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Robustness of Theta Method for Nonexpansive Mappings

  • Rahul ShuklaEmail author
  • Rajendra Pant
Research Paper
  • 12 Downloads
Part of the following topical collections:
  1. Mathematics
  2. Mathematics
  3. Mathematics
  4. Mathematics

Abstract

In this paper, we approximate fixed points of nonexpansive mappings in Hilbert spaces using an implicit method. More precisely, we study weak and strong convergence results for \(\theta\)-method with small perturbations. Some illustrative examples and numerical computations show the usefulness of our theorems. We also present an application of our results to integral equations.

Keywords

Nonexpansive mapping Theta method Nearest projection 

Mathematics Subject Classification

Primary 47H10 47H09 47J05 47J25 54H25 

Notes

Acknowledgements

We are very much thankful to the reviewers for their constructive comments and suggestions which have been useful for the improvement of this paper.

References

  1. Alvarez F (2003) Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J Optim 14(3):773–782MathSciNetCrossRefzbMATHGoogle Scholar
  2. Boţ R, Csetnek ER, Hendrich C (2015) Inertial Douglas–Rachford splitting for monotone inclusion problems. Appl Math Comput 256:472–487MathSciNetzbMATHGoogle Scholar
  3. Browder FE (1965a) Fixed-point theorems for noncompact mappings in Hilbert space. Proc Natl Acad Sci USA 53:1272–1276MathSciNetCrossRefzbMATHGoogle Scholar
  4. Browder FE (1965b) Nonexpansive nonlinear operators in a Banach space. Proc Natl Acad Sci USA 54:1041–1044MathSciNetCrossRefzbMATHGoogle Scholar
  5. Browder FE (1967) Convergence theorems for sequences of nonlinear operators in Banach spaces. Math Z 100:201–225MathSciNetCrossRefzbMATHGoogle Scholar
  6. Byrne C (2004) A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl 20(1):103–120MathSciNetCrossRefzbMATHGoogle Scholar
  7. Combettes PL (2001) On the numerical robustness of the parallel projection method in signal synthesis. IEEE Signal Process Lett 8:45–47CrossRefGoogle Scholar
  8. Eckstein J, Bertsekas DP (1992) On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math Program 55(3, Ser. A):293–318MathSciNetCrossRefzbMATHGoogle Scholar
  9. Engl HW, Leitão A (2001) A Mann iterative regularization method for elliptic Cauchy problems. Numer Funct Anal Optim 22(7–8):861–884MathSciNetCrossRefzbMATHGoogle Scholar
  10. Kim T-H, Xu H-K (2007) Robustness of Mann’s algorithm for nonexpansive mappings. J Math Anal Appl 327(2):1105–1115MathSciNetCrossRefzbMATHGoogle Scholar
  11. Mann WR (1953) Mean value methods in iteration. Proc Am Math Soc 4:506–510MathSciNetCrossRefzbMATHGoogle Scholar
  12. Marino G, Xu H-K (2004) Convergence of generalized proximal point algorithms. Commun Pure Appl Anal 3(4):791–808MathSciNetCrossRefzbMATHGoogle Scholar
  13. Martinet B (1970) Régularisation d’inéquations variationnelles par approximations successives. Rev Française Informat Recherche Opérationnelle 4(Sér. R–3):154–158MathSciNetzbMATHGoogle Scholar
  14. Opial Z (1967) Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Am Math Soc 73:591–597MathSciNetCrossRefzbMATHGoogle Scholar
  15. Podilchuk CI, Mammone RJ (1990) Image recovery by convex projections using a least-squares constraint. J Opt Soc Am A 7:517–521CrossRefGoogle Scholar
  16. Rahul S, Rajendra P (2018) Approximating solution of split equality and equilibrium problems by viscosity approximation algorithms. Comput Appl Math 37(4):5293–5314MathSciNetCrossRefzbMATHGoogle Scholar
  17. Rockafellar RT (1976) Monotone operators and the proximal point algorithm. SIAM J Control Optim 14(5):877–898MathSciNetCrossRefzbMATHGoogle Scholar
  18. Tan K-K, Xu H-K (1993) Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J Math Anal Appl 178(2):301–308MathSciNetCrossRefzbMATHGoogle Scholar
  19. Viorel B (1976) Nonlinear semigroups and differential equations in Banach spaces. Editura Academiei Republicii Socialiste România, Bucharest. Noordhoff International Publishing, Leiden Translated from the RomanianzbMATHGoogle Scholar
  20. Xu H-K, Alghamdi MA, Shahzad N (2016) The theta method for nonexpansive mappings. J Nonlinear Convex Anal 17(10):2029–2038MathSciNetzbMATHGoogle Scholar
  21. Yamada I, Ogura N (2004) Adaptive projected subgradient method for asymptotic minimization of sequence of nonnegative convex functions. Numer Funct Anal Optim 25(7–8):593–617MathSciNetzbMATHGoogle Scholar
  22. Zaslavski AJ (2016) Approximate solutions of common fixed-point problems. Springer, BerlinCrossRefzbMATHGoogle Scholar

Copyright information

© Shiraz University 2019

Authors and Affiliations

  1. 1.Department of MathematicsVisvesvaraya National Institute of TechnologyNagpurIndia
  2. 2.Department of Pure MathematicsUniversity of JohannesburgAuckland ParkSouth Africa

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