Abstract
We introduce a new implicit iterative scheme with perturbation for finding the approximate solutions of a hierarchical variational inequality, that is, a variational inequality over the common fixed point set of a finite family of nonexpansive mappings. We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme. In particular, necessary and sufficient conditions for the strong convergence of the sequence are obtained.
1. Introduction
Let be a real Hilbert space with inner product and norm and a nonempty closed convex subset of . For a given nonlinear operator , the classical variational inequality problem (VIP) [1] is to find such that The set of solutions of VIP is denoted by . If the set is replaced by the set of fixed points of a mapping ; then the VIP is called a hierarchical variational inequality problem (HVIP). The signal recovery [2], the power control problem [3], and the beamforming problem [4] can be written in the form of a hierarchical variational inequality problem. In the recent past, several authors paid their attention toward this kind of problem and developed different kinds of solution methods with applications; see [2, 5–11] and the references therein.
Let be -strongly monotone (i.e., if there exists a constant such that , for all ) and -Lipschitz continuous (i.e., if there exists a constant such that , for all ). Assume that is the intersection of the sets of fixed points of nonexpansive mappings . For an arbitrary initial guess , Yamada [10] proposed the following hybrid steepest-descent method: Here, , for every integer , with the mod function taking values in the set ; that is, if for some integers and , then if and if . Moreover, and the sequence of parameters satisfies the following conditions:(i);(ii);(iii) is convergent.
Under these conditions, Yamada [10] proved the strong convergence of the sequence to the unique element of .
Xu and Kim [12] replaced the condition (iii) by the following condition:(iii)', or equivalently,
and proved the strong convergence of the sequence to the unique element of .
On the other hand, let be a nonempty convex subset of , and let be a finite family of nonexpansive self-maps on K. Xu and Ori [13] introduced the following implicit iteration process: for and , the sequence is generated by the following process: where we use the convention . They also studied the weak convergence of the sequence generated by the above scheme to a common fixed point of the mappings under certain conditions. Subsequently, Zeng and Yao [14] introduced another implicit iterative scheme with perturbation for finding the approximate common fixed points of a finite family of nonexpansive self-maps on .
Motivated and inspired by the above works, in this paper, we propose a new implicit iterative scheme with perturbation for finding the approximate solutions of the hierarchical variational inequalities, that is, variational inequality problem over the common fixed point set of a finite family of nonexpansive self-maps on . We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme with perturbation. In particular, necessary and sufficient conditions for strong convergence of the sequence generated by the proposed implicit iterative scheme with perturbation are obtained.
2. Preliminaries
Throughout the paper, we write to indicate that the sequence converges weakly to in a Banach space . Meanwhile, implies that converges strongly to . For a given sequence , denotes the weak -limit set of , that is, A Banach space is said to satisfy Opial’s property if whenever a sequence in . It is well known that every Hilbert space satisfies Opial’s property; see for example [15].
A mapping is said to be hemicontinuous if for any , the mapping , defined by , is continuous in the weak topology of the Hilbert space . The metric projection onto a nonempty, closed and convex set , denoted by , is defined by, for all and .
Proposition 2.1. Let be a nonempty closed and convex set and monotone and hemicontinuous. Then,(a)[1] ,(b)[1] when is bounded,(c)[16, Lemma??2.24] for all , where stands for the identity mapping on ,(d)[16, Theorem??2.31] consists of one point if is strongly monotone and Lipschitz continuous.
On the other hand, it is well known that the metric projection onto a given nonempty closed and convex set is nonexpansive with [17, Theorem??3.1.4 (i)]. The fixed point set of a nonexpansive mapping has the following properties.
Proposition 2.2. Let be a nonempty closed and convex subset and a nonexpansive map.(a)[18, Proposition??5.3] is closed and convex.(b)[18, Theorem??5.1] when is bounded.
The following proposition provides an example of a nonexpansive mapping in which the set of fixed points is equal to the solution set of a monotone variational inequality.
Proposition 2.3 (see [6, Proposition??2.3]). Let be a nonempty closed and convex set and an -inverse-strongly monotone operator. Then, for any given , the mapping , defined by is nonexpansive and .
The following lemmas will be used in the proof of the main results of this paper.
Lemma 2.4 (see [18, Demiclosedness Principle]). Assume that is a nonexpansive self-mapping on a closed convex subset of a Hilbert space . If has a fixed point, then is demiclosed, that is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that , where is the identity operator of .
Lemma 2.5 (see [19, page 80]). Let , , and be sequences of nonnegative real numbers satisfying the inequality If and , then exists. If in addition has a subsequence, which converges to zero, then .
Let be a nonexpansive mapping and ??-Lipschitz continuous and -strongly monotone for some constants , . For any given numbers and , we define the mapping by
Lemma 2.6 (see [12]). If and , then where .
3. An Iterative Scheme and Convergence Results
Let be a finite family of nonexpansive self-maps on . Let be an -inverse-strongly monotone mapping (i.e., if there exists a constant such that , for all ). Let be -Lipschitz continuous and -strongly monotone for some constants , . Let , , , and take a fixed number . We introduce the following implicit iterative scheme with perturbation . For an arbitrary initial point , the sequence is generated by the following process: Here, we use the convention . If , then the implicit iterative scheme (3.1) reduces to the implicit iterative scheme studied in [14].
Let be an -inverse-strongly monotone mapping and . By Lemma 2.6, for every and , the mapping defined by satisfies where , , and . By Banach’s contraction principle, there exists a unique such that This shows that the implicit iterative scheme (3.1) with perturbation is well defined and can be employed for finding the approximate solutions of the variational inequality problem over the common fixed point set of a finite family of nonexpansive self-maps on .
We now state and prove the main results of this paper.
Theorem 3.1. Let be a real Hilbert space, an -inverse-strongly monotone mapping, and a -Lipschitz continuous and -strongly monotone mapping for some constants , . Let be nonexpansive self-maps on with a nonempty common fixed point set . Suppose . Denote by for . Let , , , , and be such that , and , for some . Then, the sequence , defined by
converges weakly to an element of .
If, in addition, , then converges weakly to an element of .
Proof. Notice first that the following identity:
holds for all and all . Let be an arbitrary element of . Observe that
Since is -inverse strongly monotone and , we have
By Lemma 2.6, we have
It follows
This together with (3.7) yields
and so,
Since converges, by Lemma 2.5, exists. As a consequence, the sequence is bounded. Moreover, we have
Therefore,
Obviously, it is easy to see that for each . Now observe that
Also note that the boundedness of implies that and are both bounded. Thus, we have
This implies
Consequently,
and hence, for each ,
This shows that for each . Therefore,
On the other hand, since is bounded, it has a subsequence , which converges weakly to some , and so, we have . From Lemma 2.4, it follows that is demiclosed at zero. Thus, . Since is an arbitrary element in the finite set , we get .
Now, let be an arbitrary element of . Then, there exists another subsequence of , which converges weakly to . Clearly, by repeating the same argument, we get . We claim that . Indeed, if , then by the Opial’s property of , we conclude that
This leads to a contradiction, and so, we get . Therefore, is a singleton set. Hence, converges weakly to a common fixed point of the mappings , denoted still by .
Assume that . Let be arbitrary but fixed. Then, it follows from the nonexpansiveness of each and the monotonicity of that
which implies that
where . Note that , and . Thus, for any , there exists an integer such that for all . Consequently, for all . Since , we have as . Therefore, from the arbitrariness of , we deduce that for all . Proposition 2.1 (a) ensures that
that is, .
Corollary 3.2 (see [14], Theorem??2.1). Let be a real Hilbert space and -Lipschitz continuous and -strongly monotone for some constants , . Let be nonexpansive self-maps on such that . Let , , , and be such that and , for some . Then, the sequence , defined by converges weakly to a common fixed point of the mappings .
Proof. In Theorem 3.1, putting , we can see readily that for any given positive number , is an -inverse-strongly monotone mapping. In this case, we have Hence, for any given sequence with , the implicit iterative scheme (3.5) reduces to (3.25). Therefore, by Theorem 3.1, we obtain the desired result.
Lemma 3.3. In the setting of Theorem 3.1, we have(a) exists for each ,(b) exists, where ,(c),where .
Proof. Conclusion (a) follows from (3.12), and conclusion (c) follows from (3.18). We prove conclusion (b). Indeed, for each , This together with (3.12) implies that and hence, where Since and both and are bounded, it is known that . On account of Lemma 2.5, we deduce that exists, that is, conclusion (b) holds.
Finally, we give necessary and sufficient conditions for the strong convergence of the sequence generated by the implicit iterative scheme (3.5) with perturbation .
Theorem 3.4. In the setting of Theorem 3.1, the sequence converges strongly to an element of if and only if where .
Proof. From (3.28), we derive for each
where . Put . Then .
Suppose that the sequence converges strongly to a common fixed point of the family . Then, . Since
we have .
Conversely, suppose that . Then, by Lemma 3.3 (b), we deduce that . Thus, for arbitrary , there exists a positive integer such that
Furthermore, the condition implies that there exists a positive integer such that
Choose . Observe that (3.31) yields
Note that and . Thus, for all and all , we have from (3.35) that
Taking the infimum over all , we obtain
and hence, . This shows that is a Cauchy sequence in . Let as . Then, we derive from (3.20) that for each ,
Therefore, for each , and hence, .
On the other hand, choose a positive sequence such that as . For each , from the definition of , it follows that there exists a point such that
Since and as , it is clear that as . Note that
Hence, we get
Furthermore, for each , the mapping is defined as follows:
From Proposition 2.3, we deduce that is nonexpansive and
From Proposition 2.2 (a), we conclude that is closed and convex. Thus, from the condition , it is known that is a nonempty closed and convex set. Since lies in and converges strongly to , we must have .
Remark 3.5. Setting in Lemma 3.3 and Theorem 3.4 above, we shall derive Lemma??2.1 and Theorem??2.2 in [14] as direct consequences, respectively.
Acknowledgment
This paper is partially supported by the Taiwan NSC Grants 99-2115-M-110-007-MY3 and 99-2221-E-037-007-MY3.