Generalized 𝐻(⋅,⋅,⋅)-𝜂-Cocoercive Operators and Generalized Set-Valued Variational-Like Inclusions

Abstract

We investigate a new class of cocoercive operators named generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operators in Hilbert spaces. We prove that generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with 𝐻(⋅,⋅)-cocoercive operators to the generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operators. Some examples are given to justify the definition of generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operators. Further, we consider a generalized set-valued variational-like inclusion problem involving generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operator. In terms of the new resolvent operator technique, we give the approximate solution and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.

1. Introduction

Variational inclusions, as the generalization of variational inequalities, have been widely studied in recent years. One of the most interesting and important problems in the theory of variational inclusions include variational, quasi-variational, and variational-like inequalities as special cases. For applications of variational inclusions, see, for example, [1]. Various kinds of iterative methods have been studied to solve the variational inclusions. Among these methods, the resolvent operator technique for the study of variational inclusions has been widely used by many authors. For details, we refer to [2–16].

Recently Fang and Huang, Kazmi and Khan, and Lan et al. investigated several resolvent operators for generalized operators such as 𝐻-monotone [5], 𝐻-accretive [6], (𝑃,𝜂)-proximal point [11], (𝑃,𝜂)-accretive [12], (𝐻,𝜂)-monotone [7], (𝐴,𝜂)-accretive [13], and mappings. Very recently, Zou and Huang [16] introduced and studied 𝐻(⋅,⋅)-accretive operators, Kazmi et al. [8–10] introduced and studied generalized 𝐻(⋅,⋅)-accretive operators, 𝐻(⋅,⋅)-𝜂-proximal point mapping, Xu and Wang [15] introduced and studied (𝐻(⋅,⋅),𝜂)-monotone operators, and Ahmad et al. [2] introduced and studied 𝐻(⋅,⋅)-cocoercive operators.

Motivated by the recent work going in this direction, we consider a class of cocoercive operators called generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive, a natural generalization of monotone (accretive) operators in Hilbert (Banach) spaces. For details, we refer to [2, 5–7, 13–16]. We prove that generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with 𝐻(⋅,⋅)-cocoercive operators to the generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operators. Further, we consider the generalized set-valued variational-like inclusion problem involving generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operator in Hilbert spaces. Using new a resolvent operator technique, we prove the existence of solutions and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as an extension and generalization of some known results [2, 15, 16]. For illustration of Definitions 2 and 7 and examples 2.3 and 3.2 are given, respectively.

2. Preliminaries

Throughout this paper, we suppose that 𝑋 is a real Hilbert space endowed with a norm ‖⋅‖ and an inner product ⟨⋅,⋅⟩, 2^𝑋 (resp., CB(𝑋)) is the family of all the nonempty (resp., closed and bounded) subsets of 𝑋, and 𝒟(⋅,⋅) is the Hausdorff metric on CB(𝑋) defined by 𝒟(𝑃,𝑄)=maxsup_𝑥∈𝑃𝑑(𝑥,𝑄),sup_𝑦∈𝑄𝑑𝑃,𝑦,∀𝑃,𝑄∈CB(𝑋), (1) where 𝑑(𝑥,𝑄)=inf_𝑦∈𝑄‖𝑥−𝑦‖ and 𝑑(𝑃,𝑦)=inf_𝑥∈𝑃‖𝑥−𝑦‖.

In the sequel, let us recall some concepts.

Definition 1 (see [4, 17]). Let 𝑃∶𝑋→𝑋 and 𝜂∶𝑋×𝑋→𝑋 be two mappings. Then, 𝑃 is said to be(i)𝜂-monotone if 𝑃(𝑥)−𝑃𝑦,𝜂𝑥,𝑦≥0,∀𝑥,𝑦∈𝑋, (2)(ii)𝛿₁-𝜂-strongly monotone if there exists a constant 𝛿₁>0 such that 𝑃(𝑥)−𝑃𝑦,𝜂𝑥,𝑦≥𝛿₁𝑥−𝑦²,∀𝑥,𝑦∈𝑋, (3)(iii)𝜇₁-𝜂-cocoercive if there exists a constant 𝜇₁>0 such that 𝑃(𝑥)−𝑃𝑦,𝜂𝑥,𝑦≥𝜇₁𝑃(𝑥)−𝑃𝑦²,∀𝑥,𝑦∈𝑋, (4)(iv)𝛾₁-𝜂-relaxed cocoercive if there exists a constant 𝛾₁>0 such that 𝑃(𝑥)−𝑃𝑦,𝜂𝑥,𝑦≥−𝛾₁𝑃(𝑥)−𝑃𝑦²,∀𝑥,𝑦∈𝑋, (5)(v)𝜆_𝑃-Lipschitz continuous, if there exists a constant 𝜆_𝑃>0 such that 𝑃(𝑥)−𝑃𝑦≤𝜆_𝑃𝑥−𝑦,∀𝑥,𝑦∈𝑋, (6)(vi)𝛼-expansive, if there exists a constant 𝛼>0 such that 𝑃(𝑥)−𝑃𝑦≥𝛼𝑥−𝑦,∀𝑥,𝑦∈𝑋, (7) if 𝛼=1, then it is expansive;(vii)𝜂 is said to be 𝜏-Lipschitz continuous, if there exists a constant 𝜏>0 such that 𝜂𝑥,𝑦≤𝜏𝑥−𝑦,∀𝑥,𝑦∈𝑋. (8)

If 𝜂(𝑥,𝑦)=𝑥−𝑦, for all 𝑥,𝑦∈𝑋, then definitions (i) to (iv) reduce to the Definitions of monotonicity, strong monotonicity [18], cocoercivity [19], and relaxed cocoercive, respectively.

Definition 2. Let 𝑃,𝑄,𝑅∶𝑋→𝑋, 𝜂∶𝑋×𝑋→𝑋, and 𝐻∶𝑋×𝑋×𝑋→𝑋 be the single-valued mappings. Then,(i)𝐻(𝑃,⋅,⋅) is said to be 𝜇-𝜂-cocoercive with respect to 𝑃 if there exists a constant 𝜇>0 such that 𝐻(𝑃𝑥,𝑢,𝑢)−𝐻𝑃𝑦,𝑢,𝑢,𝜂𝑥,𝑦≥𝜇𝑃𝑥−𝑃𝑦²,∀𝑥,𝑦,𝑢∈𝑋, (9)(ii)𝐻(⋅,𝑄,⋅) is said to be 𝛾-𝜂-relaxed cocoercive with respect to 𝑄 if there exists a constant 𝛾>0 such that 𝐻(𝑢,𝑄𝑥,𝑢)−𝐻𝑢,𝑄𝑦,𝑢,𝜂𝑥,𝑦≥−𝛾𝑄𝑥−𝑄𝑦²,∀𝑥,𝑦,𝑢∈𝑋, (10)(iii)𝐻(⋅,⋅,𝑅) is said to be 𝛿-𝜂-strongly monotone with respect to 𝑅 if there exists a constant 𝛿>0 such that 𝐻(𝑢,𝑢,𝑅𝑥)−𝐻𝑢,𝑢,𝑅𝑦,𝜂𝑥,𝑦≥𝛿𝑥−𝑦²,∀𝑥,𝑦,𝑢∈𝑋, (11)(iv)𝐻(𝑃,⋅,⋅) is said to be 𝑟₁-Lipschitz continuous with respect to 𝑃 if there exists a constant 𝑟₁>0 such that 𝐻(𝑃𝑥,⋅,⋅)−𝐻𝑃𝑦,⋅,⋅≤𝑟₁𝑥−𝑦,∀𝑥,𝑦∈𝑋, (12)(v)𝐻(⋅,𝑄,⋅) is said to be 𝑟₂-Lipschitz continuous with respect to 𝑄 if there exists a constant 𝑟₂>0 such that 𝐻(⋅,𝑄𝑥,⋅)−𝐻⋅,𝑄𝑦,⋅≤𝑟₂𝑥−𝑦,∀𝑥,𝑦∈𝑋, (13)(vi)𝐻(⋅,⋅,𝑅) is said to be 𝑟₃-Lipschitz continuous with respect to 𝑅 if there exists a constant 𝑟₃>0 such that 𝐻(⋅,⋅,𝑅𝑥)−𝐻⋅,⋅,𝑅𝑦≤𝑟₃𝑥−𝑦,∀𝑥,𝑦∈𝑋. (14)

If 𝜂(𝑥,𝑦)=𝑥−𝑦, for all 𝑥,𝑦∈𝐻 and 𝐻(⋅,⋅,⋅)=𝐻(⋅,⋅), then Definitions (i)-(ii) are reduced to the definition of cocoercivity and relaxed cocoercive [2], respectively, and (iii) reduces to strong accretivity [16].

Example 3. Let 𝑋=ℝ² with usual inner product. Let 𝜂∶ℝ²×ℝ²→ℝ²,𝑃,𝑄,𝑅∶ℝ²→ℝ² be defined by 𝜂𝑥,𝑦=𝑚𝑥₁+𝑛𝑥₂−𝑚𝑦₁−𝑛𝑦₂,𝑃𝑥=𝑚𝑥₁−𝑚𝑥₂−𝑚𝑥₁+𝑛𝑥₂,𝑄𝑦=−𝑚𝑦₁+𝑚𝑦₂−𝑛𝑦₂,𝑅𝑧=𝑛𝑧₁𝑚𝑧₂, (15) for all scalers 𝑛,𝑚∈ℝ with 𝑛>𝑚 and for all 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂),𝑧=(𝑧₁,𝑧₂)∈ℝ².
Suppose that 𝐻(𝑃𝑥,𝑄𝑦,𝑅𝑧)=𝑃𝑥+𝑄𝑦+𝑅𝑧. Then, 𝐻(𝑃,𝑄,𝑅) is (1/2)-𝜂-cocoercive with respect to 𝑃, (1/2)-𝜂-relaxed cocoercive with respect to 𝑄, 𝑚²-𝜂-strongly monotone with respect to 𝑅, and 𝜂(𝑥,𝑦) is 𝑛-Lipschitz continuous.

Indeed, let for any 𝑢∈𝑋, 𝐻()==𝑃𝑥,𝑢,𝑢−𝐻𝑃𝑦,𝑢,𝑢,𝜂𝑥,𝑦𝑃𝑥−𝑃𝑦,𝜂𝑥,𝑦𝑚𝑥₁−𝑚𝑥₂,−𝑚𝑥₁+𝑛𝑥₂−𝑚𝑦₁−𝑚𝑦₂,−𝑚𝑦₁+𝑛𝑦₂,𝑚𝑥₁−𝑦₁,𝑛𝑥₂−𝑦₂=𝑚𝑥₁−𝑦₁−𝑚𝑥₂−𝑦₂,−𝑚𝑥₁−𝑦₁+𝑛𝑥₂−𝑦₂,𝑚𝑥₁−𝑦₁,𝑛𝑥₂−𝑦₂=𝑚²𝑥₁−𝑦₁²−𝑚(𝑚+𝑛)𝑥₁−𝑦₁𝑥₂−𝑦₂+𝑛²𝑥₂−𝑦₂²,𝑃𝑥−𝑃𝑦²==𝑃𝑥−𝑃𝑦,𝑃𝑥−𝑃𝑦𝑚𝑥₁−𝑚𝑥₂,−𝑚𝑥₁+𝑛𝑥₂−𝑚𝑦₁−𝑦₂,−𝑚𝑦₁+𝑛𝑦₂,𝑚𝑥₁−𝑚𝑥₂,−𝑚𝑥₁+𝑛𝑥₂−𝑚𝑦₁−𝑦₂,−𝑚𝑦₁+𝑛𝑦₂=2𝑚²𝑥₁−𝑦₁²−2𝑚(𝑚+𝑛)𝑥₁−𝑦₁𝑥₂−𝑦₂+𝑚²+𝑛²𝑥₂−𝑦₂²≤2𝑚²𝑥₁−𝑦₁²−2𝑚(𝑚+𝑛)𝑥₁−𝑦₁𝑥₂−𝑦₂+2𝑛²𝑥₂−𝑦₂²𝐻(),𝐻()≥1=2𝑃𝑥,𝑢,𝑢−𝐻𝑃𝑦,𝑢,𝑢,𝜂𝑥,𝑦𝑃𝑥,𝑢,𝑢−𝐻𝑃𝑦,𝑢,𝑢,𝜂𝑥,𝑦2𝑃𝑥−𝑃𝑦², (16) that is, 𝐻(𝑃,𝑄,𝑅) is (1/2)-𝜂-cocoercive with respect to 𝑃. Consider 𝐻()==𝑢,𝑄𝑥,𝑢−𝐻𝑢,𝑄𝑦,𝑢,𝜂𝑥,𝑦𝑄𝑥−𝑄𝑦,𝜂𝑥,𝑦−𝑚𝑥₁+𝑚𝑥₂,−𝑛𝑥₂−−𝑚𝑦₁+𝑚𝑦₂,−𝑛𝑦₂,𝑚𝑥₁−𝑦₁,𝑛𝑥₂−𝑦₂=−𝑚𝑥₁−𝑦₁+𝑚𝑥₂−𝑦₂,−𝑛𝑥₂−𝑦₂,𝑚𝑥₁−𝑦₁,𝑛𝑥₂−𝑦₂=−𝑚²𝑥₁−𝑦₁²+𝑚²𝑥₁−𝑦₁𝑥₂−𝑦₂−𝑛²𝑥₂−𝑦₂²,𝑄𝑥−𝑄𝑦²==𝑄𝑥−𝑄𝑦,𝑄𝑥−𝑄𝑦−𝑚𝑥₁+𝑚𝑥₂,−𝑛𝑥₂−−𝑚𝑦₁+𝑚𝑦₂,−𝑛𝑦₂,−𝑚𝑥₁+𝑚𝑥₂,−𝑛𝑥₂−−𝑚𝑦₁−𝑚𝑦₂,−𝑛𝑦₂=𝑚²𝑥₁−𝑦₁²−2𝑚²𝑥₁−𝑦₁𝑥₂−𝑦₂+𝑚²+𝑛²𝑥₂−𝑦₂²≤2𝑚²𝑥₁−𝑦₁²−2𝑚²𝑥₁−𝑦₁𝑥₂−𝑦₂+2𝑛²𝑥₂−𝑦₂²=(𝐻()𝜂,𝐻()1−2)𝑢,𝑄𝑥,𝑢−𝐻𝑢,𝑄𝑦,𝑢𝑥,𝑦𝑢,𝑄𝑥,𝑢−𝐻𝑢,𝑄𝑦,𝑢,𝜂𝑥,𝑦≥−2𝑄𝑥−𝑄𝑦², (17) that is, 𝐻(𝑃,𝑄,𝑅) is (1/2)-𝜂-relaxed cocoercive with respect to 𝑄. In addition, 𝐻()==𝑢,𝑢,𝑅𝑥−𝐻𝑢,𝑢,𝑅𝑦,𝜂𝑥,𝑦𝑅𝑥−𝑅𝑦,𝜂𝑥,𝑦𝑛𝑥₁,𝑚𝑥₂−𝑛𝑦₂,𝑚𝑦₂,𝑚𝑥₁−𝑦₁,𝑛𝑥₂−𝑦₂=𝑚𝑛𝑥₁−𝑦₁²+𝑚𝑛𝑥₂−𝑦₂²≥𝑚²𝑥−𝑦²,𝐻(𝑢,𝑢,𝑅𝑥)−𝐻𝑢,𝑢,𝑅𝑦,𝜂𝑥,𝑦≥𝑚²𝑥−𝑦², (18) that is, 𝐻(𝑃,𝑄,𝑅) is 𝑚²-𝜂-strongly monotone with respect to 𝑅. Moreover, 𝜂𝑥,𝑦=𝑚𝑥₁−𝑦₁,𝑛𝑥₂−𝑦₂=𝑚²𝑥₁−𝑦₁²+𝑛²𝑥₂−𝑦₂²≤𝑛²𝑥₁−𝑦₁²+𝑛²𝑥₂−𝑦₂²≤𝑛𝑥−𝑦, (19) that is, 𝜂(𝑥,𝑦) is 𝑛-Lipschitz continuous.

Definition 4. A set-valued mapping 𝑀∶𝑋→2^𝑋 is said to be 𝑚-𝜂-relaxed monotone if there exists a constant 𝑚>0 such that 𝑢−𝑣,𝜂𝑥,𝑦≥(−𝑚)𝑥−𝑦²(𝑥)𝑦.,∀𝑥,𝑦∈𝑋,𝑢∈𝑀,𝑣∈𝑀 (20)

Definition 5. A set-valued mapping 𝑇∶𝑋→CB(𝑋) is said to be 𝒟-Lipschitz continuous, if there exists a constant 𝑙>0 such that 𝒟𝑇𝑥,𝑇𝑦≤𝑙𝑥−𝑦,∀𝑥,𝑦∈𝑋. (21)

Definition 6. Let 𝑆,𝑇∶𝑋→CB(𝑋) be the set-valued mappings. A mapping 𝑁∶𝑋×𝑋→𝑋 is said to be(i)𝜖₁-Lipschitz continuous in the first argument with respect to 𝑆, if there exists a constant 𝜖₁>0 such that 𝑁𝑤₁,⋅−𝑁𝑤₂,⋅≤𝜖₁𝑤₁−𝑤₂,∀𝑢₁,𝑢₂𝑤∈𝑋,₁∈𝑆𝑢₁,𝑤₂∈𝑆𝑢₂, (22)(ii)𝜖₂-Lipschitz continuous in the second argument with respect to 𝑇, if there exists a constant 𝜖₂>0 such that 𝑁⋅,𝑣₁−𝑁⋅,𝑣₂≤𝜖₂𝑣₁−𝑣₂,∀𝑢₁,𝑢₂𝑣∈𝑋,₁∈𝑇𝑢₁,𝑣₂∈𝑇𝑢₂. (23)

3. Generalized 𝐻(⋅,⋅,⋅)-𝜂-Cocoercive Operators

Definition 7. Let 𝐻∶𝑋×𝑋×𝑋→𝑋, 𝑃,𝑄,𝑅∶𝑋→𝑋, and 𝜂∶𝑋×𝑋→𝑋 be the single-valued mappings. Let 𝐻(𝑃,𝑄,𝑅) be 𝜇-𝜂-cocoercive with respect to 𝑃, 𝛾-𝜂-relaxed cocoercive with respect to 𝑄, and 𝛿-𝜂-strongly monotone with respect to 𝑅. Then, the set-valued mapping 𝑀∶𝑋→2^𝑋 is said to be a generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive with respect to the mappings 𝑃,𝑄, and 𝑅, if(i)𝑀 is 𝑚-𝜂-relaxed monotone;(ii)(𝐻(𝑃,𝑄,𝑅)+𝜆𝑀)(𝑋)=𝑋, for all 𝜆>0.

Example 8. Let 𝑋,𝐻,𝑃,𝑄,𝑅, and 𝜂 be the same as in Example 3, and let 𝑀∶ℝ²→ℝ² be defined by 𝑀(𝑥)=(−𝑛𝑥₁,−𝑚𝑥₂), for all 𝑥=(𝑥₁,𝑥₂)∈ℝ².

We claim that 𝑀 is 𝑛²-𝜂-relaxed monotone mapping. Indeed, for any 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈ℝ², we have =𝑀𝑥−𝑀𝑦,𝜂𝑥,𝑦−𝑛𝑥₁−𝑚𝑥₂−−𝑛𝑦₁−𝑚𝑦₂,𝑚𝑥₁−𝑦₁,𝑛𝑥₂−𝑦₂=−𝑛𝑥₁−𝑦₁,−𝑚𝑥₂−𝑦₂,𝑚𝑥₁−𝑦₁,𝑛𝑥₂−𝑦₂=−𝑚𝑛𝑥₁−𝑦₁²+𝑥₂−𝑦₂²≥−𝑛²𝑥₁−𝑦₁²+𝑥₂−𝑦₂²=−𝑛²𝑥−𝑦²,𝑀𝑥−𝑀𝑦,𝜂𝑥,𝑦≥−𝑛²𝑥−𝑦². (24) Furthermore, 𝑀 is also a generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operator since (𝐻(𝑃,𝑄,𝑅)+𝜆𝑀)(ℝ²)=ℝ² for any 𝜆>0.

Remark 9. If 𝐻(𝑃,𝑄,𝑅)=𝐻(𝑃,𝑄), 𝑃 is 𝛼-strongly monotone, and 𝑄 is 𝛽-relaxed monotone, then generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operator reduces to 𝐻(⋅,⋅)-𝜂-monotone operator introduced and studied by Xu and Wang [15].

Proposition 10. Let set-valued mapping 𝑀∶𝑋→2^𝑋 be a generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operator with respect to the mappings 𝑃,𝑄, and 𝑅. If 𝑃 is 𝛼-expansive, 𝑄 is 𝛽-Lipschitz continuous, and 𝜇>𝛾, 𝛼>𝛽 with 𝑟=𝜇𝛼²−𝛾𝛽²+𝛿>𝑚, then the following inequality 𝑢−𝑣,𝜂𝑥,𝑦≥0 (25) holds for all (𝑦,𝑣)∈Graph(𝑀) and implies 𝑢∈𝑀𝑥, where Graph(𝑀)={(𝑎,𝑏)∈𝑋×𝑋∶𝑏∈𝑀(𝑎)}. (26)

Proof. Suppose on the contrary that there exists (𝑥₀,𝑢₀)∉Graph𝑀 such that 𝑢₀−𝑣,𝜂𝑥₀,𝑦≥0,∀𝑦,𝑣∈Graph(𝑀). (27) Since 𝑀 is a generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive, we know that (𝐻(𝑃,𝑄,𝑅)+𝜆𝑀)(𝑋)=𝑋 holds for all 𝜆>0, and so there exists (𝑥₁,𝑢₁)∈Graph(𝑀) such that 𝐻𝑃𝑥₀,𝑄𝑥₀,𝑅𝑥₀+𝜆𝑢₀=𝐻𝑃𝑥₁,𝑄𝑥₁,𝑅𝑥₁+𝜆𝑢₁∈𝑋. (28) Now, 𝜆𝑢₀−𝜆𝑢₁=𝐻(𝑃𝑥₁,𝑄𝑥₁,𝑅𝑥₁)−𝐻(𝑃𝑥₀,𝑄𝑥₀,𝑅𝑥₀)∈𝑋𝜆𝑢₀−𝜆𝑢₁,𝜂𝑥₀,𝑥₁=−𝐻𝑃𝑥₀,𝑄𝑥₀,𝑅𝑥₀−𝐻𝑃𝑥₁,𝑄𝑥₁,𝑅𝑥₁,𝜂𝑥₀,𝑥₁. (29) Setting (𝑦,𝑣)=(𝑥₁,𝑢₁) in (27), and then from the resultant, (28) and 𝑚-𝜂-relaxed monotonicity of 𝑀, we obtain −𝑚𝑥₀−𝑥₁²≤𝜆𝑢₀−𝑢₁,𝜂𝑥₀,𝑥₁=−𝐻𝑃𝑥₀,𝑄𝑥₀,𝑅𝑥₀−𝐻𝑃𝑥₁,𝑄𝑥₁,𝑅𝑥₁,𝜂𝑥₀,𝑥₁=−𝐻𝑃𝑥₀,𝑄𝑥₀,𝑅𝑥₀−𝐻𝑃𝑥₁,𝑄𝑥₀,𝑅𝑥₀,𝜂𝑥₀,𝑥₁−𝐻𝑃𝑥₁,𝑄𝑥₀,𝑅𝑥₀−𝐻𝑃𝑥₁,𝑄𝑥₁,𝑅𝑥₀,𝜂𝑥₀,𝑥₁−𝐻𝑃𝑥₁,𝑄𝑥₁,𝑅𝑥₀−𝐻𝑃𝑥₁,𝑄𝑥₁,𝑅𝑥₁,𝜂𝑥₀,𝑥₁. (30) Since 𝐻(𝑃,𝑄,𝑅) is 𝜇-𝜂-cocoercive with respect to 𝑃, 𝛾-𝜂-relaxed cocoercive with respect to 𝑄, 𝛿-𝜂-strongly monotone with respect to 𝑅, and 𝑃 is 𝛼-expansive, 𝑄 is 𝛽-Lipschitz continuous, thus (30) becomes −𝑚𝑥₀−𝑥₁²≤−𝜇𝑃𝑥₀−𝑃𝑥₁²+𝛾𝑄𝑥₀−𝑄𝑥₁²−𝛿𝑥₀−𝑥₁²≤−𝜇𝛼²−𝛾𝛽²+𝛿𝑥₀−𝑥₁²=−𝑟𝑥₀−𝑥₁²(𝑥≤0,≤−𝑟−𝑚)₀−𝑥₁²≤0,where𝑟=𝜇𝛼²−𝛾𝛽²+𝛿, (31) which gives 𝑥₀=𝑥₁ since 𝑟>𝑚. By (27), we have 𝑢₀=𝑢₁, a contradiction. This completes the proof.

Theorem 11. Let set-valued mapping 𝑀∶𝑋→2^𝑋 be a generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operator with respect to the mappings 𝑃,𝑄, and 𝑅. If 𝑃 is 𝛼-expansive, 𝑄 is 𝛽-Lipschitz continuous, and 𝜇>𝛾, 𝛼>𝛽 with 𝑟=𝜇𝛼²−𝛾𝛽²+𝛿>𝜆𝑚, then (𝐻(𝑃,𝑄,𝑅)+𝜆𝑀)⁻¹ is single-valued.

Proof. For any given 𝑥∈𝑋, let 𝑢,𝑣∈(𝐻(𝑃,𝑄,𝑅)+𝜆𝑀)⁻¹(𝑥). It follows that ()()−𝐻𝑃𝑢,𝑄𝑢,𝑅𝑢+𝑥∈𝜆𝑀𝑢,−𝐻𝑃𝑣,𝑄𝑣,𝑅𝑣+𝑥∈𝜆𝑀𝑣. (32) Since 𝑀 is 𝑚-𝜂-relaxed monotone, we have −𝑚‖𝑢−𝑣‖²≤1𝜆⟨()(())(,‖‖−𝐻𝑃𝑢,𝑄𝑢,𝑅𝑢+𝑥−−𝐻𝑃𝑣,𝑄𝑣,𝑅𝑣+𝑥,𝜂𝑢,𝑣)−𝜆𝑚𝑢−𝑣²≤()(())(𝐻()()(𝐻()()(−𝐻()()(−𝐻()()(.−𝐻𝑃𝑢,𝑄𝑢,𝑅𝑢+𝑥−−𝐻𝑃𝑣,𝑄𝑣,𝑅𝑣+𝑥,𝜂𝑢,𝑣)=−𝑃𝑢,𝑄𝑢,𝑅𝑢−𝐻𝑃𝑣,𝑄𝑣,𝑅𝑣,𝜂𝑢,𝑣)=−𝑃𝑢,𝑄𝑢,𝑅𝑢−𝐻𝑃𝑣,𝑄𝑢,𝑅𝑢,𝜂𝑢,𝑣)𝑃𝑣,𝑄𝑢,𝑅𝑢−𝐻𝑃𝑣,𝑄𝑣,𝑅𝑢,𝜂𝑢,𝑣)𝑃𝑣,𝑄𝑣,𝑅𝑢−𝐻𝑃𝑣,𝑄𝑣,𝑅𝑣,𝜂𝑢,𝑣) (33) Since 𝐻(𝑃,𝑄,𝑅) is 𝜇-𝜂-cocoercive with respect to 𝑃, 𝛾-𝜂-relaxed cocoercive with respect to 𝑄, 𝛿-𝜂-strongly monotone with respect to 𝑅, and 𝑃 is 𝛼-expansive, 𝑄 is 𝛽-Lipschitz continuous, thus (33) becomes −𝜆𝑚‖𝑢−𝑣‖²≤−𝜇‖𝑃𝑢−𝑃𝑣‖²+𝛾‖𝑄𝑢−𝑄𝑣‖²−𝛿‖𝑢−𝑣‖²≤−𝜇𝛼²−𝛾𝛽²+𝛿‖𝑢−𝑣‖²=−𝑟‖𝑢−𝑣‖²(‖≤0,≤−𝑟−𝜆𝑚)‖𝑢−𝑣²≤0,where𝑟=𝜇𝛼²−𝛾𝛽²+𝛿, (34) since 𝑟>𝜆𝑚. Hence, it follows that ‖𝑢−𝑣‖≤0. This implies that 𝑢=𝑣, and so (𝐻(𝑃,𝑄,𝑅)+𝜆𝑀)⁻¹ is single-valued.

Definition 12. Let set-valued mapping 𝑀∶𝑋→2^𝑋 be a generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operator with respect to the mappings 𝑃,𝑄, and 𝑅. If 𝑃 is 𝛼-expansive, 𝑄 is 𝛽-Lipschitz continuous, and 𝜇>𝛾, 𝛼>𝛽 and 𝑟=𝜇𝛼²−𝛾𝛽²+𝛿>𝜆𝑚, then the resolvent operator 𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀∶𝑋→𝑋 is defined by 𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑢)=(𝐻(𝑃,𝑄,𝑅)+𝜆𝑀)⁻¹(𝑢),∀𝑢∈𝑋. (35)

Now, we prove that the resolvent operator defined by (35) is Lipschitz continuous.

Theorem 13. Let set-valued mapping 𝑀∶𝑋→2^𝑋 be a generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operator with respect to the mappings 𝑃,𝑄, and 𝑅. If 𝑃 is 𝛼-expansive, 𝑄 is 𝛽-Lipschitz continuous, 𝜂 is 𝜏-Lipschitz continuous, and 𝜇>𝛾, 𝛼>𝛽 with 𝑟=𝜇𝛼²−𝛾𝛽²+𝛿>𝜆𝑚, then the resolvent operator 𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀∶𝑋→𝑋 is (𝜏/(𝑟−𝜆𝑚))-Lipschitz continuous, that is, 𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑢)−𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑣≤𝜏)‖‖𝑟−𝜆𝑚𝑢−𝑣,∀𝑢,𝑣∈𝑋. (36)

Proof. Let 𝑢,𝑣∈𝑋 be any given points. It follows from (35) that 𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑢)=(𝐻(𝑃,𝑄,𝑅)+𝜆𝑀)⁻¹(𝑢),R^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑣)=(𝐻(𝑃,𝑄,𝑅)+𝜆𝑀)⁻¹(𝑣).1𝜆𝑢−𝐻𝑃𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑢),𝑄𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑢),𝑅𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑢)∈𝑀𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑢,1)𝜆𝑣−𝐻𝑃𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑣),𝑄𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑣),𝑅𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑣)∈𝑀𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑣). (37) Let 𝑧₁=𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑢) and 𝑧₂=𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑣).
Since 𝑀 is 𝑚-𝜂-relaxed monotone, we have 1𝜆(𝑢−𝐻𝑃𝑧₁,𝑄𝑧₁,𝑅𝑧₁−𝑣−𝐻𝑃𝑧₂,𝑄𝑧₂,𝑅𝑧₂,𝜂𝑧₁,𝑧₂≥−𝑚𝑧₁−𝑧₂²,1𝜆𝑢−𝑣−𝐻𝑃𝑧₁,𝑄𝑧₁,𝑅𝑧₁+𝐻𝑃𝑧₂,𝑄𝑧₂,𝑅𝑧₂,𝜂𝑧₁,𝑧₂≥−𝑚𝑧₁−𝑧₂², (38) which implies 𝑢−𝑣,𝜂𝑧₁,𝑧₂≥𝐻𝑃𝑧₁,𝑄𝑧₁,𝑅𝑧₁−𝐻𝑃𝑧₂,𝑄𝑧₂,𝑅𝑧₂,𝜂𝑧₁,𝑧₂−𝜆𝑚𝑧₁−𝑧₂². (39) Further, we have 𝜏‖𝑢−𝑣‖𝑧₁−𝑧₂≥‖𝑢−𝑣‖𝜂𝑧₁,𝑧₂≥𝑢−𝑣,𝜂𝑧₁,𝑧₂≥𝐻𝑃𝑧₁,𝑄𝑧₁,𝑅𝑧₁−𝐻𝑃𝑧₂,𝑄𝑧₂,𝑅𝑧₂,𝜂𝑧₁,𝑧₂−𝜆𝑚𝑧₁−𝑧₂²=𝐻𝑃𝑧₁,𝑄𝑧₁,𝑅𝑧₁−𝐻𝑃𝑧₂,𝑄𝑧₁,𝑅𝑧₁,𝜂𝑧₁,𝑧₂+𝐻𝑃𝑧₂,𝑄𝑧₁,𝑅𝑧₁−𝐻𝑃𝑧₂,𝑄𝑧₂,𝑅𝑧₁,𝜂𝑧₁,𝑧₂+𝐻𝑃𝑧₂,𝑄𝑧₂,𝑅𝑧₁−𝐻𝑃𝑧₂,𝑄𝑧₂,𝑅𝑧₂,𝜂𝑧₁,𝑧₂−𝜆𝑚𝑧₁−𝑧₂²≥𝜇𝑃𝑧₁−𝑃𝑧₂²−𝛾𝑄𝑧₁−𝑄𝑧₂²+𝛿𝑅𝑧₁−𝑅𝑧₂²−𝜆𝑚𝑧₁−𝑧₂²=𝜇𝛼²−𝛾𝛽²+𝛿𝑧₁−𝑧₂²−𝜆𝑚𝑧₁−𝑧₂²=(𝑟−𝜆𝑚)𝑧₁−𝑧₂²,where𝑟=𝜇𝛼²−𝛾𝛽²+𝛿, (40) and hence, 𝜏‖𝑢−𝑣‖𝑧₁−𝑧₂≥(𝑟−𝜆𝑚)𝑧₁−𝑧₂², (41) that is, 𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑢)−𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑣≤𝜏)‖‖,𝑟−𝜆𝑚𝑢−𝑣∀𝑢,𝑣∈𝑋. (42) This completes the proof.

4. An Application of 𝐻(⋅,⋅,⋅)-𝜂-Cocoercive Operators for Solving Generalized Variational Inclusions

In this section, we will show that under suitable assumptions, the generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive operator can also play important roles for solving the variational inclusion problem in Hilbert spaces.

Let 𝑁∶𝑋×𝑋→𝑋, 𝜂∶𝑋×𝑋→𝑋, 𝐻∶𝑋×𝑋×𝑋→𝑋, 𝑃,𝑄,𝑅∶𝑋→𝑋, and 𝑝∶𝑋→𝑋 be the single-valued mappings, and 𝑆,𝑇∶𝑋→CB(𝑋),𝑀∶𝑋→2^𝑋 be the set-valued mappings such that 𝑀 is generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive with respect to 𝑃,𝑄, and 𝑅 and range (𝑝)∩ dom 𝑀≠∅. Then we consider the problem to find 𝑢∈𝑋,𝑤∈𝑆(𝑢),𝑣∈𝑇(𝑢) such that 0∈𝑁(𝑤,𝑣)+𝑀𝑝(𝑢). (43)

The problem (43) is called generalized set-valued variational-like inclusion problem. The problem of type (43) was introduced and studied by Chidume et al. [3] by applying 𝜂-proximal mapping. If 𝑇=0 and 𝜂(𝑢,𝑣)=𝑢−𝑣, for all 𝑢,𝑣∈𝑋, and 𝑁(⋅,⋅)=𝑆(⋅), where 𝑆∶𝑋→CB(𝑋) is a set-valued mapping. Then, problem (43) reduces to the problem of finding 𝑢∈𝑋,𝑤∈𝑆(𝑢) such that 0∈𝑤+𝑀𝑝(𝑢). (44) The problem of type (44) was studied by Ahmad et al. [2] by applying 𝐻(⋅,⋅)-cocoercive operators.

If 𝑆,𝑇=0, 𝑁(⋅,⋅)=0, and 𝜂(𝑢,𝑣)=𝑢−𝑣, for all 𝑢,𝑣∈𝑋, then problem (43) reduces to the problem of finding 𝑢∈𝑋 such that 0∈𝑀𝑝(𝑢). (45) The problem of type (45) was studied by Verma [14] in the setting of Banach spaces when 𝑀 is 𝐴-maximal-relaxed accretive.

Lemma 14. The (𝑢,𝑤,𝑣), where 𝑢∈𝑋,𝑤∈𝑆(𝑢),𝑣∈𝑇(𝑢) is a solution of the problem (43), if and only if (𝑢,𝑤,𝑣) satisfies the following relation: 𝑝(𝑢)=𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀𝐻𝑃𝑝(𝑢)𝑝(𝑢)𝑝(𝑢)(,,𝑄,𝑅−𝜆𝑁𝑤,𝑣) (46) where 𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀(𝑢)=(𝐻(𝑃,𝑄,𝑅)+𝜆𝑀)⁻¹(𝑢) and 𝜆>0 is a constant.

Proof. By using the definitions of resolvent operators 𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀, the conclusion follows directly.

Using Lemma 14 and using the technique of Chidume et al. [3] and Nadler [20], we develop an iterative algorithm for finding the approximate solution of problem (43) as follows.

Algorithm 15. Let 𝑁∶𝑋×𝑋→𝑋, 𝜂∶𝑋×𝑋→𝑋, 𝐻∶𝑋×𝑋×𝑋→𝑋, 𝑃,𝑄,𝑅∶𝑋→𝑋, 𝑝∶𝑋→𝑋, and 𝑆,𝑇∶𝑋→CB(𝑋) be such that, for each 𝑢∈𝑋, 𝐹(𝑢)⊆𝑝(𝑋), where 𝐹∶𝑋→2^𝑋 is the set-valued mappings defined by 𝐹(𝑢)=_{𝑤∈𝑆(𝑢)𝑣∈𝑇(𝑢)}𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀𝐻𝑃𝑝(𝑢)𝑝(𝑢)𝑝(𝑢)(),,𝑄,𝑅−𝜆𝑁𝑤,𝑣 (47) where 𝑀∶𝑋→2^𝑋 is a set-valued mapping such that, 𝑀 is generalized 𝐻(⋅,⋅,⋅)-𝜂-cocoercive with respect to the mappings 𝑃,𝑄, and 𝑅.
For given 𝑢₀∈𝑋, take 𝑤₀∈𝑆(𝑢₀) and 𝑣₀∈𝑇(𝑢₀). Let 𝑥₀=𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀𝐻𝑃𝑝𝑢₀,𝑄𝑝𝑢₀,𝑅𝑝𝑢₀−𝜆𝑁𝑤₀,𝑣₀∈𝐹𝑢₀⊆𝑝(𝑋). (48) Hence, there exists 𝑢₁∈𝑋 such that 𝑥₀=𝑝(𝑢₁). Since 𝑤₀∈𝑆(𝑢₀)∈CB(𝑋) and 𝑣₀∈𝑇(𝑢₀)∈CB(𝑋), then, by Nadler’s result [20], there exist 𝑤₁∈𝑆(𝑢₁) and 𝑣₁∈𝑇(𝑢₁) such that 𝑤₀−𝑤₁≤1+1⁻¹𝒟𝑆𝑢₀,𝑆𝑢₁,𝑣₀−𝑣₁≤1+1⁻¹𝒟𝑇𝑢₀,𝑇𝑢₁. (49) Let 𝑥₁=𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀𝐻𝑃𝑝𝑢₁,𝑄𝑝𝑢₁,𝑅𝑝𝑢₁−𝜆𝑁𝑤₁,𝑣₁∈𝐹𝑢₁⊆𝑝(𝑋). (50) Hence, there exists 𝑢₂∈𝑋 such that 𝑥₁=𝑝(𝑢₂). By induction, we can define iterative sequences {𝑢_𝑛}, {𝑝(𝑢_𝑛)}, {𝑤_𝑛}, and {𝑣_𝑛} as follows: 𝑝𝑢_𝑛+1=𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀𝐻𝑃𝑝𝑢_𝑛,𝑄𝑝𝑢_𝑛,𝑅𝑝𝑢_𝑛−𝜆𝑁𝑤_𝑛,𝑣_𝑛,𝑤_𝑛∈𝑆𝑢_𝑛,𝑤_𝑛−𝑤_𝑛+1≤11+𝒟𝑆𝑢𝑛+1^𝑛,𝑆𝑢_𝑛+1,𝑣_𝑛∈𝑇𝑢_𝑛,𝑣_𝑛−𝑣_𝑛+1≤11+𝒟𝑇𝑢𝑛+1^𝑛,𝑇𝑢_𝑛+1, (51)(52)(53) for all 𝑛=0,1,2,…, and 𝜆>0 is a constant.

If 𝑁(𝑤,𝑣)=𝑤, for all 𝑤,𝑣∈𝑋, then Algorithm 15 reduces to the following algorithm for solving the problem (44).

Algorithm 16. For any 𝑢₀∈𝑋, and 𝑤₀∈𝑆(𝑢₀), compute the sequence {𝑢_𝑛} and {𝑤_𝑛} by the following: 𝑝𝑢_𝑛=𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀𝐻𝑃𝑝𝑢_𝑛,𝑄𝑝𝑢_𝑛,𝑅𝑝𝑢_𝑛−𝜆𝑤_𝑛,𝑤_𝑛∈𝑆𝑢_𝑛,𝑤_𝑛−𝑤_𝑛+1≤11+𝒟𝑆𝑢𝑛+1^𝑛,𝑆𝑢_𝑛+1, (54) for all 𝑛=0,1,2,…, and 𝜆>0 is a constant.

If 𝑁(𝑤,𝑣)=0, for all 𝑤,𝑣∈𝑋, then Algorithm 15 reduces to the following algorithm for solving the problem (45).

Algorithm 17. For any 𝑢₀∈𝑋, compute the sequence {𝑢_𝑛} by the following: 𝑝𝑢_𝑛=𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀𝐻𝑃𝑝𝑢_𝑛,𝑄𝑝𝑢_𝑛,𝑅𝑝𝑢_𝑛, (55) for all 𝑛=0,1,2,…, and 𝜆>0 is a constant.

Now, we prove the existence of a solution of problem (43) and the convergence of Algorithm 15.

Theorem 18. Let 𝑋 be a real Hilbert space. Let 𝜂,𝑁∶𝑋×𝑋→𝑋, 𝐻∶𝑋×𝑋×𝑋→𝑋, 𝑃,𝑄,𝑅∶𝑋→𝑋, and 𝑝∶𝑋→𝑋 be single-valued mappings, and 𝑆,𝑇∶𝑋→CB(𝑋) and 𝑀∶𝑋→2^𝑋 be the set-valued mappings such that 𝑀 is generalized 𝐻(𝑃,𝑄,𝑅)-𝜂-cocoercive operator with respect to the mappings 𝑃,𝑄, and 𝑅 and range (𝑝)∩  dom 𝑀≠∅, and, for each 𝑢∈𝑋, let 𝐹(𝑢)⊆𝑝(𝑋), where 𝐹 is defined by (47). Assume that(i)𝑆,𝑇 are 𝒟-Lipschitz continuous with constants 𝑙₁ and 𝑙₂, respectively;(ii)𝜂 is 𝜏-Lipschitz continuous, 𝑃 is 𝛼-expansive, and 𝑄 is 𝛽-Lipschitz continuous;(iii)𝑝 is 𝜆_𝑝-Lipschitz continuous and 𝜉-strongly monotone;(iv)𝐻(𝑃,𝑄,𝑅) is 𝑟₁-Lipschitz continuous with respect to 𝑃, 𝑟₂-Lipschitz continuous with respect to 𝑄, and 𝑟₃-Lipschitz continuous with respect to 𝑅;(v)𝑁 is 𝜖₁-Lipschitz continuous in the first argument with respect to 𝑆 and 𝜖₂-Lipschitz continuous in the second argument with respect to 𝑇.
In addition, 𝑟₁+𝑟₂+𝑟₃𝜆_𝑝𝜏+𝜆𝜖₁𝑙₁+𝜖₂𝑙₂<𝜉(𝑟−𝜆𝑚), (56) where 𝑟=𝜇𝛼²−𝛾𝛽²+𝛿>𝜆𝑚 and 𝜇>𝛾, 𝛼>𝛽, 𝛿>0.
Then, generalized set-valued variational inclusion problem (43) has a solution (𝑢,𝑤,𝑣)∈𝑋, and the iterative sequences {𝑢_𝑛}, {𝑝(𝑢_𝑛)}, {𝑤_𝑛}, and {𝑣_𝑛} generated by Algorithm 15 converge strongly to 𝑢, 𝑝(𝑢), 𝑤, and 𝑣, respectively.

Proof. Since 𝑆,𝑇 are 𝒟-Lipschitz continuous with constants 𝑙₁ and 𝑙₂, respectively, it follows from (52) and (53) that 𝑤_𝑛+1−𝑤_𝑛≤11+𝒟𝑆𝑢𝑛+1^𝑛+1,𝑆𝑢_𝑛≤11+𝑙𝑛+1¹𝑢_𝑛+1−𝑢_𝑛,𝑣_𝑛+1−𝑣_𝑛≤11+𝒟𝑇𝑢𝑛+1^𝑛+1,𝑇𝑢_𝑛≤11+𝑙𝑛+1²𝑢_𝑛+1−𝑢_𝑛, (57) for 𝑛=0,1,2,….
It follows from (51) and Theorem 13 that 𝑝𝑢_𝑛+1−𝑝𝑢_𝑛=𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀𝐻𝑃𝑝𝑢_𝑛,𝑄𝑝𝑢_𝑛,𝑅𝑝𝑢_𝑛−𝜆𝑁𝑤_𝑛,𝑣_𝑛−𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀𝐻𝑃𝑝𝑢_𝑛−1,𝑄𝑝𝑢_𝑛−1,𝑅𝑝𝑢_𝑛−1−𝜆𝑁𝑤_𝑛−1,𝑣_𝑛−1+𝜏𝐻𝑃𝑝𝑢𝑟−𝜆𝑚^𝑛,𝑄𝑝𝑢_𝑛,𝑅𝑝𝑢_𝑛−𝐻𝑃𝑝𝑢_𝑛−1,𝑄𝑝𝑢_𝑛−1,𝑅𝑝𝑢_𝑛−1+𝜏𝜆𝑁𝑤𝑟−𝜆𝑚^𝑛,𝑣_𝑛−𝑁𝑤_𝑛−1,𝑣_𝑛−1. (58) Since 𝐻(𝑃,𝑄,𝑅) is 𝑟₁-Lipschitz continuous with respect to 𝑃, 𝑟₂-Lipschitz continuous with respect to 𝑄, and 𝑟₃-Lipschitz continuous with respect to 𝑅, and 𝑝 is 𝜆_𝑝-Lipschitz continuous, we have 𝐻𝑃𝑝𝑢_𝑛,𝑄𝑝𝑢_𝑛,𝑅𝑝𝑢_𝑛−𝐻𝑃𝑝𝑢_𝑛−1,𝑄𝑝𝑢_𝑛−1,𝑅𝑝𝑢_𝑛−1=𝐻𝑃𝑝𝑢_𝑛,𝑄𝑝𝑢_𝑛,𝑅𝑝𝑢_𝑛−𝐻𝑃𝑝𝑢_𝑛−1,𝑄𝑝𝑢_𝑛,𝑅𝑝𝑢_𝑛+𝐻𝑃𝑝𝑢_𝑛−1,𝑄𝑝𝑢_𝑛,𝑅𝑝𝑢_𝑛−𝐻𝑃𝑝𝑢_𝑛−1,𝑄𝑝𝑢_𝑛−1,𝑅𝑝𝑢_𝑛+𝐻𝑃𝑝𝑢_𝑛−1,𝑄𝑝𝑢_𝑛−1,𝑅𝑝𝑢_𝑛−𝐻𝑃𝑝𝑢_𝑛−1,𝑄𝑝𝑢_𝑛−1,𝑅𝑝𝑢_𝑛−1≤𝑟₁+𝑟₂+𝑟₃𝜆_𝑝𝑢_𝑛−𝑢_𝑛−1. (59)
Since 𝑁 is 𝜖₁-Lipschitz continuous in the first argument with respect to 𝑆 and 𝜖₂- Lipschitz continuous in the second argument with respect to 𝑇, and 𝑆,𝑇 are 𝒟-Lipschitz continuous with constants 𝑙₁ and 𝑙₂, respectively, we have 𝑁𝑤_𝑛,𝑣_𝑛−𝑁𝑤_𝑛−1,𝑣_𝑛−1≤𝑁𝑤_𝑛,𝑣_𝑛−𝑁𝑤_𝑛−1,𝑣_𝑛+𝑁𝑤_𝑛−1,𝑣_𝑛−𝑁𝑤_𝑛−1,𝑣_𝑛−1≤𝜖₁𝑤_𝑛−𝑤_𝑛−1+𝜖₂𝑣_𝑛−𝑣_𝑛−1≤𝜖₁11+𝑛𝒟𝑆𝑢_𝑛,𝑆𝑢_𝑛−1+𝜖₂11+𝑛𝒟𝑇𝑢_𝑛,𝑇𝑢_𝑛−1≤𝜖₁11+𝑛𝑙₁𝑢_𝑛−𝑢_𝑛−1+𝜖₂11+𝑛𝑙₂𝑢_𝑛−𝑢_𝑛−1=𝜖₁𝑙₁+𝜖₂𝑙₂11+𝑛𝑢_𝑛−𝑢_𝑛−1. (60)
Using (59), (60) in (58), we have 𝑝𝑢_𝑛+1−𝑝𝑢_𝑛≤𝑟₁+𝑟₂+𝑟₃𝜆_𝑝𝜏+𝜆𝜖₁𝑙₁+𝜖₂𝑙₂(1+(1/𝑛))×𝑢𝑟−𝜆𝑚_𝑛−𝑢_𝑛−1. (61)
Using the 𝜉-strong monotonicity of 𝑝, we have 𝑝𝑢_𝑛+1−𝑝𝑢_𝑛𝑢_𝑛+1−𝑢_𝑛≥𝑝𝑢_𝑛+1−𝑝𝑢_𝑛,𝑢_𝑛+1−𝑢_𝑛≥𝜉𝑢_𝑛+1−𝑢_𝑛, (62) which implies that 𝑢_𝑛+1−𝑢_𝑛≤1𝜉𝑝𝑢_𝑛+1−𝑝𝑢_𝑛. (63) Combining (61) and (63), we have 𝑢_𝑛+1−𝑢_𝑛≤𝜃_𝑛𝑢_𝑛−𝑢_𝑛−1, (64) where 𝜃_𝑛=𝑟₁+𝑟₂+𝑟₃𝜆_𝑝𝜏+𝜆𝜖₁𝑙₁+𝜖₂𝑙₂(1+(1/𝑛))𝜉().𝑟−𝜆𝑚 (65) Let 𝑟𝜃=¹+𝑟₂+𝑟₃𝜆_𝑝𝜏+𝜆𝜖₁𝑙₁+𝜖₂𝑙₂𝜉().𝑟−𝜆𝑚 (66)
From (56), it is easy to see that 𝜃<1. Therefore, (64) implies that {𝑢_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is a Hilbert space, there exists 𝑢∈𝑋 such that 𝑢_𝑛→𝑢 as 𝑛→∞. From (57), {𝑤_𝑛} and {𝑣_𝑛} are also Cauchy sequences in 𝑋; thus, there exist 𝑤,𝑣∈𝑋 such that 𝑤_𝑛→𝑤 and 𝑣_𝑛→𝑣 as 𝑛→∞. By the continuity of 𝑝,𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀,𝐻,𝜂,𝑁,𝑃,𝑄,𝑅, and (51) of Algorithm 15, we have 𝑝(𝑢)=𝑅^{𝐻(⋅,⋅,⋅)-𝜂}_𝜆,𝑀𝐻𝑃𝑝(𝑢),𝑄𝑝(𝑢),𝑅𝑝(𝑢)−𝜆𝑁(𝑤,𝑣). (67)
Now, we prove that 𝑤∈𝑆(𝑢). In fact, since 𝑤_𝑛∈𝑆(𝑢_𝑛), we have 𝑑(𝑤,𝑆(𝑢))≤𝑤−𝑤_𝑛+𝑑𝑤_𝑛(𝑢)≤,𝑆𝑤−𝑤_𝑛+𝒟𝑆𝑢_𝑛(𝑢)≤,𝑆𝑤−𝑤_𝑛+𝑙₁𝑢_𝑛−𝑢⟶0as𝑛⟶∞, (68) which implies that 𝑑(𝑤,𝑆(𝑢))=0. Since 𝑆(𝑢)∈CB(𝑋), it follows that 𝑤∈𝑆(𝑢). Similarly, it is easy to see that 𝑣∈𝑇(𝑢). By Lemma 14, (𝑢,𝑤,𝑣) is the solution of problem (43). This completes the proof.

Based on Lemma 14 and Algorithm 16, Theorem 18 reduced to the following result for solving problem (44).

Theorem 19. Let 𝑋 be a real Hilbert space. Let 𝜂∶𝑋×𝑋→𝑋, 𝐻∶𝑋×𝑋×𝑋→𝑋, 𝑃,𝑄,𝑅∶𝑋→𝑋, and 𝑝∶𝑋→𝑋 be single-valued mappings, and let 𝑆∶𝑋→CB(𝑋) and 𝑀∶𝑋→2^𝑋 be the set-valued mappings such that 𝑀 is generalized 𝐻(𝑃,𝑄,𝑅)-𝜂-cocoercive operator with respect to the mappings 𝑃,𝑄, and 𝑅 and range (𝑝)∩  dom 𝑀≠∅, and, for each 𝑢∈𝑋, let 𝐹(𝑢)⊆𝑝(𝑋), where 𝐹 is defined by (47). Assume that(i)𝑆 is 𝒟-Lipschitz continuous with constants 𝑙;(ii)𝜂 is 𝜏-Lipschitz continuous, 𝑃 is 𝛼-expansive, and 𝑄 is 𝛽-Lipschitz continuous;(iii)𝑝 is 𝜆_𝑝-Lipschitz continuous and 𝜉-strongly monotone;(iv)𝐻(𝑃,𝑄,𝑅) is 𝑟₁-Lipschitz continuous with respect to 𝑃, 𝑟₂-Lipschitz continuous with respect to 𝑄, and 𝑟₃-Lipschitz continuous with respect to 𝑅.
In addition, 𝑟₁+𝑟₂+𝑟₃𝜆_𝑝𝜏+𝜆𝑙<𝜉(𝑟−𝜆𝑚), (69) where 𝑟=𝜇𝛼²−𝛾𝛽²+𝛿>𝜆𝑚 and 𝜇>𝛾, 𝛼>𝛽, 𝛿>0.
Then, generalized set-valued variational inclusion problem (44) has a solution (𝑢,𝑤)∈𝑋, and the iterative sequences {𝑢_𝑛}, {𝑝(𝑢_𝑛)}, and {𝑤_𝑛} generated by Algorithm 16 converge strongly to 𝑢, 𝑝(𝑢) and 𝑤, respectively.

Based on Lemma 14 and Algorithm 17, Theorem 18 reduced to the following result for solving problem (45).

Theorem 20. Let 𝑋 be a real Hilbert space. Let 𝜂∶𝑋×𝑋→𝑋, 𝐻∶𝑋×𝑋×𝑋→𝑋, 𝑃,𝑄,𝑅∶𝑋→𝑋, and 𝑝∶𝑋→𝑋 be single-valued mappings, and let 𝑀∶𝑋→2^𝑋 be the set-valued mappings such that 𝑀 is generalized 𝐻(𝑃,𝑄,𝑅)-𝜂-cocoercive operator with respect to the mappings 𝑃,𝑄, and 𝑅 and range (𝑝)∩  dom 𝑀≠∅, and, for each 𝑢∈𝑋, let 𝐹(𝑢)⊆𝑝(𝑋), where 𝐹 is defined by (47). Assume that(i)𝜂 is 𝜏-Lipschitz continuous, 𝑃 is 𝛼-expansive, and 𝑄 is 𝛽-Lipschitz continuous;(ii)𝑝 is 𝜆_𝑝-Lipschitz continuous and 𝜉-strongly monotone;(iii)𝐻(𝑃,𝑄,𝑅) is 𝑟₁-Lipschitz continuous with respect to 𝑃, 𝑟₂-Lipschitz continuous with respect to 𝑄, and 𝑟₃-Lipschitz continuous with respect to 𝑅.
In addition, 𝑟₁+𝑟₂+𝑟₃𝜆_𝑝𝜏<𝜉(𝑟−𝜆𝑚), (70) where 𝑟=𝜇𝛼²−𝛾𝛽²+𝛿>𝜆𝑚 and 𝜇>𝛾, 𝛼>𝛽, 𝛿>0.
Then, generalized set-valued variational inclusion problem (45) has a solution 𝑢∈𝑋, and the iterative sequence {𝑢_𝑛} and {𝑝(𝑢_𝑛)} generated by Algorithm 17 converge strongly to 𝑢 and 𝑝(𝑢), respectively.

Acknowledgments

The authors are grateful to the editor and referees for valuable comments and suggestions.