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Iterative algorithms for a system of generalized variational inequalities in Hilbert spaces
Fixed Point Theory and Applications volume 2012, Article number: 232 (2012)
Abstract
In this paper, a new system of generalized nonlinear variational inequalities involving three operators is introduced. A three-step iterative algorithm is considered for the system of generalized nonlinear variational inequalities. Strong convergence theorems of the three-step iterative algorithm are established.
MSC:47H05, 47H09, 47J25.
1 Introduction
Variational inequalities are among the most interesting and intensively studied classes of mathematical problems and have wide applications in the fields of optimization and control, economics, transportation equilibrium and engineering sciences. There exists a vast amount of literature (see, for instance, [1–26]) on the approximation solvability of nonlinear variational inequalities as well as operator equations.
Iterative algorithms have played a central role in the approximation solvability, especially of nonlinear variational inequalities as well as of nonlinear equations, in several fields such as applied mathematics, mathematical programming, mathematical finance, control theory and optimization, engineering sciences and others. Projection methods have played a significant role in the numerical resolution of variational inequalities based on their convergence analysis. However, the convergence analysis does require some sort of strong monotonicity besides the Lipschitz continuity. There have been some recent developments where convergence analysis for projection methods under somewhat weaker conditions such as cocoercivity [28] and partial relaxed monotonicity [24] is achieved.
Recently, Chang et al. [17] introduced a two-step iterative algorithm for a system of nonlinear variational inequalities and established strong convergence theorems. Huang and Noor [16] introduced the so-called explicit two-step iterative algorithm for a system of nonlinear variational inequalities involving two different nonlinear operators and established strong convergence theorems.
In this paper, we consider, based on the projection method, the approximate solvability of a new system of generalized nonlinear variational inequalities involving three different nonlinear operators in the framework of Hilbert spaces. The results presented in this paper extend and improve the corresponding results announced in Huang and Noor [16], Chang et al. [17], Verma [24–26] and many others.
Let H be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let C be a nonempty closed convex subset of H and be the metric projection from H onto C.
Given nonlinear operators and , we consider the problem of finding such that
where is a constant. The variational inequality (1.1) is called the generalized variational inequality involving three operators.
We see that an element is a solution to the generalized variational inequality (1.1) if and only if is a fixed point of the mapping
where I is the identity mapping. This equivalence plays an important role in this work.
If , then the generalized variational inequality (1.1) is equivalent to the following.
Find such that
Further, if , then the problem (1.2) is reduced to finding such that
which is known as the classical variational inequality originally introduced and studied by Stampacchia [27].
Let be a mapping. Recall the following definitions.
-
(1)
T is said to be monotone if
-
(2)
T is called δ-strongly monotone if there exists a constant such that
This implies that
that is, T is δ-expansive.
-
(3)
T is said to be γ-cocoercive if there exists a constant such that
Clearly, every γ-cocoercive mapping A is -Lipschitz continuous.
-
(4)
T is said to be relaxed γ-cocoercive if there exists a constant such that
-
(5)
T is said to be relaxed -cocoercive if there exist two constants such that
Let , and be nonlinear mappings for each . Consider a system of generalized nonlinear variational inequality (SGNVI) as follows.
Find such that for all ,
One can easily see SGNVI (1.4) is equivalent to the following projection problem:
Next, we consider some special classes of SGNVI (1.4) as follows.
-
(I)
If , then SGNVI (1.4) is reduced to the following.
Find such that for all ,
We see that the problem (1.6) is equivalent to the following projection problem:
-
(II)
If , then SGNVI (1.4) is reduced to the following.
Find such that for all ,
We see that the problem (1.8) is equivalent to the following projection problem:
-
(III)
If , then SGNVI (1.4) is reduced to the following.
Find such that for all ,
One can easily get that the problem (1.10) is equivalent to the following projection problem:
-
(IV)
If , and are univariate mappings, then SGNVI (1.4) is reduced to the following.
Find such that for all ,
One can easily see that the problem (1.12) is equivalent to the following projection problem:
2 Preliminaries
In this section, to study the approximate solvability of the problems (1.4), (1.6), (1.8), (1.10) and (1.12), we introduce the following three-step algorithms.
Algorithm 2.1 For any , compute the sequences , and by the following iterative process:
where are three constants and is a sequence in .
If , then Algorithm 2.1 is reduced to the following.
Algorithm 2.2 For any , compute the sequences , and by the following iterative process:
where are three constants and is a sequence in .
If , the identity mapping, then Algorithm 2.1 is reduced to the following.
Algorithm 2.3 For any , compute the sequences , and by the following iterative process:
where are three constants and is a sequence in .
If , the identity mapping, then Algorithm 2.1 is reduced to the following.
Algorithm 2.4 For any , compute the sequences , and by the following iterative process:
where are three constants and is a sequence in .
-
(IV)
If , and are univariate mappings, then Algorithm 2.1 is reduced to the following.
Algorithm 2.5 For any , compute the sequences , and by the following iterative process:
where are three constants and is a sequence in .
In order to prove our main results, we also need the following lemma and definitions.
Lemma 2.6 [29]
Assume that is a sequence of nonnegative real numbers such that
where is a nonnegative integer, is a sequence in with and , then .
Definition 2.7 A mapping is said to be relaxed -cocoercive if there exist constants such that for all ,
Definition 2.8 A mapping is said to be β-Lipschitz continuous in the first variable if there exists a constant such that for all ,
3 Main results
Theorem 3.1 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let be a relaxed -cocoercive and -Lipschitz continuous mapping in the first variable, be a relaxed -cocoercive and -Lipschitz continuous mapping and be a relaxed -cocoercive and -Lipschitz continuous mapping for each . Suppose that is a solution to the problem (1.4). Let , and be the sequences generated by Algorithm 2.1. Assume that the following conditions are satisfied:
-
(a)
;
-
(b)
, ;
-
(c)
,
where
and
Then the sequences , and converge strongly to , and , respectively.
Proof In view of being a solution to the problem (1.4), we see that
It follows from Algorithm (2.1) that
By the assumption that is relaxed -cocoercive and -Lipschitz continuous in the first variable, we obtain that
where . On the other hand, it follows from the assumption that is relaxed -cocoercive and -Lipschitz continuous that
where . In a similar way, we can obtain that
where . Substituting (3.2), (3.3) and (3.4) into (3.1), we arrive at
Next, we estimate . From Algorithm 2.1, we see that
By the assumption that is relaxed -cocoercive and -Lipschitz continuous in the first variable, we obtain that
where . It follows from the assumption that is relaxed -cocoercive and -Lipschitz continuous that
where . Substituting (3.7) and (3.8) into (3.6), we see that
On the other hand, we have
From the proof of (3.8), we arrive at
where . Substituting (3.9) and (3.11) into (3.10), we see that
It follows from the condition (b) that
That is,
Finally, we estimate . It follows from Algorithm 2.1 that
In a similar way, we can show that
and
where and . Substituting (3.14) and (3.15) into (3.13), we arrive at
Note that
On the other hand, we have
. Substituting (3.16) and (3.18) into (3.17), we arrive at
It follows from the condition (b) that
That is,
Combining (3.5), (3.12) with (3.19), we obtain that
Since and the condition (c), we can conclude the desired conclusion easily from Lemma 2.6. This completes the proof. □
Remark 3.2 Theorem 3.1 includes the corresponding results in Huang and Noor [16] Chang et al. [17], and Verma [24–26] as special cases.
From Theorem 3.1, we can get the following results immediately.
Corollary 3.3 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let be a relaxed -cocoercive and -Lipschitz continuous mapping in the first variable and be a relaxed -cocoercive and -Lipschitz continuous mapping for each . Suppose that is a solution to the problem (1.6). Let , and be the sequences generated by Algorithm 2.2. Assume that the following conditions are satisfied:
-
(a)
;
-
(b)
,
where
and
Then the sequences , and converge strongly to , and , respectively.
Corollary 3.4 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let be a relaxed -cocoercive and -Lipschitz continuous mapping in the first variable and be a relaxed -cocoercive and -Lipschitz continuous mapping for each . Suppose that is a solution to the problem (1.8). Let , and be the sequences generated by Algorithm 2.3. Assume that the following conditions are satisfied:
-
(a)
;
-
(b)
, ;
-
(c)
,
where
and
Then the sequences , and converge strongly to , and , respectively.
Corollary 3.5 Let C be a nonempty, closed and 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 the problem (1.10). Let , and be the sequences generated by Algorithm 2.4. Assume that the following conditions are satisfied:
-
(a)
;
-
(b)
,
where
and
Then the sequences , and converge strongly to , and , respectively.
Corollary 3.6 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let be a relaxed -cocoercive and -Lipschitz continuous mapping, be a relaxed -cocoercive and -Lipschitz continuous mapping and be a relaxed -cocoercive and -Lipschitz continuous mapping for each . Suppose that is a solution to the problem (1.12). Let , and be the sequences generated by Algorithm 2.5. Assume that the following conditions are satisfied:
-
(a)
;
-
(b)
, ;
-
(c)
,
where
and
Then the sequences , and converge strongly to , and , respectively.
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The author is grateful to the editors, Kejifazhan (NO. 112400430123), and Jichuheqianyanjishuyanjiu (NO. 112400430123), Henan.
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Zhang, M. Iterative algorithms for a system of generalized variational inequalities in Hilbert spaces. Fixed Point Theory Appl 2012, 232 (2012). https://doi.org/10.1186/1687-1812-2012-232
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DOI: https://doi.org/10.1186/1687-1812-2012-232