We use strongly pseudocontraction to regularize the following ill-posed monotone variational inequality: finding a point with the property such that , where , are two pseudocontractive self-mappings of a closed convex subset of a Hilbert space with the set of fixed points . Assume the solution set of (VI) is nonempty. In this paper, we introduce one implicit scheme which can be used to find an element . Our results improve and extend a recent result of (Lu et al. 2009).
Let be a real Hilbert space with inner product and norm , respectively, and let be a nonempty closed convex subset of . Let be a nonlinear mapping. A variational inequality problem, denoted , is to find a point with the property
If the mapping is a monotone operator, then we say that is monotone. It is well known that if is Lipschitzian and strongly monotone, then for small enough , the mapping is a contraction on and so the sequence of Picard iterates, given by () converges strongly to the unique solution of the . Hybrid methods for solving the variational inequality were studied by Yamada , where he assumed that is Lipschitzian and strongly monotone.
In this paper, we devote to consider the following monotone variational inequality: finding a point with the property
where are two nonexpansive mappings with the set of fixed points . Let denote the set of solutions of VI (1.2) and assume that is nonempty.
We next briefly review some literatures in which the involved mappings and are all nonexpansive.
First, we note that Yamada's methods do not apply to VI (1.2) since the mapping fails, in general, to be strongly monotone, though it is Lipschitzian. As a matter of fact, the variational inequality (1.2) is, in general, ill-posed, and thus regularization is needed. Recently, Moudafi and Maingé  studied the VI (1.2) by regularizing the mapping and defined as the unique fixed point of the equation
Since Moudafi and Maingé's regularization depends on , the convergence of the scheme (1.3) is more complicated. Very recently, Lu et al.  studied the VI (1.2) by regularizing the mapping and defined as the unique fixed point of the equation
In this paper, we will extend Lu et al.'s result to a general case. We will further study the strong convergence of the algorithm (1.4) for solving VI (1.2) under the assumption that the mappings are all pseudocontractive. As far as we know, this appears to be the first time in the literature that the solutions of the monotone variational inequalities of kind (1.2) are investigated in the framework that feasible solutions are fixed points of a pseudocontractive mapping .
Let be a nonempty closed convex subset of a real Hilbert space . Recall that a mapping is called strongly pseudocontractive if there exists a constant such that , for all . A mapping is a pseudocontraction if it satisfies the property
We denote by the set of fixed points of ; that is, . Note that is always closed and convex (but may be empty). However, for VI (1.2), we always assume . It is not hard to find that is a pseudocontraction if and only if satisfies one of the following two equivalent properties:
(a) for all , or
(b) is monotone on : for all .
Below is the so-called demiclosedness principle for pseudocontractive mappings.
Lemma 2.1 (see ).
Let be a closed convex subset of a Hilbert space . Let be a Lipschitz pseudocontraction. Then, is a closed convex subset of , and the mapping is demiclosed at 0; that is, whenever is such that and , then .
We also need the following lemma.
Lemma 2.2 (see ).
Let be a nonempty closed convex subset of a real Hilbert space . Assume that the mapping is monotone and weakly continuous along segments; that is, weakly as . Then, the variational inequality
is equivalent to the dual variational inequality
3. Main Results
In this section, we introduce an implicit algorithm and prove this algorithm converges strongly to which solves the VI (1.2). Let be a nonempty closed convex subset of a real Hilbert space . Let be a strongly pseudocontraction. Let be two Lipschitz pseudocontractions. For , we define the following mapping
It easy to see that the mapping is strongly pseudocontractive; that is, , for all . So, by Deimling , has a unique fixed point which is denoted ; that is,
Below is our main result of this paper which displays the behavior of the net as and successively.
Let be a nonempty closed convex subset of a real Hilbert space . Let be a strongly pseudocontraction. Let be two Lipschitz pseudocontractions with . Suppose that the solution set of VI (1.2) is nonempty. Let, for each , be defined implicitly by (3.2). Then, for each fixed , the net converges in norm, as , to a point . Moreover, as , the net converges in norm to the unique solution of the following VI:
Hence, for each null sequence in , there exists another null sequence in , such that the sequence in norm as .
We divide our details proofs into several lemmas as follows. Throughout, we assume all conditions of Theorem 3.1 are satisfied.
For each fixed , the net is bounded.
Take any to derive that, for all ,
It follows that
It follows that for each fixed , is bounded, so are the nets , , and .
We will use to denote possible constant appearing in the following.
From (3.2), we have
Next, we show that, for each fixed , the net is relatively norm compact as . It follows from (3.2) that
It turns out that
Assume that is such that as . By (3.8), we obtain immediately that
Since is bounded, without loss of generality, we may assume that as , converges weakly to a point . From (3.6), we get . So, Lemma 2.1 implies that . We can then substitute for in (3.9) to get
Consequently, the weak convergence of to actually implies that strongly. This has proved the relative norm compactness of the net as .
Now, we return to (3.9) and take the limit as to get
In particular, solves the following variational inequality
or the equivalent dual variational inequality (see Lemma 2.2)
Next, we show that as , the entire net converges in norm to . We assume where . Similarly, by the above proof, we deduce which solves the following variational inequality
In (3.13), we take to get
In (3.14), we take to get
Adding up (3.15) and (3.16) yields
At the same time, we note that
It follows that
Hence, we conclude that the entire net converges in norm to as .
The net is bounded.
In (3.13), we take any to deduce
By virtue of the monotonicity of and the fact that , we have
It follows from (3.21) and (3.22) that
The net which solves the variational inequality (3.3).
First, we note that the solution of the variational inequality VI (3.3) is unique.
We next prove that ; namely, if is a null sequence in such that weakly as , then . To see this, we use (3.13) to get
However, since is monotone,
Combining the last two relations yields
Letting as in (3.29), we get
which is equivalent to its dual variational inequality
Namely, is a solution of VI (1.2); hence, . We further prove that , the unique solution of VI (3.3). As a matter of fact, we have by (3.25),
Therefore, the weak convergence to of right implies that that in norm. Now, we can let in (3.23) to get
It turns out that solves VI (3.3). By uniqueness, we have . This is sufficient to guarantee that in norm, as . The proof is complete.
Yamada, I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu D, Censor Y, Reich S (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), Studies in Computational Mathematics, vol. 8, pp. 473–504. North-Holland, Amsterdam, The Netherlands (2001)
Lu, X, Xu, H-K, Yin, X: Hybrid methods for a class of monotone variational inequalities. Nonlinear Analysis: Theory, Methods & Applications. 71(3-4), 1032–1041 (2009). PubMed Abstract | Publisher Full Text
Cianciaruso, F, Colao, V, Muglia, L, Xu, H-K: On an implicit hierarchical fixed point approach to variational inequalities. Bulletin of the Australian Mathematical Society. 80(1), 117–124 (2009). Publisher Full Text
Moudafi, A: Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Problems. 23(4), 1635–1640 (2007). Publisher Full Text
Marino, G, Colao, V, Muglia, L, Yao, Y: Krasnoselski-Mann iteration for hierarchical fixed points and equilibrium problem. Bulletin of the Australian Mathematical Society. 79(2), 187–200 (2009). Publisher Full Text
Zhou, H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications. 70(11), 4039–4046 (2009). PubMed Abstract | Publisher Full Text
Deimling, K: Zeros of accretive operators. Manuscripta Mathematica. 13, 365–374 (1974). Publisher Full Text