×

zbMATH — the first resource for mathematics

Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications. (English) Zbl 1201.65091
Summary: Some properties of the generalized \(f\)-projection operator are proved in Banach spaces. Using these results, the strong convergence theorems for relatively nonexpansive mappings are studied in Banach spaces. As applications, the strong convergence of general \(H\)-monotone mappings in Banach spaces is also given. The results presented in this paper generalize and improve the main results of S. Matsushita and W. Takahashi [J. Approximation Theory 134, No. 2, 257–266 (2005; Zbl 1071.47063)].

MSC:
65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alber, Ya., Generalized projection operators in Banach spaces: properties and applications, (), 1-21 · Zbl 0882.47046
[2] Alber, Ya., Metric and generalized projection operators in Banach spaces: properties and applications, (), 15-50 · Zbl 0883.47083
[3] Li, J., The generalized projection operator on reflexive Banach spaces and its application, J. math. anal. appl., 306, 377-388, (2005)
[4] Wu, K.Q.; Huang, N.J., The generalised \(f\)-projection operator with an application, Bull. aust. math. soc., 73, 307-317, (2006) · Zbl 1104.47053
[5] Wu, K.Q.; Huang, N.J., Properties of the generalized \(f\)-projection operator and its applications in Banach spaces, Comput. math. appl., 54, 399-406, (2007) · Zbl 1151.47057
[6] Wu, K.Q.; Huang, N.J., The generalized \(f\)-projection operator and set-valued variational inequalities in Banach spaces, Nonlinear anal. TMA, 71, 2481-2490, (2009) · Zbl 1217.47108
[7] Fan, J.H.; Liu, X.; Li, J.L., Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces, Nonlinear anal. TMA, 70, 3997-4007, (2009) · Zbl 1219.47110
[8] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. math. anal. appl., 279, 372-379, (2003) · Zbl 1035.47048
[9] Matsushita, S.; Takahashi, W., A strong convergence theorems for relatively nonexpansive mappings in a Banach space, J. approx. theory, 134, 257-266, (2005) · Zbl 1071.47063
[10] Reich, S., A weak convergence theorem for the alternating method with Bregman distance, (), 313-318 · Zbl 0943.47040
[11] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin, Heidelberg, New York, Tokyo · Zbl 0559.47040
[12] Mamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. optim., 13, 938-945, (2002)
[13] Xia, F.Q.; Huang, N.J., Variational inclusions with a general \(H\)-monotone operator in Banach spaces, Comput. math. appl., 54, 24-30, (2007) · Zbl 1131.49011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.