On an Iterative Method for Finding a Zero to the Sum of Two Maximal Monotone Operators

In this paper we consider a problem that consists of finding a zero to the sum of two monotone operators. One method for solving such a problem is the forward-backward splittingmethod.We present some new conditions that guarantee the weak convergence of the forward-backward method. Applications of these results, including variational inequalities and gradient projection algorithms, are also considered.


Introduction
It is well known that monotone inclusions problems play an important role in the theory of nonlinear analysis.This problem consists of finding a zero of maximal monotone operators.However, in some examples such as convex programming and variational inequality problems, the operator is needed to be decomposed of the sum of two monotone operators (see, e.g., [1][2][3][4][5][6]).In this way, one needs to find  ∈ H so that where  and  are two monotone operators on a Hilbert space H.To solve such problem, the splitting method, such as Peaceman-Rachford algorithm [7] and Douglas-Rachford algorithm [8], is usually considered.We consider a special case whenever  : H → 2 H is multivalued and  : H → H is single-valued.A classical way to solve problem (1) under our assumption is the forward-backward splitting (FBS) (see [2,9]).Starting with an arbitrary initial  0 ∈ H, the FBS generates a sequence (  ) satisfying where  is some properly chosen real number.Then the FBS converges weakly to a solution of problem (1) whenever such point exists.
On the other hand, we observe that problem (1) is equivalent to the fixed point equation: for the single-valued operator ( + ) −1 ( − ).Moreover, if  is properly chosen, the operator ( + ) −1 ( − ) should be nonexpansive.Motivated by this assumption, by using the techniques of the fixed point theory for nonexpansive operators, we try to investigate and study various monotone inclusion problems.The rest of this paper is organized as follows.In Section 2, some useful lemmas are introduced.In Section 3, we consider the modified forward-backward splitting method and prove its weak convergence under some new conditions.In Section 4, some applications of our results in finding a solution of the variational inequality problem are included.

Preliminary and Notation
Throughout the paper,  denotes the identity operator, Fix() the set of the fixed points of an operator , and ∇ the gradient of the functional  : H → R. The notation " → " denotes strong convergence and "⇀" weak convergence.Denote by   (  ) the set of the cluster points of (  ) in the weak topology (i.e., the set { : ∃   ⇀ }, where (   ) means a subsequence of (  )).
The following lemma is known as the demiclosedness principle for nonexpansive mappings.

Lemma 2. Let 𝐶 be a nonempty closed convex subset of H and 𝑆 a nonexpansive operator with
-inverse strongly monotone (-ism), if there exists a constant  > 0 so that and maximal monotone if it is monotone and its graph () = {(, ) :  ∈ } is not properly contained in the graph of any other monotone operator.
In what follows, we shall assume that (i)  : H → H is single-valued and -ism; (ii)  : H → 2 H is multivalued and maximal monotone.
Hereafter, if no confusion occurs, we denote by the resolvent of  for any given  > 0. It is known that   is single-valued and firmly nonexpansive; moreover dom( + ) = H (see [11]).

2
Journal of Applied MathematicsLet  be a nonempty closed convex subset of H. Denote by   the projection from H onto ; namely, for  ∈ H,    is the unique point in  with the property 2(∀,  ∈ H) ; Definition 4. Assume that (  ) is a sequence in H and that (  ) is a real sequence with ∑    < ∞.Then (  ) is called quasi Fejér monotone w.r.t., if      +1 −      ≤       −      +   (∀ ∈ ) .