A System of Generalized Variational Inclusions Involving a New Monotone Mapping in Banach Spaces

Definition 1 (see [7].)A single-valued mapping η : E × E → E is said to be k-Lipschitz continuous if there exists a constant k > 0 such that

Definition 2 (see [8].)A Banach space E is called smooth if, for every x ∈ E with ∥x∥ = 1, there exists a unique f ∈ E^* such that ∥f∥ = f(x) = 1. The modulus of smoothness of E is the function ρ_E : [0, ∞)→[0, ∞), defined by

Definition 3 (see [8].)The Banach space E is said to be

• (i)

  uniformly smooth if

  $$ ()

• (ii)

  q-uniformly smooth, for q > 1, if there exists a constant c > 0 such that

  $$ ()

Lemma 4 (see [8].)Let q > 1 be a real number and let E be a smooth Banach space and $$ the normalized duality mapping. Then, E is q-uniformly smooth if and only if there exists a constant c_q > 0 such that for every x, y ∈ E,

Definition 5. A single-valued mapping g : E → E is said to be (γ, μ)-relaxed cocoercive if there exist j_q(x − y) ∈ J_q(x − y) and γ, μ > 0 such that

Definition 6. Let n ≥ 3 and $$ be a multivalued mapping, f_i : E → E_i, i = 1,2, …, n, and η : E × E → E single-valued mappings.

• (i)

  For each 1 ≤ i ≤ n, M(…, f_i, …) is said to be α_i-strongly η-monotone with respect to f_i (in the ith argument) if there exists a constant α_i > 0 such that

  $$ ()

• (ii)

  For each 1 ≤ i ≤ n,   M(…, f_i, …) is said to be β_i-relaxed η-monotone with respect to f_i (in the ith argument) if there exists a constant β_i > 0 such that

  $$ ()

• (iii)

  By assumption that n is an even number, M is said to be α₁β₂α₃β₄ ⋯ α_n−1β_n-symmetric η-monotone with respect to f₁, f₂, …, f_n if, for each i ∈ {1,3, …, n − 1}, M(…, f_i, …) is α_i-strongly η-monotone with respect to f_i (in the ith argument) and for each j ∈ {2,4, …, n}, M(…, f_j, …) is β_j-relaxed η-monotone with respect to f_j (in the jth argument) with

  $$ ()

• (iv)

  By assumption that n is an odd number, M is said to be α₁β₂α₃β₄ ⋯ α_n−1β_n-symmetric η-monotone with respect to f₁, f₂, …, f_n if, for each i ∈ {1,3, …, n}, M(…, f_i, …) is α_i-strongly η-monotone with respect to f_i (in the ith argument) and for each j ∈ {2,4, …, n − 1}, M(…, f_j, …) is β_j-relaxed η-monotone with respect to f_j (in the jth argument) with

  $$ ()

Definition 7 (see [10].)Let E be a Banach space. A multivalued mapping A : E⇉CB(E_i) is said to be H-Lipschitz continuous if there exists a constant t > 0 such that

where H(·, ·) is the Hausdorff metric on CB(E_i).

Definition 8. Let, for each i = 1,2, …, n,   T_i : E⇉CB(E_i) be a multivalued mapping. A single-valued mapping $$ is said to be $$-Lipschitz continuous in the ith argument with respect to T_i  (i = 1,2, …, n) if there exists a constant $$ such that

Definition 9. Let E be a Banach space with the dual space E^*and  η : E × E → E single-valued mappings; C_n : E → E^* is said to be η-monotone mapping if

Definition 10. Let E be a Banach space with the dual space E^*. Let n ≥ 3 and f_i : E → E_i,   i = 1,2, …, n,   C_n : E → E^* be single-valued mappings and $$ a multivalued mapping.

• (i)

  In case that n is an even number, M is said to be a C_n-η-monotone mapping if M is α₁β₂α₃β₄ ⋯ α_n−1β_n-symmetric η-monotone with respect to f₁, f₂, …, f_n and (C_n + λM(f₁, f₂, …, f_n))(E) = E^*, for every λ > 0.

• (ii)

  In case that n is an odd number, M is said to be a C_n-η-monotone mapping if M is α₁β₂α₃β₄ ⋯ β_n−1α_n-symmetric η-monotone with respect to f₁, f₂, …, f_n and (C_n + λM(f₁, f₂, …, f_n))(E) = E^*, for every λ > 0.

Remark 11. (i) If M(f₁, f₂, …, f_n) = M, η(y, x) = y − x, and M is monotone, then the C_n-η-monotone mapping reduces to the general H-monotone mapping considered in [4].

(ii) If M(f₁, f₂, …, f_n) = M(f₁, f₂), η(y, x) = y − x, then the C_n-η-monotone mapping reduces to the B-monotone mapping considered in [5].

(iii) If M(f₁, f₂, …, f_n) = M, η(y, x) = y − x, and M are m-relaxed monotone, then the C_n-η-monotone mapping reduces to the A-monotone mapping considered in [11].

(iv) If M reduces to $$,  η(y, x) = y − x, and f₁, f₂, …, f_n reduce to E → E, then the C_n-η-monotone mapping reduces to the C_n-monotone mapping considered in [6].

Example 12. Let E = l² and then E^* = l²; and assume n is an even number; let E_i = (E, ∥·∥_i),   i = 1,2, …, n, where ∥·∥_i is the equivalent norm on l² space, $$, for ∀x ∈ E; let f₁(x) = α₁x + e₁ ∈ E₁, f₂(x) = −β₂x + e₂ ∈ E₂, f₃(x) = α₃x + e₃ ∈ E₃, f₄(x) = −β₄x + e₄ ∈ E₄, …, f_n−1(x) = α_n−1x + e_n−1 ∈ E_n−1, f_n(x) = −β_nx + e_n ∈ E_n, where α₁, β₂, α₃, β₄, …, α_n−1, β_n > 0 are constants such that

Let M(u₁, u₂, …, u_n) = (u₁  −  e₁)  +  (u₂  −  e₂)  +   ⋯   +  (u_n  −  e_n), where u_i ∈ E_i, i = 1,2, …, n; let C_n(x) = x + e_n+1, x ∈ E, η(x, y) = x − y, ∀x, y ∈ E. Then M is a C_n-η-monotone mapping.

Lemma 13. Let η : E × E → E, f_i : E → E_i, i = 1,2, …, n, be single-valued mappings; $$ a α₁β₂α₃β₄ ⋯ α_n−1β_n-symmetric η-monotone with respect to f₁, f₂, …, f_n. Then for ∀x, y ∈ E one has

where K_n = (α₁ + α₃ + ⋯+α_n−1)−(β₂ + β₄ + ⋯+β_n).

Proof. Setting  ω₁ ∈ M(f₁x, f₂y, …, f_ny), ω₂ ∈ M(f₁x, f₂x, …, f_ny), …, ω_n−1 ∈ M(f₁x, …, f_n−1x, f_ny). From Definition 10, we have

where K_n = (α₁ + α₃ + ⋯+α_n−1)−(β₂ + β₄ + ⋯+β_n).

This completes the proof.

Theorem 14. Let E be a Banach space with the dual space E^*. Let n ≥ 3 and f_i : E → E_i, i = 1,2, …, n, η : E × E → E single-valued mappings, C_n : E → E^* a η-monotone mapping, and $$ a C_n-η-monotone mapping. Then, (C_n + λM(f₁, f₂, …, f_n)) ⁻¹  is a single-valued mapping.

Proof. Suppose, on the contrary, that there exists x₁, x₂ ∈ E, y^* ∈ E^*, such that

then Now, by using Lemma 13 and since C_n is a η-monotone mapping, we have Thus, we have x₁ = x₂, which implies that (C_n + λM(f₁, f₂, …, f_n)) ⁻¹ is single valued. This completes the proof.

Definition 15. Let E be a Banach space with the dual space E^*. Let n ≥ 3 and f_i : E → E_i, i = 1,2, …, n, be single-valued mappings, C_n : E → E^* a η-monotone mapping, and $$ a C_n-η-monotone mapping. A proximal mapping $$ is defined by

Theorem 16. Let E be a Banach space with the dual space E^*. Let η : E × E → E be a k-Lipschitz continuous mapping. Let n ≥ 3 and f_i : E → E_i, i = 1,2, …, n, be single-valued mappings, C_n : E → E^* a η-monotone mapping, and $$ a C_n-η-monotone mapping. Then, the proximal mapping $$ is k/λK_n-Lipschitz continuous, where K_n = (α₁ + α₃ + ⋯+α_n−1)−(β₂ + β₄ + ⋯+β_n).

Proof. Let x^*, y^* ∈ E^* be any given points. It follows from Definition 15 that

Setting This implies that By using Lemma 13, we have since η is K-Lipschitz continuous, we have thus that is, where K_n = (α₁ + α₃ + ⋯+α_n−1)−(β₂ + β₄ + ⋯+β_n).

This completes the proof.

Theorem 17. Let n ≥ 3 and A : E → E^*, p : E → E, f_n : E → E_i, i = 1,2, …, n, $$ be single-valued mappings and let T_i : E⇉CB(E_i), i = 1,2, …, n, be multivalued mappings. Let C_n : E → E^* be a η-monotone mapping and $$ a C_n-η-monotone mapping with respect to f₁, f₂, …, f_n. Then, (x, t₁, t₂, …, t_n) is a solution of problem (30) if and only if

where t₁ ∈ T₁(x), t₂ ∈ T₂(x), …, t_n ∈ T_n(x), and λ > 0 is a constant.

Proof. Let (x, t₁, t₂, …, t_n) be a solution of problem (30); then we have

then thus Setting x^* = C_n(x) − λA(x − p(x)) + λa + λF(t₁, t₂, …, t_n), from the definition of $$, we have Conversely, let $$; then thus we have

This completes the proof.

Iterative Algorithm  1 For any given x₀ ∈ E, we choose t_1,0 ∈ T₁(x₀),   t_2,0 ∈ T₂(x₀), …, t_n,0 ∈ T_n(x₀) and compute {x_m}, {t_1,m}, {t_2,m}, …, {t_n,m} by iterative schemes

for all m = 0,1, 2, ….

Theorem 18. Let E be a q-uniformly smooth Banach space with q > 1 and E^* the dual space of E. Let η : E × E → E  k-Lipschitz continuous. Let n ≥ 3 and f_i : E → E_i, i = 1,2, …, n, be single-valued mappings, C_n : E → E^* a η-monotone and δ-Lipschitz continuous mapping, p : E → E a (γ, μ)-relaxed cocoercive and λ_p-Lipschitz continuous mapping, and $$ a C_n-η-monotone mapping. Let A : E → E^* be a τ-Lipschitz continuous mapping and, for each i = 1,2, …, n, let T_i : E⇉CB(E_i) be H-Lipschitz continuous with constant $$. Suppose that $$ is $$-Lipschitz continuous in the ith argument with respect to T_i  (i = 1,2, …, n) and the following condition is satisfied:

where

Proof. By using Algorithm 4.1 and Theorem 16, we have

From the Lipschitz continuity of C_n, p, A, and (γ, μ)-relaxed cocoercivity of p and Lemma 4, we have where $$ is the normalized duality mapping.

Since F, T₁, T₂, …, T_n are Lipschitz continuous, we have

It follows from (41)–(45) that where Letting m → ∞, we obtain θ_m → θ, where From condition (39), we know that 0 < θ < 1, and hence {x_m} is a Cauchy sequence in E. Thus, there exists x ∈ E such that x_m → x, as m → ∞. Now, we prove that t_1,m → t₁ ∈ T₁(x). In fact, it follows from the Lipschitz continuity of T₁ and Algorithm 4.1 that From (49), we know that {t_1,m} is also a Cauchy sequence. In a similar way, {t_2,m}, {t_3,m}, …, {t_n,m} are Cauchy sequences. Thus, there exist t₁ ∈ E₁, t₂ ∈ E₂, …, t_n ∈ E_n such that t_1,m → t₁, t_2,m → t₂, …, t_n,m → t_n, as m → ∞. Furthermore, Since T₁(x) is closed, we have t₁ ∈ T₁(x). In a similar way, we can show that t₂ ∈ T₂(x), t₃ ∈ T₃(x), …, t_n ∈ T_n(x). By continuity of $$ and Algorithm 4.1, we have By Theorem 17, (x, t₁, t₂, …, t_n) is a solution of problem (30). This completes the proof.