×

A forward-backward splitting algorithm for quasi-Bregman nonexpansive mapping, equilibrium problems and accretive operators. (English) Zbl 1493.47082

The paper is devoted to the study of a forward-backward splitting algorithm for finding a zero point of sum of a finite family of \(m\)-accretive operators and \(\alpha\)-inverse strongly accretive operators, solution of equilibrium problems and fixed points problems of quasi-Bregman nonexpansive mappings. Quasi-Bregman nonexpansive mappings has been studied by several authors (see, for example, [V. Martín-Márquez et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 14, 5448–5465 (2012; Zbl 1408.47011)] or [G. C. Ugwunnadi et al., Fixed Point Theory Appl. 2014, Paper No. 231, 16 p. (2014; Zbl 1346.47065)]). The authors of the present paper prove the weak convergence of sequences generated by an algorithm in reflexive Banach space and extends so recent results. Finally, a numerical example is presented to illustrate the convergence of the algorithm.

MSC:

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

References:

[1] E. Asplund and R. T. Rockafellar,Gradient of Convex Function, Trans. Am. Math. Soc.228443-467, 1969. · Zbl 0181.41901
[2] Bashir Ali and M. S. LawanSplitmonotone variational inclution, mixed equilibrium problem and common fixed point for finite families of demicontractive mappings, J. Abstract and Compt. Math.4, 116-135. 2019.
[3] Bashir Ali, J. N. Ezeora and M. S. Lawan,,Inertial algorithm for solving fixed point and generalized mixed equilibrium problems in Banach spaces,PanAmerican Mathematical Society,3, 29, 64-83, 2019.
[4] Bashir Ali, G. C. Ugwunnadi and M. S. Lawan,Split common fixed point problem for Bregman demigeneralized mappings in Banach spaces with applications,J. Nonlinear Sci. Appl.,13, 270-283, 2020.
[5] H. H. Bauschke and J. M. Borwein,Legendre Function and the Method of Bregman Projections,J. Convex Anal.427-67, 1997. · Zbl 0894.49019
[6] H. H. Bauschke and J. M. Borwein,Joint and Separate Convexity of the Bregman Distance, Studies in Computational Mathematics, DOI:10.1016/S1570-579X(01) 80004-5,823-36, 2001. · Zbl 1160.65319
[7] H. H. Bauschke, P. L. Combettes and J.M. Borwein,Essential Smoothness,Essential Strict Convexity, and Legendre functions in Banach Spaces, Commun. Contemp. Math.3615-647, 2001. · Zbl 1032.49025
[8] E. Blum and W. Oettli,From optimization and variational inequalities to equilibrium problems, Math.Stud.63123-145, 1994. · Zbl 0888.49007
[9] J. F. Bonnans and A. Shapiro,Analysis of Optimazation Problems, Springer, New York, 2000. · Zbl 0966.49001
[10] F. E Browder,Perturbation Nonlinear mappings of nonexpansive and accretive type in Banach spaces,Bull. Amer. Math. Soc.73875-882, 1967. · Zbl 0176.45302
[11] F. E. Browder,Nonlinear operators and nonlinear equations of evolution in Banach spaces,Proc. Symp. Pure. Math.181976.
[12] L. M. Bregman,The Relazation Method for Finding The Common Point of Convex Set and its Application to Solution of Convex programming, USSR Comput. Math. Phys.7200-217, 1967.
[13] D. Butnariu and A. N. Iusem,Totally Convex Functions For Fixed Point Computation and Infinite Dimentional Optmization, Kluwer Academic,Dordrecht, 2000. · Zbl 0960.90092
[14] D. Butnariu and E. Resmerita,Bregman distances, totally convex functionsand a method of solving operator equtions in Banach spaces, Abstr. Appl. Anal.2006 1-39, 2006. · Zbl 1130.47046
[15] Y. Censor and A. Lent,An Iterative row-action Method Interval Convex Programming, J. Optim. Theory Appl.34321-353, 1981. · Zbl 0431.49042
[16] S. Y. Cho, X. Qin and L. Wang,Strong convergence of splitting algorithm for treating monotone operators,Fixed Point Theory and Application942014, 2014.
[17] P. L. Combettes and S. A. Hirstoaga,Equillibrium programing in Hilbert space, Nonlinear Convex Anal.6117-136, 2005 .
[18] L. O. Jalaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo,Inertial extragradient method via viscosity approximation approach for solving equilibrium problem in Hilbert space,Optimization, DOI:10.1080/02331934.2020.1716752, 2020. · Zbl 1459.65097
[19] L. O. Jalaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo,Parallel combination extrgradient method with Armijo line searching for finding common solution of finite families of equilibrium and fixed point problem,Rend. Circ. Mat. Palermo II, DOI:10.1007/s12215-019-00431-2, 2019.
[20] Hiriart-Urruty and J.B. Lemarchal,Convex Analysis and Minimization Algorithsm II,Grudlehren der Mathematischen Wissenchaften,3061993.
[21] T. Kato,Nonlinear semigroup and evolution equations,Journal of Mathematical Society Japan,19508-520, 1967. · Zbl 0163.38303
[22] F. Kohsaka, W. Takahashi,Proximal point algorithms with Bregman functions in Banach spaces, J. Nonlinear Convex Anal.6(3) 505-523, 2005. · Zbl 1105.47059
[23] M. S. Lawan, Bashir Ali, M. H. Harbau and G. C. Ugwunnadi, Approximation of common fixed points for finite families of Bregman quasi-total asymptotically nonexpansive mappings,J. of Nig. math. Soc.35282 - 302, 2016.
[24] V. Martin-Marquez, S. Reich and S. Sabach,Right bregman nonexpansive operators in Banach space, Nonlinear Anal.755448-5465, 2012. · Zbl 1408.47011
[25] E. Naraghirad and J. C. Yao,Bregman weak relatively nonexpansive mappings in Banach spaces,Fixed Point Theory and Application.141(2013), 2013. · Zbl 1423.47046
[26] S. Reich and S. Sabach,A Strong Convergence Theorm for a Proximal-type Algorithsm in Reflexive Banach Spaces, J. Nonlinear Convex Anal.10471-485, 2009. · Zbl 1180.47046
[27] S. Reich and S. Sabach,Two strong convergence theorems for a proximal methos in reflexive Banach spaces, Numer. funct. Anal. Optim.3122-44, 2010.
[28] S. Reich and S. Sabach,Two strong convergence theorems for Bregman stronglynonexpansive operators in reflexive Banach spaces, Nonlinear Anal.73122-135, 2010. · Zbl 1226.47089
[29] R. T. Rockafellar,Monotone operators and proximal point algorithm,SIAM J. Control Optim.14877-898, 1976. · Zbl 0358.90053
[30] A. Taiwo, L. O. Jolaoso, O. T. Mewomo,Parallel hybrid algorithm for solving pseudomonotone equilibrium and Split Common Fixed point problems, Bull. Malays. Math. Sci. Soc.,431893-1918, 2020. · Zbl 1480.47100
[31] S. Takahashi and W. Takahashi,Viscosity approximation method for equillibrium problems and fixed pont problems in Hilbert space, J. Maths. Anal. Appl.133 372-379, 2003.
[32] W. Takahashi and K. Zembayashi,Strong and weak convergence theorem for equillibrium problems and relarively nonexpansive mapping in Banach spaces, Nonlinear Anal, Theory Methods Appl.7045-57, 2009. · Zbl 1170.47049
[33] G. C. Ugwunnadi, Bashir Ali, Ma’aruf S. Minjibir and Ibrahim Idris,Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in reflexive Banach spaces, Fixed Point Theory Applications, 2311-16, 2014. · Zbl 1346.47065
[34] L. Wei, Y. Sheng and R. Tan,A new iterative scheme for the sum of infinite maccretive mappings and inversely strongly accretive mappings and its application, J. Nonlinear Var. Anal.1345-356, 2017.
[35] Su Yongfu and Xu Yongchun,New hybrid shrinking projection algorithm for common fixed points of a family of countable quasi-Bregman strictly pseudocontractive mappings with equilibrium and variational inequality and optimization problems, Fixed Point Theory and Applications,951-22, 2015. · Zbl 1477.47079
[36] C. Zalinescu,Convex Analysis in General Vector Spaces,World Scientific, River Edge · 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.