Relaxed Extragradient Methods with Regularization for General System of Variational Inequalities with Constraints of Split Feasibility and Fixed Point Problems

Proposition 1 (see [18], 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 2 (see [18].)There hold the following statements:

• (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

  $$ ()
  where $$;

• (iii)

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

Theorem 3 (see [18], Theorem  3.1.)Let S : C → C be a nonexpansive mapping such that Fix (S)∩Γ ≠ ∅. Let {x_n} and {y_n} the sequences in C generated by the following extragradient algorithm:

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

Lemma 4 (see [23].)For given $$ is a solution of problem (14) if and only if $$ is a fixed point of the mapping G : C → C defined by

where $$.

In particular, if the mapping B_i : C → ℋ is β_i-inverse strongly monotone for i = 1,2, then the mapping G is nonexpansive provided μ_i ∈ (0,2β_i) for i = 1,2.

Theorem 5 (see [33], 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 B_i : C → ℋ be β_i-inverse strongly monotone for i = 1,2. Let S : C → C be a k-strictly pseudo-contractive 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 $$ be generated by the relaxed extragradient iterative scheme:

where μ_i ∈ (0,2β_i) for i = 1,2, and the following conditions hold for five sequences {λ_n}⊂(0, α] and {α_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1]:

• (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 ≤ limsup _n→∞β_n < 1 and liminf _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 ≤ limsup _n→∞λ_n < α and lim _n→∞|λ_n+1 − λ_n| = 0.

Then the sequences $$ converge strongly to the same point $$ if and only if lim _n→∞∥u_n+1 − u_n∥ = 0. Furthermore, $$ is a solution of the GSVI (14), where $$.

Algorithm 6. 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 following relaxed extragradient iterative scheme with regularization:

Under mild assumptions, it is proven that the sequences $$ converge strongly to the same point $$ if and only if lim _n→∞∥u_n+1 − u_n∥ = 0. Furthermore, $$ is a solution of the GSVI (14), where $$.

Algorithm 7. Let μ_i ∈ (0,2β_i) for i = 1,2, {α_n}⊂(0, ∞), {λ_n}⊂(0, 1/∥A∥²) and {σ_n}, {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 following relaxed extragradient iterative scheme with regularization:

Also, under appropriate conditions, it is shown that the sequences {x_n}, {y_n}, {z_n} converge strongly to the same point $$ if and only if lim _n→∞∥z_n+1 − z_n∥ = 0. Furthermore, $$ is a solution of the GSVI (14), where $$.

Note that both [6, Theorem  5.7] and [18, Theorem  3.1] are weak convergence results for solving the SFP (1). Beyond question our strong convergence results are very interesting and quite valuable. Because our relaxed extragradient iterative schemes (17) and (18) with regularization involve a contractive self-mapping Q, a k-strictly pseudo-contractive self-mapping S and several parameter sequences, they are more flexible and more subtle than the corresponding ones in [6, Theorem  5.7] and [18, Theorem  3.1], respectively. Furthermore, the relaxed extragradient iterative scheme (16) is extended to develop our relaxed extragradient iterative schemes (17) and (18) with regularization. All in all, our results represent the modification, supplementation, extension, and improvement of [6, Theorem  5.7], [18, Theorem  3.1], and [33, Theorem  3.1].

Proposition 8 (see [34].)For given x ∈ ℋ and z ∈ K:

• (i)

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

• (ii)

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

• (iii)

  $$, 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, [35, 36].

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 → ℋis 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 [7].)Let T : ℋ → ℋ be a given mapping.

• (i)

  T is 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 [7], [37].)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 [39].)Let {x_n} and {y_n} be bounded sequences in a Banach space X and let {β_n} be a sequence in [0,1] with 0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1. Suppose x_n+1 = (1 − β_n)y_n + β_nx_n for all integers n ≥ 0 and limsup _n→∞(∥y_n+1 − y_n∥−∥x_n+1 − x_n∥) ≤ 0. Then, lim _n→∞∥y_n − x_n∥ = 0.

Lemma 15 (see [38], Proposition  2.1.)Let C be a nonempty closed convex subset of a real Hilbert space ℋ and S : C → C a mapping.

• (i)

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

  $$ ()

• (ii)

  If S is a k-strict pseudo-contractive 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 pseudo-contraction, 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 [34].)Let {a_n} be a sequence of nonnegative real numbers satisfying the property

where {s_n}⊂(0,1] and {t_n} are such that

• (i)

  $$;

• (ii)

  either limsup _n→∞t_n ≤ 0 or $$;

• (iii)

  $$ where r_n ≥ 0,   ∀ n ≥ 0. Then, lim _n→∞a_n = 0.

Lemma 17 (see [26].)Let C be a nonempty closed convex subset of a real Hilbert space ℋ. Let S : C → C be a k-strictly pseudo-contractive mapping. Let γ and δ be two nonnegative real numbers such that (γ + δ)k ≤ γ. Then

Lemma 18. In a real Hilbert space ℋ, there holds the inequality

Theorem 19. Let C be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let A ∈ B(ℋ₁, ℋ₂), and let B_i : C → ℋ₁ be β_i-inverse strongly monotone for i = 1,2. Let S : C → C be a k-strictly pseudo-contractive mapping such that Fix (S)∩Ξ∩Γ ≠ ∅. Let Q : C → C be a ρ-contraction with ρ ∈ [0, 1/2). For given x₀ ∈ C arbitrarily, let the sequences $$ be generated by the relaxed 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 _n→∞σ_n = 0 and $$;

• (iv)

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

• (v)

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

• (vi)

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

Then the sequences $$ converge strongly to the same point $$ if and only if lim _n→∞∥u_n+1 − u_n∥ = 0. Furthermore, $$ is a solution of the GSVI (14), where $$.

Proof. First, taking into account 0 < liminf _n→∞λ_n ≤ limsup _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 $$ is ζ_n-averaged with This immediately implies that $$ 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 an arbitrary p ∈ Fix (S)∩Ξ∩Γ. Then, we get Sp = p,  P_C(I − λ∇f)p = p for λ ∈ (0, 2/∥A∥²), and

From (17) it follows that Utilizing Lemma 18 we also have For simplicity, we write for each n ≥ 0. Then $$ for each n ≥ 0. Since B_i : C → ℋ₁ is β_i-inverse strongly monotone for i = 1,2 and 0 < μ_i < 2β_i for i = 1,  2, we know that for all n ≥ 0, Furthermore, by Proposition 8(ii), we have Further, by Proposition 8(i), we have So, from (45) we obtain Hence it follows from (48) and (51) that Since (γ_n + δ_n)k ≤ γ_n for all n ≥ 0, utilizing Lemma 17 we obtain from (52) Now, we claim that As a matter of fact, if n = 0, then it is clear that (54) is valid, that is, Assume that (54) holds for n ≥ 1, that is, Then, we conclude from (53) and (56) that By induction, we conclude that (54) is valid. Hence, {x_n} is bounded. Since $$ and B₂ are Lipschitz continuous, it is easy to see that $$ and $$ are bounded, where $$ for all n ≥ 0.

Step  2.   lim_n→∞∥x_n+1 − x_n∥ = 0.

Indeed, define x_n+1 = β_nx_n + (1 − β_n)w_n for all n ≥ 0. It follows that

Since (γ_n + δ_n)k ≤ γ_n for all n ≥ 0, utilizing Lemma 17 we have Next, we estimate ∥y_n+1 − y_n∥. Observe that and hence Combining (60) with (61), we get This together with (62) implies that Hence it follows from (58), (59), and (64) that From (60), we deduce from condition (vi) that $$. Since {x_n}, {u_n}, $$ and $$ are bounded, it follows from conditions (i), (iii), (v), and (vi) that Hence by Lemma 14 we get lim _n→∞∥w_n − x_n∥ = 0. Thus,

Step  3. lim_n→∞∥B₂x_n − B₂p∥ = 0,  $$, and $$, where q = P_C(p − μ₂B₂p).

Indeed, utilizing Lemma 17 and the convexity of ∥·∥², we obtain from (17), (48), and (51) that

Therefore, Since α_n → 0, σ_n → 0,  ∥x_n − x_n+1∥→0,  liminf _n→∞(γ_n + δ_n) > 0 and {λ_n}⊂[a, b] for some a, b ∈ (0, 1/∥A∥²), it follows that

Step  4. lim_n→∞∥Sy_n − y_n∥ = 0.

Indeed, observe that

This together with $$ implies that $$ and hence $$. By firm nonexpansiveness of P_C, we have that is, Moreover, using the argument technique similar to the above one, we derive that is, Utilizing (46), (73), and (75), we have Thus from (17) and (76) it follows that which hence implies that Since limsup _n→∞β_n < 1,  {λ_n}⊂[a, b],  α_n → 0,  σ_n → 0,  ∥B₂x_n − B₂p∥→0,  $$,  $$ and ∥x_n+1 − x_n∥→0, it follows from the boundedness of $$, and $$ that Consequently, it immediately follows that This together with $$ implies that Since it follows that

Step  5. $$ where $$.

Indeed, since {x_n} is bounded, there exists a subsequence $$ of {x_n} such that

Also, since H is reflexive and {y_n} is bounded, without loss of generality we may assume that $$ weakly for some $$. First, it is clear from Lemma 15 that $$. Now let us show that $$. We note that where G : C → C is defined as such that in Lemma 4. According to Lemma 15 we obtain $$. Further, let us show that $$. As a matter of fact, since $$ and ∥x_n − y_n∥→0, we deduce that $$ weakly and $$ weakly. Let where N_Cv = {w ∈ ℋ₁ : 〈v − u, w〉≥0,   ∀ u ∈ C}. Then, T is maximal monotone and 0 ∈ Tv if and only if v ∈ VI (C, ∇f); see [40] for more details. Let (v, w) ∈ Gph(T). Then, we have and hence So, we have On the other hand, from we have and, hence, Therefore, from we have Hence, we get Since T is maximal monotone, we have $$, and, hence, $$. Thus it is clear that $$. Therefore, $$. Consequently, in terms of Proposition 8(i) we obtain from (84) that

Step 6. $$.

Indeed, from (48) and (51) it follows that

Note that Utilizing Lemmas 17 and 18, we obtain from (48) and the convexity of ∥·∥² Note that liminf _n→∞(1 − 2ρ)(γ_n + δ_n) > 0. It follows that $$. It is clear that because $$ and lim _n→∞∥x_n − y_n∥ = 0. In addition, note also that {λ_n}⊂[a, b],  $$ and $$ is bounded. Hence we get $$. Therefore, all conditions of Lemma 16 are satisfied. Consequently, we immediately deduce that $$ as n → ∞. This completes the proof.

Corollary 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 pseudo-contractive mapping such that Fix (S)∩Ξ∩Γ ≠ ∅. For fixed u ∈ C and given x₀ ∈ C arbitrarily, let the sequences $$ be generated iteratively by

where μ_i ∈ (0,2β_i) for i = 1,2, and the following conditions hold for six sequences {α_n}⊂(0, ∞), {λ_n}⊂(0, 1/∥A∥²) and {σ_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1]:

• (i)

  $$;

• (ii)

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

• (iii)

  lim _n→∞σ_n = 0 and $$;

• (iv)

  0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1 and liminf _n→∞δ_n > 0;

• (v)

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

• (vi)

  0 < liminf _n→∞λ_n ≤ limsup _n→∞λ_n < 1/∥A∥²  and lim _n→∞ | λ_n+1 − λ_n | = 0.

Then the sequences $$ converge strongly to the same point $$ if and only if lim _n→∞∥u_n+1 − u_n∥ = 0. Furthermore, $$ is a solution of the GSVI (14), where $$.

Corollary 21. 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 fixed u ∈ C and given x₀ ∈ C arbitrarily, let the sequences $$ be generated iteratively by

where the following conditions hold for four sequences {α_n}⊂(0, ∞), {λ_n}⊂(0, 1/∥A∥²) and {σ_n}, {β_n}⊂[0,1]:

• (i)

  $$;

• (ii)

  lim _n→∞σ_n = 0 and $$;

• (iii)

  0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1;

• (iv)

  0 < liminf _n→∞λ_n ≤ limsup _n→∞λ_n < 1/∥A∥² and lim _n→∞ | λ_n+1 − λ_n | = 0.

Then the sequences $$ converge strongly to the same point $$ if and only if lim _n→∞∥u_n+1 − u_n∥ = 0.

Proof. In Corollary 20, put B₁ = B₂ = 0 and γ_n = 0. Then, Ξ = C,  β_n + δ_n = 1 for all n ≥ 0, and the iterative scheme (101) is equivalent to

This is is equivalent to (102). Since S is a nonexpansive mapping, S must be a k-strictly pseudo-contractive mapping with k = 0. In this case, it is easy to see that conditions (i)–(vi) in Corollary 20 all are satisfied. Therefore, in terms of Corollary 20, we obtain the desired result.

Theorem 22. 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 ≠ ∅. 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}, {z_n} be generated by the relaxed extragradient iterative algorithm (18) 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)

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

• (iii)

  lim _n→∞σ_n = 0 and $$;

• (iv)

  0 < liminf _n→∞τ_n ≤ limsup _n→∞τ_n < 1 and lim _n→∞ | τ_n+1 − τ_n | = 0;

• (v)

  0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1 and liminf _n→∞δ_n > 0;

• (vi)

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

• (vii)

  0 < liminf _n→∞λ_n ≤ limsup _n→∞λ_n < 1/∥A∥² and lim _n→∞ | λ_n+1 − λ_n | = 0.

Then the sequences {x_n}, {y_n}, {z_n} converge strongly to the same point $$ if and only if lim _n→∞∥z_n+1 − z_n∥ = 0. Furthermore, $$ is a solution of the GSVI (14), where $$.

Proof. First, taking into account 0 < liminf _n→∞λ_n ≤ limsup _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 19, we can show that P_C(I − λ∇f_α) is ζ-averaged for each λ ∈ (0, 2/(α + ∥A∥²)), where ζ = (2 + λ(α + ∥A∥²))/4. Further, repeating the same argument as that in the proof of Theorem 19, we can also show that for each integer $$ 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

Utilizing the arguments similar to those of (45) and (46) in the proof of Theorem 19, from (18) we can obtain 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, utilizing the argument similar to that of (48) in the proof of Theorem 19, we can obtain that for all n ≥ 0, Furthermore, utilizing Proposition 8(i)-(ii) and the argument similar to that of (51) in the proof of Theorem 19, from (105) we obtain Hence it follows from (105), (108), and (109) that Since (γ_n + δ_n)k ≤ γ_n for all n ≥ 0, utilizing Lemma 17 we obtain from (110) Repeating the same argument as that of (54) in the proof of Theorem 19, by induction we can prove that Thus, {x_n} is bounded. Since $$ and B₂ are Lipschitz continuous, it is easy to see that $$, and $$ are bounded, where $$ for all n ≥ 0.

Step  2. lim _n→∞∥x_n+1 − x_n∥ = 0.

Indeed, define x_n+1 = β_nx_n + (1 − β_n)w_n for all n ≥ 0. Then, utilizing the arguments similar to those of (58)–(61) in the proof of Theorem 19, we can obtain that

(due to Lemma 17) So, we have This together with (114) implies that Hence it follows from (113), and (117) that Since {x_n}, {y_n}, {z_n}, {u_n}, and $$ are bounded, it follows from conditions (i), (iii), (iv), (vi), and (vii) that Hence by Lemma 14 we get lim _n→∞∥w_n − x_n∥ = 0. Thus, Step  3. lim _n→∞∥B₂z_n − B₂p∥ = 0,  $$, and lim _n→∞∥x_n − z_n∥ = 0, where q = P_C(p − μ₂B₂p).

Indeed, utilizing Lemma 17 and the convexity of ∥·∥², we obtain from (18) and (106)–(109) that

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

Step  4. lim _n→∞∥Sy_n − y_n∥ = 0.

Indeed, observe that

This together with ∥z_n − x_n∥→0 implies that $$ and hence $$. Utilizing the arguments similar to those of (73) and (75) in the proof of Theorem 19 we can prove that Utilizing (106), (109), (125), we have

Thus from (18) and (126) it follows that

which hence implies that Since limsup _n→∞β_n < 1,  limsup _n→∞τ_n < 1,  {λ_n}⊂[a, b],  α_n → 0,  σ_n → 0,  ∥B₂z_n − B₂p∥→0,  $$ and ∥x_n+1 − x_n∥→0, it follows from the boundedness of {x_n}, {u_n}, {z_n}, and $$ that Consequently, it immediately follows that Also, note that This together with $$ implies that Since it follows that

Step  5. $$ where $$.

Indeed, since {x_n} is bounded, there exists a subsequence $$ of {x_n} such that

Also, since H is reflexive and {x_n} is bounded, without loss of generality we may assume that $$ weakly for some $$. 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

where G : C → C is defined as that in Lemma 4. 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 Observe that Utilizing the arguments similar to those of Step  5 in the proof of Theorem 19 we can prove that Since T is maximal monotone, we have $$, and, hence, $$. Thus it is clear that $$. Therefore, $$. Consequently, in terms of Proposition 8 (i) we obtain from (135) that

Step  6. $$.

Indeed, observe that

Utilizing Lemmas 17 and 18, we obtain from (106)–(109) and the convexity of ∥·∥² that Note that liminf _n→∞(1 − 2ρ)(γ_n + δ_n) > 0. It follows that $$. It is clear that because $$ and lim _n→∞∥x_n − y_n∥ = 0. In addition, note also that $$ and {z_n} is bounded. Hence we get $$. Therefore, all conditions of Lemma 16 are satisfied. Consequently, we immediately deduce that $$ as n → ∞. In the meantime, taking into account that ∥x_n − y_n∥→0 and ∥x_n − z_n∥→0 as n → ∞, we infer that This completes the proof.

Corollary 23. Let C be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let A ∈ B(ℋ₁, ℋ₂) and let 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 fixed u ∈ C and 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 {σ_n}, {τ_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1] such that

• (i)

  $$;

• (ii)

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

• (iii)

  lim _n→∞σ_n = 0 and $$;

• (iv)

  0 < liminf _n→∞τ_n ≤ limsup _n→∞τ_n < 1 and lim _n→∞ | τ_n+1 − τ_n | = 0;

• (v)

  0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1 and liminf _n→∞δ_n > 0;

• (vi)

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

• (vii)

  0 < liminf _n→∞λ_n ≤ limsup _n→∞λ_n < 1/∥A∥² and lim _n→∞ | λ_n+1 − λ_n | = 0.

Then the sequences {x_n}, {y_n}, {z_n} converge strongly to the same point $$ if and only if lim _n→∞∥z_n+1 − z_n∥ = 0. Furthermore, $$ is a solution of the GSVI (14), where $$.

Corollary 24. Let C be a nonempty closed convex subset of a real Hilbert space ℋ₁. Let A ∈ B(ℋ₁, ℋ₂) and let B_i : C → ℋ₁ be β_i-inverse strongly monotone for i = 1,2. Let S : C → C be a nonexpansive mapping such that Fix (S)∩Ξ∩Γ ≠ ∅. 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}, {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 {σ_n}, {τ_n}, {β_n}, {γ_n}, {δ_n}⊂[0,1] such that

• (i)

  $$;

• (ii)

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

• (iii)

  lim _n→∞σ_n = 0 and $$;

• (iv)

  0 < liminf _n→∞τ_n ≤ limsup _n→∞τ_n < 1 and lim _n→∞ | τ_n+1 − τ_n | = 0;

• (v)

  0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1 and liminf _n→∞δ_n > 0;

• (vi)

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

• (vii)

  0 < liminf _n→∞λ_n ≤ limsup _n→∞λ_n < 1/∥A∥² and lim _n→∞ | λ_n+1 − λ_n | = 0.

Then the sequences {x_n}, {y_n}, {z_n} converge strongly to the same point $$ if and only if lim _n→∞∥z_n+1 − z_n∥ = 0. Furthermore, $$ is a solution of the GSVI (14), where $$.

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

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

• (i)

  $$;

• (ii)

  σ_n + τ_n ≤ 1 for all n ≥ 0;

• (iii)

  lim _n→∞σ_n = 0 and $$;

• (iv)

  0 < liminf _n→∞τ_n ≤ limsup _n→∞τ_n < 1 and lim _n→∞ | τ_n+1 − τ_n | = 0;

• (v)

  0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1;

• (vi)

  0 < liminf _n→∞λ_n ≤ limsup _n→∞λ_n < 1/∥A∥² and lim _n→∞ | λ_n+1 − λ_n | = 0.

Then the sequences {x_n}, {z_n} converge strongly to the same point $$ if and only if lim _n→∞∥z_n+1 − z_n∥ = 0.

Proof. In Corollary 23, put B₁ = B₂ = 0 and γ_n = 0. Then, Ξ = C, β_n + δ_n = 1, P_C[P_C(z_n − μ₂B₂z_n) − μ₁B₁P_C(z_n − μ₂B₂z_n)] = z_n, and the iterative scheme (146) is equivalent to

This is equivalent to (148). 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 conditions (i)–(vii) in Corollary 23 all are satisfied. Therefore, in terms of Corollary 23, we obtain the desired result.

Remark 26. Our Theorems 19 and 22 improve, extend, and develop [6, Theorem  5.7], [18, Theorem  3.1], and [33, Theorem  3.1] in the following aspects.

• (i)

  Because both [6, Theorem  5.7] and [18, Theorem  3.1] are weak convergence results for solving the SFP, beyond question, our Theorems 19 and 22 as strong convergence results are very interesting and quite valuable.

• (ii)

  The problem of finding an element of Fix (S)∩Ξ∩Γ in our Theorems 19 and 22 is more general than the corresponding problems in [6, Theorem  5.7] and [18, Theorem  3.1], respectively.

• (iii)

  The relaxed extragradient iterative method for finding an element of Fix (S)∩Ξ∩VI (C, A) in [33, Theorem  3.1] is extended to develop the relaxed extragradient method with regularization for finding an element of Fix (S)∩Ξ∩Γ in our Theorem 19.

• (iv)

  The proof of our Theorems 19 and 22 is very different from that of [33, Theorem  3.1] because our argument technique depends on Lemma 16, the restriction on the regularization parameter sequence {α_n}, and the properties of the averaged mappings $$ to a great extent.

• (v)

  Because our iterative schemes (17) and (18) involve a contractive self-mapping Q, a k-strictly pseudo-contractive self-mapping S, and several parameter sequences, they are more flexible and more subtle than the corresponding ones in [6, Theorem  5.7] and [18, Theorem  3.1], respectively.