Strong convergence theorems for common zeros of a family of accretive operators
© Huang and Ma; licensee Springer. 2014
Received: 1 January 2014
Accepted: 25 March 2014
Published: 6 May 2014
In this paper, common zeros of a family of accretive operators are investigated based on the Kirk-like proximal point algorithm. A strong convergence theorem is established in a reflexive Banach space.
In the real world, many important problems have reformulations which require finding common zero (fixed) points of nonlinear operators, for instance, image recovery, inverse problems, transportation problems and optimization problems. It is well known that the convex feasibility problem is a special case of the common zero (fixed) points of nonlinear operators. In 1971, Kirk  introduced a parallel iterative process for finding a family of nonexpansive mappings. Common fixed point theorems were established in a Banach space; for more details, see . For studying zero points of monotone operators, the most well-known algorithm is the proximal point algorithm; see [2, 3] and the references therein. It is known that Rockfellar’s proximal point algorithm is, in general, weak convergence; see  and the references therein.
Recently, many authors have been devoted to investigating the strong convergence of a proximal point algorithm. Strong convergence theorems for zero points of accretive operators were established; see, for example, [5–29] and the references therein.
In this paper, we are concerned with the problem of finding a common zero of a family of accretive operators based on the Kirk-like proximal point algorithm. A strong convergence theorem is established in a reflexive Banach space.
Let . E is said to be smooth or to have a Gâteaux differentiable norm if the limit 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 to have a uniformly Fréchet differentiable norm if the limit is attained uniformly for .
It is well known that (uniform) Fréchet differentiability of the norm of E implies (uniform) Gâteaux differentiability of the norm of E. 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. In the sequel, we use j to denote the single-valued normalized duality mapping.
for , and implies that .
Recall that a closed convex subset C of a Banach space E is said to have a normal structure if for each bounded closed convex subset K of C which contains at least two points, there exists an element x of K which is not a diametral point of K, i.e., , where is the diameter of K.
Let C be a nonempty closed convex subset of E. Let be a mapping. In this paper, we use to denote the set of fixed points of T. Recall that T is said to be nonexpansive iff , . For the existence of fixed points of a nonexpansive mapping, we refer readers to .
Let I denote the identity operator on E. An operator with the domain and the range is said to be accretive if for each and , , there exists such that . An accretive operator A is said to be m-accretive if for all . In this paper, we use to denote the set of zero points of A. For an accretive operator A, we can define a nonexpansive single-valued mapping by for each , which is called the resolvent of A.
Next we give the following lemmas which play an important role in this article.
Lemma 2.1 
Lemma 2.2 
Let C be a closed convex subset of a strictly convex Banach space E. Let and be two nonexpansive mappings. Suppose that is nonempty. Then the mapping , where is a real number, is well defined nonexpansive with .
Lemma 2.3 
Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm and the normal structure, and let C be a nonempty closed convex subset of E. Let be a nonexpansive mapping with a fixed point. Let be a sequence generated by the following , where and is a fixed element. Then converges strongly as to a fixed point of T, which is the unique solution in to the following variational inequality , .
Lemma 2.4 
Let , , and be three nonnegative real sequences satisfying , , where is some positive integer, is a number sequence in such that , is a number sequence such that , and is a positive number sequence such that . Then .
3 Main results
where u is a fixed element in C and . Then the sequence converges strongly to , which is the unique solution to the following variational inequality , .
Proof The proof is split into five steps.
Step 1. Show that is bounded.
This proves Step 1.
Step 2. Show that .
where . In view of Lemma 2.4, we conclude Step 2.
we conclude Step 3.
Since ϵ is arbitrarily chosen, we find that . This implies that . This proves Step 4.
Step 5. Show that as .
where . We, therefore, find that . From Lemma 2.4, we find the desired conclusion. This proves the proof. □
Remark 3.2 Theorem 3.1 is still valid in the framework of the space which is uniformly convex and the norm is uniformly Gâteaux differentiable.
Now, we are in a position to give the result on the variational inequality.
where u is a fixed element in C. Then the sequence converges strongly to , which is the unique solution to the following variational inequality , .
that is, . This implies that . In light of Theorem 3.1, we draw the desired conclusion immediately. □
The authors are grateful to the reviewers for their suggestions which improved the contents of the article.
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