We prove the equivalence and the strong convergence of iterative processes involving generalized strongly asymptotically -pseudocontractive mappings in uniformly smooth Banach spaces.
Throughout this paper, we assume that is a uniformly convex Banach space and is the dual space of . Let denote the normalized duality mapping form into given by for all , where denotes the generalized duality pairing. It is well known that if is uniformly smooth, then is single valued and is norm to norm uniformly continuous on any bounded subset of . In the sequel, we will denote the single valued duality mapping by .
In 1967, Browder  and Kato , independently, introduced accretive operators (see, for details, Chidume ). Their interest is connected with the existence of results in the theory of nonlinear equations of evolution in Banach spaces.
In 1972, Goebel and Kirk  introduced the class of asymptotically nonexpansive mappings as follows.
Definition 1.1. Let be a subset of a Banach space . A mapping is said to be asymptotically nonexpansive if for each
where is a sequence of real numbers converging to 1.
This class is more general than the class of nonexpansive mappings as the following example clearly shows.
Example 1.2 (see ). If is the unit ball of and is defined as
where is such that , it satisfies.
In 1974, Deimling , studying the zeros of accretive operators, introduced the class of -strongly accretive operators.
Definition 1.3. An operator defined on a subset of a Banach space is said, -strongly accretive if
where is a strictly increasing function such that .
Note that in the special case in which , , we obtain a strongly accretive operator.
Osilike , among the others, proved that in is -strongly accretive where but not strongly accretive.
Since an operator is a strongly accretive operator if and only if is a strongly pseudocontractive mapping (i.e., , ), taking in to account Definition 1.3, it is natural to study the class of -pseudocontractive mappings, that is, the maps such that
where is a strictly increasing function such that . Of course, the set of fixed points for these mappings contains, at most, only one point.
Recently, has been also studied the following class of maps.
Definition 1.4. A mapping is a generalized -strongly pseudocontractive mapping if
where is a strictly increasing function such that .
Choosing , we obtain Definition 1.3. In , Xiang remarked that it is a open problem if every generalized -strongly pseudocontractive mapping is -pseudocontractive mapping. In the same paper, Xiang obtained a fixed-point theorem for continuous and generalized -strongly pseudocontractive mappings in the setting of the Banach spaces.
In 1991, Schu  introduced the class of asymptotically pseudocontractive mappings.
Definition 1.5 (see ). Let be a normed space, and . A mapping is said to be asymptotically pseudocontractive with the sequence if and only if , and for all and all , there exists such that
where is the normalized duality mapping.
Obviously every asymptotically nonexpansive mapping is asymptotically pseudocontractive, but the converse is not valid; it is well known that defined by is not Lipschitz but asymptotically pseudocontractive .
Theorem 1.6 (see ). Let be a Hilbert space and closed and convex; ; completely continuous, uniformly Lipschitzian, and asymptotically pseudocontractive with sequence ; for all ; ; ; for all , some and some ; ; for all , define
then converges strongly to some fixed point of .
Until 2009, no results on fixed-point theorems for asymptotically pseudocontractive mappings have been proved. First, Zhou in  completed this lack in the setting of Hilbert spaces proving a fixed-point theorem for an asymptotically pseudocontractive mapping that is also uniformly -Lipschitzian and uniformly asymptotically regular and that the set of fixed points of is closed and convex. Moreover, Zhou proved the strong convergence of a CQ-iterative method involving this kind of mappings.
In this paper, our attention is on the class of the generalized strongly asymptotically -pseudocontraction defined as follows.
Definition 1.7. If is a Banach space and is a subset of , a mapping is said to be a generalized asymptotically -strongly pseudocontraction if
where is converging to one and is strictly increasing and such that .
One can note that (i)if has fixed points, then it is unique. In fact, if are fixed points for , then for every ,
so passing to , it results that
Since is strictly increasing and , then . (ii)the mapping , where , is generalized asymptotically strongly -pseudocontraction with , for all and . However, is not strongly pseudocontractive; see .
We study the equivalence between three kinds of iterative methods involving the generalized asymptotically strongly -pseudocontractions.
Moreover, we prove that these methods are equivalent and strongly convergent to the unique fixed point of the generalized strongly asymptotically -pseudocontraction , under suitable hypotheses.
We will briefly introduce some of the results in the same line of ours. In 2001,  Chidume and Osilike proved the strong convergence of the iterative method
where , ( a -strongly accretive operator), and , to a solution of the equation .
In 2003, Chidume and Zegeye  studied the following iterative method:
where is a Lipschitzian pseudocontractive map with fixed points. The authors proved the strong convergence of the method to a fixed point of under suitable hypotheses on the control sequences .
Taking in to account Chidume and Zegeye  and Chang , we introduce the modified Mann and Ishikawa iterative processes as follows: for any given , the sequence is defined by
where , , , and are four sequences in satisfying the conditions and for all .
In particular, if for all , we can define a sequence by
which is called the modified Mann iteration sequence.
We also introduce an implicit iterative process as follows:
where are two real sequences in satisfying and for all , is a sequence in , and is an initial point.
The algorithm is well defined. Indeed, if is a asymptotically strongly -pseudocontraction, one can observe that, for every fixed , the mapping defined by is such that
that is, is a strongly pseudocontraction, for every fixed , then (see Theorem 13.1 in ) there exists a unique fixed point of for each .
These kind of iterative processes (also called by Chang iterative processes with errors) have been developed in [15–18], while equivalence theorem for Mann and Ishikawa methods has been studied, in [19, 20], among the others.
In , Huang established equivalences between convergence of the modified Mann iteration process with errors (1.15) and convergence of modified Ishikawa iteration process with errors (1.14) for strongly successively -pseudocontractive mappings in uniformly smooth Banach space.
In the next section, we prove that, in the setting of the uniformly smooth Banach space, if is an asymptotically strongly -pseudocontraction, not only (1.14) and (1.15) are equivalent but also (1.16) is equivalent to the others. Moreover, we prove also that (1.14), (1.15), and (1.16) strongly converge to the unique fixed point of , if it exists.
We recall some definitions and conclusions.
Definition 2.1. is said to be a uniformly smooth Banach space if the smooth module of
Lemma 2.2 (see ). Let be a Banach space, and let be the normalized duality mapping, then for any , one has
The next lemma is one of the main tools for our proofs.
Lemma 2.3 (see ). Let be a strictly increasing function with , and let , and be nonnegative real sequences such that
Suppose that there exists an integer such that
Proof. The proof is the same as in , but we substitute with , in (2.4).
Lemma 2.4 (see ). Let , , , and be sequences such that
for all . Assume that , then the following results hold:(1)if (where ), then is a bounded sequence;(2)if one has and , then as .
Remark 2.5. If in Lemma 2.3 choosing , for all , (), then the inequality (2.4) becomes
Setting and and by the hypotheses of Lemma 2.3, we get as , , and . That is, we reobtain Lemma 2.4 in the case of .
3. Main Results
The ideas of the proofs of our main Theorems take in to account the papers of Chang and Chidume et al. [11, 13, 24].
Theorem 3.1. Let be a uniformly smooth Banach space, and let be generalized strongly asymptotically -pseudocontractive mapping with fixed point and bounded range. Let and be the sequences defined by (1.14) and (1.15), respectively, where ,,, satisfy(H1) and ,(H2),and the sequences ,, are bounded in , then for any initial point , the following two assertions are equivalent: (i)the modified Ishikawa iteration sequence with errors (1.14) converges to ; (ii)the modified Mann iteration sequence with errors (1.15) converges to .
Proof. First of all, we note that by boundedness of the range of , of the sequences and by Lemma 2.4, it results that and are bounded sequences. So, we can set
By Lemma 2.2, we have
where . Using (1.14) and (1.15), we have
In view of the uniformly continuity of , we obtain that as . Furthermore, it follows from the definition of that for all
Therefore, we have
where . By (H1), we have that as . If for an infinite number of indices, we can extract a subsequence such that . For this subsequence, , as . In this case, we can prove that , that is, the thesis. Firstly, we note that substituting (3.4) into (3.2), we have
where . Moreover, we observe that
Thus, for every fixed , there exists such that for all
Since , , , , and are null sequences (and in particular ), for the previous fixed , there exists an index such that, for all ,
for all . Take such that for a certain . We prove, by induction, that , for every . Let . Suppose that . By (3.6), we have
Thus, . Since is strictly increasing, . From (3.8), we obtain that
One can note that
In the same manner,
So we have , which contradicts . By the same idea, we can prove that and then, by inductive step, , for all . This is enough to ensure that . If there are only finite indices for which , then definitively . By the strict increasing function , we have definitively
Again substituting (3.4) and (3.18) into (3.2) and simplifying, we have
Suppose that . It follows from , , , and the hypothesis that we have, and , as . By virtue of Lemma 2.3, we obtain that . Hence, .
Theorem 3.2. Let be a uniformly smooth Banach space, and let be generalized strongly asymptotically -pseudocontractive mapping with fixed point and bounded range. Let and be the sequences defined by (1.15) and (1.16), respectively, where , are null sequences satisfying (H1) and , (H2), of Theorem 3.1 and such that , for every . Suppose moreover that the sequences , are bounded in , then for any initial point , the following two assertions are equivalent: (i)the modified Mann iteration sequence with errors (1.15) converges to the fixed point , (ii)the implicit iteration sequence with errors (1.16) converges to the fixed point .
Proof. As in Theorem 3.1, by the boundedness of the range of and by Lemma 2.4, one obtains that our schemes are bounded. We define
By the iteration schemes (1.15) and (1.16), we have
where . By (1.15), we get
It follows from (H1) that as , which implies that as . Moreover, for all ,
Again by the boundedness of all components, we have that
Hence, we have that , where . Note that as . As in proof of Theorem 3.1, we distinguish two cases:(i)the set of indices for which contains infinite terms; (ii)the set of indices for which contains finite terms. In the first case, (i) we can extract a subsequence such that , as . Substituting (3.23) in (3.21), we have that
where . Again by (3.23), for every , there exists an index such that if ,
By hypotheses on the control sequence, with the same , there exists an index such that definitively
So take with for a certain . We can prove that as proving that, for every , the result is . Let . If we suppose that , it results that
so . In consequence of this, . In (3.26), we note that
hence in (3.26) remains
as in Theorem 3.1. This is a contradiction. By the same idea, and using the inductive hypothesis, we obtain that , for every . This ensures that . In the second case (ii), definitively, , then from the strictly increasing function , we have
Substituting (3.33) and (3.23) into (3.21) and simplifying, we have
By virtue of Lemma 2.3, we obtain that .
Theorem 3.3. Let be a uniformly smooth Banach space, and let be generalized strongly asymptotically -pseudocontractive mapping with fixed point and bounded range. Let be the sequences defined by (1.15) where satisfy(i), (ii), .and the sequence is bounded on , then for any initial point , the sequence strongly converges to .
Proof. Firstly, we observe that, by the boundedness of the range of , of the sequence , and by Lemma 2.4, we have that is bounded. By Lemma 2.2, we observe that
where . Let
so we can observe that(1) as . Indeed from the inequality
and since is norm to norm uniformly continuous, then , as , (2). Indeed, if we supposed that , by the monotonicity of ,
Thus, by (1) and by the hypotheses on and , the value is definitively negative. In this case, we conclude that there exists such that for every ,
In the same way we obtain that