𝐻(⋅,⋅)-Cocoercive Operator and an Application for Solving Generalized Variational Inclusions

Abstract

The purpose of this paper is to introduce a new 𝐻(⋅,⋅)-cocoercive operator, which generalizes many existing monotone operators. The resolvent operator associated with 𝐻(⋅,⋅)-cocoercive operator is defined, and its Lipschitz continuity is presented. By using techniques of resolvent operator, a new iterative algorithm for solving generalized variational inclusions is constructed. Under some suitable conditions, we prove the convergence of iterative sequences generated by the algorithm. For illustration, some examples are given.

1. Introduction

Various concepts of generalized monotone mappings have been introduced in the literature. Cocoercive mappings which are generalized form of monotone mappings are defined by Tseng [1], Magnanti and Perakis [2], and Zhu and Marcotte [3]. The resolvent operator techniques are important to study the existence of solutions and to develop iterative schemes for different kinds of variational inequalities and their generalizations, which are providing mathematical models to some problems arising in optimization and control, economics, and engineering sciences. In order to study various variational inequalities and variational inclusions, Fang and Huang, Lan, Cho, and Verma investigated many generalized operators such as 𝐻-monotone [4], 𝐻-accretive [5], (𝐻,𝜂)-accretive [6], (𝐻,𝜂)-monotone[7, 8], (𝐴,𝜂)-accretive mappings [9]. Recently, Zou and Huang [10] introduced and studied 𝐻(⋅,⋅)-accretive operators and Xu and Wang [11] introduced and studied (𝐻(⋅,⋅),𝜂)-monotone operators.

Motivated and inspired by the excellent work mentioned above, in this paper, we introduce and discuss new type of operators called 𝐻(⋅,⋅)-cocoercive operators. We define resolvent operator associated with 𝐻(⋅,⋅)-cocoercive operators and prove the Lipschitz continuity of the resolvent operator. We apply 𝐻(⋅,⋅)-cocoercive operators to solve a generalized variational inclusion problem. Some examples are constructed for illustration.

2. Preliminaries

Throughout the paper, we suppose that 𝑋 is a real Hilbert space endowed with a norm ‖⋅‖ and an inner product ⟨⋅,⋅⟩, 𝑑 is the metric induced by the norm ‖⋅‖,2^𝑋 (resp., 𝐶𝐵(𝑋)) is the family of all 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,(2.1) where 𝑑(𝑥,𝑄)=inf_𝑦∈𝑄𝑑(𝑥,𝑦)and𝑑(𝑃,𝑦)=inf_𝑥∈𝑃𝑑(𝑥,𝑦).

Definition 2.1. A mapping 𝑔∶𝑋→𝑋 is said to be (i)Lipschitz continuous if there exists a constant 𝜆_𝑔>0 such that 𝑔(𝑥)−𝑔𝑦≤𝜆_𝑔𝑥−𝑦,∀𝑥,𝑦∈𝑋;(2.2)(ii)monotone if 𝑔(𝑥)−𝑔𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝑋;(2.3)(iii)strongly monotone if there exists a constant 𝜉>0 such that 𝑔(𝑥)−𝑔𝑦,𝑥−𝑦≥𝜉𝑥−𝑦²,∀𝑥,𝑦∈𝑋;(2.4)(iv)𝛼-expansive if there exists a constant 𝛼>0 such that 𝑔(𝑥)−𝑔𝑦≥𝛼𝑥−𝑦,∀𝑥,𝑦∈𝑋,(2.5) if 𝛼=1, then it is expansive.

Definition 2.2. A mapping 𝑇∶𝑋→𝑋 is said to be cocoercive if there exists a constant 𝜇′>0 such that 𝑇(𝑥)−𝑇𝑦,𝑥−𝑦≥𝜇′𝑇(𝑥)−𝑇𝑦²,∀𝑥,𝑦∈𝑋.(2.6)

Note 1. Clearly 𝑇 is 1/𝜇′-Lipschitz continuous and also monotone but not necessarily strongly monotone and Lipschitz continuous (consider a constant mapping). Conversely, strongly monotone and Lipschitz continuous mappings are cocoercive, and it follows that cocoercivity is an intermediate concept that lies between simple and strong monotonicity.

Definition 2.3. A multivalued mapping 𝑀∶𝑋→2^𝑋 is said to be cocoercive if there exists a constant 𝜇^′′>0 such that 𝑢−𝑣,𝑥−𝑦≥𝜇^′′‖𝑢−𝑣‖²,∀𝑥,𝑦∈𝑋,𝑢∈𝑀(𝑥),𝑣∈𝑀𝑦.(2.7)

Definition 2.4. A mapping 𝑇∶𝑋→𝑋 is said to be relaxed cocoercive if there exists a constant 𝛾′>0 such that 𝑇(𝑥)−𝑇𝑦,𝑥−𝑦≥−𝛾^′𝑇(𝑥)−𝑇𝑦²,∀𝑥,𝑦∈𝑋.(2.8)

Definition 2.5. Let 𝐻∶𝑋×𝑋→𝑋 and 𝐴,𝐵∶𝑋→𝑋 be the mappings. (i)𝐻(𝐴,⋅) is said to be cocoercive with respect to 𝐴 if there exists a constant 𝜇>0 such that 𝐻(𝐴𝑥,𝑢)−𝐻𝐴𝑦,𝑢,𝑥−𝑦≥𝜇𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝑋;(2.9)(ii)𝐻(⋅,𝐵) is said to be relaxed cocoercive with respect to 𝐵 if there exists a constant 𝛾>0 such that 𝐻(𝑢,𝐵𝑥)−𝐻𝑢,𝐵𝑦,𝑥−𝑦≥−𝛾𝐵𝑥−𝐵𝑦²,∀𝑥,𝑦∈𝑋;(2.10)(iii)𝐻(𝐴,⋅) is said to be 𝑟₁-Lipschitz continuous with respect to 𝐴 if there exists a constant 𝑟₁>0 such that 𝐻(𝐴𝑥,⋅)−𝐻𝐴𝑦,⋅≤𝑟₁𝑥−𝑦,∀𝑥,𝑦∈𝑋;(2.11)(iv)𝐻(⋅,𝐵) is said to be 𝑟₂-Lipschitz continuous with respect to 𝐵 if there exists a constant 𝑟₂>0 such that 𝐻(⋅,𝐵𝑥)−𝐻⋅,𝐵𝑦≤𝑟₂𝑥−𝑦,∀𝑥,𝑦∈𝑋.(2.12)

Example 2.6. Let 𝑋=ℝ² with usual inner product. Let 𝐴,𝐵∶ℝ²→ℝ² be defined by 𝐴𝑥=2𝑥₁−2𝑥₂,−2𝑥₁+4𝑥₂,𝐵𝑦=−𝑦₁+𝑦₂,−𝑦₂,∀𝑥₁,𝑥₂,𝑦₁,𝑦₂∈ℝ².(2.13) Suppose that 𝐻(𝐴,𝐵)∶ℝ²×ℝ²→ℝ² is defined by 𝐻𝐴𝑥,𝐵𝑦=𝐴𝑥+𝐵𝑦,∀𝑥,𝑦∈ℝ².(2.14) Then 𝐻(𝐴,𝐵) is (1/6)-cocoercive with respect to 𝐴 and (1/2)-relaxed cocoercive with respect to 𝐵 since 𝐻()==𝐴𝑥,𝑢−𝐻𝐴𝑦,𝑢,𝑥−𝑦𝐴𝑥−𝐴𝑦,𝑥−𝑦2𝑥₁−2𝑥₂,−2𝑥₁+4𝑥₂−2𝑦₁−2𝑦₂,−2𝑦₁+4𝑦₂,𝑥₁−𝑦₁,𝑥₂−𝑦₂=2𝑥₁−𝑦₁−2𝑥₂−𝑦₂,−2𝑥₁−𝑦₁+4𝑥₂−𝑦₂,𝑥₁−𝑦₁,𝑥₂−𝑦₂=2𝑥₁−𝑦₁²+4𝑥₂−𝑦₂²−4𝑥₁−𝑦₁𝑥₂−𝑦₂,𝐴𝑥−𝐴𝑦²=2𝑥₁−2𝑥₂,−2𝑥₁+4𝑥₂−2𝑦₁−2𝑦₂,−2𝑦₁+4𝑦₂,2𝑥₁−2𝑥₂,−2𝑥₁+4𝑥₂−2𝑦₁−2𝑦₂,−2𝑦₁+4𝑦₂=8𝑥₁−𝑦₁²+20𝑥₂−𝑦₂²−24𝑥₁−𝑦₁𝑥₂−𝑦₂≤12𝑥₁−𝑦₁²+24𝑥₂−𝑦₂²−24𝑥₁−𝑦₁𝑥₂−𝑦₂=62𝑥₁−𝑦₁²+4𝑥₂−𝑦₂²−4𝑥₁−𝑦₁𝑥₂−𝑦₂=6𝐻(𝑢,𝐴𝑥)−𝐻𝑢,𝐴𝑦,𝑥−𝑦,(2.15) which implies that 𝐻()≥1𝐴𝑥,𝑢−𝐻𝐴𝑦,𝑢,𝑥−𝑦6𝐴𝑥−𝐴𝑦²,(2.16) That is, 𝐻(𝐴,𝐵) is (1/6)-cocoercive with respect to 𝐴. 𝐻()==𝑢,𝐵𝑥−𝐻𝑢,𝐵𝑦,𝑥−𝑦𝐵𝑥−𝐵𝑦,𝑥−𝑦−𝑥₁+𝑥₂,−𝑥₂−−𝑦₁+𝑦₂,−𝑦₂,𝑥₁−𝑦₁,𝑥₂−𝑦₂=−𝑥₁−𝑦₁+𝑥₂−𝑦₂,−𝑥₂−𝑦₂,𝑥₁−𝑦₁,𝑥₂−𝑦₂=−𝑥₁−𝑦₁²−𝑥₂−𝑦₂²+𝑥₁−𝑦₁𝑥₂−𝑦₂=−𝑥₁−𝑦₁²+𝑥₂−𝑦₂²−𝑥₁−𝑦₁𝑥₂−𝑦₂,𝐵𝑥−𝐵𝑦²=−𝑥₁−𝑦₁+𝑥₂−𝑦₂,−𝑥₂−𝑦₂,−𝑥₁−𝑦₁+𝑥₂−𝑦₂,−𝑥₂−𝑦₂=𝑥₁−𝑦₁²+2𝑥₂−𝑦₂²−2𝑥₁−𝑦₁𝑥₂−𝑦₂≤2𝑥₁−𝑦₁²+𝑥₂−𝑦₂²−𝑥₁−𝑦₁𝑥₂−𝑦₂=2(−1)𝐻(𝐵𝑥,𝑢)−𝐻𝐵𝑦,𝑢,𝑥−𝑦(2.17) which implies that 𝐻()1𝑢,𝐵𝑥−𝐻𝑢,𝐵𝑦,𝑥−𝑦≥−2𝐵𝑥−𝐵𝑦²,(2.18) that is, 𝐻(𝐴,𝐵) is (1/2)-relaxed cocoercive with respect to 𝐵.

3. 𝐻(⋅,⋅)-Cocoercive Operator

In this section, we define a new 𝐻(⋅,⋅)-cocoercive operator and discuss some of its properties.

Definition 3.1. Let 𝐴,𝐵∶𝑋→𝑋,𝐻∶𝑋×𝑋→𝑋 be three single-valued mappings. Let 𝑀∶𝑋→2^𝑋 be a set-valued mapping. 𝑀 is said to be 𝐻(⋅,⋅)-cocoercive with respect to mappings 𝐴 and 𝐵 (or simply 𝐻(⋅,⋅)-cocoercive in the sequel) if 𝑀 is cocoercive and (𝐻(𝐴,𝐵)+𝜆𝑀)(𝑋)=𝑋, for every 𝜆>0.

Example 3.2. Let 𝑋, 𝐴, 𝐵, and 𝐻 be the same as in Example 2.6, and let 𝑀∶ℝ²→ℝ² be define by 𝑀(𝑥₁,𝑥₂)=(0,𝑥₂),∀(𝑥₁,𝑥₂)∈ℝ². Then it is easy to check that 𝑀 is cocoercive and (𝐻(𝐴,𝐵)+𝜆𝑀)(ℝ²)=ℝ²,∀𝜆>0, that is, 𝑀 is 𝐻(⋅,⋅)-cocoercive with respect to 𝐴 and 𝐵.

Remark 3.3. Since cocoercive operators include monotone operators, hence our definition is more general than definition of 𝐻(⋅,⋅)-monotone operator [10]. It is easy to check that 𝐻(⋅,⋅)-cocoercive operators provide a unified framework for the existing 𝐻(⋅,⋅)-monotone, 𝐻-monotone operators in Hilbert space and 𝐻(⋅,⋅)-accretive, 𝐻-accretive operators in Banach spaces.
Since 𝐻(⋅,⋅)-cocoercive operators are more general than maximal monotone operators, we give the following characterization of 𝐻(⋅,⋅)-cocoercive operators.

Proposition 3.4. Let 𝐻(𝐴,𝐵) be 𝜇-cocoercive with respect to 𝐴, 𝛾-relaxed cocoercive with respect to 𝐵, 𝐴 is 𝛼-expansive, 𝐵 is 𝛽-Lipschitz continuous, and 𝜇>𝛾, 𝛼>𝛽. Let 𝑀∶𝑋→2^𝑋 be 𝐻(⋅,⋅)-cocoercive operator. If the following inequality x−𝑦,𝑢−𝑣≥0(3.1) holds for all (𝑣,𝑦)∈Graph(𝑀), then 𝑥∈𝑀𝑢, where Graph(𝑀)={(𝑥,𝑢)∈𝑋×𝑋∶𝑢∈𝑀(𝑥)}.(3.2)

Proof. Suppose that there exists some (𝑢₀,𝑥₀) such that 𝑥₀−𝑦,𝑢₀−𝑣≥0,∀𝑣,𝑦∈Graph(𝑀).(3.3) Since 𝑀 is 𝐻(⋅,⋅)-cocoercive, we know that (𝐻(𝐴,𝐵)+𝜆𝑀)(𝑋)=𝑋 holds for every 𝜆>0, and so there exists (𝑢₁,𝑥₁)∈Graph(𝑀) such that 𝐻𝐴𝑢₁,𝐵𝑢₁+𝜆𝑥₁=𝐻𝐴𝑢₀,𝐵𝑢₀+𝜆𝑥₀∈𝑋.(3.4) It follows from (3.3) and (3.4) that 0≤𝜆𝑥₀+𝐻𝐴𝑢₀,𝐵𝑢₀−𝜆𝑥₁−𝐻𝐴𝑢₁,𝐵𝑢₁,𝑢₀−𝑢₁,0≤𝜆𝑥₀−𝑥₁,𝑢₀−𝑢₁=−𝐻𝐴𝑢₀,𝐵𝑢₀−𝐻𝐴𝑢₁,𝐵𝑢₁,𝑢₀−𝑢₁=−𝐻𝐴𝑢₀,𝐵𝑢₀−𝐻𝐴𝑢₁,𝐵𝑢₀,𝑢₀−𝑢₁−𝐻𝐴𝑢₁,𝐵𝑢₀−𝐻𝐴𝑢₁,𝐵𝑢₁,𝑢₀−𝑢₁≤−𝜇𝐴𝑢₀−𝐴𝑢₁²+𝛾B𝑢₀−𝐵𝑢₁²≤−𝜇𝛼²𝑢₀−𝑢₁²+𝛾𝛽²𝑢₀−𝑢₁²=−𝜇𝛼²−𝛾𝛽²𝑢₀−𝑢₁²≤0,(3.5) which gives 𝑢₁=𝑢₀ since 𝜇>𝛾,𝛼>𝛽. By (3.4), we have 𝑥₁=𝑥₀. Hence (𝑢₀,𝑥₀)=(𝑢₁,𝑥₁)∈Graph(𝑀) and so 𝑥₀∈𝑀𝑢₀.

Theorem 3.5. Let 𝑋 be a Hilbert space and 𝑀∶𝑋→2^𝑋 a maximal monotone operator. Suppose that 𝐻∶𝑋×𝑋→𝑋 is a bounded cocoercive and semicontinuous with respect to 𝐴 and 𝐵. Let 𝐻∶𝑋×𝑋→𝑋 be also 𝜇-cocoercive with respect to 𝐴 and 𝛾-relaxed cocoercive with respect to 𝐵. The mapping 𝐴 is 𝛼-expansive, and 𝐵 is 𝛽-Lipschitz continuous. If 𝜇>𝛾 and 𝛼>𝛽, then 𝑀 is 𝐻(⋅,⋅)-cocoercive with respect to 𝐴 and 𝐵.

Proof. For the proof we refer to [10].

Theorem 3.6. Let 𝐻(𝐴,𝐵) be a 𝜇-cocoercive with respect to 𝐴 and 𝛾-relaxed cocoercive with respect to 𝐵, 𝐴 is 𝛼-expansive, and 𝐵 is 𝛽-Lipschitz continuous, 𝜇>𝛾 and 𝛼>𝛽. Let 𝑀 be an 𝐻(⋅,⋅)-cocoercive operator with respect to 𝐴 and 𝐵. Then the operator (𝐻(𝐴,𝐵)+𝜆𝑀)⁻¹ is single-valued.

Proof. For any given 𝑢∈𝑋, let 𝑥,𝑦∈(𝐻(𝐴,𝐵)+𝜆𝑀)⁻¹(𝑢). It follows that −𝐻(𝐴𝑥,𝐵𝑥)+𝑢∈𝜆𝑀𝑥,−𝐻𝐴𝑦,𝐵𝑦+𝑢∈𝜆𝑀𝑦.(3.6) As 𝑀 is cocoercive (thus monotone), we have ()𝐻()𝐻()𝐻()−𝐻.0≤−𝐻𝐴𝑥,𝐵𝑥+𝑢−−𝐻𝐴𝑦,𝐵𝑦+𝑢,𝑥−𝑦=−𝐴𝑥,𝐵𝑥−𝐻𝐴𝑦,𝐵𝑦,𝑥−𝑦=−𝐴𝑥,𝐵𝑥−𝐻𝐴𝑦,𝐵𝑥+𝐻𝐴𝑦,𝐵𝑥−𝐻𝐴𝑦,𝐵𝑦,𝑥−𝑦=−𝐴𝑥,𝐵𝑥−𝐻𝐴𝑦,𝐵𝑥,𝑥−𝑦𝐴𝑦,𝐵𝑥−𝐻𝐴𝑦,𝐵𝑦,𝑥−𝑦(3.7) Since 𝐻 is 𝜇-cocoercive with respect to 𝐴 and 𝛾-relaxed cocoercive with respect to 𝐵, 𝐴 is 𝛼-expansive and 𝐵 is 𝛽-Lipschitz continuous, thus (3.7) becomes 0≤−𝜇𝛼²𝑥−𝑦²+𝛾𝛽²𝑥−𝑦²=−𝜇𝛼²−𝛾𝛽²𝑥−𝑦²≤0(3.8) since 𝜇>𝛾,𝛼>𝛽. Thus, we have 𝑥=𝑦 and so (𝐻(𝐴,𝐵)+𝜆𝑀)⁻¹ is single-valued.

Definition 3.7. Let 𝐻(𝐴,𝐵) be 𝜇-cocoercive with respect to 𝐴 and 𝛾-relaxed cocoercive with respect to 𝐵, 𝐴 is 𝛼-expansive, 𝐵 is 𝛽-Lipschitz continuous, and 𝜇>𝛾, 𝛼>𝛽. Let 𝑀 be an 𝐻(⋅,⋅)-cocoercive operator with respect to 𝐴 and 𝐵. The resolvent operator 𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀∶𝑋→𝑋 is defined by 𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑢)=(𝐻(𝐴,𝐵)+𝜆𝑀)⁻¹(𝑢),∀𝑢∈𝑋.(3.9)

Now, we prove the Lipschitz continuity of resolvent operator defined by (3.9) and estimate its Lipschitz constant.

Theorem 3.8. Let 𝐻(𝐴,𝐵) be 𝜇-cocoercive with respect to 𝐴, 𝛾-relaxed cocoercive with respect to 𝐵, 𝐴 is 𝛼-expansive, 𝐵 is 𝛽-Lipschitz continuous, and 𝜇>𝛾, 𝛼>𝛽. Let 𝑀 be an 𝐻(⋅,⋅)-cocoercive operator with respect to 𝐴 and 𝐵. Then the resolvent operator 𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀∶𝑋→𝑋 is 1/𝜇𝛼²−𝛾𝛽²-Lipschitz continuous, that is, 𝑅^H(⋅,⋅)_𝜆,𝑀(𝑢)−𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑣≤1)𝜇𝛼²−𝛾𝛽²‖𝑢−𝑣‖,∀𝑢,𝑣∈𝑋.(3.10)

Proof. Let 𝑢 and 𝑣 be any given points in 𝑋. It follows from (3.9) that 𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑢)=(𝐻(𝐴,𝐵)+𝜆𝑀)⁻¹(𝑢),𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑣)=(𝐻(𝐴,𝐵)+𝜆𝑀)⁻¹(𝑣).(3.11) This implies that 1𝜆𝑢−𝐻𝐴𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑢),𝐵𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑢)∈𝑀𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑢),1𝜆𝑣−𝐻𝐴𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑣),𝐵𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑣)∈𝑀𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑣).(3.12) For the sake of clarity, we take 𝑃𝑢=𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑢),𝑃𝑣=𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑣).(3.13) Since 𝑀 is cocoercive (hence monotone), we have 1𝜆⟨(𝐴()(−((𝐴()(⟩1𝑢−𝐻𝑃𝑢,𝐵𝑃𝑢))𝑣−𝐻𝑃𝑣,𝐵𝑃𝑣))),𝑃𝑢−𝑃𝑣≥0,𝜆⟨𝑢−𝑣−𝐻(𝐴(𝑃𝑢),𝐵(𝑃𝑢))+𝐻(𝐴(𝑃𝑣),𝐵(𝑃𝑣)),𝑃𝑢−𝑃𝑣⟩≥0,(3.14) which implies that ⟨𝑢−𝑣,𝑃𝑢−𝑃𝑣⟩≥⟨𝐻(𝐴(𝑃𝑢),𝐵(𝑃𝑢))−𝐻(𝐴(𝑃𝑣),𝐵(𝑃𝑣)),𝑃𝑢−𝑃𝑣⟩.(3.15) Further, we have ‖‖≥⟨⟩≥⟨𝐻(𝐴()((𝐴()(⟩=⟨𝐻(𝐴()((𝐴()((𝐴()((𝐴()(⟩=⟨𝐻(𝐴()((𝐴()(⟩+⟨𝐻(𝐴()((𝐴()(⟩‖𝐴()(𝑢−𝑣‖‖𝑃𝑢−𝑃𝑣𝑢−𝑣,𝑃𝑢−𝑃𝑣𝑃𝑢,𝐵𝑃𝑢))−𝐻𝑃𝑣,𝐵𝑃𝑣)),𝑃𝑢−𝑃𝑣𝑃𝑢,𝐵𝑃𝑢))−𝐻𝑃𝑣,𝐵𝑃𝑢))+𝐻𝑃𝑣,𝐵𝑃𝑢))−𝐻𝑃𝑣,𝐵𝑃𝑣)),𝑃𝑢−𝑃𝑣𝑃𝑢,𝐵𝑃𝑢))−𝐻𝑃𝑣,𝐵𝑃𝑢)),𝑃𝑢−𝑃𝑣𝑃𝑣,𝐵𝑃𝑢))−𝐻𝑃𝑣,𝐵𝑃𝑣)),𝑃𝑢−𝑃𝑣≥𝜇𝑃𝑢−𝐴𝑃𝑣)‖²−𝛾‖𝐵(𝑃𝑢)−𝐵(𝑃𝑣)‖²≥𝜇𝛼²‖𝑃𝑢−𝑃𝑣‖²−𝛾𝛽²‖𝑃𝑢−𝑃𝑣‖²,(3.16) and so ‖𝑢−𝑣‖‖𝑃𝑢−𝑃𝑣‖≥𝜇𝛼²−𝛾𝛽²‖𝑃𝑢−𝑃𝑣‖²,(3.17) thus, ‖‖≤1𝑃𝑢−𝑃𝑣𝜇𝛼²−𝛾𝛽²‖𝑢−𝑣‖,thatis𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑢)−𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀(𝑣≤1)𝜇𝛼²−𝛾𝛽²‖𝑢−𝑣‖,∀𝑢,𝑣∈𝑋.(3.18) This completes the proof.

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

We apply 𝐻(⋅,⋅)-cocoercive operators for solving a generalized variational inclusion problem.

We consider the problem of finding 𝑢∈𝑋 and 𝑤∈𝑇(𝑢) such that 0∈𝑤+𝑀𝑔(𝑢),(4.1) where 𝑔∶𝑋→𝑋,𝑀∶𝑋→2^𝑋, and 𝑇∶𝑋→𝐶𝐵(𝑋) are the mappings. Problem (4.1) is introduced and studied by Huang [12] in the setting of Banach spaces.

Lemma 4.1. The (𝑢,𝑤), where 𝑢∈𝑋,𝑤∈𝑇(𝑢), is a solution of the problem (4.1), if and only if (𝑢,𝑤) is a solution of the following: 𝑔(𝑢)=𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀𝐻𝐴𝑔𝑢,𝐵𝑔𝑢−𝜆𝑤,(4.2) where 𝜆>0 is a constant.

Proof. By using the definition of resolvent operator 𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀, the conclusion follows directly.

Based on (4.2), we construct the following algorithm.

Algorithm 4.2. For any 𝑢₀∈𝑋,𝑤₀∈𝑇(𝑢₀), compute the sequences {𝑢_𝑛} and {𝑤_𝑛}by iterative schemes such that 𝑔𝑢_𝑛+1=𝑅^H(⋅,⋅)_𝜆,𝑀𝐻𝐴𝑔𝑢_𝑛,𝐵𝑔𝑢_𝑛−𝜆𝑤_𝑛,𝑤_𝑛∈𝑇𝑢_𝑛,𝑤_𝑛−𝑤_𝑛+1≤11+𝒟𝑇𝑢𝑛+1^𝑛,𝑇𝑢_𝑛+1,(4.3) for all 𝑛=0,1,2,…, and 𝜆>0 is a constant.

Theorem 4.3. Let 𝑋 be a real Hilbert space and 𝐴,𝐵,𝑔∶𝑋→𝑋,𝐻∶𝑋×𝑋→𝑋 the single-valued mappings. Let 𝑇∶𝑋→𝐶𝐵(𝑋) be a multi-valued mapping and 𝑀∶𝑋→2^𝑋 the multi-valued 𝐻(⋅,⋅)-cocoercive operator. Assume that (i)𝑇 is 𝛿-Lipschitz continuous in the Hausdorff metric 𝒟(⋅,⋅);(ii)𝐻(𝐴,𝐵) is 𝜇-cocoercive with respect to 𝐴 and 𝛾-relaxed cocoercive with respect to 𝐵;(iii)𝐴 is 𝛼-expansive; (iv)𝐵 is 𝛽-Lipschitz continuous; (v)𝑔 is 𝜆_𝑔-Lipschitz continuous and 𝜉-strongly monotone; (vi)𝐻(𝐴,𝐵) is 𝑟₁-Lipschitz continuous with respect to 𝐴 and 𝑟₂-Lipschitz continuous with respect to 𝐵; (vii)(𝑟₁+𝑟₂)𝜆_𝑔<[(𝜇𝛼²−𝛾𝛽²)𝜉−𝜆𝛿];𝜇>𝛾,𝛼>𝛽.

Then the generalized variational inclusion problem (4.1) has a solution (𝑢,𝑤) with 𝑢∈𝑋,𝑤∈𝑇(𝑢), and the iterative sequences {𝑢_𝑛} and {𝑤_𝑛} generated by Algorithm 4.2 converge strongly to 𝑢 and 𝑤, respectively.

Proof. Since 𝑇 is 𝛿-Lipschitz continuous, it follows from Algorithm 4.2 that 𝑤_𝑛−𝑤_𝑛+1≤11+𝒟𝑇𝑢𝑛+1^𝑛,𝑇𝑢_𝑛+1≤11+𝛿𝑢𝑛+1^𝑛−𝑢_𝑛+1,(4.4) for 𝑛=0,1,2,….
Using the 𝜉-strong monotonicity of 𝑔, we have 𝑔𝑢_𝑛+1−𝑔𝑢_𝑛𝑢_𝑛+1−𝑢_𝑛≥𝑔𝑢_𝑛+1−𝑔𝑢_𝑛,𝑢_𝑛+1−𝑢_𝑛≥𝜉𝑢_𝑛+1−𝑢_𝑛²(4.5) which implies that 𝑢_𝑛+1−𝑢_𝑛≤1𝜉𝑔𝑢_𝑛+1−𝑔𝑢_𝑛.(4.6) Now we estimate ‖𝑔(𝑢_𝑛+1)−𝑔(𝑢_𝑛)‖ by using the Lipschitz continuity of 𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀, 𝑔𝑢_𝑛+1−𝑔𝑢_𝑛=𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀𝐻𝐴𝑔𝑢_𝑛,𝐵𝑔𝑢_𝑛−𝜆𝑤_𝑛−𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀𝐻𝐴𝑔𝑢_𝑛−1,𝐵𝑔𝑢_𝑛−1−𝜆𝑤_𝑛−1≤1𝜇𝛼²−𝛾𝛽²𝐻𝐴𝑔𝑢_𝑛,𝐵𝑔𝑢_𝑛−𝐻𝐴𝑔𝑢_𝑛−1,𝐵𝑔𝑢_𝑛−1+𝜆𝜇𝛼²−𝛾𝛽²𝑤_𝑛−𝑤_𝑛−1≤1𝜇𝛼²−𝛾𝛽²𝐻𝐴𝑔𝑢_𝑛,𝐵𝑔𝑢_𝑛−𝐻𝐴𝑔𝑢_𝑛−1,𝐵𝑔𝑢_𝑛+1𝜇𝛼²−𝛾𝛽²𝐻𝐴𝑔𝑢_𝑛−1,𝐵𝑔𝑢_𝑛−𝐻𝐴𝑔𝑢_𝑛−1,𝐵𝑔𝑢_𝑛−1+𝜆𝜇𝛼²−𝛾𝛽²𝑤_𝑛−𝑤_𝑛−1.(4.7) Since 𝐻(𝐴,𝐵) is 𝑟₁-Lipschitz continuous with respect to 𝐴 and 𝑟₂-Lipschitz continuous with respect to 𝐵, 𝑔 is 𝜆_𝑔-Lipschitz continuous and using (4.4), (4.7) becomes 𝑔𝑢_𝑛+1−𝑔𝑢_𝑛≤𝑟₁𝜆_𝑔𝜇𝛼²−𝛾𝛽²𝑢_𝑛−𝑢_𝑛−1+𝑟₂𝜆_𝑔𝜇𝛼²−𝛾𝛽²𝑢_𝑛−𝑢_𝑛−1+𝜆𝜇𝛼²−𝛾𝛽²11+𝑛𝛿𝑢_𝑛−𝑢_𝑛−1(4.8) or 𝑔u_n+1−𝑔𝑢_𝑛≤𝑟₁𝜆_𝑔𝜇𝛼²−𝛾𝛽²+𝑟₂𝜆_𝑔𝜇𝛼²−𝛾𝛽²+𝜆𝜇𝛼²−𝛾𝛽²11+𝑛𝛿𝑢_𝑛−𝑢_𝑛−1.(4.9) Using (4.9), (4.6) becomes 𝑢_𝑛+1−𝑢_𝑛≤𝜃_𝑛𝑢_𝑛−𝑢_𝑛−1,(4.10) where 𝜃_𝑛=𝑟₁+𝑟₂𝜆_𝑔+𝜆𝛿(1+1/𝑛)𝜇𝛼²−𝛾𝛽²𝜉.(4.11) Let 𝑟𝜃=¹+𝑟₂𝜆_𝑔+𝜆𝛿𝜇𝛼²−𝛾𝛽²𝜉.(4.12) We know that 𝜃_𝑛→𝜃 and 𝑛→∞. From assumption (vii), it is easy to see that 𝜃<1. Therefore, it follows from (4.10) 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 (4.4), we know that {𝑤_𝑛} is also a Cauchy sequence in 𝑋, thus there exists 𝑤∈𝑋 such that 𝑤_𝑛→𝑤 and 𝑛→∞. By the continuity of 𝑔,𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀𝐻,𝐴,𝐵, and 𝑇 and Algorithm 4.2, we have 𝑔(𝑢)=𝑅^{𝐻(⋅,⋅)}_𝜆,𝑀𝐻𝐴𝑔𝑢,𝐵𝑔𝑢−𝜆𝑤.(4.13) Now, we prove that 𝑤∈𝑇(𝑢). In fact, since 𝑤_𝑛∈𝑇(𝑢_𝑛), we have 𝑑(𝑤,𝑇(𝑢))≤𝑤−𝑤_𝑛+𝑑𝑤_𝑛(𝑢)≤,𝑇𝑤−𝑤_𝑛+𝒟𝑇𝑢_𝑛(𝑢)≤,𝑇𝑤−𝑤_𝑛+𝛿𝑢_𝑛−𝑢⟶0,as𝑛⟶∞,(4.14) which implies that 𝑑(𝑤,𝑇(𝑢))=0. Since 𝑇(𝑢)∈𝐶𝐵(𝑋), it follows that 𝑤∈𝑇(𝑢). By Lemma 4.1, we know that (𝑢,𝑤) is a solution of problem (4.1). This completes the proof.

Acknowledgments

This work is partially done during the visit of the first author to National Sun-Yat Sen University, Kaohsiung, Taiwan. The first and second authors are supported by Department of Science and Technology, Government of India under Grant no. SR/S4/MS: 577/09. The fourth author is supported by Grant no. NSC 99-2221-E-037-007-MY3.