Previous |  Up |  Next


Banach limit; dual convergence theorem; duality mapping; Ishikawa iteration process; nonexpansive mapping
In this paper we establish a dual weak convergence theorem for the Ishikawa iteration process for nonexpansive mappings in a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm, and then apply this result to study the problem of the weak convergence of the iteration process.
[1] R. E.  Bruck and S. Reich: Accretive operators, Banach limits and dual ergodic theorems. Bull. Acad. Polon. Sci. 29 (1981), 585–589. MR 0654218
[2] L. Deng: Convergence of the Ishikawa iteration process for nonexpansive mappings. J.  Math. Anal. Appl. 199 (1996), 769–775. MR 1386604 | Zbl 0856.47041
[3] K.  Goebel and S.  Reich: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Marcel Dekker, New York and Basel, 1984. MR 0744194
[4] S.  Ishikawa: Fixed point by a new iteration method. Proc. Amer. Math. Soc. 44 (1974), 147–150. DOI 10.1090/S0002-9939-1974-0336469-5 | MR 0336469
[5] K. S.  Ha and J.  S.  Jung: Strong convergence theorems for accretive operators in Banach spaces. J.  Math. Anal. Appl. 104 (1990), 330–339. DOI 10.1016/0022-247X(90)90351-F | MR 1050208
[6] G.  G.  Lorentz: A contribution to the theory of divergent series. Acta Math. 80 (1948), 167–190. DOI 10.1007/BF02393648 | MR 0027868
[7] W. R.  Mann: Mean value methods in iteration. Proc. Amer. Math. Soc. 4 (1953), 506–510. DOI 10.1090/S0002-9939-1953-0054846-3 | MR 0054846 | Zbl 0050.11603
[8] S.  Reich: Weak convergence theorem for nonexpansive mappings in Banach spaces. J.  Math. Anal. Appl. 67 (1979), 274–276. DOI 10.1016/0022-247X(79)90024-6 | MR 0528688
[9] S.  Reich: Product formulas, nonlinear semigroups and accretive operators. J.  Functional Analysis 36 (1980), 147–168. DOI 10.1016/0022-1236(80)90097-X | MR 0569251 | Zbl 0437.47048
[10] K.  K.  Tan and H.  K.  Xu: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J.  Math. Anal. Appl. 178 (1993), 301–308. DOI 10.1006/jmaa.1993.1309 | MR 1238879
Partner of
EuDML logo