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Strong convergence of modified Mann iterations. (English) Zbl 1091.47055
Let $$X$$ be a real Banach space with a norm $$\|\cdot\|$$ and let $$C$$ be a nonempty, closed and convex subset of $$X$$. A mapping $$T:C\to C$$ is nonexpansive provided that $$\| Tx- Ty\|\leq\| x-y\|$$ for all $$x,y\in C$$. Assume that $$T$$ has at least one fixed point in $$C$$. The authors consider the following iteration sequence $$\{x_n\}$$ for $$T: x_0= x\in C$$, $$y_n= \alpha_nx_n+ (1-\alpha_n)Tx_n$$, $$x_{n+1}= \beta_n u+(1- \beta_n)y_n$$, where $$u$$ is an arbitrary fixed element in $$C$$ and $$\{\alpha_n\}$$, $$\{\beta_n\}$$ are two sequences in the interval $$(0,1)$$ converging to $$0$$ and such that $$\sum\alpha_n= \sum \beta_n= \infty$$. Moreover, $$\sum|\alpha_{n+1}- \alpha_n|< \infty$$, $$\sum|\beta_{n+1}- \beta_n|< \infty$$. Under the assumption that $$X$$ is uniformly smooth, it is shown that the sequence $$\{x_n\}$$ converges strongly to a fixed point of $$T$$. An analogous result is proved for accretive operators.

MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 46B25 Classical Banach spaces in the general theory 47H06 Nonlinear accretive operators, dissipative operators, etc.
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References:
 [1] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, (1976), Noordhoff Leiden [2] Browder, F.E.; Petryshyn, W.V., Construction of fixed points of nonlinear mappings, J. math. anal. appl., 20, 197-228, (1967) · Zbl 0153.45701 [3] Bruck, R.E., Nonexpansive projections on subsets of Banach spaces, Pacific J. math., 47, 341-355, (1973) · Zbl 0274.47030 [4] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse problems, 20, 103-120, (2004) · Zbl 1051.65067 [5] Cioranescu, I., Geometry of Banach spaces, duality mappings and nonlinear problems, (1990), Kluwer Dordrecht · Zbl 0712.47043 [6] Dominguez Benavides, T.; Lopez Acedo, G.; Xu, H.K., Iterative solutions for zeros of accretive operators, Math. nachr., 248-249, 62-71, (2003) · Zbl 1028.65060 [7] Genel, A.; Lindenstrass, J., An example concerning fixed points, Israel J. math., 22, 81-86, (1975) · Zbl 0314.47031 [8] Goebel, K.; Reich, S., Uniform convexity, hyperbolic geometry, and nonexpansive mappings, (1984), Marcel Dekker New York · Zbl 0537.46001 [9] Kamimura, S.; Takahashi, W., Weak and strong convergence of solutions to accretive operator inclusions and applications, Set-valued anal., 8, 361-374, (2000) · Zbl 0981.47036 [10] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. math. anal. appl., 279, 372-379, (2003) · Zbl 1035.47048 [11] Podilchuk, C.I.; Mammone, R.J., Image recovery by convex projections using a least-squares constraint, J. opt. soc. am. A, 7, 517-521, (1990) [12] Reich, S., Asymptotic behavior of contractions in Banach spaces, J. math. anal. appl., 44, 57-70, (1973) · Zbl 0275.47034 [13] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. math. anal. appl., 67, 274-276, (1979) · Zbl 0423.47026 [14] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. math. anal. appl., 75, 287-292, (1980) · Zbl 0437.47047 [15] Sezan, M.I.; Stark, H., Applications of convex projection theory to image recovery in tomography and related areas, (), 415-462 [16] 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 [17] Xu, H.K., Iterative algorithms for nonlinear operators, J. London math. soc., 66, 240-256, (2002) · Zbl 1013.47032 [18] Xu, H.K., An iterative approach to quadratic optimization, J. optimiz. theory appl., 116, 659-678, (2003) · Zbl 1043.90063 [19] Youla, D., Mathematical theory of image restoration by the method of convex projections, (), 29-77 [20] Youla, D., On deterministic convergence of iterations of relaxed projection operators, J. visual comm. image representation, 1, 12-20, (1990)
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