An inertial-based hybrid and shrinking projection methods for solving split common fixed point problems in real reflexive spaces

Document Type : Research Paper

Authors

1 Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria

2 Department of Mathematics, Chukwuemeka Odumegwu Ojukwu University, Anambra State, Nigeria

Abstract

We introduce and study an inertial-based iterative algorithm for solving the split common fixed point problem involving a finite family of Bregman quasi-strictly pseudocontractive mappings in real reflexive Banach spaces. Strong convergence of the proposed algorithm is obtained under mild assumptions.

Keywords

[1] A. Ambrosetti, and G. Prodi, A Primer of Nonlinear Analysis, Cambridge University, Press, Cambridge, 1993.
[2] E. Asplund, and R.T. Rockafellar, Gradient of convex function, Trans. Amer. Math. Soc. 228 (1969), 443—467.
[3] A. Bashir, J.N. Ezeora and M.S. Lawal, Inertial algorithm for solving fixed point and generalized mixed equilibrium problems in Banach spaces, Pan Amer. Math. J. 29 (2019), no. 3, 64–83.
[4] H.H. Bauschke, J.M. Borwein and P.L. Combettes, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math. 3 (2001), 615-–647.
[5] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAMJ. Imag. Sci. 2 (2009), 183-–202.
[6] J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problem, Springer, New York, 2000.
[7] L.M. Bregman, The relaxation method for finding the common point of convex set and its application to solution of convex programming , USSR Comput. Math. Phys. 7 (1967), 200–217.
[8] D. Butnariu and AN. Iusem,Totally Convex Functions For Fixed Point Computation and Infinite Dimentional Optimization,. Kluwer Academic, Dordrecht, 2000.
[9] D. Butnariu and E. Resmerita, Bregman distances, totally convex functions and a method of solving operator equations in Banach spaces, Abstr. Appl. Anal. 2006 (2006), 1–39.
[10] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Prob. 18 (2002), 441—453.
[11] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Prob. 20 (2004), 103—120.
[12] C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal. 13 (2012), 759—775.
[13] Y. Censor and T. Elfving, A multi projection algorithm using Bregman projections in a product space Numer. Algorithms 8 (1994), 221–239.
[14] Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its application, Inverse Prob. 21 (2005), 2071—2084.
[15] Y. Censor, A. Gibali and S. Reich,Algorithms for the split variational inequality problems Numer. Algorithms 59 (2012),301—323.
[16] Y. Censor and A. Segal, The split common fixed point problem for directed operators J. Convex Anal. 16 (2009), 587—600.
[17] U. Hiriart and J.B. Lemarchal, Convex Analysis and Minimization Algorithms II, Grudlehren der Mathematischen Wissenchaften, 1993.
[18] D. Lorenz and T. Pock, An inertial forward-backward algorithm for monotone inclusions J. Math. Imag. Vis. 51 (2015), 311—325.
[19] P.E. Maing´e,Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math. 219 (2008), 223—236.
[20] E. Masad and S. Reich, A note on the multiple-set split convex feasibility problem in Hilbert space, J. Nonlinear Convex Anal. 8 (2007), 367–371.
[21] A. Moudafi, The split common fixed point problem for demicontractive mappings, Inverse Prob. 26 (2010), 055007.
[22] E. Naraghirad and J.C. Yao, Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2013 (2013), 141.
[23] B.T. Polyak, Some method of speeding up the convergence of the iteration methods, USSR Comput. Math. Phys. 4 (1964), 1–17.
[24] X. Qin, L. Wang and J.C. Yao, Inertial splitting method for maximal monotone mappings J. Nonlinear Convex Anal. 21 (2020), no. 10, 2325–2333.
[25] S. Reich and S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces, Nonlinear Anal. 73 (2010), 122–135.
[26] S. Reich and S. Sabach,A strong convergence theorem for a proximal-type algorithms in reflexive Banach spaces, J. Nonlinear Convex Anal. 10 (2009), 471–485.
[27] S. Reich and S. Sabach, Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces Springer, New York, (2011), 301–316.
[28] S. Reich, and T.M. Tuyen, Two projection algorithms for solving the split common fixed point problem, J. Optim. Theory Appl. 186 (2020), no. 1, 148–168.
[29] E. Resmerita, On total convexity, Bregman projections and stability in Banach spaces, J. Convex Anal. 11 (2004),1—16.
[30] R.T. Rockafellar, Level sets and continuity of conjugate convex functions, Trans. Amer. Math. Soc. 123 (1966),46-–63.
[31] Y. Shehu, O.S. Iyiola and C.D. Enyi, An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces, Numer. Algorithms 72 (2016), 835—864.
[32] S. Takahashi and W. Takahashi, The split common null point problem and the shrinking projection method in Banach space, Optim. 65 (2016), 281—287.
[33] W. Takahashi, The split common null point problem in Banach spaces, Arch. Math. 104 (2015), 357—365.
[34] T.M. Tuyen, A strong convergence theorem for the split common null point problem in Banach spaces, Appl. Math. Optim. 79 (2019), 207—227.
[35] T.M. Tuyen, and N.S. Ha, A strong convergence theorem for solving the split feasibility and fixed point problems in Banach spaces, J. Fixed Point Theory Appl. 20 (2018), no. 4, 1–17.
[36] T.M. Tuyen, N.S. Ha and N.T.T. Thuy,A shrinking projection method for solving the split common null point problem in Banach spaces, Numer. Algorithms 81 (2019), 813-–832.
[37] G.C. Ugwunnadi, A. Bashir, S.M Ma’aruf and I. Ibrahim, Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in reflexive Banach spaces, Fixed Point Theory Appl. 231 (2014), 1–16.
[38] C. Zalinescu,Convex Analysis in General Vector Spaces, World Scientific, River Edge, Guarantee, 2002.
[39] H.H. Bauschke, P.L. Combettes and J.M. Borwein, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math. 3 (2001), 615–647.
Volume 14, Issue 1
January 2023
Pages 2541-2556
  • Receive Date: 16 October 2021
  • Revise Date: 28 September 2022
  • Accept Date: 03 December 2022