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The convergence analysis of the projection methods for a system of generalized relaxed cocoercive variational inequalities in Hilbert spaces
Fixed Point Theory and Applications volume 2013, Article number: 189 (2013)
Abstract
In this paper, the approximate solvability for a system of generalized relaxed cocoercive nonlinear variational inequalities in Hilbert spaces is studied, based on the convergence of the projection methods. The results presented in this paper extend and improve the main results of Refs. (Verma in Comput. Math. Appl. 41:1025-1031, 2001; Verma in Int. J. Differ. Equ. Appl. 6:359-367, 2002; Verma in J. Optim. Theory Appl. 121(1):203-210, 2004; Verma in Appl. Math. Lett. 18(11):1286-1292, 2005; Chang, Lee and Chan in Appl. Math. Lett. 20:329-334, 2007).
MSC:90C33, 65K15, 58E36.
1 Introduction
Variational inequalities are one of the most interesting and intensively studied classes of mathematical problems and there exists a considerable amount of literature [1–15] on the approximate solvability of nonlinear variational inequalities. In this paper, we consider, based on the projection methods, the approximate solvability for a system of generalized relaxed cocoercive nonlinear variational inequalities in Hilbert spaces. The results presented in this paper extend and improve the main results in [1–5].
Throughout this paper, we assume that H is a real Hilbert space with the inner product and the induced norm . Let C be a nonempty closed convex subset of H, and let be the metric projection of H onto C. Let , and be relaxed cocoercive mappings for each . We consider a system of generalized nonlinear variational inequality (SGNVI) problem as follows: find an element such that
SGNVI problem (1.1) is equivalent to the following projection problem:
Next we consider some special cases of SGNVI problem (1.1), where I is the identity mapping.
-
(1)
If , then SGNVI problem (1.1) is reduced to the following: find an element such that
(1.3) -
(2)
If , then SGNVI problem (1.1) is reduced to the following: find an element such that
(1.4) -
(3)
If , then SGNVI problem (1.1) is reduced to the following: find an element such that
(1.5) -
(4)
If and are univariate mappings, then SGNVI problem (1.1) is reduced to the following: find an element such that
(1.6) -
(5)
If , and are univariate mappings, then SGNVI problem (1.1) is reduced to the following: find an element such that
(1.7) -
(6)
If and , then SGNVI problem (1.1) is reduced to the following: find an element such that
(1.8)
2 Preliminaries
In order to prove our main results in the next section, we recall several definitions and lemmas.
Definition 2.1 Let be a mapping.
-
(1)
T is said to be β-Lipschitz continuous if there exists a constant such that
-
(2)
T is said to be monotone if
-
(3)
T is said to be δ-strongly monotone if there exists a constant such that
This implies that
that is, T is δ-expansive.
-
(4)
T is said to be γ-cocoercive if there exists a constant such that
Clearly, every γ-cocoercive mapping T is -Lipschitz continuous.
-
(5)
T is said to be relaxed γ-cocoercive if there exists a constant such that
-
(6)
T is said to be relaxed -cocoercive if there exist two constants such that
Definition 2.2 A mapping is said to be relaxed -cocoercive if there exist two constants such that for all ,
Definition 2.3 A mapping is said to be β-Lipschitz continuous in the first variable if there exists a constant such that for all ,
Definition 2.4 is called a metric projection if for every point , there exists a unique nearest point in C, denoted by , such that
Lemma 2.1 is a metric projection, then is a nonexpansive mapping, i.e.,
Lemma 2.2 [5]
Let , and be three nonnegative real sequences such that
where is some nonnegative integer, is a sequence in with , and . Then .
3 Main results
In this section, we present the projection methods and give the convergence analysis of SGNVI problem (1.1) involving relaxed -cocoercive and β-Lipschitz continuous mappings in Hilbert spaces.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a relaxed -cocoercive and -Lipschitz continuous mapping in the first variable, let be a relaxed -cocoercive and -Lipschitz continuous mapping, and let be a relaxed -cocoercive and -Lipschitz continuous mapping for each . Suppose that is a solution to SGNVI problem (1.1). For any , the sequences and are generated by
where and the following conditions are satisfied:
where
Then the sequences and converge strongly to and , respectively.
Proof Since is a solution to SGNVI problem (1.1), it follows that
For , we have
Since is a relaxed -cocoercive and -Lipschitz continuous mapping, we have
where . In a similar way, we can obtain that
where . By the assumption that is a relaxed -cocoercive and -Lipschitz continuous mapping in the first variable, we have
where . According to (3.2), (3.3), (3.4) and (3.5), we obtain that
Next, we estimate . From (3.1), we see that
Similarly, we obtain that
where , and
where , and
where . According to (3.7), (3.8), (3.9) and (3.10), we obtain that
that is,
By conditions (1), (4) and (5), we have
Substituting (3.13) into (3.12), we have
According (3.6) and (3.14), for , we have
From conditions (1), (2), (3) and (6), we get
and
The conditions in Lemma 2.2 are satisfied, then (as ), i.e., (as ).
On the one hand, from condition (3) we know that (as ). On the other hand, from (3.14) and the result that (as ), we can get (as ), i.e., (as ).
This completes the proof of Theorem 3.1. □
When , , then we have , , respectively. And from Theorem 3.1 we can get the following results immediately.
Corollary 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a relaxed -cocoercive and -Lipschitz continuous mapping in the first variable, and let be a relaxed -cocoercive and -Lipschitz continuous mapping for each . Suppose that is a solution to problem (1.3). For any , the sequences and are generated by
where and the following conditions are satisfied:
where
Then the sequences and converge strongly to and , respectively.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a relaxed -cocoercive and -Lipschitz continuous mapping in the first variable, and let be a relaxed -cocoercive and -Lipschitz continuous mapping for each . Suppose that is a solution to problem (1.4). For any , the sequences and are generated by
where and the following conditions are satisfied:
where
Then the sequences and converge strongly to and , respectively.
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a relaxed -cocoercive and -Lipschitz continuous mapping in the first variable for each . Suppose that is a solution to problem (1.5). For any , the sequences and are generated by
where and the following conditions are satisfied:
where
Then the sequences and converge strongly to and , respectively.
Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a relaxed -cocoercive and -Lipschitz continuous mapping, let be a relaxed -cocoercive and -Lipschitz continuous mapping, and let be a relaxed -cocoercive and -Lipschitz continuous mapping for each . Suppose that is a solution to problem (1.6). For any , the sequences and are generated by
where and the following conditions are satisfied:
where
Then the sequences and converge strongly to and , respectively.
Corollary 3.5 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a relaxed -cocoercive and -Lipschitz continuous mapping for each . Suppose that is a solution to problem (1.7). For any , the sequences and are generated by
where and the following conditions are satisfied:
where
Then the sequences and converge strongly to and , respectively.
For , in Corollary 3.3, we arrive at the following result.
Corollary 3.6 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a relaxed -cocoercive and -Lipschitz continuous mapping in the first variable. Suppose that is a solution to problem (1.8). For any , the sequence is generated by
where and the following conditions are satisfied:
where
Then the sequence converges strongly to .
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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, 11201074), the Fujian Natural Science Foundation (2013J01006) and the Scientific Research Special Fund Project of Fujian University (JK2013060).
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Ke, Y., Ma, C. The convergence analysis of the projection methods for a system of generalized relaxed cocoercive variational inequalities in Hilbert spaces. Fixed Point Theory Appl 2013, 189 (2013). https://doi.org/10.1186/1687-1812-2013-189
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DOI: https://doi.org/10.1186/1687-1812-2013-189