Shrinking projection methods for split common fixed-point problems in Hilbert spaces. (English) Zbl 1473.47050

Wu, Huan-chun; Cheng, Cao-zong

Abstr. Appl. Anal. 2014, Article ID 817367, 4 p. (2014).

Summary: Inspired by A. Moudafi [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 12, 4083–4087 (2011; Zbl 1232.49017)] and W. Takahashi et al. [J. Math. Anal. Appl. 341, No. 1, 276–286 (2008; Zbl 1134.47052)], we present the shrinking projection method for the split common fixed-point problem in Hilbert spaces, and we obtain the strong convergence theorem. As a special case, the split feasibility problem is also considered.

MSC:

47J26

Fixed-point iterations

Keywords:

shrinking projection method; split common fixed-point problem; Hilbert spaces; strong convergence; split feasibility problem

Citations:

Zbl 1232.49017; Zbl 1134.47052

References:

[1]	Censor, Y.; Elfving, T., A multiprojection algorithm using Bregman projections in a product space, Numerical Algorithms, 8, 2, 221-239 (1994) · Zbl 0828.65065
[2]	Xu, H.-K., Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26, 10 (2010) · Zbl 1213.65085
[3]	Byrne, C., Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18, 2, 441-453 (2002) · Zbl 0996.65048
[4]	Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20, 1, 103-120 (2004) · Zbl 1051.65067
[5]	Censor, Y.; Bortfeld, T.; Martin, B.; Trofimov, A., A unified approach for inversion problems in intensity-modulated radiation therapy, Physics in Medicine and Biology, 51, 10, 2353-2365 (2006)
[6]	Censor, Y.; Herman, G. T., On some optimization techniques in image reconstruction from projections, Applied Numerical Mathematics, 3, 5, 365-391 (1987) · Zbl 0655.65140
[7]	Itoh, S.; Takahashi, W., The common fixed point theory of singlevalued mappings and multivalued mappings, Pacific Journal of Mathematics, 79, 2, 493-508 (1978) · Zbl 0371.47042
[8]	Kocourek, P.; Takahashi, W.; Yao, J.-C., Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese Journal of Mathematics, 14, 6, 2497-2511 (2010) · Zbl 1226.47053
[9]	Yamada, I.; Ogura, N., Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numerical Functional Analysis and Optimization, 25, 7-8, 619-655 (2004) · Zbl 1095.47049
[10]	Combettes, P. L.; Pesquet, J. C., Image restoration subject to a total variation constraint, IEEE Transactions on Image Processing, 13, 9, 1213-1222 (2004)
[11]	Kim, G. E., Weak and strong convergence theorems of quasi-nonexpansive mappings in a Hilbert spaces, Journal of Optimization Theory and Applications, 152, 3, 727-738 (2012) · Zbl 1254.90298
[12]	Censor, Y.; Segal, A., The split common fixed point problem for directed operators, Journal of Convex Analysis, 16, 2, 587-600 (2009) · Zbl 1189.65111
[13]	Moudafi, A., A note on the split common fixed-point problem for quasi-nonexpansive operators, Nonlinear Analysis: Theory, Methods & Applications, 74, 12, 4083-4087 (2011) · Zbl 1232.49017
[14]	Takahashi, W.; Takeuchi, Y.; Kubota, R., Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 341, 1, 276-286 (2008) · Zbl 1134.47052
[15]	Marino, G.; Xu, H.-K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, Journal of Mathematical Analysis and Applications, 329, 1, 336-346 (2007) · Zbl 1116.47053
[16]	Lin, L.-J.; Chuang, C.-S.; Yu, Z.-T., Fixed point theorems for some new nonlinear mappings in Hilbert spaces, Fixed Point Theory and Applications, 2011, article 51 (2011) · Zbl 1315.47044
[17]	Martinez-Yanes, C.; Xu, H.-K., Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Analysis: Theory, Methods & Applications, 64, 11, 2400-2411 (2006) · Zbl 1105.47060