Projection methods of iterative solutions in Hilbert spaces
- Feng Gu1 and
- Jing Lu1Email author
https://doi.org/10.1186/1687-1812-2012-162
© Gu and Lu; licensee Springer 2012
Received: 27 June 2012
Accepted: 6 September 2012
Published: 25 September 2012
Abstract
In this paper, zero points of the sum of two monotone mappings, solutions of a monotone variational inequality, and fixed points of a nonexpansive mapping are investigated based on a hybrid projection iterative algorithm. Strong convergence of the purposed iterative algorithm is obtained in the framework of real Hilbert spaces without any compact assumptions.
MSC:47H05, 47H09, 47J25, 90C33.
Keywords
1 Introduction and preliminaries
Throughout this paper, we always assume that H is a real Hilbert space with an inner product and a norm . Let C be a nonempty, closed, and convex subset of H. Let be a nonlinear mapping. stands for the fixed point set of S; that is, .
If C is a bounded, closed, and convex subset of H, then is not empty, closed, and convex; see [1].
For such a case, A is also said to be α-inverse-strongly monotone.
In this paper, we use to denote the solution set of (1.1). It is known that is a solution of (1.1) if and only if x 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. If A is α-inverse-strongly monotone and , then the mapping is nonexpansive; see [2] for more details. It follows that is closed and convex.
Monotone variational inequality theory has emerged as a fascinating branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, ecology, social, regional, pure, and applied sciences. In recent years, much attention has been given to developing efficient numerical methods for treating solution problems of monotone variational inequality. The gradient-projection method is a powerful tool for solving constrained convex optimization problems and has extensively been studied; see [3–5] and the references therein. It has recently been applied to solving split feasibility problems which find applications in image reconstructions and the intensity modulated radiation theory; see [6–9] and the references therein. However, the gradient-projection method requires the operator to be strongly monotone and Lipschitz continuous. These strong conditions rule out many applications. Extra gradient-projection method which was first introduce by Korpelevich [10] in the finite dimensional Euclidean space has been studied recently for relaxing the strong monotonicity of operators; see [11–13] and the references therein.
Recall that a set-valued mapping is said to be monotone iff, for all , and imply . A monotone mapping is maximal iff the graph of R is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if, for any , , for all implies .
For a maximal monotone operator M on H and , we may define the single-valued resolvent , where denotes the domain of M. It is known that is firmly nonexpansive, and , where and .
Recently, variational inequalities, fixed point problems, and zero point problems have been investigated by many authors based on iterative methods; see, for example, [14–32] and the references therein. In this paper, zero point problems of the sum of a maximal monotone operator and an inverse-strongly monotone mapping, solution problems of a monotone variational inequality, and fixed point problems of a nonexpansive mapping are investigated. A hybrid iterative algorithm is considered for analyzing the convergence of iterative sequences. Strong convergence theorems are established in the framework of real Hilbert spaces without any compact assumptions.
In order to prove our main results, we also need the following definitions and lemmas.
Lemma 1.2[1]
Let C be a nonempty, closed, and convex subset of H. Letbe a nonexpansive mapping. Then the mappingis demiclosed at zero, that is, ifis a sequence in C such thatand, then.
Lemma 1.3 Let C be a nonempty, closed, and convex subset of H, be a mapping, andbe a maximal monotone operator. Then.
This completes the proof. □
Lemma 1.4[33]
Then W is maximal monotone andif and only if.
2 Main results
Now, we are in a position to give our main results.
- (a)
;
- (b)
;
- (c)
,
where a, b, c, d, and e are real constants. Then the sequenceconverges strongly to.
It is clear that . This shows that is closed and convex for each .
Since is bounded, we find that there exists a subsequence of such that . From Lemma 1.2, we easily conclude that .
Since W is maximal monotone, we conclude that . This proves that .
This implies that , that is, . This completes . Assume that there exists another subsequence of which converges weakly to . We can easily conclude from Opial’s condition that .
which yields that . It follows that converges strongly to . This completes the proof. □
If , then Theorem 2.1 is reduced to the following.
- (a)
;
- (b)
;
- (c)
,
where a, b, c, and d are real constants. Then the sequenceconverges strongly to.
If , then . Corollary 2.2 is reduced to the following.
- (a)
;
- (b)
,
where a, b, and c are real constants. Then the sequenceconverges strongly to.
If , then Theorem 2.1 is reduced to the following.
- (a)
;
- (b)
,
where a, b, and c are real constants. Then the sequenceconverges strongly to.
From Rockafellar [34], we know that ∂f is maximal monotone. It is not hard to verify that if and only if .
In the light of the above, the following is not hard to derive from Corollary 2.4.
- (a)
;
- (b)
,
where a, b, and c are real constants. Then the sequenceconverges strongly to.
3 Applications
First, we consider the problem of finding a minimizer of a proper convex lower semicontinuous function.
whereis a positive sequence such that, where a is a real constant. Then the sequenceconverges strongly to.
Proof Putting , , and , we can immediately draw the desired conclusion from Theorem 2.1. □
Second, we consider the problem of approximating a common fixed point of a pair of nonexpansive mappings.
- (a)
;
- (b)
,
where a, b, and c are real constants. Then the sequenceconverges strongly to.
Proof Putting , , and , we see that B is -inverse-strongly monotone. We also have = and . In view of (2.18), we can immediately obtain the desired result. □
To study the equilibrium problem (3.1), we may assume that F satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A4) for each , is convex and lower semi-continuous.
Putting for every , we see that the equilibrium problem (3.3) is reduced to the variational inequality (1.1).
The following lemma can be found in [35] and [37].
- (a)
is single-valued;
- (b)is firmly nonexpansive; that is,
- (c)
;
- (d)
is closed and convex.
Lemma 3.4[30]
whereis defined as in (3.3).
Finally, we consider finding a solution of the equilibrium problem.
whereis defined as (3.3), andis a positive sequence such that, where a is a real constant Then the sequenceconverges strongly to.
Proof Putting , and , we immediately reach the desired conclusion from Lemma 3.4. □
Authors’ information
Author’s information
Declarations
Acknowledgements
The first author was supported by the National Natural Science Foundation of China (11071169, 11271105), the Natural Science Foundation of Zhejiang Province (Y6110287, Y12A010095).
Authors’ Affiliations
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