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).
We denote by the set of fixed points of . Recall that a mapping is said to be
(i)monotone if , for all ;
(ii) -Lipschitz if there exists a constant such that , for all ;
It is obvious that any -inverse-strongly monotone mapping is monotone and -Lipschitz continuous.
where is a potential function for (i.e., for ).
where the sequence is in the interval .
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).
where is an -inverse strongly monotone mapping, and satisfy some parameters controlling conditions. They showed that if is nonempty, then the sequence generated by (1.14) converges strongly to some .
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 ).
Let and be bounded sequences in a Banach space and let be a sequence in with Suppose for all integers and Then,
Lemma 2.3 (see ).
where is a sequence in and is a sequence in such that
Lemma 2.4 (see ).
Assume that is a strongly positive linear bounded self-adjoint operator on a Hilbert space with coefficient and . Then .
Throughout this paper, we will assume that , for all . Concerning defined by (1.15), we have the following lemmas which are important to prove our main result.
Lemma 2.5 (see ).
Let be a nonempty closed convex subset of a Hilbert space , let be a family of infinitely nonexpansive mapping with , and let be a real sequence such that , for all . Then
(1) is nonexpansive and for each ;
(2)for each and for each positive integer , the limit exists;
is a nonexpansive mapping satisfying and it is called the -mapping generated by and
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:
which gives that the sequence is bounded, and so are and .
where is a constant such that . Similarly, there exists such that .
Putting , we get, .
Banach's Contraction Mapping Principle guarantees that has a unique fixed point, say . That is, .
Next we prove that .
First, we prove that .
This is a contradiction, which shows that .
Since is maximal monotone, we have , and hence .
The conclusion is proved.
Using (C1), (3.54), and (3.55), we get . Now applying Lemma 2.3 to (3.58), we conclude that . This completes the proof.
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.
where , , , and are sequences in satisfying the following conditions:
We have and . Applying Theorem 3.1, we obtain the desired result.
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 .
Let and be as in Lemma 4.3. Suppose that has a common fixed point in . Then .
This shows that is -inverse-strongly monotone.
where , , , and are the sequences in satisfying the following conditions:
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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