×

Approximating fixed points of non-self nonexpansive mappings in Banach spaces. (English) Zbl 1089.47058

Let \(K\) be a nonempty closed convex subset of a real uniformly convex Banach space \(E\), which is also a nonexpansive retract of \(E\) \((P\) is a nonexpansive retraction of \(E\) onto \(K)\). Let \(\{x_n\}\) be the sequence defined by \(x_1=x\in K\), \(x_{n+1}=P((1-\alpha_n)x_n+ \alpha_nTP[(1-\beta_n)x_n+\beta_nTx_n])\), \(n\geq 1\), where \(T:E\to K\) is a nonexpansive mapping and \(\{\alpha_n\}\), \(\{\beta_n\}\) are sequences in \([\varepsilon,1-\varepsilon]\) for some \(\varepsilon\in(0,1)\). In the present paper, the author proves the following:
(1) If \(F(T)\neq \emptyset\) and the dual \(E^*\) of \(E\) has the Kadec-Klee property, then \(\{x_n\}\) convergence weakly to some fixed point of \(T\).
(2) If \(F(T)\neq \emptyset\) and if there is a nondecreasing function \(f:[0,+\infty) \to[0,+\infty)\) with \(f(0)=0\) and \(f(r)>0\) for all \(r>0\) such that for all \(x\in K\), \(\| x-Tx\|\geq f(d(x,F(T)))\), then \(\{x_n\}\) converges strongly to some fixed point of \(T\).

MSC:

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

References:

[1] Browder, F. E., Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc., 74, 660-665 (1968) · Zbl 0164.44801
[2] W.J. Davis, P. Enflo, Contractive projections on \(l_p\); W.J. Davis, P. Enflo, Contractive projections on \(l_p\)
[3] J. Diestel, Geometry of Banach Spaces-Selected Topics, Lecture Notes in Mathematics, vol. 485, Springer, New York, 1975.; J. Diestel, Geometry of Banach Spaces-Selected Topics, Lecture Notes in Mathematics, vol. 485, Springer, New York, 1975. · Zbl 0307.46009
[4] Falset, J. G.; Kaczor, W.; Kuczumow, T.; Reich, S., Weak convergence theorems for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal., 43, 377-401 (2001) · Zbl 0983.47040
[5] Ishikawa, S., Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59, 65-71 (1976) · Zbl 0352.47024
[6] Ishikawa, S., Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44, 147-150 (1974) · Zbl 0286.47036
[7] Kaczor, W., Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups, J. Math. Anal. Appl., 272, 565-574 (2002) · Zbl 1058.47049
[8] Kaczor, W.; Prus, S., Asymptotical smoothness and its applications, Bull. Austral. Math. Soc., 66, 405-418 (2002) · Zbl 1031.47037
[9] Lim, T. C., A fixed point theorem for families of nonexpansive mappings, Pacific J. Math., 53, 487-493 (1974) · Zbl 0291.47032
[10] Mann, W. R., Mean value methods in iteration, Proc. Amer. Math. Soc., 4, 506-510 (1953) · Zbl 0050.11603
[11] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67, 274-276 (1979) · Zbl 0423.47026
[12] Schu, J., Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43, 153-159 (1991) · Zbl 0709.47051
[13] Senter, H. F.; Dotson, W. G., Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44, 375-380 (1974) · Zbl 0299.47032
[14] Takahashi, W.; Tamura, T., Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal., 5, 45-56 (1998) · Zbl 0916.47042
[15] Tan, K. K.; Xu, H. K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178, 301-308 (1993) · Zbl 0895.47048
[16] Xu, H. K., Inequalities in Banach spaces with applications, Nonlinear Anal., 16, 1127-1138 (1991) · Zbl 0757.46033
[17] Zeng, L. C., A note on approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 226, 245-250 (1998) · Zbl 0916.47047
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.