Abass, Hammed Anuoluwapo; Oyewole, Olawale Kazeem; Aphane, Maggie On split equality monotone variational inclusion and fixed point problems in reflexive Banach spaces. (English) Zbl 07959973 Topol. Methods Nonlinear Anal. 64, No. 1, 317-338 (2024). Summary: In this paper, motivated by the works of Akbar and Shahrosvand [Filomat 32 (2018), no. 11, 3917-3932], Ogbuisi and Izuchukwu [Numer. Funct. Anal. Optim. 41 (2020), no. 2, 322-343], and some other related results in the literature, we introduce a Halpern iterative algorithm and employ a Bregman distance approach for approximating a solution of split equality monotone variational inclusion problem and fixed point problem of Bregman relatively nonexpansive mapping in reflexive Banach spaces. Under suitable condition, we state and prove a strong convergence result for approximating a common solution of the aforementioned problems. Furthermore, we give an application of our main result to variational inequality problems and provide some numerical examples to illustrate the convergence behavior of our result. The result presented in this paper extends and complements many related results in literature. MSC: 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 Keywords:Bregman relatively nonexpansive mappings; fixed point problem; iterative scheme; monotone operators; split equality problem × Cite Format Result Cite Review PDF Full Text: DOI References: [1] 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 spaces, Fixed Point Theory 21 (2020), no. 2, 397-412. · Zbl 07285133 [2] 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. Ser. A 80 (2018), no. 1, 175-190. · Zbl 1424.47135 [3] H.A. Abass, C.C. Okeke and O.T. Mewomo, On split equality mixed equilibrium and fixed point problems of generalized \(k_i\)-strictly pseudo-contractive multivalued mappings, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 25 (2018), no. 6, 369-395. · Zbl 06994569 [4] R. Ahmad and Q.R. Ansari, An iterative algorithm for generalized nonlinear variational inclusions, Appl. Math. Lett. 13 (2000), no. 5, 23-26. · Zbl 0954.49006 [5] R. Ahmad, Q.H. Ansari and S.S. Irfan, Generalized variational inclusion and generalized resolvent equations in Banach spaces, Comput. Math. Appl. 29 (2005), 1825-1835. · Zbl 1081.49004 [6] A. Akbar and E. Shahrosvand, Split equality common null point problem for Bregman quasi-nonexpansive mappings, Filomat 32 (2018), no. 11, 3917-3932. · Zbl 1498.47119 [7] B. Ali and M. H. Harbau, Convergence theorems for Bregman \(K\)-mappings and mixed equilibrium problems in reflexive Banach spaces, J. Funct. Spaces, vol. 2016, Article ID 5161682, 18 pp. · Zbl 1459.47025 [8] K. Aoyama, Y. Kamimura, W. Takahashi and M. Toyoda, Approximation of common fixed point of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), 2350-2360. · Zbl 1130.47045 [9] H. Attouch, A. Cabot, F. Frankel and J. Peypouquet, Alternating proximal algorithms for constrained variational inequalities. Applications to domain decomposition for PDEs, Nonlinear Anal. 74 (2011), 7455-7473. · Zbl 1228.65100 [10] H.H. Bauschke, J.M. Borwein, Legendre functions and method of random Bregman functions, J. Convex Anal. 4 (1997), 2-67. · Zbl 0894.49019 [11] H.H. Bauschke, J.M. Borwein and P.L. Combettes, Essentially smoothness, essentially strict convexity and Legendre functions in Banach spaces, Commun. Contemp Math. 3 (2001), 615-647. · Zbl 1032.49025 [12] E. Blum and W. Oettli, From Optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994), 123-145. · Zbl 0888.49007 [13] A. Bowers and N.J. Kalton, An Introductory Course in Functional Analysis, New York, 2014. · Zbl 1328.46001 [14] L.M. Bregman, The relaxation method for finding the common point of convex sets and its application to solution of problems in convex programming, USSR Comput. Math. Phys. 7 (1967), 200-217. [15] C. Bryne, Iterative oblique projection onto convex subsets and the split feasibility problems, Inverse Probl. 18 (2002), 441-453. · Zbl 0996.65048 [16] M. Burwein, S. Reich and S. Sabach, A characterization of Bregman firmly nonexpansive opertors using a new monotonicity concept, J. Nonlinear Convex Anal. 12 (2011), 161-184. · Zbl 1221.26019 [17] D. Butnariu and A.N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer Academic Publishers, Dordrecht, 2000. · Zbl 0960.90092 [18] D. Butnairu, S. Reich and A.J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal. 7 (2001), 151-174. · Zbl 1010.47032 [19] D. Butnairu and E. Resmerita, Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. 2006 (2006), Art. ID 84919, 1-39. · Zbl 1130.47046 [20] L.C. Ceng, G. Mastroeni and J.C. Yao, Hybrid proximal-point method for common solutions of equilibrium problems and zeroes of maximal monotone operators , J. Optim. Theory Appl. 142 (2009), no. 3, 431-449. · Zbl 1180.90334 [21] Y. Censor and T. Elfving, A multiprojection algorithmsz using Bregman projections in a product space, Numer. Algor. 8 (1994), 221-239. · Zbl 0828.65065 [22] Y. Censor and S. Reich, Itertaions of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization 37 (1996), 323-339. · Zbl 0883.47063 [23] P. Cholamjiak, P. Sunthrayuth, A Halpern-type iteration for solving the split feasibility problem and fixed point problem of Bregman relatively nonexpansive semigroup in Banach spaces, Filomat 32 (2018), no. 9, 3211-3227. · Zbl 1497.47092 [24] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, 1990. zbMATH: 0712.47043 · Zbl 0712.47043 [25] J. Eckstein and B.F. Svaiter, A family of projective splitting methods for the sum of two maximal monotone operators, Math. Program. III (2008), 173-199. · Zbl 1134.47048 [26] G.Z. Eskandani, M. Raeisi and T. M. Rassias, A hybrid extragradient method for solving pseudomonotone equilibrium problem using Bregman distance, J. Fixed Point Theory Appl. 37 (2018), article no. 132, 20 pp. · Zbl 06969123 [27] Z. Jouymandi and F. Moradlou, Retraction algorithms for solving variational inequalities, pseudomonotone equilibrium problems and fixed point problems in Banach spaces, Numer. Algorithms 78 (2018), 1153-1182. · Zbl 1394.65042 [28] K.R. Kazmi, R. Ali and S. Yousuf, Generalized equilibrium and fixed point problems for Bregman relatively nonexpansive mappings in Banach spaces, J. Fixed Point Theory Appl. (2018), article no. 151, 20 pp. · Zbl 1401.47006 [29] Y. Kimura and S. Saejung, Strong convergence for a common fixed points of two different generalizations of cutter operators, Linear Nonlinear Anal. 1 (2015), 53-65. · Zbl 1326.47089 [30] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Society for Industrial and Applied Mathematics, Philadelphia, 2000. · Zbl 0988.49003 [31] G.M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekon. Mat. Metody 12 (1976), 747-756. · Zbl 0342.90044 [32] 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 [33] V. Martin-Marquez, S. Reich and S. Sabach, Bregman strongly nonexpansive operators in reflexive Banach spaces, J. Math. Anal. Appl. 400 (2013), 597-614. · Zbl 1284.47033 [34] A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl. 150 (2011), 275-283. · Zbl 1231.90358 [35] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006), 191-201. · Zbl 1130.90055 [36] 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. Optim. 41 (2020), no. 3, 322-343. · Zbl 1513.47125 [37] F.U. Ogbuisi and O.T. Mewomo, Iterative solution of split variational inclusion problem in a real Banach spaces, Afr. Mat. 28 (2017), 295-309. · Zbl 1368.49017 [38] C.C. Okeke, A.U. Bello, C. Izuchukwu and O.T. Mewomo, Split equality for monotone inclusion problem and fixed problem in real Banach spaces, Austr. J. Math. Anal. Appl. 12 (2017), no. 2, 1-20. [39] D. Reem and S. Reich, Solutions to inexact resolvent inclusion problems with applications to nonlinear analysis and optimization, Rend. Circ. Mat. Palermo 67 (2018), 337-371. · Zbl 1401.90233 [40] D. Reem, S. Reich and A. De Pierro, Re-examination of Bregman functions and new properties of their divergences, Optimization 68 (2019), 279-348. · Zbl 1407.52008 [41] S. Reich, A weak convergence theorem for the alternating method with Bregman distances, Theory and Applications of Nonlinear operators of Accretive and Monotone Type, Marcel Dekker, New York, 1996, pp. 313-318. · Zbl 0943.47040 [42] S. Reich and S. Sabach, A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Optim. Theory Appl. 10 (2009), 471-485. · Zbl 1180.47046 [43] S. Reich and S. Sabach, Two strong convergence theorems for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim. 31 (2010), 24-44. · Zbl 1200.47085 [44] F. Schopfer, T. Schuster and A.K. Louis, Iterative regularization method for the solution of the split feasibilty problem in Banach spaces, Inverse Probl. 24 (2008), no. 5, 055008. · Zbl 1153.46308 [45] Y. Shehu and F.U. Ogbuisi, An iterative method for solving split monotone variational inclusion and fixed point problem, Cienc. Exatas Fis. Nat. Ser. A Mat. RACSAM 110 (2015), 503-518. · Zbl 1382.47031 [46] Y. Shehu, F.U. Ogbuisi and O.S. Iyiola, Convergence analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces, Optimization 65 (2016), 299-323. · Zbl 1347.49014 [47] J.V. Tie, Convex Analysis: An Introductory Text, Wiley, New York, 1984. · Zbl 0565.49001 [48] S. Timnak, E. Naraghirad and N. Hussain, Strong convergence of Halpern iteration for products of finitely many resolvents of maximal monotone operators in Banach spaces, Filomat 31 (2017), no. 15, 4673-4693. · Zbl 1483.47103 [49] F.Q. Xia and N.J. Huang, Variational inclusions with a general \(H\)-monotone operators in Banach spaces, Comput. Math. Appl. 54 (2010), no. 1, 24-30. · Zbl 1131.49011 [50] C. Zalinescu, Convex Analysis in General Vector spaces, World Scientific Publishing Co. Inc., River Edge NJ, 2002. · Zbl 1023.46003 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.