We introduce and study a new system of generalized H(·, ·) − η-cocoercive operator inclusions in Banach spaces. Using the resolvent operator technique associated with H(·, ·) − η-cocoercive operators, we suggest and analyze a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method. We highlight the applicability of our results by examples in function spaces.

1. Introduction

The resolvent operator technique is a powerful tool to study the approximation solvability of nonlinear variational inequalities and variational inclusions, which have been applied widely to optimization and control, mechanics and physics, economics and transportation equilibrium, and engineering sciences, see, for example, [1–4] and the references therein.

In a series of papers [5–8], the authors investigated (A, η)-accretive and H(·, ·)-accretive operators for solving variational inclusions in Banach spaces. Convergence and stability of iterative algorithms for the systems of (A, η)-accretive operators have been studied in [9, 10]. The notion of (H, ϕ) − η-monotone operators has been introduced and investigated by the authors in [11]. Generalized mixed variational inclusions involving (H(·, ·), η)-monotone operators have been discussed in [12]. Some results on H((·, ·), η)-accretive operators and application for solving set-valued variational inclusions in Banach spaces have been proved in [7]. Some other related articles on the variational inclusion problems can be found in [13–22].

Very recently, Ahmad et al. [23] introduced a new H(·, ·) − η-cocoercive operator and its resolvent operator in the setting of Banach spaces. The authors proposed concrete examples in support of H(·, ·) − η-cocoercive operators and they also proved the Lipschitz continuity of resolvent operator associated with H(·, ·) − η-cocoercive operator. Motivated and inspired by the research works mentioned above, in this paper, we introduce and study a new system of H(·, ·) − η-cocoercive mapping inclusions in Banach spaces. Using the resolvent operator associated with H(·, ·) − η-cocoercive mapping, we suggest and analyze a new general algorithm and establish the existence and uniqueness of solutions for this system of H(·, ·) − η-cocoercive mappings.

2. Preliminaries

Throughout this paper, we denote the set of positive integers by ℕ. Let X be a Banach space with the norm ∥·∥ and the dual space X^*. For any x ∈ X, we denote the value of x^* ∈ X^* at x by 〈x, x^*〉. When {x_n} is a sequence in X, we denote the strong convergence of {x_n} to x ∈ X by x_n → x as n → ∞. We denote by 2^X the family of all nonempty subsets of X. Let CB(X) be the family of all nonempty, closed, and bounded subsets of X. The Hausdörff metric on CB(X) [24] is defined by

()

where d(x, B) = inf _b∈B∥x − b∥ and d(A, y) = inf _a∈A∥a − y∥.

Definition 1 (see [25].)A continuous and strictly increasing function ϕ : [0, +∞)→[0, ∞) such that ϕ(0) = 0 and lim _t→∞ϕ(t) = ∞ is called a gauge function.

Definition 2 (see [25].)Let X be a Banach space. Given a gauge function ϕ, the mapping corresponding to ϕ defined by

()

is called the duality mapping with gauge function ϕ.

In particular, if ϕ(t) = t, the duality map J = J_ϕ is called the normalized duality mapping.

Lemma 3 (see [26].)Let X be a real Banach space and be the normalized duality mapping. Then, for any x, y ∈ X,

()

for all j(x + y) ∈ J(x + y).

Definition 4. Let X be a Banach space. Let A : X → X and η : X × X → X be two mappings and be the normalized duality mapping. Then, A is called

(i)
η-cocoercive, if there exists a constant μ₁ > 0 such that
()
(ii)
η-accretive, if
()
(iii)
η-strongly accretive, if there exists a constant β₁ > 0 such that
()
(iv)
η-relaxed cocoercive, if there exists a constant γ₁ > 0 such that
()
(v)
Lipschitz continuous, if there exists a constant λ_A > 0 such that
()
(vi)
α-expansive, if there exists a constant α > 0 such that
()
(vii)
η is said to be Lipschitz continuous, if there exists a constant τ > 0 such that
()

Definition 5. Let X be a Banach space. Let A, B : X → X, H : X × X → X, η : X × X → X be four single-valued mappings and be the normalized duality mapping. Then,

(i)
H(A, ·) is said to be η-cocoercive with respect to A, if there exists a constant μ > 0 such that
()
(ii)
H(·, B) is said to be η-relaxed cocoercive with respect to B, if there exists a constant γ > 0 such that
()
(iii)
H(A, ·) is said to be r₁-Lipschitz continuous with respect to A, if there exists a constant r₁ > 0 such that
()
(iv)
H(·, B) is said to be r₂-Lipschitz continuous with respect to B, if there exists a constant r₂ > 0 such that
()

Definition 6. Let X be a Banach space. A set-valued mapping M : X → 2^X is said to be η-cocoercive, if there exists a constant μ₂ > 0 such that

()

Definition 7. Let X be a Banach space. A mapping T : X → CB(X) is said to be 𝒟-Lipschitz continuous, if there exists a constant λ_T > 0 such that

()

Definition 8. Let X be a Banach space. Let T, Q : X → CB(X) be the mappings. A mapping N : X × X → X is said to be

(i)
Lipschitz continuous in the first argument with respect to T, if there exists a constant t₁ > 0 such that
()
(ii)
Lipschitz continuous in the second argument with respect to Q, if there exists a constant t₂ > 0 such that
()
(iii)
η-relaxed Lipschitz in the first argument with respect to T, if there exists a constant τ₁ > 0 such that
()
(iv)
η-relaxed Lipschitz in the second argument with respect to Q, if there exists a constant τ₂ > 0 such that
()

Definition 9. Let X be a Banach space. Let A, B : X → X, H : X × X → X, η : X × X → X be four single-valued mappings. Let M : X → 2^X be a set-valued mapping. M is said to be H(·, ·) − η-cocoercive operator with respect to A and B, if M is η-cocoercive and (H(A, B) + λM)(X) = X, for every λ > 0.

Example 10. Let X = ℝ × ℝ and A, B : X → X be defined by

()

Assume now that H(A, B), η : X × X → X are defined by

()

Let M = I, where I is the identity mapping. Then, M is H(·, ·) − η-cocoercive with respect to A and B.

Example 11. Let X = C[0,1], the space of all real valued continuous functions defined on closed interval [0,1] with the norm

()

Let A, B : X → X be defined by

()

Let H(A, B) : X × X → X be defined by

()

Suppose that M(f) = f², where f²(t) = f(t)f(t) for all t ∈ [0,1]. Then, for λ = 1, we conclude that

()

This proves that 0 ∉ (H(A, B) + M)(X) and M is not H(A, B) − η-cocoercive with respect to A and B.

Proposition 12 (see [23].)Let H(A, B) be η-cocoercive with respect to A with constant μ > 0 and η-relaxed cocoercive with respect to B with constant γ > 0, A be α-expansive and B be β-Lipschitz continuous μ > γ and α > β. Let M : X → 2^X be H(A, B) − η-cocoercive operator. Suppose that

()

Then, x ∈ Mu, where Graph(M) = {(u, x) ∈ X × X : x ∈ Mu}.

Theorem 13 (see [23].)Let H(A, B) be η-cocoercive with respect to A with constant μ > 0 and η-relaxed cocoercive with respect to B with constant γ > 0, A be α-expansive and B be β-Lipschitz continuous, μ > γ and α > β. Let M be an H(·, ·) − η-cocoercive operator with respect to A and B. Then, for each λ > 0, the operator (H(A, B) + λM) ⁻¹ is single-valued.

Definition 14. Let X be a Banach space. Let H(A, B) be η-cocoercive with respect to A with constant μ > 0 and η-relaxed cocoercive with respect to B with constant γ > 0, A be α-expansive B be β-Lipschitz continuous and η be β-Lipschitz continuous, μ > γ, and α > β. Let M be a H(·, ·) − η-cocoercive operator with respect to A and B. Then, the resolvent is defined by

()

Theorem 15 (see [23].)Let X be a Banach space. Let H(A, B) be η-cocoercive with respect to A with constant μ > 0 and η-relaxed cocoercive with respect to B with constant γ > 0, A be α-expansive B be β-Lipschitz continuous, and η be ρ-Lipschitz continuous; μ > γ and α > β. Let M be H(·, ·) − η-cocoercive operator with respect to A and B. Then, the resolvent operator is ρ/(μα² − γβ²)-Lipschitz continuous, that is,

()

3. Strong Convergence Theorem

In this section, using the resolvent operator technique associated with H(·, ·) − η-cocoercive operators, we propose a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method.

For i = 1,2, let X_i be real Banach spaces with the norm ∥·∥_i. Let A_i, B_i : X_i → X_i, H_i : X_i × X_i → X_i, η_i : X_i × X_i → X_i, F : X₁ × X₂ → X₁, and G : X₁ × X₂ → X₂ be single-valued mappings, and T : X₁ → CB(X₁), Q : X₂ → CB(X₂) be set-valued mappings. Let , be H₁(·, ·) − η₁-cocoercive and H₂(·, ·) − η₂-cocoercive operators with respect to (A₁, B₁) and (A₂, B₂), respectively. We consider the following problem.

Find (x, y) ∈ X₁ × X₂, w ∈ T(x), and v ∈ Q(y) such that

()

We call problem (30) a system of generalized H(·, ·) − η-cocoercive operator inclusions.

Under the assumptions mentioned above, we have the following key and simple lemma.

Lemma 16. (x, y) ∈ X₁ × X₂, w ∈ T(x), v ∈ Q(y) is a solution of problem (30) if and only if

()

where

, and λ₁, λ₂ > 0 are constants.

Proof . This is an easy and direct consequence of Definition 14.

Algorithm 17. For i = 1,2, let X_i be real Banach spaces with the norm ∥·∥_i. Let A_i, B_i : X_i → X_i, H_i : X_i × X_i → X_i, η_i : X_i × X_i → X_i, F : X₁ × X₂ → X₁, and G : X₁ × X₂ → X₂ be single-valued mappings, and T : X₁ → CB(X₁), Q : X₂ → CB(X₂) be set-valued mappings. Let , be such that, for each fixed x ∈ X₁, y ∈ X₂, M(·, x) and N(·, y) are H₁(·, ·) − η₁-cocoercive and H₂(·, ·) − η₂-cocoercive operators with respect to (A₁, B₁) and (A₂, B₂), respectively. For any given constants λ_i > 0(i = 1,2), define the mappings S₁ : X₁ × X₂ → X₁ and S₂ : X₁ × X₂ → X₂ by

()

For any given (x₀, y₀) ∈ X₁ × X₂, w₀ ∈ T(x₀), v₀ ∈ Q(y₀), let

()

Since w₀ ∈ T(x₀) ⊂ CB(X₁) and v₀ ∈ Q(y₀) ⊂ CB(X₂), in view of Nadler′s theorem [24], there exist w₁ ∈ T(x₁) and v₁ ∈ Q(y₁) such that

()

By induction, we define iterative sequences {x_n}, {y_n}, {w_n}, and {v_n} as follows:

()

where n = 0,1, 2, …, and λ₁, λ₂ > 0 are constants.

Theorem 18. For i = 1,2, let X_i be real Banach spaces with the norm ∥·∥_i. Let A_i, B_i : X_i → X_i, H_i : X_i × X_i → X_i, η_i : X_i × X_i → X_i, F : X₁ × X₂ → X_1, and G : X₁ × X₂ → X₂ be single-valued mappings, and T : X₁ → CB(X₁), Q : X₂ → CB(X₂) be set-valued mappings. Let and be such that, for each fixed x ∈ X₁, y ∈ X₂, M(·, x) and N(·, y) are H₁(·, ·) − η₁-cocoercive and H₂(·, ·) − η₂-cocoercive operators with respect to (A₁, B₁) and (A₂, B₂), respectively. Suppose that the following conditions are satisfied.

(i)
H_i(A_i, B_i) is η_i-cocoercive with respect to A_i with constant μ_i and η_i-relaxed cocoercive with respect to B_i with constant γ_i, i = 1,2.
(ii)
A_i is α_i-expansive and B_i is β_i-Lipschitz continuous, i = 1, 2.
(iii)
H_i(A_i, B_i) is r_i-Lipschitz continuous with respect to A_i and s_i-Lipschitz continuous with respect to B_i, i = 1, 2.
(iv)
T is D-Lipschitz continuous with constant λ_T and Q is D-Lipschitz continuous with constant λ_Q.
(v)
F is t₁-Lipschitz continuous with respect to T in the first argument and t₂-Lipschitz continuous with respect to Q in the second argument.
(vi)
G is l₁-Lipschitz continuous with respect to T in the first argument and l₂-Lipschitz continuous with respect to Q in the second argument.
(vii)
η_i is ρ_i-Lipschitz continuous, i = 1, 2.
(viii)
F is η₁-relaxed Lipschitz continuous with respect to T in the first argument and η₁-relaxed Lipschitz continuous with respect to Q in the second argument with constants τ₁ and τ₂, respectively.
(ix)
G is η₂-relaxed Lipschitz continuous with respect to T in the first argument and η₂-relaxed Lipschitz continuous with respect to Q in the second argument with constants ϵ₁ and ϵ₂, respectively. Furthermore, assume that there exist constants σ₁, σ₂ > 0 such that
()

()

and λ₁, λ₂ > 0 are constants satisfying the following conditions:

()

Then, the iterative sequences {x_n}, {y_n}, {w_n}, and {v_n} generated by Algorithm 17 converge strongly to x, y, w, and v, respectively, and (x, y, w, v) is a solution of problem (30).

Proof. In view of Theorem 13, the resolvent operator is θ₁-Lipschitz continuous. This, together with Algorithm 17 and (36), implies that

()

Since F is t₁-Lipschitz continuous with respect to T in the first argument and t₂-Lipschitz continuous in the second argument, T is λ_T-Lipschitz continuous, and Q is λ_Q-Lipschitz continuous, by Algorithm 17, we get

()

As H₁(·, ·) is r₁-Lipschitz continuous with respect to A₁, we obtain

()

Since η₁ is ρ₁-Lipschitz continuous, we conclude that

()

Since H₁(·, ·) is η₁-relaxed Lipschitz continuous with respect to T and η₂-relaxed Lipschitz continuous with respect to Q in the first and second arguments with constants τ₁ and τ₂, respectively, we have

()

Employing Lemma 3 and taking into account (39)–(44), we obtain

()

This implies that

()

where

()

Using s₁-Lipschitz continuity of H₁(·, ·) with respect to B₁, we deduce that

()

In view of (41), (46), (48), (39) becomes

()

Similarly, we have

()

where

()

In view of (49) and (50), we obtain

()

where c_n = σ₁ + θ₁p_n + θ₁s₁ + λ₂θ₂l₁λ_T(1 + 1/n), d_n = σ₂ + θ₂q_n + θ₂s₂ + λ₁θ₁t₂λ_Q(1 + 1/n), and k_n = max {c_n, d_n}.

Letting n → ∞, we obtain k_n → k, where

()

Next, we define the norm ∥·∥ on X₁ × X₂ by

()

One can easily check that (X₁ × X₂, ∥·∥) is a Banach space.

Define a_n+1 = (x_n+1, y_n+1). Then, we have

()

In view of (38), we conclude that 0 < k < 1. This implies that there exist n₀ ∈ ℕ and k₀ ∈ (0,1) such that k_n ≤ k₀ for all n ≥ n₀. It follows from (52) and (54) that

()

In view of (56), we obtain

()

This implies that for any m ≥ n ≥ n₀,

()

Since 0 < k₀ < 1, it follows from (58) that

and n → ∞. This proves that {x_n} is a Cauchy sequence in X₁. Similarly, we conclude that {y_n} is a Cauchy sequence in X₂. Thus, there exist x ∈ X₁ and y ∈ X₂ such that x_n → x and y_n → y as n → ∞.

Next, we prove that w_n → w ∈ T(x) and v_n → v ∈ Q(y). In view of Lipschitz continuity of T and Q and Algorithm 17, we obtain

()

From (59), we deduce that {w_n}, {v_n} are Cauchy sequences in X₁ and X₂, respectively. Thus, there exist w ∈ T(x) and v ∈ Q(y) such that w_n → w and v_n → v as n → ∞. Since T is D-Lipschitz continuous with constant λ_T, it is obvious that

()

By the closedness of T(x), we conclude that w ∈ T(x). Similarly, we have v ∈ Q(y).

Assume now that

()

Then, we have

()

Since x_n → x, w_n → w, and v_n → v as n → ∞, it follows from (62) that

()

and hence x₀ = x.

A similar argument shows that y₀ = y. Therefore,

()

In view of Lemma 16, we conclude that (x, y, w, v) is a solution of problem (30), which completes the proof.

At the end of this paper, we include the following simple example in support of Theorem 18.

Example 19. Let X = ℝ² with the usual inner product. We define two mappings A, B : ℝ² → ℝ² by

()

Let a mapping H : ℝ² × ℝ² → ℝ² be defined by

()

By similar arguments, as in Example 4.1 of [27], we can prove the following.

(1)
H(A, B) is 4/17-cocoercive with respect to A and 1-relaxed cocoercive with respect to B.
(2)
A is -expansive, for n = 4,5.
(3)
B is -expansive, for n = 1,2.
(4)
H(A, B) is -Lipschitz continuous with constant with respect to A and B, for n = 1,2, …, 15,16.
(5)
Let f, g : ℝ² → ℝ² be defined by
()
(6)
Now, we define a mapping M : ℝ² × ℝ² → ℝ² by
()

Let R, S, T : ℝ² → ℝ² be the identity mappings. It is obvious that these mappings are D-Lipschitz continuous.

(7)
Assume that F, G : ℝ² → ℝ² are defined by
()

It could easily be seen that all the aspects of the hypotheses of Theorem 18 are satisfied, so we have the desired conclusion.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by King Abdulaziz University, under Grant no. 23-130-1433-HiCi. The authors, therefore, acknowledge the technical and financial support of KAU. The authors would like to thank the referees for their sincere evaluation and constructive comments which improved the paper considerably.

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Strong Convergence of Iterative Algorithm for a New System of Generalized H(·, ·) − η-Cocoercive Operator Inclusions in Banach Spaces

Abstract

1. Introduction

2. Preliminaries

3. Strong Convergence Theorem

Conflict of Interests

Acknowledgments

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Strong Convergence of Iterative Algorithm for a New System of Generalized H(·, ·) − η-Cocoercive Operator Inclusions in Banach Spaces

Abstract

1. Introduction

2. Preliminaries

3. Strong Convergence Theorem

Conflict of Interests

Acknowledgments

References

References

Related

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