Convergence theorems of iterative algorithms for continuous pseudocontractive mappings. (English) Zbl 1126.47054

In the setting of a reflexive and strictly convex Banach sapce with a uniformly Gâteaux differentiable norm, the autors discuss the convergence of the viscosity iteration process to a solution of the following variational inequality problem: find \(x^*\in F(T)\) such that \[ \langle (f(x^*)-x^*,j(x-x^*)\rangle\leq 0\quad\forall x\in F(T), \] where \(j\) is the normlized dual mapping, \(f\) is a Lipschitz strongly pseudoconstractive mapping, and \(F(T)\) is the fixed point set of a continuous pseudocontractive mapping. The authors also discuss the convergence of a modified implicit iteration to solve the above problem.


47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
47H10 Fixed-point theorems
Full Text: DOI


[1] Reich, S., An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal., 2, 85-92 (1978) · Zbl 0375.47032
[2] Xu, H.-K.; Ori, R. G., An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim., 22, 767-773 (2001) · Zbl 0999.47043
[3] Chen, R.; Song, Y.; Zhou, H. Y., Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings, J. Math. Anal. Appl., 701-709 (2006) · Zbl 1086.47046
[4] Takahashi, W., Nonlinear Functional Analysis — Fixed Point Theory and its Applications (2000), Yokohama Publishers Inc: Yokohama Publishers Inc Yokohama, (in Japanese) · Zbl 0997.47002
[5] Takahashi, W.; Ueda, Y., On Reich’s strong convergence for resolvents of accretive operators, J. Math. Anal. Appl., 104, 546-553 (1984) · Zbl 0599.47084
[6] Kim, T.-H.; Xu, H.-K., Strong convergence of modified Mann iterations, Nonlinear Anal., 61, 51-60 (2005) · Zbl 1091.47055
[7] Xu, H.-K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298, 279-291 (2004) · Zbl 1061.47060
[8] Xu, H.-K., An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116, 659-678 (2003) · Zbl 1043.90063
[9] Deimling, K., Zero of accretive operators, Manuscripta Math., 13, 365-374 (1974) · Zbl 0288.47047
[10] Martin, R. H., Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc., 179, 399-414 (1973) · Zbl 0293.34092
[11] Megginson, R. E., An Introduction to Banach Space Theory (1998), Springer-Verlag, New York, Inc. · Zbl 0910.46008
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.