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On the Mann-type iteration and the convex feasibility problem. (English) Zbl 1135.65027
Let $C$ be a closed convex subset of a Hilbert space $H$; and $T:C\to C$ be a nonlinear map with nonempty fixed point set $F(T)$ in $C$ fulfilling (a) $T$ is $p$-demicontractive on $C$, (b) $I-T$ is demiclosed at zero. Let $(x_k)$ be the Mann-type iterative process $$x_{k+1}=(1-t_k)x_k+t_kT(x_k); \quad k\ge 0,$$ where $x_0\in C$ and $(t_k)\subset \Bbb R^+$. Then (i) if $(x_k)$ remains in $C$ and $0< a\le t_k\le b< 1-p$, $\forall k$, then $(x_k)$ converges weakly to an element of $F(T)$; (ii) if in addition $\langle x-Tx,h\rangle\le 0$, for all $x\in C$ and some $h\in C$, $h\ne 0$, then $(x_k)$ converges strongly to an element of $F(T)$.

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 47J25 Iterative procedures (nonlinear operator equations) 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
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##### References:
 [1] Agmon, S.: The relaxation method for linear inequalities, Canad. J. Math. 6, 382-392 (1954) · Zbl 0055.35001 · doi:10.4153/CJM-1954-037-2 [2] Badriev, I. B.; Zadvornov, O. A.: A decomposition method for variational inequalities of the second kind with strongly inverse-monotone operators, Differential equations 39, No. 7, 936-944 (2003) · Zbl 1064.65048 · doi:10.1023/B:DIEQ.0000009189.91279.93 [3] Badriev, I. B.; Zadvornov, O. A.; Ismagilov, L. N.: On iterative regularization methods for variational inequalities of the second kind with pseudomonotone operators, Comput. methods appl. Math. 3, No. 2, 223-234 (2003) · Zbl 1040.65051 · http://www.cmam.info/issues/?Vol=3&Num=2&ItID=63 [4] Badriev, I. B.; Zadvornov, O. A.; Saddek, A. M.: Convergence analysis of iterative methods for some variational inequalities with pseudomonotone operators, Differential equations 37, No. 7, 934-942 (2001) · Zbl 1010.47041 · doi:10.1023/A:1011901503460 [5] Bauschke, H. H.; Borwein, J. M.: On projection algorithms for solving convex feasibility problems, SIAM rev. 38, No. 3, 367-426 (1996) · Zbl 0865.47039 · doi:10.1137/S0036144593251710 [6] Bauschke, H. H.; Combettes, P. L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces, Math. oper. Res. 26, No. 2, 248-264 (2001) · Zbl 1082.65058 · doi:10.1287/moor.26.2.248.10558 [7] H.H. Bauschke, S.G. Kruk, The method of reflection-projection for convex feasibility problems with an obtuse cone, Technical Report, Oakland University, Rochester, MI, February, 2002. · Zbl 1136.90432 [8] Bregman, L. M.: The method of successive projection for finding a common point of convex sets, Soviet math. Dokl. 6, 688-692 (1965) · Zbl 0142.16804 [9] Chidume, C. E.: The solution by iteration of equation in certain Banach spaces, J. nigerian math. Soc. 3, 57-62 (1984) · Zbl 0609.47069 [10] Chidume, C. E.: An iterative method for nonlinear demiclosed monotone-type operators, Dynam. systems appl. 3, No. 3, 349-355 (1994) · Zbl 0814.47072 [11] Combettes, P. L.; Pennanen, T.: Generalized Mann iterates for constructing fixed points in Hilbert spaces, J. math. Anal. appl. 275, No. 2, 521-536 (2002) · Zbl 1032.47034 · doi:10.1016/S0022-247X(02)00221-4 [12] Diaz, J. B.; Metcalf, F. T.: On the set of subsequential limit points of successive approximations, Trans. amer. Math. soc. 135, 459-485 (1969) · Zbl 0174.25904 · doi:10.2307/1995027 [13] Jr., W. G. Dotson: On the Mann iterative process, Trans. amer. Math. soc. 149, 65-73 (1970) · Zbl 0203.14801 · doi:10.2307/1995659 [14] Eremin, I. I.: Feher mappings and convex programming, Siberian math. J. 10, 762-772 (1969) [15] Gubin, L. G.; Polyac, B. T.; Raik, E. V.: The method of projections for finding the common point of convex sets, USSR comput. Math. phys. 7, 1-24 (1967) · Zbl 0199.51002 · doi:10.1016/0041-5553(67)90113-9 [16] Jakubowich, V. A.: Finite convergent iterative algorithm for solving system of inequalities, Dokl. akad. Nauk. SSSR. 166, 1308-1311 (1966) [17] Liu, L. S.: Ishikawa and Mann iterative process with errors for strongly accretive operator equations, J. math. Anal. appl. 194, 114-125 (1995) · Zbl 0872.47031 [18] Maruster, St.: The solution by iteration of nonlinear equations in Hilbert spaces, Proc. amer. Math. soc. 63, No. 1, 69-73 (1977) · Zbl 0355.47037 · doi:10.2307/2041067 [19] C. Moore, Iterative approximation of fixed points of demicontractive maps, The Abdus Salam Intern. Centre for Theoretical Physics,Trieste, Italy, Scientific Report, IC/98/214, November, 1998. [20] Motzkin, T. S.; Schoenberg, I. J.: The relaxation method for linear inequalities, Canad. J. Math. 6, 393-404 (1954) · Zbl 0055.35002 · doi:10.4153/CJM-1954-038-x [21] Petryshyn, W. V.; Williamson, T. E.: Strong and weak convergence of the sequence of successaive approximations for quasi-nonexpansive mappings, J. math. Anal. appl. 43, 459-497 (1973) · Zbl 0262.47038 · doi:10.1016/0022-247X(73)90087-5 [22] Senter, H. F.; Dotson, W. G.: Approximating fixed points of nonexpansive mappings, Proc. amer. Math. soc. 44, 375-380 (1974) · Zbl 0299.47032 · doi:10.2307/2040440 [23] Tricomi, F.: Una teorema sulla convergenza delle successioni formate deele successive iterate di una funzione di una variabile reale, Giorn. mat. Battaglini 54, 1-9 (1916) · Zbl 46.0439.03 [24] Weng, X.: The iterative solution nonlinear equations in certain Banach spaces, J.nigerian math. Soc. 11, No. 1, 1-7 (1992) [25] Xu, Y.; Ishikawa: Mann iterative process with errors for strongly accretive operator equations, J. math. Anal. appl. 224, 91-101 (1998) · Zbl 0936.47041 · doi:10.1006/jmaa.1998.5987