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Approximating solutions of maximal monotone operators in Hilbert spaces. (English) Zbl 0992.47022
This article deals with iterates $$x_{n+1}= \alpha_n x+(1- \alpha_n)J_{r_n} x_n\qquad (n= 1,2,\dots)\tag 1$$ and $$x_{n-1}= \alpha_n x_n+ (1- \alpha_n) J_{r_n} x_n\qquad (n= 1,2,\dots),\tag 2$$ where $J_r= (I+ rT)^{-1}$, $\{\alpha_n\}$ is a sequence from $[0,1]$, $\{r_n\}$ a sequence from $(0,\infty)$, $T: H\to 2^H$ a maximal monotone operator in a real Hilbert space. The basic results are (a) a theorem about strong convergence of iterates (1) to $Px$, where $P$ is the metric projection onto $T^{-1}0$; (b) a theorem about weak convergence of iterates (2) to $v\in T^{-1}0= \lim_{n\to\infty} Px_n$, where $P$ is the metric projection onto $T^{-1}0$. In the end of the article the special case when $T=\partial f$ is considered, where $f$ is a proper lower-semicontinuous convex function. The corresponding results is interpreted as theorems of finding a minimizer of $f$.

##### MSC:
 47H05 Monotone operators (with respect to duality) and generalizations 47J25 Iterative procedures (nonlinear operator equations)
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##### References:
 [1] Atsushiba, S.; Takahashi, W.: Approximating common fixed points of nonexpansive semigroups by the Mann iteration process. Ann. univ. Mariae Curie-sklodowska sect. A 51, 1-16 (1997) · Zbl 1012.47033 [2] Brézis, H.; Lions, P. L.: Produits infinis de resolvants. Israel J. Math. 29, 329-345 (1978) · Zbl 0387.47038 [3] Bruck, R. E.: A strongly convergent iterative solution of 0$inU(x)$ for a maximal monotone operator U in Hilbert space. J. math. Anal. appl. 48, 114-126 (1974)\$ · Zbl 0288.47048 [4] Bruck, R. E.; Reich, S.: A general convergence principle in nonlinear functional analysis. Nonlinear anal. 5, 939-950 (1980) · Zbl 0454.65043 [5] Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control optim. 29, 403-419 (1991) · Zbl 0737.90047 [6] Halpern, B.: Fixed points of nonexpanding maps. Bull. amer. Math. soc. 73, 957-961 (1967) · Zbl 0177.19101 [7] Jung, J. S.; Takahashi, W.: Dual convergence theorems for the infinite products of resolvents in Banach spaces. Kodai math. J. 14, 358-364 (1991) · Zbl 0755.47037 [8] Khang, D. B.: On a class of accretive operators. Analysis 10, 1-16 (1990) · Zbl 0719.47039 [9] Mann, W. R.: Mean value methods in iteration. Proc. amer. Math. soc. 4, 506-510 (1953) · Zbl 0050.11603 [10] Minty, G. J.: On the monotonicity of the gradient of a convex function. Pacific J. Math. 14, 243-247 (1964) · Zbl 0123.10601 [11] Nevanlinna, O.; Reich, S.: Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces. Israel J. Math. 32, 44-58 (1979) · Zbl 0427.47049 [12] Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. math. Anal. appl. 67, 274-276 (1979) · Zbl 0423.47026 [13] Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. math. Anal. appl. 75, 287-292 (1980) · Zbl 0437.47047 [14] Rockafellar, R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control optim. 14, 877-898 (1976) · Zbl 0358.90053 [15] Takahashi, W.: Nonlinear functional analysis. (1988) · Zbl 0647.90052 [16] Takahashi, W.; Ueda, Y.: On reich’s strong convergence theorems for resolvents of accretive operators. J. math. Anal. appl. 104, 546-553 (1984) · Zbl 0599.47084 [17] 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 [18] Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. math. 58, 486-491 (1992) · Zbl 0797.47036