Iterative algorithms with errors for zero points of m-accretive operators
© Qin et al.; licensee Springer. 2013
Received: 20 March 2013
Accepted: 21 May 2013
Published: 6 June 2013
In this paper, we study the convergence of paths for continuous pseudocontractions in a real Banach space. As an application, we consider the problem of finding zeros of m-accretive operators based on an iterative algorithm with errors. Strong convergence theorems for zeros of m-accretive operators are established in a real Banach space.
MSC:47H05, 47H09, 47J25, 65J15.
1 Introduction and preliminaries
exists for each . E is said to have a uniformly Gâteaux differentiable norm if for each , the limit is attained uniformly for all . E is said to be uniformly smooth or is said to have a uniformly Fréchet differentiable norm if the limit is attained uniformly for . It is known that if the norm of E is uniformly Gâteaux differentiable, then the duality mapping J is single-valued and uniformly norm to weak∗ continuous on each bounded subset of E.
where is the diameter of K. It is well known that a closed convex subset of a uniformly convex Banach space has the normal structure and a compact convex subset of a Banach space has the normal structure; see  for more details.
In the case that T enjoys a fixed point, Browder  proved that if E is a Hilbert space, then converges strongly to a fixed point of T. Reich  extended Browder’s result to the setting of Banach spaces and proved that if E is a uniformly smooth Banach space, then converges strongly to a fixed point of T.
Let D be a nonempty subset of C. Let . Q is said to be contraction if ; sunny if for each and , we have ; sunny nonexpansive retraction if Q is sunny, nonexpansive, and contraction. K is said to be a nonexpansive retract of C if there exists a nonexpansive retraction from C onto D.
Q is sunny and nonexpansive;
, , .
Can one extend the framework of the space from uniformly smooth Banach spaces to a more general Banach space?
Can one extend the viscosity approximation method by considering strong pseudocontractions instead of contractions?
Do Xu’s results still hold for a larger class of nonlinear mappings?
In Section 2, we give an affirmative answer to the above questions.
whose solutions correspond to the equilibrium points of the system (1.1). Consequently, considerable research efforts have been devoted, especially within the past 40 years or so, to methods for finding approximate solutions (when they exist) of equation (1.2). An early fundamental result in the theory of accretive operators, due to Browder , states that the initial value problem (1.1) is solvable if A is locally Lipschitz and accretive on E.
For an accretive operator A, we can define a nonexpansive single-valued mapping by for each , which is called the resolvent of A.
where and is a sequence of positive real numbers.
As we know, many well-known problems arising in various branches of science can be studied by using algorithms which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set in which the required solution lies. The problem of finding a point in the intersection of the sets is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such a point.
where is a sequence in the interval . If T is a nonexpansive mapping with a fixed point and the control sequence is chosen so that , then the sequence generated in the normal Mann iterative process converges weakly to a fixed point of T. In an infinite-dimensional Hilbert space, the normal Mann iteration process has only weak convergence, in general, even for nonexpansive mappings. Therefore, many authors try to modify the normal Mann iterative process to have strong convergence for nonexpansive mappings; see, e.g., [10–24] and the references therein.
where u is a fixed element in C and . They proved that the sequence generated in the above iterative process converges strongly to a zero of A.
In this paper, we study the convergence of paths for continuous pseudocontractions in a real Banach space by viscosity approximation methods. As applications, we consider the problem of finding zeros of m-accretive operators based on an iterative process with errors. Strong convergence theorems of zeros are established in a real Banach space.
In order to prove our main results, we also need the following lemmas.
Lemma 1.1 
Lemma 1.2 
Let C be a nonempty, bounded, closed, and convex subset of a reflexive Banach space E which also has the normal structure. Let T be a nonexpansive mapping of C into itself. Then is nonempty.
Lemma 1.3 
Let C be a nonempty, closed, and convex subset of a Banach space E, and let be a continuous and strong pseudocontraction. Then T has a unique fixed point in C.
Lemma 1.4 
Set , let denote the Banach space of all bounded real-valued functions on A with a supremum norm and let X be a subspace of .
Lemma 1.5 
Let C be a nonempty, closed, and convex subset of a Banach space E. Suppose that the norm of E is uniformly Gâteaux differentiable. Let be a bounded set in E, and . Let be a mean on X. Then if and only if for all .
Lemma 1.6 
where and .
Lemma 1.7 
2 Main results
Now, we are in a position to prove the strong convergence of paths for continuous pseudocontractions.
Then D is a nonempty bounded closed convex subset of C. Next, we show that there exits a point such that .
This implies that there exists a subnet of such that .
This completes the proof. □
improves the framework of spaces from uniformly smooth Banach spaces to the Banach space with the uniformly Gâteaux differentiable norm;
improves the mapping f from the class of contractions to the class of strongly pseudocontractions;
improves the mapping T from the class of nonexpansive mappings to the class of pseudocontractions.
It is not hard to see that Q is a sunny nonexpansive retraction from onto .
Next, we prove strong convergence of iterative processes with errors for m-accretive operators.
for each ;
for each and .
Put , , and for every . In view of Lemma 1.1, we can obtain the desired conclusion easily. □
As an application of Theorem 2.5, we have the following results.
for each and .
Then the sequence converges strongly to a zero of A.
Remark 2.7 Corollary 2.6 improves Theorem 2.1 of Qin and Su . To be more precise, we partially relax the restrictions on the parameters and extend the framework of the space; see  for more details.
Remark 2.8 It is of interest to design an explicit iterative process to approximate zeros of accretive operators by Moudafi’s viscosity approximation method with continuous strong pseudocontractions.
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