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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 417234, 13 pages
Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces
Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China
Received 14 October 2011; Accepted 11 December 2011
Academic Editor: Rudong Chen
Copyright © 2012 Dan Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be a real Hilbert space. Consider on a nonexpansive semigroup with a common fixed point, a contraction with the coefficient , and a strongly positive linear bounded self-adjoint operator with the coefficient > 0. Let /. It is proved that the sequence generated by the iterative method converges strongly to a common fixed point , where denotes the common fixed point of the nonexpansive semigroup. The point solves the variational inequality for all .
1. Introduction and Preliminaries
Let be a real Hilbert space and be a nonlinear mapping with the domain . A point is a fixed point of provided . Denote by the set of fixed points of ; that is, . Recall that is said to be nonexpansive if
Recall that a family of mappings from into itself is called a one-parameter nonexpansive semigroup if it satisfies the following conditions:(i), ; (ii), and ;(iii), and ;(iv)for all , is continuous.
We denote by the set of common fixed points of , that is, . It is known that is closed and convex; see . Let be a nonempty closed and convex subset of . One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping; see [2, 3]. More precisely, take and define a contraction by where is a fixed point. Banach’s contraction mapping principle guarantees that has a unique fixed point in . If enjoys a nonempty fixed point set, Browder  proved the following well-known strong convergence theorem.
Theorem B. Let be a bounded closed convex subset of a Hilbert space and let be a nonexpansive mapping on . Fix and define for . Then as , converges strongly to a element of nearest to .
As motivated by Theorem B, Halpern  considered the following explicit iteration: and proved the following theorem.
Theorem H. Let be a bounded closed convex subset of a Hilbert space and let be a nonexpansive mapping on . Define a real sequence in by . Define a sequence by (1.3). Then converges strongly to the element of nearest to .
In 1977, Lions  improved the result of Halpern , still in Hilbert spaces, by proving the strong convergence of to a fixed point of where the real sequence satisfies the following conditions:(C1); (C2); (C3).
It was observed that both Halpern’s and Lions’s conditions on the real sequence excluded the canonical choice . This was overcome in 1992 by Wittmann , who proved, still in Hilbert spaces, the strong convergence of to a fixed point of if satisfies the following conditions:(C1); (C2); (C3).
Recall that a mapping is an -contraction if there exists a constant such that
Recall that an operator is strongly positive on if there exists a constant such that
Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [7–13] and the references therein. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space : where is a linear bounded operator on and is a given point in . In , it is proved that the sequence defined by the iterative method below, with the initial guess chosen arbitrarily, strongly converges to the unique solution of the minimization problem (1.6) provided that the sequence satisfies certain conditions.
Recently, Marino and Xu  studied the following continuous scheme: where is an -contraction on a real Hilbert space , is a bounded linear strongly positive operator and is a constant. They showed that strongly converges to a fixed point of . Also in , they introduced a general explicit iterative scheme by the viscosity approximation method: and proved that the sequence generated by (1.9) converges strongly to a unique solution of the variational inequality
which is the optimality condition for the minimization problem
where is a potential function for (i.e., for ).
In this paper, motivated by Li et al. , Marino and Xu , Plubtieng and Punpaeng , Shioji and Takahashi , and Shimizu and Takahashi , we consider the mapping defined as follows: where is a constant, is an -contraction, is a bounded linear strongly positive self-adjoint operator and is a positive real divergent net. If for each , one can see that is a -contraction. So, by Banach’s contraction mapping principle, there exists an unique solution of the fixed point equation We show that the sequence generated by above continuous scheme strongly converges to a common fixed point , which is the unique point in solving the variational inequality for all . Furthermore, we also study the following explicit iterative scheme: We prove that the sequence generated by (1.14) converges strongly to the same .
The results presented in this paper improve and extend the corresponding results announced by Marino and Xu , Plubtieng and Punpaeng , Shioji and Takahashi , and Shimizu and Takahashi .
In order to prove our main result, we need the following lemmas.
Lemma 1.1 (see ). Let be a nonempty bounded closed convex subset of a Hilbert space and let be a nonexpansive semigroup on . Then, for any ,
Lemma 1.2 (see ). Let be a Hilbert space, a closed convex subset of , and a nonexpansive mapping with . Then is demiclosed, that is, if is a sequence in weakly converging to and if strongly converges to , then .
Lemma 1.3 (see ). Let be a nonempty closed convex subset of a real Hilbert space and let be the metric projection from onto ( i.e., for is the only point in such that ). Given and . Then if and only if there holds the relations
Lemma 1.4. Let be a Hilbert space, a -contraction, and a strongly positive linear bounded self-adjoint operator with the coefficient . Then, for , That is, is strongly monotone with coefficient .
Proof. From the definition of strongly positive linear bounded operator, we have On the other hand, it is easy to see Therefore, we have for all . This completes the proof.
Remark 1.5. Taking and , the identity mapping, we have the following inequality: Furthermore, if is a nonexpansive mapping in Remark 1.5, we have
Lemma 1.6 (see ). Assume is a strongly positive linear bounded self-adjoint operator on a Hilbert space with coefficient and . Then .
Lemma 1.7 (see ). Let be a sequence of nonnegative real numbers satisfying the following condition: where is a sequence in and is a sequence of real numbers such that(i) and ,(ii)either or . Then converges to zero.
2. Main Results
Lemma 2.1. Let a real Hilbert space and a nonexpansive semigroup on such that . Let be a continuous net of positive real numbers such that . Let be an -contraction, a strongly positive linear bounded self-adjoint operator of into itself with coefficient . Assume that . Let be a sequence defined by (1.13). Then (i) is bounded for all ;(ii) for all ;(iii) defines a continuous curve from into .
Proof. (i) Taking , we have
It follows that
This implies that is not only bounded, but also that is contained in of center and radius , for all fixed . Moreover for and ,
(ii) Observe that Taking as in Lemma 1.1 and passing to in (2.4), we can obtain (ii) immediately.
(iii) Taking and fixing , we see that Thus applying (2.3), we arrive at It follows that where and This inequality, together with the continuity of the net , gives the continuity of the curve .
Theorem 2.2. Let be a real Hilbert space and a nonexpansive semigroup such that . Let be a net of positive real numbers such that . Let be an -contraction and let be a strongly positive linear bounded self-adjoint operator on with the coefficient . Assume that . Then sequence defined by (1.13) strongly converges as to , which solves the following variational inequality: Equivalently, one has
Proof. The uniqueness of the solution of the variational inequality (2.10) is a consequence of the strong monotonicity of (Lemma 1.4) and it was proved in . Next, we will use to denote the unique solution of (2.10). To prove that , we write, for a given , Using to make inner product, we obtain that It follows that which yields that Since is a Hilbert space and is bounded as , we have that if is a sequence in such that and . By (2.15), we see . Moreover, by (ii) of Lemma 2.1 we have . We next prove that solves the variational inequality (2.10). From (1.13), we arrive at For , it follows from (1.22) that Passing to , since is a bounded sequence, we obtain that is, satisfies the variational inequality (2.10). By the uniqueness it follows . In a summary, we have shown that each cluster point of (as ) equals . Therefore, as . The variational inequality (2.10) can be rewritten as This, by Lemma 1.3, is equivalent to This completes the proof.
Remark 2.3. Theorem 2.2 which include the corresponding results of Shioji and Takahashi  as a special case is reduced to Theorem 3.1 of Plubtieng and Punpaeng  when , the identity mapping and .
Theorem 2.4. Let be a real Hilbert space and a nonexpansive semigroup such that . Let be a positive real divergent sequence and let and be sequences in satisfying the following conditions and . Let be an -contraction and let be a strongly positive linear bounded self-adjoint operator with the coefficient . Assume that . Then sequence defined by (1.14) strongly converges to , which solves the variational inequality (2.10).
Proof. We divide the proof into three parts.Step 1. Show the sequence is bounded.
Noticing that , we may assume, with no loss of generality, that for all . From Lemma 1.6, we know that . Picking , we have By simple inductions, we see that which yields that the sequence is bounded.Step 2. Show that where is obtained in Theorem 2.2 and .
Putting , from (2.22) we see that the closed ball of center and radius is -invariant for each and contain . Therefore, we assume, without loss of generality, is a nonexpansive semigroup on . It follows from Lemma 1.1 that for all . Taking a suitable subsequence of , we see that Since the sequence is also bounded, we may assume that . From the demiclosedness principle, we have . Therefore, we have On the other hand, we have From the assumption , we see that which combines with (2.26) gives that Step 3. Show as .
Note that It follows that By using Lemma 1.7, we can obtain the desired conclusion easily.
Corollary 2.6. Let be a real Hilbert space and a nonexpansive semigroup such that . Let be a positive real divergent sequence and let be a sequence in satisfying the following conditions and . Let be a -contraction and let be a strongly positive linear bounded self-adjoint operator with the coefficient . Assume that . Let be a sequence generated by the following manner: Then the sequence defined by above iterative algorithm converges strongly to , which solves the variational inequality (2.10).
The present studies were supported by the National Natural Science Foundation of China (11071169), (11126334) and the Natural Science Foundation of Zhejiang Province (Y6110287).
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