Top

Numerical Algorithms

Published in:

10-05-2020 | Original Paper

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

Authors: A. Taiwo, T. O. Alakoya, O. T. Mewomo

Published in: Numerical Algorithms | Issue 4/2021

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

next article On equivalence of three-parameter iterative methods for singular symmetric saddle-point problem

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

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)

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)

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

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)

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)

Chambolle, A., Pock, T.: An introduction to continuous optimization for imaging. Acta Numerica 25, 161–319 (2016). https://doi.org/10.1017/S096249291600009XMathSciNetCrossRef

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

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

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)MathSciNetCrossRef

11.

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

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

13.

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

14.

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

15.

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

16.

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

17.

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

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/s00009-013-0330-9MathSciNetCrossRefMATH

20.

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

21.

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

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)MathSciNetCrossRef

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)MathSciNetCrossRef

24.

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

25.

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

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)MathSciNetCrossRef

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)MathSciNetCrossRef

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)MathSciNetMATH

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)MathSciNetCrossRef

35.

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

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)MathSciNetCrossRef

37.

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

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

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)MathSciNetCrossRef

41.

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

42.

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

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)

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)MathSciNetCrossRef

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/10.1080/02331934.2019.1566328MathSciNetCrossRef

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)MathSciNetCrossRef

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

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/s13663-015-0324-3MathSciNetCrossRef

56.

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

57.

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

58.

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

Title: Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces
Authors: A. Taiwo
T. O. Alakoya
O. T. Mewomo
Publication date: 10-05-2020
Publisher: Springer US
Published in: Numerical Algorithms / Issue 4/2021
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI: https://doi.org/10.1007/s11075-020-00937-2

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Other articles of this Issue 4/2021

A novel alternating-direction implicit spectral Galerkin method for a multi-term time-space fractional diffusion equation in three dimensions

Analysis of two new parareal algorithms based on the Dirichlet-Neumann/Neumann-Neumann waveform relaxation method for the heat equation

Accurate sampling formula for approximating the partial derivatives of bivariate analytic functions

Unconditional optimal error estimates of linearized backward Euler Galerkin FEMs for nonlinear Schrödinger-Helmholtz equations

On the circumcentered-reflection method for the convex feasibility problem

Minimum residual modified HSS iteration method for a class of complex symmetric linear systems

Premium Partner