Outer approximation method for zeros of sum of monotone operators and fixed point problems in Banach spaces. (English) Zbl 07457420

Summary: In this paper, we investigate a hybrid algorithm for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators which is also a common fixed point problem for finite family of relatively quasi-nonexpansive mappings and split feasibility problem 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 minimization problem and convexly constrained linear inverse problem.


47H06 Nonlinear accretive operators, dissipative operators, etc.
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J05 Equations involving nonlinear operators (general)
47J25 Iterative procedures involving nonlinear operators
Full Text: Link


[1] H.A. Abass, F.U. Ogbuisi and O.T. Mewomo,Common solution of split equilibrium problem with no prior knowledge of operator norm, UPB Sci. Bull., Series A,80(2018), 175-190. · Zbl 1424.47135
[2] H.A. Abass, C. Izuchukwu, O.T. Mewomon and Q.L. Dong,Strong convergence of an inertial forward-backward splitting method for accretive operators in real Banach space, Fixed Point Theory,21(2020), 397-412. · Zbl 07285133
[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 applicationsin: Kartsatos, A.G (Ed). Theory and Applications of Nonlinear Operators and Accretive and Monotone Type. Lecture Notes Pure Appl. Math.,178, Dekker, New York (1996), 15-50. · Zbl 0883.47083
[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(1994), 39-54. · Zbl 0851.47043
[6] Y. Alber and L. Ryazantseva,Nonlinear ill-posed problems of monotone type, Springer, Dordrecht (2006). xiv+410 pp. ISBN:978-1-4020-4395-6, 1-4020-4395-3. · Zbl 1086.47003
[7] K. Aoyama and F. Koshaka,Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings, Fixed Point Theory Appl.,95(2014), pp.13. · Zbl 1332.47027
[8] K. Avetisyan, O. Djordjevic and M. Pavlovic,Littlewood-Paley inequalities in uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl.,336(2007), 31-43. · Zbl 1213.42060
[9] V. Barbu,Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei, R.S.R, Bucharest, English transl. Noordhof, Leyden, 1976. · Zbl 0328.47035
[10] Y. Censor and T. Elfving,A multiprojection algorithm using Bregman projection in a product space, Numer. Algor.,8(1994), 221-239. · Zbl 0828.65065
[11] 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. · Zbl 1089.65046
[12] Q.L. Dong, D. Jiang, P. Cholmjiak and Y. Shehu,A strong convergence result involving an inertial forward-backward splitting algorithm for monotone inclusions, J. Fixed Theory Appl.,19(2017), 3097-3118. · Zbl 1482.47118
[13] B. Eicke,Iteration methods for convexly constrained ill-posed problems in Hilbert space, Numer. Funct. Anal. Optim.,13(1992), 413-429. · Zbl 0769.65026
[14] H.W. Engl, M. Hanke and A. Neubauer,Regularization of inverse problems, Kluwer Academic Publishers Group, Dordrecht, 1996. · Zbl 0859.65054
[15] 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, Mathematik Vesnik,71(2019), 338-350. · Zbl 1474.47124
[16] 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. · Zbl 1394.65042
[17] S. Kamimura and W. Takahashi,Strong convergence of a proximal-type algorithm in Banach space, SIAM J. Optim.,13(2002), 938-945. · Zbl 1101.90083
[18] J.K. Kim, A.H. Dar and Salahuddin,Existence theorems for the generalized relaxed pseudomonotone variational inequalities,Nonlinear Funct. Anal. and Appl.,25(1) (2020), 25-34. doi.org/10.22771/nfaa.2020.25.01.03 · Zbl 1451.49015
[19] J.K. Kim, T.M. Tuyen and M.T. Ngoc Ha,Two projection methods for solving the split common fixed point problem with multiple output sets in Hilbert spaces,Numer. Funct. Anal. Optimiz.,42(8) (2021), 973-988, https://doi.org/10.1080/01630563.2021.1933528 · Zbl 1494.47122
[20] J.K. Kim and T.M. Tuyen,A parallel iterative method for a finite family of Bregman strongly nonexpansive mappings in reflexive Banach spaces,J. Korean Math. Soc.,57(3) (2020), 617-640. https://doi.org/10.4134/JKMS.j190268 · Zbl 07197878
[21] J.K. Kim and T.M. Tuyen,Parallel iterative method for solving the split common null point problem in Banach spaces,J. Nonlinear and Convex Anal.,20(10) (2019), 20752093. · Zbl 1473.49021
[22] P.L. Lions and B. Mercier,Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal.,16(1979), 964-979. · Zbl 0426.65050
[23] Z. Ma, L. Wang and S.S. Chang,On the split feasibility problem and fixed point problem of quasi-φ-nonexpansive in Banach spaces, Numer. Algor.,80(4) (2019), 1203-1218. https://doi.org/10.1007/s11075-018-0523-1. · Zbl 07042046
[24] S. Matsushita and W. Takahashi,A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory.134(2005), 257-266. · Zbl 1071.47063
[25] A.A. Mebawondu,Proximal Point Algorithms for Finding Common Fixed Points of a Finite Family of Nonexpansive Multivalued Mappings in Real Hilbert Spaces, Khayyam Journal of Mathematics,5(2), 113-123. · Zbl 1438.47114
[26] A. Moudafi and B.S. Thakur,Solving proximal split feasibility problems without prior knowledge of operator norms, Optim. Letter,8(2014), 2099-2110. · Zbl 1317.49019
[27] 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.
[28] J. Peypouquet,Convex optimization in Normed spaces. Theory, Methods and Examples. With a foreword by H. Attouch. Springer briefs in optimization. Springer, Cham., 2015. xiv+124 pp. ISBN: 978-3-319-13709-4, 978-3-319-13710-0. · Zbl 1322.90004
[29] 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. · Zbl 1165.65027
[30] R.T. Rockfellar,Monotone operators and the proximal point algorithm, SIAM J. Control Optim.,14(1977), 877-808.
[31] Y. Shehu,Convergence results of forward-backward algorithms for sum of monotone operators in Banach spaces, Results Math.,74(2019), 138. · Zbl 07099993
[32] Y. Shehu,Iterative approximation for zeros of sum of accretive operators in Banach spaces, J. Funct. Spaces, (2015), Article ID 5973468, 9 pages.
[33] Y. Shehu,Iterative approximation method for finite family of relatively quasinonexpansive mapping and systems of equilibrium problem, J. Glob. Optim., (2014), DOI.10.1007/s10898-010-9619-4.
[34] 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. · Zbl 1493.47100
[35] W. Takahashi,Nonlinear Functional Analysis,Fixed Theory Appl.,YokohamaPublishers, 2000. · Zbl 0997.47002
[36] Ng.T.T. Thuy, P.T.T. Hoai and Ng.T.T. Hoa,Explicit iterative methods for maximal monotone operators in Hilbert spaces, Nonlinear Funct. Anal. Appl.,25(4) (2020), 753767. doi.org/10.22771/nfaa.2020.25.04.09.
[37] 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, Optimization,64(2015), 2321-2341. · Zbl 1329.65129
[38] K. Wattanawitoon and P. Kuman,Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasinonexpansive mappings, Nonlinear Anal. Hybrid Syst.,3(2009), 11-20. · Zbl 1166.47060
[39] H.K. Xu,Inequalities in Banach spaces with applications, Nonlinear Anal. Theory Methods Appl.,16(1991), 1127-1138. · Zbl 0757.46033
[40] J.C. Yao,Variational inequalities with generalized monotone operators, Math. Oper. Res.,19(1994), 691-705. · Zbl 0813.49010
[41] H. Zhang and L. Ceng,Projection splitting methods for sums of maximal monotone operators with applications, J. Math. Anal. Appl.,406(2013), 323-334. · Zbl 1310.47109
[42] 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: 803 (2016). https://doi.org/10
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.