A System of Mixed Equilibrium Problems, a General System of Variational Inequality Problems for Relaxed Cocoercive, and Fixed Point Problems for Nonexpansive Semigroup and Strictly Pseudocontractive Mappings

Remark 1.1 (see [3], Remark  1.1 pages 135-136.)If B : C → H is a L_B-Lipschitz continuous and relaxed (c, d)-cocoercive mapping with $$ and $$, then I − τB satisfies the following:

Similarly, if D : C → H  is  L_D-Lipschitz continuous and relaxed (c′, d′)-cocoercive mapping with $$ and $$, then the mapping I − δD satisfies the following:

Lemma 2.1 (see [35].)Let V : C → H be a k-strict pseudo-contraction, then

• (1)

  the fixed point set F(V) of V is closed convex so that the projection P_F(V) is well defined;

• (2)

  define a mapping T : C → H by

  $$ ()
  If t ∈ [k, 1), then T is a nonexpansive mapping such that F(V) = F(T).

Lemma 2.2 (see [36].)Let C be a nonempty closed convex subset of a real Hilbert space H. Let T₁, T₂, … be nonexpansive mappings of C into itself such that $$ is nonempty, let μ₁, μ₂, … be real numbers such that 0 ≤ μ_n ≤ b < 1 for every n ≥ 1. Then,

• (1)

  W_n is nonexpansive and $$, for all n ≥ 1;

• (2)

  for every x ∈ C and k ∈ ℕ, the limit lim⁡_n→∞U_n,kx exists;

• (3)

  a mapping W : C → C defined by

  $$ ()
  is a nonexpansive mapping satisfying $$ and it is called the W-mapping generated by T₁, T₂, … and μ₁, μ₂, ….

Lemma 2.3 (see [37].)Let C be a nonempty closed convex subset of a Hilbert space H, {T_i : C → C} a countable family of nonexpansive mappings with $$, {μ_i} a real sequence such that 0 < μ_i ≤ b < 1,   for  all  i ≥ 1. If D is any bounded subset of C, then

Lemma 2.4 (see [38].)Each Hilbert space H satisfies Opial’s condition, that is, for any sequence {x_n} ⊂ H with x_n⇀x, the inequality

Lemma 2.5 (see [39].)Assume A is a strongly positive linear bounded operator on H with coefficient $$ and 0 < ρ ≤ ∥A∥⁻¹. Then, $$.

Lemma 2.6 (see [42].)Let H be a real Hilbert space and let ϕ be a lower semicontinuous and convex functional from H to ℝ. Let Θ be a bifunction from H × H to ℝ satisfying (H1)–(H3). Assume that

• (i)

  η : H × H → H is λ-Lipschitz continuous with constant λ > 0 such that

  • (a)

    η(x, y) + η(y, x) = 0,   for  all  x, y ∈ H,

  • (b)

    η(·, ·) is affine in the first variable,

  • (c)

    for each fixed x ∈ H, y ↦ η(x, y) is sequentially continuous from the weak topology to the weak topology;

• (ii)

  K : H → ℝ is η-strongly convex with constant σ > 0 and its derivative K^′ is sequentially continuous from the weak topology to the strong topology;

• (iii)

  for each x ∈ H, there exist bounded subsets E_x ⊂ H and z_x ∈ H such that for any y ∈ H∖E_x,

  $$ ()
  For given r > 0, let $$ be the mapping defined by
  $$ ()

• (1)

  $$ is single-valued.

• (2)

  $$, where MEP⁡(Θ, ϕ) is the set of solution of the mixed equilibrium problem,

  $$ ()

• (3)

  MEP⁡(Θ, ϕ) is closed and convex.

Lemma 2.7 (see [43].)Let {x_n} and {v_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)v_n + β_nx_n for all integers n ≥ 0 and limsup⁡_n→∞(∥v_n+1 − v_n∥−∥x_n+1 − x_n∥) ≤ 0. Then, lim⁡_n→∞∥v_n − x_n∥ = 0.

Lemma 2.8 (see [44].)Assume {x_n} is a sequence of nonnegative real numbers such that

• (1)

  $$,

• (2)

  limsup⁡_n→∞(b_n/a_n) ≤ 0 or $$.

Lemma 2.9 (see [45].)Let C be a nonempty closed convex subset of a real Hilbert space H and g : C → ℝ ∪ {∞} a proper lower-semicontinuous differentiable convex function. If z is a solution to the minimization problem

Lemma 2.10 (see [46].)Let C be a nonempty bounded closed convex subset of a Hilbert space H and let 𝒮 = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C, then for any h ≥ 0,

Lemma 2.11 (see [47].)Let C be a nonempty bounded closed convex subset of H, {x_n} a sequence in C, and 𝒮 = {S(s) : 0 ≤ s < ∞} a nonexpansive semigroup on C. If the following conditions are satisfied:

• (i)

  x_n⇀z;

• (ii)

  limsup⁡_s→∞limsup⁡_n→∞∥S(s)x_n − x_n∥ = 0, then z ∈ 𝒮.

Lemma 2.12 (see [25].)For given x^*, y^* ∈ C and (x^*, y^*) is a solution of the problem (1.20) if and only if x^* is a fixed point of the mapping G : C → C is defined by

Lemma 2.13 (see [32].)Let G : C → C be defined in Lemma 2.12. If B : H → H is a L_B-Lipschitzian and relaxed (c, d)-cocoercive mapping and D : H → H is a L_D-Lipschitz and relaxed (c′, d′)-cocoercive mapping where $$ and $$, then G is nonexpansive.

Lemma 2.14 (demiclosedness principle [48]). Assume that S is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space H. If S has a fixed point, then I − S is demiclosed; that is, whenever {x_n} is a sequence in C converging weakly to some x ∈ C (for short, x_n⇀x ∈ C), and the sequence {(I − S)x_n} converges strongly to some y (for short, (I − S)x_n → y), it follows that (I − S)x = y. Here I is the identity operator of H.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H which C + C ⊂ C and let f be a contraction of C into itself with α ∈ (0,1). Let ϕ be a lower semicontinuous and convex functional from H to ℝ and let {Θ_k : H × H → ℝ,   k = 1,2, …, N} be a finite family of equilibrium functions satisfying conditions (H1)–(H3). Let 𝒮 = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C and let {t_n} be a positive real divergent sequence. Let $$ be a countable family of uniformly k-strict pseudo-contractions, let $$ be the countable family of nonexpansive mappings defined by T_ix = tx + (1 − t)V_ix,   for  all  x ∈ C,   for  all  i ≥ 1,   t ∈ [k, 1), let W_n be the W-mapping defined by (2.12), and let W be a mapping defined by (2.13) with F(W) ≠ ∅. Let A be a strongly positive linear bounded operator on H with coefficient $$ and let $$, B : H → H be a L_B-Lipschitz continuous and relaxed (c, d)-cocoercive mapping with $$, and let D : H → H be a L_D-Lipschitz continuous and relaxed (c′, d′)-cocoercive mapping with $$. Suppose that Ω : = F(𝒮)∩F(W)∩𝔉∩SVI(C, B, D) ≠ ∅, where $$. Let μ > 0, γ > 0 and r_k > 0,   k = 1,2, …, N, which are constants. For given x₁ ∈ H arbitrarily and fixed u ∈ H, suppose {x_n}, {y_n}, {z_n} and $$ are the sequences generated iteratively by

• (C1)

  η : H × H → H is λ-Lipschitz continuous with constant λ > 0 such that

  • (a)

    η(x, y) + η(y, x) = 0,   for  all  x, y ∈ H,

  • (b)

    x ↦ η(x, y) is affine,

  • (c)

    for each fixed y ∈ H, y ↦ η(x, y) is sequentially continuous from the weak topology to the weak topology;

• (C2)

  K : H → ℝ is η-strongly convex with constant σ > 0 and its derivative K^′ is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant ν > 0 such that σ > λν;

• (C3)

  for each k ∈ {1,2, …, N} and for all x ∈ H, there exist bounded subsets E_x ⊂ H and z_x ∈ H such that for any y ∈ H∖E_x,

  $$ ()

• (C4)

  lim⁡_n→∞α_n = 0 and $$;

• (C5)

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

• (C6)

  $$ and $$.

Proof. By the condition (C4) and (C5), we may assume, without loss of generality, that α_n ≤ (1 − β_n)(1 + μ∥A∥) ⁻¹ for all n ∈ ℕ. Indeed, A is a strongly positive bounded linear operator on H, we have

Remark 3.2. For example, of the control conditions (C4)–(C6), we set α_n = 1/10n,   β_n = n/(n + 1). We set B, D is a 1-Lipschitz continuous and relaxed (0,1)-cocoercive mapping, (i.e., L_B = 1 = L_D and c = 0 = c^′,   d = 1 = d^′).

Then, we can choose τ ∈ (0,2) and δ ∈ (0,2) which satisfies the condition (C6) in Theorem 3.1.

Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H which C + C ⊂ C and let f be a contraction of C into itself with α ∈ (0,1). Let ϕ be a lower semicontinuous and convex functional from H to ℝ and let Θ : H × H → ℝ be a finite family of equilibrium functions satisfying conditions (H1)–(H3). Let 𝒮 = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C and let {t_n} be a positive real divergent sequence. Let $$ be a countable family of uniformly k- strict pseudo-contractions, let $$ be the countable family of nonexpansive mappings defined by T_ix = tx + (1 − t)V_ix,   for  all  x ∈ C,   for  all  i ≥ 1, t ∈ [k, 1), let W_n be the W-mapping defined by (2.12), and let W be a mapping defined by (2.13) with F(W) ≠ ∅. Let A be a strongly positive linear bounded operator on H with coefficient $$ and let $$, B : H → H be a L_B-Lipschitz continuous and relaxed (c, d)-cocoercive mapping with $$, and let D : H → H be a L_D-Lipschitz continuous and relaxed (c^′, d^′)-cocoercive mapping with $$. Suppose that Ω : = F(𝒮)∩F(W)∩MEP⁡(Θ, ϕ)∩SVI(C, B, D) ≠ ∅. Let μ > 0, γ > 0 and r > 0, which are constants. For given x₁ ∈ H arbitrarily and fixed u ∈ H, suppose {x_n}, {y_n},{z_n}, and{u_n} are the sequences generated iteratively by

Proof. Taking N = 1 in Theorem 3.1. Hence, the conclusion follows. This completes the proof.

Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H which C + C ⊂ C and let f be a contraction of C into itself with α ∈ (0,1). Let 𝒮 = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C and let {t_n} be a positive real divergent sequence. Let $$ be a countable family of uniformly k-strict pseudo-contractions, let $$ be the countable family of nonexpansive mappings defined by T_ix = tx + (1 − t)V_ix,   for  all  x ∈ C,   for  all  i ≥ 1, t ∈ [k, 1), let W_n be the W-mapping defined by (2.12), and let W be a mapping defined by (2.13) with F(W) ≠ ∅. Let A be a strongly positive linear bounded operator on H with coefficient $$ and let $$, B : H → H be a L_B-Lipschitz continuous and relaxed (c, d)-cocoercive mapping with $$, and let D : H → H be a L_D-Lipschitz continuous and relaxed (c^′, d^′)-cocoercive mapping with $$. Suppose that Ω : = F(𝒮)∩F(W)∩SVI(C, B, D) ≠ ∅. Let μ > 0 and γ > 0, which are constants. For given x₁ ∈ H arbitrarily and fixed u ∈ H, suppose {x_n}, {y_n}, and{z_n} are the sequences generated iteratively by

Proof. Put Θ(x, y) ≡ ϕ(x) ≡ 0 for all x, y ∈ H and r = 1. Take K(x) = ∥x∥²/2 and η(y, x) = y − x, for all x, y ∈ H. Then, we get u_n = P_Cx_n = x_n in Corollary 3.3. Hence, the conclusion follows. This completes the proof.

Corollary 3.5. Let C be a nonempty closed convex subset of a real Hilbert space H and let f be a contraction of H into itself with α ∈ (0,1). Let 𝒮 = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C and let {t_n} be a positive real divergent sequence. Let A be a strongly positive linear bounded operator on H with coefficient $$ and let $$, B : H → H be a L_B-Lipschitz continuous and relaxed (c, d)-cocoercive mapping with $$. Suppose that Ω : = F(𝒮)∩B⁻¹0 ≠ ∅. Let μ > 0 and γ > 0, which are constants. For given x₁ ∈ H arbitrarily and fixed u ∈ H, suppose the {x_n}, {y_n}, and {z_n} are the sequences generated iteratively by

Proof. Setting τ = δ, C ≡ H, D ≡ B and W_n ≡ P_H ≡ I in Corollary 3.4, it follows from the proof of Theorem  4.1 in [25] that B⁻¹0 = VI⁡(H, B). Hence, the conclusion follows. This completes the proof.