Projection Methods for a System of Nonlinear Mixed Variational Inequalities in Banach Spaces

Abstract

The existence and uniqueness of solution for a system of nonlinear mixed variational inequality in Banach spaces is given firstly. A Mann iterative sequences with errors for this system of nonlinear mixed variational inequalities in Banach spaces is studied, by using the generalized f-projection operator 𝜋^𝑓_𝐾. Our results extend the main results in (Verma (2005); Verma (2001)) from Hilbert spaces to Banach spaces.

1. Introduction

Throughout this paper, unless otherwise stated, we let 𝐵 be a real Banach space with the norm ‖⋅‖, we let 𝐵^∗ be the topological dual space of 𝐵, we let ⟨⋅,⋅⟩ be the duality between 𝐵, and we let 𝐵^∗ and 𝐾⊂𝐵 be a nonempty closed convex set.

Let 𝐴₁,𝐴₂,𝑆₁,𝑆₂∶𝐾→𝐵^∗ be single-valued mappings. Let 𝐽∶𝐵→𝐵^∗ be the normalized duality mapping, and let 𝑓∶𝐾⊂𝐵→𝑅∪{+∞} be a proper lower semicontinuous mapping. We will consider the following system of nonlinear mixed variational inequalities problems (SNMVIP): find (𝑥^∗,𝑦^∗)∈𝐾×𝐾 satisfying 𝜌₁𝐴₁𝑦^∗+𝜌₁𝑆₁𝑦^∗+𝐽𝑥^∗−𝐽𝑦^∗,𝑥−𝑥^∗+𝑓(𝑥)−𝑓𝑥^∗≥0,∀𝑥∈𝐾,⟨𝜂₁𝐴₂𝑥^∗+𝜂₁𝑆₂𝑥^∗+𝐽𝑦^∗−𝐽𝑥^∗,𝑥−𝑦^∗⟩+𝑓(𝑥)−𝑓𝑦^∗≥0,∀𝑥∈𝐾, (1) where 𝜌₁>0 and 𝜂₁>0.

If 𝐵=𝐻 is a real Hilbert space, 𝑆₁=𝑆₂=𝑆, 𝐴₁=𝐴₂=0, and 𝑓=0, then SNMVIP (1) reduces to the following problem: 𝜌₁𝑆𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝜂≥0,∀𝑥∈𝐾,₁𝑆𝑥^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐾. (2) The problem (2) was introduced and studied by Verma [1, 2]. We note that for 𝜌₁=0 the problem (2) reduces to the nonlinear variational inequalities problem: determine an element 𝑥^∗∈𝐾 such that 𝑆𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐾. (3)

In brief, the system of nonlinear mixed variational inequalities (1) is more general and includes many systems of variational inequalities and variational inequalities as special cases.

It is well known that the projection method and its variant forms have represented an important tool for solving variational inequalities and the system of variational inequalities; see [1–21] and the references therein. Wu and Huang [10] introduced and studied a new class of generalized 𝑓-projection operators in Banach spaces, which extends the definition of the generalized projection operators introduced and studied by Alber [3, 4] and Li [7]. Some properties of the generalized 𝑓-projection operator are given in [10]. By using the generalized projection operators and the generalized 𝑓-projection operators, some authors studied the existence theorems of solution and some iterative algorithms of approximating solutions for variational inequalities; see [3, 4, 6–8, 10–12].

In addition, Verma [1, 2] applies the metric projection operator technique and the fixed theorem to suggest some iterative algorithms for the system of variational inequalities with the monotone mappings in Hilbert spaces. By using the sunny nonexpansive retraction, Yao et al. [21] prove that the suggested two-step projection method converges strongly to the solution of a system of variational inequality, which extend the main results in Verma [1] from Hilbert spaces to Banach spaces. Recently, Wang et al. show that the Lipschitz continuity of the generalized 𝑓-projection operator and the normalized duality mapping and study a class of system of generalized mixed variational inequalities in Banach spaces.

Motivated and inspired by the research work going on this field, in this paper, we firstly show the existence and uniqueness of solution for a system of nonlinear mixed variational inequality in Banach spaces. By using the generalized 𝑓-projection operator 𝜋^𝑓_𝐾, we study Mann iterative sequences with errors for this system of nonlinear mixed variational inequalities in Banach spaces. Our results extend and improve the main results in Verma [1, 2] from Hilbert spaces to Banach spaces.

2. Preliminaries

Recall that 𝐵 is said to be strictly convex if ‖(𝑥+𝑦)/2‖<1 for all 𝑥, 𝑦∈𝐵 with ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. It is also said to be uniformly convex if lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0 for any two sequences {𝑥_𝑛}, {𝑦_𝑛} in 𝐵 with ‖𝑥_𝑛‖=‖𝑦_𝑛‖=1 and lim_𝑛→∞‖(𝑥_𝑛+𝑦_𝑛)/2‖=1. The function 𝛿_𝐵(𝜖)=inf1−𝑥+𝑦2∶‖𝑥‖=1,𝑦=1,𝑥−𝑦≥𝜖 (4) is called the modulus of convexity of 𝐵. It is well known that 𝐵 is 2-uniformly convex if and only if there exists a constant 𝑘_𝐵>0 such that 𝛿(𝜖)≥𝑘_𝐵𝜖² for all 𝜖∈(0,2]. Recall that 𝐵 is said to be smooth if lim_𝑡→0(‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡 exists for all 𝑥,𝑦∈𝐵 with ‖𝑥‖=‖𝑦‖=1. It is also said to be uniformly smooth if the limit is attained uniformly for ‖𝑥‖=‖𝑦‖=1. The function 𝜌_𝐵(𝑡)+=sup𝑥+𝑦𝑥−𝑦2−1∶‖𝑥‖=1,𝑦≤𝑡 (5) is called the modulus of smoothness of 𝐵. It is known that 𝐵 is said to be 2-uniformly smooth if and only if there exists a constant 𝑐_𝐵>0 such that 𝜌_𝐵(𝑡)≤𝑐_𝐵𝑡². Moreover, 𝐵^∗ is uniformly convex if and only if 𝐵 is uniformly smooth. In this case, 𝐵 is reflexive by the Milman theorem.

Next we recall the concept of the normalized duality mapping. The normalized duality mapping 𝐽∶𝐵→2^𝐵∗ is defined by 𝐽(𝑥)=𝑗(𝑥)∈𝐵^∗∶𝑗(𝑥)=𝑗(𝑥𝑥‖=‖𝑥‖,𝑥)‖²=𝑗(𝑥)²,∀𝑥∈𝐵. (6) Many properties of the normalized duality mapping 𝐽 can be found in [18].

We list some properties of 𝐽 as follows:(a₁)if 𝐵 is a smooth Banach space, then 𝐽 is single-valued and continuous from the strong topology of 𝐵 to the weak ∗ topology of 𝐵^∗;(a₂)𝐽 is the identity operator in Hilbert spaces;(a₃)if 𝐵 is a reflexive, smooth, and strictly convex Banach space, and 𝐽^∗∶𝐵^∗→𝐵 is the normalized duality mapping on 𝐵^∗, then 𝐽⁻¹=𝐽^∗, 𝐽𝐽^∗=𝐼_𝐵^∗ and 𝐽^∗𝐽=𝐼_𝐵.

For any fixed 𝜌>0, let 𝐺∶𝐵^∗×𝐾→(−∞,+∞] be a function defined as follows: 𝐺𝜑,𝑥=‖𝑥‖²−2𝜑,𝑥+𝜑²+2𝜌𝑓(𝑥),∀𝜑∈𝐵^∗,∀𝑥∈𝐾, (7) where 𝜑∈𝐵^∗, 𝑥∈𝐾 and 𝑓∶𝐾⊂𝐵→𝑅∪{+∞} is proper, convex, and lower semicontinuous. It is easy to see that 𝐺(𝜑,𝑥)≥(‖𝑥‖−‖𝜑‖)²+2𝜌𝑓(𝑥) for all 𝜑∈𝐵^∗ and 𝑥∈𝐾.

Definition 1. We say that 𝐵 has the property (ℎ) if 𝑥_𝑛⇀𝑥 weakly and ‖𝑥_𝑛‖→‖𝑥‖ implies 𝑥_𝑛→𝑥.

Remark 2. It is well known that any locally uniformly convex Banach space has the property (ℎ) and 𝐵^∗ has a Fréchet differentiable norm if and only if 𝐵 is reflexive and strictly convex and has the property (ℎ) (see, e.g., [19]).

Definition 3 (see [10]). We say that 𝜋^𝑓_𝐾∶𝐵^∗→2^𝐾 is a generalized 𝑓-projection operator if 𝜋^𝑓_𝐾𝜑=𝑢∈𝐾∶𝐺𝜑,𝑢=inf_𝑦∈𝐾𝐺𝜑,𝑦,∀𝜑∈𝐵^∗. (8)

Remark 4. (i) If 𝑓(𝑥)=0 for all 𝑥∈𝐾, then the generalized 𝑓-projection operator 𝜋^𝑓_𝐾 reduces to the generalized projection operator 𝜋_𝑘 defined by Alber [4] and Li [7]; that is, 𝜋_𝐾𝜑=𝑢∈𝐾∶𝐺₁𝜑,𝑢=inf_𝑦∈𝐾𝐺₁𝜑,𝑦,∀𝜑∈𝐵^∗, (9) where 𝐺₁(𝜑,𝑥)=‖𝑥‖²−2⟨𝜑,𝑥⟩+‖𝜑‖² for all 𝜑∈𝐵^∗ and 𝑥∈𝐾.
(ii) If 𝑓(𝑥)=0 for all 𝑥∈𝐾 and 𝐵=ℋ is a Hilbert space, then the generalized 𝑓-projection operator 𝜋^𝑓_𝐾 is equivalent to the following metric projection operator: 𝑃_𝐾𝜑=𝑢∈𝐾∶𝑢−𝜑=inf_𝑦∈𝐾𝑦−𝜑,∀𝜑∈ℋ. (10)

Definition 5. Let 𝐵 be a reflexive, smooth, and strictly convex Banach space with the dual space 𝐵^∗, and 2^𝐵∗ denote the family of all nonempty subsets of 𝐵^∗. 𝑇∶𝐵→2^𝐵∗ is said to be 𝐽-strongly monotone if there exists a constant 𝑆>0 satisfying 𝑢−𝑣,𝐽^∗𝐽(𝑥)−𝐽𝑦≥𝑆𝑥−𝑦²,∀𝑢∈𝑇(𝑥),∀𝑣∈𝑇𝑦. (11)

Remark 6. If 𝐵=ℋ is a Hilbert space, then 𝐽-strongly monotonicity reduce to strongly monotonicity in [15].

Theorem 7 7 (see [6, 10]). If 𝐵 is a reflexive Banach space with dual space 𝐵^∗ and 𝐾 is a nonempty closed convex subset of 𝐵, then the following conclusions hold:(i)for any given 𝜑∈𝐵^∗, 𝜋^𝑓_𝐾𝜑 is a nonempty, closed, and convex subset of 𝐾;(ii)if B is smooth, then for any given 𝜑∈𝐵^∗, 𝑥∈𝜋^𝑓_𝐾𝜑 if and only if 𝜑−𝐽(𝑥),𝑥−𝑦+𝜌𝑓𝑦−𝜌𝑓(𝑥)≥0,∀𝑦∈𝐾; (12)(iii)if 𝐵 is strictly convex, then the operator 𝜋^𝑓_𝐾∶𝐵^∗→𝐾 is single-valued.

Lemma 8 (see [13]). 𝐵 is 2-uniformly smooth if and only if there exists a constant 𝐶_𝐵>0 such that, for all 𝑥,𝑦∈𝐵, 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽𝑥+𝐶_𝐵𝑦². (13)

Lemma 9 9 (see [20]). Let {𝑎_𝑛}, {𝑏_𝑛}, and {𝑐_𝑛} be three sequences of nonnegative numbers satisfying the following conditions: there exists 𝑛₀ such that 𝑎_𝑛+1≤1−𝑡_𝑛𝑎_𝑛+𝑏_𝑛𝑡_𝑛+𝑐_𝑛,∀𝑛≥𝑛₀, (14) where 𝑡_𝑛∈[0,1],^∞_𝑛=0𝑡_𝑛=+∞;lim_𝑛→∞𝑏_𝑛=0,_∞𝑛=0𝑐_𝑛<+∞. (15) Then 𝑎_𝑛→0 as 𝑛→+∞.

Proposition 10 10 (see [14]). Let 𝐵 be a reflexive, smooth, and strictly convex Banach space with the dual space 𝐵^∗. Then for any 𝑥,𝑦∈𝐵, 𝐽𝑥−𝐽𝑦,𝑥−𝑦≥2𝐶²₁𝛿_𝐵𝑥−𝑦2𝐶₁, (16) where 𝐶₁=(‖𝑥‖²+‖𝑦‖²)/2.

Proposition 11 11 (see [14]). Let 𝐵 be a real uniformly convex and uniformly smooth Banach space. Then 𝐽𝑥−𝐽𝑦≤8𝑀₂ℎ_𝐵4𝑥−𝑦𝑀₂,∀𝑥,𝑦∈𝐵, (17) where 𝑀₂=(‖𝑥‖²+‖𝑦‖²)/2 and ℎ_𝐵(𝜏)=𝜌_𝐵(𝜏)/𝜏.

Proposition 12. Let 𝐵 be a reflexive, smooth, and strictly convex Banach space with the dual space 𝐵^∗, let 𝐵 have the property (h), and let 𝐾 be a nonempty closed convex subset of 𝐵. For any 𝑥^∗₁,𝑥^∗₂∈𝐵^∗, let 𝑥₁=𝜋^𝑓_𝐾𝑥^∗₁ and 𝑥₂=𝜋^𝑓_𝐾𝑥^∗₂. Then 𝑥^∗₁−𝑥^∗₂,𝑥₁−𝑥₂≥2𝐶²₁𝛿_𝐵𝑥₁−𝑥₂2𝐶₁, (18) where 𝐶₁=(‖𝑥₁‖²+‖𝑥₂‖²)/2.

Proof. According to Proposition 10, 𝐽𝑥₁−𝐽𝑥₂,𝑥₁−𝑥₂≥2𝐶²₁𝛿_𝐵𝑥₁−𝑥₂2𝐶₁, (19) where 𝐶₁=(‖𝑥₁‖²+‖𝑥₂‖²)/2. Since 𝑥₁=𝜋^𝑓_𝐾𝑥^∗₁ and 𝑥₂=𝜋^𝑓_𝐾𝑥^∗₂, Theorem 7 yields 𝐽𝑥₁−𝑥^∗₁,𝑥₂−𝑥₁+𝜌𝑓𝑥₂−𝜌𝑓𝑥₁≥0,𝐽𝑥₂−𝑥^∗₂,𝑥₁−𝑥₂+𝜌𝑓𝑥₁−𝜌𝑓𝑥₂≥0. (20) It follows from (20) that 𝐽𝑥₁−𝑥^∗₁,𝑥₂−𝑥₁+𝐽𝑥₂−𝑥^∗₂,𝑥₁−𝑥₂≥0 (21) and so 𝑥^∗₁−𝑥^∗₂,𝑥₁−𝑥₂≥𝐽𝑥₁−𝐽𝑥₂,𝑥₁−𝑥₂. (22) By (19), we have 𝑥^∗₁−𝑥^∗₂,𝑥₁−𝑥₂≥2𝐶²₁𝛿_𝐵𝑥₁−𝑥₂2𝐶₁, (23) where 𝐶₁=(‖𝑥₁‖²+‖𝑥₂‖²)/2. This completes the proof.

Theorem 13. Let 𝐵 be a reflexive, smooth, and 2-uniformly convex Banach space with the dual space 𝐵^∗ and 𝛿(𝜖)≥𝑘_𝐵𝜖²,∀𝜖∈(0,2]for some 𝑘_𝐵>0. Then there exist a constant 𝐾_𝐵=(2/𝑘_𝐵) such that 𝜋^𝑓_𝐾𝑥^∗₁−𝜋^𝑓_𝐾𝑥^∗₂≤𝐾_𝐵𝑥^∗₁−𝑥^∗₂,∀𝑥^∗₁,𝑥^∗₂∈𝐵^∗. (24)

Proof. From Proposition 12, it follows that 𝑥^∗₁−𝑥^∗₂,𝜋^𝑓_𝐾𝑥^∗₁−𝜋^𝑓_𝐾𝑥^∗₂≥2𝐶²₁𝛿𝜋^𝑓_𝐾𝑥^∗₁−𝜋^𝑓_𝐾𝑥^∗₂2𝐶₁, (25) where 𝐶₁=(‖𝜋^𝑓_𝐾𝑥^∗₁‖²+‖𝜋^𝑓_𝐾𝑥^∗₂‖²)/2. Since 𝛿_𝐵(𝜖)≥𝑘_𝐵𝜖², (25) implies that 𝑥^∗₁−𝑥^∗₂,𝜋^𝑓_𝐾𝑥^∗₁−𝜋^𝑓_𝐾𝑥^∗₂≥2𝐶²₁𝛿_𝐵𝜋^𝑓_𝐾𝑥^∗₁−𝜋^𝑓_𝐾𝑥^∗₂2𝐶₁≥𝑘_𝐵2𝜋^𝑓_𝐾𝑥^∗₁−𝜋^𝑓_𝐾𝑥^∗₂², (26) and so 𝜋^𝑓_𝐾𝑥^∗₁−𝜋^𝑓_𝐾𝑥^∗₂≤2𝑘_𝐵𝑥^∗₁−𝑥^∗₂=𝐾_𝐵𝑥^∗₁−𝑥^∗₂. (27) This completes the proof.

Theorem 14. Let 𝐵 be a real 2-uniformly smooth and uniformly convex Banach space with the dual space 𝐵^∗ and 𝜌_𝐵(𝑡)≤𝑐_𝐵𝑡² for some 𝑐_𝐵>0. Then there exist a constant 𝐶=32𝑐_𝐵>0 such that 𝐽𝑥−𝐽𝑦≤𝐶𝑥−𝑦,∀𝑥,𝑦∈𝐵. (28)

Proof. From Proposition 11, it follows that 𝐽𝑥−𝐽𝑦≤8𝑀₂ℎ_𝐵4𝑥−𝑦𝑀₂,𝑥,𝑦∈𝐵, (29) where 𝑀₂=(‖𝑥‖²+‖𝑦‖²)/2 and ℎ_𝐵(𝜏)=𝜌_𝐵(𝜏)/𝜏. Since 𝜌_𝐵(𝑡)≤𝑐_𝐵𝑡², 𝐽𝑥−𝐽𝑦≤8𝑀₂ℎ_𝐵4𝑥−𝑦𝑀₂≤32𝑐_𝐵𝑥−𝑦=𝐶𝑥−𝑦. (30) This completes the proof.

3. Main Results

From Theorem 7, we know that the following theorem holds.

Theorem 15. Let 𝐵 be a 2-uniformly smooth and 2-uniformly convex Banach space. (𝑥^∗,𝑦^∗) is a solution of the 𝑆𝑁𝑀𝑉𝐼𝑃 (1) if and only if (𝑥^∗,𝑦^∗)∈𝐾×𝐾 satisfying 𝑥^∗=𝜋^𝑓_𝐾𝐽𝑦^∗−𝜌₁𝐴₁𝑦^∗+𝜌₁𝑆₁𝑦^∗,𝑦^∗=𝜋^𝑓_𝐾𝐽𝑥^∗−𝜂₁𝐴₂𝑥^∗+𝜂₁𝑆₂𝑥^∗. (31)

In the sequel, we shall show the existence and uniqueness of solution for the problems (1), (2), and (3), respectively. Next, we construct some new iterative algorithms for the problems (1), (2), and (3). We also give the convergence analysis of the iterative sequences generated by the algorithms.

Theorem 16. Let 𝐵 be a 2-uniformly smooth and 2-uniformly convex Banach space with 𝜌_𝐵(𝑡)≤𝑐_𝐵𝑡² for some 𝑐_𝐵>0 and 𝛿(𝜖)≥𝑘_𝐵𝜖²,∀𝜖∈(0,2] for some 𝑘_𝐵>0. Let 𝐴₁,𝐴₂,𝑆₁,𝑆₂∶𝐾→𝐵^∗ be single-valued mappings. Let 𝐽∶𝐵→𝐵^∗ and 𝐽^∗∶𝐵^∗→𝐵 be the normalized duality mapping, and let 𝑓∶𝐾⊂𝐵→𝑅∪{+∞} be a proper lower semicontinuous mapping. Suppose that the following conditions are satisfied: (i)for each 𝑖∈{1,2}, 𝑆_𝑖 is 𝐽-strongly monotone with constants 𝑘_𝑖 and 𝑠_𝑖-Lipschitz continuous;(ii)for each 𝑖∈{1,2}, 𝐴_𝑖 is 𝑙_𝑖-Lipschitz continuous; and that there exist constants 𝜌₁,𝜂₁>0 satisfying 𝐾_𝐵=2𝑘_𝐵,𝐶=32𝑐_𝐵,Θ₁=𝐾_𝐵𝐶²−2𝜂₁𝑘₂+𝐶_𝐵𝜂²₁𝑠²₂+𝜂₁𝑙₂Θ<1,₂=𝐾_𝐵𝐶²−2𝜌₁𝑘₁+𝜌²₁𝑠²₁𝐶_𝐵+𝜌₁𝑙₁<1. (32) Then SNMVIP (1) has a unique solution (𝑥^∗,𝑦^∗).

Proof. First, we prove the existence of the solution. Define a mapping 𝐹∶𝐾→𝐾 as follows: 𝐹(𝑥)=𝜋^𝑓_𝐾𝐽𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝜌₁𝐴₁+𝑆₁×𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥). (33) For any 𝑥,𝑦∈𝐾, by Theorem 13, we know that there exists 𝐾_𝐵=(2/𝑘_𝐵)>0 such that 𝐹(𝑥)𝑦=𝜋−𝐹_𝑓𝐾𝐽𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝜌₁𝐴₁+𝑆₁𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝜋^𝑓_𝐾𝐽𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦−𝜌₁𝐴₁+𝑆₁×𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦.≤𝐾_𝐵𝐽𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥𝜋)−𝐽^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦−𝜌₁𝑆₁𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝑆₁𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦+𝜌₁𝐴₁𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝐴₁𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦. (34) Theorem 14 implies that there exists 𝐶=32𝑐_𝐵>0 such that 𝐽(𝑥)−𝐽𝑦≤𝐶𝑥−𝑦,∀𝑥,𝑦∈𝐾. (35) From condition (i) and Lemma 8, it follows that 𝐽𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝐽𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦−𝜌₁𝑆₁𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝑆₁𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦²=𝐽𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝐽𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦²+𝜌²₁𝐶_𝐵𝑆₁𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝑆₁𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦²−2𝜌₁𝑆₁𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝑆₁𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦,𝐽^∗𝐽𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝐽𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦≤𝐶²−2𝜌₁𝑘₁+𝜌²₁𝑠²₁𝐶_𝐵×𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦². (36) By condition (ii) and Theorem 13, we know that 𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦.≤𝐾_𝐵𝐽(𝑥)−𝐽𝑦−𝜂₁𝑆₂(𝑥)−𝑆₂𝑦+𝜂₁𝐴₂(𝑥)−𝐴₂𝑦≤𝐾_𝐵𝐽(𝑥)−𝐽𝑦−𝜂₁𝑆₂(𝑥)−𝑆₂𝑦+𝜂₁𝑙₂𝑥−𝑦. (37) By (35), condition (i), and Lemma 8, we have 𝐽(𝑥)−𝐽𝑦−𝜂₁𝑆₂(𝑥)−𝑆₂𝑦²≤𝐽(𝑥)−𝐽𝑦²−2𝜂₁×𝑆₂(𝑥)−𝑆₂𝑦,𝐽^∗𝐽(𝑥)−𝐽𝑦+𝐶_𝐵𝜂²₁𝑆₂(𝑥)−𝑆₂𝑦²≤𝐶²𝑥−𝑦²−2𝜂₁𝑘₂𝑥−𝑦²₁+𝐶_𝐵𝜂²₁𝑠²₂𝑥−𝑦²=𝐶²−2𝜂₁𝑘₂+𝐶_𝐵𝜂²₁𝑠²₂𝑥−𝑦². (38) Thus 𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦.≤𝐾_𝐵𝐽(𝑥)−𝐽𝑦−𝜂₁𝑆₂(𝑥)−𝑆₂𝑦+𝜂₁𝑙₂𝑥−𝑦≤𝐾_𝐵𝐶²−2𝜂₁𝑘₂+𝐶_𝐵𝜂²₁𝑠²₂+𝜂₁𝑙₂𝑥−𝑦. (39) Condition (ii) and (39) imply that 𝐴₁𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝐴₁𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦≤𝑙₁𝜋^𝑓_𝐾𝐽(𝑥)−𝜂₁𝐴₂(𝑥)+𝑆₂(𝑥)−𝜋^𝑓_𝐾𝐽𝑦−𝜂₁𝐴₂𝑦+𝑆₂𝑦≤𝑙₁𝐾_𝐵𝐶²−2𝜂₁𝑘₂+𝐶_𝐵𝜂²₁𝑠²₂+𝜂₁𝑙₂𝑥−𝑦. (40) From (34)–(40), we have 𝐹(𝑥)−𝐹𝑦≤𝐾²_𝐵𝐶²−2𝜂₁𝑘₂+𝐶_𝐵𝜂²₁𝑠²₂+𝜂₁𝑙₂×𝐶²−2𝜌₁𝑘₁+𝜌²₁𝑠²₁𝐶_𝐵+𝜌₁𝑙₁𝑥−𝑦=Θ₁Θ₂𝑥−𝑦. (41) It follows from (32) that 0<Θ₁<1 and 0<Θ₂<1. Thus, (41) implies that 𝐹 is a contractive mapping and so there exists a point 𝑥^∗∈𝐾 such that 𝑥^∗=𝐹(𝑥^∗). Let 𝑦^∗=𝜋^𝑓_𝐾𝐽𝑥^∗−𝜂₁𝐴₂𝑥^∗+𝜂₁𝑆₂𝑥^∗. (42) From the definition of 𝐹, we have 𝑥^∗=𝜋^𝑓_𝐾𝐽𝑦^∗−𝜌₁𝐴₁𝑦^∗+𝜌₁𝑆₁𝑦^∗,𝑦^∗=𝜋^𝑓_𝐾𝐽𝑥^∗−𝜂₁𝐴₂𝑥^∗+𝜂₁𝑆₂𝑥^∗. (43) By Theorem 15, we know that (𝑥^∗,𝑦^∗) is a solution of SNMVIP (1).
Next, we show the uniqueness of the solution. Let (𝑢^∗𝑣^∗) be another solution of SNMVIP (1). It follows from Theorem 15 that 𝑢^∗=𝜋^𝑓_𝐾𝐽𝑣^∗−𝜌₁𝐴₁𝑣^∗+𝜌₁𝑆₁𝑣^∗,𝑣^∗=𝜋^𝑓_𝐾𝐽𝑢^∗−𝜂₁𝐴₂𝑢^∗+𝜂₁𝑆₂𝑢^∗. (44) As the proof of (41), we have 𝑥^∗−𝑢^∗≤Θ₁Θ₂𝑥^∗−𝑢^∗. (45) Since 0<Θ₁<1 and 0<Θ₂<1, it follows that 𝑢^∗=𝑥^∗ and so 𝑣^∗=𝑦^∗. This completes the proof.

It is clear that if 𝐵 is a Hilbert space, then we can take 𝐾_𝐵=𝐶=𝐶_𝐵=1 in Theorem 16. According to Theorem 16, it is easy to get the following corollaries.

Corollary 17. Let 𝐻 be a real Hilbert space, let 𝐾⊂𝐻 be a nonempty closed convex set, and let 𝑆∶𝐾→𝐻 be strongly monotone with constants 𝑘 and 𝑠-Lipschitz continuous. There exist 𝜌₁,𝜂₁>0 satisfying 1−2𝜂₁𝑘+𝜂²₁𝑠²<1,1−2𝜌₁𝑘+𝜌²₁𝑠²<1. (46) Then the problem (2) has a unique solution (𝑥^∗,𝑦^∗).

Corollary 18. Let 𝐻 be a real Hilbert space, let 𝐾⊂𝐻 be a nonempty closed convex set, and let 𝑆∶𝐾→𝐻 be strongly monotone with constants 𝑘 and 𝑠-Lipschitz continuous. There exists 𝜂₁>0 satisfying 1−2𝜂₁𝑘+𝜂²₁𝑠²<1. (47) Then the problem (3) has a unique solution (𝑥^∗,𝑦^∗).

Algorithm 19. For any given 𝑥₀∈𝐾, define the Mann iterative sequences {𝑥_𝑛} and {𝑦_𝑛} with errors as follows: 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝜋^𝑓_𝐾×𝐽𝑦_𝑛−𝜌₁𝐴₁𝑦_𝑛+𝑆₁𝑦_𝑛+𝛼_𝑛𝑢_𝑛+𝑤_𝑛,𝑦_𝑛=𝜋^𝑓_𝐾𝐽𝑥_𝑛−𝜂₁𝐴₂𝑥_𝑛+𝑆₂𝑥_𝑛+𝑣_𝑛, (48) where {𝛼_𝑛} is a sequence in [0,1] and {𝑢_𝑛}, {𝑣_𝑛}, and {𝑤_𝑛} are three sequences in 𝐵 satisfying the following conditions: ^∞_𝑛=0𝛼_𝑛=+∞,lim^∞_{𝑛→∞𝑛=0}𝑤_𝑛<+∞,lim_𝑛→∞𝑢_𝑛=lim_𝑛→∞𝑣_𝑛=0. (49)

If 𝐵=𝐻 is a real Hilbert space, 𝑆₁=𝑆₂=𝑆, 𝐴₁=𝐴₂=0, and 𝑓=0, then Algorithm 19 reduces to the following iterative algorithm.

Algorithm 20 (see [2, Algorithm 2.1]). For any given 𝑥₀∈𝐾, define the iterative sequences {𝑥_𝑛} and {𝑦_𝑛} as follows: 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑃_𝐾𝑦_𝑛−𝜌₁𝑆𝑦_𝑛,𝑦_𝑛=𝑃_𝐾𝑥_𝑛−𝜂₁𝑆𝑥_𝑛, (50) where {𝛼_𝑛} is a sequence in [0,1] satisfying the following conditions: ^∞_𝑛=0𝛼_𝑛=+∞. (51)

If 𝐵=𝐻 is a real Hilbert space, 𝑆₁=𝑆₂=𝑆, 𝐴₁=𝐴₂=0, 𝜂₁=0, and 𝑓=0, then Algorithm 19 reduces to the following iterative algorithm.

Algorithm 21 21 (see [1, Algorithm 2.2]). For any given 𝑥₀∈𝐾, define the iterative sequences {𝑥_𝑛} as follows: 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑃_𝐾𝑥_𝑛−𝜂₁𝑆𝑥_𝑛, (52) where {𝛼_𝑛} is a sequence in [0,1] satisfying the following conditions: ^∞_𝑛=0𝛼_𝑛=+∞. (53)

Theorem 22. Let 𝐴₁, 𝐴₂, 𝑆₁, 𝑆₂, 𝐽, and 𝐽^∗ be the same as in Theorem 16, and let {𝑥_𝑛,𝑦_𝑛} be the iterative sequence generated by Algorithm 19. If the condition (32) holds, then (𝑥_𝑛,𝑦_𝑛) converges strongly to the unique solution (𝑥^∗;𝑦^∗) of SNMVIP (1).

Proof. By Theorem 16, we know that SNMVIP (1) has a unique solution (𝑥^∗;𝑦^∗). It follows from Theorem 15 that 𝑥^∗=𝜋^𝑓_𝐾𝐽𝑦^∗−𝜌₁𝐴₁𝑦^∗+𝜌₁𝑆₁𝑦^∗,𝑦^∗=𝜋^𝑓_𝐾𝐽𝑥^∗−𝜂₁𝐴₂𝑥^∗+𝜂₁𝑆₂𝑥^∗. (54) By Lipschitz property of 𝐴₁, Theorem 13, (48), and (54), we have 𝑥_𝑛+1−𝑥^∗=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝜋^𝑓_𝐾×𝐽𝑦_𝑛−𝜌₁𝐴₁𝑦_𝑛+𝑆₁𝑦_𝑛+𝛼_𝑛𝑢_𝑛+𝑤_𝑛−𝑥^∗≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝜋^𝑓_𝐾𝐽𝑦_𝑛−𝜌₁𝐴₁𝑦_𝑛+𝑆₁𝑦_𝑛−𝜋^𝑓_𝐾𝐽𝑦^∗−𝜌₁𝐴₁𝑦^∗+𝜌₁𝑆₁𝑦^∗+𝛼_𝑛𝑢_𝑛+𝑤_𝑛≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝐾_𝐵𝐽𝑦_𝑛−𝐽𝑦^∗−𝜌₁𝑆₁𝑦_𝑛−𝑆₁𝑦^∗−𝜌₁𝐴₁𝑦_𝑛−𝐴₁𝑦^∗+𝛼_𝑛𝑢_𝑛+𝑤_𝑛≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝐾_𝐵×𝐽𝑦_𝑛−𝐽𝑦^∗−𝜌₁𝑆₁𝑦_𝑛−𝑆₁𝑦^∗+𝐾_𝐵𝜌₁𝑙₁𝑦_𝑛−𝑦^∗+𝛼_𝑛𝑢_𝑛+𝑤_𝑛. (55) Since 𝑆₁ is 𝐽-strongly monotone with constants 𝑘₁ and 𝑠₁-Lipschitz continuous, by Theorem 14 and Lemma 8, we obtain 𝐽𝑦_𝑛−𝐽𝑦^∗−𝜌₁𝑆₁𝑦_𝑛−𝑆₁𝑦^∗≤𝐶²−2𝜌₁𝑘₁+𝐶_𝐵𝜌²₁𝑠²₁𝑦_𝑛−𝑦^∗. (56) Combining (55) and (56), we obtain 𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛Θ₂𝑦_𝑛−𝑦^∗+𝛼_𝑛𝑢_𝑛+𝑤_𝑛. (57)
From Lipschitz property of 𝐴₂, (48), and (54), it follows that 𝑦_𝑛−𝑦^∗=𝜋^𝑓_𝐾𝐽𝑥_𝑛−𝜂₁𝐴₂𝑥_𝑛+𝑆₂𝑥_𝑛+𝑣_𝑛−𝜋^𝑓_𝐾𝐽𝑥^∗−𝜂₁𝐴₂𝑥^∗+𝜂₁𝑆₂𝑥^∗≤𝐾_𝐵𝐽𝑥_𝑛−𝐽𝑥^∗−𝜂₁𝑆₂𝑥_𝑛−𝑆₂𝑥^∗−𝜂₁𝐴₂𝑥_𝑛−𝐴₂𝑥^∗+𝑣_𝑛≤𝐾_𝐵𝐽𝑥_𝑛−𝐽𝑥^∗−𝜂₁𝑆₂𝑥_𝑛−𝑆₂𝑥^∗+𝑣_𝑛+𝐾_𝐵𝜂₁𝑙₂𝑥_𝑛−𝑥^∗. (58) Since 𝑆₂ is 𝐽-strongly monotone with constants 𝑘₂ and 𝑠₂-Lipschitz continuous, Theorem 14 and Lemma 8 imply that 𝐽𝑥_𝑛−𝐽𝑥^∗−𝜂₁𝑆₂𝑥_𝑛−𝑆₂𝑥^∗≤𝐶²−2𝜂₁𝑘₂+𝐶_𝐵𝜂²₁𝑠²₂𝑥_𝑛−𝑥^∗. (59) It follows from (57)–(59) that 𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛Θ₂Θ₁𝑥_𝑛−𝑥^∗+𝛼_𝑛Θ₂𝑣_𝑛+𝛼_𝑛𝑢_𝑛+𝑤_𝑛=1−𝛼_𝑛1−Θ₂Θ₁𝑥_𝑛−𝑥^∗+𝑤_𝑛+𝛼_𝑛1−Θ₂Θ₁11−Θ₂Θ₁Θ₂𝑣_𝑛+𝑢_𝑛. (60) Taking 𝑡_𝑛=𝛼_𝑛(1−Θ₂Θ₁), 𝑎_𝑛=‖𝑥_𝑛+1−𝑥^∗‖, 𝑏_𝑛=1/(1−Θ₂Θ₁)(Θ₂‖𝑣_𝑛‖+‖𝑢_𝑛‖), and 𝑐_𝑛=‖𝑤_𝑛‖, Lemma 9 implies that 𝑎_𝑛→0 and so 𝑥_𝑛→𝑥^∗ as 𝑛→∞. Since 𝑥_𝑛→𝑥^∗, by (58) and (59), we know that 𝑦_𝑛→𝑦^∗ as 𝑛→∞. This completes the proof.

Corollary 23 (see [2, Theorem 2.1]). Let 𝐻, 𝐾, and 𝑆 be the same as in Corollary 17, and let {𝑥_𝑛} and {𝑦_𝑛} be the iterative sequence generated by Algorithm 20. If there exist constants 𝜌₁,𝜂₁>0 satisfying 1−2𝜂₁𝑘+𝜂²₁𝑠²<1,1−2𝜌₁𝑘+𝜌²₁𝑠²<1, (61) then (𝑥_𝑛,𝑦_𝑛) converges strongly to the unique solution (𝑥^∗,𝑦^∗) of the problem (2).

Corollary 24 (see [1, Theorem 3.3]). Let 𝐻, 𝐾, and 𝑆 be the same as in Corollary 18, and let {𝑥_𝑛} be the iterative sequence generated by Algorithm 21. If there exists 𝜂₁>0 satisfying 1−2𝜂₁𝑘+𝜂²₁𝑠²<1, (62) then 𝑥_𝑛 converges strongly to the unique solution 𝑥^∗ of the problem (3).

Conflict of Interests

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

Acknowledgments

The second author was partially supported by the Guangxi Natural Science Foundation (2013GXNSFBA019015), the Scientific Research Foundation of Guangxi University for Nationalities (2012QD015), and the Open fund of Guangxi key laboratory of hybrid computation and IC design analysis (2013HCIC06). The third author was partially supported by the Key Program of the Fundamental Research Funds for the Central Universities (A0920502051202-111).