Abstract
In this paper, we investigate a splitting algorithm for treating monotone operators. Strong convergence theorems are established in the framework of Hilbert spaces.
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1 Introduction and preliminaries
In this article, we always assume that H is a real Hilbert space with inner product and norm and C is a nonempty, closed, and convex subset of H.
Let be a mapping. stands for the fixed point set of S. 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 [1] and the references therein. S is said to be strictly pseudocontractive iff there exists a constant such that
The class of strictly pseudocontractive mapping was introduced by Browder and Petryshyn [2]. It is clear that the class of strictly pseudocontractive mappings include the class of nonexpansive mappings as a special case. It is also not hard to see that strictly pseudocontractive mapping is continuous.
Let be a mapping. Recall that A is said to be monotone iff
A is said to be strongly monotone iff there exists a constant such that
A is said to be inverse-strongly monotone iff there exists a constant such that
A is inverse-strongly monotone iff the inverse of A is strongly monotone. It is not hard to see that every inverse-strongly monotone mapping is monotone and continuous. Let I be the identity mapping on H. From [2], we know that is inverse-strongly monotone iff S is strictly pseudocontractive; for more details, see [2] and the references therein.
The classical variational inequality problem is formulated as finding a point such that
Such a point is called a solution of the variational inequality. In this paper, we use to denote the solution set of the variational inequality. It is known that x is a solution of the variational inequality iff x is a fixed point of the mapping , where is the metric projection from H onto C, I is the identity and r is some positive real number. Recently, many authors studied solutions of inverse-strongly monotone variational inequalities based on the equivalence; see [3–13].
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 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 known that is firmly nonexpansive, and .
One of the most important techniques for solving zero point problem of monotone operators goes back to the work of Browder [14]. Many important problems have reformulations which require finding zero points, for instance, evolution equations, complementarity problems, mini-max problems, variational inequalities and fixed point problems. It is well known that minimizing a convex function f can be reduced to finding zero points of the subdifferential mapping . 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. The property of the resolvent ensures that the Picard iterative algorithm converge weakly to a fixed point of , which is necessarily a zero point of A. Rockafellar introduced this iteration method and call it the proximal point algorithm (PPA); for more details, see [15] and [16] and the references therein.
It is known that PPA is only convergent and it was also pointed in [17] that it is often impractical since, in many cases, to solve the fixed point problem exactly is either impossible or of the same difficult as the original zero point problem. Therefore, one of the most interesting and important problems in the theory of monotone operators is to find an efficient iterative algorithm to compute their zero points. In many disciplines, including economics [18], image recovery [19], quantum physics [20], and control theory [21], problems arises in infinite dimension spaces. In such problems, strong convergence (norm convergence) is often much more desirable than weak convergence, for it translates the physically tangible property that the energy of the error between the iterate and the solution x eventually becomes arbitrarily small. The important of strong convergence is also underlined in [22], where a convex function f is minimized via the proximal point algorithm: it is shown that the rate of convergence of the value sequence is better when converges strongly that it converges weakly. Such properties have a direct impact when the process is executed directly in the underlying infinite dimensional space.
To improve the weak convergence of PPA, many authors considered lots of different modifications; see [23–36] the references therein. One of the classic results was established by Solodov and Svaiter [33]. They obtained strong convergence theorems in Hilbert space without any compact assumption but with the aid of the metric projection.
In this paper, we are concerned with the problem of finding an element in the zero point set of the sum of two operators which are inverse-strongly monotone and a maximal monotone and in the fixed point set of a mapping which is strictly pseudocontractive. Strong convergence theorems are established without the aid of the metric projections. The organization of this paper is as follows. In Section 1, we provide an introduction and some necessary preliminaries. In Section 2, a regularization iterative algorithm is investigated. A strong convergence theorem is established without the aid of metric projections. In Section 3, applications of the main results are discussed.
In order to prove our main results, we also need the following lemmas.
Lemma 1.1 [36]
Let be a mapping, and a maximal monotone operator. Then .
Lemma 1.2 [37]
Let E be a Banach space and let A be an m-accretive operator. For , , and , we have , where and .
Lemma 1.3 [38]
Let and be bounded sequences in a Banach space E, and be a sequence in with . Suppose that , and
Then .
Lemma 1.4 [39]
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 1.5 [40]
Let be a strictly pseudocontractive mapping with the constant . Then S is Lipschitz continuous and is demiclosed at zero. Define a mapping by for each . Then, as , T is nonexpansive such that .
2 Convergence analysis
Theorem 2.1 Let be an α-inverse-strongly monotone mapping and let B be a maximal monotone operator on H. Let be a strictly pseudocontractive mapping with the constant and let be a contractive mapping with the constant . Assume that and is not empty. Let and let be a sequence generated in the following process: and
where , and are real number sequences in and is a positive real number sequence in . Assume that the control sequences satisfy the following restrictions:
-
(a)
, ;
-
(b)
;
-
(c)
, ;
-
(d)
and ;
-
(e)
,
where a, b and c are three real numbers. 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 (d), we find that is nonexpansive. Fixing , we find that
Putting for each , we see from Lemma 1.5 that is nonexpansive with for each . It follows that
This proves that the sequence is bounded, so are and . Notice that
Putting , we find that
It follows from Lemma 1.2 that
where
This implies that
In view of the restrictions (a), (c), (d), and (e), we find that
It follows from Lemma 1.3 that
This in turn implies that
Notice that
where . It follows that
In view of the restrictions (a), (b), (c), (d), and (e), we find from (2.2) that
Since is firmly nonexpansive, we find that
It follows that
This implies that
It follows that
In view of the restrictions (a), (b), and (e), we find from (2.2) and (2.3) that . This in turn implies that
Notice that
It follows from (2.4) that
On the other hand, we have
Since S is Lipschitz continuous, we find from (2.1) and (2.5) that
Notice that
That is,
In view of (2.5) and (2.6), we find from the restriction (c) that
Since is contractive, we see that there exists a unique fixed point, say . Next, we show that . To show it, we can choose a subsequence of such that
Since is bounded, we can choose a subsequence of which converges weakly to some point x. We may assume, without loss of generality, that converges weakly to x. In view of (2.4), we find that also converges weakly to x. It follows from Lemma 1.5 that .
Now, we are in a position to show that . Notice that . It follows that
That is,
Since B is monotone, we get, for any ,
Replacing n by and letting , we obtain from (2.4) that
This gives , that is, . This proves that . This complete the proof that . It follows that
Finally, we show that . Notice that
This implies that
It follows that
where . It follows from (2.8) that
In view of the restriction (a), (b), and (e), we find from Lemma 1.4 that . This completes the proof. □
If S is nonexpansive and , then we have the following result immediately.
Corollary 2.2 Let be an α-inverse-strongly monotone mapping and let B be a maximal monotone operator on H. Let be a nonexpansive mapping and let be a contractive mapping with the constant . Assume that and is not empty. Let and let be a sequence generated in the following process: and
where and are real number sequences in and is a positive real number sequence in . Assume that the control sequences satisfy the following restrictions:
-
(a)
, ;
-
(b)
;
-
(c)
and ;
-
(d)
,
where b and c are two real numbers. Then converges strongly to a point , where .
3 Applications
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 real word 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 inclusion problem when the governing maximal monotone F is of the form ; for more details, see [41] and the references therein.
First, we give the following result.
Theorem 3.1 Let be an α-inverse-strongly monotone mapping and let B be a maximal monotone operator on H. Let be a contractive mapping with the constant . Assume that and is not empty. Let and let be a sequence generated in the following process: and
where and are real number sequences in and is a positive real number sequence in . Assume that the control sequences satisfy the following restrictions:
-
(a)
, ;
-
(b)
;
-
(c)
and ;
-
(d)
,
where b and c are two real numbers. Then converges strongly to a point , where .
Proof Put , the identity on H. The desired conclusion can be obtained immediately.
Let H be a Hilbert space and a proper convex lower semicontinuous function. Then the subdifferential ∂f of f is defined as follows:
From Rockafellar [16], we know that ∂f is maximal monotone. It is easy to verify that if and only if . Let be the indicator function of C, i.e.,
Since is a proper lower semicontinuous convex function on H, we see that the subdifferential of is a maximal monotone operator. Then , , . □
Theorem 3.2 Let be an α-inverse-strongly monotone mapping. Let be a strictly pseudocontractive mapping with the constant and let be a contractive mapping with the constant . Assume that is not empty. Let be a sequence generated in the following process: and
where , and are real number sequences in and is a positive real number sequence in . Assume that the control sequences satisfy the following restrictions:
-
(a)
, ;
-
(b)
;
-
(c)
, ;
-
(d)
and ;
-
(e)
,
where a, b, and c are three real numbers. Then converges strongly to a point , where .
Proof Put . Next, we show that . Notice that
We can conclude the desired conclusion immediately. □
If , the identity on H, then we find from Theorem 3.1 the following result immediately.
Corollary 3.3 Let be an α-inverse-strongly monotone mapping. Let be a strictly pseudocontractive mapping with the constant and let be a contractive mapping with the constant . Assume that is not empty. Let be a sequence generated in the following process: and
where and are real number sequences in and is a positive real number sequence in . Assume that the control sequences satisfy the following restrictions:
-
(a)
, ;
-
(b)
;
-
(c)
and ;
-
(d)
,
where b and c are three real numbers. Then converges strongly to a point , where .
Let F be a bifunction of into ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem:
In this paper, we use to denote the solution set of the equilibrium problem (3.2).
To study the equilibrium problems (3.2), we may assume that F satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each ,
(A4) for each , is convex and weakly lower semicontinuous.
Putting for every , we see that the equilibrium problem (3.2) is reduced to a variational inequality.
Let C be a nonempty closed convex subset of H and let be a bifunction satisfying (A1)-(A4). Then, for any and , there exists such that
Further, define
for all and . Then, the following hold:
-
(a)
is single-valued;
-
(b)
is firmly nonexpansive, i.e., for any ,
-
(c)
;
-
(d)
is closed and convex.
Lemma 3.5 [44]
Let C be a nonempty closed convex subset of a real Hilbert space H, F a bifunction from to ℝ which satisfies (A1)-(A4) and a multivalued mapping of H into itself defined by
Then is a maximal monotone operator with the domain , and
where is defined as in (3.3).
Theorem 3.6 Let be an α-inverse-strongly monotone mapping and let be a bifunction from to ℝ which satisfies (A1)-(A4). Let be a strictly pseudocontractive mapping with the constant and let be a contractive mapping with the constant . Assume that is not empty. Let be a sequence generated in the following process: and
where , , and are real number sequences in and is a positive real number sequence in . Assume that the control sequences satisfy the following restrictions:
-
(a)
, ;
-
(b)
;
-
(c)
, ;
-
(d)
and ;
-
(e)
,
where a, b, and c are three real numbers. Then converges strongly to a point , where .
If , the identity on H, then we find from Theorem 3.6 the following result on the equilibrium problem immediately.
Corollary 3.7 Let be an α-inverse-strongly monotone mapping and let be a bifunction from to ℝ which satisfies (A1)-(A4). Let be a contractive mapping with the constant . Assume that is not empty. Let be a sequence generated in the following process: and
where and are real number sequences in and is a positive real number sequence in . Assume that the control sequences satisfy the following restrictions:
-
(a)
, ;
-
(b)
;
-
(c)
and ;
-
(d)
,
where b and c are two real numbers. Then converges strongly to a point , where .
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The authors are grateful to the referees for useful suggestions which improved the contents of the paper.
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The main idea of this paper was proposed by SYC. XQ 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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Cho, S.Y., Qin, X. & Wang, L. Strong convergence of a splitting algorithm for treating monotone operators. Fixed Point Theory Appl 2014, 94 (2014). https://doi.org/10.1186/1687-1812-2014-94
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DOI: https://doi.org/10.1186/1687-1812-2014-94