A New System of Generalized Mixed Quasivariational Inclusions with Relaxed Cocoercive Operators and Applications

Example 2.1 (see [4].)All Hilbert spaces, L^p(or  l^p), and the Sobolev spaces $$, (p ≥ 2) are 2-uniformly smooth, while L^p(or  l^p) and $$ spaces (1 < p ≤ 2) are p-uniformly smooth.

The generalized duality mapping $$ defined as follows:

for all x ∈ E. As we know that J = J₂ is the usual normalized duality mapping, and J_q(x) = ∥x∥^q−2J(x) for x ≠ 0, J_q(tx) = t^q−1J_q(x), and J_q(−x) = −J_q(x) for all x ∈ E and t ∈ [0, +∞), and J_q is single-valued if E is smooth (see, e.g., [28]). If E is a Hilbert space, then J = I, where I is the identity operator. Let g_j : E → E, let A_j : E × E → E be single-valued mappings, and let M_j : E → 2^E be set-valued mappings for all j ∈ {1,2, …, n}. We consider the system of generalized mixed quasivariational inclusions problem (for short, (SGMQVI)) as follows: find $$ such that where ρ_i  (i = 1,2, …, n) are positive constants. Denote the set of solutions to (SGMQVI) by Ξ.

Special cases are as follows:

(I) If g_j(x) = x and A_j(x, y) = T_j(x) + S_j(x) for all x, y ∈ E and j = 1,2, …, n, where T_j, S_j : E → E are single-valued mappings, then the problem (SGMQVI) is equivalent to find $$ such that

where ρ_i  (i = 1,2, …, n) are positive constants, which is called the system of generalized nonlinear mixed variational inclusions problem [8].

(II) If n = 2,   A₁ = A₂ = A,   E = H is a Hilbert space, g₁(x) = g₂(x) = x and M₁(x) = M₂(x) = ∂ϕ(x) for all x ∈ E, where ϕ : E → R ∪ {+∞} is a proper, convex, and lower semicontinuous functional, and ∂ϕ denotes the subdifferential operator of ϕ, then the problem (SGMQVI) is equivalent to find (x^*, y^*) ∈ E × E such that

where ρ_i  (i = 1,2) are positive constants, which is called the generalized system of relaxed cocoercive mixed variational inequality problem [29].

(III) If n = 2, E = H is a Hilbert space, and K is a closed convex subset of E, and ϕ(x) = δ_K(x) for all x ∈ K, where δ_K is the indicator function of K defined by

then the problem (SGMQVI) is equivalent to find (x^*, y^*) ∈ K × K such that where ρ_i  (i = 1,2) are positive constants, which is called the system of general variational inequalities problem [27].

(IV) If n = 2, A₁ = A₂ = A, E = H is a Hilbert space, and K is a closed convex subset of E, g₁(y) = g₂(y) = y, and ϕ(x) = δ_K(x) for all x ∈ K, y ∈ E, where δ_K is the indicator function of K defined by

then the problem (SGMQVI) is equivalent to find (x^*, y^*) ∈ K × K such that where ρ_i  (i = 1,2) are positive constants, which is called the generalized system of relaxed cocoercive variational inequality problem [30].

(V) If for each i ∈ {1,2},   z ∈ E, A_i(x, z) = Ψ_i(x), and g_i(x) = x for all x ∈ E, where Ψ_i : E → E, then the problem (SGMQVI) is equivalent to find (x^*, y^*) ∈ E × E such that

where ρ_i  (i = 1,2) are positive constants, which is called the system of quasivariational inclusion [19, 20].

(VI) If n = 2, for each i ∈ {1,2}, z ∈ E, A_i(x, z) = Ψ(x), and g_i(x) = x for all x ∈ E, where Ψ : E → E and M₁(x) = M₂(x) = M, then the problem (SGMQVI) is equivalent to find (x^*, y^*) ∈ E × E such that

where ρ_i(i = 1,2) are positive constants, which is called the system of quasivariational inclusion [20].

We first recall some definitions and lemmas which are needed in our main results.

Definition 2.2. Let M : dom (M) ⊂ E → 2^E be a set-valued mapping, where dom (M) is the effective domain of the mapping M. M is said to be

• (i)

  accretive if, for any x, y ∈ dom (M), u ∈ M(x) and v ∈ M(y), there exists j_q(x − y) ∈ J_q(x − y) such that

  $$ ()

• (ii)

  m-accretive (maximal-accretive) if M is accretive and (I + ρM)dom (M) = E holds for every ρ > 0, where I is the identity operator on E.

Remark 2.3. If E is a Hilbert space, then accretive operator and m-accretive operator are reduced to monotone operator and maximal monotone operator, respectively.

Definition 2.4 (see [24], [31].)Let T : E → E be a single-valued mapping. T is said to be

• (i)

  γ-Lipschitz continuous mapping if there exists a constant γ > 0 such that

  $$ ()

• (ii)

  (a, b)-relaxed cocoercive if there exist two constants a ≥ 0 and b > 0 such that

Remark 2.5 (see [24].)(1) If γ = 1, then a γ-Lipschitz continuous mapping reduces to a nonexpansive mapping.

(2) If γ ∈ (0,1), then a γ-Lipschitz continuous mapping reduces to a contractive mapping.

(3) It is easy to see that the identity operator I : E → E is (0,1) relaxed cocoercive, where I(x) = x for all x ∈ E.

Definition 2.6 (see [24].)Let A : E × E → E be a mapping. A is said to be

• (i)

  τ-Lipschitz continuous in the first variable if there exists a constant τ > 0 such that, for $$,

  $$ ()

• (ii)

  α-strongly accretive if there exists a constant α > 0 such that

  $$ ()
  or equivalently,
  $$ ()

• (iii)

  (μ, ν) relaxed cocoercive if there exist two constants μ ≥ 0 and ν > 0 such that

Remark 2.7. (1) Every α-strongly accretive mapping is a (μ, α) relaxed cocoercive for any positive constant μ. But the converse is not true in general.

(2) The conception of the cocoercivity is applied in several directions, especially for solving variational inequality problems by using the auxiliary problem principle and projection methods [14]. Several classes of relaxed cocoercive variational inequalities have been investigated in [5, 22, 28, 30].

Definition 2.8 (see [9], [32].)Let the set-valued mapping M : dom (M) ⊂ E → 2^E be m-accretive. For any positive number ρ > 0, the mapping $$ defined by

is called the resolvent operator associated with M and ρ, where I is the identity operator on E.

Remark 2.9. Let C ⊂ E be a nonempty closed convex set. If E is a Hilbert space and M = ∂ϕ, the subdifferential of the indicator function ϕ, that is,

then R_(ρ,M) = P_C, the metric projection operator from E onto C.

Lemma 2.10 (see [9], [32].)Let the set-valued mapping M : dom (M) ⊂ E → 2^E be m-accretive. Then the resolvent operator $$ is single-valued and nonexpansive for all ρ > 0.

Lemma 2.11 (see [33].)Let {B_n}, {C_n}, and {D_n} be three nonnegative real sequences satisfying the following conditions:

for some n₀ ∈ N, {λ_n}⊂(0,1) with $$, C_n = 0(λ_n) and $$. Then lim _n→∞B_n = 0.

Lemma 2.12 (see [34].)Let E be a real q-uniformly Banach space. Then there exists a constant c_q > 0 such that

Theorem 3.1. Let $$, M_i : E → 2^E  (i = 1,2, …, n) be maximal accretive. Then $$ is a solution of the problem (SGMQVI) if and only if

where ρ_i  (i = 1,2, …, n) are positive constants.

Proof. Let $$ be a solution of the problem (SGMQVI). Then, for any given positive constants ρ_i  (i = 1,2, …, n), the problem (SGMQVI) is equivalent to

From Definition 2.8 and Lemma 2.10, it yields that (3.2) is equivalent to (3.1). This completes the proof.

Theorem 3.2. Let E be a real q-uniformly smooth Banach space. Let j ∈ {1,2, …, n}, M_j : E → 2^E be m-accretive mapping, Let A_j : E × E → E be (μ_j, ν_j)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τ_j, and Let g_j : E → E be (a_j, b_j)-relaxed cocoercive and Lipschitz continuous with constant ι_j. Then, for each j ∈ {1,2, …, n}, x ∈ E, the mapping $$ has at most one fixed point. If

then the implicit function y_j(x) determined by is continuous on E.

Proof. Let x ∈ E. We show by contradiction that $$ has at most one fixed point. Suppose to the contrary that $$ and $$ such that

Since A_j is Lipschitz continuous in the first variable with constant τ_j, then which is a contradiction. Therefore, the mapping $$ has at most one fixed point.

Next, we show that the implicit function y_j(x) is continuous on E. For any sequence, {x_n} ⊂ E, x₀ ∈ E, x_n → x₀ as n → ∞. Since A_j : E × E → E is (μ_j, ν_j)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τ_j and g_j : E → E is (a_j, b_j)-relaxed cocoercive and Lipschitz continuous with constant ι_j, one has

Therefore, from Lemma 2.10, we get From (3.3), it follows that the implicit function y_j(x) is continuous on E. This completes the proof.

Corollary 3.3 (see [24].)Let E be a real q-uniformly smooth Banach space. Let M₂ : E → 2^E be m-accretive mapping; Let A₂ : E × E → E be (μ₂, ν₂)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τ₂. Then, for each x ∈ E, the mapping $$ has at most one fixed point. If

then the implicit function y(x) determined by is continuous on E.

Theorem 3.4. Let E be a real q-uniformly smooth Banach space. Let M_j : E → 2^E be m-accretive mapping, Let A_j : E × E → E be (μ_j, ν_j)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τ_j, and let g_j : E → E be (a_j, b_j)-relaxed cocoercive and Lipschitz continuous with constant ι_j for j ∈ {1,2, …, n}. Assume that

Then the problem (SGMQVI) has a solution. Moreover, the solutions set Ξ of (SGMQVI) is a singleton.

Proof. By Theorem 3.1, the problem (SGMQVI) has a solution if and only if (3.1) holds. For the convenience, we define a mapping F : E → E by

Since A_j : E × E → E are (μ_j, ν_j)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τ_j and g_j : E → E are (a_j, b_j)-relaxed cocoercive and Lipschitz continuous with constant ι_j for j ∈ {1,2, …, n}, by Theorem 3.2, we know that F(x) and y_i(x)(i = 2,3, …, n) are continuous on E. For any x, z ∈ E, From Lemma 2.10, it yields that Note that, for each j ∈ {2,3, …, n − 1}, Therefore, we obtain It follows from (3.12) that the mapping F is contractive. By banach’s contraction principle, there exists a unique $$ such that $$. Therefore, by Theorem 3.2, there exists an unique $$ such that $$ is a solution of the problem (SGMQVI), where $$ for i = 2,3, …, n, that is, $$. This completes the proof.

Algorithm 4.1. Let ρ_j be positive constants for all j = 1,2, …, n. For any given points x_1,0 ∈ E, define sequences {x_j,k}j = 1,2, …, n in E by the following implicit algorithm:

where {e_k} ⊂ E and {α_k} is a real sequence in [0,1].

Algorithm 4.2. Let ρ_j be positive constants for all j = 1,2, …, n. For any given points x_1,0 ∈ E, define sequences {x_j,k}j = 1,2, …, n in E by the following implicit algorithm

where {α_k} is a real sequence in [0,1].

Algorithm 4.3. Let ρ_j be positive constants for all j = 1,2, …, n. For any given points (x_1,0, x_2,0, …, x_n−1,0, x_n,0) ∈ Eⁿ, define sequences {x_j,k}(j = 1,2, …, n) in E by the following explicit algorithm

where {α_k} is a real sequence in [0,1].

Remark 4.4. If n = 2, E = H is a Hilbert space, and K is a closed convex subset of E, ϕ(x) = δ_K(x) for all x ∈ K, and M₁(x) = M₂(x) = ∂ϕ(x) for all x ∈ E, where δ_K is the indicator function of K, and ∂ϕ denotes the subdifferential operator of ϕ, then, from Remark 2.9, Algorithms 4.1 and 4.3 are reduced to the Algorithms 4.5 and 4.6 for solving the system of general variational inequalities problem (2.7).

Algorithm 4.5. Let ρ_j be positive constants for all j = 1,2. For any given points x_1,0 ∈ E, define sequences {x_j,k}j = 1,2 in E by the following implicit algorithm:

where {e_k} ⊂ E and {α_k} is a real sequence in [0,1].

Algorithm 4.6. Let ρ_j be positive constants for all j = 1,2. For any given points (x_1,0, x_2,0) ∈ E², define sequences {x_j,k}(j = 1,2) in E by the following explicit algorithm:

where {α_k} is a real sequence in [0,1].

Theorem 4.7. Let E be a real q-uniformly smooth Banach space, and let M_j, A_j, and g_j  (j = 1,2, …, n) be the same as in Theorem 3.4. Assume that {α_n} is a real sequence in (0,1] and satisfy the following conditions:

• (i)

  $$;

• (ii)

  $$;

• (iii)

  $$;

• (iv)

  $$.

Then the sequences {x_j,k}(j = 1,2, …, n) generated by Algorithm 4.1 converge strongly to $$, respectively, such that $$.

Proof. By Theorem 3.4, we know that there exist an unique point $$ such that $$. Then, from Theorem 3.1, one has

Therefore, from both (4.1) and (4.6), we have Since A_j : E × E → E are (μ_j, ν_j)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τ_j and g_j : E → E are (a_j, b_j)-relaxed cocoercive and Lipschitz continuous with constant ι_j for j ∈ {1,2, …, n}, we can conclude Noticing that, for each i ∈ {1,2, …, n − 2}, As a consequence, we have Putting $$, $$, and D_k = ∥e_k∥. Then B_k+1 ≤ (1 − λ_k)B_k + C_k + D_k. From the conditions (i)–(iv), it follows that Therefore, by Lemma 2.11 and (4.11), one has that is, $$. Again from (iii), this shows that and so, That is, $$ as k → ∞ for j = 2,3, …, n. Thus, (x_1,k, x_2,k, …, x_n,k) converges strongly to $$. This completes the proof.

Theorem 4.8. Let E be a real q-uniformly smooth Banach space, and let M_j, A_j and g_j  (j = 1,2, …, n) be the same as in Theorem 3.4. Assume that {α_n} is a real sequence in (0,1] and satisfies the following conditions:

• (i)

  $$;

• (ii)

  $$,  j = 1,2, …, n;

• (iii)

  $$.

Then the sequences {x_j,k}  (j = 1,2, …, n) generated by Algorithm 4.2 converge strongly to $$(j = 1,2, …, n), respectively, such that $$.

Proof. It directly follows from Theorem 4.7, and so the proof is omitted. This completes the proof.

Theorem 4.9. Let E be a real q-uniformly smooth Banach space, and let M_j, A_j and g_j(j = 1,2, …, n) be the same as in Theorem 3.4. Assume that {α_n} is a real sequence in (0,1] and satisfy the following conditions:

• (i)

  $$;

• (ii)

  $$;

• (iii)

  $$.

Then the sequences {x_j,k}(j = 1,2, …, n) generated by Algorithm 4.3 converge strongly to $$(j = 1,2, …, n), respectively, such that $$.

Proof. As in the proof of Theorem 4.7, we know that there exists an unique point $$ such that $$, and so

Note that Since A_j : E × E → E are (μ_j, ν_j)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τ_j and g_j : E → E are (a_j, b_j)-relaxed cocoercive and Lipschitz continuous with constant ι_j for j ∈ {1,2, …, n}, we can conclude that Noticing that, for each i ∈ {1,2, …, n − 2}, Consequently, we have Taking $$, C_k = D_k = 0, and $$, then B_k+1 ≤ (1 − λ_k)B_k + C_k + D_k. It follows from the conditions (i)–(iii) that Therefore, by Lemma 2.11 and (4.20), one has lim _k→∞B_k = 0. By the same argument of Theorem 4.7, we get that is, $$ as k → ∞ for j = 1,2, …, n. Thus, (x_1,k, x_2,k, …, x_n,k) converges strongly to $$. This completes the proof.

Corollary 4.10. Let K be a closed convex subset of a real Hilbert space E, and let A_j and g_j  (j = 1,2) be the same as in Theorem 3.4. Assume that {α_n} is a real sequence in (0,1] and satisfies the following conditions:

• (i)

  $$;

• (ii)

  $$;

• (iii)

  $$;

• (iv)

  $$.

Then the sequences {x_j,k}  (j = 1,2) generated by Algorithm 4.5 converge strongly to $$, respectively, such that $$ is the unique solution of the system of general variational inequalities problem (2.7).

Corollary 4.11. Let K be a closed convex subset of a real Hilbert space E, and let A_j and g_j(j = 1,2) be the same as in Theorem 3.4. Assume that {α_n} is a real sequence in (0,1] and satisfies the following conditions:

• (i)

  $$;

• (ii)

  $$, j = 1,2;

• (iii)

  $$.

Then the sequences {x_j,k}  (j = 1,2) generated by Algorithm 4.6 converge strongly to $$, respectively, such that $$ is the unique solution of the system of general variational inequalities problem (2.7).

Lemma 5.1. Let (x^*, y^*) ∈ K₁ × K₂. Then (x^*, y^*) is a solution of the problem (BVI) if and only if $$, where $$ and ρ is a positive constant.

Lemma 5.2. Let K₁ and K₂ be nonempty closed convex subsets of a Hilbert space E. Let A : E × E → E be (μ, ν)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τ, and let g : E → E be (a₂, b₂)-relaxed cocoercive and Lipschitz continuous with constant c₂. Assume that $$ and

Then, for each x ∈ K₁, the parametric variational inequalities (5.2) have a uniquely solution. Further, the solution mapping y(x) of the parametric variational inequalities (5.2) is continuous on K₁.

Proof. It directly follows from Theorems 3.2 and 3.4. This completes the proof.

Theorem 5.3. Let K₁ and K₂ be nonempty closed convex subsets of a Hilbert space E. Let A : E × E → E be (μ, ν)-relaxed cocoercive and Lipschitz continuous in the first variable with constant τ, h : E → E a (a₁, b₁)-relaxed cocoercive and Lipschitz continuous with constant c₁, and let g : E → E be a (a₂, b₂)-relaxed cocoercive and Lipschitz continuous with constant c₂. Assume that {α_k} is a real sequence in (0,1] and satisfies the following conditions:

• (i)

  $$;

• (ii)

  $$;

• (iii)

  $$.

The sequences {x_k} and {y_k} generated by the following algorithm: where ρ is a positive constant. Then the sequences {x_k} and {y_k} converge strongly to x^* and y^*, respectively, such that (x^*, y^*) is a solution of the (BVI).

Proof. The proof is similar to Remark 2.9 and Theorem 4.7, and so the proof is omitted. This completes the proof.