Abstract
Using a sequence of subsets of an index set for a family of quasinonexpansive mappings, we propose an iterative scheme generated by the shrinking projection method for finding their common fixed point. We prove strong convergence of this scheme under appropriate conditions.
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References
Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers Group, Dordrecht (1993)
Kimura, Y.: Convergence of a sequence of sets in a Hadamard space and the shrinking projection method for a real Hilbert ball. Abstr. Appl. Anal., Art. ID 582,475, 11 (2010)
Kimura, Y., Nakajo, K., Takahashi, W.: Strongly convergent iterative schemes for a sequence of nonlinear mappings. J. Nonlinear Convex Anal. 9(3), 407–416 (2008)
Kimura, Y., Takahashi, W.: On a hybrid method for a family of relatively nonexpansive mappings in a Banach space. J. Math. Anal. Appl. 357, 356–363 (2009)
Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. in Math. 3, 510–585 (1969)
Plubtieng, S., Ungchittrakool, K.: Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl., Art. ID 583,082, 19 (2008)
Qin, X., Cho, Y.J., Kang, S.M.: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 225(1), 20–30 (2009)
Takahashi, W., Takeuchi, Y., Kubota, R.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)
Takahashi, W., Zembayashi, K.: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theory Appl., Art. ID 528,476, 11 (2008)
Tsukada, M.: Convergence of best approximations in a smooth Banach space. J. Approx. Theory 40(4), 301–309 (1984)
Wattanawitoon, K., Kumam, P.: Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings Nonlinear. Anal. Hybrid Syst. 3(1), 11–20 (2009)
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Kimura, Y. (2011). Shrinking Projection Method for a Family of Quasinonexpansive Mappings with a Sequence of Subsets of an Index Set. In: Li, S., Wang, X., Okazaki, Y., Kawabe, J., Murofushi, T., Guan, L. (eds) Nonlinear Mathematics for Uncertainty and its Applications. Advances in Intelligent and Soft Computing, vol 100. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22833-9_45
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DOI: https://doi.org/10.1007/978-3-642-22833-9_45
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22832-2
Online ISBN: 978-3-642-22833-9
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