Ergodic-type method for a system of split variational inclusion and fixed point problems in Hilbert spaces

Volume 10, Issue 6, pp 3046--3058 http://dx.doi.org/10.22436/jnsa.010.06.19

Publication Date: June 25, 2017 Submission Date: March 31, 2017

Download PDF

Download XML

1722 Downloads
3096 Views

Authors

Dao-Jun Wen - College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China. Yi-An Chen - College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China. Ying-Ling Lu - College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China.

Abstract

In this paper, we introduce an ergodic-type method for solving a system of split variational inclusion and fixed point problems of a family of nonexpansive mappings with averaged resolvent operator. We prove that the sequence generated by the proposed algorithm converges strongly to a common element of the set of solutions of a system of split variational inclusion and the set of fixed points of a family of nonexpansive mappings in Hilbert spaces, from which the minimum norm solution is deduced as a special case. Moreover, a numerical example is given to illustrate the operational reliability and convergence of the presented method and results, which may be viewed as a refinement and improvement of the previously known results.

Share and Cite

ISRP Style

Dao-Jun Wen, Yi-An Chen, Ying-Ling Lu, Ergodic-type method for a system of split variational inclusion and fixed point problems in Hilbert spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 6, 3046--3058

AMA Style

Wen Dao-Jun, Chen Yi-An, Lu Ying-Ling, Ergodic-type method for a system of split variational inclusion and fixed point problems in Hilbert spaces. J. Nonlinear Sci. Appl. (2017); 10(6):3046--3058

Chicago/Turabian Style

Wen, Dao-Jun, Chen, Yi-An, Lu, Ying-Ling. "Ergodic-type method for a system of split variational inclusion and fixed point problems in Hilbert spaces." Journal of Nonlinear Sciences and Applications, 10, no. 6 (2017): 3046--3058

Keywords

Split variational inclusion
nonexpansive mapping
ergodic-type iteration
fixed point
minimum norm solution.

MSC

47H10
41A29
47J22
65D15

References

[1] H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, With a foreword by Hédy Attouch, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York (2011)
- View Article
- MathSciNet

[2] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441–453.

[3] C. Byrne, Y. Censor, A. Gibali, S. Reich, The split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759–775.
- Google Scholar

[4] L.-C. Ceng, Q. H. Ansari, J.-C. Yao, Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Anal., 75 (2012), 2116–2125.

[5] Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353–2365.
- View Article
- Google Scholar

[6] Y. Censor, A. Gibali, S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301–323.
- View Article
- Google Scholar

[7] P. L. Combettes, The convex feasibility problem in image recovery, Adv. Imag. Elect. Phys., 95 (1996), 155–270.
- View Article
- Google Scholar

[8] J. U. Jeong, Generalized viscosity approximation methods for mixed equilibrium problems and fixed point problems, Appl. Math. Comput., 283 (2016), 168–180.

[9] K. R. Kazmi, S. H. Rizvi, An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping, Optim. Lett., 8 (2014), 1113–1124.

[10] A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), 275–283.
- View Article
- Google Scholar

[11] X.-L. Qin, S. Y. Cho, S. M. Kang, Iterative algorithms for variational inequality and equilibrium problems with applications, J. Global Optim., 48 (2010), 423–445.

[12] T. Shimizu, W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl., 211 (1997), 71–83.

[13] K. Sitthithakerngkiet, J. Deepho, P. Kumam, A hybrid viscosity algorithm via modify the hybrid steepest descent method for solving the split variational inclusion in image reconstruction and fixed point problems, Appl. Math. Comput., 250 (2015), 986–1001.

[14] W. Takahashi, Nonlinear functional analysis, Fixed point theory and its applications, Yokohama Publishers, Yokohama (2000)
- Google Scholar

[15] S.-H. Wang, X.-Y. Gong, A. A. N. Abdou, Y. J. Cho, Iterative algorithm for a family of split equilibrium problems and fixed point problems in Hilbert spaces with applications, Fixed Point Theory Appl., 2016 (2016), 22 pages.

[16] D.-J. Wen, Y.-A. Chen, Iterative methods for split variational inclusion and fixed point problem of nonexpansive semigroup in Hilbert spaces, J. Inequal. Appl., 2015 (2015), 14 pages.

[17] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.
- View Article
- Google Scholar

[18] L. Yang, F.-H. Zhao, J. K. Kim, Hybrid projection method for generalized mixed equilibrium problem and fixed point problem of infinite family of asymptotically quasi-\(\phi\)-nonexpansive mappings in Banach spaces, Appl. Math. Comput., 218 (2012), 6072–6082.

[19] Y.-H. Yao, R. P. Agarwal, M. Postolache, Y.-C. Liou, Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem, Fixed Point Theory Appl., 2014 (2014), 14 pages.