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Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces

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Abstract

In this paper, we study the split common fixed point and monotone variational inclusion problem in uniformly convex and 2-uniformly smooth Banach spaces. We propose a Halpern-type algorithm with two self-adaptive stepsizes for obtaining solution of the problem and prove strong convergence theorem for the algorithm. Many existing results in literature are derived as corollary to our main result. In addition, we apply our main result to split common minimization problem and fixed point problem and illustrate the efficiency and performance of our algorithm with a numerical example. The main result in this paper extends and generalizes many recent related results in the literature in this direction.

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References

  1. Alakoya, T.O., Jolaoso, L.O., Mewomo, O.T.: Modified inertia subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems. Optimization. https://doi.org/10.1080/02331934.2020.1723586 (2020)

  2. Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes in Pure and Appl. Math., vol. 178, pp 15–50. Marcel Dekker, New York (1996)

  3. Ansari, Q.H., Rehan, A.: Iterative methods for generalized split feasibility problems in Banach spaces. Carpathian J. Math. 33(1), 9–26 (2017)

    MathSciNet  MATH  Google Scholar 

  4. Aremu, K.O., Izuchukwu, C., Ogwo, G.N., Mewomo, O.T.: Multi-step Iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. J. Ind. Manag Optim. https://doi.org/10.3934/jimo.2020063 (2020)

  5. Aremu, K.O., Jolaoso, L.O., Izuchukwu, C., Mewomo, O.T.: Approximation of common solution of finite family of monotone inclusion and fixed point problems for demicontractive multivalued mappings in CAT(0) Spaces Ricerche Mat. https://doi.org/10.1007/s11587-019-00446-y (2019)

  6. Chambolle, A., Pock, T.: An introduction to continuous optimization for imaging. Acta Numerica 25, 161–319 (2016). https://doi.org/10.1017/S096249291600009X

    Article  MathSciNet  Google Scholar 

  7. Aoyama, K., Kohsaka, F.: Existence of fixed points of firmly nonexpansive-like mappings in Banach spaces. Fixed Point Theory Appl., 2010(2010), Art. ID 512751, p. 15. https://doi.org/10.1155/2010/512751

  8. Bertero, M., Boccacci, P.: Introduction to Inverse Problem in Imaging. Institute of Physics Publishing, Bristol (1998)

    Book  Google Scholar 

  9. Borwein, J., Luke, D.: Duality and convex programming. In: Scherzer, O. (ed.) Handbook of Mathematical Methods in Imaging, vol. 2015, pp 257–304. Springer, New York (2015)

  10. Byrne, C.: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Problems 18(2), 441–453 (2002)

    Article  MathSciNet  Google Scholar 

  11. Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13(4), 759–775 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics, vol. 2057. Springer, Berlin (2012). ISBN 978-3-642-30900-7

    Google Scholar 

  13. Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in product space. Numer. Algor. 8, 221–239 (1994)

    Article  MathSciNet  Google Scholar 

  14. Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algor. 59, 301–323 (2012)

    Article  MathSciNet  Google Scholar 

  15. Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Chambolle, A., Pock, T.: An introduction to continuous optimization for imaging. Acta Numer. 25, 161–319 (2016)

    Article  MathSciNet  Google Scholar 

  17. Chidume, C.E.: Geometric Properties of Banach Spaces and Nonlinear Iterations, Lecture Notes in Mathematics, vol. 1965. Springer, London (2009)

    Google Scholar 

  18. Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul., 4(4) (2005). https://doi.org/10.1137/050626090

  19. Eslamian, M.: Halpern-type iterative algorithm for an infinite family of relatively quasi-nonexpansive multivalued mappings and equilibrium problem in Banach spaces. Mediterr. J. Math. 11(2), 713–727 (2019). https://doi.org/10.1007/s00009-013-0330-9

    Article  MathSciNet  MATH  Google Scholar 

  20. Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  21. Halpern, B.: Fixed points of nonexpansive maps. Bull. Amer. Math. Soc. 73, 957–961 (1967)

    Article  MathSciNet  Google Scholar 

  22. Hendrickx, J.M., Olshevsky, A.: Matrix P-norms are np-hard to approximate if \(P\neq 1, 2, \infty \). SIAM J. Matrix Anal. Appl. 31, 2802–2812 (2010)

    Article  MathSciNet  Google Scholar 

  23. Hsu, M.-H, Takahashi, W., Yao, J.-C.: Generalized hybrid mappings in Hilbert spaces and Banach spaces. Taiwanese J. Math. 16(1), 129–149 (2012)

    Article  MathSciNet  Google Scholar 

  24. Ibaraki, T., Takahashi, W.: A new projection and convergence theorems for projections in Banach spaces. J. Approx. Theory 149(1), 1–14 (2007)

    Article  MathSciNet  Google Scholar 

  25. Iiduka, H.: Fixed point optimization algorithm and its application to network bandwidth allocation. J. Comput. Appl. Math. 236, 1733–1742 (2012)

    Article  MathSciNet  Google Scholar 

  26. Izuchukwu, C., Aremu, K.O., Mebawondu, A.A., Mewomo, O.T.: A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space. Appl. Gen. Topol. 20(1), 193–210 (2019)

    Article  MathSciNet  Google Scholar 

  27. Izuchukwu, C., Mebawondu, A.A., Aremu, K.O., Abass, H.A., Mewomo, O.T.: Viscosity iterative techniques for approximating a common zero of monotone operators in a Hadamard space. Rend. Circ. Mat Palermo (2). https://doi.org/10.1007/s12215-019-00415-2 (2019)

  28. Izuchukwu, C., Okeke, C.C., Isiogugu, F.O.: A viscosity iterative technique for split variational inclusion and fixed point problems between a Hilbert space and a Banach space. J. Fixed Point Theory Appl. 20(157) (2018). https://doi.org/10.1007/s11784-018-0632-4

  29. Jolaoso, L.O., Alakoya, T.O., Taiwo, A., Mewomo, O.T.: A parallel combination extragradient method with Armijo line searching for finding common solution of finite families of equilibrium and fixed point problems. Rend. Circ. Mat Palermo II. https://doi.org/10.1007/s12215-019-00431-2 (2019)

  30. Jolaoso, L.O., Alakoya, T.O., Taiwo, A., Mewomo, O.T.: Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space. Optimization. https://doi.org/10.1080/02331934.2020.1716752 (2020)

  31. Jolaoso, L.O., Taiwo, A., Alakoya, T.O., Mewomo, O.T.: A strong convergence theorem for solving variational inequalities using an inertial viscosity subgradient extragradient algorithm with self adaptive stepsize. Demonstr. Math. 52 (1), 183–203 (2019)

    Article  MathSciNet  Google Scholar 

  32. Jolaoso, L.O., Taiwo, A., Alakoya, T.O., Mewomo, O.T.: A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem. Comput. Appl Math. https://doi.org/10.1007/s40314-019-1014-2(2019)

  33. Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13(2), 938–945 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Kassay, G., Riech, S., Sabach, S.: Iterative methods for solving systems of variational inequalities in Reflexive Banach spaces. SIAM J. Optim. 21, 1319–1344 (2011)

    Article  MathSciNet  Google Scholar 

  35. Kohsaka, F., Takahashi, W.: Proximal point algorithms with Bregman functions in Banach spaces. J. Nonlinear Convex Anal. 6(3), 505–523 (2005)

    MathSciNet  MATH  Google Scholar 

  36. Kohsaka, F., Takahashi, W.: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Optim. 19(2), 824–835 (2008)

    Article  MathSciNet  Google Scholar 

  37. Levi, L.: Fitting a bandlimited signal to given points. IEEE Trans. Inform. Theory 11, 372–376 (1965)

    Article  MathSciNet  Google Scholar 

  38. Lions, P.L.: Approximation de points fixes de contractions. Comptes Rendus de l’Académie des Sciences. Serie A-B 284(21), A1357–A1359 (1977)

    MathSciNet  Google Scholar 

  39. Luo, C., Ji, H., Li, Y.: Utility-based multi-service bandwidth allocation in the 4G heterogeneous wireless networks. IEEE Wireless Communication and Networking Conference. https://doi.org/10.1109/WCNC.2009.4918017 (2009)

  40. Martìn-Màrquez, V., Reich, S., Sabach, S.: Bregman strongly nonexpansive operators in reflexive Banach spaces. J. Math. Anal. Appl. 400, 597–614 (2013)

    Article  MathSciNet  Google Scholar 

  41. Matsushita, S., Takahashi, W.: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J. Approx. Theory 134, 257–266 (2005)

    Article  MathSciNet  Google Scholar 

  42. Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)

    Article  MathSciNet  Google Scholar 

  43. Ogwo, G.N., Izuchukwu, C., Aremu, K.O., Mewom, O.T.: A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space. Bull. Belg. Math. Soc. Simon Stevin, (2019), (accepted, to apeear)

  44. Peypouquet, J.: Convex optimization in normed spaces: Theory, methods and examples, SpringerBriefs in Optimization, (2015). ISBN 978-3-319-13709-4 ISBN 978-3-319-13710-0 (eBook) https://doi.org/10.1007/978-3-319-13710-0

  45. Promluang, K., Kuman, P.: Viscosity approximation method for split common null point problems between Banach spaces and Hilbert spaces. J. Inform. Math. Sci. 9(1), 27–44 (2017)

    Google Scholar 

  46. Reich, S.: A weak convergence theorem for the alternating method with Bregman distances. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes in Pure and Appl. Math., vol. 178, pp 313–318. Marcel Dekker, New York (1996)

  47. Reich, S., Sabach, S.: Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp 299–314. Springer, New York (2010)

  48. Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Amer. Math. Soc. 149, 75–288 (1970)

    Article  MathSciNet  Google Scholar 

  49. Suantai, S., Witthayarat, U., Shehu, Y., Cholamjiak, P.: Iterative methods for the split feasibility problem and the fixed point problem in Banach spaces. Optimization 68(5), 955–980 (2019). https://doi.org/10.1080/02331934.2019.1566328

    Article  MathSciNet  Google Scholar 

  50. Taiwo, A., Jolaoso, L.O., Mewomo, O.T.: A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces. Comput. Appl. Math. 38(2), Article 77 (2019)

    Article  MathSciNet  Google Scholar 

  51. Taiwo, A., Jolaoso, L.O., Mewomo, O.T.: Parallel hybrid algorithm for solving pseudomonotone equilibrium and Split Common Fixed point problems. Bull. Malays. Math. Sci Soc. https://doi.org/10.1007/s40840-019-00781-1 (2019)

  52. Taiwo, A., Jolaoso, L.O., Mewomo, O.T.: General alternative regularization method for solving Split Equality Common Fixed Point Problem for quasi-pseudocontractive mappings in Hilbert spaces. Ricerche Mat. https://doi.org/10.1007/s11587-019-00460-0 (2019)

  53. Taiwo, A., Jolaoso, L.O., Mewomo, O.T.: Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert Spaces. J. Ind. Manag. Optim., (2020), (accepted, to appear)

  54. Takahashi, W.: Nonlinear Functional Analysis, Fixed Point Theory and Applications. Yokohama Publisher, Yokohama (2000)

    MATH  Google Scholar 

  55. Takahashi, W., Yao, J.-C.: Strong convergence theorems by hybrid methods for the split common null point problem in Banach spaces. Fixed Point Theory Appl. 2015, 87 (2015). https://doi.org/10.1186/s13663-015-0324-3

    Article  MathSciNet  Google Scholar 

  56. Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58(5), 486–491 (1992)

    Article  MathSciNet  Google Scholar 

  57. Xu, H.-K : Another control condition in an iterative method for nonexpansive mappings. Bull. Austral. Math. Soc. 65(1), 109–113 (2002)

    Article  MathSciNet  Google Scholar 

  58. Xu, H.-K.: Strong convergence of approximating fixed point sequences for nonexpansive mappings. Bull. Austral. Math. Soc 74, 143–151 (2006)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors sincerely thank the anonymous reviewers for their careful reading, constructive comments, and fruitful suggestions that substantially improved the manuscript.

Funding

The first author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. The third author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the IMU and NRF.

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

    A. Taiwo, T. O. Alakoya & O. T. Mewomo

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  1. A. Taiwo
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  3. O. T. Mewomo
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Correspondence to O. T. Mewomo.

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Taiwo, A., Alakoya, T.O. & Mewomo, O.T. Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces. Numer Algor 86, 1359–1389 (2021). https://doi.org/10.1007/s11075-020-00937-2

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