Throughout this article, we always assume that H is a real Hilbert space with the inner product ⟨ ·, ·⟩, and the norm ||·|| and that C is a nonempty closed convex subset of H.

Let A : C → H be a mapping. Recall that A is said to be monotone if

⟨ A x - A y , x - y ⟩ ≥ 0 , ∀ x , y ∈ C .

A is said to be inverse strongly-monotone if there exists a constant α > 0 such that

⟨ A x - A y , x - y ⟩ ≥ α A x - A y 2 , ∀ x , y ∈ C .

For such a case, A is also said to be α-inverse strongly monotone.

Let M : H → 2^H be a set-valued mapping. The set D(M) defined by D(M) = {x ∈ H: Mx ≠ ∅} is said to be the domain of M. The set R(M) defined by R ( M ) = ⋃ x ∈ H M x is said to be the range of M. The set G(M) defined by G(M) = {(x, y) ∈ H × H: x ∈ D(M), y ∈ R(M)} is said to be the graph of M.

Recall that M is said to be monotone if

⟨ x - y , f - g ⟩ ≥ 0 , ∀ ( x , f ) , ( y , g ) ∈ G ( M ) .

M is said to be maximal monotone if it is not properly contained in any other monotone operator. Equivalently, M is maximal monotone if R(I + rM) = H for all r > 0. The class of monotone mappings is one of the most important classes of mappings. Within the past several decades, many authors have been devoting to the studies on the existence and convergence of zero points for maximal monotone mappings, see 123456789101112131415 and the references therein. For a maximal monotone operator M on H and r > 0, we may define the single-valued resolvent J_r = (I + rM)^-1 : H → D(M). It is known that J_r is firmly nonexpansive and M^-1 (0) = F(J_r), where F (J_r) denotes the fixed point set of J_r.

Let S : C → C be a nonlinear mapping. In this study, we use F (S) to denote the fixed point set of S. Recall that the mapping S is said to be nonexpansive if

S x - S y ≤ x - y , ∀ x , y ∈ C .

S is said to be κ-strictly pseudocontractive if there exists a constant κ ∈ [0, 1) such that

S x - S y 2 ≤ x - y 2 + κ ( x - S x ) - ( y - S y ) 2 , ∀ x , y ∈ C .

The class of strictly pseudocontractive mappings was introduced by Browder and Petryshyn 16. If κ = 0, the class of strictly pseudocontractive mappings is reduced to the class of nonexpansive mappings. In case that κ = 1, we call S a pseudocontractive mapping. Marino and Xu 17 proved that fixed point sets of strictly pseudocontractive mappings are closed and convex. They also proved that I - S is demi-closed at zero. To be more precise, if {x_n} is a sequence in C with x_n⇀ x and x_n - Sx_n → 0, then x ∈ F (S).

Let A : C → H be an inverse strongly-monotone mapping. Recall that the classical variational inequality problem is to find x ∈ C such that

⟨ A x , y - x ⟩ ≥ 0 , ∀ y ∈ C .

Denote by V I(C, A) of the solution set of (1.1). It is known that x ∈ C is a solution to (1.1) if and only if x is a fixed point of the mapping P_C (I - λA), where λ > 0 is a constant and I is the identity mapping. In 3, Iiduka and Takahashi showed that if λ ∈ [0, 2α], then I - λA is nonexpansive.

Let F be a bifunction from C × C to ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem.

Find x ∈ C such that F ( x , y ) ≥ 0 , ∀ y ∈ C .

To study the equilibrium problems (1.2), we may assume that F satisfies the following conditions:

(A1) F (x, x) = 0 for all x ∈ C;

(A2) F is monotone, i.e., F (x, y) + F (y, x) ≤ 0 for all x, y ∈ C;

(A3) for each x, y, z ∈ C,

lim sup t ↓ 0 F ( t z + ( 1 - t ) x , y ) ≤ F ( x , y ) ;

(A4) for each x ∈ C, y ↦ F (x, y) is convex and lower semi-continuous.

Putting F (x, y) = ⟨Ax, y - x⟩ for every x, y ∈ C, we see that the equilibrium problem (1.2) is reduced to the variational inequality (1.1).

Recently, many authors considered the convergence of iterative sequences for the variational inequality (1.1), the equilibrium problem (1.2) and fixed point problems of nonlinear mappings; see, for example, 245671112131415181920212223242526.

In 2003, Takahashi and Toyoda 13 proved the following weak convergence theorem.

Theorem 1.1. Let C be a closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly-monotone mapping from C into H and S be a non-expansive mapping from C into itself such that F(S) ∩ VI (C, A) ≠ ∅. Let {x_n} be a sequence generated by

x 0 ∈ C , x n + 1 = α n x n + ( 1 - α n ) S P C ( x n - λ n A x n ) , ∀ n ≥ 0 ,

where λ_n ∈ [a, b] for some a, b ∈ (0, 2α) and α_n ∈ [c, d] for some c, d ∈ (0, 1). Then, {x_n} converges weakly to z ∈ F(S) ∩ VI (C, A), where z = lim_n→∞ P_{F(S)∩VI (C, A)}x_n.

In 2007, Tada and Takahashi 11 obtained the following weak convergence theorem.

Theorem 1.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C to ℝ satisfying (A1)-(A4) and S be a nonexpansive mapping from C into H such that F (S) ∩ EP (F) ≠ ∅. Let {x_n} and {u_n} be sequences generated by x₁ = x ∈ H and let

u n ∈ C s u c h t h a t F ( u n , u ) + 1 r n ⟨ u - u n , u n - x n ⟩ ≥ 0 , ∀ u ∈ C , x n + 1 = α n x n + ( 1 - α n ) S u n

for each n ≥ 1, where {α_n} ⊂ [a, b] for some a, b ∈ (0, 1) and {r_n} ⊂ (0, ∞) satisfies lim inf_n→∞ r_n> 0. Then, {x_n} converges weakly to w ∈ F(S)∩EP(F) where w = lim_n→∞ P_F(S)∩EP(F)x_n.

A very common problem in diverse areas of mathematics and physical sciences consists of trying to find a point in the intersection of convex sets. This problem is referred to as the convex feasibility problem; its precise mathematical formulation is as follows. Find an x ∈ ⋂ m = 1 N C r , where N ≥ 1 is an integer and each C_m is a nonempty closed convex subset of H. There is a considerable investigation on the convex feasibility problem in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomography, and radiation therapy treatment planning.

Let K be an integer, S : C → C a strict pseudocontraction, A_m : C → H be an α_m-inverse strongly-monotone mapping and M_m : H → 2^H be a maximal monotone operator such that D(M_m) ⊂ C, where D(M_m) is the domain of M_m, where m ∈ {1, 2, ..., K}. In this article, motivated by Theorems 1.1 and 1.2, we consider the problem of finding a common element in the following set: F ( S ) ∩ ⋂ m = 1 K ( A m + M m ) - 1 ( 0 ) , where F (S) is the fixed point set of S and (A_m + M_m)^-1 (0) is the zero point set of A_m + M_m. Weak convergence theorems of common elements are established in real Hilbert spaces. The results presented in this article improve and extend the corresponding results announced by Tada and Takahashi 11 and Takahshi and Toyoda 13.

In order to prove our main results, we also need the following lemmas.

Lemma 1.3. 16 Let C be a nonempty closed convex subset of a real Hilbert space H and S : C → C be a κ-strict pseudo-contraction with a fixed point. Define S : C → C by S_ax = ax + (1 - a)Sx for each x ∈ C. If a ∈ [κ, 1), then S_a is nonexpansive with F (S_a) = F (S).

Lemma 1.4. 17 Let C be a nonempty closed convex subset of a real Hilbert space H and S : C → C be a κ-strict pseudocontraction. Then

(a) S is 1 + κ 1 - κ -Lipschitz;

(b) I - S is demi-closed, this is, if {x_n} is a sequence in C with x_n ⇀ x and x_n - Sx_n → 0, then x ∈ F (S).

Lemma 1.5. 27 Let H be a real Hilbert space and 0 < p ≤ t_n ≤ q < 1 for all n ≥ 1. Suppose that {x_n} and {y_n} are sequences in H such that

lim sup n → ∞ x n ≤ r , lim sup n → ∞ y n ≤ r

and

lim n → ∞ t n x n + ( 1 - t n ) y n = r

hold for some r ≥ 0. Then lim_n→∞ ||x_n-y_n|| = 0.

Lemma 1.6. 13 Let C be a nonempty closed convex subset of a real Hilbert space H and P_C be the metric projection from H onto C. Let {x_n} be a sequence in H. Suppose that, for all y ∈ C

x n + 1 - y ≤ x n - y , ∀ n ≥ 1 .

Then {P_C x_n} converges strongly to some z ∈ C.

Lemma 1.7. 28 Let C be a nonempty closed convex subset of a real Hilbert space H, A : C → H be a mapping and M : H → 2^H be a maximal monotone mapping. Then

F ( J r ( I - r A ) ) = ( A + M ) - 1 ( 0 ) , ∀ r > 0 .

Lemma 1.8. 29 Let H be a Hilbert space and suppose {x_n} converges weakly to x. Then

lim inf n → ∞ x n - x < lim inf n → ∞ x n - y

for all y ∈ H with x ≠ y.

Theorem 2.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S : C → C be a κ-strict pseudocontraction, A : C → H be an α-inverse strongly monotone mapping and B : C → H be a β-inverse strongly monotone mapping. Let M : H → 2^H and W : H → 2^H be maximal monotone operators such that D(M) ⊂ C and D(W) ⊂ C. Assume that F : = F ( S ) ∩ ( A + M ) - 1 ( 0 ) ∩ ( B + W ) - 1 ( 0 ) ≠ ∅ . Let {x_n} be a sequence generated in the following manner:

x 0 ∈ C , y n = γ n J r n ( x n - r n A x n ) + ( 1 - γ n ) J s n ( x n - s n B x n ) , x n + 1 = α n x n + ( 1 - α n ) ( β n y n + ( 1 - β n ) S y n ) , n ≥ 0 ,

where J r n = ( I + r n M ) - 1 , J s n = ( I + s n W ) - 1 , {r_n} is a sequence in (0, 2α), {s_n} is a sequence in (0, 2β) and {α_n}, {β_n}, and {γ_n} are sequences in (0, 1). Assume that the following restrictions are satisfied

(a) 0 < a ≤ r_n ≤ b < 2α and 0 < c ≤ s_n ≤ d < 2β;

(b) 0 ≤ κ ≤ β_n < e < 1, 0 < h ≤ α_n ≤ i < 1 and 0 < j ≤ γ_n ≤ k < 1,

where a, b, c, d, e, h, i, j, k are real numbers. Then the sequence {x_n} converges weakly to x ̄ ∈ F , where x ̄ = lim n → ∞ P F x n .

Proof. Note that (I - r_nA) and (I - s_nB) are nonexpansive for each fixed n ≥ 0.

Indeed, for any x, y, ∈ C, we see from the restriction (a) that

( I - r n A ) x - ( I - r n A ) y 2 = x - y 2 - 2 r n ⟨ x - y , A x - A y ⟩ + r n 2 A x - A y 2 ≤ x - y 2 - r n ( 2 α - r n ) A x - A y 2 ≤ x - y 2 .

This shows that (I - r_nA) is nonexpansive for each fixed n ≥ 0, so is (I - s_nB).

Put

S n x = β n x + ( 1 - β n ) S x , ∀ x ∈ C .

In the restriction (b), we obtain from Lemma 1.3 that S_n is nonexpansive for each fixed n ≥ 0. Fixing p ∈ F and since J r n , J s n , I - r_nA, and I - s_nB are nonexpansive, we see that

y n - p ≤ γ n J r n ( x n - r n A x n ) - p + ( 1 - γ n ) J s n ( x n - s n B x n ) - p ≤ x n - p .

Since S_n is nonexpansive, we see that

x n + 1 - p | ≤ α n x n - p + ( 1 - α n ) S n y n - p ≤ α n x n - p + ( 1 - α n ) y n - p ≤ x n - p .

Hence, the limit of the sequence {||x_n - p||} exists. This shows that the sequence {x_n} is bounded, so is {y_n}. Without loss of generality, we may assume that lim_n→∞ ||x_n-p|| = d > 0. Notice that

y n - p 2 ≤ γ n J r n ( x n - r n A x n ) - p 2 + ( 1 - γ n ) J s n ( x n - s n B x n ) - p 2 ≤ γ n ( x n - r n A x n ) - p 2 + ( 1 - γ n ) ( x n - s n B x n ) - p 2 ≤ γ n x n - p 2 - r n ( 2 α - r n ) A x n - A p 2 + ( 1 - γ n ) ( x n - p 2 - s n ( 2 β - s n ) B x n - B p 2 ) ≤ x n - p 2 - r n γ n ( 2 α - r n ) A x n - A p 2 - s n ( 1 - γ n ) ( 2 β - s n ) B x n - B p 2 .

This in turn implies that

x n + 1 - p 2 ≤ α n x n - p 2 + ( 1 - α n ) S n y n - p 2 ≤ α n x n - p 2 + ( 1 - α n ) y n - p 2 ≤ x n - p 2 - ( 1 - α n ) r n γ n ( 2 α - r n ) A x n - A p 2 - ( 1 - α n ) s n ( 1 - γ n ) ( 2 β - s n ) B x n - B p 2 .

It follows from the restrictions (a) and (b) that

( 1 - i ) a j ( 2 α - b ) A x n - A p 2 ≤ x n - p 2 - x n + 1 - p 2 .

Since lim_n→∞ ||x_n-p|| = d, we see that

lim n → ∞ A x n - A p = 0 .

In view of (2.2), we see from the restrictions (a) and (b) that

( 1 - i ) c ( 1 - k ) ( 2 β - d ) B x n - B p 2 ≤ x n - p 2 - x n + 1 - p 2 .

Since lim_n→∞ ||x_n-p|| = d, we see that

lim n → ∞ B x n - B p = 0 .

Notice that Jrn is firmly nonexpansive. Putting u n = J r n ( x n - r n A x n ) and v n = J s n ( x n - s n B x n ) , we see that

u n - p 2 = J r n ( x n - r n A x n ) - J r n ( p - r n A p ) 2 ≤ ⟨ u n - p , ( x n - r n A x n ) - ( p - r n A p ) ⟩ = 1 2 u n - p 2 + ( x n - r n A x n ) - ( p - r n A p ) 2 - ( u n - p ) - ( ( x n - r n A x n ) - ( p - r n A p ) ) 2 ≤ 1 2 u n - p 2 + x n - p 2 - u n - x n + r n ( A x n - A p ) 2 = 1 2 u n - p 2 + x n - p 2 - u n - x n 2 - r n 2 A x n - A p 2 - 2 r n ⟨ u n - x n , A x n - A p ⟩ ≤ 1 2 u n - p 2 + x n - p 2 - u n - x n 2 + 2 r n u n - x n A x n - A p .

This in turn implies that

u n - p 2 ≤ x n - p 2 - u n - x n 2 + 2 r n u n - x n A x n - A p .

In a similar way, we can obtain that

v n - p 2 ≤ x n - p 2 - v n - x n 2 + 2 s n v n - x n B x n - B p .

Combining (2.5) with (2.6) yields that

x n + 1 - p 2 ≤ α n x n - p 2 + ( 1 - α n ) S n y n - p 2 ≤ α n x n - p 2 + ( 1 - α n ) y n - p 2 ≤ α n x n - p 2 + ( 1 - α n ) ( γ n u n - p 2 + ( 1 - γ n ) v n - p 2 ) ≤ x n - p 2 - ( 1 - α n ) γ n u n - x n 2 + 2 r n u n - x n A x n - A p - ( 1 - α n ) ( 1 - γ n ) v n - x n 2 + 2 s n v n - x n B x n - B p .

It follows that

( 1 - α n ) γ n u n - x n 2 ≤ x n - p 2 - x n + 1 - p 2 + 2 r n u n - x n A x n - A p + 2 s n v n - x n B x n - B p .

In view of (2.3) and (2.4), we see from the restrictions (a) and (b) that

lim n → ∞ u n - x n = 0 .

It also follows from (2.7) that

( 1 - α n ) ( 1 - γ n ) v n - x n 2 ≤ x n - p 2 - x n + 1 - p 2 + 2 r n u n - x n A x n - A p + 2 s n v n - x n B x n - B p .

In view of (2.3) and (2.4), we see from the restrictions (a) and (b) that

lim n → ∞ v n - x n = 0

Notice that

y n - x n ≤ u n - x n + v n - x n .

It follows from (2.8) and (2.9) that

lim n → ∞ y n - x n = 0 .

On the other hand, we have

lim n → ∞ α n ( x n - p ) + ( 1 - α n ) ( S n y n - p ) = d .

Notice that

S n y n - p ≤ y n - p ≤ x n - p .

This implies that

lim sup n → ∞ S n y n - p ≤ d .

In view of Lemma 1.5, we arrive at

lim n → ∞ S n y n - x n = 0 .

Note that

S y n - x n = S n y n - x n 1 - β n + β n ( x n - y n ) 1 - β n .

From (2.10), (2.11) and the restriction (b), we get that

lim n → ∞ S y n - x n = 0 .

On the other hand, we see from Lemma 1.4 that

S x n - x n ≤ S x n - S y n + S y n - x n ≤ 1 + κ 1 - κ x n - y n + S y n - x n .

It follows from (2.10) and (2.12) that

lim n → ∞ S x n - x n = 0 .

Since {x_n} is bounded, we see that there exists a subsequence { x n i } of {x_n} which converges weakly to x ̄ . By virtue of Lemma 1.4, we obtain that x ̄ ∈ F ( S ) . Next, we show that x ̄ ∈ ( A + M ) - 1 ( 0 ) . Notice that

x n - r n A x n ∈ u n + r n M u n .

Let μ ∈ Mν. Since M is monotone, we have

x n - u n r n - A x n - μ , u n - ν ≥ 0 .

In view of the restriction (a), we see from (2.8) that

⟨ - A x ̄ - μ , x ̄ - ν ⟩ ≥ 0 .

This implies that - A x ̄ ∈ M x ̄ , that is, x ̄ ∈ ( A + M ) - 1 ( 0 ) . In a similar way, we can obtain that x ̄ ∈ ( B + W ) - 1 ( 0 ) . This proves that x ̄ ∈F .

Assume that there exists another subsequence { x n j } of {x_n} such that {xnj} converges weakly to x'. By the above proof, we also have x ′ ∈ F . If x ̄ ≠ x ′ , we get from Lemma 1.8 that

lim n → ∞ x n - x ̄ = lim inf i → ∞ x n i - x ̄ < lim inf i → ∞ x n i - x ′ = lim n → ∞ x n - x ′ = lim inf j → ∞ x n j - x ′ < lim inf j → ∞ x n j - x ̄ = lim n → ∞ x n - x ̄ .

This derives a contradiction. Hence, we have x ̄ = x ′ . This implies that x n ⇀ x ̄ ∈ F . Let e n = P F x n . In view of (2.1), we obtain from Lemma 1.6 that {e_n} converges strongly to some e ∈ F . On the other hand, we see from x ̄ ∈F that ⟨ x n - e n , e n - x ̄ ⟩ ≥ 0 . Note that {x_n} converges weakly to x ̄ . It follows that

⟨ x ̄ - e , e - x ̄ ⟩ ≥ 0 .

This implies that x ̄ = e = lim n → ∞ P F x n . The proof is completed. □

From Theorem 2.1, we can obtain the following immediately.

Theorem 2.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S : C → C be a κ-strict pseudocontraction, A_m : C → H be an α_m-inverse strongly monotone mapping and M_m : H → 2^H be a maximal monotone operator such that D(M_m) ⊂ C, where m ∈ {1, 2, ..., K}. Assume that F : = F ( S ) ∩ ⋂ m = 1 N ( A m + M m ) - 1 ( 0 ) ≠ ∅ . Let {x_n} be a sequence generated in the following manner:

x 0 ∈ C , y n = ∑ m = 1 K γ n , m J r n , m ( x n - r n , m A m x n ) , x n + 1 = α n x n + ( 1 - α n ) ( β n y n + ( 1 - β n ) S y n ) , n ≥ 0 ,

where J r n , m = ( I + r n , m M m ) - 1 , {r_n,m} is a sequence in (0, 2α_m), {α_n}, {β_n}, and {γ_n,m} are sequences in (0, 1). Assume that the following restrictions are satisfied

(a) 0 < a_m ≤ r_n,m ≤ b_m < 2α_m for each m ∈ {1, 2, ..., K};

(b) ∑ m = 1 K γ n , m = 1 ;

(c) 0 ≤ k ≤ β_n < c < 1, 0 < d ≤ α_n ≤ e < 1 and 0 < h_m ≤ γ_n,m ≤ i_m < 1, where a₁, a₂, ..., a_K, b₁, b₂, ..., b_K, c, d, e, h₁, h₂, ..., h_K, i₁, i₂, ..., i_K are real numbers. Then the sequence {x_n} converges weakly to x ̄ ∈F , where x ̄ =limn→∞PFxn .

If S = I, where I denotes the identity, then Theorem 2.2 is reduced to the following.

Corollary 2.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A_m : C → H be an α_m-inverse strongly monotone mapping and M_m : H → 2^H be a maximal monotone operator such that D(M_m) ⊂ C, where m ∈ {1, 2, ..., K}. Assume that F : = ⋂ m = 1 N ( A m + M m ) - 1 ( 0 ) ≠ ∅ . Let {x_n} be a sequence generated in the following manner:

x 0 ∈ C , x n + 1 = α n x n + ( 1 - α n ) ∑ m = 1 K γ n , m J r n , m ( x n - r n , m A m x n ) , n ≥ 0 ,

where J r n , m = ( I + r n , m M m ) - 1 , {r_n,m} is a sequence in (0, 2α_m) and {α_n}, {β_n} and {γ_n,m} are sequences in (0, 1). Assume that the following restrictions are satisfied

(a) 0 < a_m ≤ r_n,m ≤ b_m < 2α_m for each m ∈ {1, 2, ..., K};

(b) ∑ m = 1 K γ n , m = 1 ;

(c) 0 < c ≤ α_n < d < 1 and 0 < h_m ≤ γ_n,m ≤ i_m < 1,

where a₁, a₂, ..., a_K, b₁, b₂, ..., b_K, c, d, h₁, h₂, ..., h_K, i₁, i₂, ..., i_K are real numbers. Then the sequence {x_n} converges weakly to x ̄ ∈F , where x ̄ =limn→∞PFxn .

Let H be a Hilbert space and f : H → (-∞, +∞] a proper convex lower semicontinuous function. Then the subdifferential ∂f of f is defined as follows:

∂ f ( x ) = { y ∈ H : f ( z ) ≥ f ( x ) + ⟨ z - x , y ⟩ , z ∈ H } , ∀ x ∈ H .

From Rockafellar 930, we know that ∂f is maximal monotone. It is easy to verify that 0 ∈ ∂f(x) if and only if f(x) = min_y∈H f(y).

First, we consider the problem of finding common minimizers of proper convex lower semicontinuous functions.

Theorem 3.1. Let H be a real Hilbert space. Let f : H → (-∞, +∞] and g : H → (-∞, +∞] be proper convex lower semi-continuous functions. Assume that F : = ( ∂ f ) - 1 ( 0 ) ∩ ( ∂ g ) - 1 ( 0 ) ≠ ∅ . Let {x_n} be a sequence generated in the following manner:

x 0 ∈ H , z n = arg min z ∈ H { g ( z ) + z - x n 2 2 s n } , y n = arg min z ∈ H { f ( z ) + z - x n 2 2 r n } , x n + 1 = α n x n + ( 1 - α n ) ( γ n y n + ( 1 - γ n ) z n ) , n ≥ 0 ,

where {α_n}, {β_n}, and {γ_n} are sequences in (0, 1). Assume that the following restrictions are satisfied

(a) 0 < a ≤ r_n ≤ b < ∞ and 0 < c ≤ s_n ≤ d < ∞;

(b) 0 < h ≤ α_n ≤ i < 1 and 0 < j ≤ γ_n ≤ k < 1,

where a, b, c, d, h, i, j, k are real numbers. Then the sequence {x_n} converges weakly to x ̄ ∈F , where x ̄ =limn→∞PFxn .

Proof. Putting A = B = 0 and S = I, the identity mapping, we can conclude from Theorem 2.1 the desired conclusion immediately. □

Let I_C be the indicator function of C, i.e.,

I C ( x ) = 0 , x ∈ C , + ∞ , x ∉ C .

Since I_C is a proper lower semicontinuous convex function on H, we see that the subdifferential ∂I_C of I_C is a maximal monotone operator.

Lemma 3.2. 12 Let C be a nonempty closed convex subset of a real Hilbert space H. Let P_C be the metric projection from H onto C, ∂I_C be the subdifferential of I_C, where I_C is as defined in (3.1) and J_r = (I + r∂I_C )^-1. Then

y = J r x ⇔ y = P C x , x ∈ H , y ∈ C .

Second, we consider the variation inequality (1.1).

Theorem 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H and P_C be the metric projection from H onto C. Let S : C → C be a κ-strict pseudocontraction, A : C → H be an α-inverse strongly monotone mapping and B : C → H be a β-inverse strongly monotone mapping. Assume that F : = F ( S ) ∩ V I ( C , A ) ∩ V I ( C , B ) ≠ ∅ . Let {x_n} be a sequence generated in the following manner:

x 0 ∈ C , y n = γ n P C ( x n - r n A x n ) + ( 1 - γ n ) P C ( x n - s n B x n ) , x n + 1 = α n x n + ( 1 - α n ) ( β n y n + ( 1 - β n ) S y n ) , n ≥ 0 ,

where {r_n} is a sequence in (0, 2α), {s_n} is a sequence in (0, 2β) and {α_n}, {β_n} and {γ_n} are sequences in (0, 1). Assume that the following restrictions are satisfied

(a) 0 < a ≤ r_n ≤ b < 2α and 0 < c ≤ s_n ≤ d < 2β;

(b) 0 ≤ κ ≤ β_n < e < 1, 0 < h ≤ α_n ≤ i < 1 and 0 < j ≤ γ_n ≤ k < 1,

where a, b, c, d, e, h, i, j, k are real numbers. Then the sequence {x_n} converges weakly to x ̄ ∈F , where x ̄ =limn→∞PFxn .

Proof. Put M = W = ∂I_C. Next, we show that V I(C, A) = (A + ∂I_C)^-1(0) and VI(C, B) = (B + ∂I_C)^-1(0), respectively. Notice that

x ∈ ( A + ∂ I C ) - 1 ( 0 ) ⇔ 0 ∈ A x + ∂ I C x ⇔ - A x ∈ ∂ I C x ⇔ ⟨ A x , y - x ⟩ ≥ 0 ⇔ x ∈ V I ( C , A )

In the same way, we can obtain that x ∈ (B + ∂I_C)^-1 ⇔ (0) x ∈ V I(C, B). From Lemma 3.2, we can conclude the desired conclusion immediately. □

Remark 3.1. Let S be a nonexpansive mapping, A = B, M = W and β_n = 0 in Theorem 3.3. Then Theorem 3.3 is reduced to Theorem 1.1 in Section 1.

Third, we consider the problem of finding common fixed points of three strict pseudocontractions.

Theorem 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S : C → C be a κ-strict pseudocontraction, T : C → C be an α-strict pseudocontraction and R : C → C be a β-strict pseudocontraction. Assume that F : = F ( R ) ∩ F ( S ) ∩ F ( T ) ≠ ∅ . Let {x_n} be a sequence generated in the following manner:

x 0 ∈ C , y n = γ n ( ( 1 - r n ) x n + r n T x n ) + ( 1 - γ n ) ( ( 1 - s n ) x n + s n R x n ) , x n + 1 = α n x n + ( 1 - α n ) ( β n y n + ( 1 - β n ) S y n ) , n ≥ 0 ,

where {r_n} is a sequence in (0, 1 - α), {s_n} is a sequence in (0, 1 - β) and {α_n}, {β_n} and {γ_n} are sequences in (0, 1). Assume that the following restrictions are satisfied

(a) 0 < a ≤ r_n ≤ b < 1 - α and 0 < c ≤ s_n ≤ d < 1 - β;

(b) 0 ≤ κ ≤ β_n < e < 1, 0 < h ≤ α_n ≤ i < 1 and 0 < j ≤ γ_n ≤ k < 1,

where a, b, c, d, e, h, i, j, k are real numbers. Then the sequence {x_n} converges weakly to x ̄ ∈F , where x ̄ =limn→∞PFxn .

Proof. Putting A = I - T, we see that A is 1 - α 2 -inverse-strongly monotone. We also have F (T) = V I(C, A) and P_C (x_n - r_nAx_n) = (1 - r_n)x_n + r_nTx_n. Putting B = I - R, we see that B is 1 - β 2 -inverse-strongly monotone. We also have F(R) = V I(C, B) and P_C (x_n - s_nBx_n) = (1 - s_n)x_n + s_nRu_n. In view of Theorem 3.2, we can obtain the desired result immediately. □

The following lemma can be found in 3132.

Lemma 3.5. Let C be a nonempty closed convex subset of a real Hilbert space H and let F be a bifunction from C × C to ℝ which satisfies (A1)-(A4). Then, for any r > 0 and x∈ H, there exists z∈ C such that

F ( z , y ) + 1 r ⟨ y - z , z - x ⟩ ≥ 0 , ∀ y ∈ C .

Further, define

T r x = z ∈ C : F ( z , y ) + 1 r ⟨ y - z , z - x ⟩ ≥ 0 , ∀ y ∈ C

for all r > 0 and x ∈ H. Then, the following hold:

(a) T_r is single-valued;

(b) T_r is firmly nonexpansive, i.e., for any x, y ∈ H.,

T r x - T r y 2 ≤ ⟨ T r x - T r y , x - y ⟩ ;