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Robustness of Mann’s algorithm for nonexpansive mappings. (English) Zbl 1110.47057
The authors prove robustness results of three somewhat different versions (corresponding to three different settings) of Mann’s algorithm to obtain a fixed-point of a suitable mapping. We state the version in the first setting (Theorem 3.3) in extenso. Let $X$ be a uniformly convex Banach space, where either $X$ satisfies Opial’s property or where the dual of $X$ has the Kadec--Klee property. Let $T:X\to X$ be a nonexpansive mapping having a nonempty set of fixed points. Starting from some point $x(0)$ in $X$, let the sequence $x(n)$ be generated by the following perturbed Mann algorithm: $x(n+1)=(1-\alpha(n))x(n) +\alpha(n) (Tx (n)+e(n))$, where $\{\alpha(n)\}$ and $\{e(n)\}$ are sequences in $(0,1)$ and in $X$, respectively, satisfying the following properties: $\Sigma\alpha(n)(1-\alpha(n))=\infty$ and $\Sigma\alpha(n)\|e(n)\|< \infty$. Then the sequence $\{x(n)\}$ converges weakly to a fixed point of $T$. In the second setting, the authors consider a nonexpansive map $T$ defined on a closed convex subset of a real Hilbert space (Theorem 4.1). Finally, in the third setting (Theorem 5.1), the map is an $m$-accretive operator in a uniformly convex Banach space $X$, where $X$ has the same additional properties as in the first setting.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
65J15Equations with nonlinear operators (numerical methods)
Full Text: DOI
[1] Browder, F. E.: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 100, 201-225 (1967) · Zbl 0149.36301
[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.: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel J. Math. 32, 107-116 (1979) · Zbl 0423.47024
[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] Combettes, P. L.: The convex feasibility problem in image recovery. Advances in imaging and electron physics, vol. 95 95, 155-270 (1996)
[6] Combettes, P. L.: On the numerical robustness of the parallel projection method in signal synthesis. IEEE signal process. Lett. 8, 45-47 (2001)
[7] Van Dulst, D.: Equivalent norms and the fixed point property for nonexpansive mappings. J. London math. Soc. (2) 25, 139-144 (1982) · Zbl 0453.46017
[8] Engl, H. W.; Leitao, A.: A Mann iterative regularization for elliptic Cauchy problems. Numer. funct. Anal. optim. 22, 861-884 (2001) · Zbl 0998.65114
[9] Engl, H. W.; Scherzer, O.: Convergence rates results for iterative methods for solving nonlinear ill-posed problems. Surveys on solution methods for inverse problems, 7-34 (2000) · Zbl 0998.65058
[10] Garcia-Falset, J.; Kaczor, W.; Kuczumow, T.; Reich, S.: Weak convergence theorems for asymptotically nonexpansive mappings and semigroups. Nonlinear appl. 43, 377-401 (2001) · Zbl 0983.47040
[11] Magnanti, T. L.; Perakis, G.: Solving variational inequality and fixed point problems by line searches and potential optimization, algorithms. Math. program. Ser. A 101, 435-461 (2004) · Zbl 1073.90051
[12] Marino, G.; Xu, H. K.: Convergence of generalized proximal point algorithms. Comm. appl. Anal. 3, 791-808 (2004) · Zbl 1095.90115
[13] Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. amer. Math. soc. 73, 595-597 (1967) · Zbl 0179.19902
[14] 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)
[15] Sezan, M. I.; Stark, H.: Applications of convex projection theory to image recovery in tomography and related areas. Image recovery theory and applications, 415-462 (1987)
[16] 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
[17] 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
[18] Xu, H. K.: Inequalities in Banach spaces with applications. Nonlinear anal. 16, 1127-1138 (1991) · Zbl 0757.46033
[19] Yamada, I.; Ogura, N.: Adaptive projected subgradient method for asymptotic minimization of sequence of nonnegative convex functions. Numer. funct. Anal. optim. 25, 593-617 (2004) · Zbl 1156.90428
[20] Yamada, I.; Ogura, N.: Hybrid steepest descent method for variational inequality problems over the fixed point set of certain quasi-nonexpansive mappings. Numer. funct. Anal. optim. 25, 619-655 (2004) · Zbl 1095.47049
[21] Youla, D.: Mathematical theory of image restoration by the method of convex projections. Image recovery theory and applications, 29-77 (1987)
[22] Youla, D.: On deterministic convergence of iterations of relaxed projection operators. J. visual comm. Image representation 1, 12-20 (1990)