Strong convergence algorithm for the split problem of variational inclusions, split generalized equilibrium problem and fixed point problem

Authors

  • Shamshad Husain Aligarh Muslim University
  • Mohd Asad Aligarh Muslim University
  • Mubashshir Khairoowala Aligarh Muslim University

DOI:

https://doi.org/10.52737/18291163-2021.13.7-1-32

Keywords:

Non expansive mapping, Split feasibility problem, Averaged mapping, Split variational inclusion problem, Split generalized equilibrium problem, Fixed point problem

Abstract

The purpose of this paper is to recommend an iterative scheme to approximate a common element of the solution sets of the split problem of variational inclusions, split generalized equilibrium problem and fixed point problem for non-expansive mappings. We prove that the sequences generated by the recommended iterative scheme strongly converge to a common element of solution sets of stated split problems. In the end, we provide a numerical example to support and justify our main result. The result studied in this paper generalizes and extends some widely recognized results in this direction.

References

M. Abbas, Y. Ibrahim, A.R. Khan, and M. De la Sen, Split variational inclusion problem and fixed point problem for a class of multivalued mappings in CAT(0) spaces. Mathematics, 7 (2019), no. 8, article number 749. https://doi.org/10.3390/math7080749

M. Abbas, Y. Ibrahim, A.R. Khan, and M. De la Sen, Strong convergence of a system of generalized mixed equilibrium problem, split variational inclusion problem and fixed point problem in Banach spaces. Symmetry, 11 (2019), no. 5, article number 722. https://doi.org/10.3390/sym11050722

Q.H. Ansari and A. Rehan, An iterative method for split hierarchical monotone variational inclusions. Fixed Point Theory Appl., 2015 (2015), article number 121. https://doi.org/10.1186/s13663-015-0368-4

H. H. Bauschke and J. M Borwein, On projection algorithms for solving convex feasibility problems. SIAM review, 38 (1996), no. 3, pp. 367-426. https://doi.org/10.1137/s0036144593251710

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems, 18 (2002), no. 2, pp. 441-453. https://doi.org/10.1088/0266-5611/18/2/310

C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems, 20 (2004), no. 1, pp. 103-120. https://doi.org/10.1088/0266-5611/20/1/006

Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol., 51 (2006), no. 10, pp. 2353-2365. https://doi.org/10.1088/0031-9155/51/10/001

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space. Numerical Algorithms, 8 (1994), no. 2, pp. 221-239. https://doi.org/10.1007/bf02142692

Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems. Inverse problems, 21 (2005), no. 6, pp. 2071-2084. https://doi.org/10.1088/0266-5611/21/6/017

Y. Censor, A. Gibali, and S. Reich. Algorithms for the split variational inequality problem. Numerical Algorithms, 59 (2012), no. 2, pp. 301-323. https://doi.org/10.1007/s11075-011-9490-5

H. Chadli and O. Mahdioui, On a system of generalized mixed equilibrium problems involving variational like inequalities in Banach spaces: existence and algorithmic aspects. Adv. Oper. Res., 2021 (2012), Article ID 843486. https://doi.org/10.1155/2012/843486

C. Combettes and P.L. Byrne, Solving monotone inclusions via compositions of non-expansive averaged operators. Optimizations, 53 (2004), no. 5-6, pp. 475-504. https://doi.org/10.1080/02331930412331327157

B. Fan, A hybrid iterative with averaged mappings for hierarchical fixed points and variational inequalities. Numer. Algorithms, 70 (2015), no. 3, pp. 451-467. https://doi.org/10.1007/s11075-014-9956-3

T. Inchan, I. Jitpeera and P. Kumam, A general iterative algorithm combining viscosity method with parallel method for mixed equilibrium problems for a family of strict pseudocontractions. J. Appl. math. Inform., 29 (2011), no. 3-40, pp. 621-639.

S. Jahedi and M.A. Payvand, System of generalized mixed equilibrium problems, variational inequality, and fixed point problems. Fixed Point Theory Appl. 2016 (2016) article number 93. https://doi.org/10.1186/s13663-016-0583-7

K. Kirk and W.A. Goebel, Topics in metric fixed point theory, Cambridge University Press, Cambridge 28 (1990). https://doi.org/10.1017/CBO9780511526152

P. Majee and C. Nahak, A modified iterative method for capturing a common solution of split generalized equilibrium problem and fixed point problem. RACSAM, 112 (2017), no. 4, pp. 1327-1348. https://doi.org/10.1007/s13398-017-0428-1

P. Majee and C. Nahak, A modified iterative method for split problem of variational inclusion and fixed point problems. J. Comput. and Appl. Math., 37 (2018), no. 4, pp. 4710-4729. https://doi.org/10.1007/s40314-018-0596-4

F. Marino, G. Cianciaruso, L. Muglia, Y. Yao, and M.A. Khamsi, A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem. Fixed Point Theory Appl., 2010 (2009), article number 383740. https://doi.org/10.1155/2010/383740

A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems. Nonlinear Anal. Theory Methods Appl., 79 (2013), pp. 117-121. https://doi.org/10.1016/j.na.2012.11.013

A. Moudafi. Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), no. 2, pp. 275-283. https://doi.org/10.1007/s10957-011-9814-6

J.W. Peng, Y.C. Liou and J.-C. Yao, An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions. Fixed Point Theory Appl. 2009 (2009) article number 794178. https://doi.org/10.1155/2009/794178

S. Punpaeng and R. Plubtieng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl., 336 (2007), no. 1, pp. 455-469.

T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one parameter non-expansive semigroups without Bochner integrals. J. Math. Anal. Appl., 305 (2005), no. 1, pp. 227-239. https://doi.org/10.1016/j.jmaa.2004.11.017

W. Takahashi, Split feasibility problem in Banach spaces. J. Nonlinear Convex Anal. 15 (2014), pp. 1349-1355.

S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl., 331 (2007), no. 1, pp. 506-515. https://doi.org/10.1016/j.jmaa.2006.08.036

G. Xu and H.K. Marino, A general iterative method for non-expansive mappings in Hilbert spaces. J. Math. Anal. Appl., 318 (2006), no. 1, pp. 43-52.

H.K. Xu, An iterative approach to quadratic optimization. J. Optim. Theory Appl., 116 (2003), no. 3, pp. 659-678.

H.K. Xu. A variable Krasnosel'skiĭ-Mann algorithm and the multiple-set split feasibility problem. Inverse Problems, 22 (2006), no. 6, pp. 2021-2034. https://doi.org/10.1088/0266-5611/22/6/007

H.K. Xu, Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl., 150 (2011), no. 2, pp. 360-378. https://doi.org/10.1007/s10957-011-9837-z

Q. Yang, The relaxed CQ-algorithm solving the split feasibility problem. Inverse Problems, 20 (2004), no. 4, pp. 1261-1265. https://doi.org/10.1088/0266-5611/20/4/014

Y. Yao, Y.C. Liou and J.C. Yao, Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithm construction. Fixed Point Theory and Appl., 1 (2015), pp. 1-19. https://doi.org/10.1186/s13663-015-0376-4

H. Zhou, Convergence theorems of fixed points for k-strict pseudocontractions in Hilbert spaces. Nonlinear Anal. Theory Methods Appl., 69 (2008), no. 2, pp. 456-462. https://doi.org/10.1016/j.na.2007.05.032

L.J. Zhu, Y.C. Liou, Y. Yao, and C.C. Chyu, Algorithmic and analytical approaches to the split feasibility problems and fixed point problems. Taiwan. J. Math. 17 (2013), no. 5, pp. 1839-1853. https://doi.org/10.11650/tjm.17.2013.3175

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Published

2021-11-03

How to Cite

Husain, S., Asad, M., & Khairoowala, M. (2021). Strong convergence algorithm for the split problem of variational inclusions, split generalized equilibrium problem and fixed point problem. Armenian Journal of Mathematics, 13(7), 1–32. https://doi.org/10.52737/18291163-2021.13.7-1-32