Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
권호 표지
DOI QR코드

DOI QR Code

ORBITAL CONTRACTION IN METRIC SPACES WITH APPLICATIONS OF FRACTIONAL DERIVATIVES

  • Haitham Qawaqneh (Department of Mathematics, Faculty of Science and Information Technology, Al-Zaytoonah University of Jordan) ;
  • Waseem G. Alshanti (Department of Mathematics, Faculty of Science and Information Technology, Al-Zaytoonah University of Jordan) ;
  • Mamon Abu Hammad (Department of Mathematics, Faculty of Science and Information Technology, Al-Zaytoonah University of Jordan) ;
  • Roshdi Khalil (Department of Mathematics, Faculty of Science, The University of Jordan)
  • Received : 2023.09.09
  • Accepted : 2024.05.18
  • Published : 2024.09.15
Download PDF
⟨ Previous Next ⟩

Abstract

This paper explores the significance and implications of fixed point results related to orbital contraction as a novel form of contraction in various fields. Theoretical developments and theorems provide a solid foundation for understanding and utilizing the properties of orbital contraction, showcasing its efficacy through numerous examples and establishing stability and convergence properties. The application of orbital contraction in control systems proves valuable in designing resilient and robust control strategies, ensuring reliable performance even in the presence of disturbances and uncertainties. In the realm of financial modeling, the application of fixed point results offers valuable insights into market dynamics, enabling accurate price predictions and facilitating informed investment decisions. The practical implications of fixed point results related to orbital contraction are substantiated through empirical evidence, numerical simulations, and real-world data analysis. The ability to identify and leverage fixed points grants stability, convergence, and optimal system performance across diverse applications.

Keywords

Acknowledgement

The author thanks for the support of Al-Zaytoonah University of Jordan.

References

  1. B. Ahmad, A. Alsaedi and S.K. Ntouyas, Existence and uniqueness results for nonlinear fractional differential equations with Caputo and Liouville-Caputo fractional derivatives, Appl. Anal., 100(5) (2021), 1-15. 
  2. B. Ahmad and J.J. Nieto, Fractional-order Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Springer, 2017. 
  3. R.P. Agarwal, N. Hussain and M.T. Mahmood, Existence and uniqueness of solutions for nonlinear fractional differential equations with integral boundary conditions, Appl. Math. Comput., 396(2) (2021), 1-12. 
  4. R. Almeida, N.R. Bastos and M.T.T. Monteiro, Modeling some real phenomena by fractional differential equations, Math. Meth. Appl. Sci., 39(2016), 4846-4855, https://doi.org/10.1002/mma.3818. 
  5. K. Balachandran, Controllability of Generalized Fractional Dynamical Systems, Nonlinear Funct. Anal. Appl., 28(4) (2023), 1115-1125. 
  6. S. Banach, Sur les op'erations dans les ensembles abstraits et leur applications aux S. equations integrales, Fund. Math., 3 (1922), 133-181. 
  7. I.M. Batiha, S.A. Njadat, R.M. Batyha, A. Zraiqat, A. Dababneh and Sh. Momani, Design Fractional-order PID Controllers for Single-Joint Robot Arm Model, Int. J. Adv. Soft Comput. its Appl., 14(2) (2022), 96-114. 
  8. I.M. Batiha, J. Oudetallah, A. Ouannas, A.A. Al-Nana, I.H. Jebril Tuning the Fractional-order PID-Controller for Blood Glucose Level of Diabetic Patients, Int. J. Adv. Soft Comput. its Appl., 13(2) (2021), 1-10. 
  9. I. Beg and M. Abbas, Common fixed points and best approximation in convex metric spaces, Soochow J. Math., 33(4) (2077), 729-738. 
  10. D. Burago, Yu. Burago and S. Ivanov, A course in metric geometry, Grad. Stud. Math. Amer. Math. Soc., Providence, RI, 33 (2001). 
  11. M. Caputo, Linear model of dissipation whose Q is almost frequency independent-II, Geophys. J. R. Astron. Soc., 13 (1967), 529-539. 
  12. L. Changpin, Ch. YangQuan and K. Jrgen, Fractional calculus and its applications, Phil. Trans. R. Soc. A., 3 (2013), 371: 20130037. 
  13. E.V. Denardo, Contraction Mappings in the Theory Underlying Dynamic Programming, SIAM Review, 9(2) (1967), 165-177. 
  14. T. Hamadneh, M. Ali and H. AL-Zoubi, Linear Optimization of Polynomial Rational Functions: Applications for Positivity Analysis, Mathematics, 2:283 (2020), https://doi.org/10.3390/math8020283. 
  15. J. Inciura, Invariant Sets For Certain Linard Equations With Delay, Dynamical Systems, Academic Press, (1977), 431-433. 
  16. H. Jafari, R.M. Ganji, N.S. Nkomo and Y.P. Lv, A numerical study of fractional order population dynamics model, Results in Physics, 27:104456 (2021), https://doi.org/10.1016/j.rinp.2021.104456. 
  17. S. A.M. Jameel, S. A. Rahman and A. A. Hamoud, Analysis of Hilfer fractional Volterra-Fredholm system, Nonlinear Funct. Anal. Appl., 29(1) (2024), 259-273. 
  18. M.D. Johansyah, A.K. Supriatna, E. Rusyaman and J. Saputra, Application of fractional differential equation in economic growth model: A systematic review approach, AIMS, Mathematics, 6(9) (2021), 10266-10280. 
  19. R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71-76. 
  20. R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A New Definition of Fractional Derivative, J. Comput. Appl. Math., 264 (2014), 65-70, http://dx.doi.org/10.1016/j.cam.2014.01.002. 
  21. U. Khristenko and B. Wohlmuth, Solving time-fractional differential equations via rational approximation, IMA J. Num. Anal., 43(3) (2023), 1263-1290, https://doi.org/10.1093/imanum/drac022. 
  22. A.E. Matouk and I. Khan, Complex dynamics and control of a novel physical model using nonlocal fractional differential operator with singular kernel, J. Adv. Res., 24 (2020), 463-474, https://doi.org/10.1016/j.jare.2020.05.003. 
  23. M.E. Newman, Networks: An Introduction, Oxford University Press, 2010. 
  24. S.N.T. Polat and A. Turan Dincel, Euler Wavelet Method as a Numerical Approach for the Solution of Nonlinear Systems of Fractional Differential Equations, Fractal Fract., 7(246) (2023), https://doi.org/10.3390/fractalfract7030246. 
  25. D. Pumplun, The metric completion of convex sets and modules, Result. Math., 41 (2002), 346-360, https://doi:10.1007/BF0332277. 
  26. H. Qawaqneh, Fractional analytic solutions and fixed point results with some applications, Adv. Fixed Point Theory, 14(1) (2024), https://doi.org/10.28919/afpt/8279. 
  27. H. Qawaqneh, M.S.M. Noorani, H. Aydi, A. Zraiqat and A.H. Ansari, On fixed point results in partial b-metric spaces, J. Funct. Spaces, (2021), https://doi.org/10.1155/2021/8769190. 
  28. H. Qawaqneh, M.S.M. Noorani and W. Shatanawi, Common Fixed Point Theorems for Generalized Geraghty (α, ψ, φ)-Quasi Contraction Type Mapping in Partially Ordered Metric-like Spaces, Axioms, 7 (2018). 
  29. W. Rudin, Principles of Mathematical Analysis (3rd ed.), McGraw-Hilln, 1976. 
  30. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integral and Derivative. Gordon and Breach, London, 1993. 
  31. M.H. Shahsavaran and F. Fattahzadeh, Existence and uniqueness of solutions for a class of nonlinear integral equations on Banach spaces, J. Nonlinear Sci. Appl.,14(1) (2021), 116-124. 
  32. H.M. Srivastava and R.K. Saxena, Operators of Fractional Integration and Their Applications, Math. Comput., 118 (2001), 1-52. 
  33. M.I. Troparevsky, S.A. Seminara and M.A. Fabio, A Review on Fractional Differential Equations and a Numerical Method to Solve Some Boundary Value Problems, Intech Open, (2020), https://doi:10.5772/intechopen.86273.