The Mann-Type Extragradient Iterative Algorithms with Regularization for Solving Variational Inequality Problems, Split Feasibility, and Fixed Point Problems

Lemma 1 (see [4].)For given $$, $$ is a solution of problem (3) if and only if $$ is a fixed point of the mapping G : C → C defined by

where $$.

Theorem 2 (see [14], Theorem 3.1.)Let C be a nonempty closed convex subset of a real Hilbert space ℋ. Let A : C → ℋ be α-inverse strongly monotone, and let B_i : C → ℋ be β_i-inverse strongly monotone for i = 1,2. Let S : C → C be a k-strictly pseudocontractive mapping such that Fix (S)∩Ξ∩VI (C, A) ≠ ∅. Let Q : C → C be a ρ-contraction with ρ ∈ [0, 1/2). For given x₀ ∈ C arbitrarily, let the sequences {x_n}, {y_n}, and {z_n} be generated iteratively by

where μ_i ∈ (0,2β_i) for i = 1,2, {λ_n}⊂(0,2α] and {α_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1] such that

• (i)

  β_n + γ_n + δ_n = 1 and (γ_n + δ_n)k ≤ γ_n for all n ≥ 0;

• (ii)

  lim _n→∞α_n = 0 and $$;

• (iii)

  0 < lim  inf _n→∞β_n ≤ lim  sup _n→∞β_n < 1 and lim  inf _n→∞δ_n > 0;

• (iv)

  lim _n→∞(γ_n+1/(1 − β_n+1) − γ_n/(1 − β_n)) = 0;

• (v)

  0 < lim  inf _n→∞λ_n ≤ lim  sup _n→∞λ_n < 2α and lim _n→∞ | λ_n+1 − λ_n | = 0.

Then the sequence {x_n} generated by (5) converges strongly to $$ and $$ is a solution of GSVI (3), where $$.

Proposition 3 (see [31], Proposition 3.1.)Given x^* ∈ ℋ₁, the following statements are equivalent:

• (i)

  x^*solves the SFP;

• (ii)

  x^*solves the fixed point equation

  $$ ()
  where λ > 0, ∇f = A^*(I − P_Q)A and A^* is the adjoint of A;

• (iii)

  x^*solves the variational inequality problem (VIP) of finding x^* ∈ C such that

  $$ ()

Proposition 4 (see [31].)The following statements hold:

• (i)

  the gradient

  $$ ()
  is (α + ∥A∥²)-Lipschitz continuous and α-strongly monotone;

• (ii)

  the mapping P_C(I − λ∇f_α) is a contraction with coefficient

  $$ ()
  where 0 < λ ≤ α/(∥A∥² + α) ²;

• (iii)

  if the SFP is consistent, then the strong lim _α→0x_α exists and is the minimum-norm solution of the SFP.

Theorem 5 (see [31], Theorem 3.1.)Let S : C → C be a nonexpansive mapping such that Fix (S)∩Γ ≠ ∅. Let {x_n} and {y_n} be the sequences in C generated by the following extragradient algorithm:

where $$, {λ_n}⊂[a, b] for some a, b ∈ (0, 1/∥A∥²) and {β_n}⊂[c, d] for some c, d ∈ (0, 1). Then, both sequences {x_n} and {y_n} converge weakly to an element $$.

Algorithm 6. Let μ_i ∈ (0, 2β_i) for i = 1,2, {α_n}⊂(0, ∞), {λ_n}⊂(0, 1/∥A∥²) and {σ_n}, {τ_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1] such that σ_n + τ_n ≤ 1 and β_n + γ_n + δ_n = 1 for all n ≥ 0. For given x₀ ∈ C arbitrarily, let {x_n}, {y_n}, {z_n} be the sequences generated by the Mann-type extragradient iterative scheme with regularization

Under appropriate assumptions, it is proven that all the sequences {x_n}, {y_n}, {z_n} converge weakly to an element $$. Furthermore, $$ is a solution of the GSVI (3), where $$.

Algorithm 7. Let μ_i ∈ (0, 2β_i) for i = 1,2, {α_n}⊂(0, ∞), {λ_n}⊂(0, 1/∥A∥²) and {σ_n}, {β_n}, {γ_n}, {δ_n}⊂[0, 1] such that β_n + γ_n + δ_n = 1 for all n ≥ 0. For given x₀ ∈ C arbitrarily, let $$ be the sequences generated by the Mann-type extragradient iterative scheme with regularization

Also, under mild conditions, it is shown that all the sequences {x_n}, {u_n}, $$ converge weakly to an element $$. Furthermore, $$ is a solution of the GSVI (3), where $$.

Observe that both [20, Theorem 5.7] and [31, Theorem 3.1] are weak convergence results for solving the SFP and so are our results as well. But our problem of finding an element of Fix (S)∩Ξ∩Γ is more general than the corresponding ones in [20, Theorem 5.7] and [31, Theorem 3.1], respectively. Hence, there is no doubt that our weak convergence results are very interesting and quite valuable. Because the Mann-type extragradient iterative schemes (16) and (17) with regularization involve two inverse strongly monotone mappings B₁ and B₂, a k-strictly pseudocontractive self-mapping S and several parameter sequences, they are more flexible and more subtle than the corresponding ones in [20, Theorem 5.7] and [31, Theorem 3.1], respectively. Furthermore, the hybrid extragradient iterative scheme (5) is extended to develop the Mann-type extragradient iterative schemes (16) and (17) with regularization. In our results, the Mann-type extragradient iterative schemes (16) and (17) with regularization lack the requirement of boundedness for the domain in which various mappings are defined; see, for example, Yao et al. [7, Theorem 3.2]. Therefore, our results represent the modification, supplementation, extension, and improvement of [20, Theorem 5.7], [31, Theorem 3.1], [14, Theorem 3.1], and [7, Theorem 3.2].

Proposition 8. For given x ∈ ℋ and z ∈ K:

• (i)

  z = P_Kx⇔〈x − z, y − z〉 ≤ 0, for all y ∈ K;

• (ii)

  z = P_Kx⇔∥x − z∥² ≤ ∥x − y∥² − ∥y − z∥², for  all y ∈ K;

• (iii)

  〈P_Kx − P_Ky, x − y〉≥∥P_Kx − P_Ky∥², for all y ∈ ℋ, which hence implies that P_K is nonexpansive and monotone.

Definition 9. A mapping T : ℋ → ℋ is said to be

• (a)

  nonexpansive if

  $$ ()

• (b)

  firmly nonexpansive if 2T − I is nonexpansive, or equivalently,

  $$ ()
  alternatively, T is firmly nonexpansive if and only if T can be expressed as
  $$ ()
  where S : ℋ → ℋ is nonexpansive; projections are firmly nonexpansive.

Definition 10. Let T be a nonlinear operator with domain D(T)⊆ℋ and range R(T)⊆ℋ.

• (a)

  T is said to be monotone if

  $$ ()

• (b)

  Given a number β > 0, T is said to be β-strongly monotone if

  $$ ()

• (c)

  Given a number ν > 0, T is said to be ν-inverse strongly monotone (ν-ism) if

  $$ ()

It can be easily seen that if S is nonexpansive, then I − S is monotone. It is also easy to see that a projection P_K is 1-ism.

Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields, for instance, in traffic assignment problems; see, for example, [33, 34].

Definition 11. A mapping T : ℋ → ℋ is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping, that is,

where α ∈ (0, 1) and S : ℋ → ℋ is nonexpansive. More precisely, when the last equality holds, we say that T is α-averaged. Thus, firmly nonexpansive mappings (in particular, projections) are 1/2-averaged maps.

Proposition 12 (see [21].)Let T : ℋ → ℋ be a given mapping.

• (i)

  Tis nonexpansive if and only if the complement I − T is 1/2-ism.

• (ii)

  If T is ν-ism, then for γ > 0, γT is ν/γ-ism.

• (iii)

  T is averaged if and only if the complement I − T is ν-ism for some ν > 1/2. Indeed, for α ∈ (0,1), T is α-averaged if and only if I − T is 1/2α-ism.

Proposition 13 (see [21], [35].)Let S, T, V : ℋ → ℋ be given operators.

• (i)

  If T = (1 − α)S + αV for some α ∈ (0,1) and if S is averaged and V is nonexpansive, then T is averaged.

• (ii)

  T is firmly nonexpansive if and only if the complement I − T is firmly nonexpansive.

• (iii)

  If T = (1 − α)S + αV for some α ∈ (0,  1) and if S is firmly nonexpansive and V is nonexpansive, then T is averaged.

• (iv)

  The composite of finitely many averaged mappings is averaged. That is, if each of the mappings $$ is averaged, then so is the composite T₁∘T₂∘···∘T_N. In particular, if T₁ is α₁-averaged and T₂ is α₂-averaged, where α₁, α₂ ∈ (0,  1), then the composite T₁∘T₂ is α-averaged, where α = α₁ + α₂ − α₁α₂.

• (v)

  If the mappings $$ are averaged and have a common fixed point, then

  $$ ()
  The notation Fix (T) denotes the set of all fixed points of the mapping T, that is, Fix (T) = {x ∈ ℋ : Tx = x}.

Lemma 14 (see [36].)Let ℋ be a real Hilbert space. Then, for all x, y ∈ ℋ and λ ∈ [0, 1],

Lemma 15 (see [37], Proposition 2.1.)Let C be a nonempty closed convex subset of a real Hilbert space ℋ and S : C → C be a mapping.

• (i)

  If S is a k-strict pseudocontractive mapping, then S satisfies the Lipschitz condition

  $$ ()

• (ii)

  If S is a k-strict pseudocontractive mapping, then the mapping I − S is semiclosed at 0, that is, if {x_n} is a sequence in C such that $$ weakly and (I − S)x_n → 0 strongly, then $$.

• (iii)

  If S is k-(quasi-)strict pseudocontraction, then the fixed point set Fix (S) of S is closed and convex so that the projection P_Fix (S) is well defined.

Lemma 16 (see [38], p. 80.)Let $$, $$, and $$ be sequences of nonnegative real numbers satisfying the inequality

If $$ and $$, then lim _n→∞a_n exists. If, in addition, $$ has a subsequence which converges to zero, then lim _n→∞a_n = 0.

Corollary 17 (see [39], p. 303.)Let $$ and $$ be two sequences of nonnegative real numbers satisfying the inequality

If $$ converges, then lim _n→∞a_n exists.

Lemma 18 (see [7].)Let C be a nonempty closed convex subset of a real Hilbert space ℋ. Let S : C → C be a k-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers such that (γ + δ)k ≤ γ. Then

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

Theorem 20. Let C be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let A ∈ B(ℋ₁, ℋ₂) and B_i : C → ℋ₁ be β_i-inverse strongly monotone for i = 1,2. Let S : C → C be a k-strictly pseudocontractive mapping such that Fix (S)∩Ξ∩Γ ≠ ∅. For given x₀ ∈ C arbitrarily, let {x_n}, {y_n}, {z_n} be the sequences generated by the Mann-type extragradient iterative algorithm (16) with regularization, where μ_i ∈ (0, 2β_i) for i = 1,2, {α_n}⊂(0, ∞), {λ_n}⊂(0, 1/∥A∥²) and {σ_n}, {τ_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1] such that

• (i)

  $$;

• (ii)

  β_n + γ_n + δ_n = 1 and (γ_n + δ_n)k ≤ γ_n for all n ≥ 0;

• (iii)

  σ_n + τ_n ≤ 1 for all n ≥ 0;

• (iv)

  0 < lim  inf _n→∞τ_n ≤ lim  sup _n→∞(σ_n + τ_n) < 1;

• (v)

  0 < lim  inf _n→∞β_n ≤ lim  sup _n→∞β_n < 1 and lim  inf _n→∞δ_n > 0;

• (vi)

  0 < lim  inf _n→∞λ_n ≤ lim  sup _n→∞λ_n < 1/∥A∥².

Then all the sequences {x_n}, {y_n}, {z_n} converge weakly to an element $$. Furthermore, $$ is a solution of GSVI (3), where $$.

Proof. First, taking into account 0 < lim  inf _n→∞λ_n ≤ lim  sup _n→∞λ_n < 1/∥A∥², without loss of generality we may assume that {λ_n}⊂[a, b] for some a, b ∈ (0, 1/∥A∥²).

Now, let us show that P_C(I − λ∇f_α) is ζ-averaged for each λ ∈ (0, 2/(α + ∥A∥²)), where

Indeed, it is easy to see that ∇f = A^*(I − P_Q)A is 1/∥A∥²-ism, that is,

Observe that

Hence, it follows that ∇f_α = αI + A^*(I − P_Q)A is 1/(α + ∥A∥²)-ism. Thus, λ∇f_α is 1/λ(α + ∥A∥²)-ism according to Proposition 12(ii). By Proposition 12(iii), the complement I − λ∇f_α is λ(α + ∥A∥²)/2-averaged. Therefore, noting that P_C is 1/2-averaged and utilizing Proposition 13(iv), we know that for each λ ∈ (0, 2/(α + ∥A∥²)), P_C(I − λ∇f_α) is ζ-averaged with

This shows that P_C(I − λ∇f_α) is nonexpansive. Furthermore, for {λ_n}⊂[a, b] with a, b ∈ (0, 1/∥A∥²), we have Without loss of generality, we may assume that Consequently, it follows that for each integer n ≥ 0, $$ is ζ_n-averaged with

This immediately implies that $$ is nonexpansive for all n ≥ 0.

Next we divide the remainder of the proof into several steps.

Step 1. {x_n} is bounded.

Indeed, take p ∈ Fix (S)∩Ξ∩Γ arbitrarily. Then Sp = p, P_C(I − λ∇f)p = p for λ ∈ (0, 2/∥A∥²), and

From (16), it follows that Utilizing Lemma 19, we also have For simplicity, we write q = P_C(p − μ₂B₂p), $$, for each n ≥ 0. Then $$ for each n ≥ 0. Since B_i : C → ℋ₁ is β_i-inverse strongly monotone and 0 < μ_i < 2β_i for i = 1,2, we know that for all n ≥ 0, Furthermore, by Proposition 8(ii), we have Further, by Proposition 8(i), we have So, from (46), we obtain Hence, it follows from (46), (49), and (52) that Since (γ_n + δ_n)k ≤ γ_n for all n ≥ 0, utilizing Lemma 18, we obtain from (53) Since $$, it is clear that $$. Thus, by Corollary 17, we conclude that and the sequence {x_n} is bounded. Taking into account that P_C, $$, B₁ and B₂ are Lipschitz continuous, we can easily see that {z_n}, {u_n}, $$, {y_n}, and $$ are bounded, where $$ for all n ≥ 0.

Step 2. Consider lim _n→∞∥B₂z_n − B₂p∥ = 0, $$ and lim _n→∞∥x_n − z_n∥ = 0, where q = P_C(p − μ₂B₂p).

Indeed, utilizing Lemma 18 and the convexity of ∥·∥², we obtain from (16) and (47)–(52) that

Therefore, Since α_n → 0, lim _n→∞∥x_n − p∥ exists, lim  inf _n→∞(γ_n + δ_n) > 0, {λ_n}⊂[a, b] and 0 < lim  inf _n→∞τ_n ≤ lim  sup _n→∞(σ_n + τ_n) < 1, it follows that

Step 3. Consider   lim _n→∞ ∥Sy_n − y_n ∥ = 0.

Indeed, observe that

This together with ∥z_n − x_n∥ → 0 implies that $$ and hence $$. By firm nonexpansiveness of P_C, we have

that is,

Moreover, using the argument technique similar to the previous one, we derive

that is,

Utilizing (47), (52), (61), and (63), we have

Thus, utilizing Lemma 14, from (16) and (64) it follows that which hence implies that Since 0 < lim  inf _n→∞β_n ≤ lim  sup _n→∞β_n < 1, lim  sup _n→∞(σ_n + τ_n) < 1, {λ_n}⊂[a, b], α_n → 0, ∥B₂z_n − B₂p∥→0, $$ and lim _n→∞∥x_n − p∥ exists, it follows from the boundedness of {u_n}, {z_n} and $$ that $$, Consequently, it immediately follows that Also, note that This together with $$ implies that Since we have

Step 4. {x_n}, {y_n},   and {z_n} converge weakly to an element $$.

Indeed, since {x_n} is bounded, there exists a subsequence $$ of {x_n} that converges weakly to some $$. We obtain that $$. Taking into account that ∥x_n − y_n∥→0 and ∥x_n − z_n∥→0 as n → ∞, we deduce that $$ weakly and $$ weakly. First, it is clear from Lemma 15 and ∥Sy_n − y_n∥→0 that $$. Now let us show that $$. Note that

as n → ∞ where G : C → C is defined as that in Lemma 1. According to Lemma 15, we get $$. Further, let us show that $$. As a matter of fact, define where N_Cv = {w ∈ ℋ₁ : 〈v − u, w〉≥0, ∀u ∈ C}. Then, T is maximal monotone and 0 ∈ Tv if and only if v ∈ VI (C, ∇f); see [40] for more details. Let (v, w) ∈ Gph(T). Then, we have and hence So, we have On the other hand, from we have and hence, Therefore, from we have Hence, we get Since T is maximal monotone, we have $$, and hence, $$. Thus, it is clear that $$. Therefore, $$.

Let $$ be another subsequence of {x_n} such that $$ converges weakly to $$. Then, $$. Let us show that $$. Assume that $$. From the Opial condition [41], we have

This leads to a contradiction. Consequently, we have $$. This implies that {x_n} converges weakly to $$. Further, from ∥x_n − y_n∥→0 and ∥x_n − z_n∥→0, it follows that both {y_n} and {z_n} converge weakly to $$. This completes the proof.

Corollary 21. Let C be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let A ∈ B(ℋ₁, ℋ₂) and B_i : C → ℋ₁ be β_i-inverse strongly monotone for i = 1,  2. Let S : C → C be a k-strictly pseudocontractive mapping such that Fix (S)∩Ξ∩Γ ≠ ∅. For given x₀ ∈ C arbitrarily, let the sequences {x_n}, {y_n}, {z_n} be generated iteratively by

where μ_i ∈ (0, 2β_i) for i = 1,  2, {α_n}⊂(0, ∞), {λ_n}⊂(0, 1/∥A∥²) and {τ_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1] such that

• (i)

  $$;

• (ii)

  β_n + γ_n + δ_n = 1 and (γ_n + δ_n)k ≤ γ_n for all n ≥ 0;

• (iii)

  0 < lim  inf _n→∞τ_n ≤ lim  sup _n→∞τ_n < 1;

• (iv)

  0 < lim  inf _n→∞β_n ≤ lim  sup _n→∞β_n < 1 and lim  inf _n→∞δ_n > 0;

• (v)

  0 < lim  inf _n→∞λ_n ≤ lim  sup _n→∞λ_n < 1/∥A∥².

Then all the sequences {x_n}, {y_n}, {z_n} converge weakly to an element $$. Furthermore, $$ is a solution of the GSVI (3), where $$.

Proof. In Theorem 20, put σ_n = 0 for all n ≥ 0. Then, in this case, Theorem 20 reduces to Corollary 21.

Corollary 22. Let C be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let A ∈ B(ℋ₁, ℋ₂) and S : C → C be a nonexpansive mapping such that Fix (S)∩Γ ≠ ∅. For given x₀ ∈ C arbitrarily, let the sequences {x_n}, {y_n}, {z_n} be generated iteratively by

where {α_n}⊂(0, ∞), {λ_n}⊂(0, 1/∥A∥²) and {τ_n}, {β_n}⊂[0,1] such that

• (i)

  $$;

• (ii)

  0 < lim  inf _n→∞τ_n ≤ lim  sup _n→∞τ_n < 1;

• (iii)

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

• (iv)

  0 < lim  inf _n→∞λ_n ≤ lim  sup _n→∞λ_n < 1/∥A∥².

Then all the sequences {x_n}, {y_n}, {z_n} converge weakly to an element $$.

Proof. In Corollary 21, put B₁ = B₂ = 0 and γ_n = 0. Then, Ξ = C, β_n + δ_n = 1 for all n ≥ 0, and the iterative scheme (85) is equivalent to

This is equivalent to (86). Since S is a nonexpansive mapping, S must be a k-strictly pseudocontractive mapping with k = 0. In this case, it is easy to see that all the conditions (i)–(v) in Corollary 21 are satisfied. Therefore, in terms of Corollary 21, we obtain the desired result.

Theorem 23. Let C be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let A ∈ B(ℋ₁, ℋ₂) and B_i : C → ℋ₁ be β_i-inverse strongly monotone for i = 1,2. Let S : C → C be a k-strictly pseudocontractive mapping such that Fix (S)∩Ξ∩Γ ≠ ∅. For given x₀ ∈ C arbitrarily, let $$ be the sequences generated by the Mann-type extragradient iterative algorithm (17) with regularization, where μ_i ∈ (0, 2β_i) for i = 1,2, {α_n}⊂(0, ∞), {λ_n}⊂(0, 1/∥A∥²) and {σ_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1] such that

• (i)

  $$;

• (ii)

  β_n + γ_n + δ_n = 1 and (γ_n + δ_n)k ≤ γ_n for all n ≥ 0;

• (iii)

  lim  sup _n→∞σ_n < 1;

• (iv)

  0 < lim  inf _n→∞β_n ≤ lim  sup _n→∞β_n < 1 and lim  inf _n→∞δ_n > 0;

• (v)

  0 < lim  inf _n→∞λ_n ≤ lim  sup _n→∞λ_n < 1/∥A∥².

Then the sequences {x_n}, {u_n}, $$ converge weakly to an element $$. Furthermore, $$ is a solution of the GSVI (3), where $$.

Proof. First, taking into account 0 < lim  inf _n→∞λ_n ≤ lim  sup _n→∞λ_n < 1/∥A∥², without loss of generality, we may assume that {λ_n}⊂[a, b] for some a, b ∈ (0, 1/∥A∥²). Repeating the same argument as that in the proof of Theorem 20, we can show that P_C(I − λ∇f_α) is ζ-averaged for each λ ∈ (0, 1/(α + ∥A∥²)), where ζ = (2 + λ(α + ∥A∥²))/4. Further, repeating the same argument as that in the proof of Theorem 20, we can also show that for each integer n ≥ 0, $$ is ζ_n-averaged with ζ_n = (2 + λ_n(α_n + ∥A∥²))/4 ∈ (0, 1).

Next we divide the remainder of the proof into several steps.

Step 1.  {x_n} is bounded.

Indeed, take p ∈ Fix (S)∩Ξ∩Γ arbitrarily. Then Sp = p, P_C(I − λ∇f)p = p for λ ∈ (0, 2/∥A∥²), and

For simplicity, we write for each n ≥ 0. Then $$ for each n ≥ 0. Utilizing the arguments similar to those of (46) and (47) in the proof of Theorem 20, from (17) we can obtain Since B_i : C → ℋ₁ is β_i-inverse strongly monotone and 0 < μ_i < 2β_i for i = 1,2, utilizing the argument similar to that of (49) in the proof of Theorem 20, we can obtain that for all n ≥ 0, Utilizing the argument similar to that of (52) in the proof of Theorem 20, from (90) we can obtain Hence, it follows from (92) and (93) that Since (γ_n + δ_n)k ≤ γ_n for all n ≥ 0, by Lemma 18 we can readily see from (94) that Since $$, it is clear that $$. Thus, by Corollary 17 we conclude that and the sequence {x_n} is bounded. Since P_C, $$, B₁ and B₂ are Lipschitz continuous, it is easy to see that {u_n}, $$, $$, {y_n} and $$ are bounded, where $$ for all n ≥ 0.

Step 2. Consider lim _n→∞∥B₂x_n − B₂p∥ = 0, $$ and $$, where q = P_C(p − μ₂B₂p).

Indeed, utilizing Lemma 18 and the convexity of ∥·∥², we obtain from (17), (92), and (93) that

Therefore, Since α_n → 0, lim  sup _n→∞σ_n < 1, lim _n→∞∥x_n − p∥ exists, lim  inf _n→∞(γ_n + δ_n) > 0 and {λ_n}⊂[a, b] for some a, b ∈ (0, 1/∥A∥²), it follows from the boundedness of $$ that

Step 3. Consider lim _n→∞∥Sy_n − y_n∥ = 0.

Indeed, utilizing the Lipschitz continuity of $$, we have

This together with $$ implies that $$ and hence $$. Utilizing the arguments similar to those of (61) and (63) in the proof of Theorem 20, we get Utilizing (91) and (101), we have Thus, utilizing Lemma 14, from (17) and (102), it follows that which hence implies that Since 0 < lim  inf _n→∞β_n ≤ lim  sup _n→∞β_n < 1, lim  sup _n→∞σ_n < 1, {λ_n}⊂[a, b], α_n → 0, ∥B₂x_n − B₂p∥→0, $$, $$ and lim _n→∞∥x_n − p∥ exists, it follows from the boundedness of {x_n}, $$, {u_n}, $$ and $$ that $$: Consequently, it immediately follows that This together with $$ implies that Since we have

Step 4.  {x_n}, {u_n} and $$ converge weakly to an element $$.

Indeed, since {x_n} is bounded, there exists a subsequence $$ of {x_n} that converges weakly to some $$. We obtain that $$. Taking into account that ∥x_n − u_n∥→0 and $$ and ∥x_n − y_n∥→0, we deduce that $$ weakly and $$ weakly.

First, it is clear from Lemma 15 and ∥Sy_n − y_n∥→0 that $$. Now let us show that $$. Note that

as n → ∞, where G : C → C is defined as that in Lemma 1. According to Lemma 15, we get $$. Further, let us show that $$. As a matter of fact, define where N_Cv = {w ∈ ℋ₁ : 〈v − u, w〉≥0, ∀u ∈ C}. Utilizing the argument similar to that of Step 4 in the proof of Theorem 20, from the relation we can easily conclude that It is easy to see that $$. Therefore, $$. Finally, utilizing the Opial condition [41], we infer that {x_n} converges weakly to $$. Further, from ∥x_n − u_n∥→0 and $$, it follows that both {u_n} and $$ converge weakly to $$. This completes the proof.

Corollary 24. Let C be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let A ∈ B(ℋ₁, ℋ₂) and B_i : C → ℋ₁ be β_i-inverse strongly monotone for i = 1,2. Let S : C → C be a k-strictly pseudocontractive mapping such that Fix (S)∩Ξ∩Γ ≠ ∅. For given x₀ ∈ C arbitrarily, let the sequences {x_n}, {u_n}, $$ be generated iteratively by

where μ_i ∈ (0,2β_i) for i = 1,2, {α_n}⊂(0, ∞), {λ_n}⊂(0, 1/∥A∥²) and {β_n}, {γ_n}, {δ_n}⊂[0,1] such that

• (i)

  $$;

• (ii)

  β_n + γ_n + δ_n = 1 and (γ_n + δ_n)k ≤ γ_n for all n ≥ 0;

• (iii)

  0 < lim  inf _n→∞β_n ≤ lim  sup _n→∞β_n < 1 and lim  inf _n→∞δ_n > 0;

• (iv)

  0 < lim  inf _n→∞λ_n ≤ lim  sup _n→∞λ_n < 1/∥A∥².

Then the sequences {x_n}, {u_n}, $$ converge weakly to an element $$. Furthermore, $$ is a solution of GSVI (3), where $$.

Corollary 25. Let C be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let A ∈ B(ℋ₁, ℋ₂) and S : C → C be a nonexpansive mapping such that Fix (S)∩Γ ≠ ∅. For given x₀ ∈ C arbitrarily, let the sequences {x_n}, $$ be generated iteratively by

where {α_n}⊂(0, ∞), {λ_n}⊂(0, 1/∥A∥²) and {β_n}⊂[0,1] such that

• (i)

  $$;

• (ii)

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

• (iii)

  0 < lim  inf _n→∞λ_n ≤ lim  sup _n→∞λ_n < 1/∥A∥².

Then, both the sequences {x_n} and $$ converge weakly to an element $$.

Proof. In Corollary 24, put B₁ = B₂ = 0 and γ_n = 0. Then, Ξ = C, β_n + δ_n = 1 for all n ≥ 0, and the iterative scheme (114) is equivalent to

This is equivalent to (115). Since S is a nonexpansive mapping, S must be a k-strictly pseudocontractive mapping with k = 0. In this case, it is easy to see that all the conditions (i)–(iv) in Corollary 24 are satisfied. Therefore, in terms of Corollary 24, we obtain the desired result.

Remark 26. Compared with the Ceng and Yao [31, Theorem 3.1], our Corollary 25 coincides essentially with [31, Theorem 3.1]. This shows that our Theorem 23 includes [31, Theorem 3.1] as a special case.

Remark 27. Our Theorems 20 and 23 improve, extend, and develop [20, Theorem 5.7], [31, Theorem 3.1], [7, Theorem 3.2], and [14, Theorem 3.1] in the following aspects.

• (i)

  Compared with the relaxed extragradient iterative algorithm in [7, Theorem 3.2], our Mann-type extragradient iterative algorithms with regularization remove the requirement of boundedness for the domain C in which various mappings are defined.

• (ii)

  Because [31, Theorem 3.1] is the supplementation, improvement, and extension of [20, Theorem 5.7] and our Theorem 23 includes [31, Theorem 3.1] as a special case, beyond question our results are very interesting and quite valuable.

• (iii)

  The problem of finding an element of Fix (S)∩Ξ∩Γ in our Theorems 20 and 23 is more general than the corresponding problems in [20, Theorem 5.7] and [31, Theorem 3.1], respectively.

• (iv)

  The hybrid extragradient method for finding an element of Fix (S)∩Ξ∩VI (C, A) in [14, Theorem 3.1] is extended to develop our Mann-type extragradient iterative algorithms (16) and (17) with regularization for finding an element of Fix (S)∩Ξ∩Γ.

• (v)

  The proof of our results are very different from that of [14, Theorem 3.1] because our argument technique depends on the Opial condition, the restriction on the regularization parameter sequence {α_n}, and the properties of the averaged mappings $$ to a great extent.

• (vi)

  Because our iterative algorithms (16) and (17) involve two inverse strongly monotone mappings B₁ and B₂, a k-strictly pseudocontractive self-mapping S and several parameter sequences, they are more flexible and more subtle than the corresponding ones in [20, Theorem 5.7] and [31, Theorem 3.1], respectively.