A strong convergence theorem for relatively nonexpansive mappings in a Banach space. (English) Zbl 1071.47063

The present paper is concerned with the problem of finding a fixed point of a relatively nonexpansive mapping defined on a closed convex subset of a Banach space. To this end, the authors investigate under what conditions a sequence constructed by the so-called “hybrid method in mathematical programming” converges strongly to a fixed point of the mapping. The main result (Theorem 3.1) goes as follows:
Theorem. Let \(E\) be a uniformly convex and uniformly smooth Banach space, let \(C\) be a nonempty closed convex subset of \(E\), let \(T\) be a relatively nonexpansive mapping from \(C\) into itself, and let \(\{\alpha(n)\}\) be a sequence of real numbers such that \(0\leq \alpha(n)\leq 1\) and \(\limsup\alpha(n)< 1\) when \(n\to\infty\). When the set \(F(T)\) of the fixed points of \(T\) is nonempty, then the sequence \(x(n)\) constructed by the hybrid method converges strongly to the point that is the generalised projection from \(C\) onto \(F(T)\).
As special cases, they obtain analogous strong convergence results for a nonexpansive mapping on a Hilbert space (using the metric projection) and for a maximal monotone operator on a Banach space (using the generalised projection).


47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H05 Monotone operators and generalizations
Full Text: DOI


[1] Alber, Ya.I., Metric and generalized projection operators in Banach spacesproperties and applications, (), 15-50 · Zbl 0883.47083
[2] Alber, Ya.I.; Reich, S., An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. math. J., 4, 39-54, (1994) · Zbl 0851.47043
[3] Butnariu, D.; Reich, S.; Zaslavski, A.J., Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. appl. anal., 7, 151-174, (2001) · Zbl 1010.47032
[4] Butnariu, D.; Reich, S.; Zaslavski, A.J., Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numer. funct. anal. optim., 24, 489-508, (2003) · Zbl 1071.47052
[5] Censor, Y.; Reich, S., Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization, 37, 323-339, (1996) · Zbl 0883.47063
[6] Cioranescu, I., Geometry of Banach spaces, duality mappings and nonlinear problems, (1990), Kluwer Dordrecht · Zbl 0712.47043
[7] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. optim., 13, 938-945, (2002) · Zbl 1101.90083
[8] F. Kohsaka, W. Takahashi, Strong convergence of an iterative sequence for maximal monotone operators in a Banach space, Abstr. Appl. Anal. (2004) 239-249. · Zbl 1064.47068
[9] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. math. anal. appl., 279, 372-379, (2003) · Zbl 1035.47048
[10] Ohsawa, S.; Takahashi, W., Strong convergence theorems for resolvents of maximal monotone operators in Banach spaces, Arch. math., 81, 439-445, (2003) · Zbl 1067.47080
[11] Reich, S., Constructive techniques for accretive and monotone operators, applied nonlinear analysis, (), 335-345
[12] Reich, S., Review of geometry of Banach spaces, duality mappings and nonlinear problems by ioana cioranescu, kluwer Academic publishers, dordrecht, 1990, Bull. amer. math. soc., 26, 367-370, (1992)
[13] Reich, S., A weak convergence theorem for the alternating method with Bregman distance, (), 313-318 · Zbl 0943.47040
[14] Rockafellar, R.T., On the maximality of sums of nonlinear monotone operators, Trans. amer. math. soc., 149, 75-88, (1970) · Zbl 0222.47017
[15] Solodov, M.V.; Svaiter, B.F., Forcing strong convergence of proximal point iterations in Hilbert space, Math. program., 87, 189-202, (2000) · Zbl 0971.90062
[16] Takahashi, W., Convex analysis and approximation fixed points, (2000), Yokohama-Publishers, (Japanese)
[17] Takahashi, W., Nonlinear functional analysis, (2000), Yokohama-Publishers
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