On a System of Nonlinear Variational Inclusions with 𝐻_ℎ,𝜂-Monotone Operators

Abstract

This paper is concerned mainly with the existence and iterative approximation of solutions for a system of nonlinear variational inclusions involving the strongly 𝐻_ℎ,𝜂-monotone operators in Hilbert spaces. The results presented in this paper extend, improve, and unify many known results in the literature.

1. Introduction

Recently, a few authors introduced and studied several classes of systems of nonlinear variational inequalities and inclusions in Hilbert spaces and established the existence of solutions or the approximate solutions for these systems of nonlinear variational inequalities and inclusions [1–19]. Using projection methods, Liu et al. [12], Verma [14–17], Rhoades and Verma [18], and Wu et al. [19] suggested some iterative algorithms for approximating the solutions of several classes of systems of variational inequalities involving relaxed monotone operators, strongly monotone operators, relaxed cocoercive operators, and pseudocontractive operators, respectively. Applying the resolvent operator techniques, Liu et al. [11] and Nie et al. [13] discussed the existence and uniqueness of solutions and suggested iterative algorithms for a system of general quasivariational-like inequalities and a system of nonlinear variational inequalities, respectively, and gave the convergence analysis for the iterative algorithms. Utilizing the resolvent operator method associated with (𝐻,𝜂)-monotone operators, Fang et al. [5] investigated the existence and uniqueness of solutions for a class of system of variational inclusions in Hilbert spaces, constructed an iterative algorithm for approximating the solution of this system of variational inclusions, and discussed the convergence of iterative sequences generated by the algorithm.

Motivated and inspired by the results [1–19], in this paper, we introduce two new classes of strictly 𝐻_ℎ,𝜂-monotone and strongly 𝐻_ℎ,𝜂-monotone operators and investigate the existence and Lipschitz continuity of the resolvent operators with respect to the strictly 𝐻_ℎ,𝜂-monotone and strongly 𝐻_ℎ,𝜂-monotone operators. We introduce also a class of system of nonlinear variational inclusions involving strongly 𝐻_ℎ,𝜂-monotone operators in Hilbert spaces and suggest two new iterative algorithms for approximating solutions of the system of nonlinear variational inclusions by using the resolvent operator technique associated with 𝐻_ℎ,𝜂-monotone operators. The convergence criteria of iterative sequences generated by the algorithms are established. The results presented in this paper extend, improve, and unify some known results in the literature.

This paper is organized as follows. In Section 2, we recall and introduce some definitions, notation, and a lemma. In Section 3, we study properties of the resolvent operators with respect to strictly 𝐻_ℎ,𝜂-monotone and strongly 𝐻_ℎ,𝜂-monotone operators, respectively, in Hilbert spaces. In Section 4, we introduce a new class of system of nonlinear variational inclusions in Hilbert spaces and use the resolvent operator technique with respect to strongly 𝐻_ℎ,𝜂-monotone operators to investigate the existence and uniqueness of solution and suggest two new iterative algorithms for the system of nonlinear variational inclusions. The convergence criteria of the sequences generated by the iterative algorithms are also given under certain conditions.

2. Preliminaries and Lemmas

In this section, we recall and introduce some notation, definitions, and a lemma, which will be used in this paper. Let 𝐻 be a real Hilbert space endowed with an inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. 𝐼 stands for the identity mapping on 𝐻 and ℝ=(−∞,+∞). Let 𝑇∶𝐻→2^𝐻 be a set-valued operator. The graph of 𝑇, denoted by Graph(𝑇), is defined as follows: Graph(𝑇)=𝑥,𝑦∶𝑥∈𝐻,𝑦∈𝑇(𝑥). (2.1)

Definition 2.1. Let 𝜂∶𝐻×𝐻→𝐻 and 𝑀∶𝐻→2^𝐻 be operators. The operator 𝑀 is said to be(1)monotone if 𝑢−𝑣,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐻,𝑢∈𝑀(𝑥),𝑣∈𝑀𝑦; (2.2)(2)𝜂-monotone if 𝑢−𝑣,𝜂𝑥,𝑦≥0,∀𝑥,𝑦∈𝐻,𝑢∈𝑀(𝑥),𝑣∈𝑀𝑦; (2.3)(3)strictly monotone if 𝑢−𝑣,𝑥−𝑦>0,∀𝑥,𝑦∈𝐻,𝑥≠𝑦,𝑢∈𝑀(𝑥),𝑣∈𝑀𝑦; (2.4)(4)strictly 𝜂-monotone if 𝑢−𝑣,𝜂𝑥,𝑦>0,∀𝑥,𝑦∈𝐻,𝑥≠𝑦,𝑢∈𝑀(𝑥),𝑣∈𝑀𝑦; (2.5)(5)strongly monotone if there exists a constant 𝑟>0 satisfying 𝑢−𝑣,𝑥−𝑦≥𝑟𝑥−𝑦²,∀𝑥,𝑦∈𝐻,𝑢∈𝑀(𝑥),𝑣∈𝑀𝑦; (2.6)(6)strongly 𝜂-monotone if there exists a constant 𝑟>0 satisfying 𝑢−𝑣,𝜂𝑥,𝑦≥𝑟𝑥−𝑦²,∀𝑥,𝑦∈𝐻,𝑢∈𝑀(𝑥),𝑣∈𝑀𝑦; (2.7)(7)maximal monotone (resp., maximally strictly monotone, maximally strongly monotone) if 𝑀 is monotone (resp., strictly monotone, strongly monotone) and (𝐼+𝜆𝑀)(𝐻)=𝐻 for any 𝜆>0;(8)maximal 𝜂-monotone (resp., maximally strictly 𝜂-monotone, maximally strongly 𝜂-monotone) if 𝑀 is 𝜂-monotone (resp., strictly 𝜂-monotone, strongly 𝜂-monotone) and (𝐼+𝜆𝑀)(𝐻)=𝐻 for any 𝜆>0;(9)𝐻_ℎ-monotone (resp., strictly 𝐻_ℎ-monotone, strongly 𝐻_ℎ-monotone) if 𝑀 is monotone (resp., strictly monotone, strongly monotone) and (ℎ+𝜆𝑀)(𝐻)=𝐻 for any 𝜆>0;(10)𝐻_ℎ,𝜂-monotone (resp., strictly 𝐻_ℎ,𝜂-monotone, strongly 𝐻_ℎ,𝜂-monotone) if 𝑀 is 𝜂-monotone (resp., strictly 𝜂-monotone, strongly 𝜂-monotone) and (ℎ+𝜆𝑀)(𝐻)=𝐻 for any 𝜆>0.

Definition 2.2. Let 𝑔∶𝐻→𝐻 and 𝜂∶𝐻×𝐻→𝐻 be operators. The operator 𝑔 is called(1)Lipschitz continuous if there exists a constant 𝑟>0 satisfying 𝑔(𝑥)−𝑔𝑦≤𝑟𝑥−𝑦,∀𝑥,𝑦∈𝐻; (2.8)(2)monotone if 𝑔(𝑥)−𝑔𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐻; (2.9)(3)𝜂-monotone if 𝑔(𝑥)−𝑔𝑦,𝜂𝑥,𝑦≥0,∀𝑥,𝑦∈𝐻; (2.10)(4)strongly monotone if there exists a constant 𝑟>0 satisfying 𝑔(𝑥)−𝑔𝑦,𝑥−𝑦≥𝑟𝑥−𝑦²,∀𝑥,𝑦∈𝐻; (2.11)(5)strongly 𝜂-monotone if there exists a constant 𝑟>0 satisfying 𝑔(𝑥)−𝑔𝑦,𝜂𝑥,𝑦≥𝑟𝑥−𝑦²,∀𝑥,𝑦∈𝐻; (2.12)(6)relaxed Lipschitz if there exists a constant 𝑟>0 satisfying 𝑔(𝑥)−𝑔𝑦,𝑥−𝑦≤−𝑟𝑥−𝑦²,∀𝑥,𝑦∈𝐻. (2.13)

Definition 2.3. Let 𝑁∶𝐻×𝐻→𝐻 and 𝑎,𝑔∶𝐻→𝐻 be operators. The operator 𝑁 is said to be(1)strongly monotone with respect to 𝑎 and 𝑔 in the first argument if there exists a constant 𝑟>0 satisfying 𝑁(𝑎(𝑥),𝑧)−𝑁𝑎𝑦,𝑧,𝑔(𝑥)−𝑔𝑦≥𝑟𝑥−𝑦²,∀𝑥,𝑦,𝑧∈𝐻; (2.14)(2)Lipschitz continuous in the first argument if there exists a constant 𝑟>0 satisfying 𝑁(𝑥,𝑤)−𝑁𝑦,𝑤≤𝑟𝑥−𝑦,∀𝑥,𝑦,𝑤∈𝐻. (2.15)
Similarly, we could define the strong monotonicity of 𝑁 with respect to 𝑎 and 𝑔 in the second argument and the Lipschitz continuity of 𝑁 in the second argument.

Remark 2.4. For ℎ=𝐼, the definition of the 𝐻_ℎ-monotone (resp., strictly 𝐻_ℎ-monotone, strongly 𝐻_ℎ-monotone) operator reduces to the definition of the maximal monotone (resp., maximally strictly monotone, maximally strongly monotone) operator.

Remark 2.5. Notice that 𝑀 is maximal monotone (resp., maximally strictly monotone, maximally strongly monotone) if and only if 𝑀 is monotone (resp., strictly monotone, strongly monotone) and there is no other monotone (resp., strictly monotone, strongly monotone) operator whose graph contains strictly the graph Graph(𝑀) of 𝑀.

Lemma 2.6 (see [20]). Let {𝛼_𝑛}_𝑛≥0, {𝛽_𝑛}_𝑛≥0 and {𝛾_𝑛}_𝑛≥0 be nonnegative sequences satisfying 𝛼_𝑛+1≤1−𝜆_𝑛𝛼_𝑛+𝛽_𝑛𝜆_𝑛+𝛾_𝑛,∀𝑛≥0, (2.16) where {𝜆_𝑛}_𝑛≥0⊂[0,1], ∑^∞_𝑛=0𝜆_𝑛=∞,lim_𝑛→∞𝛽_𝑛=0 and ∑^∞_𝑛=0𝛾_𝑛<∞. Then lim_𝑛→∞𝛼_𝑛=0.

3. The Properties of Strictly 𝐻_ℎ,𝜂-Monotone Operators and Strongly 𝐻_ℎ,𝜂-Monotone Operators

In this section, we discuss some properties of the set-valued strictly 𝐻_ℎ,𝜂-monotone operators and set-valued strongly 𝐻_ℎ,𝜂-monotone operators, respectively, dealing with a 𝜂-monotone operator ℎ in Hilbert spaces.

Theorem 3.1. Let 𝐻 be a real Hilbert space and let ℎ∶𝐻→𝐻,𝜂∶𝐻×𝐻→𝐻 and 𝑀∶𝐻→2^𝐻 be operators such that ℎ is 𝜂-monotone and 𝑀 is strictly 𝐻_ℎ,𝜂-monotone. Then(i)𝑀is maximally strictly 𝜂-monotone;(ii)(ℎ+𝜆𝑀)⁻¹∶𝐻→𝐻 is a single-valued operator for each 𝜆>0.

Proof. (i) Since 𝑀 is strictly 𝐻_ℎ,𝜂-monotone, it follows that 𝑀 is strictly 𝜂-monotone. Suppose that there exists a strictly 𝜂-monotone set-valued operator 𝐴∶𝐻→2^𝐻 satisfying Graph(𝐴)^⊃_≠ Graph(𝑀), that is, there exists (𝑢₀,𝑥₀)∈ Graph(𝐴)⧵Graph(𝑀) such that 𝑥₀−𝑦,𝜂𝑢₀,𝑣>0,∀𝑣,𝑦∈Graph(𝑀)with𝑣≠𝑢₀. (3.1) Notice that ℎ(𝑢₀)+𝜆𝑥₀∈𝐻=(ℎ+𝜆𝑀)(𝐻) for any 𝜆>0. Thus there exists (𝑣₀,𝑦₀)∈ Graph(𝑀) satisfying ℎ𝑣₀+𝜆𝑦₀=ℎ𝑢₀+𝜆𝑥₀,∀𝜆>0. (3.2) Suppose that 𝑢₀=𝑣₀. Equation (3.2) means that 𝑥₀=𝑦₀. Therefore, Graph(𝑀)∋𝑣₀,𝑦₀=𝑢₀,𝑥₀∈Graph(𝐴)⧵Graph(𝑀), (3.3) which is impossible. Suppose that 𝑢₀≠𝑣₀. It follows from (3.1), (3.2), and the 𝜂-monotonicity of ℎ and strict 𝜂-monotonicity of 𝐴 that 0<𝑥₀−𝑦₀,𝜂𝑢₀,𝑣₀=−𝜆⁻¹ℎ𝑢₀−ℎ𝑣₀,𝜂𝑢₀,𝑣₀≤0, (3.4) which is a contradiction. Hence 𝑀 is maximally strictly 𝜂-monotone.
(ii) Suppose that there exists some 𝑢∈𝐻 such that (ℎ+𝜆𝑀)⁻¹(𝑢) contains at least two different elements 𝑥 and 𝑦. Since 𝑀 is strictly 𝐻_ℎ,𝜂-monotone, ℎ is 𝜂-monotone, and 𝜆⁻¹(𝑢−ℎ(𝑥))∈𝑀(𝑥), 𝜆⁻¹(𝑢−ℎ(𝑦))∈𝑀(𝑦), it follows that 0<𝜆⁻¹(𝑢−ℎ(𝑥))−𝜆⁻¹𝑢−ℎ𝑦,𝜂𝑥,𝑦=−𝜆⁻¹ℎ(𝑥)−ℎ𝑦,𝜂𝑥,𝑦≤0, (3.5) which is a contradiction. Consequently, the operator (ℎ+𝜆𝑀)⁻¹ is single valued. This completes the proof.

Definition 3.2. Let 𝐻 be a real Hilbert space and let ℎ∶𝐻→𝐻,𝜂∶𝐻×𝐻→𝐻 and 𝑀∶𝐻→2^𝐻 be operators such that ℎ is 𝜂-monotone and 𝑀 is strictly 𝐻_ℎ,𝜂-monotone. Then for each 𝜆>0, the resolvent operator 𝐽^ℎ_𝑀,𝜆∶𝐻→𝐻 is defined by 𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)=(ℎ+𝜆𝑀)⁻¹(𝑥),∀𝑥∈𝐻. (3.6)

Theorem 3.3. Let 𝐻 be a real Hilbert space and let 𝜂∶𝐻×𝐻→𝐻 be Lipschitz continuous with constant 𝐿_𝜂. Assume that ℎ∶𝐻→𝐻 is 𝜂-monotone and 𝑀∶𝐻→2^𝐻 is strongly 𝐻_ℎ,𝜂-monotone with constant 𝑟. Then for every 𝜆>0, the resolvent operator 𝐽^ℎ,𝜂_𝑀,𝜆∶𝐻→𝐻 is Lipschitz continuous with constant 𝐿_𝜂/𝜆𝑟.

Proof. Since 𝑀 is strongly 𝐻_ℎ,𝜂-monotone with constant 𝑟, it follows that 𝑀 is strictly 𝐻_ℎ,𝜂-monotone. Let 𝑥,𝑦 be in 𝐻. In view of 𝜆⁻¹(𝑥−ℎ(𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)))∈𝑀(𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)) and 𝜆⁻¹(𝑦−ℎ(𝐽^ℎ,𝜂_𝑀,𝜆(𝑦)))∈𝑀(𝐽^ℎ,𝜂_𝑀,𝜆(𝑦)) and the strong monotonicity of 𝑀, we deduce that 𝜆⁻¹𝑥−𝑦,𝜂𝐽^ℎ,𝜂_𝑀,𝜆(𝑥),𝐽^ℎ,𝜂_𝑀,𝜆𝑦−𝜆⁻¹ℎ𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−ℎ𝐽^ℎ,𝜂_𝑀,𝜆𝑦,𝜂𝐽^ℎ,𝜂_𝑀,𝜆(𝑥),𝐽^ℎ,𝜂_𝑀,𝜆𝑦=𝜆⁻¹𝑥−ℎ𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝜆⁻¹𝑦−ℎ𝐽^ℎ,𝜂_𝑀,𝜆𝑦,𝜂𝐽^ℎ,𝜂_𝑀,𝜆(𝑥),𝐽^ℎ,𝜂_𝑀,𝜆𝑦≥𝑟𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝐽^ℎ,𝜂_𝑀,𝜆𝑦². (3.7) This leads to 𝐿_𝜂𝑥−𝑦𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝐽^ℎ,𝜂_𝑀,𝜆𝑦≥𝑥−𝑦,𝜂𝐽^ℎ,𝜂_𝑀,𝜆(𝑥),𝐽^ℎ,𝜂_𝑀,𝜆𝑦≥ℎ𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−ℎ𝐽^ℎ,𝜂_𝑀,𝜆𝑦,𝜂𝐽^ℎ,𝜂_𝑀,𝜆(𝑥),𝐽^ℎ,𝜂_𝑀,𝜆𝑦+𝜆𝑟𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝐽^ℎ,𝜂_𝑀,𝜆𝑦²≥𝜆𝑟𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝐽^ℎ,𝜂_𝑀,𝜆𝑦², (3.8) which yields that 𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝐽^ℎ,𝜂_𝑀,𝜆𝑦≤𝐿_𝜂.𝜆𝑟𝑥−𝑦 (3.9) This completes the proof.

Theorem 3.4. Let 𝐻 be a real Hilbert space and let 𝜂∶𝐻×𝐻→𝐻 be Lipschitz continuous with constant 𝐿_𝜂. Assume that ℎ∶𝐻→𝐻 is strongly 𝜂-monotone with constant 𝑠 and 𝑀∶𝐻→2^𝐻 is strongly 𝐻_ℎ,𝜂-monotone with constant 𝑟. Then for every 𝜆>0, the resolvent operator 𝐽^ℎ,𝜂_𝑀,𝜆∶𝐻→𝐻 is Lipschitz continuous with constant 𝐿_𝜂/(𝑠+𝜆𝑟).

Proof. As in the proof of Theorem 3.3, by the strong monotonicity of ℎ and 𝑀, we infer that for any 𝑥,𝑦∈𝐻𝐿_𝜂𝑥−𝑦𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝐽^ℎ,𝜂_𝑀,𝜆𝑦≥𝑥−𝑦𝜂𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝐽^ℎ,𝜂_𝑀,𝜆𝑦≥𝑥−𝑦,𝜂𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝐽^ℎ,𝜂_𝑀,𝜆𝑦≥ℎ𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−ℎ𝐽^ℎ,𝜂_𝑀,𝜆𝑦,𝜂𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝐽^ℎ,𝜂_𝑀,𝜆𝑦+𝜆𝑟𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝐽^ℎ,𝜂_𝑀,𝜆𝑦²≥(𝑠+𝜆𝑟)𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝐽^ℎ,𝜂_𝑀,𝜆𝑦², (3.10) which means that 𝐽^ℎ,𝜂_𝑀,𝜆(𝑥)−𝐽^ℎ,𝜂_𝑀,𝜆𝑦≤𝐿_𝜂.𝑠+𝜆𝑟𝑥−𝑦 (3.11) This completes the proof.

The following example shows that Theorem 3.3 is different from Lemma 2.2 in [5].

Example 3.5. Let 𝐻=ℝ with the usual norm. Define single-valued and set-valued operators ℎ∶𝐻→𝐻,𝜂∶𝐻×𝐻→𝐻 and 𝑀∶𝐻→2^𝐻 by ℎ(𝑥)=3𝑥³𝑀(𝑥)={}−2,∀𝑥∈𝐻,𝜂𝑥,𝑦=2𝑥−𝑦,∀𝑥,𝑦∈𝐻,3𝑥+1for[]𝑥<0,1,5for{}𝑥=0,5+4𝑥for𝑥>0. (3.12) It is clear that 𝜂 is Lipschitz continuous with constant 2 and ℎ(𝑥)−ℎ𝑦,𝜂𝑥,𝑦=6𝑥−𝑦²𝑥²+𝑥𝑦+𝑦²≥0,∀𝑥,𝑦∈𝐻, (3.13) which yields that ℎ is 𝜂-monotone. On the other hand, for any 𝑟>0, there exist (𝑥_𝑟,𝑦_𝑟)=(√𝑟/10,√𝑟/20)∈𝐻×𝐻 satisfying ℎ𝑥_𝑟−ℎ𝑦_𝑟,𝜂𝑥_𝑟,𝑦_𝑟=21𝑟²<𝑟80000²𝑥400=𝑟_𝑟−𝑦_𝑟², (3.14) that is, ℎ is not strongly 𝜂-monotone. Now we claim that 𝑀 is strongly 𝐻_ℎ,𝜂-monotone. Let 𝑥,𝑦 be in 𝐻 with 𝑦>𝑥. We have to consider the following five cases.

Case 1. Suppose that 𝑥<𝑦<0. It follows that 𝑀(𝑥)−𝑀𝑦,𝜂𝑥,𝑦=6𝑥−𝑦²≥2𝑥−𝑦². (3.15)

Case 2. Suppose that 𝑥<0=𝑦. For any 𝑧∈[1,5], we get that 𝑀(𝑥)−𝑧,𝜂𝑥,𝑦=2(3𝑥+1−𝑧)𝑦−𝑥=2𝑦−𝑥²+2𝑥(1+2𝑥−𝑧)=2𝑥−𝑦². (3.16)

Case 3. Suppose that 𝑥<0<𝑦. Clearly, we have 𝑀(𝑥)−𝑀𝑦,𝜂𝑥,𝑦=23𝑥−4−4𝑦𝑥−𝑦=8𝑥−𝑦²+2(4+𝑥)𝑦−𝑥≥2𝑥−𝑦². (3.17)

Case 4. Suppose that 𝑥=0<𝑦. For each 𝑧∈[0,1], we deduce that 𝑧−𝑀𝑦,𝜂𝑥,𝑦=2𝑧−5−4𝑦𝑥−𝑦≥8𝑥−𝑦²+2(5−𝑧−𝑥)≥2𝑥−𝑦². (3.18)

Case 5. Suppose that 0<𝑥<𝑦. It is easy to verify that 𝑀(𝑥)−𝑀𝑦,𝜂𝑥,𝑦=8𝑥−𝑦²≥2𝑥−𝑦². (3.19) Hence 𝑀 is strongly 𝜂-monotone with constant 2.

Let 𝜆>0. In order to show that (ℎ+𝜆𝑀)(𝐻)=𝐻, we need only to verify that (ℎ+𝜆𝑀)(𝐻)⊇𝐻. Assume that 𝑡∈[𝜆−2,5𝜆−2]⊆𝐻. We know that 𝑡∈(ℎ+𝜆𝑀)(0)=[𝜆−2,5𝜆−2]. Assume that 𝑡∈(−∞,𝜆−2)⊆𝐻. Define a function 𝑓∶ℝ→ℝ by 𝑓(𝑥)=3𝑥³+3𝜆𝑥+𝜆−2−𝑡,∀𝑥∈ℝ. (3.20) Notice that the function 𝑓 is continuous and 𝑓(0)𝑓−𝜆−2−𝑡𝜆=(𝜆−2−𝑡)−3𝜆−2−𝑡𝜆³−2(𝜆−2−𝑡)²<0. (3.21) It follows that there exists 𝑥₀∈(−(𝜆−2−𝑡)/𝜆,0) such that 𝑓(𝑥₀)=0, that is, 𝑡=3𝑥³₀+3𝜆𝑥₀+𝜆−2=(ℎ+𝜆𝑀)𝑥₀. (3.22) Assume that 𝑡∈(5𝜆−2,+∞)⊆𝐻. Define 𝑔∶ℝ→ℝ by 𝑔(𝑥)=3𝑥³+4𝜆𝑥+5𝜆−2−𝑡,∀𝑥∈ℝ. (3.23) Obviously, 𝑔 is continuous and 𝑔(0)𝑔𝑡−5𝜆+2𝜆=3(5𝜆−2−𝑡)𝑡−5𝜆+2𝜆³+𝑡−5𝜆+2<0, (3.24) which implies that there exists 𝑥₀∈(0,(𝑡−5𝜆+2)/𝜆) with 𝑔(𝑥₀)=0. Consequently, 𝑡=3𝑥³₀+4𝜆𝑥₀+5𝜆−2=(ℎ+𝜆𝑀)𝑥₀. (3.25) Therefore, 𝐻⊆(ℎ+𝜆𝑀)(𝐻). Thus 𝑀 is strongly 𝐻_ℎ,𝜂-monotone. It follows from Theorem 3.3 that the resolvent operator 𝐽^ℎ,𝜂_𝑀,𝜆 is Lipschitz continuous with constant 1/𝜆. However, we cannot invoke Lemma 2.2 in [5] to prove the Lipschitz continuity of the resolvent operator 𝐽^ℎ,𝜂_𝑀,𝜆 because ℎ is not strongly 𝜂-monotone.

4. A System of Nonlinear Variational Inclusions

In this section, we investigate a new class of system of nonlinear variational inclusions involving strongly 𝐻_ℎ,𝜂-monotone operators in Hilbert spaces and suggest two new iterative algorithms for approximating solutions of the system of nonlinear variational inclusions by using the resolvent operator technique.

Let 𝑎_𝑖,𝑏_𝑖,𝑓_𝑖,𝑔_𝑖,ℎ_𝑖∶𝐻→𝐻,𝜂_𝑖,𝑁_𝑖∶𝐻×𝐻→𝐻 be single-valued operators and 𝑀_𝑖∶𝐻→2^𝐻 set-valued operators and 𝑝_𝑖∈𝐻 for 𝑖∈{1,2}. We now consider the following system of nonlinear variational inclusions:

find (𝑥,𝑦)∈𝐻×𝐻 such that 𝑝₁∈𝑁₁𝑎₁(𝑥),𝑏₁𝑦+𝑀₁𝑓₁(𝑥)−𝑔₁(𝑥),𝑝₂∈𝑁₂𝑎₂(𝑥),𝑏₂𝑦+𝑀₂𝑓₂𝑦−𝑔₂𝑦. (4.1)

It is easy to see that the system of nonlinear variational inclusions (4.1) includes a lot of variational inequalities, quasivariational inequalities, variational-like inequalities, variational inclusions, and systems of variational inequalities, quasivariational inequalities, variational-like inequalities, and variational inclusions in [1–16] and the references therein as special cases.

Now we use the resolvent operator technique to establish the equivalence between the existence of solutions for the system of nonlinear variational inclusions (4.1) and the existence of fixed points for the single-valued operator 𝑄 defined by (4.3) below.

Lemma 4.1. Let 𝐻 be a real Hilbert space. Let 𝑡_𝑖∈(0,1) and 𝜌_𝑖>0 be constants, 𝑎_𝑖,𝑏_𝑖,𝑓_𝑖,𝑔_𝑖,ℎ_𝑖∶𝐻→𝐻,𝜂_𝑖,𝑁_𝑖∶𝐻×𝐻→𝐻 and 𝑀_𝑖∶𝐻→2^𝐻 be single-valued and set-valued operators, respectively, such that ℎ_𝑖 is 𝜂_𝑖-monotone and 𝑀_𝑖 is strictly 𝐻_{ℎ𝑖,𝜂𝑖}-monotone for 𝑖∈{1,2}. Then the following statements are pairwise equivalent.(a)The system of nonlinear variational inclusions (4.1) has a solution (𝑥,𝑦)∈𝐻×𝐻.(b)There exists (𝑥,𝑦)∈𝐻×𝐻 satisfying 𝑓₁(𝑥)−𝑔₁(𝑥)=𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁(𝑥)−𝑔₁(𝑥)+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁(𝑥),𝑏₁𝑦,𝑓₂𝑦−𝑔₂𝑦=𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂𝑦−𝑔₂𝑦+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂(𝑥),𝑏₂𝑦. (4.2)(c)The single-valued operator 𝑄∶𝐻×𝐻→𝐻×𝐻 defined by 𝑄(𝑢,𝑣)=𝐺₁(𝑢,𝑣),𝐺₂(𝑢,𝑣),∀(𝑢,𝑣)∈𝐻×𝐻, (4.3)where 𝐺₁(𝑢,𝑣)=1−𝑡₁𝑢+𝑡₁𝑢−𝑓₁(𝑢)−𝑔₁(𝑢)+𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁(𝑢)−𝑔₁(𝑢)+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁(𝑢),𝑏₁(𝑣)()𝐺,∀𝑢,𝑣∈𝐻×𝐻,₂(𝑢,𝑣)=1−𝑡₂𝑣+𝑡₂𝑣−𝑓₂(𝑣)−𝑔₂(𝑣)+𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂(𝑣)−𝑔₂(𝑣)+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂(𝑢),𝑏₂(𝑣),∀(𝑢,𝑣)∈𝐻×𝐻 (4.4) has a fixed point (𝑥,𝑦)∈𝐻×𝐻.

Proof. It follows from Theorem 3.1, (4.3), and (4.4) that (𝑥,𝑦)∈𝐻×𝐻 is a solution of the system of nonlinear variational inclusions (4.1) if and only if 𝑝₁∈𝑁₁𝑎₁(𝑥),𝑏₁𝑦+𝑀₁𝑓₁(𝑥)−𝑔₁(𝑥),𝑝₂∈𝑁₂𝑎₂(𝑥),𝑏₂𝑦+𝑀₂𝑓₂𝑦−𝑔₂𝑦⟺ℎ₁𝑓₁(𝑥)−𝑔₁(𝑥)+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁(𝑥),𝑏₁𝑦∈ℎ₁+𝜌₁𝑀₁𝑓₁(𝑥)−𝑔₁(𝑥),⟺ℎ₂𝑓₂𝑦−𝑔₂𝑦+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂(𝑥),𝑏₂𝑦∈ℎ₂+𝜌₂𝑀₂𝑓₂𝑦−𝑔₂𝑦,⟺𝑓₁(𝑥)−𝑔₁(𝑥)=𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁(𝑥)−𝑔₁(𝑥)+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁(𝑥),𝑏₁𝑦,⟺𝑓₂𝑦−𝑔₂𝑦=𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂𝑦−𝑔₂𝑦+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂(𝑥),𝑏₂𝑦,⟺𝑥=1−𝑡₁𝑥+𝑡₁𝑥−𝑓₁(𝑥)−𝑔₁(𝑥)+𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁(𝑥)−𝑔₁(𝑥)+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁(𝑥),𝑏₁𝑦=𝐺₁𝑥,𝑦,⟺𝑦=1−𝑡₂𝑦+𝑡₂𝑦−𝑓₂𝑦−𝑔₂𝑦+𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂𝑦−𝑔₂𝑦+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂(𝑥),𝑏₂𝑦=𝐺₂,=𝐺𝑥,𝑦⟺𝑄𝑥,𝑦₁𝑥,𝑦,𝐺₂𝑥,𝑦=𝑥,𝑦. (4.5) This completes the proof.

Based on Lemma 4.1, we suggest the following new iterative algorithms for the system of nonlinear variational inclusions (4.1).

Algorithm 4.2 (The Mann iteration method with errors). For any 𝑥₀,𝑦₀∈𝐻, compute {𝑥_𝑛}_𝑛≥0,{𝑦_𝑛}_𝑛≥0⊂𝐻 by 𝑥_𝑛+1=1−𝜎_𝑛𝑥_𝑛+𝜎_𝑛𝑥_𝑛−𝑓₁𝑥_𝑛−𝑔₁𝑥_𝑛+𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁𝑥_𝑛−𝑔₁𝑥_𝑛𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁𝑥_𝑛,𝑏₁𝑦_𝑛+𝜉_𝑛𝑦,∀𝑛≥0,_𝑛+1=1−𝛿_𝑛𝑦_𝑛+𝛿_𝑛𝑦_𝑛−𝑓₂𝑦_𝑛−𝑔₂𝑦_𝑛+𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂𝑦_𝑛−𝑔₂𝑦_𝑛𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂𝑥_𝑛,𝑏₂𝑦_𝑛+𝜉_𝑛,∀𝑛≥0, (4.6) where {𝜎_𝑛}_𝑛≥0 and {𝛿_𝑛}_𝑛≥0 are sequences in (0,1), {𝜉_𝑛}_𝑛≥0 and {𝜁_𝑛}_𝑛≥0 are sequences in 𝐻 introduced to take into account possible in inexact computation with ∑^∞_𝑛=0(‖𝜉_𝑛‖+‖𝜁_𝑛‖)<+∞.

Algorithm 4.3 (The implicit Mann iteration method with errors). For any 𝑥₀,𝑦₀∈𝐻, compute {𝑥_𝑛}_𝑛≥0,{𝑦_𝑛}_𝑛≥0⊂𝐻 by 𝑥_𝑛+1=1−𝜎_𝑛𝑥_𝑛+𝜎_𝑛𝑥_𝑛+1−𝑓₁𝑥_𝑛+1−𝑔₁𝑥_𝑛+1+𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁𝑥_𝑛+1−𝑔₁𝑥_𝑛+1+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁𝑥_𝑛+1,𝑏₁𝑦_𝑛+1+𝜉_𝑛𝑦,∀𝑛≥0,_𝑛+1=1−𝛿_𝑛𝑦_𝑛+𝛿_𝑛𝑦_𝑛+1−𝑓₂𝑦_𝑛+1−𝑔₂𝑦_𝑛+1+𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂𝑦_𝑛+1−𝑔₂𝑦_𝑛+1+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂𝑥_𝑛+1,𝑏₂𝑦_𝑛+1+𝜁_𝑛,∀𝑛≥0, (4.7) where {𝜎_𝑛}_𝑛≥0 and {𝛿_𝑛}_𝑛≥0 are sequences in (0,1), and {𝜉_𝑛}_𝑛≥0 and {𝜁_𝑛}_𝑛≥0 are sequences in 𝐻 introduced to take into account possible in inexact computation with ∑^∞_𝑛=0(‖𝜉_𝑛‖+‖𝜁_𝑛‖)<+∞.
Now we investigate those conditions under which the approximate solutions generated by Algorithms 4.2 and 4.3 converge strongly to the exact solutions of the system of nonlinear variational inclusions (4.1).

Theorem 4.4. Let 𝐻 be a real Hilbert space and let 𝑎_𝑖,𝑏_𝑖,𝑓_𝑖,𝑔_𝑖,ℎ_𝑖∶𝐻→𝐻,𝜂_𝑖∶𝐻×𝐻→𝐻 be Lipschitz continuous with constants 𝐿_𝑎𝑖,𝐿_𝑏𝑖,𝐿_𝑓𝑖,𝐿_𝑔𝑖,𝐿_ℎ𝑖,𝐿_𝜂𝑖, respectively, ℎ_𝑖𝜂_𝑖-monotone, 𝑓_𝑖 strongly monotone with constant 𝛼_𝑖, 𝑔_𝑖 relaxed Lipschitz with constant 𝛽_𝑖 for 𝑖∈{1,2}. Let 𝑀_𝑖∶𝐻→2^𝐻 be strongly 𝐻_{ℎ𝑖,𝜂𝑖}-monotone with constant 𝑟_𝑖 for 𝑖∈{1,2}. Assume that 𝑁_𝑖∶𝐻×𝐻→𝐻 is Lipschitz continuous with constants 𝐿_𝑁𝑖1 and 𝐿_𝑁𝑖2 in the first and second arguments, respectively, 𝑁_𝑖 is strongly monotone with constant 𝛾_𝑖 with respect to 𝑎_𝑖 and 𝑓_𝑖−𝑔_𝑖 in the first argument for 𝑖∈{1,2}. Let 𝐾₁=1−2𝛼₁−𝛽₁+𝐿_𝑓1+𝐿_𝑔1²+𝐿_𝜂1𝜌₁𝑟₁𝐿_𝑓1+𝐿_𝑔11+𝐿_ℎ1+𝐿_𝑓1+𝐿_𝑔1²−2𝜌₁𝛾₁+𝐿²_𝑁11𝐿²_𝑎1,𝐾₂=1−2𝛼₂−𝛽₂+𝐿_𝑓2+𝐿_𝑔2²+𝐿_𝜂2𝜌₂𝑟₂𝐿_𝑓2+𝐿_𝑔21+𝐿_ℎ2+𝐿_𝑓2+𝐿_𝑔2²−2𝜌₂𝛾₂+𝐿²_𝑁22𝐿²_𝑏2,𝑉₁=1𝑟₁𝐿_𝜂1𝐿_𝑁12𝐿_𝑏1,𝑉₂=1𝑟₂𝐿_𝜂2𝐿_𝑁21𝐿_𝑎2. (4.8) If the following condition is fulfilled: 𝑉₁𝑉₂<1−𝐾₁1−𝐾₂,max𝐾₁,𝐾₂<1, (4.9) then the system of nonlinear variational inclusions (4.1) has a unique solution (𝑥,𝑦)∈𝐻×𝐻. Moreover, if sup1−1−𝐾₁𝜎_𝑛+𝑉₂𝛿_𝑛,1−1−𝐾₂𝛿_𝑛+𝑉₁𝜎_𝑛∶𝑛≥0<1, (4.10) then for each 𝑥₀,𝑦₀∈𝐻, the sequences {𝑥_𝑛}_𝑛≥0 and {𝑦_𝑛}_𝑛≥0 generated by Algorithm 4.2 converge strongly to 𝑥 and 𝑦, respectively.

Proof. First of all we prove that the operator 𝑄 defined by (4.3) is a contraction on the Banach space 𝐻×𝐻 with norm ‖(𝑢,𝑣)‖₁=‖𝑢‖+‖𝑣‖ for (𝑢,𝑣)∈𝐻×𝐻. It follows from (4.9) that 𝑉₁𝑉₂<1−𝐾₁1−𝐾₂⟺𝑉₂1−𝐾₁<1−𝐾₂𝑉₁⟺thereexists𝑡≠1with𝑉₂1−𝐾₁<𝑡<1−𝐾₂𝑉₁⟺thereexist𝑡₁,𝑡₂∈(0,1)with𝑡₁=𝑡(1−𝑡)for(),𝑡𝑡∈0,11+𝑡for𝑡𝑡>1,₂=1−𝑡for(),1𝑡∈0,11+𝑡for𝑡>1 (4.11) such that 𝑉₂𝑡₂<1−𝐾₁𝑡₁,𝑉₁𝑡₁<1−𝐾₂𝑡₂⟺thereexist𝑡₁,𝑡₂∈(0,1)satisfying(4.11)andmax1−1−𝐾₁𝑡₁+𝑡₂𝑉₂,1−1−𝐾₂𝑡₂+𝑡₁𝑉₁<1. (4.12)(4.13) Put (𝑢₁,𝑣₁),(𝑢₂,𝑣₂)∈𝐻×𝐻. By the Lipschitz continuity and various monotonicity of 𝑎_𝑖,𝑏_𝑖,𝑓_𝑖,𝑔_𝑖,ℎ_𝑖,𝑀_𝑖,𝑁_𝑖 for 𝑖∈{1,2}, (4.4) and (4.9), we deduce that 𝐺₁𝑢₁,𝑣₁−𝐺₁𝑢₂,𝑣₂≤1−𝑡₁𝑢₁−𝑢₂+𝑡₁𝑢₁−𝑢₂−𝑓₁𝑢₁−𝑔₁𝑢₁−𝑓₁𝑢₂−𝑔₁𝑢₂+𝑡₁𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁𝑢₁−𝑔₁𝑢₁+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁𝑢₁,𝑏₁𝑣₁−𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁𝑢₂−𝑔₁𝑢₂+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁𝑢₂,𝑏₁𝑣₂≤1−𝑡₁1−1−2𝛼₁−𝛽₁+𝐿_𝑓1+𝐿_𝑔1²𝑢₁−𝑢₂+𝑡₁𝐿_𝜂1𝜌₁𝑟₁ℎ₁𝑓₁𝑢₁−𝑔₁𝑢₁−ℎ₁𝑓₁𝑢₂−𝑔₁𝑢₂+𝑓₁𝑢₁−𝑔₁𝑢₁−𝑓₁𝑢₂−𝑔₁𝑢₂+𝑓₁𝑢₁−𝑔₁𝑢₁−𝑓₁𝑢₂−𝑔₁𝑢₂−𝜌₁𝑁₁𝑎₁𝑢₁,𝑏₁𝑣₁−𝑁₁𝑎₁𝑢₂,𝑏₁𝑣₁+𝜌₁𝑁₁𝑎₁𝑢₂,𝑏₁𝑣₁−𝑁₁𝑎₁𝑢₂,𝑏₁𝑣₂≤1−𝑡₁1−1−2𝛼₁−𝛽₁+𝐿_𝑓1+𝐿_𝑔1²𝑢₁−𝑢₂+𝑡₁𝐿_𝜂1𝜌₁𝑟₁1+𝐿_ℎ1𝐿_𝑓1+𝐿_𝑔1+𝐿_𝑓1+𝐿_𝑔1²−2𝜌₁𝛾₁+𝜌²₁𝐿²_𝑁11𝐿²_𝑎1𝑢₁−𝑢₂+𝑡₁𝐿_𝜂1𝑟₁𝐿_𝑁12𝐿_𝑏1𝑣₁−𝑣₂=1−1−𝐾₁𝑡₁𝑢₁−𝑢₂+𝑡₁𝑉₁𝑣₁−𝑣₂,𝐺₂𝑢₁,𝑣₁−𝐺₂𝑢₂,𝑣₂≤1−𝑡₂𝑣₁−𝑣₂+𝑡₂𝑣₁−𝑣₂−𝑓₂𝑣₁−𝑔₂𝑣₁−𝑓₂𝑣₂−𝑔₂𝑣₂+𝑡₂𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂𝑣₁−𝑔₂𝑣₁+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂𝑢₁,𝑏₂𝑣₁−𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂𝑣₂−𝑔₂𝑣₂+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂𝑢₂,𝑏₂𝑣₂≤1−𝑡₂1−1−2𝛼₂−𝛽₂+𝐿_𝑓2+𝐿_𝑔2²𝑣₁−𝑣₂+𝑡₂𝐿_𝜂2𝜌₂𝑟₂ℎ₂𝑓₂𝑣₁−𝑔₂𝑣₁−ℎ₂𝑓₂𝑣₂−𝑔₂𝑣₂+𝑓₂𝑣₁−𝑔₂𝑣₁−𝑓₂𝑣₂−𝑔₂𝑣₂+𝑓₂𝑣₁−𝑔₂𝑣₁−𝑓₂𝑣₂−𝑔₂𝑣₂−𝜌₂𝑁₂𝑎₂𝑢₁,𝑏₂𝑣₁−𝑁₂𝑎₂𝑢₁,𝑏₂𝑣₂+𝜌₂𝑁₂𝑎₂𝑢₁,𝑏₂𝑣₂−𝑁₂𝑎₂𝑢₂,𝑏₂𝑣₂≤1−𝑡₂1−1−2𝛼₂−𝛽₂+𝐿_𝑓2+𝐿_𝑔2²𝑣₁−𝑣₂+𝑡₂𝐿_𝜂2𝜌₂𝑟₂1+𝐿_ℎ2𝐿_𝑓2+𝐿_𝑔2+𝐿_𝑓2+𝐿_𝑔2²−2𝜌₂𝛾₂+𝜌²₂𝐿²_𝑁22𝐿²_𝑏2𝑣₁−𝑣₂+𝑡₂𝐿_𝜂2𝑟₂𝐿_𝑁21𝐿_𝑎2𝑢₁−𝑢₂=1−1−𝐾₂𝑡₂𝑣₁−𝑣₂+𝑡₂𝑉₂𝑢₁−𝑢₂. (4.14) In light of (4.3) and (4.8)–(4.14), we infer that 𝑄𝑢₁,𝑣₁−𝑄𝑢₂,𝑣₂₁=𝐺₁𝑢₁,𝑣₁−𝐺₁𝑢₂,𝑣₂+𝐺₂𝑢₁,𝑣₁−𝐺₂𝑢₂,𝑣₂≤1−1−𝐾₁𝑡₁+𝑡₂𝑉₂𝑢₁−𝑢₂+1−1−𝐾₂𝑡₂+𝑡₁𝑉₁𝑣₁−𝑣₂≤max1−1−𝐾₁𝑡₁+𝑡₂𝑉₂,1−1−𝐾₂𝑡₂+𝑡₁𝑉₁×𝑢₁−𝑢₂+𝑣₁−𝑣₂=𝐾𝑢₁,𝑣₁−𝑢₂,𝑣₂₁, (4.15) where 𝐾=max1−1−𝐾₁𝑡₁+𝑡₂𝑉₂,1−1−𝐾₂𝑡₂+𝑡₁𝑉₁<1, (4.16) that is, 𝑄 is a contraction on (𝐻×𝐻,‖⋅‖₁). Therefore, the operator 𝑄 has a unique fixed point (𝑥,𝑦)∈𝐻×𝐻, which is the unique solution of the system of nonlinear variational inclusions (4.1) by Lemma 4.1. It follows that 𝑥=1−𝜎_𝑛𝑥+𝜎_𝑛𝑥−𝑓₁(𝑥)−𝑔₁(𝑥)+𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁(𝑥)−𝑔₁(𝑥)+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁(𝑥),𝑏₁𝑦,∀𝑛≥0,𝑦=1−𝛿_𝑛𝑦+𝛿_𝑛𝑦−𝑓₂𝑦−𝑔₂𝑦+𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂𝑦−𝑔₂𝑦+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂(𝑥),𝑏₂𝑦,∀𝑛≥0. (4.17) In view of Algorithm 4.2 and (4.17), we deduce that for any 𝑛≥0𝑥_𝑛+1≤−𝑥1−𝜎_𝑛𝑥_𝑛−𝑥+𝜎_𝑛𝑥_𝑛−𝑥−𝑓₁𝑥_𝑛−𝑔₁𝑥_𝑛−𝑓₁(𝑥)−𝑔₁(𝑥)+𝜎_𝑛𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁𝑥_𝑛−𝑔₁𝑥_𝑛+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁𝑥_𝑛,𝑏₁𝑦_𝑛−𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁(𝑥)−𝑔₁(𝑥)+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁(𝑥),𝑏₁𝑦+𝜉_𝑛≤1−𝜎_𝑛1−1−2𝛼₁−𝛽₁+𝐿_𝑓1+𝐿_𝑔1²𝑥_𝑛+𝜎−𝑥_𝑛𝐿_𝜂1𝜌₁𝑟₁ℎ₁𝑓₁𝑥_𝑛−𝑔₁𝑥_𝑛−ℎ₁𝑓₁(𝑥)−𝑔₁(𝑥)+𝑓₁𝑥_𝑛−𝑔₁𝑥_𝑛−𝑓₁(𝑥)−𝑔₁(𝑥+𝑓₁)𝑥_𝑛−𝑔₁𝑥_𝑛−𝑓₁(𝑥)−𝑔₁(𝑥)−𝜌₁𝑁₁𝑎₁𝑥_𝑛,𝑏₁𝑦_𝑛−𝑁₁𝑎₁(𝑥),𝑏₁𝑦_𝑛+𝜌₁𝑁₁𝑎₁(𝑥),𝑏₁𝑦_𝑛−𝑁₁𝑎₁(𝑥),𝑏₁𝑦+𝜉_𝑛≤1−𝜎_𝑛1−1−2𝛼₁−𝛽₁+𝐿_𝑓1+𝐿_𝑔1²𝑥_𝑛+𝜎−𝑥_𝑛𝐿_𝜂1𝜌₁𝑟₁1+𝐿_ℎ1𝐿_𝑓1+𝐿_𝑔1+𝐿_𝑓1+𝐿_𝑔1²−2𝜌₁𝛾₁+𝜌²₁𝐿²_𝑁11𝐿²_𝑎1𝑥_𝑛+𝜎−𝑥_𝑛𝐿_𝜂1𝑟₁𝐿_𝑁12𝐿_𝑏1𝑦_𝑛−𝑦+𝜉_𝑛=1−1−𝐾₁𝜎_𝑛𝑥_𝑛−𝑥+𝜎_𝑛𝑉₁𝑦_𝑛−𝑦+𝜉_𝑛,𝑦_𝑛+1≤−𝑦1−𝛿_𝑛𝑦_𝑛−𝑦+𝛿_𝑛𝑦_𝑛−𝑦−𝑓₂𝑦_𝑛−𝑔₂𝑦_𝑛−𝑓₂𝑦−𝑔₂𝑦+𝛿_𝑛𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂𝑦_𝑛−𝑔₂𝑦_𝑛+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂𝑥_𝑛,𝑏₂𝑦_𝑛−𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂(𝑥)−𝑔₂(𝑥)+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂(𝑥),𝑏₂𝑦+𝜁_𝑛≤1−1−𝐾₂𝛿_𝑛𝑦_𝑛−𝑦+𝛿_𝑛𝑉₂𝑥_𝑛−𝑥+𝜁_𝑛. (4.18) By virtue of (4.18), we have 𝑥_𝑛+1−𝑥+𝑦_𝑛+1−𝑦≤1−1−𝐾₁𝜎_𝑛𝑥_𝑛−𝑥+𝜎_𝑛𝑉₁𝑦_𝑛+−𝑦1−1−𝐾₂𝛿_𝑛𝑦_𝑛−𝑦+𝛿_𝑛𝑉₂𝑥_𝑛−𝑥+𝜉_𝑛+𝜁_𝑛≤max1−1−𝐾₁𝜎_𝑛+𝑉₂𝛿_𝑛,1−1−𝐾₂𝛿_𝑛+𝑉₁𝜎_𝑛×𝑥_𝑛−𝑥+𝑦_𝑛−𝑦+𝜉_𝑛+𝜁_𝑛≤𝐾^∗𝑥_𝑛−𝑥+𝑦_𝑛−𝑦+𝜉_𝑛+𝜁_𝑛,∀𝑛≥0, (4.19) where 𝐾^∗=sup1−1−𝐾₁𝜎_𝑛+𝑉₂𝛿_𝑛,1−1−𝐾₂𝛿_𝑛+𝑉₁𝜎_𝑛∶𝑛≥0<1 (4.20) by (4.10). Lemma 2.6 and (4.19) ensure that the sequences {𝑥_𝑛}_𝑛≥0 and {𝑥_𝑛}_𝑛≥0 generated by Algorithm 4.2 converge strongly to 𝑥 and 𝑦, respectively. This completes the proof.

Theorem 4.5. Let 𝐻,𝑎_𝑖,𝑏_𝑖,𝑓_𝑖,𝑔_𝑖,ℎ_𝑖,𝜂_𝑖,𝑀_𝑖,𝑁_𝑖,𝐾_𝑖,𝑉_𝑖 for 𝑖∈{1,2} be as in Theorem 4.4. If (4.9) is satisfied, then the system of nonlinear variational inclusions (4.1) has a unique solution (𝑥,𝑦)∈𝐻×𝐻. Moreover, if sup𝐾₁𝜎_𝑛+𝑉₂𝛿_𝑛,𝑉₁𝜎_𝑛+𝐾₂𝛿_𝑛∶𝑛≥0<1,sup1−𝜎_𝑛1−𝐾₁𝜎_𝑛,1−𝛿_𝑛1−𝐾₂𝛿_𝑛𝑉∶𝑛≥0+sup¹𝜎_𝑛1−𝐾₁𝜎_𝑛,𝑉₂𝛿_𝑛1−𝐾₂𝛿_𝑛∶𝑛≥0<1 (4.21)(4.22) hold, then for each 𝑥₀,𝑦₀∈𝐻, the sequences {𝑥_𝑛}_𝑛≥0 and {𝑥_𝑛}_𝑛≥0 generated by Algorithm 4.3 converge strongly to 𝑥 and 𝑦, respectively.

Proof. It follows from Theorem 4.4 that the system of nonlinear variational inclusions (4.1) has a unique solution (𝑥,𝑦)∈𝐻×𝐻. Now we prove that Algorithm 4.3 is well defined. Let 𝑛≥0 be a fixed integer. For each (𝑥_𝑛,𝑦_𝑛)∈𝐻×𝐻, define 𝑆₁,𝑆₂∶𝐻×𝐻→𝐻 and 𝑆∶𝐻×𝐻→𝐻×𝐻 by 𝑆₁(𝑢,𝑣)=1−𝜎_𝑛𝑥_𝑛+𝜎_𝑛𝑢−𝑓₁(𝑢)−𝑔₁(𝑢)+𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁(𝑢)−𝑔₁(𝑢)+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁(𝑢),𝑏₁(𝑣)+𝜉_𝑛𝑆,∀𝑢,𝑣∈𝐻,₂(𝑢,𝑣)=1−𝛿_𝑛𝑦_𝑛+𝛿_𝑛𝑢−𝑓₂(𝑢)−𝑔₂(𝑢)+𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂(𝑢)−𝑔₂(𝑢)+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂(𝑢),𝑏₂(𝑣)+𝜁_𝑛𝑆()=𝑆,∀𝑢,𝑣∈𝐻𝑢,𝑣₁(𝑢,𝑣),𝑆₂(𝑢,𝑣),∀𝑢,𝑣∈𝐻. (4.23) As in the proof of Theorem 4.4, we infer that for any (𝑢₁,𝑣₁),(𝑢₂,𝑣₂)∈𝐻×𝐻𝑆₁𝑢₁,𝑣₁−𝑆₁𝑢₂,𝑣₂≤𝜎_𝑛𝑢₁−𝑢₂−𝑓₁𝑢₁−𝑔₁𝑢₁−𝑓₁𝑢₂−𝑔₁𝑢₂+𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁𝑢₁−𝑔₁𝑢₁+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁𝑢₁,𝑏₁𝑣₁−𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁𝑢₂−𝑔₁𝑢₂+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁𝑢₂,𝑏₁𝑣₂≤𝐾₁𝜎_𝑛𝑢₁−𝑢₂+𝑉₁𝜎_𝑛𝑣₁−𝑣₂,𝑆₂𝑢₁,𝑣₁−𝑆₂𝑢₂,𝑣₂≤𝐾₂𝛿_𝑛𝑣₁−𝑣₂+𝑉₂𝛿_𝑛𝑢₁−𝑢₂, (4.24) which yield that 𝑆𝑢₁,𝑣₁−𝑆𝑢₂,𝑣₂₁=𝑆₁𝑢₁,𝑣₁−𝑆₁𝑢₂,𝑣₂+𝑆₂𝑢₁,𝑣₁−𝑆₂𝑢₂,𝑣₂≤𝐾₁𝜎_𝑛+𝑉₂𝛿_𝑛𝑢₁−𝑢₂+𝑉₁𝜎_𝑛+𝐾₂𝛿_𝑛𝑣₁−𝑣₂≤max𝐾₁𝜎_𝑛+𝑉₂𝛿_𝑛,𝑉₁𝜎_𝑛+𝐾₂𝛿_𝑛𝑢₁,𝑣₁−𝑢₂,𝑣₂₁. (4.25) (4.21) and (4.25) ensure that 𝑆 is a contraction on 𝐻×𝐻. Thus 𝑆 has a unique fixed point (𝑥_𝑛+1,𝑦_𝑛+1)∈𝐻×𝐻. Consequently, for each (𝑥_𝑛,𝑦_𝑛)∈𝐻×𝐻, there exists a unique element (𝑥_𝑛+1,𝑦_𝑛+1)∈𝐻×𝐻 satisfying 𝑥_𝑛+1=1−𝜎_𝑛𝑥_𝑛+𝜎_𝑛𝑥_𝑛+1−𝑓₁𝑥_𝑛+1−𝑔₁𝑥_𝑛+1+𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁𝑥_𝑛+1−𝑔₁𝑥_𝑛+1+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁𝑥_𝑛+1,𝑏₁𝑦_𝑛+1+𝜉_𝑛,𝑦_𝑛+1=1−𝛿_𝑛𝑦_𝑛+𝛿_𝑛𝑦_𝑛+1−𝑓₂𝑦_𝑛+1−𝑔₂𝑦_𝑛+1+𝐽^ℎ2,𝜂2_𝑀2,𝜌2ℎ₂𝑓₂𝑦_𝑛+1−𝑔₂𝑦_𝑛+1+𝜌₂𝑝₂−𝜌₂𝑁₂𝑎₂𝑥_𝑛+1,𝑏₂𝑦_𝑛+1+𝜁_𝑛. (4.26) That is, Algorithm 4.3 is well defined. Next we show the convergence of the sequences {𝑥_𝑛}_𝑛≥0 and {𝑥_𝑛}_𝑛≥0 generated by Algorithm 4.3. In light of (4.17) and (4.26), we conclude that for any 𝑛≥0𝑥_𝑛+1≤−𝑥1−𝜎_𝑛𝑥_𝑛−𝑥+𝜎_𝑛𝑥_𝑛+1−𝑥−𝑓₁𝑥_𝑛+1−𝑔₁𝑥_𝑛+1−𝑓₁(𝑥)−𝑔₁(𝑥)+𝜎_𝑛𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁𝑥_𝑛+1−𝑔₁𝑥_𝑛+1+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁𝑥_𝑛+1,𝑏₁𝑦_𝑛+1−𝐽^ℎ1,𝜂1_𝑀1,𝜌1ℎ₁𝑓₁(𝑥)−𝑔₁(𝑥)+𝜌₁𝑝₁−𝜌₁𝑁₁𝑎₁(𝑥),𝑏₁𝑦+𝜉_𝑛≤1−𝜎_𝑛𝑥_𝑛−𝑥+𝐾₁𝜎_𝑛𝑥_𝑛+1−𝑥+𝑉₁𝜎_𝑛𝑦_𝑛+1−𝑦+𝜉_𝑛, (4.27) which means that 𝑥_𝑛+1−𝑥≤1−𝜎_𝑛1−𝐾₁𝜎_𝑛𝑥_𝑛+𝑉−𝑥¹𝜎_𝑛1−𝐾₁𝜎_𝑛𝑦_𝑛+1+1−𝑦1−𝐾₁𝜎_𝑛𝜉_𝑛,∀𝑛≥0. (4.28) Similarly, we have 𝑦_𝑛+1−𝑦≤1−𝛿_𝑛1−𝐾₂𝛿_𝑛𝑦_𝑛+𝑉−𝑦²𝛿_𝑛1−𝐾₂𝛿_𝑛𝑥_𝑛+1+1−𝑥1−𝐾₂𝛿_𝑛𝜁_𝑛,∀𝑛≥0. (4.29) In view of (4.28) and (4.29), we deduce that 𝑥_𝑛+1−𝑥+𝑦_𝑛+1−𝑦≤max1−𝜎_𝑛1−𝐾₁𝜎_𝑛,1−𝛿_𝑛1−𝐾₂𝛿_𝑛𝑥_𝑛−𝑥+𝑦_𝑛𝑉−𝑦+max¹𝜎_𝑛1−𝐾₁𝜎_𝑛,𝑉₂𝛿_𝑛1−𝐾₂𝛿_𝑛𝑥_𝑛+1−𝑥+𝑦_𝑛+11−𝑦+max1−𝐾₁𝜎_𝑛,11−𝐾₂𝛿_𝑛𝜉_𝑛+𝜁_𝑛,∀𝑛≥0, (4.30) which implies that 𝑥_𝑛+1−𝑥+𝑦_𝑛+1≤−𝑦max1−𝜎_𝑛/1−𝐾₁𝜎_𝑛,1−𝛿_𝑛/1−𝐾₂𝛿_𝑛1−max𝑉₁𝜎_𝑛/1−𝐾₁𝜎_𝑛,𝑉₂𝛿_𝑛/1−𝐾₂𝛿_𝑛𝑥_𝑛−𝑥+𝑦_𝑛+−𝑦max1/1−𝐾₁𝜎_𝑛,1/1−𝐾₂𝛿_𝑛1−max𝑉₁𝜎_𝑛/1−𝐾₁𝜎_𝑛,𝑉₂𝛿_𝑛/1−𝐾₂𝛿_𝑛𝜉_𝑛+𝜁_𝑛≤𝑎𝑥1−𝑏^𝑛−𝑥+𝑦_𝑛+−𝑦max1/1−𝐾₁,1/1−𝐾₂𝜉1−𝑏^𝑛+𝜁_𝑛,∀𝑛≥0, (4.31) where 𝑎=sup1−𝜎_𝑛1−𝐾₁𝜎_𝑛,1−𝛿_𝑛1−𝐾₂𝛿_𝑛,𝑉∶𝑛≥0𝑏=sup¹𝜎_𝑛1−𝐾₁𝜎_𝑛,𝑉₂𝛿_𝑛1−𝐾₂𝛿_𝑛∶𝑛≥0. (4.32) Note that (4.22) yields that 𝑎/(1−𝑏)<1. Consequently, Lemma 2.6 and (4.31) yield that the sequences {𝑥_𝑛}_𝑛≥0 and {𝑥_𝑛}_𝑛≥0 generated by Algorithm 4.3 converge strongly to 𝑥 and 𝑦, respectively. This completes the proof.

Remark 4.6. Lemma 4.1 and Theorems 4.4 and 4.5 extend, improve, and unify the corresponding results in [2–5, 11–19].

Acknowledgments

This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2002165).