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CONVERGENCE THEOREMS OF PROXIMAL TYPE ALGORITHM FOR A CONVEX FUNCTION AND MULTIVALUED MAPPINGS IN HILBERT SPACES

  • Aggarwal, Sajan (Department of Mathematics Jamia Millia Islamia) ;
  • Uddin, Izhar (Department of Mathematics Jamia Millia Islamia) ;
  • Pakkaranang, Nuttapol (Department of Mathematics, Faculty of Science King Mongkut's University of Technology Thonburi) ;
  • Wairojjana, Nopparat (Applied Mathematics Program, Faculty of Science and Technology Valaya Alongkorn Rajabhat University under the Royal Patronage) ;
  • Cholamjiak, Prasit (School of Science, University of Phayao)
  • 투고 : 2020.05.30
  • 심사 : 2020.12.12
  • 발행 : 2021.03.15

초록

In this paper we study the weak and strong convergence to minimizers of convex function of proximal point algorithm SP-iteration of three multivalued nonexpansive mappings in a Hilbert space.

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참고문헌

  1. L. Ambrosio, N. Gigli and G. Savare, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zurich Birkhauser Verlag, Basel, (2008).
  2. F.E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1272-1276.
  3. S.S. Chang, D.P. Wu, L. Wang and G. Wang, Proximal point algorithms involving fixed point of nonspreading-type multivalued mappings in Hilbert spaces, J. Nonlinear Sci. Appl., 9 (2016), 5561-5569. https://doi.org/10.22436/jnsa.009.10.06
  4. P. Cholamjiak, T. Thianwan, L. Kittiratanawasin and C. Chairatsiripong, Modified SP-iteration algorithms for solving fixed point problems of continuous functions on a arbitrary interval, Nonlinear Funct. Anal. Appl., 24(4) (2019), 735-745.
  5. O. Guler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), 403-419. https://doi.org/10.1137/0329022
  6. J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comment. Math. Helv., 70 (1995), 659-673. https://doi.org/10.1007/BF02566027
  7. S. Kamimura, F. Kohsaka and W. Takahashi, Weak and strong convergence theorems for maximal monotone operators in a Banach space, Set-Valued Anal., 12 (2004), 417-429. https://doi.org/10.1007/s11228-004-8196-4
  8. G. Kassay, The proximal point algorithm for reflexive Banach spaces, Studia Univ. Babes-Bolyai Math., 30 (1985), 9-17.
  9. K. Lerkchaiyaphum and W. Phuengrattana, Proximal point algorithms for numerical reckoning fixed points of hybrid-type multivalued mappings in Hilbert spaces, Khayyam J. Math., 3 (2017), 81-89.
  10. B. Martinet, Regularisation d'inequations variationnelles par approximations successives, Rev. Francaise Informat. Recherche Operationnelle, 4 (1970), 154-158.
  11. Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597. https://doi.org/10.1090/S0002-9904-1967-11761-0
  12. N. Pakkaranang, P. Kumam and Y.J. Cho, Proximal point algorithms for solving convex minimization problem and common fixed points problem of asymptotically quasinonexpansive mappings in CAT(0) spaces with convergence analysis, Numer. Algorithms, 78(3) (2018), 827-845. https://doi.org/10.1007/s11075-017-0402-1
  13. W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math., 235 (2011), 3006-3014. https://doi.org/10.1016/j.cam.2010.12.022
  14. R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898. https://doi.org/10.1137/0314056
  15. S. Suantai and W. Phuengrattana, Existence and convergence theorems for λ-hybrid mappings in Hilbert spaces, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 22 (2015), 177-188.