Hybrid Algorithms for Solving Variational Inequalities, Variational Inclusions, Mixed Equilibria, and Fixed Point Problems

Theorem 1 (see [35], Theorem 3.2.)Let X be a strictly convex and reflexive Banach space with a uniformly Gateaux differentiable norm, let C be a nonempty closed convex subset of X, let $$ be an infinite family of nonexpansive self-mappings on C such that the common fixed point set $$, and let f : C → C be a ρ-contraction with the contraction coefficient ρ ∈ (1/2, 1). Let $$ be a sequence of positive numbers in (0, b] for some b ∈ (0,1). For any given x₁ ∈ C, let $$ be the iterative sequence defined by

where $$ and $$ are two sequences in (0,1) with α_n + β_n ≤ 1(n ≥ 1), $$ is a sequence in [0,1], and W_n is the W-mapping defined by (14). Assume that

• (i)

  lim⁡_n→∞α_n = 0, $$ and 0 < lim⁡ inf⁡_n→∞β_n ≤ lim⁡ sup⁡_n→∞β_n < 1,

• (ii)

  lim⁡_n→∞ | γ_n+1 − γ_n | = 0 and lim⁡ sup⁡_n→∞γ_n < 1.

Then there hold the following:

• (I)

  lim⁡_n→∞∥x_n+1 − x_n∥ = 0;

• (II)

  the sequence $$ converges strongly to some $$, provided lim⁡_n→∞γ_n = 0 and β_n ≡ β for some fixed β ∈ (0,1), which is the unique solution of the VIP:

  $$ (16)

•  

  where J is the normalized duality mapping of X.

Furthermore, let λ_n,1, λ_n,2, …, λ_n,N ∈ (0,1], n ≥ 1. Given the nonexpansive mappings S₁, S₂, …, S_N on H, Atsushiba and Takahashi [36] defined, for each n ≥ 1, mappings U_n,1, U_n,2, …, U_n,N by

The W_n is called the W-mapping generated by S₁, …, S_N and λ_n,1, λ_n,2, …, λ_n,N. Note that the nonexpansivity of S_i implies the nonexpansivity of W_n.

In 2008, Colao et al. [37] introduced and studied an iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space H. Subsequently, combining Yamada’s hybrid steepest-descent method [38] and Colao et al.’s hybrid viscosity approximation method [37], Ceng et al. [7] proposed and analyzed the following hybrid iterative method for finding a common element of the set of solutions of GMEP⁡ (5) and the set of fixed points of a finite family of nonexpansive mappings $$.

Theorem 2 (see [7], Theorem 3.1.)Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ : C × C → R be a bifunction satisfying assumptions (A1)–(A4) and let φ : C → R be a lower semicontinuous and convex function with restriction (B1) or (B2). Let the mapping A : H → H be δ-inverse strongly monotone, and let $$ be a finite family of nonexpansive mappings on H such that $$. Let F : H → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f : H → H a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let 0 < μ < 2η/κ² and 0 ≤ γρ < τ, where $$. Suppose that {α_n} and {β_n} are two sequences in (0,1), {γ_n} is a sequence in (0,2δ], and $$ is a sequence in [a, b] with 0 < a ≤ b < 1. For every n ≥ 1, let W_n be the W-mapping generated by S₁, …, S_N and λ_n,1, λ_n,2, …, λ_n,N. Given x₁ ∈ H arbitrarily, suppose that the sequences {x_n} and {u_n} are generated iteratively by

where the sequences {α_n}, {β_n}, {r_n} and the finite family of sequences $$ satisfy the following conditions:

• (i)

  lim⁡_n→∞α_n = 0 and $$;

• (ii)

  0 < lim⁡ inf⁡_n→∞β_n ≤ lim⁡ sup⁡_n→∞β_n < 1;

• (iii)

  0 < lim⁡ inf⁡_n→∞r_n ≤ lim⁡ sup⁡_n→∞r_n < 2δ and lim⁡_n→∞(r_n+1 − r_n) = 0;

• (iv)

  lim⁡_n→∞(λ_n+1,i − λ_n,i) = 0 for all i = 1,2, …, N.

Then, both {x_n} and {u_n} converge strongly to $$, where $$ is a unique solution of the variational inequality:

On the other hand, whenever C = H a real Hilbert space, Yao et al. [4] very recently introduced and analyzed an iterative algorithm for finding a common element of the set of solutions of GMEP⁡ (5), the set of solutions of the variational inclusion (11), and the set of fixed points of an infinite family of nonexpansive mappings.

Theorem 3 (see [4], Theorem 3.2.)Let φ : H → R be a lower semicontinuous and convex function and let Θ : H × H → R be a bifunction satisfying conditions (A1)–(A4) and (B1). Let V be a strongly positive bounded linear operator with coefficient μ > 0 and let R : H → 2^H be a maximal monotone mapping. Let the mappings A, B : H → H be α-inverse strongly monotone and β-inverse strongly monotone, respectively. Let f : H → H be a ρ-contraction. Let r > 0, γ > 0, and λ > 0 be three constants such that r < 2α, λ < 2β, and 0 < γ < μ/ρ. Let $$ be a sequence of positive numbers in (0, b] for some b ∈ (0,1) and $$ an infinite family of nonexpansive self-mappings on H such that $$. For arbitrarily given x₁ ∈ H, let the sequence {x_n} be generated by

where {α_n}, {β_n} are two real sequences in [0,1] and W_n is the W-mapping defined by (14) (with X = H and C = H). Assume that the following conditions are satisfied:

• (C1)

  lim⁡_n→∞α_n = 0 and $$;

• (C2)

  0 < lim⁡ inf⁡_n→∞β_n ≤ lim⁡ sup⁡_n→∞β_n < 1.

Then, the sequence {x_n} converges strongly to x^* ∈ Ω, where x^* = P_Ω(γf(x^*)+(I − V)x^*) is a unique solution of the VIP:

Proposition 4 (see [31], [39].)For given x ∈ H and z ∈ C:

• (i)

  z = P_Cx⇔〈x − z, y − z〉 ≤ 0, ∀y ∈ C;

• (ii)

  z = P_Cx⇔∥x−z∥² ≤ ∥x−y∥² − ∥y−z∥², ∀y ∈ C;

• (iii)

  $$, ∀y ∈ H.

Consequently, P_C is nonexpansive and monotone.

If A is an α-inverse strongly monotone mapping (α-ism) of C into H, then it is obvious that A is 1/α-Lipschitzian. We also have that, for all u, v ∈ C and λ > 0,

So, if λ ≤ 2α, then I − λA is a nonexpansive mapping from C to H.

It is also easy to see that a projection P_C is 1-ism. Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.

A set-valued mapping R : D(R) ⊂ H → 2^H is called monotone if, for all x, y ∈ D(R), f ∈ R(x) and g ∈ R(y) imply

A set-valued mapping R is called maximal monotone if R is monotone and (I + λR)D(R) = H for each λ > 0, where I is the identity mapping of H. We denote by G(R) the graph of R. It is known that a monotone mapping R is maximal if and only if, for (x, f) ∈ H × H, 〈f − g, x − y〉≥0 for every (y, g) ∈ G(R) implies f ∈ R(x).

Let A : C → H be a monotone, k-Lipschitzian mapping and let N_Cv be the normal cone to C at v ∈ C; that is,

Defining then, T is maximal monotone and 0 ∈ Tv if and only if v ∈ VI (C, A); see [11].

Assume that R : D(R) ⊂ H → 2^H is a maximal monotone mapping. Then, for λ > 0, associated with R, the resolvent operator J_R,λ can be defined as

In terms of Huang [12] (see also [13]), there holds the following property for the resolvent operator $$.

Lemma 5. J_R,λ is single-valued and firmly nonexpansive; that is,

Consequently, J_R,λ is nonexpansive and monotone.

Lemma 6. Let C be a nonempty closed convex subset of H and A : C → H a monotone mapping. In the context of the VIP (12), there holds the following relation:

Lemma 7 (see [17].)Let R be a maximal monotone mapping with D(R) = C. Then, for any given λ > 0, u ∈ C is a solution of problem (11) if and only if u ∈ C satisfies

Lemma 8 (see [13].)Let R be a maximal monotone mapping with D(R) = C and let B : C → H be a strongly monotone, continuous, and single-valued mapping. Then, for each z ∈ H, the equation z ∈ (B + λR)x has a unique solution x_λ for λ > 0.

Lemma 9 (see [17].)Let R be a maximal monotone mapping with D(R) = C and B : C → H a monotone, continuous, and single-valued mapping. Then (I + λ(R + B))C = H for each λ > 0. In this case, R + B is maximal monotone.

Lemma 10 ([40], see also [10]). Assume that Θ : C × C → R satisfies (A1)–(A4) and let φ : C → R be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r > 0 and x ∈ H, define a mapping $$ as follows:

for all x ∈ H. Then the following hold:

• (i)

  for each x ∈ H, $$;

• (ii)

  $$ is single-valued;

• (iii)

  $$ is firmly nonexpansive, that is, for any x, y ∈ H,

  $$ (33)

• (iv)

  $$;

• (v)

  MEP⁡(Θ, φ) is closed and convex.

Proposition 11 (see [5], Proposition 2.1.)Let C, H, Θ, φ, and $$ be as in Lemma 10. Then the following inequality holds:

for all s, t > 0 and x ∈ H.

Remark 12. From the conclusion of Proposition 11, it immediately follows that

for all s, t > 0 and x ∈ H.

Lemma 13 (see [41].)Let {x_n} and {z_n} be bounded sequences in a Banach space X and {β_n} a sequence in [0,1] with

Suppose that x_n+1 = (1 − β_n)z_n + β_nx_n for each n ≥ 1 and Then lim⁡_n→∞∥z_n − x_n∥ = 0.

Lemma 14 (see [42], Lemma 3.2.)Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let $$ be a sequence of nonexpansive self-mappings on C such that $$ and let $$ be a sequence of positive numbers in (0, b] for some b ∈ (0,1). Then, for every x ∈ C and k ≥ 1, the limit lim⁡_n→∞U_n,kx exists.

Lemma 15 (see [42], Lemma 3.3.)Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let $$ be a sequence of nonexpansive self-mappings on C such that $$ and let $$ be a sequence of positive numbers in (0, b] for some b ∈ (0,1). Then, $$.

Remark 16. Using Lemma 14, we can define the mapping W : C → C as follows:

Such a W is called the W-mapping generated by the sequences $$ and $$. As pointed out in [43], if {x_n} is a bounded sequence in C, then we have Throughout this paper, we always assume that $$ is a sequence of positive numbers in (0, b] for some b ∈ (0,1).

Lemma 17 (see [44].)Let {s_n} be a sequence of nonnegative numbers satisfying the conditions

where {α_n} and {β_n} are sequences of real numbers such that

• (i)

  {α_n}⊂[0,1] and $$, or equivalently,

  $$ (41)

• (ii)

  lim⁡ sup⁡_n→∞β_n ≤ 0, or $$.

Then lim⁡_n→∞s_n = 0.

Lemma 18 (see [39], demiclosedness principle.)Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C with Fix⁡(T) ≠ ∅. Then I − T is demiclosed. That is, whenever {x_n} is a sequence in C weakly converging to some x ∈ C and the sequence {(I − T)x_n} strongly converges to some y, it follows that (I − T)x = y. Here I is the identity operator of H.

Lemma 19. In a real Hilbert space H, there holds the inequality

Let C be a nonempty closed convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in (0,1] and let μ > 0. Associating with a nonexpansive mapping T : C → C, we define the mapping T^λ : C → H by

where F : C → H is an operator such that, for some positive constants κ, η > 0, F is κ-Lipschitzian and η-strongly monotone on C; that is, F satisfies the conditions for all x, y ∈ C.

Lemma 20 (see [44], Lemma 3.1.)T^λ is a contraction provided 0 < μ < 2η/κ²; that is,

where $$.

Remark 21. (i) Since F is κ-Lipschitzian and η-strongly monotone on C, we get 0 < η ≤ κ. Hence, whenever 0 < μ < 2η/κ², we have

which implies So, $$.

(ii) In Lemma 20, put F = (1/2)I and μ = 2. Then we know that κ = η = 1/2, 0 < μ = 2 < 2η/κ² = 4 and

Theorem 22. Let C be a nonempty closed convex subset of a real Hilbert space H. Let M, N be two integers. Let Θ_k be a bifunction from C × C to R satisfying (A1)–(A4) and let φ_k : C → R ∪ {+∞} be a proper lower semicontinuous and convex function, where k ∈ {1,2, …, M}. Let B_k : H → H and A_i : C → H be μ_k-inverse strongly monotone and η_i-inverse strongly monotone, respectively, where k ∈ {1,2, …, M}, i ∈ {1,2, …, N}. Let F : C → H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ, η > 0 and let f : H → H be a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let R : C → 2^H be a maximal monotone mapping and let the mapping B : C → H be β-inverse strongly monotone. Let 0 < λ < 2β, 0 < μ < 2η/κ², and 0 ≤ γρ < τ, where $$. Let $$ be a sequence of positive numbers in (0, b] for some b ∈ (0,1) and $$ an infinite family of nonexpansive self-mappings on C such that $$. For arbitrarily given x₁ ∈ H, let the sequence {x_n} be generated by

where {α_n}, {β_n} are two real sequences in [0,1] and W_n is the W-mapping defined by (14). Assume that the following conditions are satisfied:

• (i)

  lim⁡_n→∞α_n = 0 and $$;

• (ii)

  0 < lim⁡ inf⁡_n→∞β_n ≤ lim⁡ sup⁡_n→∞β_n < 1;

• (iii)

  {λ_i,n}⊂[a_i, b_i]⊂(0,2η_i) and lim⁡_n→∞ | λ_i,n+1 − λ_i,n | = 0 for all i ∈ {1,2, …, N};

• (iv)

  {r_k,n}⊂[e_k, f_k]⊂(0,2μ_k) and lim⁡_n→∞ | r_k,n+1 − r_k,n | = 0 for all k ∈ {1,2, …, M}.

Assume that either (B1) or (B2) holds. Then the sequence {x_n} converges strongly to x^* ∈ Ω, where x^* = P_Ω(I − μF + γf)x^* is a unique solution of the VIP:

Proof. Let Q = P_Ω. Note that F : C → H is a κ-Lipschitzian and η-strongly monotone operator with positive constants κ, η > 0 and f : H → H is a ρ-Lipschitzian mapping with constant ρ ≥ 0. Then, we have

where $$, and hence Since 0 ≤ γρ < τ ≤ 1, it is known that 1 − (τ − γρ)∈[0,1). Therefore, Q(I − μF + γf) is a contraction of C into itself, which implies that there exists a unique element x^* ∈ C such that x^* = Q(I − μF + γf)x^* = P_Ω(I − μF + γf)x^*.

We divide the remainder of the proof into several steps.

Step  1. Let us show that {x_n} is bounded.

Indeed, taking into account the control conditions (i) and (ii), we may assume, without loss of generality, that α_n ≤ 1 − β_n for all n ≥ 1. Put

for all k ∈ {1,2, …, M} and n ≥ 1, for all i ∈ {1,2, …, N} and n ≥ 1, and $$, where I is the identity mapping on H. Then we have that $$ and $$. Take p ∈ Ω arbitrarily. Then from (24) and Lemma 10 we have Similarly, we have Combining (55) and (56), we have Since the mapping B : C → H is β-inverse strongly monotone with 0 < λ < 2β, we have It is clear that, if 0 ≤ λ ≤ 2β, then I − λB is nonexpansive. Set y_n = J_R,λ(v_n − λBv_n) for each n ≥ 1. It follows that which, together with (57), yields Utilizing Lemma 20, from (49) we obtain By induction, we get Therefore, {x_n} is bounded and hence {u_n}, {v_n}, {y_n}, {W_ny_n}, {FW_ny_n}, and {f(x_n)} are also bounded.

Step  2. Let us show that ∥x_n+1 − x_n∥ → 0 as n → ∞.

Indeed, define x_n+1 = β_nx_n + (1 − β_n)z_n for each n ≥ 1. Then from the definition of z_n we obtain

It follows that Utilizing (14) and the nonexpansivity of T_i and U_n,i, we obtain that for each n ≥ 1 where $$ for some $$. Note that Note that where $$ for some $$. Also, utilizing Lemma 10 and Proposition 11 we deduce that where $$ is a constant such that for each n ≥ 1 Combining (64)–(68), we get Consequently, it follows from (70), {λ_n}⊂(0, b], and conditions (i)–(iv) that Hence, by Lemma 13 we have Consequently

Step  3. Let us show that ∥Bv_n − Bp∥ → 0, $$, and $$, k ∈ {1,2, …, M}, i ∈ {1,2, …, N}.

Indeed, we can rewrite (49) as follows:

It follows that that is, This, together with α_n → 0 and (73), implies that Also, from (24) it follows that for all i ∈ {1,2, …, N} and k ∈ {1,2, …, M} So, from (57), (58), and (78), it follows that By (49) and Lemma 20, we obtain From (79) and (80), it follows that and so Since {λ_i,n}⊂[a_i, b_i]⊂(0,2η_i) and {r_k,n}⊂[e_k, f_k]⊂(0,2μ_k) for all i ∈ {1,2, …, N} and k ∈ {1,2, …, M}, by (73), (77), and (82) we conclude immediately that for all i ∈ {1,2, …, N} and k ∈ {1,2, …, M}.

Step  4. Let us show that ∥x_n − Wx_n∥ → 0.

Indeed, by Lemma 10 (iii) we obtain that for each k ∈ {1,2, …, M}

which implies that Also, by Proposition 4 (iii), we obtain that for each i ∈ {1,2, …, N} which implies that Since J_R,λ is 1-inverse strongly monotone, we have which implies that Thus, from (85)–(89) we get Substituting (90) into (80), we have that is, So, from α_n → 0, (73), (77), and (83) we immediately get for all i ∈ {1,2, …, N} and k ∈ {1,2, …, M}. Note that Thus, from (93) we have It is easy to see that as n → ∞ Also, observe that Hence, we have Since it follows from Remark 16 that This, together with ∥x_n − y_n∥ → 0, implies that

Step  5. Let us show that limsup⁡_n→∞〈(γf − μF)x^*, x_n − x^*〉≤0 where x^* = P_Ω(I − μF + γf)x^*.

Indeed, as previously noted, it is known that P_Ω(I − μF + γf) is contractive and so P_Ω(I − μF + γf) has a unique fixed point, denoted by x^* ∈ C. This implies that x^* = P_Ω(I − μF + γf)x^*.

First, we show that ω_w(x_n) ⊂ Ω. As a matter of fact, we note that there exists a subsequence $$ of {x_n} such that

Since $$ is bounded, there exists a subsequence $$ of $$ which converges weakly to w. Without loss of generality, we may assume that $$. Note that ∥Wx_n − x_n∥ → 0. Then, by the demiclosedness principle for nonexpansive mappings, we obtain $$. Furthermore, from (93) and (96), we have that $$, $$, $$, $$, and $$, where k ∈ {1,2, …, M} and m ∈ {1,2, …, N}.

Now we prove that $$. Let

where m ∈ {1,2, …, N}. Let (v, u) ∈ G(T_m). Since u − A_mv ∈ N_Cv and $$, we have On the other hand, from $$ and v ∈ C, we have and hence Therefore, we have From (93) and since A_m is Lipschitzian, we obtain that $$. From $$, {λ_m,n}⊂[a_m, b_m]⊂(0,2η_m), ∀m ∈ {1,2, …, N}, and (93), we have Since T_m is maximal monotone, we have $$ and hence w ∈ VI⁡(C, A_m), m = 1,2, …, N, which implies that $$.

Next we prove that $$. Since $$, n ≥ 1, k ∈ {1,2, …, M}, we have

By (A2), we have Let z_t = ty + (1 − t)w for all t ∈ (0,1] and y ∈ C. This implies that z_t ∈ C. Then, we have By (93) and the fact that B_k is Lipschitzian, we have $$ as n → ∞. In addition, by the monotonicity of B_k, we obtain $$. Then, by (A4) and (112) we obtain Utilizing (A1), (A4), and (113), we obtain and hence Letting t → 0, we have, for each y ∈ C, This implies that w ∈ GMEP⁡(Θ_k, φ_k, B_k) and hence $$.

Further, we prove that w ∈ I(B, R). In fact, since B is β-inverse strongly monotone, B is monotone and Lipschitzian. It follows from Lemma 9 that R + B is maximal monotone. Let (v, g) ∈ G(R + B); that is, g − Bv ∈ Rv. Again, since $$, we have $$; that is, $$. By virtue of the monotonicity of R, we have

and so Since ∥v_n − y_n∥ → 0, ∥Bv_n − By_n∥ → 0, and $$, we have It follows from the maximal monotonicity of B + R that 0 ∈ (R + B)w; that is, w ∈ I(B, R). Therefore, w ∈ Ω. This shows that ω_w({x_n}) ⊂ Ω. Consequently, it follows from (103) that

Step  6. Let us show that x_n → x^* as n → ∞.

Indeed, from (49) and Lemma 20 it follows that

which immediately yields where It is easy to see that $$ and lim⁡ sup⁡_n→∞σ_n ≤ 0. Hence, by Lemma 17 we conclude that the sequence {x_n} converges strongly to x^*. This completes the proof.

Corollary 23. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C × C to R satisfying (A1)–(A4) and let φ : C → R ∪ {+∞} be a proper lower semicontinuous and convex function. Let ℬ : H → H and A_i : C → H be ζ-inverse strongly monotone and η_i-inverse strongly monotone, respectively, where i = 1,2. Let F : C → H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ, η > 0 and let f : H → H be a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let R : C → 2^H be a maximal monotone mapping and let the mapping B : C → H be β-inverse strongly monotone. Let 0 < λ < 2β, 0 < μ < 2η/κ², and 0 ≤ γρ < τ, where $$. Let $$ be a sequence of positive numbers in (0, b] for some b ∈ (0,1) and let $$ be an infinite family of nonexpansive self-mappings on C such that $$. For arbitrarily given x₁ ∈ H, let the sequence {x_n} be generated by

where {α_n}, {β_n} are two real sequences in [0,1] and W_n is the W-mapping defined by (14). Assume that the following conditions are satisfied:

• (i)

  lim⁡_n→∞α_n = 0 and $$;

• (ii)

  0 < lim⁡ inf⁡_n→∞β_n ≤ lim⁡ sup⁡_n→∞β_n < 1;

• (iii)

  {λ_i,n}⊂[a_i, b_i]⊂(0,2η_i) and lim⁡_n→∞|λ_i,n+1 − λ_i,n| = 0 for i = 1,2;

• (iv)

  {r_n}⊂[e, f]⊂(0,2ζ) and lim⁡_n→∞|r_n+1 − r_n| = 0.

Assume that either (B1) or (B2) holds. Then the sequence {x_n} converges strongly to x^* ∈ Ω, where x^* = P_Ω(I − μF + γf)x^* is a unique solution of the VIP:

Corollary 24. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C × C to R satisfying (A1)–(A4) and let φ : C → R ∪ {+∞} be a proper lower semicontinuous and convex function. Let ℬ : H → H and 𝒜 : C → H be ζ-inverse strongly monotone and ξ-inverse strongly monotone, respectively. Let F : C → H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ, η > 0 and let f : H → H be a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let R : C → 2^H be a maximal monotone mapping and let the mapping B : C → H be β-inverse strongly monotone. Let 0 < λ < 2β, 0 < μ < 2η/κ², and 0 ≤ γρ < τ, where $$. Let $$ be a sequence of positive numbers in (0, b] for some b ∈ (0,1) and let $$ be an infinite family of nonexpansive self-mappings on C such that $$. For arbitrarily given x₁ ∈ H, let the sequence {x_n} be generated by

where {α_n}, {β_n} are two real sequences in [0,1] and W_n is the W-mapping defined by (14). Assume that the following conditions are satisfied:

• (i)

  lim⁡_n→∞α_n = 0 and $$;

• (ii)

  0 < lim⁡ inf⁡_n→∞β_n ≤ lim⁡ sup⁡_n→∞β_n < 1;

• (iii)

  {ρ_n}⊂[a, b]⊂(0,2ξ) and lim⁡_n→∞|ρ_n+1 − ρ_n| = 0;

• (iv)

  {r_n}⊂[e, f]⊂(0,2ζ) and lim⁡_n→∞|r_n+1 − r_n| = 0.

Assume that either (B1) or (B2) holds. Then the sequence {x_n} converges strongly to x^* ∈ Ω, where x^* = P_Ω(I − μF + γf)x^* is a unique solution of the VIP:

Corollary 25. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C × C to R satisfying (A1)–(A4) and let φ : C → R ∪ {+∞} be a proper lower semicontinuous and convex function. Let 𝒜 : C → H be ξ-inverse strongly monotone and f : H → H a ρ-contractive mapping with constant ρ ∈ [0,1). Let R : C → 2^H be a maximal monotone mapping and let the mapping B : C → H be β-inverse strongly monotone. Let 0 < λ < 2β. Let $$ be a sequence of positive numbers in (0, b] for some b ∈ (0,1) and let $$ be an infinite family of nonexpansive self-mappings on C such that $$. For arbitrarily given x₁ ∈ H, let the sequence {x_n} be generated by

where {α_n}, {β_n} are two real sequences in [0,1] and W_n is the W-mapping defined by (14). Assume that the following conditions are satisfied:

• (i)

  lim⁡_n→∞α_n = 0 and $$;

• (ii)

  0 < lim⁡ inf⁡_n→∞β_n ≤ lim⁡ sup⁡_n→∞β_n < 1;

• (iii)

  {ρ_n}⊂[a, b]⊂(0,2ξ) and lim⁡_n→∞|ρ_n+1 − ρ_n| = 0;

• (iv)

  {r_n}⊂[e, f]⊂(0, ∞) and lim⁡_n→∞|r_n+1 − r_n| = 0.

Assume that either (B1) or (B2) holds. Then the sequence {x_n} converges strongly to x^* ∈ Ω, where x^* = P_Ωf(x^*) is a unique solution of the VIP:

Proof. In Corollary 24, put ℬ = 0, F = (1/2)I, μ = 2, and γ = 1. Then from Remark 16 (ii), we get τ = 1. Moreover, for {r_n}⊂[e, f]⊂(0, ∞), we can choose a positive constant ζ > 0 such that {r_n}⊂[e, f]⊂(0,2ζ). It is easy to see that ℬ is ζ-inverse strongly monotone. In addition, for the contraction f : H → H, we have 0 ≤ γρ < τ. Hence, all the conditions of Corollary 24 are satisfied. Thus, in terms of Corollary 24, we obtain the desired result.

Corollary 26. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C × C to R satisfying (A1)–(A4) and let φ : C → R ∪ {+∞} be a proper lower semicontinuous and convex function. Let 𝒜 : C → H be ξ-inverse strongly monotone. Let F : C → H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ, η > 0 and let f : H → H be a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let R : C → 2^H be a maximal monotone mapping and let the mapping B : C → H be β-inverse strongly monotone. Let 0 < λ < 2β, 0 < μ < 2η/κ², and 0 ≤ γρ < τ, where $$. Let $$ be a sequence of positive numbers in (0, b] for some b ∈ (0,1) and let $$ be an infinite family of nonexpansive self-mappings on C such that $$. For arbitrarily given x₁ ∈ H, let the sequence {x_n} be generated by

where {α_n}, {β_n} are two real sequences in [0,1] and W_n is the W-mapping defined by (14). Assume that the following conditions are satisfied:

• (i)

  lim⁡_n→∞α_n = 0 and $$;

• (ii)

  0 < lim⁡ inf⁡_n→∞β_n ≤ lim⁡ sup⁡_n→∞β_n < 1;

• (iii)

  {ρ_n}⊂[a, b]⊂(0,2ξ) and lim⁡_n→∞|ρ_n+1 − ρ_n| = 0;

• (iv)

  {r_n}⊂[e, f]⊂(0, ∞) and lim⁡_n→∞|r_n+1 − r_n| = 0.

Assume that either (B1) or (B2) holds. Then the sequence {x_n} converges strongly to x^* ∈ Ω, where x^* = P_Ω(I − μF + γf)x^* is a unique solution of the VIP:

Corollary 27. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C × C to R satisfying (A1)–(A4) and let φ : C → R ∪ {+∞} be a proper lower semicontinuous and convex function. Let ℬ : H → H and 𝒜 : C → H be ζ-inverse strongly monotone and ξ-inverse strongly monotone, respectively. Let F : C → H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ, η > 0 and let f : H → H be a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let R : C → 2^H be a maximal monotone mapping and let the mapping B : C → H be β-inverse strongly monotone. Let 0 < λ < 2β, 0 < μ < 2η/κ², and 0 ≤ γρ < τ, where $$. Assume that Ω : = GMEP⁡(Θ, φ, ℬ)∩VI⁡(C, 𝒜)∩ I (B, R) ≠ ∅. For arbitrarily given x₁ ∈ H, let the sequence {x_n} be generated by

where {α_n}, {β_n} are two real sequences in [0,1]. Assume that the following conditions are satisfied:

• (i)

  lim⁡_n→∞α_n = 0 and $$;

• (ii)

  0 < lim⁡ inf⁡_n→∞β_n ≤ lim⁡ sup⁡_n→∞β_n < 1;

• (iii)

  {ρ_n}⊂[a, b]⊂(0,2ξ) and lim⁡_n→∞|ρ_n+1 − ρ_n| = 0;

• (iv)

  {r_n}⊂[e, f]⊂(0,2ζ) and lim⁡_n→∞|r_n+1 − r_n| = 0.

Assume that either (B1) or (B2) holds. Then the sequence {x_n} converges strongly to x^* ∈ Ω, where x^* = P_Ω(I − μF + γf)x^* is a unique solution of the VIP:

Corollary 28. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C × C to R satisfying (A1)–(A4). Let ℬ : H → H and 𝒜 : C → H be ζ-inverse strongly monotone and ξ-inverse strongly monotone, respectively. Let F : C → H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ, η > 0 and let f : H → H be a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let R : C → 2^H be a maximal monotone mapping and let the mapping B : C → H be β-inverse strongly monotone. Let 0 < λ < 2β, 0 < μ < 2η/κ², and 0 ≤ γρ < τ, where $$. Let $$ be a sequence of positive numbers in (0, b] for some b ∈ (0,1) and let $$ be an infinite family of nonexpansive self-mappings on C such that $$. For arbitrarily given x₁ ∈ H, let the sequence {x_n} be generated by

where {α_n}, {β_n} are two real sequences in [0,1] and W_n is the W-mapping defined by (14). Assume that the following conditions are satisfied:

• (i)

  lim⁡_n→∞α_n = 0 and $$;

• (ii)

  0 < lim⁡ inf⁡_n→∞β_n ≤ lim⁡ sup⁡_n→∞β_n < 1;

• (iii)

  {ρ_n}⊂[a, b]⊂(0,2ξ) and lim⁡_n→∞|ρ_n+1 − ρ_n| = 0;

• (iv)

  {r_n}⊂[e, f]⊂(0,2ζ) and lim⁡_n→∞|r_n+1 − r_n| = 0.

Assume that either (B1) or (B2) holds. Then the sequence {x_n} converges strongly to x^* ∈ Ω, where x^* = P_Ω(I − μF + γf)x^* is a unique solution of the VIP:

Corollary 29. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C × C to R satisfying (A1)–(A4). Let ℬ : H → H and 𝒜 : C → H be ζ-inverse strongly monotone and ξ-inverse strongly monotone, respectively. Let F : C → H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ, η > 0 and let f : H → H be a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let R : C → 2^H be a maximal monotone mapping and let the mapping B : C → H be β-inverse strongly monotone. Let 0 < λ < 2β, 0 < μ < 2η/κ², and 0 ≤ γρ < τ, where $$. Assume that Ω : = GEP⁡(Θ, ℬ)∩VI⁡(C, 𝒜)∩ I (B, R) ≠ ∅. For arbitrarily given x₁ ∈ H, let the sequence {x_n} be generated by

where {α_n}, {β_n} are two real sequences in [0,1]. Assume that the following conditions are satisfied:

• (i)

  lim⁡_n→∞α_n = 0 and $$;

• (ii)

  0 < lim⁡ inf⁡_n→∞β_n ≤ lim⁡ sup⁡_n→∞β_n < 1;

• (iii)

  {ρ_n}⊂[a, b]⊂(0,2ξ) and lim⁡_n→∞|ρ_n+1 − ρ_n| = 0;

• (iv)

  {r_n}⊂[e, f]⊂(0,2ζ) and lim⁡_n→∞|r_n+1 − r_n| = 0.

Assume that either (B1) or (B2) holds. Then the sequence {x_n} converges strongly to x^* ∈ Ω, where x^* = P_Ω(I − μF + γf)x^* is a unique solution of the VIP:

Corollary 30. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C × C to R satisfying (A1)–(A4). Let ℬ : H → H and 𝒜 : C → H be ζ-inverse strongly monotone and ξ-inverse strongly monotone, respectively. Let f : H → H be a ρ-contractive mapping with constant ρ ∈ [0,1). Let R : C → 2^H be a maximal monotone mapping and let the mapping B : C → H be β-inverse strongly monotone. Let 0 < λ < 2β. Assume that Ω : = GEP⁡(Θ, ℬ)∩VI⁡(C, 𝒜)∩ I (B, R) ≠ ∅. For arbitrarily given x₁ ∈ H, let the sequence {x_n} be generated by

where {α_n}, {β_n} are two real sequences in [0,1]. Assume that the following conditions are satisfied:

• (i)

  lim⁡_n→∞α_n = 0 and $$;

• (ii)

  0 < lim⁡ inf⁡_n→∞β_n ≤ lim⁡ sup⁡_n→∞β_n < 1;

• (iii)

  {ρ_n}⊂[a, b]⊂(0,2ξ) and lim⁡_n→∞|ρ_n+1 − ρ_n| = 0;

• (iv)

  {r_n}⊂[e, f]⊂(0,2ζ) and lim⁡_n→∞|r_n+1 − r_n| = 0.

Assume that either (B1) or (B2) holds. Then the sequence {x_n} converges strongly to x^* ∈ Ω, where x^* = P_Ωf(x^*) is a unique solution of the VIP:

Remark 31. Theorem 22 extends, improves, and supplements [4, Theorem 3.2] in the following aspects.

• (i)

  The problem of finding a point $$ in Theorem 22 is very different from the problem of finding a point $$ in [4, Theorem 3.2] (i.e., Theorem 3 in this paper). There is no doubt that the problem of finding a point $$ is more general and more subtle than the problem of finding a point $$ in [4, Theorem 3.2].

• (ii)

  If, in Corollary 24, C = H, 𝒜 = 0, r_n = r > 0 (∀n ≥ 1), μF = V is a strongly positive bounded linear operator, and f is a contraction, then Corollary 24 reduces essentially to [4, Theorem 3.2]. This shows that Theorem 22 includes [4, Theorem 3.2] as a special case.

• (iii)

  The iterative scheme in [4, Algorithm 3.1] is extended to develop the iterative scheme in Theorem 22 by virtue of Korpelevič’s extragradient method and hybrid steepest-descent method [38]. The iterative scheme in Theorem 22 is more advantageous and more flexible than the iterative scheme in [4, Algorithm 3.1] because it involves solving four problems: a finite family of GMEPs, a finite family of VIPs, the variational inclusion (11), and the fixed point problem of an infinite family of nonexpansive self-mappings.

• (iv)

  The iterative scheme in Theorem 22 is very different from the iterative scheme in [4, Algorithm 3.1] because the iterative scheme in Theorem 22 involves Korpelevič’s extragradient method and hybrid steepest-descent method.

• (v)

  The proof of Theorem 22 combines the proof for viscosity approximation method in [4, Theorem 3.2], the proof for Korpelevič’s extragradient method in [8, Theorem 3.1], and the proof for hybrid steepest-descent method in [44, Theorem 3.1].