Iterative approximation of fixed points of nonexpansive mappings. (English) Zbl 1095.47034

Let \(K\) be a nonempty closed convex subset of a real Banach space \(E\) which has a uniformly Gâteaux differentiable norm and let \(T: K\rightarrow K\) be a nonexpansive mapping with nonempty set of all fixed points \(F(T)\). For a sequence \(\{\alpha_n\}\) of real numbers in \([0,1]\) and an arbitrary \(u\in K\), the sequence \(\{x_n\}\) in \(K\) defined by \(x_0\in K\) and \[ x_{n+1}=\alpha_n u+ (1-\alpha_n) T x_n,\,\,n\geq 0, \] was introduced by B. Halpern [Bull. Am. Math. Soc. 73, 957–961 (1967; Zbl 0177.19101)] and subsequently studied by several authors in order to approximate the fixed points of \(T\), see, e.g., the reviewer’s recent monograph [V. Berinde, “Iterative approximation of fixed points” (Efemeride, Baia Mare) (2002; Zbl 1036.47037)].
In the present paper, the authors use the same kind of iterative scheme but with the averaged map \(S\), given by \(Sx:=(1-\delta)x+\delta Tx,\,\,\delta\in (0,1)\), instead of \(T\), and prove a strong convergence theorem for approximating the fixed points of \(T\).


47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: DOI


[1] Baillon, J.B., Quelques aspects de la theorie des points fixes dans LES espaces de Banach I, II, (), 45, (in French) · Zbl 0414.47040
[2] Browder, F.E., Convergence of approximates to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. rational mech. anal., 24, 82-90, (1967) · Zbl 0148.13601
[3] Halpern, B., Fixed points of nonexpansive maps, Bull. amer. math. soc., 73, 957-961, (1967) · Zbl 0177.19101
[4] Kirk, W.A., Locally nonexpansive mappings in Banach spaces, Lecture notes in math., vol. 886, (1981), Springer Berlin, pp. 178-198 · Zbl 0475.47040
[5] Lions, P.-L., Approximation de points fixes de contractions, C. R. acad. sci. Paris Sér. A-B, 284, A1357-A1359, (1977)
[6] Morales, C.H.; Jung, J.S., Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. amer. math. soc., 128, 3411-3419, (2000) · Zbl 0970.47039
[7] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. math. anal. appl., 75, 287-292, (1980) · Zbl 0437.47047
[8] Reich, S., Some problems and results in fixed point theory, Contemp. math., 21, 179-187, (1983) · Zbl 0531.47048
[9] Shioji, N.; Takahashi, W., Strong convergence of approximated sequences for nonexpansive mapping in Banach spaces, Proc. amer. math. soc., 125, 3641-3645, (1997) · Zbl 0888.47034
[10] Suziki, T., Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. math. anal. appl., 305, 227-239, (2005) · Zbl 1068.47085
[11] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math. (basel), 58, 486-491, (1992) · Zbl 0797.47036
[12] Xu, H.K., Another control condition in an iterative method for nonexpansive mappings, Bull. astral. math. soc., 65, 109-113, (2002) · Zbl 1030.47036
[13] Xu, H.K., Iterative algorithms for nonlinear operators, J. London math. soc., 2, 240-256, (2002) · Zbl 1013.47032
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.