Abstract
An inclusion problem and a fixed point problem are investigated based on a hybrid projection method. The strong convergence of the hybrid projection method is obtained in the framework of Hilbert spaces. Variational inequalities and fixed point problems of quasi-nonexpansive mappings are also considered as applications of the main results.
MSC:47H05, 47H09, 47J25.
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1 Introduction and preliminaries
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. Splitting methods for linear equations were introduced by Peaceman and Rachford [1] and Douglas and Rachford [2]. Extensions to nonlinear equations in Hilbert spaces were carried out by Kellogg [3] and Lions and Mercier [4]. The central problem is to iteratively find a zero of the sum of two monotone operators A and B in a Hilbert space H. In this paper, we consider the problem of finding a solution to the following problem: find an x in the fixed point set of the mapping S such that
where A and B are two monotone operators. The problem has been addressed by many authors in view of the applications in image recovery and signal processing; see, for example, [5–9] and the references therein.
Throughout this paper, we always assume that H is a real Hilbert space with the inner product and norm , respectively. Let C be a nonempty closed convex subset of H and be the metric projection from H onto C. Let be a mapping. In this paper, we use to denote the fixed point set of S; that is, .
Recall that 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 [10].
S is said to be quasi-nonexpansive iff and
It is easy to see that nonexpansive mappings are Lipschitz continuous; however, the quasi-nonexpansive mapping is discontinuous on its domain generally. Indeed, the quasi-nonexpansive mapping is only continuous in its fixed point set.
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
For such a case, A is also said to be α-strongly monotone. 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. Notice that
clearly shows that A is -Lipschitz continuous.
Recall that the classical variational inequality is to find an such that
In this paper, we use to denote the solution set of (1.1). It is known that is a solution to (1.1) iff is a fixed point of the mapping , where is a constant, I stands for the identity mapping, and stands for the metric projection from H onto C.
A multivalued operator with the domain and the range is said to be monotone if for , , , and , we have . A monotone operator T is said to be maximal if its graph is not properly contained in the graph of any other monotone operator. Let I denote the identity operator on H and be a maximal monotone operator. Then we can define, for each , a nonexpansive single-valued mapping by . It is called the resolvent of T. We know that for all and is firmly nonexpansive.
The Mann iterative algorithm is efficient to study fixed point problems of nonlinear operators. Recently, many authors have studied the common solution problem, that is, find a point in a solution set and a fixed point (zero) point set of some nonlinear problems; see, for example, [11–30] and the references therein.
In [11], Kamimura and Takahashi investigated the problem of finding zero points of a maximal monotone operator by considering the following iterative algorithm:
where is a sequence in , is a positive sequence, is a maximal monotone, and . They showed that the sequence generated in (1.2) converges weakly to some provided that the control sequence satisfies some restrictions. Further, using this result, they also investigated the case that , where is a proper lower semicontinuous convex function. Convergence theorems are established in the framework of real Hilbert spaces.
In [12], Takahashi an Toyoda investigated the problem of finding a common solution of the variational inequality problem (1.1) and a fixed point problem involving nonexpansive mappings by considering the following iterative algorithm:
where is a sequence in , is a positive sequence, is a nonexpansive mapping, and is an inverse-strongly monotone mapping. They showed that the sequence generated in (1.3) converges weakly to some provided that the control sequence satisfies some restrictions.
The above convergence theorems are weak. In this paper, motivated by the above results, we consider the problem of finding a common solution to the zero point problems and fixed point problems based on hybrid iterative methods with errors. Strong convergence theorems are established in the framework of Hilbert spaces.
To obtain our main results in this paper, we need the following lemmas and definitions.
Let C be a nonempty, closed, and convex subset of H. Let be a mapping. Then the mapping is demiclosed at zero, that is, if is a sequence in C such that and , then .
Lemma [9]
Let C be a nonempty, closed, and convex subset of H, be a mapping, and be a maximal monotone operator. Then .
2 Main results
Theorem 2.1 Let C be a nonempty closed convex subset of a real Hilbert space H, be an α-inverse-strongly monotone mapping, be a quasi-nonexpansive mapping such that is demiclosed at zero, and B be a maximal monotone operator on H such that the domain of B is included in C. Assume that . Let be a positive real number sequence. Let be a real number sequence in . Let be a sequence in C generated in the following iterative process:
where . Suppose that the sequences and satisfy the following restrictions:
-
(a)
;
-
(b)
.
Then the sequence converges strongly to .
Proof First, we show that is closed and convex. Notice that is closed and convex. Suppose that is closed and convex for some . We show that is closed and convex for the same i. Indeed, for any , we see that
is equivalent to
Thus is closed and convex. This shows that is closed and convex.
Next, we prove that is a nonexpansive mapping. Indeed, we have
In view of the restriction (b), we obtain that is nonexpansive. Next, we show that for each . From the assumption, we see that . Assume that for some . For any , we find from Lemma that
Put . Since and are nonexpansive, we have
It follows from (2.1) that
This shows that . This proves that . Notice that . For every , we have
In particular, we have
This implies that is bounded. Since and , we arrive at
It follows that
This implies that exists. On the other hand, we have
It follows that
Notice that . It follows that
This in turn implies that
In view of (2.2), we obtain that
On the other hand, we have
It follows from (2.3) that
For any , we see that
Notice that
Substituting (2.5) into (2.6), we see that
It follows that
This implies from (2.3) that
On the other hand, we have
It follows that
Substituting (2.8) into (2.6), we see that
It follows that
In view of the restriction (a), we obtain from (2.7) that
Since is bounded, we may assume that there is a subsequence of converging weakly to some point . It follows from (2.9) that converges weakly to . Notice that
It follows from (2.4) and (2.9) that
In view of the assumption that S is demiclosed at zero, we see that .
Next, we show that . Notice that . This implies that
That is,
Since B is monotone, we get for any , that
Replacing n by and letting , we obtain from (2.10) that
This means , that is, . Hence, we get . This completes the proof that .
Notice that and , we have
On the other hand, we have
We, therefore, obtain that
This implies . Since is an arbitrary subsequence of , we obtain that as . This completes the proof. □
From Theorem 2.1, we have the following results immediately.
Corollary 2.2 Let C be a nonempty closed convex subset of a real Hilbert space H, be an α-inverse-strongly monotone mapping, and B be a maximal monotone operator on H such that the domain of B is included in C. Assume that . Let be a positive real number sequence. Let be a real number sequence in . Let be a sequence in C generated in the following iterative process:
where . Suppose that the sequences and satisfy the following restrictions:
-
(a)
;
-
(b)
.
Then the sequence converges strongly to .
Let be a proper lower semicontinuous convex function. Define the subdifferential
for all . Then ∂f is a maximal monotone operator of H into itself; see [23] for more details. Let C be a nonempty closed convex subset of H and be the indicator function of C, that is,
Furthermore, we define the normal cone of C at v as follows:
for any . Then is a proper lower semicontinuous convex function on H and is a maximal monotone operator. Let for any and . From and , we get
where is the metric projection from H into C. Similarly, we can get that . Putting in Theorem 2.1, we can see . The following is not hard to derive.
Corollary 2.3 Let C be a nonempty closed convex subset of a real Hilbert space H, be an α-inverse-strongly monotone mapping, and be a quasi-nonexpansive mapping such that is demiclosed at zero. Assume that . Let be a positive real number sequence. Let be a real number sequence in . Let be a sequence in C generated in the following iterative process:
Suppose that the sequences and satisfy the following restrictions:
-
(a)
;
-
(b)
.
Then the sequence converges strongly to .
In view of Corollary 2.3, we have the following corollary on variational inequalities.
Corollary 2.4 Let C be a nonempty closed convex subset of a real Hilbert space H and be an α-inverse-strongly monotone mapping. Assume that . Let be a positive real number sequence. Let be a real number sequence in . Let be a sequence in C generated in the following iterative process:
Suppose that the sequences and satisfy the following restrictions:
-
(a)
;
-
(b)
.
Then the sequence converges strongly to .
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Hecai, Y. On solutions of inclusion problems and fixed point problems. Fixed Point Theory Appl 2013, 11 (2013). https://doi.org/10.1186/1687-1812-2013-11
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DOI: https://doi.org/10.1186/1687-1812-2013-11