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 ).
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.
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 ).
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 ).
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 ).
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.
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).
We support our results by giving the following examples.
Doubly Picards iteration converges.
Doubly Mann's iteration converges.
4. Convergence to a Zero of Accretive Operator
Lemma 4.1 ( (the resolvent identity)).
5. Weakly Convergence Theorems
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