On split generalized equilibrium and fixed point problems of Bregman W-mappings with multiple output sets in reflexive Banach spaces
Abstract
In this paper, we introduce a Halpern iteration process for computing the common solution of split generalized equilibrium problem and fixed points of a countable family of Bregman W-mappings with multiple output sets in reflexive Banach spaces. We prove a strong convergence result for approximating the solutions of the aforementioned problems under some mild conditions. It is worth mentioning that the iterative algorithm employ in this article is designed in such a way that it does not require the prior knowledge of operator norm. We also provide some numerical examples to illustrate the performance of our proposed iterative method. The result discuss in this paper extends and complements many related results in literature.
Keyword : Bregman weak relatively nonexpansive mapping, Bregman W-mapping, Halpern method, iterative scheme, split generalized equilibrium problem
How to Cite
Abass, H. A., Ugwunnadi, G. C., Jolaoso, L. O., & Narain, O. K. (2023). On split generalized equilibrium and fixed point problems of Bregman W-mappings with multiple output sets in reflexive Banach spaces. Mathematical Modelling and Analysis, 28(4), 653–672. https://doi.org/10.3846/mma.2023.17087
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
H.A. Abass, F.U. Ogbuisi and O.T. Mewomo. On split equality mixed equilibrium and fixed point problems of generalized ki-strictly pseudo-contractive multivalued mappings. Dyn. Contin. Discrete Impuls. Syst., Series B: Applications and Algorithms., 25(6):369–395, 2018.
H.A. Abass, C.C. Okeke and O.T. Mewomo. Common solution of split equilibrium problem with no prior knowledge of operator norm. U. P. B Sci. Bull., 80(1), 2018.
K. Afassinou, O.K. Narain and O.E. Otunuga. Iterative algorithm for approximating solutions of split monotone variational inclusion, variational inequality and fixed point problems in real Hilbert space. Nonlinear Funct. Anal. and Appl., 25(3):491–510, 2020.
F. Akutsah, O.K. Narain, H.A. Abass and A.A. Mebawondu. Shrinking approximation method for solution of split monotone variational inclusion and fixed point problems in Banach spaces. International J. Nonlinear Anal. Appl., 12(2):825–842, 2020. https://doi.org/10.1155/2021/9421449
M. Alansari, M. Farid and R. Ali. An inertial iterative algorithm for generalized equilibrium problems and Bregman relatively nonexpansive mappings in Banach spaces. J. Inequal. Appll., 2022(11), 2022.
S.F. Aldosary, W. Cholamjiak, R. Ali and M. Farid. Strong convergence of an inertial iterative algorithm for generalized mixed variational-like inequality problem and Bregman relatively nonexpansive mapping in reflexive Banach space. J. Math., 2021:Art. ID 9421449, 2021. https://doi.org/10.1155/2021/9421449
K. Aoyama, Y. Kimura, W. Takahashi and M. Toyodau. Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal., 67:2350–2360, 2007. https://doi.org/10.1016/j.na.2006.08.032
S. Atsushiba and W. Takahashi. Strong convergence theorems for a finite family of nonexpansive mappings and application. Indian J. Math., 41(2):435–453, 1999.
H.H. Bauschke and J.M. Borwein. Legendre functions and method of random Bregman functions. J. Convex Anal, 4:27–67, 1997.
H.H. Bauschke, J.M. Borwein and P.L. Combettes. Essentially smoothness, essentially strict convexity and Legendre functions in Banach spaces. Commun. Contemp. Math., 3:615–647, 2001. https://doi.org/10.1142/S0219199701000524
E. Blum and W. Oettli. From optimization and variational inequalities to equilibrium problems. Math. Stud., 63:123–145, 1994.
L.M. Bregman. The relaxation method for finding the common point of convex sets and its application to solution of problems in convex programming. U.S.S.R Comput. Math. Phys., 7:200–217, 1967. https://doi.org/10.1016/0041-5553(67)90040-7
D. Butnairu and E. Resmerita. Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstract and Applied Analysis., 2006(2):1–39, Art. ID 84919, 2006. https://doi.org/10.1155/AAA/2006/84919
L.C. Ceng and J.C. Yao. SA hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. App. Math., 214:186–201, 2008. https://doi.org/10.1016/j.cam.2007.02.022
Y. Censor and T. Elfving. A multiprojection algorithms using Bregman projections in a product space. Numer. Algor., 8:221–239, 1994. https://doi.org/10.1007/BF02142692
J. Chen, Z. Wan, L. Yuan and Y. Zhang. Approximation of fixed points of weak relatively nonexpansive mappings in Banach spaces. IJMMS, p. Art. ID 420192, 2011. https://doi.org/10.1155/2011/420192
P.L. Combettes and S.A. Hirstoaga. Equilibrium programming using proximallike algorithm. J. Nonlinear Convex Anal., 6:117–136, 2005.
J. Deepho, W. Kumam and P. Kumam. A new hybrid projection algorithm for solving the generalized equilibrium problems and system of variational inequality problems. J. Math. Model. Algorithms Oper. Res., 13(4):404–423, 2014. https://doi.org/10.1007/s10852-014-9261-0
M. Farid, R. Ali and W. Cholamjiak. An inertial iterative algorithm to find common solution of a split generalized equilibrium and a variational inequality problem in Hilbert spaces. J. Math., 2021(157):Art. ID 3653807, 2021. https://doi.org/10.1155/2021/3653807
M. Farid and K.R. Kazmi. A new mapping for finding a common solution of split generalized equilibrium problem, variational inequality problem and fixed point problem. The Korean J. Math., 27(2):297–327, 2019.
H. Gazmeh and E. Naraghirad. The split common null point problem for Bregman generalized resolvents in two Banach spaces. Optimization., 2020. https://doi.org/10.1080/02331934.2020.1751157
N. Hussain, E. Naraghirad and A. Alotaibi. Existence of common fixed points using Bregman nonexpansive retracts and Bregman functions in Banach spaces. Fixed Point Theory Appl., 113:1–19, 2013. https://doi.org/10.1186/1687-1812-2013-113
C. Izuchukwu, C.C. Okeke and F. O. Isiogugu. A viscosity iterative technique for split variational inclusion and fixed point problems between a Hilbert and a Banach space. J. Fixed Point Theory Appl., 20(157), 2018. https://doi.org/10.1007/s11784-018-0632-4
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., 20(151), 2018. https://doi.org/10.1007/s11784-018-0627-1
K.R. Kazmi and S.H. Rizvi. Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed problem for nonexpansive semigroup. Math. Sci., 7(1), 2013.
Y. Kimura and S. Saejun. Strong convergence for a common fixed points of two different generalizations of cutter operators. Linear Nonlinear Anal., 1:53–65, 2015.
S.Y. Matsushika, K. Nakajo and W. Takahashi. Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces. Nonlinear Anal., 73:1466–1480, 2010. https://doi.org/10.1016/j.na.2010.04.007
E. Naraghirad and J.C. Yao. Bregman weak relatively nonexpansive mappings in Banach space. Fixed Point Theory Appl., 2013. https://doi.org/10.1186/1687-1812-2013-141
N. Naraghirad and S. Timnak. Strong convergence theorems for Bregman Wmappings with applications to convex feasibility problems in Banach spaces. Fixed Point Theory App.,149,2015. https://doi.org/10.1186/s13663-015-0395-1
F.U. Ogbuisi and O.T. Mewomo. Iterative solution of split variational inclusion problem in a real Banach spaces. Afr. Mat., 28:295–309, 2017. https://doi.org/10.1007/s13370-016-0450-z
O.K. Oyewole, H.A. Abass and O.T. Mewomo. A strong convergence algorithm for a fixed point constraint split null point problem. Rend. Circ. Mat., pp. 1–20, 2020.
W. Phuengrattana and K. Lerkchaiyaphum. On solving split generalized equilibrium problem and the fixed point problem for a countable family of nonexpansive multivalued mappings. Fixed Point Theory Appl., 6, 2018. https://doi.org/10.1186/s13663-018-0631-6
S. Reich. Weak convergence theorems for nonexpansive mappings in Banach spaces,. J. Math. Anal. Appl., 67(2):274–276, 1979. https://doi.org/10.1016/0022-247X(79)90024-6
S. Reich. On the asymptotic behavior of nonlinear semigroups and the range of accretive operators. J. Math. Anal. Appl., 79(1):113–126, 1981. https://doi.org/10.1016/0022-247X(81)90013-5
S. Reich and S. Sabach. S strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J. Nonlinear Convex Anal., 10:471–485, 2009.
S. Reich and S. Sabach. Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim., 31(1):24–44, 2010. https://doi.org/10.1080/01630560903499852
S. Reich and A.J. Zaslavski. Existence of a unique fixed point for nonlinear contractive mappings. Math., 2020. https://doi.org/10.3390/math8010055
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(15):4673–4693, 2017.
C. Zalinescu. Convex Analysis in General Vector spaces. World Scientific Publishing Co. Inc., River Edge NJ, 2002.
H.A. Abass, C.C. Okeke and O.T. Mewomo. Common solution of split equilibrium problem with no prior knowledge of operator norm. U. P. B Sci. Bull., 80(1), 2018.
K. Afassinou, O.K. Narain and O.E. Otunuga. Iterative algorithm for approximating solutions of split monotone variational inclusion, variational inequality and fixed point problems in real Hilbert space. Nonlinear Funct. Anal. and Appl., 25(3):491–510, 2020.
F. Akutsah, O.K. Narain, H.A. Abass and A.A. Mebawondu. Shrinking approximation method for solution of split monotone variational inclusion and fixed point problems in Banach spaces. International J. Nonlinear Anal. Appl., 12(2):825–842, 2020. https://doi.org/10.1155/2021/9421449
M. Alansari, M. Farid and R. Ali. An inertial iterative algorithm for generalized equilibrium problems and Bregman relatively nonexpansive mappings in Banach spaces. J. Inequal. Appll., 2022(11), 2022.
S.F. Aldosary, W. Cholamjiak, R. Ali and M. Farid. Strong convergence of an inertial iterative algorithm for generalized mixed variational-like inequality problem and Bregman relatively nonexpansive mapping in reflexive Banach space. J. Math., 2021:Art. ID 9421449, 2021. https://doi.org/10.1155/2021/9421449
K. Aoyama, Y. Kimura, W. Takahashi and M. Toyodau. Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal., 67:2350–2360, 2007. https://doi.org/10.1016/j.na.2006.08.032
S. Atsushiba and W. Takahashi. Strong convergence theorems for a finite family of nonexpansive mappings and application. Indian J. Math., 41(2):435–453, 1999.
H.H. Bauschke and J.M. Borwein. Legendre functions and method of random Bregman functions. J. Convex Anal, 4:27–67, 1997.
H.H. Bauschke, J.M. Borwein and P.L. Combettes. Essentially smoothness, essentially strict convexity and Legendre functions in Banach spaces. Commun. Contemp. Math., 3:615–647, 2001. https://doi.org/10.1142/S0219199701000524
E. Blum and W. Oettli. From optimization and variational inequalities to equilibrium problems. Math. Stud., 63:123–145, 1994.
L.M. Bregman. The relaxation method for finding the common point of convex sets and its application to solution of problems in convex programming. U.S.S.R Comput. Math. Phys., 7:200–217, 1967. https://doi.org/10.1016/0041-5553(67)90040-7
D. Butnairu and E. Resmerita. Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstract and Applied Analysis., 2006(2):1–39, Art. ID 84919, 2006. https://doi.org/10.1155/AAA/2006/84919
L.C. Ceng and J.C. Yao. SA hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. App. Math., 214:186–201, 2008. https://doi.org/10.1016/j.cam.2007.02.022
Y. Censor and T. Elfving. A multiprojection algorithms using Bregman projections in a product space. Numer. Algor., 8:221–239, 1994. https://doi.org/10.1007/BF02142692
J. Chen, Z. Wan, L. Yuan and Y. Zhang. Approximation of fixed points of weak relatively nonexpansive mappings in Banach spaces. IJMMS, p. Art. ID 420192, 2011. https://doi.org/10.1155/2011/420192
P.L. Combettes and S.A. Hirstoaga. Equilibrium programming using proximallike algorithm. J. Nonlinear Convex Anal., 6:117–136, 2005.
J. Deepho, W. Kumam and P. Kumam. A new hybrid projection algorithm for solving the generalized equilibrium problems and system of variational inequality problems. J. Math. Model. Algorithms Oper. Res., 13(4):404–423, 2014. https://doi.org/10.1007/s10852-014-9261-0
M. Farid, R. Ali and W. Cholamjiak. An inertial iterative algorithm to find common solution of a split generalized equilibrium and a variational inequality problem in Hilbert spaces. J. Math., 2021(157):Art. ID 3653807, 2021. https://doi.org/10.1155/2021/3653807
M. Farid and K.R. Kazmi. A new mapping for finding a common solution of split generalized equilibrium problem, variational inequality problem and fixed point problem. The Korean J. Math., 27(2):297–327, 2019.
H. Gazmeh and E. Naraghirad. The split common null point problem for Bregman generalized resolvents in two Banach spaces. Optimization., 2020. https://doi.org/10.1080/02331934.2020.1751157
N. Hussain, E. Naraghirad and A. Alotaibi. Existence of common fixed points using Bregman nonexpansive retracts and Bregman functions in Banach spaces. Fixed Point Theory Appl., 113:1–19, 2013. https://doi.org/10.1186/1687-1812-2013-113
C. Izuchukwu, C.C. Okeke and F. O. Isiogugu. A viscosity iterative technique for split variational inclusion and fixed point problems between a Hilbert and a Banach space. J. Fixed Point Theory Appl., 20(157), 2018. https://doi.org/10.1007/s11784-018-0632-4
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., 20(151), 2018. https://doi.org/10.1007/s11784-018-0627-1
K.R. Kazmi and S.H. Rizvi. Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed problem for nonexpansive semigroup. Math. Sci., 7(1), 2013.
Y. Kimura and S. Saejun. Strong convergence for a common fixed points of two different generalizations of cutter operators. Linear Nonlinear Anal., 1:53–65, 2015.
S.Y. Matsushika, K. Nakajo and W. Takahashi. Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces. Nonlinear Anal., 73:1466–1480, 2010. https://doi.org/10.1016/j.na.2010.04.007
E. Naraghirad and J.C. Yao. Bregman weak relatively nonexpansive mappings in Banach space. Fixed Point Theory Appl., 2013. https://doi.org/10.1186/1687-1812-2013-141
N. Naraghirad and S. Timnak. Strong convergence theorems for Bregman Wmappings with applications to convex feasibility problems in Banach spaces. Fixed Point Theory App.,149,2015. https://doi.org/10.1186/s13663-015-0395-1
F.U. Ogbuisi and O.T. Mewomo. Iterative solution of split variational inclusion problem in a real Banach spaces. Afr. Mat., 28:295–309, 2017. https://doi.org/10.1007/s13370-016-0450-z
O.K. Oyewole, H.A. Abass and O.T. Mewomo. A strong convergence algorithm for a fixed point constraint split null point problem. Rend. Circ. Mat., pp. 1–20, 2020.
W. Phuengrattana and K. Lerkchaiyaphum. On solving split generalized equilibrium problem and the fixed point problem for a countable family of nonexpansive multivalued mappings. Fixed Point Theory Appl., 6, 2018. https://doi.org/10.1186/s13663-018-0631-6
S. Reich. Weak convergence theorems for nonexpansive mappings in Banach spaces,. J. Math. Anal. Appl., 67(2):274–276, 1979. https://doi.org/10.1016/0022-247X(79)90024-6
S. Reich. On the asymptotic behavior of nonlinear semigroups and the range of accretive operators. J. Math. Anal. Appl., 79(1):113–126, 1981. https://doi.org/10.1016/0022-247X(81)90013-5
S. Reich and S. Sabach. S strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J. Nonlinear Convex Anal., 10:471–485, 2009.
S. Reich and S. Sabach. Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim., 31(1):24–44, 2010. https://doi.org/10.1080/01630560903499852
S. Reich and A.J. Zaslavski. Existence of a unique fixed point for nonlinear contractive mappings. Math., 2020. https://doi.org/10.3390/math8010055
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(15):4673–4693, 2017.
C. Zalinescu. Convex Analysis in General Vector spaces. World Scientific Publishing Co. Inc., River Edge NJ, 2002.