Existence and Uniqueness of Fixed Point in Various Abstract Spaces and Related Applications
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On Solutions of Variational Inequality Problems via Iterative Methods
Abstract
We investigate an algorithm for a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family of γinverse strongly accretive mappings. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings.
1. Introduction
Let be a subset of a real Hilbert space . Let be a nonlinear mapping. The variational inequality problem for and is to
The set of solutions of variational inequality problem is denoted by ; that is,
It is well known that variational inequality theory has emerged as an important tool in studying a wide class of numerous problems in variational inequalities, minimax problems, optimization, physics, and the Nash equilibrium problems in noncooperative games. Several numerical methods have been developed for solving variational inequalities and related optimization problems; see, for instance, [1–5] and the references therein.
A mapping is said to be inverse strongly accretive (or inverse strongly monotone) if there exists a positive real number such that
If is inverse strongly accretive, then inequality (3) implies that is Lipschitzian with constant ; that is, , for all . If in (3) we have that , then is called accretive (or monotone).
Let be a closed and convex subset of a real Hilbert space . A mapping is called a contraction mapping if there exists such that for all . If , then is called nonexpansive. A mapping is called strictly pseudocontractive of BrowderPetryshyn type [6] if and only if there exists such that is called pseudocontractive if
We note that inequalities (4) and (5) can be equivalently written as for some and respectively. We remark that is pseudocontractive if and only if is accretive. A point is a fixed point of if and we denote by the set of fixed points of ; that is, .
We observe that in a real Hilbert space a class of pseudocontractive mappings includes the class of strictly pseudocontractive mappings and hence the classes of nonexpansive and contraction mappings.
Closely related to the variational inequality problems is the problem of finding fixed points of nonexpansive mappings, strict pseudocontraction mappings or pseudocontractive mappings which is the current interest in functional analysis. Several researchers considered a unified approach that approximates a common point of fixed point of nonlinear problems and solutions of variational inequality problems and solutions of variational inequality problems; see, for example, [7–18] and the references therein.
In [19], Takahashi and Toyoda studied the problem of finding a common point of fixed points of a nonexpansive mapping and solutions of a variational inequality problem (1) by considering the following iterative algorithm: where is a sequence in , is a positive sequence, is a nonexpansive mapping, and is an inverse strongly accretive mapping. They showed that the sequence generated by (8) converges weakly to some provided that the control sequences satisfy some restrictions.
Iiduka and Takahashi [20] reconsidered the common element problem via the following iterative algorithm: where is a nonexpansive mapping, is a inversestrongly accretive mapping, is a sequence in , and is a sequence in . They proved that the sequence strongly converges to some point .
Recently, Zegeye and Shahzad [21] investigated the problem of finding a common point of fixed points of a Lipschitz pseudocontractive mapping and solutions of a variational inequality problem for inverse strongly accretive mapping by considering the following iterative algorithm: where is a metric projection from onto and are in satisfying certain conditions. Then, they proved that the sequence converges strongly to the minimumnorm point of .
A natural question arises whether we can obtain an iterative scheme which converges strongly to a common point of fixed points of a finite family of pseudocontractive mappings and solutions of a finite family of variational inequality problems for inverse strongly accretive mappings or not.
It is our purpose in this paper to introduce an algorithm and prove that the algorithm converges strongly to a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family of variational inequality problems for inverse strongly accretive mappings. The results obtained in this paper improve and extend the results of Takahashi and Toyoda [19], Iiduka and Takahashi [20], and Zegeye and Shahzad [21], Theorem 3.2 of Yao et al. [22], and some other results in this direction.
2. Preliminaries
In what follows we will make use of the following lemmas.
Lemma 1. Letting be a real Hilbert space, the following identity holds:
Lemma 2 (see [23]). Let be a nonempty closed and convex subset of a real Hilbert space . Let be a inverse strongly accretive mapping. Then, for , the mapping is nonexpansive.
Lemma 3 (see [24]). Let be a nonempty, closed, and convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto and let be an accretive operator of into . Then for all ,
Lemma 4 (see [25]). Let be a nonempty, closed, and convex subset of a real Hilbert space . Let , , be nonexpansive mappings such that . Let with . Then is nonexpansive and .
Lemma 5 (see [26]). Let be a convex subset of a real Hilbert space . Let . Then if and only if
Lemma 6 (see [27]). Let be a closed convex subset of a real Hilbert space and be a continuous pseudocontractive mapping. Then, for , the mapping is nonexpansive(i) is a closed convex subset of ;(ii) is demiclosed at zero; that is, if is a sequence in such that and , as , then .
Lemma 7 (see [28]). Let be a real Hilbert space. Then for all and for such that the following equality holds:
Lemma 8 (see [29]). Let be sequences of real numbers such that there exists a subsequence of such that for all . Then there exists an increasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers :
In fact, is the largest number in the set such that the condition holds.
Lemma 9 (see [30]). Let be a sequence of nonnegative real numbers satisfying the following relation: where and satisfying the following conditions: , and . Then, .
3. Main Result
For the rest of this paper, let , for some , and , for some , satisfy () ; () ; and () .
Theorem 10. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let , , be Lipschitz pseudocontractive mappings with Lipschitz constants , respectively. Let , for , be inverse strongly accretive mappings. Let be a contraction with constant . Assume that is nonempty. Let a sequence be generated from an arbitrary by where and , for , for with and , for . Then, converges strongly to a point which is the unique solution of the variational inequality for all .
Proof. From Lemmas 2, 4, and 3 we get that is nonexpansive mapping with . Let . Then from (17), (5), and Lemma 7 we have that
Now, substituting (18) in (19) we get that
Moreover, from (17), Lemma 7, and Lipschitz property of we get that
Substituting (21) into (20) we obtain that
But, from the hypothesis we have that
and hence inequality (22) gives that
But we have that
Substituting (25) into (24) we get that
Therefore, by induction we get that
which implies that and hence are bounded.
Let . Then, from (17), Lemmas 1 and 7, and the methods used to get (22) we obtain that
which implies that
But
Thus, substituting (31) in (30) we obtain that
Next, we consider two cases.
Case 1. Suppose that there exists such that is decreasing for all . Then, we get that is convergent. Thus, from (29) and (23) we have that
Furthermore, from (17) and (33) we obtain that
and hence Lipschitz continuity of , (34), and (33) implies that
Thus, from (33) and (35) we have that
Therefore, , as , for all , and hence
as , for all .
Now, since is bounded subset of , we can choose a subsequence of such that and . Then, from (37) and Lemma 6 we have that , for each . Hence, .
In addition, since is nonexpansive, from Lemma 6 we get that and hence by Lemmas 4 and 3 we obtain that , for each .
Therefore, by Lemma 5, we immediately obtain that
Then, it follows from (32), (38), and Lemma 9 that as . Consequently, .
Case 2. Suppose that there exists a subsequence of such that for all . Then, by Lemma 8, there exists a nondecreasing sequence such that , and for all . Now, from (29) and (23) we get that and as . Thus, following the method in Case 1, we obtain that , , and
Furthermore, from (32) and (40) we obtain that
Now, using the fact that and (41) we get that and this together with (32) implies that as . Since for all , we obtain that . Hence, from the above two cases, we can conclude that converges strongly to a point , which satisfies the variational inequality , for all . The proof is complete.
If, in Theorem 10, we assume that , a constant mapping, then we get the following corollary.
Corollary 11. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let , , be Lipschitz pseudocontractive mappings with Lipschitz constants , respectively. Let , for , be inverse strongly accretive mappings. Assume that is nonempty. Let a sequence be generated from an arbitrary by where , , for , for with , and , for . Then, converges strongly to a unique point satisfying , which is the unique solution of the variational inequality for all .
If, in Theorem 10, we assume that and , then we get the following corollary which is Theorem 3.1 of [21].
Corollary 12. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be Lipschitz pseudocontractive mappings with Lipschitz constant and an inverse strongly accretive mapping. Let be a contraction with constant . Assume that is nonempty. Let a sequence be generated from an arbitrary by where and . Then, converges strongly to a point , which is the unique solution of the variational inequality for all .
If, in Theorem 10, we assume that are strictly pseudocontractive mappins, then we get the following corollary.
Corollary 13. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let , , be strictly pseudocontractive mappings and let , for , be an inverse strongly accretive mappings. Let be a contraction with constant . Assume that is nonempty. Let a sequence be generated from an arbitrary by where , , for , for with , and , . Then, converges strongly to a point , which is the unique solution of the variational inequality for all .
If, in Theorem 10, we assume that are nonexpansive mapping, then we get the following corollary.
Corollary 14. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let , , be nonexpansive mappings and let , for , be an inverse strongly accretive mappings. Let be a contraction with constant . Assume that is nonempty. Let a sequence be generated from an arbitrary by where , , for , for with , , and . Then, converges strongly to point , which is the unique solution of the variational inequality for all .
We note that the method of proof of Theorem 10 provides the following theorem which is a convergence theorem for a minimum norm point of common fixed points of a finite family of Lipschitz pseudocontractive mappings and common solutions of a finite family of variational inequality problems for accretive mappings.
Theorem 15. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let , , be Lipschitz pseudocontractive mappings with Lipschitz constants , respectively. Let , for , be inverse strongly accretive mappings. Assume that is nonempty. Let a sequence be generated from an arbitrary by where , , for , for with , and , for . Then, converges strongly to a unique minimum norm point of i.e., , which is the unique solution of the variational inequality for all .
4. Numerical Example
Now, we give an example of two Lipschitz pseudocontractive mappings and two inverse strongly accretive mappings satisfying Theorem 10 and some numerical experiment result to explain the conclusion of the theorem as follows.
Example 1. Let with absolute value norm. Let and let be defined by
Clearly, for we have that
which show that both mappings are pseudocontractive. Next, we show that is Lipschitz with . If , then
If , then
If and , then
Thus, we get that is Lipschitz pseudocontractive with and which is not nonexpansive, since if we take and , we have that . Similarly, we can show that is Lipschitz pseudocontractive with and which is not nonexpansive.
Furthermore, for , let be defined by
Then we first show that is inverse strongly accretive mapping with .
If , then
If and , we get that
If , then we get that and hence
Therefore, is inverse strongly accretive mapping with and . Similarly, we can show that is inverse strongly accretive mapping with and .
Note that we have .
Thus, taking , , , , and , we observe that conditions of Theorem 10 are satisfied and Scheme (17) provides the following Table 1 and Figures 1(a) and 1(b) for and , respectively.
We observe that the data provides strong convergence of the sequence to the common point of fixed points of both pseudocontractive mappings and solutions of both variational inequality problems for inverse strongly accretive mappings.

(a)
(b)
Remark 2. Theorem 10 provides an iteration scheme which converges strongly to a common point of fixed points of a finite family of Lipschitzian pseudocontractive mappings and solutions of a finite family of variational inequality problems in Hilbert spaces.
Remark 3. Theorem 10 improves Theorem 3.1 of Takahashi and Toyoda [19], Iiduka and Takahashi [20], and Zegeye and Shahzad [21] and Theorem 3.2 of Yao et al. [22] in the sense that our convergence is to a common point of fixed points of a finite family of Lipschitzian pseudocontractive mappings and solutions of a finite family of variational inequality problems.
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
Acknowledgments
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. 167/130/1434. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors also thank the referees for their valuable comments and suggestions, which improved the presentation of this paper.
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Copyright © 2014 Mohammed Ali Alghamdi 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.