## 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 and generalizations 47J25 Iterative procedures involving nonlinear operators
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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∈U(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), Kindai-kagaku-sha Tokyo [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
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