https://doi.org/10.5937/matmor2101031g.
  22. D. Dey, M. Saha, Partial Cone Metric Space and Some Fixed Point Theorems, TWMS J. Appl. Eng. Math. 3 (2013), 1–9.
  23. T.L. Shateri, Common Fixed Point Results in Partial Cone Metric Spaces, arXiv:2208.06798 (2022). https://doi.org/10.48550/arXiv.2208.06798.
  24. A. Arif, M. Nazam, A. Hussain, M. Abbas, The Ordered Implicit Relations and Related Fixed Point Problems in the Cone $b$-Metric Spaces, AIMS Math. 7 (2022), 5199–5219. https://doi.org/10.3934/math.2022290.
  25. A. Arif, M. Nazam, H.H. Al-Sulami, A. Hussain, H. Mahmood, Fixed Point and Homotopy Methods in Cone A-Metric Spaces and Application to the Existence of Solutions to Urysohn Integral Equation, Symmetry 14 (2022), 1328. https://doi.org/10.3390/sym14071328.
  26. R. Kannan, Some Results on Fixed Points, Bull. Calcutta Math. Soc. 60 (1968), 71–76. https://cir.nii.ac.jp/crid/1572543024587220992.
  27. U. Ishtiaq, M. Asif, A. Hussain, K. Ahmad, I. Saleem, et al., Extension of a Unique Solution in Generalized Neutrosophic Cone Metric Spaces, Symmetry 15 (2022), 94. https://doi.org/10.3390/sym15010094.
  28. M. Younis, H. Ahmad, L. Chen, M. Han, Computation and Convergence of Fixed Points in Graphical Spaces with an Application to Elastic Beam Deformations, J. Geom. Phys. 192 (2023), 104955. https://doi.org/10.1016/j.geomphys.2023.104955.