The mixed viscosity approximation is proposed for finding fixed points of nonexpansive mappings, and the strong convergence of the scheme to a fixed point of the nonexpansive mapping is proved in a real Banach space with uniformly Gâteaux differentiable norm. The theorem about Halpern type approximation for nonexpansive mappings is shown also. Our theorems extend and improve the correspondingly results shown recently.
1. Introduction and Preliminaries
Let be a real Banach space with norm , denote the dual space of , and denote the generalized duality pairing. Let denote the normalized duality mapping defined by
It is well known that the following results: , , ,
Let be the unit sphere of Banach space , the norm of is said to be Gâteaux differentiable if the limit
exists for each . Such an is called a smooth Banach space. The norm of Banach space is said to be uniformly Gâteaux differentiable if for each , the limit is attained uniformly for . A Banach space is said to be strictly convex if
to be uniformly convex if for all , such that
It is well known that (see [1, 2]): (1) if has a uniformly Gâteaux differentiable norm, then is norm-to-weak continuous on bounded set of . (2) If a Banach space admits a sequentially continuous duality mapping from weak topology to weak star topology, then the duality mapping is single-valued. (3) Each uniformly convex Banach space is reflexive and strictly convex and has fixed point property for nonexpansive self-mappings; every uniformly smooth Banach space is a reflexive Banach space with a uniformly Gâteaux differentiable norm and has fixed point property for nonexpansive self-mappings.
Let be a nonempty closed convex subset of a Banach space . If is a nonempty subset of , then a mapping is said to be a retraction if for all . A mapping is said to be a sunny if where . A subset of is said to be a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction of onto . For more details, see [2]. Note that every closed convex subset of a Hilbert space is a nonexpansive retract. In the sequel, we always take to denote the sunny nonexpansive retraction of onto .
Let be a nonexpansive mapping; for a sequence and a fixed contractive mapping , the sequence , iteratively defined in by
is said to be viscosity approximation. If for a given , it is called Halpern approximation which was first introduced by Halpern [3] in 1967. Under the following assumption:
Xu [4] proved the strong convergence of to a fixed point of in Hilbert spaces and in uniformly smooth Banach spaces in 2004. In [3], Halpern pointed out that the condition (i) is necessary for the convergence of the Halpern approximation to a fixed point of . At the same time, he put forth the following open problem: is the condition (i) a sufficient condition for the convergence of the Halpern approximation to a fixed point of ? which was put forward by Reich in [5] also. In order to answer the open question, many authors have done extensively some works; see [6–11] and the references therein. In [7–9], the strong convergence of the Halpern approximation depends on the convergence of the path . In [6], Song got rid of the dependence on the convergence of the path , and proved the convergence of the Halpern approximation under the assumptions for and as follows:
where and is a Banach limit. Recently, many authors have studied extensively the problem of approximating a fixed point of nonexpansive nonself-mappings in a Banach space, by using the Halpern type iteration (see [12–16]) and the viscosity type iteration (see [17–23]).
In this work, on one hand, we will prove the strong convergence of the mixed viscosity iterative scheme, which is introduced as follows: for any chosen ,
where is a fixed contractive mapping, , , and are the real number sequences in , to a fixed point of the nonexpansive mapping in a real Banach space with uniformly Gâteaux differentiable norm under the following conditions:
where , is a Banach limit, and is defined by (1.9). On the other hand, we will show that the condition is not necessary for proving the strong convergence of the mixed viscosity iterative scheme. As the applications, we will show some results about mixed viscosity type approximation and Halpern type approximation for nonexpansive nonself-mappings also. Our theorems complement and generalize the corresponding results in [8, 9, 16, 24–26].
Now, we recall the following lemmas for proving our theorems firstly.
Lemma 1.1 (see [27]). Let and be bounded sequences in a Banach space , a sequence satisfying
Suppose that , for all , and
Then .
Lemma 1.2 (see [28]). Let , and, be sequences of nonnegative real numbers such that If ,
then .
2. Main Results
In this section, the mixed viscosity iterations for a contractive self-mapping for approximating to a fixed point of nonexpansive mapping are studied in a real Banach space.
Theorem 2.1. Let be a nonempty closed convex subset of a real Banach space . Let be a nonexpansive mapping with and a fixed contractive mapping with the contractive coefficient . If the sequences , , and in satisfy () and for any , the sequence is defined as follows:
then we obtain the following: (1) is bounded;(2) and
Proof. First we show that is bounded. Now let , then
Therefore, is bounded, so are , , , and . Next, we show that
For all , let
where , then
Note that
Hence, we have that
Then it follows from the boundedness of , , , and , and () that
It follow from (2.9) and Lemma 1.1 that
Since
from (), (2.11) and, the boundedness of , , and , we have
From
we obtain
Therefore, This completes the proof.
Proposition 2.2 (see [18]). Let be a nonempty closed convex subset of a real Banach space which has uniformly Gâteaux differentiable norm. Suppose that is a bounded sequence of such that and is a Banach limit. If such that , then
Let be defined by (2.1) and , it follows from Theorem 2.1 that is bounded. Let
then is convex and continuous. If is reflexive, there exists such that (see [2], Theorem ). Let
then is a closed convex subset in a reflexive Banach space .
Theorem 2.3. Let be a nonempty closed convex subset of a real Banach space which has uniformly Gâteaux differentiable norm. Suppose that is a nonexpansive mapping with , is a fixed contractive mapping with the contractive coefficient , and the sequences , and in satisfy (). If , then the sequence defined by (2.1) converges strongly to a fixed point of as .
Proof. Take . It follows from Theorem 2.1 that is bounded and For , by Proposition 2.2, we have that
Next, we show that . Since
then
Hence,
Let and ; it follows from Lemma 1.2 that converges strongly to . This completes the proof.
Proposition 2.4. Let be a real reflexive Banach space with uniformly Gâteaux differentiable norm, a nonempty closed convex subset of , a nonexpansive mapping with , and defined by (2.1). Then .
Proof. From the reflexivity of and the definition of , it follows that is a nonempty closed convex subset of . By Theorem 2.1, we know that
Claim that . Indeed, for any , we have
Therefore, and .
Since , there exists unique such that , for all . By and , we have
Hence by the uniqueness of . Thus .
By the above results, we can obtain the following theorem.
Theorem 2.5. Let be a real reflexive Banach space with uniformly Gâteaux differentiable norm, a nonempty closed convex subset of , a nonexpansive mapping with , and a fixed contractive mapping with the contractive coefficient . If the sequences , , and in satisfy (), then the sequence defined by (2.1) converges strongly to a fixed point of .
Remark 2.6. Theorem 2.5 shakes off the assumption in [26] and extends Theorem in [24] shown in uniformly smooth Banach spaces.
Theorem 2.7. Let be a real strictly convex Banach space with uniformly Gâteaux differentiable norm and a nonempty closed convex subset of which is a sunny nonexpansive retract of . Let be a nonexpansive nonself-mapping with and a fixed contractive mapping with the contractive coefficient . Suppose that the sequences , , and in satisfy (), is a sunny nonexpansive retract of , and the sequence is defined as follows:
If , then the sequence converges strongly to a fixed point of as .
Proof. It follows from [12, Lemmas 3.1 and 3.3] that , and then . By replacing by in Theorem 2.3, we can show that the conclusion holds.
Theorem 2.8. Let be a real reflexive and strictly convex Banach space with uniformly Gâteaux differentiable norm, a nonempty closed convex subset of a real Banach space which is a sunny nonexpansive retract of , a nonexpansive nonself-mapping with , and a fixed contractive mapping with the contractive coefficient . If the sequences , and in satisfy (), then the sequence defined by (2.26) converges strongly to a fixed point of .
Proof. It follows from [12, Lemmas and ] that , and then . By replacing by in Theorem 2.5, we can show that the conclusion holds.
Corollary 2.9. Let be a real uniformly convex Banach space with uniformly Gâteaux differentiable norm, , , , , and as Theorem 2.8. Then the sequence defined by (2.1) converges strongly to a fixed point of .
3. Some Applications
In this section, we introduce the following Halpern type approximation: for the given , the sequence is defined by