Proximal point algorithms for zero points of nonlinear operators
© Qing and Cho; licensee Springer. 2014
Received: 26 November 2013
Accepted: 4 February 2014
Published: 14 February 2014
A proximal point algorithm with double computational errors for treating zero points of accretive operators is investigated. Strong convergence theorems of zero points are established in a Banach space.
MSC:47H05, 47H09, 47H10, 65J15.
In this paper, we are concerned with the problem of finding zero points of an operator ; that is, finding such that . The domain domA of A is defined by the set . Many important problems have reformulations which require finding zero points, for instance, evolution equations, complementarity problems, mini-max problems, variational inequalities and optimization problems; see [1–20] and the references therein. One of the most popular techniques for solving the inclusion problem goes back to the work of Browder . One of the basic ideas in the case of a Hilbert space H is reducing the above inclusion problem to a fixed point problem of the operator defined by , which is called the classical resolvent of A. If A has some monotonicity conditions, the classical resolvent of A is with full domain and firmly nonexpansive. Rockafellar introduced the algorithm and call it the proximal point algorithm; for more detail, see  and the references therein. Regularization methods recently have been investigated for treating zero points of monotone operators; for [23–33] and the references therein. Methods for finding zero points of monotone mappings in the framework of Hilbert spaces are based on the good properties of the resolvent , but these properties are not available in the framework of Banach spaces.
In this paper, we investigate a proximal point algorithm with double computational errors based on regularization ideas in the framework of Banach spaces. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, strong convergence of the algorithm is obtained in a general Banach space. In Section 4, an application is provided to support the main results.
In what follows, we always assume that E is Banach space with the dual . Recall that a closed convex subset C of E is said to have 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. It is well known that a closed convex subset of uniformly convex Banach space has the normal structure and a compact convex subset of a Banach space has the normal structure; for more details, see  and the references therein.
for all . In the sequel, we use j to denote the single-valued normalized duality mapping. 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.
Let D be a nonempty subset of C. Let . Q is said to be a 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.
The following result, which was established in , describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
Q is sunny and nonexpansive;
, , .
Let I denote the identity operator on E. An operator with domain and 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 a real Hilbert space, an operator A is m-accretive if and only if A is maximal monotone. In this paper, we use to denote the set of zeros 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.
In order to prove our main results, we also need the following lemmas.
Lemma 2.1 
where and .
Lemma 2.2 
Let be a sequence of nonnegative numbers satisfying the condition , , where is a number sequence in such that and , is a number sequence such that , and is a positive number sequence such that . Then .
Lemma 2.3 
Lemma 2.4 
Let E a real reflexive Banach space with the uniformly Gâteaux differentiable norm and the normal structure, and C be a nonempty closed convex subset of E. Let be a nonexpansive mapping with a fixed point, and be a fixed contraction with the coefficient . Let be a sequence generated by the following , where . Then converges strongly as to a fixed point of S, which is the unique solution in to the following variational inequality , .
3 Main results
for each and .
Then the sequence converges strongly to , which is the unique solution to the following variational inequality , .
In view of Lemma 2.2, we find the desired conclusion immediately. □
If the mapping f maps any element in C into a fixed element u and , then we have the following result.
for each and .
Then the sequence converges strongly to , which is the unique solution to the following variational inequality: , .
Remark 3.3 We remark here that the algorithm (ϒ) is convergence under mild restrictions. However, it does not include the Halpern iterative algorithm as a special case because of the restriction (b). It is of interest to develop a different analysis technique for the algorithm without the restriction or under mild restrictions.
In this section, we give an application of Theorem 3.1 in the framework of Hilbert spaces.
Rockafellar  proved that ∂w is a maximal monotone operator. It is easy to verify that if and only if .
where is a sequence in H, is a bounded sequence in H, and is a positive real numbers sequence. Assume that the above control sequences satisfy the restrictions (a), (b), (d), and . Then the sequence converges strongly to , which is the unique solution to the following variational inequality: , .
Putting in Theorem 3.1, we draw the desired conclusion from Theorem 3.1 immediately. □
The authors are grateful to the reviewers for the useful suggestions which improved the contents of the article.
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