Nonlinear Functional Analysis and Applications

  • 제27권3호
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  • Pages.553-568
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  • 2022
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  • 1229-1595(pISSN)
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  • 2466-0973(eISSN)
경남대학교 수학교육과 (Kyungnam University, Department of Mathematics Eduaction)
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ACCELERATED HYBRID ALGORITHMS FOR NONEXPANSIVE MAPPINGS IN HILBERT SPACES

  • Baiya, Suparat (Department of Mathematics, Faculty of Science, Naresuan University) ;
  • Ungchittrakool, Kasamsuk (Department of Mathematics, Faculty of Science, Naresuan University, Research Center for Academic Excellence in Nonlinear Analysis and Optimization, Faculty of Science, Naresuan University)
  • 투고 : 2021.06.21
  • 심사 : 2022.04.12
  • 발행 : 2022.09.01
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초록

In this paper, we introduce and study two different iterative hybrid projection algorithms for solving a fixed point problem of nonexpansive mappings. The first algorithm is generated by the combination of the inertial method and the hybrid projection method. On the other hand, the second algorithm is constructed by the convex combination of three updated vectors and the hybrid projection method. The strong convergence of the two proposed algorithms are proved under very mild assumptions on the scalar control. For illustrating the advantages of these two newly invented algorithms, we created some numerical results to compare various numerical performances of our algorithms with the algorithm proposed by Dong and Lu [11].

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과제정보

The second author would like to thank Naresuan University and The Thailand Research Fund for financial support. Moreover, S. Baiya is also supported by The Royal Golden Jubilee Program under Grant PHD/0080/2561, Thailand.

참고문헌

  1. N. Artsawang and K. Ungchittrakool, Inertial Mann-type algorithm for a nonexpansive mapping to solve monotone inclusion and image restoration problems, Symmetry, 12(5) (2020), 750. https://doi.org/10.3390/sym12050750
  2. H.H. Bauschke and P.L. Combettes, A weak-to-strong convergence principle for Fejrmonotone methods in Hilbert spaces, Math. Oper. Res., 26(2) (2001), 248-264. https://doi.org/10.1287/moor.26.2.248.10558
  3. H.H. Bauschke and P.L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, CMS Books in Mathematics, New York: Springer, 2011.
  4. V. Berinde, Iterative approximation of fixed points, Lecture Notes in Mathematics, Berlin: Springer, 2007.
  5. R.I. Bot, E.R. Csetnek and S. Laszlo, An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions, EURO J. Comput. Optim., 4 (2016), 3-25. https://doi.org/10.1007/s13675-015-0045-8
  6. R.I. Bot, E.R. Csetnek and N. Nimana, Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data, Optim. Lett., 2017.
  7. L.C. Ceng, Q.H. Ansari and J.C. Yao, Hybrid proximal-type and hybrid shrinking projection algorithms for equilibrium problems, maximal monotone operators and relatively nonexpansive mappings, Numer. Funct. Anal. Optim., 31(7) (2010), 763-797. https://doi.org/10.1080/01630563.2010.496697
  8. C.E. Chidume, Geometric properties of Banach spaces and nonlinear iterations, Lecture Notes in Mathematics, London: Springer, 2009.
  9. W. Cholamjiak, P. Cholamjiak and S. Suantai, An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces, J. Fixed Point Theory Appl., 20(42) (2018).
  10. P. Cholamjiak and Y. Shehu, Inertial forward-backward splitting method in Banach spaces with application to compressed sensing, Appl. Math., 64 (2019), 409-435. https://doi.org/10.21136/AM.2019.0323-18
  11. Q.L. Dong and Y.Y. Lu, A new hybrid algorithm for a nonexpansive mapping, Fixed Point Theory Appl., 37 (2015).
  12. S. He and C. Yang, Boundary point algorithms for minimum norm fixed points of nonexpansive mappings, Fixed Point Theroy Appl., 56 (2014).
  13. S. He, C. Yang and P. Duan, Realization of the hybrid method for Mann iterations, Appl. Math. Comput., 217 (2010), 4239-4247.
  14. T.H. Kim and H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal., 61 (2005), 51-60. https://doi.org/10.1016/j.na.2004.11.011
  15. D. Kitkuan, P. Kumam, J. Martinez-Moreno and K. Sitthithakerngkiet, Inertial viscosity forwardbackward splitting algorithm for monotone inclusions and its application to image restoration problems, Int. J. Comput. Math., (2019), 1-19.
  16. P.E. Mainge, Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236. https://doi.org/10.1016/j.cam.2007.07.021
  17. Y.V. Malitsky and V.V. Semenov, A hybrid method without extrapolation step for solving variational inequality problems, J. Glob. Optim., 61(1) (2015), 193-202. https://doi.org/10.1007/s10898-014-0150-x
  18. W.R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc., 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  19. C. Matinez-Yanes and H.K. Xu, Strong convergence of the CQ method for fixed point processes, Nonlinear Anal., 64 (2006), 2400-2411. https://doi.org/10.1016/j.na.2005.08.018
  20. K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279 (2003), 372-379. https://doi.org/10.1016/S0022-247X(02)00458-4
  21. E. Picard, Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pures et Appl., 6 (1890), 145-210.
  22. S. Plubtieng and K. Ungchittrakool, Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2008 (2008), 17 pages, https://doi.org/10.1155/2008/583082.
  23. B.T. Polyak, Some methods of speeding up the convergence of iteration methods, U.S.S.R. Comput. Math. Math. Phys., 4(5) (1964), 1-17. https://doi.org/10.1016/0041-5553(64)90137-5
  24. S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274-276. https://doi.org/10.1016/0022-247X(79)90024-6
  25. W. Takahashi, Nonlinear functional analysis-fixed point theory and its applications, Yokohama Publishers Inc., Yokohama, 2000.
  26. K. Ungchittrakool, A strong convergence theorem for a common fixed point of two sequences of strictly pseudocontractive mappings in Hilbert spaces and applications, Abstr. Appl. Anal., 2010 (2010), 17 pages, https://doi.org/10.1155/2010/876819.
  27. Y. Xu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl., 224 (1998), 91101.
  28. C. Yang and S. He, General alternative regularization methods for nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 203 (2014).