Abstract
We define a viscosity method for continuous pseudocontractive mappings defined on closed and convex subsets of reflexive Banach spaces with a uniformly Gâteaux differentiable norm. We prove the convergence of these schemes improving the main theorems in the work by Y. Yao et al. (2007) and H. Zhou (2008).
1. Introduction
Let be a real Banach space and let be the normalized duality mapping from into defined bywhere denotes the dual space of and the generalized duality pairing between and .
Recall that ifexists for each and on the unit sphere of , the norm of is Gâteaux differentiable. Moreover, if for each the limit defined by (1.2) is uniformly attained for , we say that the norm of is uniformly Gâteaux differentiable.
Definition 1.1. A mapping is said to be k-pseudocontractive () if, for every , there exist some such thatIn the inequality (1.3), if , we say that is strongly pseudocontractive. For is called pseudocontractive mapping.
Among classes of nonlinear mappings, the class of pseudocontractions is probably one of the most important classes of mappings. This happens because of the corresponding relation between the classes of pseudocontractions and accretive operators. In fact, a mapping is accretive (i.e., , for all ) if and only if is pseudocontractive.
Let be two opportune mappings from to , where is a closed and convex subset of a Banach space . Consider the variational inequality problem of finding a fixed point of , with respect to another mapping , to satisfy the inequalityA particular case occurs when with a -contraction (i.e., for all . In this case, the method (implicit or explicit) that permits to solve the variational inequality problem is known as viscosity approximation method. It was first studied by Moudafi [1] in Hilbert spaces and further developed by Xu [2] in more general setting.
Next results, due to Morales [3] (2007), are the more general results concerning the convergence of implicit viscosity methods for continuous pseudocontractive mappings.
In particular, the author studies the convergence of the path defined asin more setting of Banach spaces and in more large class of mappings including the -contraction mappings.
Theorem 1.2 (see [3]). Let be a nonempty closed convex subset of a reflexive Banach space with a uniformly Gâteaux differentiable norm. Let and be pseudocontractive and strongly pseudocontractive continuous mappings, respectively. Suppose that every closed, bounded, and convex subset of has the fixed point property for nonexpansive self-mappings. If the sets and are bounded, then the path described by strongly converges, as , to a fixed point of which is the unique solution of the variational inequality
Corollary 1.3 (see [3]). Let be a nonempty closed convex subset of a uniformly smooth Banach space . Let be a pseudocontractive and strongly pseudocontractive continuous mappings. If is bounded and admits at least a fixed point, then the path described by strongly converges, as , to a fixed point of that is the unique solution of the variational inequality
Also in 2007, H. Zegeye et al. in [4] proved a convergence theorem of viscosity approximation methods for continuous pseudocontractive mappings in reflexive and strictly convex Banach spaces.
Theorem 1.4 (see [4]). Let be a nonempty closed and convex subset of a real Banach space reflexive, strictly convex that has uniformly Gâteaux differentiable norm. Let be a continuous pseudocontractive mapping and be a -contraction. Suppose further that . Then, strongly converges, as , to a fixed point of that is the unique solution of the variational inequality (1.8).
Another interesting implicit-type Halpern algorithm has been recently introduced by Yao, Liou, and Chen in uniformly smooth Banach spaces.
Theorem 1.5 (see [5]). Let be a closed and convex subset of a real
uniformly smooth Banach space X. Let be a continuous pseudocontractive mapping. Let ,
and be three real sequences in satisfying the following conditions:
(i);(ii);(iii)Then, for
arbitrary initial value and a fixed ,
the sequence defined by strongly converges to a fixed
point of .
In our first result (Theorem 2.1), we prove the strong convergence of the viscosity implicit approximation methodwhere is a continuous strongly pseudocontractive mapping. This result has as particular case Theorem 1.5 when is a constant mapping.
On the other hand, on the idea of the implicit scheme (1.11), Zhou in [6] defines a Halpern explicit method for suitable continuous pseudocontractive mappings. Fixing an element and an initial point , he constructs elements as follows:If for any the continuous pseudocontractive mapping admits an integer that satisfies the following condition:then he defines iteratively a sequence as follows: called the least positive integer satisfying (1.14),and he proves the convergence for this explicit method. Of course if , one reobtains the implicit method (1.11).
In our second theorem, we improve Zhou's result [6] to the viscosity setting. In both proofs, we use Morales's Theorem 1.2.
Let us conclude this section by two lemmas that are useful in many convergence results.
Following the proof of Theorem 2.3 in [7], one can show the following.
Lemma 1.6. Let be a nonempty closed convex subset of a real
Banach space with a uniformly Gâteaux differentiable norm,
let be a pseudocontractive and strongly
pseudocontractive continuous mappings with .
Let be a bounded sequence such that .
Define, for all , and let us suppose that exists.
Then,
The following lemma on real sequences can be found in Liu [8].
Lemma 1.7. Let be a sequence of nonnegative real numbers
satisfying the following inequality: where is a sequence in such that and is a summable sequence of positive numbers.
Then, converges to zero.
2. Convergence Results
In this section, we prove the convergence's theorems on implicit and explicit viscosity method.
Theorem 2.1. Let be a nonempty closed convex subset of a
reflexive Banach space with a uniformly Gâteaux differentiable norm.
Suppose that every closed, bounded, and convex subset of has the fixed point property for nonexpansive
self-mappings. Let be a continuous pseudocontractive mapping and
let be a continuous strongly pseudocontractive
mapping (with constant ) such that the sets and are bounded.
Let ,
and be three real sequences in satisfying the following conditions:
(i);(ii);(iii)For arbitrary
initial point, and a fixed ,
we construct elements as follows: Then, strongly converges to ,
where is the unique solution of (1.8)
Proof. First of all, from [3], it follows that .
Now, we verify that the sequence exists.
We prove that, for fixed with and ,
the maphas a unique fixed point. By
Deimling [9], it is
enough to show that is strongly pseudocontractive and continuous.
Now,Since ,
then
is a strongly pseudocontractive. To prove the
claim of the theorem, we show firstly that is bounded.
Picking ,
we havewhich implies
thatBy a simple induction, we get
thatMoreover, we have that .
In fact,and by boundedness of and condition (ii), it follows the statement.
Let, for every ,By Morales's Theorem 1.2, this
implicit method converges to a unique point that is the unique solution of (1.8). Next, we
show that .
By Lemma 1.6, we obtainthen, if we define the real
sequencewe can show that and .
So we concludewhich implies
thatBy Liu's Lemma 1.7 and condition
(iii), we obtain that ,
as .
In the next theorem, we consider a viscosity explicit method which extends (1.15) substituting the constant with a -contraction , and we establish a convergence's result for this scheme.
Theorem 2.2. Let be a nonempty closed convex subset of a
reflexive Banach space with a uniformly Gâteaux differentiable norm.
Suppose that every closed, bounded, and convex subset of has the fixed point property for nonexpansive
self-mappings. Let be a continuous pseudocontractive mapping and
let be a -contraction such that the set is bounded.
Let ,
and be three real sequences in satisfying the following conditions:
(i);(ii);(iii)Let be a summable sequence of positive numbers.
For arbitrary initial point and a fixed ,
we construct elements as follows: Suppose that there exists ,
the least positive integer satisfying the following condition: Then, defined as strongly converges to ,
where is the unique solution of (1.8),
Proof. We divide the proof into three
steps.
Step 1. is bounded.
Proof of Step 1. Picking ,
we havewhich implies thatBy a simple induction, we get
that
Step 2. .
Proof of Step 2. Sinceby the boundedness of ,
condition (ii), and the summability of ,
we obtain the claim.
Step 3. .
Proof of Step . As in Theorem 2.1, setwhere ,
andWe known that ;
now we show that .
In fact,which implies
thatBy Liu's Lemma 1.7, we obtain
that ,
as .
Remark 2.3. We can prove that if is a nonexpansive mapping and is defined as (2.32) of Theorem 2.2, then there always exists a positive integer satisfyingIn fact, fixed , for every , we haveIf , we are done. Otherwise, since , it follows that there exists a sufficiently large such that
It is also well known [3, 5, 10] that if is a continuous pseudocontractive mapping, defining the mapping as , we can observe that the following hold:
(1) is a nonexpansive mapping;(2);(3), for all .
By Remark 2.3 and Theorem 2.2, we have the following.
Corollary 2.4. Let be a nonempty closed convex subset of a real
reflexive Banach space with a uniformly Gâteaux differentiable norm.
Suppose that every closed, bounded, and convex subset of has the fixed point property for nonexpansive
self-mappings. Let be a continuous pseudocontractive mapping and
let be a -contraction such that the set is bounded.
Let ,
and be three real sequences in satisfying the following conditions:
(i);(ii);(iii)
Let be a summable sequence of positive numbers.
For arbitrary initial point and a fixed ,
we construct elements as follows (here, as above, ): and we define as where for every is the positive integer such that Then, converges strongly to ,
where is the unique solution of (1.8),
Acknowledgment
This work was supported in part by Ministero dell'Universitá e della Ricerca of Italy.