Abstract
In this paper, a new iterative algorithm is proposed for finding a common solution to a constrained convex minimization problem, a quasi-variational inclusion problem and the fixed point problem of a strictly pseudo-contractive mapping in a real Hilbert space. It is proved that the sequence generated by the proposed algorithm converges strongly to a common solution of the three above described problems. By applying this result to some special cases, several interesting results can be obtained.
MSC:47H09, 47H10, 49J40.
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1 Introduction
Variational inequalities, introduced by Hartman and Stampacchia [1] in the early sixties, are one of the most interesting and intensively studied classes of mathematical problems. They are a very powerful tool of the current mathematical technology and have been extended to study a considerable amount of problems arising in mechanics, physics, optimization and control, nonlinear programming, transportation equilibrium and engineering sciences (see, e.g., [2–4]). As a useful and important generalization of variational inequalities, quasi-variational inclusion problems are also further studied (see, e.g., [5–14] and the references contained therein).
Throughout this paper, we assume that H is a real Hilbert space with the inner product and the induced norm , and let C be a nonempty closed convex subset of H. denotes a fixed point set of the mapping T.
Let Φ be a single-valued mapping of C into H and M be a multi-valued mapping with domain . The quasi-variational inclusion problem is to find such that
The solution set of quasi-variational inclusion problem (1.1) is denoted by . In particular, if , where is the indicator function of C, i.e.,
then the variational inclusion problem (1.1) is equivalent to finding such that
This problem is called the Hartman-Stampacchia variational inequality problem [1]. The solution set of problem (1.2) is denoted by .
Recall that is called a k-strictly pseudo-contractive mapping if there exists a constant such that
and T is called a pseudo-contractive mapping if
It is obvious that , then the mapping T is nonexpansive, that is,
It is well known that finding fixed points of nonexpansive mappings is an important topic in the theory of nonexpansive mappings and has wide applications in a number of applied areas, such as the convex feasibility problem [15, 16], the split feasibility problem [17], image recovery and signal processing [18]. After that, as an important generalization of nonexpansive mappings, strictly pseudo-contractive mappings become one of the most interesting studied classes of nonexpansive mappings (see, e.g., [19–22]). In fact, strictly pseudo-contractive mappings have more powerful applications than nonexpansive mappings do such as in solving an inverse problem [23].
In order to find a common element of the solution set of quasi-variational inclusion problem (1.1) and the fixed point set of k-strictly pseudo-contractive mapping (1.3), which is also a solution of the following constrained convex minimization problem:
where is a real-valued convex function and assumes that problem (1.4) is consistent (i.e., its solution set is nonempty), let Ω denote its solution set. Ceng et al. [24] studied the following algorithm: take arbitrarily and
Under appropriate conditions they obtained one strong convergence theorem.
In this paper, motivated and inspired by the above facts, we propose a new algorithm as follows: take arbitrarily, set , and
and also get a strong convergence theorem under certain conditions.
The remainder of this paper is organized as follows. In Section 2, some definitions and lemmas are provided to get the main results of this paper. In Section 3, we give and prove one strong convergence theorem about our proposed algorithm. Finally, in Section 4, we apply our conclusion to some special cases.
2 Preliminaries
Let H be a real Hilbert space. It is well known that
and
for all and .
Now, we recall some definitions and useful results which will be used in the next section.
Definition 2.1 Let be a nonlinear operator.
-
(1)
T is Lipschitz continuous if there exists a constant such that
-
(2)
T is monotone if
-
(3)
T is ρ-strongly monotone if there exists a number such that
-
(4)
T is η-inverse-strongly monotone if there exists a number such that
It is easy to see that the following results hold: (i) strongly monotone is monotone; (ii) an η-inverse-strongly monotone mapping is monotone and -Lipschitz continuous; (iii) T is k-strictly pseudo-contractive, then is -inverse strongly monotone.
Definition 2.2 A multi-valued mapping is called monotone if its graph is a monotone set in , that is, M is monotone if and only if
A monotone multi-valued mapping M is called maximal if for any , for every implies .
Remark 2.1 [24]
The following results are equivalent:
-
(1)
A multi-valued mapping M is maximal monotone;
-
(2)
M is monotone and for each ;
-
(3)
M is monotone and its graph is not properly contained in the graph of any other monotone mapping in H.
Definition 2.3 is called a metric projection if for every point , there exists a unique nearest point in C, denoted by , such that
Lemma 2.1 Let C be a nonempty closed convex subset of H and let be a metric projection, then
-
(1)
, ;
-
(2)
moreover, is a nonexpansive mapping, i.e., , ;
-
(3)
, , ;
-
(4)
, , .
Definition 2.4 Let be a multi-valued maximal monotone mapping, then the single-valued mapping defined by
is called the resolvent operator associated with M, where μ is any positive number and I is the identity mapping.
The resolvent operator associated with M is single-valued and firmly nonexpansive, i.e.,
Consequently, is nonexpansive and monotone.
Lemma 2.3 [24]
Let M be a multi-valued maximal monotone mapping with . Then, for any given , is a solution of problem (1.1) if and only if satisfies
Lemma 2.4 [24]
Let M be a multi-valued maximal monotone mapping with and let be a monotone, continuous and single-valued mapping. Then is maximal monotone.
In the sequel, we use and to denote the weak convergence and strong convergence of the sequence in H, respectively.
Lemma 2.5 [26]
Let C be a nonempty closed convex subset of H and let be a k-strictly pseudocontractive mapping, then the following results hold:
-
(1)
inequality (1.3) is equivalent to
-
(2)
T is Lipschitz continuous with a constant , i.e.,
-
(3)
(Demi-closedness principle) is demi-closed on C, that is,
Lemma 2.6 Let C be a nonempty closed convex subset of H and let be an η-inverse-strongly monotone mapping, then for all and , we have
So, if , then is a nonexpansive mapping from C to H.
Let be a monotone, Lipschitz continuous mapping, and let be the normal cone to C at , i.e.,
Define
Then, T is maximal monotone and if and only if .
For the minimization problem (1.4), if f is (Frechet) differentiable, then we have the following lemma.
Lemma 2.8 [28] (Optimality condition)
A necessary condition of optimality for a point to be a solution of the minimization problem (1.4) is that solves the variational inequality
Equivalently, solves the fixed point equation
for every constant . If, in addition, f is convex, then the optimality condition (2.1) is also sufficient.
3 Main results
In this section, we prove a strong convergence theorem by an iterative algorithm for finding a solution of the constrained convex minimization problem (1.4), which is also a common solution of the quasi-variational inclusion problem (1.1) and the fixed point problem of a k-strictly pseudo-contractive mapping (1.3) in a real Hilbert space.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. For the minimization problem (1.4), assume that f is (Frechet) differentiable and the gradient ∇f is a ρ-inverse-strongly monotone mapping. Let be an η-inverse-strongly monotone mapping and M be a maximal monotone mapping with , and let be a k-strictly pseudo-contractive mapping such that . Pick any and set . Let be a sequence generated by
where the following conditions hold:
-
(i)
;
-
(ii)
for some ;
-
(iii)
.
Then the sequence converges strongly to .
Proof It is obvious that is closed for each . Since is equivalent to
we have that is convex for each . Therefore, is well defined. We divide the proof into five steps.
Step 1. We show by induction that for each .
It is obvious that . Suppose that for some . Let , we have
According to Lemma 2.3, Lemma 2.2 and Lemma 2.6, we get
Since the gradient ∇f is a ρ-inverse-strongly monotone mapping and , from Lemma 2.8, we have
From Lemma 2.1(4) and (3.4), we obtain
It is easy to see that ρ-inverse-strongly monotone mapping ∇f is -Lipschitz continuous. Further, since and by Lemma 2.1(3) we have
Substituting (3.6) into (3.5), we obtain
From (3.2), (3.3) and (3.7), we have
Hence . This implies that for all .
Step 2. We prove that and .
Let . From and , we obtain
Then is bounded. This implies that , and are also bounded. Since and , we have
Therefore exists. From , and Lemma 2.1(4), we obtain
which implies
It follows from that , and hence
From (3.10) and (3.11), we have
Step 3. We show that , and .
For , from (3.2), (3.3) and (3.7), we have
Then
Since and , from (3.12) and (3.13) we get
and
As ∇f is -Lipschitz continuous, we have
Hence
From (3.14) and (3.16), we obtain
We observe
From (3.15), we get
Combining (3.12) and (3.19), we have
Step 4. Since is bounded, there exists a subsequence which converges weakly to u. We show that
Indeed, firstly, we show . Since and , we have . From , we obtain as . By Lemma 2.5 (Demi-closedness principle), we can conclude that .
Secondly, we show . Since and by Lemma 2.1(3), we have
that is,
Let
Then, from Lemma 2.7, we know that T is maximal monotone and if and only if . Let be the graph of T and let . Then we have . Hence . So, we have
Therefore,
Since and , we have and . Then, from (3.21), we obtain as . Since T is maximal monotone, we have and hence . According to Lemma 2.8, we obtain .
Finally, let us show . Since is η-inverse-strongly monotone and M is maximal monotone, by Lemma 2.4 we know that is maximal monotone. Take a fixed arbitrarily. Then we have , that is,
Since , then
Therefore,
which hence yields
Observe that
By and , we have . Then
Let , from (3.22) we get
This implies that . Hence . Therefore,
Step 5. We show that , where .
Indeed, from , and (3.9), we have
Then
From and the Kadec-Klee property of H, we have , and hence . Since and , we have
Let , by and , we have
Hence , which implies that . This completes the proof. □
4 Applications
From Theorem 3.1, we can obtain the following theorems.
Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. For the minimization problem (1.4), assume that f is (Frechet) differentiable and the gradient ∇f is a ρ-inverse-strongly monotone mapping. Let be a k-strictly pseudo-contractive mapping such that . Pick any and set . Let be a sequence generated by
where the following conditions hold:
-
(i)
;
-
(ii)
.
Then the sequence converges strongly to .
Proof Let in Theorem 3.1, we have and . Let η be any positive number in the interval and take any sequence for some . In addition, we have
Therefore, by Theorem 3.1 we obtain the expected result. □
Theorem 4.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a nonexpansive mapping such that . Pick any and set . Let be a sequence generated by
where the following condition holds: . Then the sequence converges strongly to .
Proof Let and in Theorem 3.1. Let ρ, η be any positive number in the interval . Take any sequence which satisfies and take any sequence for some . In this case, we have
Therefore, by Theorem 3.1 we obtain the expected result. □
Theorem 4.3 Let C be a nonempty closed convex subset of a real Hilbert space H. For the minimization problem (1.4), assume that f is (Frechet) differentiable and the gradient ∇f is a ρ-inverse-strongly monotone mapping. Let be γ-strictly pseudo-contractive and let be a k-strictly pseudo-contractive mapping such that . Pick any and set . Let be a sequence generated by
where the following conditions hold:
-
(i)
;
-
(ii)
for some ;
-
(iii)
.
Then the sequence converges strongly to .
Proof Let and in Theorem 3.1, then we have that Φ is η-inverse strongly monotone with . Now, we show that . In fact, since and , we obtain
Thus,
Note that , hence . In this case, we have
Therefore, by Theorem 3.1 we obtain the expected result. □
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Acknowledgements
Changfeng Ma is grateful for the hospitality and support during his research at Chern Mathematics Institute in Nankai University in June 2013. The project is supported by the National Natural Science Foundation of China (11071041), Fujian Natural Science Foundation (2013J01006) and R&D of Key Instruments and Technologies for Deep Resources Prospecting (the National R&D Projects for Key Scientific Instruments) under grant number ZDYZ2012-1-02-04.
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Ke, Y., Ma, C. Iterative algorithm of common solutions for a constrained convex minimization problem, a quasi-variational inclusion problem and the fixed point problem of a strictly pseudo-contractive mapping. Fixed Point Theory Appl 2014, 54 (2014). https://doi.org/10.1186/1687-1812-2014-54
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DOI: https://doi.org/10.1186/1687-1812-2014-54