zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Iterative methods for strict pseudo-contractions in Hilbert spaces. (English) Zbl 1133.47050
This article deals with two iterative algorithms of finding a common fixed points for $N$ strict pseudo-contractions $\{T_i\}_{i=1}^N$ defined on a closed convex subset $C$ of a real Hilbert space $H$ (an operator $T: C \to C$ is a strict pseudo-contraction, if there exists a constant $0 \le k < 1$ such that $\Vert Tx - Ty\Vert ^2 \le \Vert x - y\Vert ^2 + k\Vert (I - T)x - (I - T)y\Vert ^2$). The first algorithm, called parallel, is defined by the formula $$x_{n+1} = \alpha_nx_n + (1 - \alpha_n) \sum_{i=1}^N \lambda_i^{(n)} T_ix_n, \ x_0 \in C,\ \lambda_i^{(n)} > 0, \ \lambda_1^{(n)} + \cdots + \lambda_N^{(n)} = 1;\tag1$$ the second one, called cyclic, by the formula $$x_{n+1} = \alpha_nx_n + (1 - \alpha) T_{[n]}x_n, \quad x_0 \in C, \quad T_{[n]} = T_i, \ i = n(\text{ mod} \, N), \ 1 \le i \le N.\tag2$$ The main results describe (provided that $F = \bigcap_{i=1}^N \text{Fix} (T_i) \ne \emptyset$) conditions on the control sequence $\{\alpha_n\}$ so that the approximations $x_n$ converge weakly to a common fixed point of $\{T_i\}_{i=1}^N$. At the end of the article, some modifications of algorithms (1) and (2) are proposed; it is proved that approximations $x_n$ for these modified algorithms converge strongly to $P_Fx_0$, where $P_F$ is the nearest point projection from $H$ onto $F$.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
65J15Equations with nonlinear operators (numerical methods)
Full Text: DOI
[1] Bauschke, H.: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. J. math. Anal. appl. 202, 150-159 (1996) · Zbl 0956.47024
[2] Browder, F. E.; Petryshyn, W. V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. math. Anal. appl. 20, 197-228 (1967) · Zbl 0153.45701
[3] Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse problems 20, 103-120 (2004) · Zbl 1051.65067
[4] Genel, A.; Lindenstrauss, J.: An example concerning fixed points. Israel J. Math. 22, 81-86 (1975) · Zbl 0314.47031
[5] Goebel, K.; Kirk, W. A.: Topics in metric fixed point theory. Cambridge studies in advanced mathematics 28 (1990) · Zbl 0708.47031
[6] Goebel, K.; Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. (1984) · Zbl 0537.46001
[7] Güler, O.: On the convergence of the proximal point algorithm for convex optimization. SIAM J. Control optim. 29, 403-419 (1991) · Zbl 0737.90047
[8] Halpern, B.: Fixed points of nonexpanding maps. Bull. amer. Math. soc. 73, 957-961 (1967) · Zbl 0177.19101
[9] Ishikawa, S.: Fixed points by a new iteration method. Proc. amer. Math. soc. 44, 147-150 (1974) · Zbl 0286.47036
[10] Kamimura, S.; Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938-945 (2003) · Zbl 1101.90083
[11] Kim, T. H.; Xu, H. K.: Strong convergence of modified Mann iterations. Nonlinear anal. 61, 51-60 (2005) · Zbl 1091.47055
[12] Kim, T. H.; Xu, H. K.: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear anal. 64, 1140-1152 (2006) · Zbl 1090.47059
[13] Lions, P. L.: Approximation de points fixes de contractions. C. R. Acad. sci. Sèr. A--B Paris 284, 1357-1359 (1977) · Zbl 0349.47046
[14] Mann, W. R.: Mean value methods in iteration. Proc. amer. Math. soc. 4, 506-510 (1953) · Zbl 0050.11603
[15] Marino, G.; Xu, H. K.: Convergence of generalized proximal point algorithms. Comm. pure appl. Anal. 3, 791-808 (2004) · Zbl 1095.90115
[16] G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. (2006), in press (doi:10.1016/j.jmaa.2006.06.055). Available online 27 July 2006
[17] Matinez-Yanes, C.; Xu, H. K.: Strong convergence of the CQ method for fixed point processes. Nonlinear anal. 64, 2400-2411 (2006) · Zbl 1105.47060
[18] Nakajo, K.; Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. math. Anal. appl. 279, 372-379 (2003) · Zbl 1035.47048
[19] O’hara, J. G.; Pillay, P.; Xu, H. K.: Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces. Nonlinear anal. 54, 1417-1426 (2003) · Zbl 1052.47049
[20] O’hara, J. G.; Pillay, P.; Xu, H. K.: Iterative approaches to convex feasibility problems in Banach spaces. Nonlinear anal. 64, 2022-2042 (2006) · Zbl 1139.47056
[21] Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. math. Anal. appl. 67, 274-276 (1979) · Zbl 0423.47026
[22] Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. math. Anal. appl. 75, 287-292 (1980) · Zbl 0437.47047
[23] Scherzer, O.: Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J. math. Anal. appl. 194, 911-933 (1991) · Zbl 0842.65036
[24] Shioji, N.; Takahashi, W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. amer. Math. soc. 125, 3641-3645 (1997) · Zbl 0888.47034
[25] Solodov, M. V.; Svaiter, B. F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. program. Ser. A 87, 189-202 (2000) · Zbl 0971.90062
[26] Tan, K. K.; Xu, H. K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. math. Anal. appl. 178, No. 2, 301-308 (1993) · Zbl 0895.47048
[27] Tan, K. K.; Xu, H. K.: Fixed point iteration processes for asymptotically nonexpansive mappings. Proc. amer. Math. soc. 122, 733-739 (1994) · Zbl 0820.47071
[28] Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. math. 58, 486-491 (1992) · Zbl 0797.47036
[29] Xu, H. K.: Iterative algorithms for nonlinear operators. J. London math. Soc. 66, 240-256 (2002) · Zbl 1013.47032
[30] Xu, H. K.: Remarks on an iterative method for nonexpansive mappings. Comm. appl. Nonlinear anal. 10, No. 1, 67-75 (2003) · Zbl 1035.47035
[31] Xu, H. K.: Strong convergence of an iterative method for nonexpansive mappings and accretive operators. J. math. Anal. appl. 314, 631-643 (2006) · Zbl 1086.47060
[32] Xu, H. K.: Strong convergence of approximating fixed point sequences for nonexpansive mappings. Bull. austral. Math. soc. 74, 143-151 (2006) · Zbl 1126.47056