zbMATH — the first resource for mathematics

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
\[ \min_{x\in F}\tfrac12 \langle Ax,x \rangle-h(x). \]

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
Full Text: DOI
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.