1. Introduction and Preliminaries

Many problems in mathematics, in physical sciences and in real-world applications of various technological innovations can be modeled as a convex feasibility problem (CFP). This is the problem of finding a point in the intersection of finitely many closed convex sets in a real Hilbert spaces H. That is,

(1.1)

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].

Throughout this paper, we always assume that H is a real Hilbert space, whose inner product and norm are denoted by 〈·, ·〉 and ||·||. Let C be a nonempty closed and convex subset of H and A: C → H a nonlinear mapping. Recall the following definitions:

  1. (a)

    A is said to be monotone if

  2. (b)

    A is said to be ρ-strongly monotone if there exists a positive real number ρ > 0 such that

  3. (c)

    A is said to be η-cocoercive if there exists a positive real number η > 0 such that

  4. (d)

    A is said to be relaxed η-cocoercive if there exists a positive real number η > 0 such that

  5. (e)

    A is said to be relaxed (η, ρ)-cocoercive if there exist positive real numbers η, ρ > 0 such that

The main purpose of this paper is to consider the following generalized variational inequality. Given nonlinear mappings A : C → H and B : C → H, find a u ∈ C such that

(1.2)

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).

It is easy to see that an element u ∈ C is a solution to the variational inequality (1.2) if and only if u ∈ C is a fixed point of the mapping P C (τB - λA), where P C denotes the metric projection from H onto C. Indeed, we have the following relations:

(1.3)

Next, we consider a special case of (1.2). If B = I, the identity mapping and τ = 1, then the generalized variational inequality (1.1) is reduced to the following. Find u ∈ C such that

(1.4)

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 Stam-pacchia [5]. In this paper, we use V I(C, A) to denote the set of solutions of the variational inequality (1.4).

Let S : C → C be a mapping. We use F(S) to denote the set of fixed points of the mapping S. Recall that S is said to be nonexpansive if

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 demi-closed at the origin if for each sequence {x n } in C, x n ⇀ x0 and Sx n → 0 imply Sx0 = 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.

Theorem ITT. Let C be a nonempty closed convex subset of a real Hilbert space H and let A be an α-cocoercive operator of H into H with V I(C, A) ≠ ∅. Let {x n } be a sequence defined as follows. x1 = x ∈ C and

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 (inverse-strongly monotone mappings) and nonexpansive mappings. They obtained a strong convergence theorem. More precisely, they proved the following theorem.

Theorem IT. Let C be a closed convex subset of a real Hilbert space H. Let S : C → C be a nonexpanisve mapping and A an α-cocoercive mapping of C into H such that F(S) ∩ V I(C, A) ≠ ∅. Suppose x1 = u ∈ C and {x n } is given by

for every n = 1, 2, ..., where {α n } is a sequence in [0, 1) and {λ n } is a sequence in [a, b].

If {α n } and {λ n } are chosen so that {λ n } ∈ [a, b] for some a, b with 0 < a < b < 2α,

then {x n } converges strongly to PF(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.

Lemma 1.1 [33]. Let {x n } and {y n } be bounded sequences in a Hilbert space H and {β n } a sequence in (0, 1) with

Suppose that xn+1= (1 - β n )y n + β n x n for all integers n ≥ 0 and

Then limn→∞||y n - x n || = 0.

Lemma 1.2 [34]. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let S1 : C → C and S2 : C → C be nonexpansive mappings on C. Suppose that F(S1) ∩ F (S2) is nonempty. Define a mapping S : C → C by

where a is a constant in (0, 1). Then S is nonexpansive with F(S) = F(S1) ∩ F (S2).

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 demi-closed at zero.

Lemma 1.4 [36]. Assume that {α n } is a sequence of nonnegative real numbers such that

where {γ n } is a sequence in (0, 1) and {δ n } is a sequence such that

  1. (a)

    ;

  2. (b)

    lim supn→∞δ n /γ n ≤ 0 or .

Then limn→∞α n = 0.

2. Main results

Theorem 2.1. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A m : C → H be a relaxed (η m , ρ m )-cocoercive and μ m -Lipschitz continuous mapping and B m : C → H a relaxed -cocoercive and -Lipschitz continuous mapping for each 1 ≤ m ≤ r. Assume that . Let {x n } be a sequence generated in the following manner:

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) satisfying the following restrictions:

  1. (a)

    ;

  2. (b)

    0 < lim infn→∞β n ≤ lim supn→∞β n < 1;

  3. (c)

    limn→∞α n = 0 and ;

  4. (d)

    limn→∞δ(m,n)= δ m ∈ (0, 1), ∀1 ≤ m ≤ r,

And , are two positive sequences such that

  1. (e)

    .

Then the sequence {x n } generated in the iterative process (ϒ) converges strongly to a common element , which uniquely solves the following variational inequality.

Proof. First, we prove that the mapping P C (τ m B m - λ m A m ) is nonexpansive for each 1 ≤ m ≤ r. For each x, y ∈ C, we have

(2.1)

It follows from the assumption that each A m is relaxed (η m , ρ m )-cocoercive and μ m -Lipschitz continuous that

where . This shows that

(2.2)

In a similar way, we can obtain that

(2.3)

where . Substituting (2.2) and (2.3) into (2.1), we from the condition (e) see that P C (τ m B m - λ m A m ) is nonexpansive for each 1 ≤ m ≤ r. Put

Fixing , we see that

It follows that

By mathematical inductions we arrive at

Since the mapping P C (τ m B m - λ m A m ) is nonexpansive for each 1 ≤ m ≤ r, we see that

(2.4)

where M is an appropriate constant such that

Put , for all n ≥ 1. That is,

(2.5)

Now, we estimate ||ln+1- l n ||. Note that

which combines with (2.4) yields that

It follows from the conditions (b), (c) and (d) that

It follows from Lemma 1.1 that limn→∞||l n - x n || = 0. In view of (2.5), we see that x n +1 x n = (1 - β n )(l n - x n ). It follows that

(2.6)

On the other hand, from the iterative algorithm (ϒ), we see that x n +1 - x n = α n (u - x n ) + γ n (y n - x n ). It follows from (2.6) and the conditions (b), (c) that

(2.7)

Next, we show that . To show it, we can choose a subsequence of {x n } such that

(2.8)

Since is bounded, we obtain that there exists a subsequence of which converges weakly to q. Without loss of generality, we may assume that . Next, we show that . Define a mapping R : C → C by

where δ m = limn→∞δ(m,n). From Lemma 1.2, we see that R is nonexpansive with

Now, we show that Rx n - x n → 0 as n → ∞. Note that

From the condition (d) and (2.7), we obtain that limn→∞||Rx n - x n || = 0. From Lemma 1.3, we see that

In view of (2.8), we arrive at

(2.9)

Finally, we show that as n - ∞. Note that

which implies that

(2.10)

From the condition (c), (2.9) and applying Lemma 1.4 to (2.10), we obtain that

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).

Corollary 2.2. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A m : C → H be a relaxed (η m , ρ m )-cocoercive and μ m -Lipschitz continuous mapping for each 1 ≤ m ≤ r. Assume that . Let {x n } be a sequence generated by the following manner:

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) satisfying the following restrictions.

  1. (a)

    ;

  2. (b)

    0 < lim infn→∞β n ≤ lim supn→∞β n < 1;

  3. (c)

    limn→∞α n = 0 and ;

  4. (d)

    limn→∞δ(m,n)= δ m ∈ (0, 1), ∀1 ≤ m ≤ r, and is a positive sequence such that

  5. (e)

    , ∀1 ≤ m ≤ r.

Then the sequence {x n } converges strongly to a common element , which uniquely solves the following variational inequality

If r = 1, then Theorem 2.1 is reduced to the following.

Corollary 2.3. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A : C → H be a relaxed (η, ρ)-cocoercive and μ-Lipschitz continuous mapping and B : C → H a relaxed -cocoercive and -Lipschitz continuous mapping. Assume that GV I(C, B, A) is not empty. Let {x n } be a sequence generated in the following manner:

where u ∈ C is a fixed point, {α n }, {β n } and {γ n } are sequences in (0, 1) satisfying the following restrictions.

  1. (a)

    α n + β n + γ n = 1, ∀ n ≥ 1;

  2. (b)

    0 < lim infn→∞β n ≤ lim supn→∞β n < 1;

  3. (c)

    limn→∞α n = 0 and

  4. (d)

    .

Then the sequence {x n } converges strongly to a common element , which uniquely solves the following variational inequality

For the variational inequality (1.4), we can obtain from Corollary 2.3 the following immediately.

Corollary 2.4. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A : C → H be a relaxed (η, ρ)-cocoercive and μ-Lipschitz continuous mapping. Assume that V I(C, A) is not empty. Let {x n } be a sequence generated in the following manner:

where u ∈ C is a fixed point, {α n }, {β n } and {γ n } are sequences in (0, 1) satisfying the following restrictions.

  1. (a)

    α n + β n + γ n = 1, ∀n ≥ 1;

  2. (b)

    0 < lim infn→∞β n ≤ lim supn→∞β n < 1;

  3. (c)

    limn→∞α n = 0 and ;

  4. (d)

    .

Then the sequence {x n } converges strongly to a common element , which uniquely solves the following variational inequality

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.