International Journal of Nonlinear Analysis and Applications
Strong convergence result for split inclusion problems in Banach spaces
Document Type : Research Paper
Author
School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur (C.G.) 492010, India
Abstract
By using Halpern’s type iteration process, an iterative algorithm is proposed to study the split inclusion problem and fixed points of a relatively nonexpansive mapping in Banach spaces. This method uses dynamic stepsize that is generated at each iteration by simple computations, which allows it to be easily implemented without the prior information of the operator norm. Then, the main result is used to study the fixed points of a countable family of relatively nonexpansive mappings and the semigroup of relatively nonexpansive mappings. Finally, a numerical example is provided to illustrate the main result.
Keywords
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[33] K. Sitthithakerngkiet, J. Deepho, J. Martinez-Moreno and P. Kumam, Convergence analysis of a general iterative algorithm for finding a common solution of split variational inclusion and optimization problems , Numer. Algorithms 79 (2018), no. 3, 801–824.
[34] S. Suantai, Y. J. Cho and P. Cholamjiak, Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces, Comput. Math. Appl. 64 (2012), no. 4, 489—499
[35] S. Suantai, Y. Shehu, P. Cholamjiak and O.S. Iyiola, Strong convergence of a self-adaptive method for the split feasibility problem in Banach spaces, J. Fixed Point Theory Appl. 20 (2018), no. 2, 1–21.
[36] S. Suthep, Y. Shehu and P. Cholamjiak, Nonlinear iterative methods for solving the split common null point problem in Banach spaces, Optim. Meth. Softw. 34 (2019), no. 4, 853–874.
[37] Y. Su, H. K. Xu, H. and X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010), no. 12, 3890–3906.
[38] S. Takahashi and W. Takahashi, The split common null point problem and the shrinking projection method in Banach spaces, Optim. 65 (2016), no. 2, 281–287.
[39] A. Taiwo, L.O. Jolaoso and O.T. Mewomo, Inertial-type algorithm for solving split common fixed point problems in Banach spaces, J. Sci. Comput. 86 (2021), no. 1, 1–30.
[40] H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127–1138.
[7] M. Asadi and E. Karapinar, Coincidence point theorem on Hilbert spaces via weak Ekeland variational principle and application to boundary value problem, Thai J. Math. 19 (2021), no. 1, 1–7.
[8] T.D. Benavides, G.L. Acedo and H.K. Xu, Construction of sunny nonexpansive retractions in Banach spaces, Bull. Aust. Math. Soc. 66 (2002), no. 1, 9—16.
[9] T. Bonesky, K.S. Kazimierski, P. Maass, F. Sch¨opfer and T. Schuster, Minimization of Tikhonov functionals in Banach spaces, Abstr. Appl. Anal. 2008 (2008), 1–19.
[10] L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys. 7 (1967), 200–217.
[11] C. Byrne, Y. Censor, A. Gibali and S. Reich, Weak and strong convergence of algorithms for the split common null point problem, J. Nonlinear Convx Anal. 13 (2012), no. 4, 759–775.
[12] Y. Censor, T. Bortfeld, B. Martin B. and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol. 51 (2006), no. 10, 2353—2365.
[13] Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms 8 (1994), no. 2, 221–239.
[14] P. Cholamjiak, S. Suantai and P. Sunthrayuth, An iterative method with residual vectors for solving the fixed point and the split inclusion problems in Banach spaces, Comput. Appl. Math. 38 (2019), no. 1, 1–25.
[15] I. Cioranescu, Duality Mappings and Nonlinear Problems, Geometry of Banach Spaces , Math. Appl. 62 (1990).
[16] J.A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396–414.
[17] M.M. Day, Uniform convexity in factor and conjugate spaces, Ann. Math. 45 (1944), no. 2, 375–385.
[18] K.J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer, New York Inc., 2000.
[19] K.R. Kazmi and H. Rizvi, An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping, Optim. Lett. 8 (2014), no. 3, 1113–1124.
[20] F. Kohsaka and W. Takahashi, Proximal point algorithm with Bregman function in Banach spaces , J. Nonlinear Convx Anal. 6 (2005), no. 3, 505–523.
[21] L.W. Kuo and D.R. Sahu, Bregman distance and strong convergence of proximal-type algorithms, Abstr. Appl. Anal. 2013 (2013), 1–12.
[22] G. L´opez, V. Martin-Marquez, F.H. Wang and H.K. Xu, Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Probl. 28 (2012), no. 8, 085004.
[23] P.E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), 899–912.
[24] S. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory 134 (2005), no. 2, 257–266.
[25] A. Moudafi, Split monotone variational inclusions, J. Optim Theory Appl. 150 (2011), no. 2, 275–283.
[26] E. Naraghirad and J.C. Yao, Bregman weak relatively non expansivemappings in Banach space, Fixed Point Theory Appl. 2013 (2013), no. 1, 1–43.
[27] F.U. Ogbuisi and O. T. Mewomo, Iterative solution of split variational inclusion problem in a real Banach spaces , Afrika Mat. 28 (2017), no. 1, 295–309.
[28] R.P. Phelps, Convex functions, monotone operators, and differentiability, Lecture Notes in Mathematics, 1364, Springer, 2009.
[29] N. Pholasa, K. Kankam and P. Cholamjiak, Solving the split feasibility problem and the fixed point problem of left Bregman firmly nonexpansive mappings via the dynamical step sizes in Banach spaces , Vietnam J. Math. 49(2021), no. 4, 1011–1026.
[30] S. Reich, Book Review: geometry of Banach spaces, duality mappings and nonlinear problems , Bull. Amer. Math. Soc. 26 (1992), no. 2, 367—370.
[31] S. Reich and S. Sabach, Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces, Fixed-point algorithms for inverse problems in science and engineering. Springer, New York, NY, 2011, pp. 301–316.
[32] F. Sch¨opfer, T. Schuster and A. . Louis, An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Probl. 24 (2008), no. 5, 1–20.
[33] K. Sitthithakerngkiet, J. Deepho, J. Martinez-Moreno and P. Kumam, Convergence analysis of a general iterative algorithm for finding a common solution of split variational inclusion and optimization problems , Numer. Algorithms 79 (2018), no. 3, 801–824.
[34] S. Suantai, Y. J. Cho and P. Cholamjiak, Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces, Comput. Math. Appl. 64 (2012), no. 4, 489—499
[35] S. Suantai, Y. Shehu, P. Cholamjiak and O.S. Iyiola, Strong convergence of a self-adaptive method for the split feasibility problem in Banach spaces, J. Fixed Point Theory Appl. 20 (2018), no. 2, 1–21.
[36] S. Suthep, Y. Shehu and P. Cholamjiak, Nonlinear iterative methods for solving the split common null point problem in Banach spaces, Optim. Meth. Softw. 34 (2019), no. 4, 853–874.
[37] Y. Su, H. K. Xu, H. and X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010), no. 12, 3890–3906.
[38] S. Takahashi and W. Takahashi, The split common null point problem and the shrinking projection method in Banach spaces, Optim. 65 (2016), no. 2, 281–287.
[39] A. Taiwo, L.O. Jolaoso and O.T. Mewomo, Inertial-type algorithm for solving split common fixed point problems in Banach spaces, J. Sci. Comput. 86 (2021), no. 1, 1–30.
[40] H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127–1138.
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Volume 14, Issue 1
January 2023Pages 2247-2263
- Receive Date: 06 October 2022
- Revise Date: 06 November 2022
- Accept Date: 13 December 2022