Solving split equality monotone inclusion problem of maximal monotone mappings in Banach spaces

Document Type : Research Paper

Author

Department of Mathematics, College of Natural Sciences, Jimma University, Jimma, Ethiopia

10.22075/ijnaa.2023.26621.3368

Abstract

A new iterative scheme for approximating a solution of the split equality monotone inclusion problem (SEMIP) of maximal monotone mappings in the setting of Banach spaces is introduced. Strong convergence of the sequence generated by the proposed scheme to a solution of the SEMIP is then derived without prior knowledge of operator norms of the linear operators involved. In addition, we provide applications of our method and provide numerical examples to illustrate the convergence of the proposed scheme. Our results generalize, improve and extend many results in the literature.

Keywords

[1] H. Attouch, J. Bolte, P. Redont, and A. Soubeyran, Alternative proximal algorithm for weakly coupled minimum problems, application to dynamic games and PDEs, J. Convex Anal. 15 (2008), 485–506.
[2] H. Attouch, A. Cabot, F. Frankel, and J. Peypouquet, Alternative proximal algorithm for constrained variational inequalities, Application to domain decomposition for PDEs, Nonlinear Anal. 74 (2011), 7455–7473.
[3] J.B. Baillon and G. Haddad, Quelques proprites des operateurs angle-bornes et cycliquement monontones, Isr. J. Math. 26 (1977), 137–150.
[4] H.H. Bauschke and J.M. Borwein, Legendre functions and the method of random Bregman projections, J. Convex Anal. 4 (1997), 27–67.
[5] H.H. Bauschke, J.M. Borwein, and P.L. combettes, Bregman monotone optimization algorithms, SIAM J. Control Optim. 42 (2003), 596–636.
[6] D. Butnariu and E. Resmerita, Bregman distance, totally convex functions and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. 2006 (2006), 1–39.
[7] D. Butnariu and A.N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Vol. 40, Kluwer Academic, Dordrecht, The Netherlands, 2000.
[8] F.E. Browder, Nonlinear mappings of nonexpancive and accretive-type in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 875–882.
[9] J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problem, Springer, New York, 2000.
[10] S.S. Chang, L. Wang, and L.J. Qin, Split equality fixed point problem for quasi-pseudo-contractive mappings with applications, Fixed Point Theory Appl. 2015 (2015), 208.
[11] Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34 (1981), no. 3, 321–353.
[12] C.E. Chidume, P. Ndambomve, and A.U. Bello, The split equality fixed point problem for demi-constractive mappings, J. Non. Ana. Optim. 6 (2015), no. 1, 61–69.
[13] K. Deimling, Zeros of accretive operators, Manuscripta Math. 13 (1974), 365–374.
[14] H. Guo, H. He, and R. Chen, Convergence theorems for the split variational inclusion problem and fixed point problems in Hilbert spaces, Fixed Point Theory Appl. 2015 (2015), Art. ID 223.
[15] L.O. Jolaoso, F.U. Ogbuisi, and O.T. Mewomo, On split equality variational inclusion problems in Banach spaces without operator norms, Int. J. Nonlinear Anal. Appl. 12 (2021) 425–446.
[16] B. Liu, Fixed point of strong duality pseudocontractive mappings and applications, Abstr. Appl. Anal. 2012 (2012), Article ID 623625.
[17] P.E. Mainge, Strong convergence of projected subgradient method for nonsmooth and nonstrictily convex minimization, Set-Valued Anal. 16 (2008), no. 7-8, 899–912.
[18] B. Martinet, Regulation diequation variationelles par approximations successives, Rev. Francaise inf. Rech. Oper. (1970) 154–159.
[19] A. Moudafi, A relaxed alternating CQ algorithm for convex feasibility problems, Nonlinear Anal. 79 (2013), 117–121.
[20] E. Naraghirad and J.C. Yao, Bergman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Appl. 141 (2013).
[21] Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Kluwer Academic Publishers, 2004.
[22] D. Pascali and S. Sburian, Nonlinear Mappings of Monotone Type, Editura Academia Bucuresti, Romania, 1978.
[23] R.P. Phelps, Convex Functions, Monotone Operators, and Differentiability, Lecture Notes in Mathematics, Vol. 1364, 2nd ed. Springer Verlag, Berlin, 1993.
[24] S. Reich and S. Sabach, Existence and approximation of fixed point of Bregman firmily nonexpansive mappings in reflexive Banach spaces, Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer, New York, 2011, pp 301–316.
[25] S. Reich and S. Sabach, Two strong convergence theorem for Bregman strongly nonexpansive operators in reflexive Banach spaces, Nonlinear Anal. TMA 73 (2010), 122–135.
[26] S. Reich and S. Sabach, A projection method for solving nonlinear problems in reflexive Banach spaces, J. Fixed Point Theory Appl. 9 (2011), 101–116.
[27] R.T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pac. J. Math. 33 (1970), 209–216.
[28] P. Senakka and P. Cholamjiak, Approximation method for solving fixed point problem of Bregman strongly nonexpansive mappings in reflexive Banach spaces, Ric. Mat. 65 (2016), 209–220.
[29] G.B. Wega and H. Zegeye, Convergence results of Forward-Backward method for a zero of the sum of maximally monotone mappings in Banach spaces, Comp. Appl. Math. 39 (2020), 1–16.
[30] G.B. Wega and H. Zegeye, Split equality methods for a solution of monotone inclusion problems in Hilbert spaces, Linear Nonlinear Anal. 5 (2020), no. 3, 495–516.
[31] H.H. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), no. 1, 279–291.
[32] Y. Su and H.K. Xu, A duality fixed point theorem and applications, Fixed Point Theory 13 (2012), no.1, 259–265.
[33] C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific, River Edge, NJ, USA, 2002.
[34] H. Zegeye and G.B. Wega, Approximation of a common f-fixed point of f-pseudo contractive mappings in Banach spaces, Rend. Circ. Mat. Palermo Ser. 2 70 (2021), no. 3, 1139–1162.
[35] H. Zegeye, Strongly convergence theorems for maximal monotone mappings in Banach spaces, J. Math. Anal. Appl. 343 (2008), 663–671.
[36] J. Zhao, Solving split equality fixed point problem of quasi-nonexpansive mappings without prior knowledge of operators norms, Optimization 64 (2015), no. 12, 2619–2630.

Articles in Press, Corrected Proof
Available Online from 22 January 2024
  • Receive Date: 16 March 2022
  • Revise Date: 10 November 2023
  • Accept Date: 05 December 2023