1. Introduction and Main Result

Let be a real Hilbert space with inner product and norm . A mapping is said to be nonexpansive if for all . The set of fixed points, , of a nonexpansive mapping is always a closed and convex subset of .

In addition to nonexpansive mappings, we are going to use contractions and -Lipschitzian and -strongly monotone operators. A self-mapping is a contraction on , if there exists a constant such that , for all . We use to denote the collection of mappings verifying the above inequality. That is, is a contraction with constant . A mapping is called -Lipschitzian if there exists a positive constant such that is said to be -strongly monotone if there exists a positive constant such that

Recently, Jung [1] introduced the following composite iterative scheme for the solution of a specific minimization problem, which involves a closed convex subset , a nonexpansive mapping , and a contraction , Therein the control sequences and satisfy certain conditions. He proved that the sequence defined by (1.3) converges strongly to , which is the unique solution of the variational inequality . The results of Jung [1] are even stronger than stated here. In fact, they hold for a finite family of nonexpansive mappings and in the setting of Banach spaces.

Very recently, Tian [2] considered the following iterative method: for nonexpansive mapping with , where is a -Lipschitzian and -strongly monotone operator. He obtained that the sequence generated by (1.4) converges to a point in , which is the unique solution of the variational inequality .

In this connection, notice that Aoyama et al. [3] proposed a Halpern approximation method for finding a common fixed point of a countable family of nonexpansive mappings. As their main result, they established the following strong convergence theorem.

Theorem 1.1 (see [3, Theorem 3.4]). Let be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, and let be a nonempty closed convex subset of . Let be a sequence of nonexpansive mappings such that for each bounded subset of . Suppose in addition that where is the nonexpansive mapping defined by . Let be a sequence satisfying the conditions (C1) , (C2) , and (C3) or . Given , define by Then converges strongly to , where is a sunny nonexpansive retraction of onto .

Inspired by Jung [1], Tian [2] and Aoyama et al. [3], the goal of this paper is to combine these three results into a single method. In this connection, observe that the iteration methods (1.3) and (1.4) just can compute a fixed point of one nonexpansive mapping (or, perhaps, finitely many), while the iteration methods (1.3) and (1.7) do not contain the -Lipschitzian and -strongly monotone operator . On the other hand, in contrast to the result of Jung [1] and Aoyama et al. [3], our result will be restricted to the setting of Hilbert spaces, which on the other hand is natural when dealing with a -Lipschitzian and -strongly monotone operator .

First of all, let us remark that condition (1.5) implies that sequence of mappings are uniformly Cauchy on each bounded subset . Hence the limiting map is well defined and is in fact the uniform limit (on ) of the maps . In other words,

Notice that condition (1.5) is quite strong so that, in general, we cannot apply the result directly to an arbitrary countable family of nonexpansive mappings. Furthermore, we have to assume the nontrivial condition (1.6) that the fixed point set of coincides with the set of common fixed points of the family .

Fortunately, as pointed out in [3, Section 4], given an arbitrary countable family of nonexpansive mappings , which have at least one common fixed point, one can find nonnegative numbers () such that the mappings are nonexpansive self-maps of satisfying (1.5) and (1.6). More specifically, the are constructed as certain convex combinations of the . (For details, see the reference.) Another possibility of a construction will be mentioned below. Furthermore, we establish the following strong convergence theorem.

Theorem 1.2. Let be a real Hilbert space. Let be a -Lipschitzian and -strongly monotone operator on with and with and . Assume that is a sequence of nonexpansive mappings from into itself such that the condition (1.5). Suppose that is defined by such that the condition (1.6). Let be sequences in satisfying the following conditions:(B1) and ; (B2) and ; (B3) with and . Then, for arbitrary , the sequence , defined by converges strongly to some , which satisfies the variational inequality , for all .

The iterative scheme (1.10) is a direct generalization of the three iteration methods considered before. Thus Theorem 1.2 complements or improves Theorem 3.2 of Jung [1], Theorem 3.2 of Tian [2], Theorem 3.4 of Aoyama et al. [3], and Theorem 3.1 of Jung [4].

An important special case is obtained for , the identity mapping. Then and we can choose (compare with iteration scheme (1.3) above).

Besides the basic conditions (B1) and (B2) on the sequences and , we have the “control condition” (B3). It can obviously be replaced by one of the following ones:(B3-1);(B3-2) for all and . Indeed, (B3-1) implies (B3) by choosing , and (B3-2) implies (B3) by choosing . In this sense, (B3) is a weaker condition than the previous condition (C3).

As has already noticed in Theorem 1.1, the assumptions (1.5) and (1.6) do not apply to arbitrary families of nonexpansive mappings. Besides the construction above, we mention another construction, which has appeared in the literature. See [5–8], the references therein, and also Remark 3.1 of Peng and Yao [9]. Proofs are given there.

Let be a countable family of nonexpansive mappings, and let be a sequence of real numbers such that , for all . For any , define a mapping as follows:

Proposition 1.3. Let be a real Hilbert space, a sequence nonexpansive mappings with , and a real sequence such that , for all . Define as above. Then(1) is a nonexpansive and for each ;(2) the mappings satisfy the condition (1.5);(3) defining by , we have

2. Proof of the Main Results

Ahead of the proof, we start with recalling some known auxiliary results.

Lemma 2.1 (see [10]). Let be a -Lipschitzian and -strongly monotone operator on a Hilbert space with , and . Then is a contraction with contractive coefficient and .

Lemma 2.2 (see [2]). Let be a Hilbert space. Assume that is defined by where is a -Lipschitzian and -strongly monotone operator on a Hilbert space with and , and is a nonexpansive mapping. Then converges strongly as to a fixed point of , which solves the variational inequality .

Some more auxiliary results are given next.

Lemma 2.3 (see [11]). Let be a Hilbert space, a closed convex subset of , and a nonexpansive mapping with . If is a sequence in which converges weakly to and if converges strongly to , then .

Lemma 2.4 (see [3, Lemma 2.3]). Let be a sequence of nonnegative real numbers, a sequence of numbers in such that , a sequence of nonnegative real numbers with , and a sequence of real numbers with . Suppose that for all . Then .

Now we are prepared to prove the main result.

Proof of Theorem 1.2. We divide the proof into seven steps.
Step 1. We claim that is bounded. Taking any point and using Lemma 2.1, we obtain By induction, it follows and hence is bounded. From this, we also obtain that , and are all bounded. In what follows, let stand for some bounded set of , which contains all of , .
Step 2. We show that .
Let . From the definition of , we obtain for all . From (1.10), we have With , we thus have
Combining (2.5) and (2.7), we conclude that where . Using the assumption (B3), we obtain with . The various assumptions imply that . Now it follows from Lemma 2.4 that .
Step 3. Next we show that In fact, from Step 1, is bounded. Moreover, it follows from that This is the first part. From condition (B2), we obtain, for sufficiently large and , which implies Hence, using Step 2 and the above, we obtain and thus .
Step 4. We show that .
Indeed, we have and now the statement follows from Step 3.
Step 5. We show that as .
Observe that and now apply Step 4 and the fact that converges uniformly on to (which is a consequence of (1.5)).
Step 6. We claim that , where with defined by (2.1) and the existence of being guaranteed by Lemma 2.2.
Assume the contrary. Then there exists a subsequence of such that the limit exists and is greater than zero. Since is bounded, there exists a subsequence of , which converges weakly to some . We can assume without loss of generality that is this subsequence. From , we obtain that converges weakly to . Now Lemma 2.3 implies that . Hence from weak convergence and Lemma 2.2, This is a contradiction.
Step 7. We show that converges strongly to . From (1.10), we have Using the general inequality , we can further estimate the above by Introducing , , and we obtain Because , , and , it follows from Lemma 2.4 that the sequence converges strongly to . This finishes the proof of the main part of the theorem.
From the definition of as (see Step 6) and from Lemma 2.2, we obtain , for all . This completes the proof.

Acknowledgment

This author is supported by the Natural Science Foundation of Yancheng Teachers University under Grant (11YCKL009).