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Numerical Algorithms

, Volume 82, Issue 1, pp 349–369 | Cite as

The Yosida approximation iterative technique for split monotone Yosida variational inclusions

  • Mijanur RahamanEmail author
  • Mohd. Ishtyak
  • Rais Ahmad
  • Imran Ali
Original Paper
  • 113 Downloads

Abstract

In this article, we introduce a new type of split monotone Yosida inclusion problem in the setting of infinite-dimensional Hilbert spaces. To calculate the approximate solutions of split monotone Yosida inclusion problem, first we develop a new iterative algorithm and then study the weak as well strong convergence analysis of iterative sequences generated by the proposed iterative algorithm by using demicontractive property, nonexpansive property, and strongly positive bounded linear property of mappings. A numerical example is formulated to explain our main result through MATLAB programming.

Keywords

Split monotone Yosida variational inclusions Yosida operator Convergence Demicontractive mapping Nonexpansive mappings 

Mathematics Subject Classification (2010)

47H05 47H09 47J25 

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References

  1. 1.
    Ahmad, R., Ishtyak, M., Rahaman, M., Ahmad, I.: Graph convergence and generalized Yosida approximation operator with an application. Math. Sci. 11 (2), 155–163 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Akram, M., Chen, J.W., Dilshad, M.: Generalized Yosida approximation operator with an application to a system of Yosida inclusions, J. Nonlinear Funct. Anal., 2018. Article ID 17 (2018)Google Scholar
  3. 3.
    Byrne, C.: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    Byrne, C., Censor, Y., Gibali, A., Reich, S.: Weak and strong convergence of algorithms for the split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cegeilski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Springer, New York (2012)Google Scholar
  6. 6.
    Ceng, L.C., Wen, C.F., Yao, Y.: Iteration approaches to hierarchical variational inequalities for infinite nonexpansive mappings and finding zero points of m-accretive operators. J. Nonlinear Var. Anal. 1, 213–235 (2017)zbMATHGoogle Scholar
  7. 7.
    Censor, Y., Bortfeld, T., Martin, N., Trofimov, A.: A unified approach for inversion problem in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)CrossRefGoogle Scholar
  8. 8.
    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8(2), 221–239 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chang, S.S., Yao, J.C., Kim, J.K., Yang, L.: Iterative approximation to convex feasibility problems in Banach space. Fixed Point Theory Appl. 2007, 46797 (2007).  https://doi.org/10.1155/2007/46797 MathSciNetzbMATHGoogle Scholar
  11. 11.
    Eiche, B.: Iteration methods for convexly constrained ill-posed problems in Hilbert spaces. Numer. Funct. Anal. Optim. 13, 413–429 (1992)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mainge, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-valued Anal. 16, 899–912 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Marino, G., Xu, H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73, 591–597 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Palta, R.J., Mackie, T.R. (eds.): Intensity-modulated Radiation Therapy: The State of Art, Medical Physics Monograph. Medical Physics Publishing, Madison (2003)Google Scholar
  17. 17.
    Pettersson, R.: Yosida approximations for multi-valued stochastic differential equations. Stochastics and Stochastic Rep. 52(1-2), 107–120 (1995)CrossRefzbMATHGoogle Scholar
  18. 18.
    Qin, X., Yao, J.C.: Projection splitting algorithms for nonself operators. J. Nonlinear Convex Anal. 18, 925–935 (2017)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Shan, S.Q., Xiao, Y.B., Huang, N.J.: A new system of generalized implicit set-valued variational inclusions in Banach spaces. Nonlinear Func. Anal. Appl. 22 (5), 1091–1105 (2017)zbMATHGoogle Scholar
  20. 20.
    Sitthithakerngkiet, K., Deepho, J., Kumam, P.: A hybrid viscocity algorithm via modify the hybrid steepest descent method for solving the split variational inclusion in image reconstruction and fixed point problems. Appl. Math. Comput. 250, 986–1001 (2015)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhang, C., Xu, Z.: Explicit iterative algorithm for solving split variational inclusion and fixed point problem for the infinite family of nonexpansive operators. Nonlinear Func. Anal. Appl. 21(4), 669–683 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Mijanur Rahaman
    • 1
    Email author
  • Mohd. Ishtyak
    • 1
  • Rais Ahmad
    • 1
  • Imran Ali
    • 1
  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia

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