A General Iterative Method for Solving the Variational Inequality Problem and Fixed Point Problem of an Infinite Family of Nonexpansive Mappings in Hilbert Spaces
© R.Wangkeeree and U. Kamraksa. 2009
Received: 3 November 2008
Accepted: 16 January 2009
Published: 10 February 2009
We introduce an iterative scheme for finding a common element of the set of common fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. As applications, at the end of the paper we utilize our results to study the problem of finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of -strictly pseudocontractive mappings. The results presented in the paper improve some recent results of Qin and Cho (2008).
where is a contraction, is a nonexpansive mapping, and is a strongly positive linear bounded self-adjoint operator, proved that, under certain appropriate assumptions on the parameters, defined by (1.12) converges strongly to a fixed point of , which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).
The existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [17–20] and the references therein). The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings (see [21, 22]). The problem of finding an optimal point that minimizes a given cost function over the common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see [18, 23]). A simple algorithmic solution to the problem of minimizing a quadratic function over the common set of fixed points of a family of nonexpansive mappings is of extreme value in many applications including set theoretic signal estimation (see [23, 24]).
where is a nonnegative real sequence with , for all , , form a family of infinitely nonexpansive mappings of into itself. Nonexpansivity of each ensures the nonexpansivity of . Such a is nonexpansive from to and it is called a -mapping generated by and .
where is a mapping defined by (1.15), is a contraction, is strongly positive linear bounded self-adjoint operator, is a -inverse strongly monotone, and we prove that under certain appropriate assumptions on the sequences , , , and , the sequences defined by (1.16) converge strongly to a common element of the set of common fixed points of a family of and the set of solutions of the variational inequality for an inverse-strongly monotone mapping, which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).
It is well known that each Hilbert space satisfies the Opial's condition.
Then is the maximal monotone and if and only if ; see .
Now we collect some useful lemmas for proving the convergence result of this paper.
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Lemma 2.5 (see ).
Lemma 2.6 (see ).
3. Main Results
Now we are in a position to state and prove the main result in this paper.
Let be a closed convex subset of a real Hilbert space , let be a contraction of into itself, let be an -inverse strongly monotone mapping of into , and let be a family of infinitely nonexpansive mappings with . Let be a strongly positive linear bounded self-adjoint operator with the coefficient such that . Assume that . Let , , , and be sequences in satisfying the following conditions:
Theorem 3.1 mainly improve the results of Qin and Cho  from a single nonexpansive mapping to an infinite family of nonexpansive mappings.
In this section, we obtain two results by using a special case of the proposed method.
Next, we will apply the main results to the problem for finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of -strictly pseudocontractive mappings.
The following lemmas can be obtained from [31, Proposition 2.6] by Acedo and Xu, easily.
Let be a Hilbert space, let be a closed convex subset of . For any integer , assume that, for each is a -strictly pseudocontractive mapping for some . Assume that is a positive sequence such that . Then is a -strictly pseudocontractive mapping with .
The conclusion of Theorem 4.5 can be obtained from Theorem 3.1.
The authors would like to thank the referee for the comments which improve the manuscript. R. Wangkeeree was supported for CHE-PhD-THA-SUP/2551 from the Commission on Higher Education and the Thailand Research Fund under Grant TRG5280011.
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