Convergence theorems of solutions of a generalized variational inequality
 Li Yu^{1}Email author and
 Ma Liang^{2}
DOI: 10.1186/16871812201119
© Yu and Liang; licensee Springer. 2011
Received: 14 November 2010
Accepted: 25 July 2011
Published: 25 July 2011
Abstract
The convex feasibility problem (CFP) of finding a point in the nonempty intersection is considered, where r ≥ 1 is an integer and each C_{ m } is assumed to be the solution set of a generalized variational inequality. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A_{ m }, B_{ m } : C → H be relaxed cocoercive mappings for each 1 ≤ m ≤ r. It is proved that the sequence {x_{ n }} generated in the following algorithm:
where u ∈ C is a fixed point, {α_{ n }}, {β_{ n }}, {γ_{ n }}, {δ_{(1,n)}}, ..., and {δ_{(r,n)}} are sequences in (0, 1) and , are positive sequences, converges strongly to a solution of CFP provided that the control sequences satisfies certain restrictions.
2000 AMS Subject Classification: 47H05; 47H09; 47H10.
Keywords
nonexpansive mapping fixed point relaxed cocoercive mapping variational inequality1. Introduction and Preliminaries
where r ≥ 1 is an integer and each C_{ m } is a nonempty closed and convex subset of H. There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [1, 2], computer tomography [3] and radiation therapy treatment planning [4].
 (a)A is said to be monotone if
 (b)A is said to be ρstrongly monotone if there exists a positive real number ρ > 0 such that
 (c)A is said to be ηcocoercive if there exists a positive real number η > 0 such that
 (d)A is said to be relaxed ηcocoercive if there exists a positive real number η > 0 such that
 (e)A is said to be relaxed (η, ρ)cocoercive if there exist positive real numbers η, ρ > 0 such that
where λ and τ are two positive constants. In this paper, we use GV I(C, B, A) to denote the set of solutions of the generalized variational inequality (1.2).
The variational inequality (1.4) emerging as a fascinating and interesting branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, social, ecology, regional, pure and applied sciences was introduced by Stampacchia [5]. In this paper, we use V I(C, A) to denote the set of solutions of the variational inequality (1.4).
It is well known that if C is nonempty bounded closed and convex subset of H, then the fixed point set of the nonexpansive mapping S is nonempty, see [6] more details. Recently, fixed point problems of nonexpansive mappings have been considered by many authors; see, for example, [7–16].
Recall that S is said to be demiclosed at the origin if for each sequence {x_{ n }} in C, x_{ n } ⇀ x_{0} and Sx_{ n } → 0 imply Sx_{0} = 0, where ⇀ and → stand for weak convergence and strong convergence.
Recently, many authors considered the variational inequality (1.4) based on iterative methods; see [17–32]. For finding solutions to a variational inequality for a cocoercive mapping, Iiduka et al. [22] proved the following theorem.
for every n = 1, 2, ..., where C is the metric projection from H onto C, {α_{ n }} is a sequence in [1, 1], and {λ_{ n }} is a sequence in [0, 2α]. If {α_{ n }} and {λ_{ n }} are chosen so that {α_{ n }} ∈ [a, b] for some a, b with 1 < a < b < 1 and {λ_{ n }} ∈ [c, d] for some c, d with 0 < c < d < 2(1 + a)α, then {x_{ n }} converges weakly to some element of V I(C, A).
Subsequently, Iiduka and Takahashi [23] further studied the problem of finding solutions of the classical variational inequality (1.4) for cocoercive mappings (inversestrongly monotone mappings) and nonexpansive mappings. They obtained a strong convergence theorem. More precisely, they proved the following theorem.
for every n = 1, 2, ..., where {α_{ n }} is a sequence in [0, 1) and {λ_{ n }} is a sequence in [a, b].
then {x_{ n }} converges strongly to P_{F(S)∩V I(C,A)}x.
In this paper, motivated by research work going on in this direction, we study the CFP in the case that each C_{ m } is a solution set of generalized variational inequality (1.2). Strong convergence theorems of solutions are established in the framework of real Hilbert spaces.
In order to prove our main results, we need the following lemmas.
Then lim_{n→∞}y_{ n }  x_{ n } = 0.
where a is a constant in (0, 1). Then S is nonexpansive with F(S) = F(S_{1}) ∩ F (S_{2}).
Lemma 1.3 [35]. Let C be a nonempty closed and convex subset of a real Hilbert space H and S : C → C a nonexpansive mapping. Then I  S is demiclosed at zero.
 (a)
;
 (b)
lim sup_{n→∞}δ_{ n }/γ_{ n } ≤ 0 or .
Then lim_{n→∞}α_{ n } = 0.
2. Main results
 (a)
;
 (b)
0 < lim inf_{n→∞}β_{ n } ≤ lim sup_{n→∞}β_{ n } < 1;
 (c)
lim_{n→∞}α_{ n } = 0 and ;
 (d)
lim_{n→∞}δ_{(m,n)}= δ_{ m }∈ (0, 1), ∀1 ≤ m ≤ r,
 (e)
.
This completes the proof.
If B_{ m } ≡ I, the identity mapping and τ_{ m } ≡ 1, then Theorem 2.1 is reduced to the following result on the classical variational inequality (1.4).
 (a)
;
 (b)
0 < lim inf_{n→∞}β_{ n } ≤ lim sup_{n→∞}β_{ n } < 1;
 (c)
lim_{n→∞}α_{ n } = 0 and ;
 (d)
lim_{n→∞}δ_{(m,n)}= δ_{ m } ∈ (0, 1), ∀1 ≤ m ≤ r, and is a positive sequence such that
 (e)
, ∀1 ≤ m ≤ r.
If r = 1, then Theorem 2.1 is reduced to the following.
 (a)
α_{ n } + β_{ n } + γ_{ n } = 1, ∀_{ n } ≥ 1;
 (b)
0 < lim inf_{n→∞}β_{ n } ≤ lim sup_{n→∞}β_{ n } < 1;
 (c)
lim_{n→∞}α_{ n } = 0 and
 (d)
.
For the variational inequality (1.4), we can obtain from Corollary 2.3 the following immediately.
 (a)
α_{ n } + β_{ n } + γ_{ n } = 1, ∀n ≥ 1;
 (b)
0 < lim inf_{n→∞}β_{ n } ≤ lim sup_{n→∞}β_{ n } < 1;
 (c)
lim_{n→∞}α_{ n }= 0 and ;
 (d)
.
Remark 2.5. In this paper, the generalized variational inequality (1.2), which includes the classical variational inequality (1.4) as a special case, is considered based on iterative methods. Strong convergence theorems are established under mild restrictions imposed on the parameters. It is of interest to extend the main results presented in this paper to the framework of Banach spaces.
Abbreviation
 CFP:

convex feasibility problem.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant no. 70871081 and Important Science and Technology Research Project of Henan province, China (102102210022).
Authors’ Affiliations
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