- Research Article
- Open access
- Published:
Degree of Convergence of Iterative Algorithms for Boundedly Lipschitzian Strong Pseudocontractions
Fixed Point Theory and Applications volume 2010, Article number: 210340 (2010)
Abstract
Let be a nonempty closed convex subset of a real Hilbert space
and let
be a boundedly Lipschitzian strong pseudo-contraction with a nonempty fixed point set. Three iterative algorithms are proposed for approximating the unique fixed point of
; one of them is for the self-mapping case, and the others are for the nonself-mapping case. Not only the strong convergence, but also the degree of convergence of the three iterative algorithms is obtained. Some numerical results corresponding to the self-mapping case are given which show advantages of our methods. As an application of our results, adopting the regularization idea, we also propose implicit and explicit algorithms for approximating a fixed point of a boundedly Lipschitzian pseudocontractive self-mapping from
into itself, respectively.
1. Introduction and Preliminaries
Let be a real Hilbert space with inner product
and norm
, and let
be a nonempty closed convex subset of
. Recall that a mapping
is said to be pseudo-contractive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ1_HTML.gif)
for every .
is said to be strongly pseudo-contractive if there exists a positive constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ2_HTML.gif)
for all . In this case, we also call
a
-strong pseudocontraction. Using (1.2), it is easy to see that every strong pseudocontraction has at most one fixed point.
is said to be Lipschitzian if there exists a positive constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ3_HTML.gif)
for all . In this case,
is also said to be
-Lipschitzian. In particular,
is said to be nonexpansive if
; and it is said to be contractive if
.
is said to be boundedly Lipschitzian if, for each bounded subset
of
, there exists a positive constant
depending only on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ4_HTML.gif)
for all .
We will denote by the set of fixed points of
, that is,
. Let
be a sequence and
a point in
. Then we use
and
to denote strong and weak convergence to
of the sequence
, respectively.
Among classes of nonlinear mappings, the class of pseudocontractions is one of the most important classes of mappings. This is mainly due to the fact that there is a precise corresponding relation between the class of pseudocontractions and the class of monotone mappings. A mapping is monotone (i.e.,
for all
) if and only if
is pseudo-contractive, where
and
denotes the identity mapping on
.
Within the past 40 years or so, mathematicians have been devoting their study to the existence and iterative construction of fixed points for pseudocontractions and of zeros for monotone mappings (see, e.g., [1–18]). However, most of these algorithms have no estimation of degree of convergence even if for strong pseudocontractions in setting of Hilbert spaces. Everyone knows that it is very important to get the degree of convergence for an algorithm in computing science.
The main purpose of this paper is to consider the iterative algorithms for approximating the unique fixed point (if the set of fixed points is not empty) of a boundedly Lipschitzian strong pseudocontraction defined on a nonempty closed convex subset of a real Hilbert space. Three iterative algorithms are proposed; one of them is for the self-mapping case, and the others are for the nonself-mapping case. Not only the strong convergence, but also the degree of convergence of the three iterative algorithms is obtained. Some numerical results corresponding to the self-mapping case are given which show advantages of our methods. As an application of our results, adopting the regularization idea, we also establish implicit and explicit algorithms for approximating a fixed point of a boundedly Lipschitzian pseudo-contractive self-mapping from into itself, respectively.
In order to give our main results, let us recall a basic existence result for fixed points for continuous strong pseudocontractions which was proved by Deimling [6] in 1974.
Theorem 1.1 (Deimling [6]).
Let be a closed subset of a real Banach space
, and let
be a continuous
-strong pseudocontraction, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ5_HTML.gif)
for each , where
denotes the distance from the point
to the subset
of
. Then
has a unique fixed point.
Corollary 1.2 (see [6]).
Let be a closed convex subset of a real Banach space
, and let
be a continuous
-strong pseudocontraction, then
has a unique fixed point.
We also need some facts which are listed as lemmas below.
Lemma 1.3 (see [7]).
Let be a real Hilbert space. Given a closed and convex subset
of
and points
and given also a real number
such that
, then the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ6_HTML.gif)
is closed and convex.
Lemma 1.4 (see, e.g., [9]).
Let be a closed convex subset of a real Hilbert space
, and let
be the (metric or nearest point) projection from
onto
(i.e., for
,
is the only point in
such that
). Given
and
, then
if and only if there holds the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ7_HTML.gif)
Lemma 1.5 (see [18]).
Let be a nonempty closed convex subset of a real Hilbert space
and
a demicontinuous pseudo-contractive self-mapping from
into itself. Then
is a closed convex subset of
and
is demiclosed at zero.
Now we are in a position to prove main results in this paper.
2. Algorithms for Strongly Pseudocontractive Self-Mappings
In this section, we propose an iterative algorithm for boundedly Lipschitzian and strongly pseudo-contractive self-mappings. Since the algorithm has nothing to do with the metric projection, it is easy to realize in practical computing.
Theorem 2.1.
Let be a nonempty closed convex subset of a real Hilbert space
, and let
be a boundedly Lipschitzian and
strong pseudocontraction. Take
arbitrarily, and let
and
. Define
recursively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ8_HTML.gif)
where is a constant such that
and
is the bounded Lipschitz constant of
upon
. Then
converges strongly to the unique fixed point
of
, and the estimation of degree of convergence is as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ9_HTML.gif)
In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.
Proof.
Firstly, it concludes by using Corollary 1.2 that has a unique fixed point, denoted by
, in
. We also assert that
holds. Indeed, since
is a
-strong pseudocontraction, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ10_HTML.gif)
holds for all . Taking
and
in (2.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ11_HTML.gif)
That is, .
Now we prove by mathematical induction that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ12_HTML.gif)
and hold for all
. For
, observing that
,
is
-Lipschitzian restricted to
, and
is
-strongly pseudo-contractive, it is easy to get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ13_HTML.gif)
hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ14_HTML.gif)
Noting that the condition implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ15_HTML.gif)
we have from (2.4), (2.7), and (2.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ16_HTML.gif)
That is, .
Suppose that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ17_HTML.gif)
Similar to (2.7), we have from that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ18_HTML.gif)
Thus (2.11) together with (2.10) leads to (2.5). On the other hand, we have from (2.4), (2.5), and (2.8) that , that is,
.
By (2.7), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ19_HTML.gif)
Consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ20_HTML.gif)
Thus (2.2) is obtained by using (2.5) and (2.13).
Finally, we show that (2.2) is the optimal estimation of degree of convergence in the sense of ignoring constant factors. For this purpose, it suffices to find an example such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ21_HTML.gif)
Indeed, taking with the usual inner product and norm and taking
, let
be a rotation operator defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ22_HTML.gif)
where such that
. Obviously,
has a unique fixed point
. Moreover,
is nonexpansive, that is, Lipschitz constant
. Since
is a linear operator, using (2.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ23_HTML.gif)
hence is a
-strong pseudocontraction.
Taking an initial value and a control parameter
such that
, it follows from direct calculating that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ24_HTML.gif)
This shows that the estimation (2.2) cannot be improved.
Remark 2.2.
If , then
reaches the minimum
when
, so
is said to be optimal control parameter of process (2.1). If
, then it is not difficult to verify that the optimal control parameter is zero. The same result also applies to all of the following algorithms.
If is Lipschitzian on the whole
, that is, there exists a positive constant
such that
for all
, then we obtain the following result as a special case of Theorem 2.1.
Theorem 2.3.
Let be a nonempty closed convex subset of a real Hilbert space
, and let
be an
-Lipschitzian and
-strong pseudocontraction. Take an initial guess
arbitrarily, and define a sequence
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ25_HTML.gif)
where is a constant such that
. Then
converges strongly to the unique fixed point
, and the estimation of degree of convergence is obtained as follow:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ26_HTML.gif)
In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.
In order to test the computing effect of the algorithm (2.1), some numerical results for the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ27_HTML.gif)
are given as follows. Using the mean value theorem, it is easy to verify that is a
-strongly pseudo-contractive and boundedly Lipcshitzian function. For each constant
, the bounded Lipschitz constant of
upon the interval
is
. Choosing the initial guess
in (2.1), it follows by using Theorem 2.1 that
,
,
, and the control parameter
such that
. Since we do not know the exact fixed point
of
, we propose the relative rate of convergence
to test the computing effect of algorithm (2.1) for
. All the numerical results are in Tables 1–3.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_IEq185_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_IEq196_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_IEq207_HTML.gif)
3. Algorithms for Strongly Pseudo-Contractive Nonself-Mappings
In this section, we turn to designing two iterative algorithms for boundedly Lipschitzian and strongly pseudo-contractive nonself-mappings. In this case, a boundedly Lipschitzian strong pseudocontraction may not have a fixed point, so we assume that the mapping has a unique fixed point (noting that each strong pseudocontraction has at most one fixed point). In addition, we will have to use the metric projection in the algorithms.
In fact, the first algorithm is a modification of process (2.1) as follows. We omit its proof, which is very similar to the proof of Theorem 2.1.
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
, and let
be a boundedly Lipschitzian and
strong pseudocontraction with a unique fixed point. Take
arbitrarily, and let
and
. Define a sequence
via the recursive formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ28_HTML.gif)
where is a constant such that
and
is the bounded Lipschitz constant of
upon
. Then
converges strongly to the unique fixed point
of
, and the estimation of degree of convergence is as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ29_HTML.gif)
In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.
If is Lipschitzian on the whole
, we have the following result as a special case of Theorem 3.1.
Theorem 3.2.
Let be a nonempty closed convex subset of a real Hilbert space
, and let
be an
-Lipschitzian and
-strong pseudocontraction. Let
has a unique fixed point
. Take an initial guess
arbitrarily, and define
recursively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ30_HTML.gif)
where is a constant such that
. Then
converges strongly to the unique fixed point
, and the estimation of degree of convergence is as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ31_HTML.gif)
In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.
Now we give the second algorithm for the strongly pseudo-contractive nonself-mapping case.
Theorem 3.3.
Let be a nonempty closed convex subset of a real Hilbert space
, and let
be a boundedly Lipschitzian and
strong pseudocontraction
. Let
have a unique fixed point
. Take
arbitrarily, and let
. Define a sequence
of
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ32_HTML.gif)
where is the bounded Lipschitz constant of
upon
and
is a constant such that
. Then
converges strongly to
. One also has the estimation of degree of convergence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ33_HTML.gif)
In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.
Proof.
By the proof of Theorem 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ34_HTML.gif)
that is, .
Now we verify by mathematical induction that and the sequence
generated by (3.5) is well defined for each
. For
, observing that
, and
is a
-strong pseudocontraction, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ35_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ36_HTML.gif)
and this means . Noting that the condition
implies that
, and by using Lemma 1.3,
is nonempty, closed, and convex. Using Lemma 1.4, there exists a unique element
such that
. Suppose that
has been obtained and
for some
. Likewise, observing that
, and
is a
-strong pseudocontraction, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ37_HTML.gif)
The definition of and (3.10) imply that
. Using Lemma 1.3 again, the condition
guarantees that
is nonempty, closed, and convex. So there exists a unique element
such that
.
Finally, we prove that (3.6) holds and converges strongly to
. Observing process (3.5),
, and
for all
, we have from an argument similar to getting (3.10) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ38_HTML.gif)
By induction step, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ39_HTML.gif)
By (3.11) and triangular inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ40_HTML.gif)
hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ41_HTML.gif)
Thus the combination of (3.12) and (3.14) leads to (3.6), and converges strongly to
due to the fact that
.
By the same argument in the proof of Theorem 2.1, we assert that (3.6) is the optimal estimation of degree of convergence.
Remark 3.4.
The formulation of process (3.1) is simpler than that of process (3.5). But process (3.5) is believed to have faster rate of convergence than that of process (3.1) due to the fact that , in general.
4. Algorithms for Pseudo-Contractive Self-Mappings
Let be a nonempty closed convex subset of a real Hilbert space
, and let
be a boundedly Lipschitzian pseudocontraction with a nonempty fixed point set
, that is,
. It follows from Lemma 1.5 that
is closed and convex, so the metric projection operator
is well defined.
In this section, adopting the regularization idea, we propose implicit and explicit algorithms for approximating a fixed point of , respectively. More precisely, given an arbitrary element
, for each
, it is easy to show that
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ42_HTML.gif)
is a boundedly Lipschitzian -strong pseudocontraction. Then we have from Corollary 1.2 that
has a unique fixed point. Denote by
the unique fixed point of
. Namely,
is the only solution of the fixed point equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ43_HTML.gif)
Firstly, we prove that converges strongly to a fixed point of
, as
. Next, we give our explicit method based on this implicit method and Theorem 2.1.
Theorem 4.1.
Let be a nonempty closed convex subset of a real Hilbert space
, and let
be a boundedly Lipschitzian pseudocontraction with a nonempty fixed point set
. Let
, and
is determined by (4.2). Then
is bounded, and
. Moreover,
.
Proof.
First we show that is bounded. Take
arbitrarily; noting that
is a pseudocontraction, we have from (4.2) and the fact
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ44_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ45_HTML.gif)
Clearly, . This says that
is bounded. Since
is boundedly Lipschitzian, so it is not difficult to show that
is also bounded. Thus we can assert that the set of weak cluster points
, where
.
Next, we prove ; namely, if
is a null sequence in
such that
as
, then
. To see this, using (4.2), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ46_HTML.gif)
Clearly, this together with the boundedness of implies that
as
. Using Lemma 1.5, we have
, that is,
. Taking
and
in (4.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ47_HTML.gif)
Taking the limit as , we see that
.
Finally, we turn to proving that . Since
is monotone, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ48_HTML.gif)
Observing , it follows from (4.2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ49_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ50_HTML.gif)
Since is continuous and
, we obtain by taking the limit that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ51_HTML.gif)
By Lemma 1.4, we get . This means that
. Thus we have proved that
.
Our following explicit method is motivated by Theorem 4.1, Theorem 2.1, and Zhou's iterative method in [17]. Given a sequence such that
as
. Denote by
the unique fixed point of the mapping
. Namely,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ52_HTML.gif)
Theorem 4.1 says that . Observe that
is boundedly Lipschitzian and
-strongly pseudo-contractive.
Theorem 4.2.
Let be a nonempty closed convex subset of a real Hilbert space
, and let
be a boundedly Lipschitzian pseudocontraction with a nonempty fixed point set
. Let
,
such that
as
. For arbitrary initial datum
. Define iteratively a sequence
in an explicit manner as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ53_HTML.gif)
where is the least positive integer satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ54_HTML.gif)
is the bounded Lipschitz constant of
upon
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ55_HTML.gif)
and the control parameter sequence such that
. Then
converges strongly to
.
Proof.
Recalling that, for each ,
is a boundedly Lipschitzian
-strong pseudocontraction, we have by using Theorem 2.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ56_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ57_HTML.gif)
Since the condition implies
, so there exists a least positive integer
satisfying condition (4.13). Using Theorem 4.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ58_HTML.gif)
In order to complete the proof, it suffices to show that as
. To this end, we estimate
. By the proof of Theorem 2.1, we assert that
and
for all
. Thus we have from (4.11)–(4.16) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F210340/MediaObjects/13663_2010_Article_1226_Equ59_HTML.gif)
Hence as
, since
as
. Consequently,
.
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This paper is supported by Fundamental Research Funds for the Central Universities (Grant no. ZXH2009D021).
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He, S., Su, Y. & Li, R. Degree of Convergence of Iterative Algorithms for Boundedly Lipschitzian Strong Pseudocontractions. Fixed Point Theory Appl 2010, 210340 (2010). https://doi.org/10.1155/2010/210340
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DOI: https://doi.org/10.1155/2010/210340