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A general iterative method for an infinite family of nonexpansive mappings. (English) Zbl 1223.47105
Summary: Let $$H$$ be a real Hilbert space. Consider the iterative sequence
$x_{n+1}=\alpha_n\gamma f(x_n)+\beta_nx_n+((1-\beta_n)I-\alpha_nA)W_nx_n,$
where $$\gamma>0$$ is some constant, $$f:H\to H$$ is a given contractive mapping, $$A$$ is a strongly positive bounded linear operator on $$H$$ and $$W_n$$ is the $$W$$-mapping generated by an infinite countable family of nonexpansive mappings $$T_1,T_2,\dots,T_n,\dots$$ and $$\lambda_1,\lambda_2,\dots,\lambda_n,\dots$$ such that the common fixed points set $$F:=\bigcap^\infty_{n=1}\text{Fix}(T_n)\neq\emptyset$$. Under very mild conditions on the parameters, we prove that $$\{x_n\}$$ converges strongly to $$p\in F$$ where $$p$$ is the unique solution in $$F$$ of the following variational inequality: $$\langle(A-\gamma f)p,p-x^*\rangle\leq 0$$ for all $$x^*\in F$$, which is the optimality condition for the minimization problem
$\min_{x\in F}\tfrac12 \langle Ax,x \rangle-h(x).$

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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##### References:
 [1] Bauschke, H.H.; Borwein, J.M., On projection algorithms for solving convex feasibility problems, SIAM rev., 38, 367-426, (1996) · Zbl 0865.47039 [2] Combettes, P.L., The foundations of set theoretic estimation, Proc. IEEE, 81, 182-208, (1993) [3] Bauschke, H.H., The approximation of fixed points of compositions of nonexpansive mappings in Hilbert spaces, J. math. anal. appl., 202, 150-159, (1996) · Zbl 0956.47024 [4] Combettes, P.L., Constrained image recovery in a product space, (), 2025-2028 [5] Deutsch, F.; Hundal, H., The rate of convergence of dykstra’s cyclic projections algorithm: the polyhedral case, Numer. funct. anal. optim., 15, 537-565, (1994) · Zbl 0807.41019 [6] Youla, D.C., Mathematical theory of image restoration by the method of convex projections, (), 29-77 [7] Iusem, A.N.; De Pierro, A.R., On the convergence of han’s method for convex programming with quadratic objective, Math. program, ser. B, 52, 265-284, (1991) · Zbl 0744.90066 [8] Xu, H.K., An iterative approach to quadratic optimization, J. optim. theory appl., 116, 659-678, (2003) · Zbl 1043.90063 [9] Chang, S.S., Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces, J. math. anal. appl., 323, 1402-1416, (2006) · Zbl 1111.47057 [10] Jung, J.S., Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. math. anal. appl., 32, 509-520, (2005) · Zbl 1062.47069 [11] Kikkawa, M.; Takahashi, W., Approximating fixed points of infinite nonexpansive mappings by the hybrid method, J. optim. theory appl., 117, 93-101, (2003) · Zbl 1033.65037 [12] O’Hara, J.G.; Pillay, P.; Xu, H.K., Iterative approaches to convex feasibility problems in Banach spaces, Nonlinear anal., 64, 2022-2042, (2006) · Zbl 1139.47056 [13] Shimoji, K.; Takahashi, W., Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. math., 5, 387-404, (2001) · Zbl 0993.47037 [14] Marino, G.; Xu, H.K., A general iterative method for nonexpansive mappings in Hilbert spaces, J. math. anal. appl., 318, 43-52, (2006) · Zbl 1095.47038 [15] Suzuki, T., Strong convergence of Krasnoselskii and mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. math. anal. appl., 305, 227-239, (2005) · Zbl 1068.47085 [16] Takahashi, W., Nonlinear functional analysis, (1988), Kindai-kagakusha Tokyo, (in Japanese)
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