Abstract
In this paper, a regularization method for treating zero points of the sum of two monotone operators is investigated. Strong convergence theorems are established in the framework of Hilbert spaces.
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1 Introduction
In the real world, many important problems have reformulations which require finding zero points of some nonlinear operator, for instance, evolution equations, complementarity problems, mini-max problems, variational inequalities and optimization problems; see [1–13] and the references therein. It is well known that minimizing a convex function f can be reduced to finding zero points of the subdifferential mapping . Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two nonlinear operators. The central problem is to iteratively find a zero point of the sum of two monotone operators; that is, . Many problems can be formulated as a problem of the above form. For instance, a stationary solution to the initial value problem of the evolution equation , , can be recast as the above inclusion problem when the governing maximal monotone F is of the form ; for more details; see [14] and the references therein.
In this paper, we study a regularization method for treating zero points of the sum of an inverse-strongly monotone and a maximal monotone operator. The main contribution of the paper is establish a strong convergence theorem for viscosity zero points under mild restrictions imposed on the control sequences. The main results include the corresponding results in Xu [15] as a special case. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a regularization method is investigated. A strong convergence theorem for zero points of the sum operator is established. In Section 4, applications of the main results are discussed.
2 Preliminaries
In what follows, we always assume that H is a real Hilbert space with inner product and norm . Let C be a nonempty, closed and convex subset of H. Let be a mapping. stands for the fixed point set of S; that is, . Recall that S is said to be contractive iff there exists a constant such that
It is well known that every contractive mapping has a unique fixed point in metric spaces. S is said to be nonexpansive iff
If C is a bounded, closed, and convex subset of H, then is not empty, closed, and convex; see [16] and the references therein.
Let be a mapping. Recall that A is said to be monotone iff
Recall that A is said to be inverse-strongly monotone iff there exists a constant such that
For such a case, A is also said to be α-inverse-strongly monotone. It is not hard to see that every inverse-strongly monotone mapping is monotone and continuous.
Recall that a set-valued mapping is said to be monotone iff, for all , and imply . In this paper, we use to stand for the zero point of B. A monotone mapping is maximal iff the graph of B is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping B is maximal if and only if, for any , , for all implies . For a maximal monotone operator B on H, and , we may define the single-valued resolvent , where denote the domain of B. It is well known that is firmly nonexpansive, and .
Recently, many authors studied zero points of monotone operators based on different regularization methods; see [17–29] and the references therein. The main motivation is from Xu [15]. We propose a regularization method for treating zero points of the sum of two monotone operators. Strong convergence theorems are established in the framework of Hilbert spaces.
In order to prove our main results, we also need the following lemmas.
Lemma 2.1 [30]
Let be a mapping, and a maximal monotone operator. Then .
Lemma 2.2 [31]
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 [32]
Let H be a Hilbert space, and A a maximal monotone operator. For , , and , we have , where and .
3 Main results
Theorem 3.1 Let be an α-inverse-strongly monotone mapping and let B be a maximal monotone operator on H. Assume that and is not empty. Let be a fixed κ-contraction and let . Let be a sequence in C in the following process: and
where is a real number sequence in , is sequence in H and is a positive real number sequence in . If the control sequences satisfy the following restrictions:
-
(a)
, and ;
-
(b)
and ;
-
(c)
,
then converges strongly to a point , where .
Proof First, we show that is bounded. Notice that is nonexpansive. Indeed, we have
In view of the restriction (b), we find that is nonexpansive. Fixing , we find that
It follows that
This proves that the sequence is bounded, and so is . Notice that
Putting , we find that
It follows from Lemma 2.3 that
where
It follows from the restrictions (a), (b), and (c) that . In view of Lemma 2.2, we find that . In view of , we find from the above that
Next, we show that
where is the unique fixed point of the mapping . To show this inequality, we choose a subsequence of such that
Since is bounded, we find that there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that .
Now, we show that . Notice that ; that is,
Let . Since B is monotone, we find that
In view of the restriction (b), we see from (3.1) that . This implies that , that is, . This proves that (3.2) holds. Notice that
It follows that . On the other hand, we have
An application of Lemma 2.2 to the above inequality yields . This completes the proof. □
4 Applications
First, we consider the problem of finding a minimizer of a proper convex lower semicontinuous function.
For a proper lower semicontinuous convex function , the subdifferential mapping ∂g of g is defined by
Rockafellar [33] proved that ∂g is a maximal monotone operator. It is easy to verify that if and only if .
Theorem 4.1 Let be a proper convex lower semicontinuous function such that is not empty. Let be a κ-contraction and let be a sequence in H in the following process: and
where is a real number sequence in , is sequence in H and is a positive real number sequence. If the control sequences satisfy the following restrictions:
-
(a)
, and ;
-
(b)
;
-
(c)
,
then converges strongly to a point , where .
Proof Since is a proper convex and lower semicontinuous function, we see that subdifferential ∂g of g is maximal monotone. Noting that
is equivalent to
It follows that
Putting , we immediately derive from Theorem 3.1 the desired conclusion. □
Next, we consider the problem of finding a solution of a classical variational inequality.
Let C be a nonempty closed and convex subset of a Hilbert space H. Let be the indicator function of C, that is,
Since is a proper lower and semicontinuous convex function on H, the subdifferential of is maximal monotone. So, we can define the resolvent of for , i.e., . Letting , we find that
where is the metric projection from H onto C and .
Theorem 4.2 Let be an α-inverse-strongly monotone mapping. Assume that is not empty. Let be a fixed κ-contraction. Let be a sequence in C in the following process: and
where is a real number sequence in , is sequence in H and is a positive real number sequence in . If the control sequences satisfy the following restrictions:
-
(a)
, and ;
-
(b)
and ;
-
(c)
,
then converges strongly to a point , where .
Proof Putting in Theorem 3.1, we find that . We can draw the desired conclusion from Theorem 3.1.
Next, we consider the problem of finding a solution of a Ky Fan inequality, which is known as an equilibrium problem in the terminology of Blum and Oettli; see [34] and [35] and the references therein.
Let F be a bifunction of into ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem:
To study the equilibrium problem (4.1), we may assume that F satisfies the following restrictions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each , ;
(A4) for each , is convex and lower semicontinuous.
□
The following lemma can be found in [35].
Lemma 4.3 Let be a bifunction satisfying (A1)-(A4). Then, for any and , there exists such that , . Further, define
for all and . Then (1) is single-valued and firmly nonexpansive; (2) is closed and convex.
Lemma 4.4 [36]
Let F be a bifunction from to ℝ which satisfies (A1)-(A4), and let be a multivalued mapping of H into itself defined by
Then is a maximal monotone operator with the domain , , where stands for the solution set of (4.1), and , , , where is defined as in (4.2).
Theorem 4.5 Let be a bifunction satisfying (A1)-(A4). Assume that is not empty. Let be a fixed κ-contraction and let . Let be a sequence in C in the following process: and
where is a real number sequence in , is sequence in H and is a positive real number sequence. If the control sequences satisfy the following restrictions:
-
(a)
, and ;
-
(b)
and ;
-
(c)
,
then converges strongly to a point , where .
Proof Putting in Theorem 3.1, we find that . From Theorem 3.1, we can draw the desired conclusion immediately.
Recall that a mapping is said to be α-strictly pseudocontractive if there exists a constant such that
The class of strictly pseudocontractive mappings was first introduced by Browder and Petryshyn [37]. It is well known that if T is α-strictly-pseudocontractive, then is -inverse-strongly monotone. □
Finally, we consider fixed point problem of α-strictly pseudocontractive mappings.
Theorem 4.6 Let be an α-strictly pseudocontractive mapping with a nonempty fixed point set and let be a fixed κ-contraction. Let be a sequence generated in the following manner: and
where is a real number sequence in and is a positive real number sequence in . If the control sequences satisfy the following restrictions:
-
(a)
, and ;
-
(b)
and ;
then converges strongly to a point , where .
Proof Putting , we find A is -inverse-strongly monotone. We also have and . In view of Theorem 3.1, we obtain the desired result. □
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The main idea of this paper was proposed by XQ. SYC and LW participate the research and performed some steps of the proof in this research. All authors read and approved the final manuscript.
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Qin, X., Cho, S.Y. & Wang, L. A regularization method for treating zero points of the sum of two monotone operators. Fixed Point Theory Appl 2014, 75 (2014). https://doi.org/10.1186/1687-1812-2014-75
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DOI: https://doi.org/10.1186/1687-1812-2014-75