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Approximating a common solution of extended split equality equilibrium and fixed point problems

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Abstract

In this paper, we study an extension of the split equality equilibrium problem called the extended split equality equilibrium problem. We give an iterative algorithm for approximating a solution of extended split equality equilibrium and fixed point problems and obtained a strong convergence result in a real Hilbert space. We further applied our result to solve extended split equality monotone variational inclusion and equilibrium problems. The result of this paper complements and extends results on split equality equilibrium problems in the literature.

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References

  1. P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optimization, 62(2013), 271-283

    Article  MathSciNet  Google Scholar 

  2. H. Attouch, Variational convergence for functions and operators. Pitman, London (1984)

    MATH  Google Scholar 

  3. H. Attouch, J. Bolte, P. Redont and A. Soubeyran, Alternating proximal algorithms for weakly coupled minimization problems. Applications to dynamical games and PDE’s. J. Convex Anal. 28(2008), 39-44.

    MathSciNet  MATH  Google Scholar 

  4. H. Attouch, P. Redont, and A. Soubeyran. A new class of alternating proximal minimization algorithms with Costs-to-Move. SIAM J. Optim. 18(2007), 1061-1081.

    Article  MathSciNet  Google Scholar 

  5. H.Bréziz; Operateur maximaux monotones, in mathematics studies vol.5, North-Holland, Amsterdam, The Netherlands,(1973).

    Google Scholar 

  6. C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.

    Article  MathSciNet  Google Scholar 

  7. Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365.

    Article  Google Scholar 

  8. Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8(1994), 221-239.

    Article  MathSciNet  Google Scholar 

  9. H. Che, H. Chen and M. Li A new simultaneous iterative method with a parameter for solving the extended split equality fixed point problem. Numer. Algorithmshttps://doi.org/10.1007/s11075-018-0482-6.

  10. S.S. Chang, H.W.J. Lee and C.K. Chan: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 70 (2009), 3307-3319.

    Article  MathSciNet  Google Scholar 

  11. G. Crombez; A hierarchical presentation of operators with fixed points on Hilbert spaces, Numer. Funct. Anal. Optim. 27, (2006), 259-277.

    Article  MathSciNet  Google Scholar 

  12. G. Crombez; A geometrical look at iterative methods for operators with fixed points.Numer. Funct. Anal. Optim. 26, (2005), 157-175.

    Article  MathSciNet  Google Scholar 

  13. P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136.

    MathSciNet  MATH  Google Scholar 

  14. Q.-L. Dong, S.-N. He and J. Zhao, Solving the split equality problem without prior knowledge of operator norms, Optimization, 64 (2014), 1887-1906.

    Article  MathSciNet  Google Scholar 

  15. M. Eslamian, Hybrid method for equilibrium problems and fixed point problems of finite families of nonexpansive semigroups, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 107 (2013), 299-307.

    Article  MathSciNet  Google Scholar 

  16. M. Eslamian, General algorithms for split common fixed point problem of demicontractive mappings, Optimization, 65(2016), 443-465.

    Article  MathSciNet  Google Scholar 

  17. M. Eslamian and A. Abkar, Viscosity iterative scheme for generalized mixed equilibrium problems and nonexpansive semigroups, TOP, 22 (2014), 554-570.

    Article  MathSciNet  Google Scholar 

  18. M. Eslamian and A. Latif, General split feasibility problems in Hilbert spaces, Abstr. Appl. Anal., 2013 (2013), 6 pages.

  19. M. Eslamian, J. Vahidi, Split common fixed point problem of nonexpansive semigroup,Mediterr. J. Math., 13 (2016),1177-1195.

    Article  MathSciNet  Google Scholar 

  20. H. Iiduka, I. Yamada, A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping, SIAM J. Optim., 19 (2009), 1881-1893.

    Article  MathSciNet  Google Scholar 

  21. P. Kocourek, W. Takahashi and J.-C. Yao, Fixed point theorems and weak convergence theorems for generalised hybrid mappingsin Hilbert spaces, Taiwanese J. Math., 14(2010), 2497-2511.

    Article  MathSciNet  Google Scholar 

  22. G. M. Korpelevic, An extragradient method for finding saddle points and for other problems, (Russian) Ékonom. i Mat. Metody, 12 (1976), 747-756.

    MathSciNet  Google Scholar 

  23. F. Kosaka, W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive type mappings in Banach spaces, SIAM. J. Optim., 19(2008), 824-835.

    Article  MathSciNet  Google Scholar 

  24. F. Kosaka, W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces., Arch. Math., Basel, 91(2008), 166-177.

    Article  MathSciNet  Google Scholar 

  25. A. Latif, J. Vahidi and M. Eslamian, Strong convergence for generalized multiple-set split feasibility problem, Filomat, 30(2016), 459-467.

    Article  MathSciNet  Google Scholar 

  26. B.Lemaire; Which fixed point does the iteration method select?,in Recent Advances in optimization,vol.452,pp. 154-157 springer,Berlin,Germany,(1997).

  27. J.-L. Lions, J. Contr\(\hat{0}\)le des systémes distribués singuliers. Gauthier-Villars, Paris (1983).

  28. G. López, V. Martín-Márquez, F.-H. Wang and H.-K. Xu, Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Problems, 27 (2012), 18 pages.

  29. P. E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515.

    Article  MathSciNet  Google Scholar 

  30. P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal.,16 (2008), 899-912.

    Article  MathSciNet  Google Scholar 

  31. A. Moudafi, Alternating CQ-algorithm for convex feasibility and split fixed-point problems. J. Nonlinear Convex Anal. 15 (2014), 809-818.

    MathSciNet  MATH  Google Scholar 

  32. A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117-121.

    Article  MathSciNet  Google Scholar 

  33. A. Moudafi and E. Al-Shemas, Simultaneous iterative methods for split equality problem, Trans. Math. Program. Appl., 1(2013), 1-11.

    Google Scholar 

  34. A. Moudafi, M. Théra, Proximal and dynamical approaches to equilibrium problems, Ill-posed variational problems and regularization techniques, Trier, (1998), Lecture Notes in Econom. and Math. Systems, Springer, Berlin, 477 (1999), 187-201.

  35. W. Takahashi, Fixed point theorems for new nolinear mappings in a Hilbert space, J. Nonlinear Convex Anal., 11(2010), 78-88.

    Google Scholar 

  36. W. Takahashi, The split common fixed point problem and the shrinking projection method in Banach spaces, J. Convex Anal. 24(2017), 1017-1026.

    MathSciNet  Google Scholar 

  37. W. Takahashi, C.-F, Wen and J.-C. Yao The Shrinking projection method for a finite family of demimetric mappings with variational inequality problem in a Hilbert space, Fixed Point Theory, 19(1)(2018), 407-420.

    Article  MathSciNet  Google Scholar 

  38. H.-K. Xu. Iterative algorithms for nonlinear operators, J. London Math.Soc., 66(2002), 240-256.

    Article  MathSciNet  Google Scholar 

  39. J. Zhao, Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms, Optimization, 64 (2014), 2619-2630.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work is based on the research supported wholly by the National Research Foundation (NRF) of South Africa (Grant Numbers: 111992). The second author acknowledges with thanks the financial support from Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS) Post-doctoral fellowship. Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF.

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Authors and Affiliations

  1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

    F. U. Ogbuisi, F. O. Isiogugu & J. M. Ngnotchouye

  2. Department of Mathematics, University of Nigeria, Nsukka, Nigeria

    F. U. Ogbuisi & F. O. Isiogugu

  3. DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa

    F. O. Isiogugu

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  1. F. U. Ogbuisi
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  2. F. O. Isiogugu
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  3. J. M. Ngnotchouye
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Correspondence to F. U. Ogbuisi.

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Communicated by T S S R K Rao.

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Ogbuisi, F.U., Isiogugu, F.O. & Ngnotchouye, J.M. Approximating a common solution of extended split equality equilibrium and fixed point problems. Indian J Pure Appl Math 52, 46–61 (2021). https://doi.org/10.1007/s13226-021-00124-6

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  • DOI: https://doi.org/10.1007/s13226-021-00124-6

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