Shrinking approximation method for solution of split monotone variational inclusion and fixed point problems in Banach spaces

Document Type : Research Paper


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

2 DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)


In this paper, we investigate a shrinking algorithm for finding a solution of split monotone variational inclusion problem which is also a common fixed point problem of relatively nonexpansive mapping in uniformly convex real Banach spaces which are also uniformly smooth. The iterative algorithm employed in this paper is design in such a way that it does not require prior knowledge of operator norm. We prove a strong convergence result for approximating the solutions of the aforementioned problems and give applications of our main result to split convex minimization problem. The result present in this paper extends and complements many related results in literature.


[1] H. A. Abass, F. U. Ogbuisi and O. T. Mewomo, Common solution of split equilibrium problem with no prior knowledge of operator norm, U. P. B Sci. Bull., Series A. 80(1) (2018) 175–190.
[2] H. A. Abass, C. Izuchukwu, O. T. Mewomo and Q. L. Dong, Strong convergence of an inertial forward-backward splitting method for accretive operators in real Banach space, Fixed Point Theory, 21(2) (2020) 397–412.
[3] H. A. Abass, K. O. Aremu, L. O. Jolaoso and O.T. Mewomo, An inertial forward-backward splitting method for approximating solutions of certain optimization problem, J. Nonlinear Funct. Anal. 2020 (2020), Article ID 6.
[4] Y.I. Alber, Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications in: Kartsatos, A.G (Ed). Theory and Applications of Nonlinear Operators and Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1996.
[5] Y. I. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. Math. J. 4(2) (1994) 39–54.
[6] Y. Alber and L. Ryazantseva, Nonlinear ill-posed problems of monotone type, Springer, Dordrecht, 2006.
[7] K. Aoyama and F. Koshaka, Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings, Fixed Point Theory Appl. 95 (2014) 13 pp.
[8] K. Aoyama and F. Koshaka, Existence of fixed points of firmly nonexpansive-like mappings in Banach spaces, Fixed Point Theory Appl. (2010) Art. ID 512751, p. 15.
[9] K. Avetisyan, O. Djordjevic and M. Pavlovic, Littlewood-Paley inequalities in uniformly convex and uniformly smooth Banach spaces, J. math. Anal. Appl. 336(1) (2007) 31–43.
[10] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, R. S. R, Bucharest, 1976.
[11] Y. Censor, A. Gibali and S. Reich, Algorithms for split variational inequality problem, Numer. Algor. 59 (2012) 301–323.
[12] Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projection in a product space, Numer. Algor., 8 (1994) 221–239.
[13] Y. Censor, T. Elfving, N. Kopf and T. Bortfield, The multiple-sets split feasibilty problem and its applications for inverse problems, Inverse Prob. 21 (2005) 2071–2084.
[14] Q. L. Dong, D. Jiang, P.Cholmjiak and Y. Shehu, A strong convergence result involving an inertial forwardbackward splitting algorithm for monotone inclusions, J. Fixed Theory Appl. 19(4) (2017) 3097–3118.
[15] B. Eicke, Iteration methods for convexly constrained ill-posed problems in Hilbert space, Numer. Funct. Anal. Optim. 13, (1992) 413–429.
[16] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers Group, Dordrecht, 1996.
[17] J. N. Ezeora and C. Izuchukwu, Iterative approximation of solution of split variational inclusion problem, Filomat, 32 (8) (2018) 2921–2932.
[18] J. N. Ezeora, H. A. Abass and C. Izuchukwu, Strong convergence of an inertial-type algorithm to a common solution of minimization and fixed point problems, Math. Vesnik, 71(4) (2019) 338-350.
[19] K. Geobel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990.
[20] G. Kassay, S. Reich and S. Sabach, Iterative methods for solving systems of variational inequalities in Reflexive Banach spaces, SIAM J. Optim. 21 (2011) 1319–1344.
[21] F. Kohsaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim. 19(2) (2008) 824–835.
[22] Z. Jouymandi and F. Moradlou, Retraction algorithms for solving variational inequalities, pseudomonotone equilibrium problems and fixed point problems in Banach spaces, Numer. Algor. 78 (2018) 1153–1182.
[23] S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in Banach space, SIAM J. Optim. 13 (2002) 938–945.
[24] P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal. 16 (1979) 964–979.
[25] Z. Ma, L. Wang and S. S. Chen, On the split feasibility problem and fixed point problem of quasi-φ-nonexpansive in Banach spaces, Numer. Algor. (2018),
[26] S. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 134 (2005) 257–266.
[27] A.A. Mebawondu, C. Izuchukwu, K.O. Aremu and O.T. Mewomo, Some fixed point results for a generalized TAC-Suzuki-Berinde type F-contractions in b-metric spaces, Appl. Math. E-Notes 19 (2019) 629-–653.
[28] A.A. Mebawondu and O.T. Mewomo, Some fixed point results for TAC-Suzuki contractive mappings, Commun. Korean Math. Soc. 34(4) (2019) 1201-–1222.
[29] A. A. Mebawondu and O.T. Mewomo, Some convergence results for Jungck-AM iterative process in hyperbolic spaces, Aust. J. Math. Anal. Appl. 16(1)(2019) 20.
[30] A. A. Mebawondu and O.T. Mewomo, Suzuki-type fixed point results in Gb-metric spaces, Asian-Eur. J. Math. (2020) DOI: 10.1142/S1793557121500704.
[31] A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl. 150 (2011) 275–283.
[32] A. Moudafi and B. S. Thakur, Solving proximal split feasibility problems without prior knowledge of operator norms, Optim. Letter, 8 (2014) 2099–2110.
[33] F.U. Ogbuisi and C. Izuchukwu, Approximating a zero of sum of two monotone operators which solves a fixed point problem in reflexive Banach spaces, Numer. Funct. Anal. 41 (3) (2020) 322-–343
[34] D. H. Peaceman and H. H. Rashford, The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math. 3 (1995) 267–275.
[35] J. Peypouquet, Convex optimization in Normed spaces. Theory, Methods and Examples, Springer Briefs in Optimization, Springer, 2015.
[36] K. Promluang and P. Kuman, Viscosity approximation method for split common null point problems betweenBanach spaces and Hilbert spaces, J. Inform. Math. Sci. 9 (1) (2017) 27–44.
[37] X. Qin, Y. J. Cho and S. M. Kang, Convergence theorems of common elements for equilibrium problem and fixed point problems in Banach spaces, J. Comput. Appl. Math. 225 (2009) 20–30.
[38] R.T. Rockfellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1977) 877–808.
[39] S. Reich and S. Sabach, 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.
[40] R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970) 75–288.
[41] Y. Shehu, Convergence results of forward-backward algorithms for sum of monotone operators in Banach spaces, Results Math. 74 (2019) 138.
[42] Y. Shehu, Iterative approximation for zeros of sum of accretive operators in Banach spaces, J. Fuct. Spac. (2015), Article ID 5973468, 9 pages.
[43] Y. Shehu, Iterative approximation method for finite family of relatively quasi-nonexpansive mapping and systems of equilibrium problem, J. Glob. Optim. (2014), DOI.10.1007/s10898-010-9619-4.
[44] Y. Shehu and O. S. Iyiola, Convergence analysis for the proximal split feasibility using an inertial extrapolation term method, J. Fixed Point Theory Appl. 19 (2017) 2483–2510.
[45] A. Taiwo, T. O. Alakoya and O. T. Mewomo, Halpern-type iterative process for solving split common fixed point and monotone variational inclusion between Banach spaces, Numer. Algor., (2020), 11075-020-00737-2.
[46] W. Takahashi, Nonlinear Functional Analysis, Fixed Theory Applications, Yokohama-Publishers, 2000.
[47] P.T. Vuong, J. J. Stroduot and V. H. Nguyen, A gradient projection method for solving split equality and split feasibility problems in Hilbert space, Optim. 64 (2015) 2321–2341.
[48] K. Wattanawitoon and P. Kuman, Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Anal. Hybrid Syst. 3 (2009) 11–20.
[49] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. Theory Meth. Appl. 16 (1991) 1127–1138.
[50] J. C. Yao, Variational inequalities with generalized monotone operators, Math. Oper. Res. 19 (1994) 691–705.
[51] H. Zhang and L. Ceng, Projection splitting methods for sums of maximal monotone operators with applications, J. Math. Anal. Appl. 406 (2013) 323–334.
[52] J. Zhang and N. Jiang, Hybrid algorithm for common solution of monotone inclusion problem and fixed point problem and applications to variational inequalities, Springer Plus, 5 (2016) 803.
Volume 12, Issue 2
November 2021
Pages 825-842
  • Receive Date: 09 February 2021
  • Accept Date: 24 April 2021