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A necessary and sufficient condition for the strong convergence of nonexpansive mappings in Banach spaces
Fixed Point Theory and Applicationsvolume 2014, Article number: 106 (2014)
In this paper, we establish a necessary and sufficient condition for the strong convergence of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. It is worth pointing out that we remove some quite restrictive conditions in the corresponding results. An appropriate example, such that all conditions of this result are satisfied and that other conditions are not satisfied, is provided.
MSC:47H05, 47H09, 47H10.
Let X be a real Banach space with the dual space . The value of at is denoted by . The normal duality mapping J from X into a family of nonempty (by the Hahn-Banach theorem) weak-star compact subsets of is defined by
Let . A Banach space X is said to be uniformly convex if for each , there exists such that for any ,
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space X is said to be smooth if the limit
exists for all . It is said to be uniformly smooth if limit (1.1) is attained uniformly for . It is well known that if X is smooth, then J is single-valued and continuous from the norm topology of X to the weak-star topology of , i.e., norm to weak∗ continuous. It is also well known that if X is uniformly smooth, then J is uniformly continuous on bounded subsets of X from the strong topology of X to the strong topology of , i.e., uniformly norm-to-norm continuous on any bounded subset of X; see [1, 2] for more details.
Also, we define a function called the modulus of smoothness of X as follows:
It is known that X is uniformly smooth if and only if . Let q be a fixed real number with . Then a Banach space X is said to be q-uniformly smooth if there exists a constant such that for all . One should note that no Banach space is q-uniformly smooth for ; see  for more details. So, in this paper, we focus on a 2-uniformly smooth Banach space. It is well known that Hilbert spaces and Lebesgue () spaces are uniformly convex and 2-uniformly smooth.
Recall that a mapping is said to be nonexpansive if
A point is a fixed point of T if . Let denote the set of fixed points of T; that is, .
A mapping is called strongly pseudo-contractive if there exists a constant and satisfying
A mapping is a contraction if there exists a constant such that
Since , we have that f is a strong pseudo-contraction.
Let , a mapping of X into X is said to be η-strongly accretive if there exists such that
for all . A mapping of X into X is said to be k-Lipschitzian if, for ,
for all . It is well known that if X is a Hilbert space, then an η-strongly accretive operator coincides with an η-strongly monotone operator.
Yamada  introduced the following hybrid iterative method for solving the variational inequality in a Hilbert space:
where F is a k-Lipschitzian and η-strongly monotone operator with , and . Let a sequence of real numbers in satisfy the conditions below:
He proved that generated by (1.2) converges strongly to the unique solution of the variational inequality
An example of sequence which satisfies conditions (A1)-(A3) is given by , where . We note that condition (A3) was first used by Lions . It was observed that Lion’s conditions on the sequence excluded the canonical choice . This was overcome in 2003 by Xu and Kim  in a Hilbert space. They proved that if satisfies conditions (A1), (A2) and (A4)
then is strongly convergent to the unique solution of the variational inequality , . It is easy to see that condition (A4) is strictly weaker than condition (A3), coupled with conditions (A1) and (A2). Moreover, (A4) includes the important and natural choice of .
In 2010, Tian  improved Yamada’s method (1.2) and established the following strong convergence theorems.
Theorem 1.1 ([, Theorem 3.1])
Let H be a Hilbert space. Let be a nonexpansive mapping with , and let be a contraction mapping with . Assume that is defined by
where F is a k-Lipschitzian and η-strongly monotone operator on a Hilbert space H with , . Let , and . Then converges strongly as to a fixed point of T, which solves the variational inequality , .
Theorem 1.2 ([, Theorem 3.1])
Let H be a Hilbert space. Let be a nonexpansive mapping with , let be a contraction mapping with , and let F be a k-Lipschitzian and η-strongly monotone operator on H with , and . For an arbitrary , let be generated by
where and satisfies
(C3) either or .
Then converges strongly to that is obtained in Theorem 1.1.
We remind the reader of the following facts: (i) The results are obtained when the underlying space is a Hilbert space in Yamada , Xu  and Tian . (ii) In order to guarantee the strong convergence of the iterative sequence , there is at least one parameter sequence converging to zero (i.e., or ) in Yamada , Xu  and Tian . (iii) The parameter sequence satisfies the condition (or ).
In this paper, we establish a necessary and sufficient condition for the strong convergence of generated by (3.7) (defined below) in a uniformly convex and 2-uniformly smooth Banach space. In the meantime, we remove the control condition (C1) and replace condition (C3) with (C3′) (defined below) in the result of Tian . It is worth pointing out that we use a new method to prove our main results. The results presented in this paper can be viewed as an improvement, supplement and extension of the results obtained in [4, 6, 7].
For the sequence in X, we write to indicate that the sequence converges weakly to x. means that converges strongly to x. In order to prove our main results, we need the following lemmas.
Lemma 2.1 ()
Let q be a given real number with , and let X be a q-uniformly smooth Banach space. Then
for all , where K is a q-uniformly smooth constant of X and is the generalized duality mapping from X into defined by
for all .
Lemma 2.2 ()
Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space X, and let T be a nonexpansive mapping of C into itself. If is a sequence of C such that and , then x is a fixed point of T.
Let be a sequence of nonnegative real numbers satisfying
where , and satisfy the following conditions: (i) and , (ii) or , (iii) (), . Then .
The following lemma is easy to prove.
Lemma 2.4 Let X be a Banach space, let be a strongly pseudo-contractive operator with , and let be a k-Lipschitzian and η-strongly accretive operator with , . Then, for ,
that is, is a strongly accretive operator with coefficient .
Lemma 2.5 Let X be a real 2-uniformly smooth Banach space. Let t be a number in , and let . Let be an operator such that, for some constant , is k-Lipschitzian and η-strongly accretive. Then is a contraction provided , that is,
Proof Using Lemma 2.1, we have
It follows from and that
for all . Therefore, we have
where . □
3 Main results
Throughout this section, let X be a uniformly convex and 2-uniformly smooth Banach space. Let be a nonexpansive mapping with . Let be a k-Lipschitzian and η-strongly accretive operator with . Let and . Let be a Lipschitzian and strongly pseudo-contractive operator with . Let t be a number in . Consider a mapping on X defined by
It is easy to see that the mapping is strongly pseudo-contractive. Indeed, from Lemma 2.5, we have
for all . Since is Lipschitzian and is contractive, hence is continuous. So, by Deimling , we can obtain that has a unique fixed point which we denoted by , that is,
Our first main result below shows that converges strongly as to a fixed point of T which solves some variational inequality.
Theorem 3.1 generated by the implicit method (3.1) converges in norm as to the unique solution of the variational inequality
Proof It is easy to see the uniqueness of a solution of variational inequality (3.2). By Lemma 2.4, is strongly accretive, so variational inequality (3.2) has only one solution. Below we use to denote the unique solution of (3.2).
Next, we prove that is bounded. Take , from (3.1) and using Lemma 2.5, we have
It follows that
Therefore, is bounded, and so are the nets and .
On the other hand, from (3.1) we obtain
Next, we show that is relatively norm-compact as . Assume that such that as . Put . It follows from (3.3) that
For a given , by (3.1) and using Lemma 2.5, we have
Since is bounded, without loss of generality, we may assume that converges weakly to a point . By (3.4) and using Lemma 2.2, we have . Then by (3.5), . This has proved the relative norm compactness of the net as .
We next show that solves variational inequality (3.2). Observe that
Since is accretive (this is due to the nonexpansiveness of T), for any , we can deduce immediately that
Therefore, for any ,
Now, replace t in (3.6) with . Noting that for as , we have
That is is a solution of (3.2), hence by uniqueness. In summary, we have shown that each cluster point of (as ) equals . Therefore, as . □
The framework of a Hilbert space is extended to a uniformly convex and 2-uniformly smooth Banach space.
The η-strongly monotone operator F is extended to the case of an η-strongly accretive operator . The contraction is extended to the case of a Lipschitzian and strongly pseudo-contractive operator .
Next we consider the following iteration process: the initial guess is selected in X arbitrarily and the th iterate is defined by
where is a contractive mapping with , is a sequence in satisfying conditions (C2) and
(C3′) with and .
Besides the basic condition (C2) on the sequence , we have the control condition (C3′). It can obviously be replaced by one of the following:
Indeed, (C3-1) implies (C3′) by choosing , and (C3-2) implies (C3′) by choosing . In this sense (C3′) is a weaker condition than the previous condition (C3).
Our second main result below shows that we have established a necessary and sufficient condition for the strong convergence of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space.
Theorem 3.3 Let be generated by algorithm (3.7) with the sequence of parameters satisfying conditions (C2) and (C3′). Then
where solves the variational inequality , .
Proof On the one hand, suppose that (). We proceed with the following steps.
Step 1. We claim that is bounded. In fact, taking , from (3.7) and using Lemma 2.5, we have
By induction, we have
Therefore, is bounded. We also obtain that and are bounded.
Step 2. We claim that . Observe that
where . By Lemma 2.3, we have .
Step 3. We claim that . Indeed, from Step 2, we have
Step 4. We claim that , where and is defined by . Since is bounded, there exists a subsequence of which converges weakly to ω. From Step 3, we obtain . From Lemma 2.2, we have . Since f is a contractive mapping, we have that γf is a Lipschitzian and strongly pseudo-contractive operator with . Hence, using Theorem 3.1, we have and
Step 5. We claim that converges strongly to . From (3.7) and using Lemma 2.5, we have
It follows that
where and . It is easy to see that and . Hence, by Lemma 2.3, the sequence converges strongly to . From and Theorem 3.1, we have that is the unique solution of the variational inequality , .
On the other hand, suppose that as , where is the unique solution of the variational inequality , . Observe that
This completes the proof. □
The framework of a Hilbert space is extended to a uniformly convex and 2-uniformly smooth Banach space.
The η-strongly monotone operator F is extended to the case of an η-strongly accretive operator .
We establish a necessary and sufficient condition for the strong convergence of nonexpansive mappings. It follows from (C1) that (). Hence, we can obtain Theorem 3.2 of Tian  immediately. Thus, our Theorem 3.3 covers Theorem 3.1 of Tian  as a special case.
The following example shows that all the conditions of Theorem 3.3 are satisfied. However, condition (C1) is not satisfied.
Example 3.5 Let be the set of real numbers. Define the mappings , and as follows:
It is easy to see that , and . By , we have and hence . Also, put . It is easy to see that . From , we have . Without loss of generality, we put . Given sequences and , , and for all . For an arbitrary , let be defined as
Observe that for all ,
Hence we have for all . This implies that converges strongly to .
Observe that , , that is, 0 is the solution of the variational inequality , .
Finally, we have
Hence there is no doubt that all the conditions of Theorem 3.3 are satisfied. Since , condition (C1): of Tian  is not satisfied.
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The authors would like to thank the referees for giving useful suggestions and comments for the improvement of this paper. This study was supported by the Natural Science Foundation of Jiangsu Province under Grant (13KJB110028), and the National Science Foundation of China (11271277).
The authors declare that they have no competing interests.
Both authors contributed equally and significantly to this research article. Both authors read and approved the final manuscript.