Construction of Fixed Points by Some Iterative Schemes
© A. El-Sayed Ahmed and A. Kamal. 2009
Received: 23 October 2008
Accepted: 23 February 2009
Published: 8 March 2009
We obtain strong convergence theorems of two modifications of Mann iteration processes with errors in the doubly sequence setting. Furthermore, we establish some weakly convergence theorems for doubly sequence Mann's iteration scheme with errors in a uniformly convex Banach space by a Frechét differentiable norm.
Reich  proved that if is a uniformly convex Banach space with a Frechét differentiable norm and if is chosen such that then the sequence defined by (1.1) converges weakly to a fixed point of However, this scheme has only weak convergence even in a Hilbert space (see ). Some attempts to modify Mann's iteration method (1.1) so that strong convergence is guaranteed have recently been made.
where denotes the metric projection from onto a closed convex subset of . They proved that if the sequence is bounded from one, then defined by (1.2) converges strongly to Their argument does not work outside the Hilbert space setting. Also, at each iteration step, an additional projection is needed to calculate.
where is an arbitrary (but fixed) element in , and and are two sequences in It is proved, under certain appropriate assumptions on the sequences and that defined by (1.3) converges to a fixed point of (see ).
The second modification of Mann's iteration method (1.1) is an adaption to (1.3) for finding a zero of an -accretive operator , for which we assume that the zero set
where for each is the resolvent of . In , it is proved, in a uniformly smooth Banach space and under certain appropriate assumptions on the sequences and , that defined by (1.4) converges strongly to a zero of
Now, we define Opial's condition in the sense of doubly sequence.
where denotes that converges weakly to
We are going to work in uniformly smooth Banach spaces that can be characterized by duality mappings as follows (see  for more details).
Lemma 2.2 (see ).
A Banach space is uniformly smooth if and only if the duality map is single-valued and norm-to-norm uniformly continuous on bounded sets of
Lemma 2.3 (see ).
If and are nonempty subsets of a Banach space such that is a nonempty closed convex subset and then the map is called a retraction from onto provided for all A retraction is sunny [1, 4] provided for all and whenever A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive. A sunny nonexpansive retraction plays an important role in our argument.
Lemma 2.4 (see ).
where and are such that
Then, converges to zero.
Lemma 2.6 (see ).
Assume that has a weakly continuous duality map with gauge . Then, is demiclosed in the sense that is closed in the product space , where is equipped with the norm topology and with the weak topology. That is, if then
Lemma 2.7 (see ).
Let be a Banach space and Then,
where the closed ball of centered at the origin with radius r, and
where is a constant.
Lemma 2.8 (see ).
Let be a closed convex subset of a uniformly convex Banach space with a Fréchet differentiable norm, and let be a sequence of nonexpansive self mapping of with a nonempty common fixed point set If and for then exists for all In particular, where and are weak limit points of
Lemma 2.9 (the demiclosedness principle of nonexpansive mappings ).
In 2005, Kim and Xu , proved the following theorem.
Let be a closed convex subset of a uniformly smooth Banach space , and let be a nonexpansive mapping such that Given a point and given sequences and in the following conditions are satisfied.
Then is strongly converges to a fixed point of .
Recently, the study of fixed points by doubly Mann iteration process began by Moore (see ). In [15, 16], we introduced the concept of Mann-type doubly sequence iteration with errors, then we obtained some fixed point theorems for some different classes of mappings. In this paper, we will continue our study in the doubly sequence setting. We propose two modifications of the doubly Mann iteration process with errors in uniformly smooth Banach spaces: one for nonexpansive mappings and the other for the resolvent of accretive operators. The two modified doubly Mann iterations are proved to have strong convergence. Also, we append this paper by obtaining weak convergence theorems for Mann's doubly sequence iteration with errors in a uniformly convex Banach space by a Fréchet differentiable norm. Our results in this paper extend, generalize, and improve a lot of known results (see, e.g., [4, 7, 8, 17]). Our generalizations and improvements are in the use of doubly sequence settings as well as by adding the error part in the iteration processes.
3. A Fixed Point of Nonexpansive Mappings
The advantage of this modification is that not only strong convergence is guaranteed, but also computations of iteration processes are not substantially increased.
Now, we will generalize and extend Theorem A by using scheme (3.1).
Let be a closed convex subset of a uniformly smooth Banach space and let be a nonexpansive mapping such that Given a point and given sequences and in the following conditions are satisfied.
Define a sequence in by (3.1). Then, converges strongly to a fixed point of
Hence, by assumptions (i)-(ii), we obtain
Now we apply Lemma 2.5, and using (3.11) we obtain that
We support our results by giving the following examples.
Let be given by Then, the modified doubly Mann's iteration process with errors converges to the fixed point , and both Picard and Mann iteration processes converge to the same point too.
Doubly Picards iteration converges.
Doubly Mann's iteration converges.
- (III)Modified doubly Mann's iteration process with errors converges because the sequence as we can see and by using (3.1), we obtain(3.25)
Let and using Theorem 3.1 ( we obtain
Let be given by Then the doubly Mann's iteration converges to the fixed point of but modified doubly Mann's iteration process with errors does not converge.
- (I)Doubly Mann's iteration converges because the sequence as we can see,(3.27)
The last inequality is true because for all and
(II)The origin is the unique fixed point of
Letting we deduce that
4. Convergence to a Zero of Accretive Operator
Lemma 4.1 ( (the resolvent identity)).
Suppose and satisfy the conditions,
Then, converges strongly to a zero of
where is a constant such that for all and By assumptions (i)–(iii) in the theorem, we have that and Hence, Lemma 2.5 is applicable to (4.12), and we conclude that
Now we apply Lemma 2.5 and using (4.20), we obtain that
5. Weakly Convergence Theorems
Let be a uniformly convex Banach space with a Frechét differentiable norm. Assume that has a weakly continuous duality map with gauge . Assume also that
Then, the scheme (5.1) converges weakly to a point in
First, we observe that for any , the sequence is nonincreasing.
In particular, is bounded, so is . Let be the set of weak limit point of the sequence
By Lemma 2.6, we conclude that that is, .
Let be a uniformly convex Banach space which either has a Frechét differentiable norm or satisfies Opial's property. Assume for some
Then, the scheme (5.1) converges weakly to a point in
For any fixed by Lemma 4.1, we have
Apply Lemma 2.9 to find out that It remains to show that is a singleton set. Towards this end, we take and distinguish the two cases.
In either case, we have shown that consists of exact one point, which is clearly the weak limit of
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