Existence and Algorithm for Solving the System of Mixed Variational Inequalities in Banach Spaces

Abstract

The purpose of this paper is to study the existence and convergence analysis of the solutions of the system of mixed variational inequalities in Banach spaces by using the generalized f projection operator. The results presented in this paper improve and extend important recent results of Zhang et al. (2011) and Wu and Huang (2007) and some recent results.

1. Introduction

Let 𝐸 be a real Banach space with norm ‖⋅‖, let 𝐶 be a nonempty closed and convex subset of 𝐸, and let 𝐸^∗ denote the dual of 𝐸. Let ⟨⋅,⋅⟩ denote the duality pairing of 𝐸^∗ and 𝐸. If 𝐸 is a Hilbert space, ⟨⋅,⋅⟩ denotes an inner product on 𝐸. It is well known that the metric projection operator 𝑃_𝐶∶𝐸→𝐶 plays an important role in nonlinear functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, and complementarity problems, and so forth (see, e.g., [1, 2] and the references therein). In 1993, Alber [3] introduced and studied the generalized projections 𝜋_𝐶∶𝐸^∗→𝐶 and Π_𝐸∶𝐸→𝐶 from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [1] presented some applications of the generalized projections to approximately solving variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li [2] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solving the variational inequality in Banach spaces. Later, Wu and Huang [4] introduced a new generalized 𝑓-projection operator in Banach spaces which extended the definition of the generalized projection operators introduced by Abler [3] and proved some properties of the generalized 𝑓-projection operator. As an application, they studied the existence of solution for a class of variational inequalities in Banach spaces. In 2007, Wu and Huang [5] proved some properties of the generalized 𝑓-projection operator and proposed iterative method of approximating solutions for a class of generalized variational inequalities in Banach spaces. In 2009, Fan et al. [6] presented some basic results for the generalized 𝑓-projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces. In 2011, Zhang et al. [7] introduced and considered the system of mixed variational inequalities in Banach spaces. Using the generalized 𝑓-projection operator technique, they introduced some iterative methods for solving the system of mixed variational inequalities and proved the convergence of the proposed iterative methods under suitable conditions in Banach spaces. Recently, many authors studied methods for solving the system of generalized (mixed) variational inequalities and the system of nonlinear variational inequalities problems (see, e.g., [8–17] and references therein).

We first introduce and consider the system of mixed variational inequalities (SMVI) which is to find 𝑥̂,𝑦,̂𝑧̂∈𝐶 such that⟨⟨𝛿𝛿₁₂𝑇₁𝑧̂+𝐽𝑥̂−𝐽𝑧,̂𝑦−𝑥̂⟩+𝑓₁(𝑦)−𝑓₁(𝑥̂)≥0,∀𝑦∈𝐶,⟨𝛿₃𝑇₂𝑥̂+𝐽𝑦̂−𝐽𝑥̂,𝑦−𝑦⟩̂+𝑓₂(𝑦)−𝑓₂(𝑦)̂≥0,∀𝑦∈𝐶,𝑇₃𝑦̂+𝐽𝑧̂−𝐽𝑦,̂𝑦−𝑧⟩̂+𝑓₃(𝑦)−𝑓₃(𝑧)̂≥0,∀𝑦∈𝐶,(1.1) where 𝛿_𝑗>0,𝑇_𝑗∶𝐶→𝐸^∗,𝑓_𝑗∶𝐶→ℝ∪{+∞} for 𝑗=1,2,3 are mappings and 𝐽 is the normalized duality mapping from 𝐸 to 𝐸^∗.

As special case of the problem (1.1), we have the following.

If 𝑓_𝑗(𝑥)=0 for 𝑗=1,2,3, for all 𝑥∈𝐶, (1.1) is equivalent to find 𝑥̂, 𝑦̂ and 𝑧̂∈𝐶 such that⟨⟨𝛿𝛿₂𝑇₁𝑧̂+𝐽𝑥̂−𝐽z,̂𝑦−𝑥̂⟩≥0,∀𝑦∈𝐶,⟨𝛿₃𝑇₂𝑥̂+𝐽𝑦̂−𝐽𝑥̂,𝑦−𝑦⟩̂≥0,∀𝑦∈𝐶,𝑇₃𝑦̂+𝐽𝑧̂−𝐽𝑦,̂𝑦−𝑧⟩̂≥0,∀𝑦∈𝐶.(1.2) The problem (1.2) is called the system of variational inequalities we denote by (SVI).

If 𝑇₂=𝑇₃,𝑓₂(𝑥)=𝑓₃(𝑥), for all 𝑥∈𝐶 and 𝑦̂=𝑧̂, then (1.1) is reduced to find 𝑥̂,𝑦̂∈𝐶 such that⟨⟨𝛿𝛿₁₂𝑇₁𝑦̂+𝐽𝑥̂−𝐽𝑦,̂𝑦−𝑥̂⟩+𝑓₁(𝑦)−𝑓₁(𝑥̂)≥0,∀𝑦∈𝐶,𝑇₂𝑥̂+𝐽𝑦̂−𝐽𝑥̂,𝑦−𝑦⟩̂+𝑓₂(𝑦)−𝑓₂(𝑦)̂≥0,∀𝑦∈𝐶,(1.3) which is studied by Zhang et al. [7].

If 𝑇=𝑇₁=𝑇₂=𝑇₃,𝑓₁(𝑥)=𝑓₂(𝑥)=𝑓₃(𝑥), for all 𝑥∈𝐶 and 𝑥̂=𝑦̂=𝑧̂, (1.1) is reduced to find 𝑥̂ such that⟨𝑇𝑥̂,𝑦−𝑥̂⟩+𝑓₁(𝑦)−𝑓₁(𝑥̂)≥0,∀𝑦∈𝐶.(1.4) This iterative method is studied by Wu and Huang [5].

If 𝑓₁(𝑥)=0, for all 𝑥∈𝐶, (1.4) is reduced to find 𝑥̂ such that⟨𝑇𝑥̂,𝑦−𝑥̂⟩≥0,∀𝑦∈𝐶,(1.5) which is studied by Alber [1, 18], Li [2], and Fan [19]. If 𝐸=𝐻 is a Hilbert space, (1.5) holds which is known as the classical variational inequality introduced and studied by Stampacchia [20].

If 𝐸=𝐻 is a Hilbert space, then (1.1) is reduced to find 𝑥̂,𝑦,̂𝑧̂∈𝐶 such that⟨⟨𝛿𝛿₁₂𝑇₁𝑧̂+𝑥̂−𝑧,̂𝑦−𝑥̂⟩+𝑓₁(𝑦)−𝑓₁(𝑥̂)≥0,∀𝑦∈𝐶,⟨𝛿₃𝑇₂𝑥̂+𝑦̂−𝑥̂,𝑦−𝑦⟩̂+𝑓₂(𝑦)−𝑓₂(𝑦)̂≥0,∀𝑦∈𝐶,𝑇₃𝑦̂+𝑧̂−𝑦,̂𝑦−𝑧⟩̂+𝑓₃(𝑦)−𝑓₃(𝑧)̂≥0,∀𝑦∈𝐶.(1.6) If 𝑓_𝑗(𝑥)=0 for 𝑗=1,2,3, for all 𝑥∈𝐶, (1.6) reduces to the following (SVI):⟨⟨𝛿𝛿₁₂𝑇₁𝑧̂+𝑥̂−𝑧,̂𝑦−𝑥̂⟩≥0,∀𝑦∈𝐶,⟨𝛿₃𝑇₂𝑥̂+𝑦̂−𝑥̂,𝑦−𝑦⟩̂≥0,∀𝑦∈𝐶,𝑇₃𝑦̂+𝑧̂−𝑦,̂𝑦−𝑧⟩̂≥0,∀𝑦∈𝐶.(1.7) The purpose of this paper is to study the existence and convergence analysis of solutions of the system of mixed variational inequalities in Banach spaces by using the generalized 𝑓-projection operator. The results presented in this paper improve and extend important recent results in the literature.

2. Preliminaries

A Banach space 𝐸 is said to be strictly convex if ‖(𝑥+𝑦)/2‖<1 for all 𝑥,𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. Let 𝑈={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸. Then, a Banach space 𝐸 is said to be smooth if the limit lim_𝑡→0(‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡 exists for each 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the limit exists uniformly in 𝑥,𝑦∈𝑈. Let 𝐸 be a Banach space. The modulus of smoothness of 𝐸 is the function 𝜌_𝐸∶[0,∞)→[0,∞) defined by 𝜌_𝐸(𝑡)=sup{((‖𝑥+𝑦‖+‖𝑥−𝑦‖)/2)−1∶‖𝑥‖=1,‖𝑦‖≤𝑡}. The modulus of convexity of 𝐸 is the function 𝜂_𝐸∶[0,2]→[0,1] defined by 𝜂_𝐸(𝜀)=inf{1−‖(𝑥+𝑦)/2‖∶𝑥,𝑦∈𝐸,‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀}. The normalized duality mapping 𝐽∶𝐸→2^𝐸∗ is defined by 𝐽(𝑥)={𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖²,‖𝑥^∗‖=‖𝑥‖}. If 𝐸 is a Hilbert space, then 𝐽=𝐼, where 𝐼 is the identity mapping.

If 𝐸 is a reflexive smooth and strictly convex Banach space and 𝐽^∗∶𝐸^∗→2^𝐸 is the normalized duality mapping on 𝐸^∗, then 𝐽⁻¹=𝐽^∗, 𝐽𝐽^∗=𝐼_𝐸^∗ and 𝐽^∗𝐽=𝐼_𝐸, where 𝐼_𝐸 and 𝐼^∗𝐸 are the identity mappings on 𝐸 and 𝐸^∗. If 𝐸 is a uniformly smooth and uniformly convex Banach space, then 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸 and 𝐽^∗ is also uniformly norm-to-norm continuous on bounded subsets of 𝐸^∗.

Let 𝐸 and 𝐹 be Banach spaces, 𝑇∶𝐷(𝑇)⊂𝐸→𝐹, the operator 𝑇 is said to be compact if it is continuous and maps the bounded subsets of 𝐷(𝑇) onto the relatively compact subsets of 𝐹; the operator 𝑇 is said to be weak to norm continuous if it is continuous from the weak topology of 𝐸 to the strong topology of 𝐹.

We also need the following lemmas for the proof of our main results.

Lemma 2.1 (Xu [21]). Let 𝑞>1 and 𝑟>0 be two fixed real numbers. Let 𝐸 be a 𝑞-uniformly convex Banach space if and only if there exists a continuous strictly increasing and convex function 𝑔∶[0,+∞)→[0,+∞), 𝑔(0)=0, such that ‖𝜆𝑥+(1−𝜆)𝑦‖^𝑞≤𝜆‖𝑥‖^𝑞+(1−𝜆)‖𝑦‖^𝑞−𝜍_𝑞(𝜆)𝑔(‖𝑥−𝑦‖)(2.1) for all 𝑥,𝑦∈𝐵_𝑟={𝑥∈𝐸∶‖𝑥‖≤𝑟} and 𝜆∈[0,1], where 𝜍_𝑞(𝜆)=𝜆(1−𝜆)^𝑞+𝜆^𝑞(1−𝜆).

For case 𝑞=2, we have‖𝜆𝑥+(1−𝜆)𝑦‖^𝑞≤𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)𝑔(‖𝑥−𝑦‖).(2.2)

Lemma 2.2 (Chang [22]). Let 𝐸 be a uniformly convex and uniformly smooth Banach space. The following holds: ‖𝜙+Φ‖²≤‖𝜙‖²+2⟨Φ,𝐽^∗(𝜙+Φ)⟩,∀𝜙,Φ∈𝐸^∗.(2.3)

Next we recall the concept of the generalized 𝑓-projection operator. Let 𝐺∶𝐸^∗×𝐶→ℝ∪{+∞} be a functional defined as follows:𝐺(𝜉,𝑥)=‖𝜉‖²−2⟨𝜉,𝑥⟩+‖𝑥‖²+2𝜌𝑓(𝑥),(2.4) where 𝜉∈𝐸^∗,𝜌 is positive number and 𝑓∶𝐶→ℝ∪{+∞} is proper, convex, and lower semicontinuous. From definitions of 𝐺 and 𝑓, it is easy to see the following properties:(1)(‖𝜉‖−‖𝑥‖)²+2𝜌𝑓(𝑥)≤𝐺(𝜉,𝑥)≤(‖𝜉‖+‖𝑥‖)²+2𝜌𝑓(𝑥);(2)𝐺(𝜉,𝑥) is convex and continuous with respect to 𝑥 when 𝜉 is fixed;(3)𝐺(𝜉,𝑥) is convex and lower semicontinuous with respect to 𝜉 when 𝑥 is fixed.

Definition 2.3. Let 𝐸 be a real Banach space with its dual 𝐸^∗. Let 𝐶 be a nonempty closed convex subset of 𝐸. It is said that Π^𝑓𝐶∶𝐸^∗→2^𝐶 is the generalized 𝑓-projection operator if ^𝑓_𝐶𝜉=𝑢∈𝐶∶𝐺(𝜉,𝑢)=inf_𝑦∈𝐶𝐺(𝜉,𝑦),∀𝜉∈𝐸^∗.(2.5)

In this paper, we fixed 𝜌=1, we have𝐺(𝜉,𝑥)=‖𝜉‖²−2⟨𝜉,𝑥⟩+‖𝑥‖²+2𝑓(𝑥).(2.6)

For the generalized 𝑓-projection operator, Wu and Hung [5] proved the following basic properties.

Lemma 2.4 (Wu and Hung [4]). Let 𝐸 be a reflexive Banach space with its dual 𝐸^∗ and 𝐶 is a nonempty closed convex subset of 𝐸. The following statements hold:(1)Π^𝑓𝐶𝜉 is nonempty closed convex subset of 𝐶 for all 𝜉∈𝐸^∗;(2)if 𝐸 is smooth, then for all 𝜉∈𝐸^∗, 𝑥∈Π^𝑓𝐶𝜉 if and only if ⟨𝜉−𝐽𝑥,𝑥−𝑦⟩+𝜌𝑓(𝑦)−𝜌𝑓(𝑥)≥0,∀𝑦∈𝐶;(2.7)(3)if 𝐸 is smooth, then for any 𝜉∈𝐸^∗, Π^𝑓𝐶𝜉=(𝐽+𝜌𝜕𝑓)⁻¹𝜉, where 𝜕𝑓 is the subdifferential of the proper convex and lower semicontinuous functional 𝑓.

Lemma 2.5 (Wu and Hung [4]). If 𝑓(𝑥)≥0 for all 𝑥∈𝐶, then for any 𝜌>0, 𝐺(𝐽𝑥,𝑦)≤𝐺(𝜉,𝑦)+2𝜌𝑓(𝑦),∀𝜉∈𝐸^∗,𝑦∈𝐶,𝑥∈^𝑓_𝐶𝜉.(2.8)

Lemma 2.6 (Fan et al. [6]). Let 𝐸 be a reflexive strictly convex Banach space with its dual 𝐸^∗ and 𝐶 is a nonempty closed convex subset of 𝐸. If 𝑓∶𝐶→ℝ∪{+∞} is proper, convex, and lower semicontinuous, then(1)Π^𝑓𝐶∶𝐸^∗→𝐶 is single valued and norm to weak continuous;(2)if 𝐸 has the property (h), that is, for any sequence {𝑥_𝑛}⊂𝐸,𝑥_𝑛⇀𝑥∈𝐸 and ‖𝑥_𝑛‖→‖𝑥‖, imply that 𝑥_𝑛→𝑥, then Π^𝑓𝐶∶𝐸^∗→𝐶 is continuous.

Defined the functional 𝐺₂∶𝐸×𝐶→ℝ∪{+∞} by𝐺₂(𝑥,𝑦)=𝐺(𝐽𝑥,𝑦),∀𝑥∈𝐸,𝑦∈𝐶.(2.9)

3. Generalized Projection Algorithms

Proposition 3.1. Let 𝐶 be a nonempty closed and convex subset of a reflexive strictly convex and smooth Banach space 𝐸. If 𝑓_𝑗∶𝐶→ℝ∪{+∞} for 𝑗=1,2,3 is proper, convex, and lower semicontinuous, then (𝑥̂,ŷ,𝑧)̂ is a solution of (SMVI) equivalent to finding 𝑥̂,𝑦,̂𝑧̂ such that 𝑥̂=^𝑓1_𝐶𝐽𝑧̂−𝛿₁𝑦̂=_𝑓2𝑇₁𝑧̂,_𝐶𝐽𝑥̂−𝛿₂𝑧̂=_𝑓3𝑇₁𝑥̂,_𝐶𝐽𝑦̂−𝛿₃𝑇₁𝑦̂.(3.1)

Proof. From Lemma 2.4 (2) and 𝐸 is a reflexive strictly convex and smooth Banach space, we known that 𝐽 is single valued and Π^𝑓𝑗_𝐶 for 𝑗=1,2,3 is well defined and single valued. So, we can conclude that Proposition 3.1 holds.

For solving the system of mixed variational inequality (1.1), we defined some projection algorithms as follow.

Algorithm 3.2. For an initial point 𝑥₀,𝑧₀∈𝐶, define the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} as follows:𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑓¹_𝐶𝐽𝑧_𝑛𝑦_𝑛+1−𝛿₁𝑇₁𝑧_𝑛,=^𝑓2_𝐶𝐽𝑥_𝑛+1𝑧_𝑛+1−𝛿₂𝑇₂𝑥_𝑛+1,=^𝑓3_𝐶𝐽𝑦_𝑛+1−𝛿₃𝑇₃𝑦_𝑛+1,(3.2) where 0<𝑎≤𝛼_𝑛≤𝑏<1.

If 𝑓_𝑗(𝑥)=0,𝑗=1,2,3, for all 𝑥∈𝐶, then Algorithm 3.2 reduces to the following iterative method for solving the system of variational inequalities (1.2).

Algorithm 3.3. For an initial point 𝑥₀,𝑧₀∈𝐶, define the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} as follows:𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛_𝐶𝐽𝑧_𝑛𝑦_𝑛+1−𝛿₁𝑇₁𝑧_𝑛,𝑧_𝑛+1=_𝐶𝐽𝑥_𝑛+1−𝛿₂𝑇₂𝑥_𝑛+1,=_𝐶𝐽𝑦_𝑛+1−𝛿₃𝑇₃𝑦_𝑛+1,(3.3) where 0<𝑎≤𝛼_𝑛≤𝑏<1.

For solving the problem (1.6), we defined the algorithm as follows:

If 𝐸=𝐻 is a Hilbert space, then Algorithm 3.2 reduces to the following.

Algorithm 3.4. For an initial point 𝑥₀,𝑧₀∈𝐶, define the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} as follows:𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑓¹_𝐶𝐽𝑧_𝑛𝑦_𝑛+1−𝛿₁𝑇₁𝑧_𝑛,=^𝑓2_𝐶𝐽𝑥_𝑛+1𝑧_𝑛+1−𝛿₂𝑇₂𝑥_𝑛+1,=^𝑓3_𝐶𝐽𝑦_𝑛+1−𝛿₃𝑇₃𝑦_𝑛+1,(3.4) where 0<𝑎≤𝛼_𝑛≤𝑏<1.

If 𝑓_𝑗(𝑥)=0,𝑗=1,2,3, for all 𝑥∈𝐶, then Algorithm 3.4 reduces to the following iterative method for solving the problem (1.7) as follows.

Algorithm 3.5. For an initial point 𝑥₀,𝑧₀∈𝐶, define the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} as follows:𝑥𝑦_𝑛+1𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑃_𝐶𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛,𝑧_𝑛+1=𝑃_𝐶𝐽𝑥_𝑛+1−𝛿₂𝑇₂𝑥_𝑛+1,=𝑃_𝐶𝐽𝑦_𝑛+1−𝛿₃𝑇₃𝑦_𝑛+1,(3.5) where 0<𝑎≤𝛼_𝑛≤𝑏<1.

4. Existence and Convergence Analysis

Theorem 4.1. Let 𝐶 be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space 𝐸 with dual space 𝐸^∗. If the mapping 𝑇_𝑗∶𝐶→𝐸^∗ and 𝑓_𝑗∶𝐶→ℝ∪{+∞} which is convex lower semicontinuous mappings for 𝑗=1,2,3 satisfying the following conditions:(i)⟨𝑇_𝑗𝑥,𝐽^∗(𝐽𝑥−𝛿_𝑗𝑇_𝑗𝑥)⟩≥0, for all 𝑥∈𝐶 for 𝑗=1,2,3;(ii)(𝐽−𝛿_𝑗𝑇_𝑗) are compact for 𝑗=1,2,3;(iii)𝑓_𝑗(0)=0 and 𝑓_𝑗(𝑥)≥0, for all 𝑥∈𝐶 and 𝑗=1,2,3; then the system of mixed variational inequality (1.1) has a solution (𝑥̂,𝑦,̂𝑧)̂ and sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} defined by Algorithm 3.2 have convergent subsequences {𝑥_𝑛𝑖},{𝑦_𝑛𝑖}, and {𝑧_𝑛𝑖} such that 𝑦𝑥_𝑛𝑖⟶𝑥̂,𝑖⟶∞,_𝑛𝑖𝑧⟶𝑦,̂𝑖⟶∞,_𝑛𝑖⟶𝑧,̂𝑖⟶∞.(4.1)

Proof. Since 𝐸 is a uniformly convex and uniform smooth Banach space, we known that 𝐽 is bijection from 𝐸 to 𝐸^∗ and uniformly continuous on any bounded subsets of 𝐸. Hence, Π^𝑓𝑗_𝐶 for 𝑗=1,2,3 is well-defined and single-value implies that {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} are well defined. Let 𝐺₂(𝑥,𝑦)=𝐺(𝐽𝑥,𝑦), for any 𝑥∈𝐶 and 𝑦=0, we have 𝐺₂(𝑥,0)==‖𝐺𝐽(𝑥𝐽‖𝑥₂,0)=‖𝐽𝑥‖₂−2⟨𝐽𝑥,0⟩+2𝑓(0)=‖𝑥‖².(4.2) By (4.2) and Lemma 2.5, we have 𝐺₂⎛⎜⎜⎝^𝑓1_𝐶𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛,0⎞⎟⎟⎠=𝐺⎛⎜⎜⎝𝐽⎛⎜⎜⎝^𝑓1_𝐶𝐽𝑧_𝑛≤𝐺𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛⎞⎟⎟⎠,0⎞⎟⎟⎠=‖‖𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛,0−𝛿₁𝑇₁𝑧_𝑛‖‖².(4.3) From Lemma 2.2, and for all 𝑥∈𝐶, ⟨𝑇₁𝑥,𝐽^∗(𝐽𝑥−𝛿₁𝑇₁𝑥)⟩≥0, so for 𝑧_𝑛∈𝐶, we obtain ‖‖𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖²≤≤‖‖‖‖𝐽𝐽𝑧𝑧_𝑛𝑛‖‖²−2𝛿₁𝑇₁𝑧_𝑛,𝐽^∗𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖²≤‖‖𝑧_𝑛‖‖².(4.4) Again by Lemma 2.2, for all 𝑥∈𝐶,⟨𝑇₂𝑥,𝐽^∗(𝐽𝑥−𝛿₂𝑇₂𝑥)⟩≥0, and for 𝑥_𝑛+1∈𝐶, we have ‖‖𝑦_𝑛+1‖‖²==𝐺𝐺₂𝐽𝑦𝑦_𝑛+1𝑛+1,0=𝐺⎛⎜⎜⎝𝐽_𝑓2,0_𝐶𝐽𝑥_𝑛+1≤𝐺𝐽𝑥_𝑛+1−𝛿₂𝑇₂𝑥_𝑛+1,0⎞⎟⎟⎠≤‖‖𝐽𝑥_𝑛+1−𝛿₂𝑇₂𝑥_𝑛+1,0−𝛿₂𝑇₂𝑥_𝑛+1‖‖²≤≤‖‖‖‖𝐽𝐽𝑥𝑥_𝑛+1𝑛+1‖‖²−2𝛿₂𝑇₂𝑥_𝑛+1,𝐽^∗𝐽𝑥_𝑛+1−𝛿₂𝑇₂𝑥_𝑛+1‖‖²≤‖‖𝑥_𝑛+1‖‖².(4.5) In similar way, for all 𝑥∈𝐶, ⟨𝑇₃𝑥,𝐽^∗(𝐽𝑥−𝛿₃𝑇₃𝑥)⟩≥0, and 𝑧_𝑛+1∈𝐶, we also have ‖‖𝑧_𝑛+1‖‖²≤=𝐺𝐺𝐽𝐽𝑧𝑦_𝑛+1𝑛+1,0=‖‖𝐽𝑦_𝑛+1−𝛿₃𝑇₃𝑦_𝑛+1,0−𝛿₃𝑇₃𝑦_𝑛+1‖‖²≤≤‖‖‖‖𝐽𝑦𝑦_𝑛+1‖‖²−2𝛿₃𝑇₃𝑦_𝑛+1,𝐽^∗𝐽𝑦_𝑛+1−𝛿₃𝑇₃𝑦_𝑛+1_𝑛+1‖‖².(4.6) It follows from (4.5) and (4.6) that ‖‖𝑧_𝑛+1‖‖²≤‖‖𝑥_𝑛+1‖‖²,∀𝑛∈ℕ.(4.7) From (4.5) and (4.6), we compute ‖‖𝑥_𝑛+1‖‖²≤1−𝛼_𝑛‖‖𝑥_𝑛‖‖+𝛼_𝑛‖‖‖‖^𝑓1_𝐶𝐽𝑧_𝑛≤1−𝛼_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖‖‖≤1−𝛼_𝑛‖‖𝑥_𝑛‖‖+𝛼_𝑛‖‖𝑧_𝑛‖‖≤1−𝛼_𝑛‖‖𝑥_𝑛‖‖+𝛼_𝑛‖‖𝑦_𝑛‖‖=‖‖𝑥_𝑛‖‖𝑥_𝑛‖‖+𝛼_𝑛‖‖𝑥_𝑛‖‖‖‖.(4.8) This implies that the sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛}, and {Π^𝑓1_𝐶(𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛)} are bounded. For a positive number 𝑟 such that {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛},{Π^𝑓1_𝐶(𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛)}∈𝐵_𝑟, by Lemma 2.1, for 𝑞=2, there exists a continuous, strictly increasing, and convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that for 𝛼_𝑛∈[0,1], we have ‖‖𝑥_𝑛+1‖‖²=‖‖‖‖1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑓¹_𝐶𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖‖‖²≤1−𝛼_𝑛‖‖𝑥_𝑛‖‖²+𝛼_𝑛‖‖‖‖^𝑓1_𝐶𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖‖‖²−𝛼_𝑛1−𝛼_𝑛𝑔‖‖‖‖𝑥_𝑛−^𝑓1_𝐶𝐽𝑧_𝑛=1−𝛼_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖‖‖‖‖𝑥_𝑛‖‖²+𝛼_𝑛𝐺₂⎛⎜⎜⎝^𝑓1_𝐶𝐽𝑧_𝑛−𝛼_𝑛−𝛿₁𝑇₁𝑧_𝑛,0⎞⎟⎟⎠1−𝛼_𝑛𝑔‖‖‖‖𝑥_𝑛−^𝑓1_𝐶𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖‖‖.(4.9) Applying (4.3), (4.4), and (4.7), we have 𝛼_𝑛1−𝛼_𝑛𝑔‖‖‖‖𝑥_𝑛−^𝑓1_𝐶𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖‖‖≤1−𝛼_𝑛‖‖𝑥_𝑛‖‖²−‖‖𝑥_𝑛+1‖‖²+𝛼_𝑛𝐺₂⎛⎜⎜⎝^𝑓1_𝐶𝐽𝑧_𝑛≤1−𝛼_𝑛−𝛿₁𝑇₁𝑧_𝑛,0⎞⎟⎟⎠‖‖𝑥_𝑛‖‖²−‖‖𝑥_𝑛+1‖‖²+𝛼_𝑛‖‖𝑥_𝑛‖‖²=‖‖𝑥_𝑛‖‖²−‖‖𝑥_𝑛+1‖‖².(4.10) Summing (4.10), for 𝑛=0,1,2,3,…,𝑘, we have ^𝑘_𝑛=0𝛼_𝑛1−𝛼_𝑛𝑔‖‖‖‖𝑥_𝑛−^𝑓1_𝐶𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖‖‖≤‖‖𝑥₀‖‖²−‖‖𝑥_𝑘+1‖‖²≤‖‖𝑥₀‖‖²,(4.11) taking 𝑘→∞, we get ^∞_𝑛=0𝛼_𝑛1−𝛼_𝑛𝑔‖‖‖‖𝑥_𝑛−^𝑓1_𝐶𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖‖‖≤‖‖𝑥₀‖‖².(4.12) This shows that series (4.12) is converge, we obtain that lim_𝑛→∞𝛼_𝑛1−𝛼_𝑛𝑔‖‖‖‖𝑥_𝑛−^𝑓1_𝐶𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖‖‖=0.(4.13) From 0<𝑎≤𝛼_𝑛≤𝑏<1 for all 𝑛, thus ∑^∞𝑛=0𝛼_𝑛(1−𝛼_𝑛)>0 and (4.13), we have lim_𝑛→∞𝑔‖‖‖‖𝑥_𝑛−^𝑓1_𝐶𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖‖‖=0.(4.14) By property of functional 𝑔, we have lim_𝑛→∞‖‖‖‖𝑥_𝑛−^𝑓1_𝐶𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖‖‖=0.(4.15) Since {𝑧_𝑛} is bounded sequence and (𝐽−𝛿₁𝑇₁) is compact on 𝐶, then sequence {𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛} has a convergence subsequence such that 𝐽𝑧_𝑛𝑖−𝛿₁𝑇₁𝑧_𝑛𝑖⟶𝑤₀∈𝐸^∗as𝑖⟶∞.(4.16) By the continuity of the Π^𝑓1_𝐶, we have lim_{𝑓𝑖→∞}¹_𝐶𝐽𝑧_𝑛𝑖−𝛿₁𝑇₁𝑧_𝑛𝑖=^𝑓1_𝐶𝑤₀.(4.17) Again since {𝑥_𝑛},{𝑦_𝑛} are bounded and (𝐽−𝛿₂𝑇₂),(𝐽−𝛿₃𝑇₃) are compact on 𝐶, then sequences {𝐽𝑥_𝑛−𝛿₂𝑇₂𝑥_𝑛} and {𝐽𝑦_𝑛−𝛿₃𝑇₃𝑦_𝑛} have convergence subsequences such that 𝐽𝐽𝑥𝑦_{𝑛𝑖𝑛𝑖}−𝛿₂𝑇₂𝑥_𝑛𝑖⟶𝑢₀∈𝐸^∗as𝑖⟶∞,−𝛿₃𝑇₃𝑦_𝑛𝑖⟶𝑣₀∈𝐸^∗as𝑖⟶∞.(4.18) By the continuity of Π^𝑓2_𝐶 and Π^𝑓3_𝐶, we have lim_{𝑓𝑖→∞}²_𝐶𝐽𝑥_𝑛𝑖−𝛿₂𝑇₂𝑥_𝑛𝑖=^𝑓2lim_{𝑓𝑖→∞3}_𝐶𝑢₀,(4.19)_𝐶𝐽𝑦_𝑛𝑖−𝛿₃𝑇₃𝑦_𝑛𝑖=^𝑓3_𝐶𝑣₀.(4.20) Let ^𝑓1_𝐶𝑤₀=𝑥̂,(4.21)_𝑓2_𝐶𝑢₀=𝑦,̂(4.22)_𝑓3_𝐶𝑣₀=𝑧.̂(4.23) By using the triangle inequality, we have ‖‖𝑥_𝑛𝑖−𝑥̂‖‖≤‖‖‖‖𝑥_𝑛𝑖−^𝑓1_𝐶𝐽𝑧_𝑛𝑖−𝛿₁𝑇₁𝑧_𝑛𝑖‖‖‖‖+‖‖‖‖^𝑓1_𝐶𝐽𝑧_𝑛𝑖−𝛿₁𝑇₁𝑧_𝑛𝑖−𝑥̂‖‖‖‖.(4.24) From (4.15) and (4.17), we have lim_𝑖→∞𝑥_𝑛𝑖=𝑥̂.(4.25) By definition of 𝑧_𝑛𝑖, we get ‖‖𝑧_𝑛𝑖−𝑧‖‖̂≤‖‖‖‖^𝑓3_𝐶𝐽𝑦_𝑛𝑖−𝛿₃𝑇₃𝑦_𝑛𝑖−𝑧‖‖‖‖̂.(4.26) It follows by (4.20) and (4.23), we obtain lim_𝑖→∞𝑧_𝑛i=𝑧.̂(4.27) In the same way, we also have lim_𝑖→∞𝑦_𝑛𝑖=𝑦.̂(4.28) By the continuity properties of (𝐽−𝛿₁𝑇₁),(𝐽−𝛿₂𝑇₂),(𝐽−𝛿₃𝑇₃), and Π^𝑓𝑗_𝐶 for 𝑗=1,2,3. We conclude that 𝑥̂=^𝑓1_𝐶𝐽𝑧̂−𝛿₁𝑦̂=_𝑓2𝑇₁𝑧̂,_𝐶𝐽𝑥̂−𝛿𝑇₂𝑧̂=_𝑓3𝑥̂,_𝐶𝐽𝑦̂−𝛿₃𝑇₃𝑦̂.(4.29) This completes of proof.

Theorem 4.2. Let 𝐶 be a nonempty compact and convex subset of a uniformly convex and uniformly smooth Banach space 𝐸 with dual space 𝐸^∗. If the mapping 𝑇_𝑗∶𝐶→𝐸^∗ and 𝑓_𝑗∶𝐶→ℝ∪{+∞} which is convex lower semicontinuous mappings for 𝑗=1,2,3 satisfy the following conditions:(i)⟨𝑇_𝑗𝑥,𝐽^∗(𝐽𝑥−𝛿_𝑗𝑇_𝑗𝑥)⟩≥0, for all 𝑥∈𝐶 for 𝑗=1,2,3;(ii)𝑓_𝑗(0)=0 and 𝑓_𝑗(𝑥)≥0, for all 𝑥∈𝐶 for 𝑗=1,2,3; then the system of mixed variational inequality (1.1) has a solution (𝑥̂,𝑦,̂𝑧)̂ and sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} defined by Algorithm 3.2 have a convergent subsequences {𝑥_𝑛𝑖},{𝑦_𝑛𝑖}, and {𝑧_𝑛𝑖} such that 𝑥𝑦_𝑛𝑖⟶𝑥̂,𝑖⟶∞,_𝑛𝑖𝑧⟶𝑦,̂𝑖⟶∞,_𝑛𝑖⟶𝑧,̂𝑖⟶∞.(4.30)

Proof. In the same way to the proof in Theorem 4.1, we have lim_𝑛→∞‖‖‖‖𝑥_𝑛−∏^𝑓1_𝐶𝐽𝑧_𝑛−𝛿₁𝑇₁𝑧_𝑛‖‖‖‖=0.(4.31) Hence, there exist subsequences {𝑥_𝑛𝑖}⊂{𝑥_𝑛} and {𝑧_𝑛𝑖}⊂{𝑧_𝑛} such that lim_𝑖→∞‖‖‖‖𝑥_𝑛𝑖−∏^𝑓1_𝐶𝐽𝑧_n𝑖−𝛿₁𝑇₁𝑧_𝑛𝑖‖‖‖‖=0.(4.32) From the compactness of 𝐶, we have that 𝑥𝑧_𝑛𝑖⟶𝑥̂as𝑖⟶∞,_𝑛𝑖⟶𝑧̂as𝑖⟶∞,(4.33) where 𝑥̂,𝑧̂ are points in 𝐶. Also, for a sequence {𝑦_𝑛}⊃{𝑦_𝑛𝑖}→𝑦̂as𝑖→∞, where 𝑦̂ is a points in 𝐶. By the continuity properties of 𝐽,𝑇₂,𝑇₃Π^𝑓2_𝐶, and Π^𝑓3_𝐶, we obtain that 𝑦̂=^𝑓2_𝐶𝐽𝑥̂−𝛿₂𝑧̂=_𝑓3𝑇₂𝑥̂,_𝐶𝐽𝑦̂−𝛿₃𝑇₃𝑦̂.(4.34) From definition of 𝑥_𝑛+1, we get ‖‖‖‖^𝑓1_𝐶𝐽𝑧_𝑛𝑖=‖‖‖‖−𝛿₁𝑇₁𝑧_𝑛𝑖−𝑥_𝑓1̂‖‖‖‖_𝐶𝐽𝑧_𝑛i−𝛿₁𝑇₁𝑧_𝑛𝑖−𝑥̂+𝑥_𝑛𝑖+1−1−𝛼_𝑛𝑥_𝑛𝑖−𝛼_𝑛𝑓¹_𝐶𝐽𝑧_𝑛𝑖=‖‖‖‖𝑥_𝑛𝑖+1−𝛿₁𝑇₁𝑧_𝑛𝑖‖‖‖‖−𝑥̂+1−𝛼_𝑛^𝑓1_𝐶𝐽𝑧_𝑛𝑖−𝛿₁𝑇₁𝑧_𝑛𝑖−𝑥_𝑛𝑖≤‖‖𝑥_𝑛𝑖+1‖‖‖‖−𝑥̂‖‖+1−𝛼_𝑛‖‖‖‖𝑥_𝑛𝑖−^𝑓1_𝐶𝐽𝑧_𝑛𝑖−𝛿₁𝑇₁𝑧_𝑛𝑖‖‖‖‖.(4.35) By (4.25) and (4.31), we have 𝑥̂=^𝑓1_𝐶𝐽𝑧̂−𝛿₁𝑇₁𝑧̂.(4.36) This completes of proof.

Corollary 4.3. Let 𝐶 be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space 𝐸 with dual space 𝐸^∗. If the mapping 𝑇_𝑗∶𝐶→𝐸^∗ for 𝑗=1,2,3 satisfy the following conditions:(i)⟨𝑇_𝑗𝑥,𝐽^∗(𝐽𝑥−𝛿_𝑗𝑇_𝑗𝑥)⟩≥0, for all 𝑥∈𝐶 for 𝑗=1,2,3;(ii)(𝐽−𝛿_𝑗𝑇_𝑗) are compact for 𝑗=1,2,3; then the system of mixed variational inequality (1.2) has a solution (𝑥̂,𝑦,̂𝑧)̂ and sequences {𝑥_𝑛},{𝑦_𝑛}, and {z_𝑛} defined by Algorithm 3.3 have convergent subsequences {𝑥_𝑛𝑖},{𝑦_𝑛𝑖}, and {𝑧_𝑛𝑖} such that 𝑥_𝑛𝑖→𝑥̂,𝑖→∞,𝑦_𝑛𝑖→𝑦,̂𝑖→∞, and 𝑧_𝑛𝑖→𝑧,̂𝑖→∞.

If 𝐸=𝐻 is a Hilbert space, then 𝐻^∗=𝐻,𝐽^∗=𝐽=𝐼, so one obtains the following corollary.

Corollary 4.4. Let 𝐶 be a nonempty closed and convex subset of a Hilbert space 𝐻. If the mapping 𝑇_𝑗∶𝐶→𝐻 and 𝑓_𝑗∶𝐶→ℝ∪{+∞} which is convex lower semicontinuous mappings for 𝑗=1,2,3 satisfy the following conditions:(i)⟨𝑇_𝑗𝑥,𝑥−𝛿_𝑗𝑇_𝑗𝑥⟩≥0 for 𝑗=1,2,3;(ii)𝑓_𝑗(0)=0 and 𝑓_𝑗(𝑥)≥0 for all 𝑥∈𝐶 for 𝑗=1,2,3; then the system of mixed variational inequality (1.6) has a solution (𝑥̂,𝑦,̂𝑧)̂ and sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} defined by Algorithm 3.4 have a convergent subsequences {𝑥_𝑛𝑖},{𝑦_𝑛𝑖}, and {𝑧_𝑛𝑖} such that 𝑥_𝑛𝑖→𝑥̂,𝑖→∞,𝑦_𝑛𝑖→𝑦,̂𝑖→∞, and 𝑧_𝑛𝑖→𝑧,̂𝑖→∞.

Corollary 4.5. Let 𝐶 be a nonempty closed and convex subset of a Hilbert space 𝐻. If the mapping 𝑇_𝑗∶𝐶→𝐻 for 𝑗=1,2,3 satisfy the conditions: ⟨𝑇_𝑗𝑥,𝑥−𝛿_𝑗𝑇_𝑗𝑥⟩≥0 for 𝑗=1,2,3; then the system of mixed variational inequality (1.7) has a solution (𝑥̂,𝑦,̂𝑧)̂ and sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} defined by Algorithm 3.5 have a convergent subsequences {𝑥_𝑛𝑖},{𝑦_𝑛𝑖}, and {𝑧_𝑛𝑖} such that 𝑥_𝑛𝑖→𝑥̂,𝑖→∞,𝑦_𝑛𝑖→𝑦,̂𝑖→∞, and 𝑧_𝑛𝑖→𝑧,̂𝑖→∞.

Remark 4.6. Theorems 4.1 and 4.2 and Corollary 4.3 extend and improve the results of Zhang et al. [7] and Wu and Huang [5].

Acknowledgments

This research was supported by grant from under the Program Strategic Scholarships for Frontier Research Network for the Joint Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand. Furthermore, we would like to thank the King Mongkuts Diamond Scholarship for Ph.D. Program at King Mongkuts University of Technology Thonburi (KMUTT) and the National Research University Project of Thailand’s Office of the Higher Education Commission (under CSEC project no.54000267) for their financial support during the preparation of this paper. P. Kumam was supported by the Commission on Higher Education and the Thailand Research Fund (Grant no. MRG5380044).