Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
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A NOVEL FIXED POINT ITERATION PROCEDURE FOR APPROXIMATING THE SOLUTION OF IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS

  • James Abah Ugboh (Department of Mathematics, University of Calabar) ;
  • Joseph Oboyi (Department of Mathematics, University of Calabar) ;
  • Austine Efut Ofem (School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal) ;
  • Godwin Chidi Ugwunnadi (Department of Mathematics, University of Eswatini, Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University) ;
  • Ojen Kumar Narain (School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal)
  • Received : 2023.11.27
  • Accepted : 2023.12.24
  • Published : 2024.09.15
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Abstract

In this research, we propose a new efficient iterative method for fixed point problems of generalized α-nonexpansive mappings. We show the weak and strong convergence analysis of the proposed method under some mild assumptions on the control parameters. We consider the application of the new method to some real world problems such as convex minimization problems, image restoration problems and impulsive fractional differential equations. We carryout a numerical experiment to show the computational advantage of our method over some well known existing methods.

Keywords

Acknowledgement

The authors wish to thank the editor and the referees for their useful comments and suggestions.

References

  1. M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat Vesn., 66 (2014), 223-234.
  2. J.A. Abuchu, A.E. Ofem, G.C. Ugwunnadi, O.K. Narain and A. Hussain, Hybrid Alternated Inertial Projection and Contraction Algorithm for Solving Bilevel Variational Inequality Problems, J. Math., 2023, Article ID 3185746, https://doi.org/10.1155/2023/3185746.
  3. R.P. Agarwal, D.O. Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8 (2007), 61-79.
  4. A. A. N. Al-Sudani and I. A. H. Al-Nuh, Investigation of a New Coupled System of Fractional Differential Equations in Frame of Hilfer-Hadamard , Nonlinear Funct. Anal. Appl., 29 (2) ( 2024), 501-515.
  5. N. Anakira, Z. Chebana, T-E. Oussaeif, I. M. Batiha and A. Ouannas, A study of a weak solution of a diffusion problem for a temporal fractional differential equation, Nonlinear Funct. Anal. Appl., 27 (3 ) ( 2022), 679-689..
  6. K. Aoyama and F. Kohsaka, Fixed point theorem for α-nonexpansive mappings in Banach spaces, Nonlinear Anal., 74 (2011), 4387-4391.
  7. I. M. Batiha, N. Alamarat, S. Alshorm, O. Y. Ababneh and S. Moman , Semi-analytical solution to a coupled linear incommensurate system of fractional differential equations, Nonlinear Funct. Anal. Appl., 28 (2) ( 2023), 449-471.
  8. H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer Publishing Company, Incorporated, 2nd Edition, 2017.
  9. A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imag. Sci., 2(1) (2009), 183-202.
  10. T.A. Faree and S.K. Panchal, Existence of solution for impulsive fractional differential equations with nonlocal conditions by topological degree theory, Results in Appl. Math., 18 (2023), 10037.
  11. F. Gusoy, A Picard-S iterative Scheme for Approximating Fixed Point of Weak-Contraction Mappings, Filomat 30(10) (2016), 2829-2845.
  12. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150.
  13. K. Janngam and S. Suantai, An accelerated forward-backward algorithm with applications to image restoration problems, Thai J. Math. 19(2) (2021), 325-339, http://thaijmath.in.cmu.ac.th.
  14. S.H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl., 2013(2013), 69.
  15. A. Kilbas and S. Marzan, Cauchy problem for differential equation with Caputo derivative, Fract. Calc. Appl. Anal., 7 (2004), 297-321.
  16. W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.
  17. M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251(1) (2000), 217-229.
  18. A.E. Ofem, J.A. Abuchu, R. George, G.C. Ugwunnadi and O.K. Narain, it Some new results on convergence, weak w2-stability and data dependence of two multivalued almost contractive mappings in hyperbolic spaces, Mathematics, 2022:10 (2022), 3720.
  19. A.E. Ofem, J.A. Abuchu, G.C. Ugwunnadi, H. I,sik and O.K. Narain, On a four-step iterative algorithm and its application to delay integral equations in hyperbolic spaces, Rend. Circ. Mat. Palermo Ser. 2, 73(2024), 189-224.).
  20. A.E. Ofem, A. Hussain, O. Joseph, M.O. Udo, U. Ishtiaq and H. Al Sulami, Solving Fractional VolterraFredholm Integro-Differential Equations via A** Iteration Method, Axioms, 11(9) (2022), 470.
  21. A.E. Ofem and D.I. Igbokwe, An efficient iterative method and its applications to a nonlinear integral equation and a delay differential equation in banach spaces, Turkish J. Ineq., 4 (2020), 79-107.
  22. A.E. Ofem and D.I. Igbokwe, A new faster four step iterative algorithm for Suzuki generalized nonexpansive mappings with an application, Adv. Theoryf Nonlinear Anal. Appl.,, 5 (2021), 482-506.
  23. A.E. Ofem, H. I,sik, F. Ali and J. Ahmad, A new iterative approximation scheme for Reich-Suzuki type nonexpansive operators with an application, J. Ineq.. Appl., 2022: 28(2022).
  24. A.E. Ofem, A.A. Mebawondu, G.C. Ugwunnadi and Q.K. Narain, A modified subgradient extragradient algorithm-type for solving quasimonotone variational inequality problems with applications, J Ineq.l. Appl. 2023(73) (2023), https://doi.org/10.1186/s13660-023-02981-7.
  25. A.E. Ofem, M.O. Udo, O. Joseph, R. George and C.F. Chikwe, Convergence Analysis of a New Implicit Iterative Scheme and Its Application to Delay Caputo Fractional Differential Equations, Fractal Fractional, 7(3) (2023), 212.
  26. A.E. Ofem, U.E. Udofia and D.I. Igbokwe, New iterative algorithm for solving constrained convex minimization problem and split feasibility problem, European J. Math. Anal., 1(2) (2021), 106-132.
  27. A.E. Ofem, U.E. Udofia and D.I. Igbokwe, A robust iterative approach for solving nonlinear Volterra Delay integro-differential equations, Ural Math. J., 7 (2021), 59-85.
  28. A.E. Ofem, G.C. Ugwunnadi, O.K. Narain and J.K. Kim, Approximating common fixed point of three multivalued mappings satisfying condition (E) in hyperbolic spaces, Nonlinear Funct. Anal. Appl., 28(3) (2023), 623-646.
  29. G.A. Okeke and A.E. Ofem, A novel iterative scheme for solving delay differential equations and nonlinear integral equations in Banach spaces, Math. Meth. Appl. Sci., 45(9) (2022), 5111-5134, https://doi.org/10.1002/mma.8095.
  30. G.A. Okeke, A.E. Ofem, T. Abdeljawad, M. A Alqudah and A. Khan, A solution of a nonlinear Volterra integral equation with delay via a faster iteration method, AIMS Mathematics, 8 (2023), 102-124.
  31. R. Pandey, R. Pant, V. Rakocevic and R. Shukla, Approximating fixed points of a general class of nonexpansive mappings in Banach spaces with applications, Results Math. 74(1) (2019), Article No. 7.
  32. R. Pant and R. Pandey, Existence and convergence results for a class of non-expansive type mappings in hyperbolic spaces, Appl. Gen. Topology, 20 (2019), 281-295.
  33. R. Pant and R. Shukla, Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim., 38(2) (2017), 248-266.
  34. N. Parikh and S. Boyd, Proximal algorithms, Found. Trends Optim., 1(3) (2014), 127-239.
  35. E. Picard, Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pure Appl., 6 (1890), 145-210.
  36. J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc., 43 (1991), 153-159.
  37. H.F. Senter and W.G. Dotson, Approximating fixed points of nonexpansive mapping, Proc. Amer. Math. Soc., 44 (1974), 375-380.
  38. S.M. Soltuz and T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive like operators. Fixed Point Theory Appl., (2008), doi:10.1155/2008/242916.
  39. T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340(2) (2008), 1088-1095.
  40. B.S. Thakur, D. Thakur and M. Postolache, A new iteration scheme for approximating fixed points of nonexpansive mappings, Filomat, 30 (2016), 2711-2720.
  41. K. Thung and P. Raveendran, A survey of image quality measures, In Proceedings of the International Conference for Technical Postgraduates (TECHPOS), Kuala Lumpur, Malaysia, 14-15, December 2009.
  42. R. Tibshirani, Regression shrinkage and selection via the lasso, J. Royal Stat. Soc., Series B (Methodological), 58(1) (1996), 267-288.
  43. U.E. Udofia, A.E. Ofem and D.I. Igbokwe, Weak and strong convergence theorems for fixed points of generalized α-nonexpansive mappings with application, Eur. J. Math. Appl., (2021)1:3.
  44. K. Ullah and M. Arshad, Numerical Reckoning Fixed Points for Suzuki's Generalized Nonexpansive Mappings via New Iteration Process, Filomat, 32(1) (2018), 187-196.