AIMS Mathematics, 2020, 5(4): 3182-3200. doi: 10.3934/math.2020205.

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A new fixed point algorithm for finding the solution of a delay differential equation

Department of Mathematics, Jamia Millia Islamia, New Delhi-110025, India

In this paper, we construct a new iterative algorithm and show that the newly introduced iterative algorithm converges faster than a number of existing iterative algorithms. We present a numerical example followed by graphs to validate our claim. We prove strong and weak convergence results for approximating fixed points of Suzuki generalized nonexpansive mappings. Again we reconfirm our results by example and table. Further, we utilize our proposed algorithm to solve delay differential equation.
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Keywords Suzuki generalized nonexpansive mappings; fixed point; contractive-like mappings; iteration process; strong and weak convergence; delay differential equation

Citation: Chanchal Garodia, Izhar Uddin. A new fixed point algorithm for finding the solution of a delay differential equation. AIMS Mathematics, 2020, 5(4): 3182-3200. doi: 10.3934/math.2020205

References

  • 1. M. Abbas, T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Matematicki Vesnik, 66 (2014), 223-234.
  • 2. A. Abkar, M. Eslamian, Fixed point theorems for Suzuki generalized nonexpansive multivalued mappings in Banach spaces, Fixed Point Theory and Applications, 2010 (2010), 457935.
  • 3. A. Abkar, M. Eslamian, A fixed point theorem for generalized nonexpansive multivalued mappings, Fixed Point Theory, 12 (2011), 241-246.
  • 4. R. P. Agarwal, D. Ó Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex A., 8 (2007), 61-79.
  • 5. S. Banach, Sur les operations dans les ensembles abstraits et leurs applications, Fund. Math., 3 (1992), 133-181.
  • 6. V. Berinde, On the convergence of the Ishikawa iteration in the class of quasi contractive operators, Acta Math. Univ. Comenianae, 73 (2004), 1-11.
  • 7. V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasicontractive operators, Fixed Point Theory and Applications, 2 (2004), 97-105.
  • 8. G. H. Coman, G. Pavel, I. Rus, et al. Introduction in the Theory of Operational Equation, Ed. Dacia, Cluj-Napoca, 1976.
  • 9. S. Dhompongsa, W. Inthakon, A. Kaewkhao, Edelstein's method and fixed point theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 350 (2009), 12-17.    
  • 10. J. Garcia-Falset, E. Llorens-Fuster, T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl., 375 (2011), 185-195.    
  • 11. F. Gursoy, V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, arXiv:1403.2546v2, 2014.
  • 12. B. Halpern, Fixed points of nonexpanding maps, B. Am. Math. Soc., 73 (1967), 957-961.    
  • 13. H. Hu, L. Xu, Existence and uniqueness theorems for periodic Markov process and applications to stochastic functional differential equations, J. Math. Anal. Appl., 466 (2018), 896-926.    
  • 14. C. Huang, S. Vandewalle, Unconditionally stable difference methods for delay partial differential equations, Numer. Math., 122 (2012), 579-601.    
  • 15. C. Huang, S. Vandewalle, Stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays, Front. Math. China, 4 (2009), 63-87.    
  • 16. C. O. Imoru, M. O. Olantiwo, On the stability of the Picard and Mann iteration processes, Carpathian J. Math., 19 (2003), 155-160.
  • 17. S. Ishikawa, Fixed points by a new iteration method, P. Am. Math. Soc., 44 (1974), 147-150.    
  • 18. V. Karakaya, N. E. H. Bouzara, K. Dǒgan, et al. On different results for a new two-step iteration method under weak-contraction mappings in banach spaces, arXiv:1507.00200v1, 2015.
  • 19. W. R. Mann, Mean value methods in iteration, P. Am. Math. Soc., 4 (1953), 506-510.    
  • 20. M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217-229.    
  • 21. W. Phuengrattana, S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SPiterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math., 235 (2011), 3006-3014.    
  • 22. W. Phuengrattana, Approximating fixed points of Suzuki-generalized nonexpansive mappings, Nonlinear Anal-Hybri., 5 (2011), 583-590.    
  • 23. D. R. Sahu, A. Petrusel, Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces, Nonlinear Anal-Theor., 74 (2011), 6012-6023.    
  • 24. J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, B. Aust. Math. Soc., 43 (1991), 153-159.    
  • 25. H. F. Senter, W. G. Dotson, Approximating fixed points of nonexpansive mappings, P. Am. Math. Soc., 44 (1974), 375-380.    
  • 26. T. Suzuki, Fixed point theorems and convergence theorems for some generalized non-expansive mapping, J. Math. Anal. Appl., 340 (2008), 1088-1095.    
  • 27. K. K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308.    
  • 28. D. Thakur, B. S. Thakur, M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings, Appl. Math. Comput., 275 (2016), 147-155.
  • 29. D. Thakur, B. S. Thakur, M. Postolache, A New iteration scheme for approximating fixed points of nonexpansive mappings, Filomat, 30 (2016), 2711-2720.    
  • 30. K. Ullah, M. Arshad, New iteration process and numerical reckoning fixed points in Banach spaces, University Politehnica of Bucharest Scientific Bulletin Series A, 79 (2017), 113-122.
  • 31. K. Ullah, M. Arshad, Numerical Reckoning Fixed Points for Suzuki's Generalized Nonexpansive Mappings via New Iteration Process, Filomat, 32 (2018), 187-196.    
  • 32. L. Xu, Z. Dai, H. Hu, Almost sure and moment asymptotic boundedness of stochastic delay differential systems, Appl. Math. Comput., 361 (2019), 157-168.    
  • 33. L. Xu, S. Sam Ge, Asymptotic behavior analysis of complex-valued impulsive differential systems with time-varying delays, Nonlinear Anal-Hybri., 27 (2018), 13-28.    
  • 34. T. Zamfirescu, Fix point theorems in metric spaces, Archiv der Mathematik, 23 (1972), 292-298.    
  • 35. Z. Zuo, Y. Cui, Iterative approximations for generalized multivalued mappings in Banach spaces, Thai Journal of Mathematics, 9 (2011), 333-342.

 

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