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Journal of Applied Mathematics
Volume 2012, Article ID 413468, 15 pages
http://dx.doi.org/10.1155/2012/413468
Research Article

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

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Received 22 December 2011; Accepted 29 January 2012

Academic Editor: Hong-Kun Xu

Copyright © 2012 Siwaporn Saewan and Poom Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝑃𝐶𝐸𝐶 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 𝜋𝐶𝐸𝐶 and Π𝐸𝐸𝐶 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., [817] and references therein).

We first introduce and consider the system of mixed variational inequalities (SMVI) which is to find ̂𝑥,̂𝑦,̂𝑧𝐶 such that𝛿1𝑇1̂𝑧+𝐽̂𝑥𝐽̂𝑧,𝑦̂𝑥+𝑓1(𝑦)𝑓1(̂𝑥)0,𝑦𝐶,𝛿2𝑇2̂𝑥+𝐽̂𝑦𝐽̂𝑥,𝑦̂𝑦+𝑓2(𝑦)𝑓2(̂𝑦)0,𝑦𝐶,𝛿3𝑇3̂𝑦+𝐽̂𝑧𝐽̂𝑦,𝑦̂𝑧+𝑓3(𝑦)𝑓3(̂𝑧)0,𝑦𝐶,(1.1) where 𝛿𝑗>0,𝑇𝑗𝐶𝐸,𝑓𝑗𝐶{+} 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 𝑓𝑗(𝑥)=0 for 𝑗=1,2,3, for all 𝑥𝐶, (1.1) is equivalent to find ̂𝑥, ̂𝑦 and ̂𝑧𝐶 such that𝛿𝑇1̂𝑧+𝐽̂𝑥𝐽̂z,𝑦̂𝑥0,𝑦𝐶,𝛿2𝑇2̂𝑥+𝐽̂𝑦𝐽̂𝑥,𝑦̂𝑦0,𝑦𝐶,𝛿3𝑇3̂𝑦+𝐽̂𝑧𝐽̂𝑦,𝑦̂𝑧0,𝑦𝐶.(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 𝑥𝐶 and ̂𝑦=̂𝑧, then (1.1) is reduced to find ̂𝑥,̂𝑦𝐶 such that𝛿1𝑇1̂𝑦+𝐽̂𝑥𝐽̂𝑦,𝑦̂𝑥+𝑓1(𝑦)𝑓1(̂𝑥)0,𝑦𝐶,𝛿2𝑇2̂𝑥+𝐽̂𝑦𝐽̂𝑥,𝑦̂𝑦+𝑓2(𝑦)𝑓2(̂𝑦)0,𝑦𝐶,(1.3) which is studied by Zhang et al. [7].

If 𝑇=𝑇1=𝑇2=𝑇3,𝑓1(𝑥)=𝑓2(𝑥)=𝑓3(𝑥), for all 𝑥𝐶 and ̂𝑥=̂𝑦=̂𝑧, (1.1) is reduced to find ̂𝑥 such that𝑇̂𝑥,𝑦̂𝑥+𝑓1(𝑦)𝑓1(̂𝑥)0,𝑦𝐶.(1.4) This iterative method is studied by Wu and Huang [5].

If 𝑓1(𝑥)=0, for all 𝑥𝐶, (1.4) is reduced to find ̂𝑥 such that𝑇̂𝑥,𝑦̂𝑥0,𝑦𝐶,(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 ̂𝑥,̂𝑦,̂𝑧𝐶 such that𝛿1𝑇1̂𝑧+̂𝑥̂𝑧,𝑦̂𝑥+𝑓1(𝑦)𝑓1(̂𝑥)0,𝑦𝐶,𝛿2𝑇2̂𝑥+̂𝑦̂𝑥,𝑦̂𝑦+𝑓2(𝑦)𝑓2(̂𝑦)0,𝑦𝐶,𝛿3𝑇3̂𝑦+̂𝑧̂𝑦,𝑦̂𝑧+𝑓3(𝑦)𝑓3(̂𝑧)0,𝑦𝐶.(1.6) If 𝑓𝑗(𝑥)=0 for 𝑗=1,2,3, for all 𝑥𝐶, (1.6) reduces to the following (SVI):𝛿1𝑇1̂𝑧+̂𝑥̂𝑧,𝑦̂𝑥0,𝑦𝐶,𝛿2𝑇2̂𝑥+̂𝑦̂𝑥,𝑦̂𝑦0,𝑦𝐶,𝛿3𝑇3̂𝑦+̂𝑧̂𝑦,𝑦̂𝑧0,𝑦𝐶.(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,𝑦𝐸 with 𝑥=𝑦=1 and 𝑥𝑦. Let 𝑈={𝑥𝐸𝑥=1} be the unit sphere of 𝐸. Then, a Banach space 𝐸 is said to be smooth if the limit lim𝑡0(𝑥+𝑡𝑦𝑥)/𝑡 exists for each 𝑥,𝑦𝑈. It is also said to be uniformly smooth if the limit exists uniformly in 𝑥,𝑦𝑈. Let 𝐸 be a Banach space. The modulus of smoothness of 𝐸 is the function 𝜌𝐸[0,)[0,) defined by 𝜌𝐸(𝑡)=sup{((𝑥+𝑦+𝑥𝑦)/2)1𝑥=1,𝑦𝑡}. The modulus of convexity of 𝐸 is the function 𝜂𝐸[0,2][0,1] defined by 𝜂𝐸(𝜀)=inf{1(𝑥+𝑦)/2𝑥,𝑦𝐸,𝑥=𝑦=1,𝑥𝑦𝜀}. The normalized duality mapping 𝐽𝐸2𝐸 is defined by 𝐽(𝑥)={𝑥𝐸𝑥,𝑥=𝑥2,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 𝐽𝐸2𝐸 is the normalized duality mapping on 𝐸, then 𝐽1=𝐽, 𝐽𝐽=𝐼𝐸 and 𝐽𝐽=𝐼𝐸, where 𝐼𝐸 and 𝐼𝐸 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, 𝑇𝐷(𝑇)𝐸𝐹, 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+(1𝜆)𝑦𝑞𝜆𝑥𝑞+(1𝜆)𝑦𝑞𝜍𝑞(𝜆)𝑔(𝑥𝑦)(2.1) for all 𝑥,𝑦𝐵𝑟={𝑥𝐸𝑥𝑟} and 𝜆[0,1], where 𝜍𝑞(𝜆)=𝜆(1𝜆)𝑞+𝜆𝑞(1𝜆).

For case 𝑞=2, we have𝜆𝑥+(1𝜆)𝑦𝑞𝜆𝑥2+(1𝜆)𝑦2𝜆(1𝜆)𝑔(𝑥𝑦).(2.2)

Lemma 2.2 (Chang [22]). Let 𝐸 be a uniformly convex and uniformly smooth Banach space. The following holds: 𝜙+Φ2𝜙2+2Φ,𝐽(𝜙+Φ),𝜙,Φ𝐸.(2.3)

Next we recall the concept of the generalized 𝑓-projection operator. Let 𝐺𝐸×𝐶{+} be a functional defined as follows:𝐺(𝜉,𝑥)=𝜉22𝜉,𝑥+𝑥2+2𝜌𝑓(𝑥),(2.4) where 𝜉𝐸,𝜌 is positive number and 𝑓𝐶{+} is proper, convex, and lower semicontinuous. From definitions of 𝐺 and 𝑓, it is easy to see the following properties:(1)(𝜉𝑥)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 Π𝑓𝐶𝐸2𝐶 is the generalized 𝑓-projection operator if 𝑓𝐶𝜉=𝑢𝐶𝐺(𝜉,𝑢)=inf𝑦𝐶𝐺(𝜉,𝑦),𝜉𝐸.(2.5)

In this paper, we fixed 𝜌=1, we have𝐺(𝜉,𝑥)=𝜉22𝜉,𝑥+𝑥2+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)Π𝑓𝐶𝜉 is nonempty closed convex subset of 𝐶 for all 𝜉𝐸;(2)if 𝐸 is smooth, then for all 𝜉𝐸, 𝑥Π𝑓𝐶𝜉 if and only if 𝜉𝐽𝑥,𝑥𝑦+𝜌𝑓(𝑦)𝜌𝑓(𝑥)0,𝑦𝐶;(2.7)(3)if 𝐸 is smooth, then for any 𝜉𝐸, Π𝑓𝐶𝜉=(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 𝑥𝐶, then for any 𝜌>0, 𝐺(𝐽𝑥,𝑦)𝐺(𝜉,𝑦)+2𝜌𝑓(𝑦),𝜉𝐸,𝑦𝐶,𝑥𝑓𝐶𝜉.(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 𝑓𝐶{+} is proper, convex, and lower semicontinuous, then(1)Π𝑓𝐶𝐸𝐶 is single valued and norm to weak continuous;(2)if 𝐸 has the property (h), that is, for any sequence {𝑥𝑛}𝐸,𝑥𝑛𝑥𝐸 and 𝑥𝑛𝑥, imply that 𝑥𝑛𝑥, then Π𝑓𝐶𝐸𝐶 is continuous.

Defined the functional 𝐺2𝐸×𝐶{+} by𝐺2(𝑥,𝑦)=𝐺(𝐽𝑥,𝑦),𝑥𝐸,𝑦𝐶.(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 𝑓𝑗𝐶{+} for 𝑗=1,2,3 is proper, convex, and lower semicontinuous, then (̂𝑥,̂y,̂𝑧) is a solution of (SMVI) equivalent to finding ̂𝑥,̂𝑦,̂𝑧 such that ̂𝑥=𝑓1𝐶𝐽̂𝑧𝛿1𝑇1,̂𝑧̂𝑦=𝑓2𝐶𝐽̂𝑥𝛿2𝑇1,̂𝑥̂𝑧=𝑓3𝐶𝐽̂𝑦𝛿3𝑇1.̂𝑦(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 Π𝑓𝑗𝐶 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,𝑧0𝐶, define the sequences {𝑥𝑛},{𝑦𝑛}, and {𝑧𝑛} as follows:𝑥𝑛+1=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛,𝑦𝑛+1=𝑓2𝐶𝐽𝑥𝑛+1𝛿2𝑇2𝑥𝑛+1,𝑧𝑛+1=𝑓3𝐶𝐽𝑦𝑛+1𝛿3𝑇3𝑦𝑛+1,(3.2) where 0<𝑎𝛼𝑛𝑏<1.

If 𝑓𝑗(𝑥)=0,𝑗=1,2,3, for all 𝑥𝐶, 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,𝑧0𝐶, define the sequences {𝑥𝑛},{𝑦𝑛}, and {𝑧𝑛} as follows:𝑥𝑛+1=1𝛼𝑛𝑥𝑛+𝛼𝑛𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛,𝑦𝑛+1=𝐶𝐽𝑥𝑛+1𝛿2𝑇2𝑥𝑛+1,𝑧𝑛+1=𝐶𝐽𝑦𝑛+1𝛿3𝑇3𝑦𝑛+1,(3.3) where 0<𝑎𝛼𝑛𝑏<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,𝑧0𝐶, define the sequences {𝑥𝑛},{𝑦𝑛}, and {𝑧𝑛} as follows:𝑥𝑛+1=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛,𝑦𝑛+1=𝑓2𝐶𝐽𝑥𝑛+1𝛿2𝑇2𝑥𝑛+1,𝑧𝑛+1=𝑓3𝐶𝐽𝑦𝑛+1𝛿3𝑇3𝑦𝑛+1,(3.4) where 0<𝑎𝛼𝑛𝑏<1.

If 𝑓𝑗(𝑥)=0,𝑗=1,2,3, for all 𝑥𝐶, 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,𝑧0𝐶, define the sequences {𝑥𝑛},{𝑦𝑛}, and {𝑧𝑛} as follows:𝑥𝑛+1=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑃𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛,𝑦𝑛+1=𝑃𝐶𝐽𝑥𝑛+1𝛿2𝑇2𝑥𝑛+1,𝑧𝑛+1=𝑃𝐶𝐽𝑦𝑛+1𝛿3𝑇3𝑦𝑛+1,(3.5) where 0<𝑎𝛼𝑛𝑏<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 𝑇𝑗𝐶𝐸 and 𝑓𝑗𝐶{+} which is convex lower semicontinuous mappings for 𝑗=1,2,3 satisfying the following conditions:(i)𝑇𝑗𝑥,𝐽(𝐽𝑥𝛿𝑗𝑇𝑗𝑥)0, for all 𝑥𝐶 for 𝑗=1,2,3;(ii)(𝐽𝛿𝑗𝑇𝑗) are compact for 𝑗=1,2,3;(iii)𝑓𝑗(0)=0 and 𝑓𝑗(𝑥)0, for all 𝑥𝐶 and 𝑗=1,2,3; then the system of mixed variational inequality (1.1) has a solution (̂𝑥,̂𝑦,̂𝑧) and sequences {𝑥𝑛},{𝑦𝑛}, and {𝑧𝑛} defined by Algorithm 3.2 have convergent subsequences {𝑥𝑛𝑖},{𝑦𝑛𝑖}, and {𝑧𝑛𝑖} such that 𝑥𝑛𝑖𝑦̂𝑥,𝑖,𝑛𝑖𝑧̂𝑦,𝑖,𝑛𝑖̂𝑧,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, Π𝑓𝑗𝐶 for 𝑗=1,2,3 is well-defined and single-value implies that {𝑥𝑛},{𝑦𝑛}, and {𝑧𝑛} are well defined. Let 𝐺2(𝑥,𝑦)=𝐺(𝐽𝑥,𝑦), for any 𝑥𝐶 and 𝑦=0, we have 𝐺2=(𝑥,0)=𝐺(𝐽𝑥,0)𝐽𝑥22𝐽𝑥,0+2𝑓(0)=𝐽𝑥2=𝑥2.(4.2) By (4.2) and Lemma 2.5, we have 𝐺2𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛𝐽,0=𝐺𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛,0𝐺𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛=,0𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛2.(4.3) From Lemma 2.2, and for all 𝑥𝐶, 𝑇1𝑥,𝐽(𝐽𝑥𝛿1𝑇1𝑥)0, so for 𝑧𝑛𝐶, we obtain 𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛2𝐽𝑧𝑛2𝛿21𝑇1𝑧𝑛,𝐽𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛𝐽𝑧𝑛2𝑧𝑛2.(4.4) Again by Lemma 2.2, for all 𝑥𝐶,𝑇2𝑥,𝐽(𝐽𝑥𝛿2𝑇2𝑥)0, and for 𝑥𝑛+1𝐶, we have 𝑦𝑛+12=𝐺2𝑦𝑛+1,0=𝐺𝐽𝑦𝑛+1𝐽,0=𝐺𝑓2𝐶𝐽𝑥𝑛+1𝛿2𝑇2𝑥𝑛+1,0𝐺𝐽𝑥𝑛+1𝛿2𝑇2𝑥𝑛+1,0𝐽𝑥𝑛+1𝛿2𝑇2𝑥𝑛+12𝐽𝑥𝑛+12𝛿22𝑇2𝑥𝑛+1,𝐽𝐽𝑥𝑛+1𝛿2𝑇2𝑥𝑛+1𝐽𝑥𝑛+12𝑥𝑛+12.(4.5) In similar way, for all 𝑥𝐶, 𝑇3𝑥,𝐽(𝐽𝑥𝛿3𝑇3𝑥)0, and 𝑧𝑛+1𝐶, we also have 𝑧𝑛+12=𝐺𝐽𝑧𝑛+1,0𝐺𝐽𝑦𝑛+1𝛿3𝑇3𝑦𝑛+1=,0𝐽𝑦𝑛+1𝛿3𝑇3𝑦𝑛+12𝐽𝑦𝑛+12𝛿23𝑇3𝑦𝑛+1,𝐽𝐽𝑦𝑛+1𝛿3𝑇3𝑦𝑛+1𝑦𝑛+12.(4.6) It follows from (4.5) and (4.6) that 𝑧𝑛+12𝑥𝑛+12,𝑛.(4.7) From (4.5) and (4.6), we compute 𝑥𝑛+121𝛼𝑛𝑥𝑛+𝛼𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛1𝛼𝑛𝑥𝑛+𝛼𝑛𝑧𝑛1𝛼𝑛𝑥𝑛+𝛼𝑛𝑦𝑛1𝛼𝑛𝑥𝑛+𝛼𝑛𝑥𝑛=𝑥𝑛.(4.8) This implies that the sequences {𝑥𝑛},{𝑦𝑛},{𝑧𝑛}, and {Π𝑓1𝐶(𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛)} are bounded. For a positive number 𝑟 such that {𝑥𝑛},{𝑦𝑛},{𝑧𝑛},{Π𝑓1𝐶(𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛)}𝐵𝑟, 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 𝛼𝑛[0,1], we have 𝑥𝑛+12=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛21𝛼𝑛𝑥𝑛2+𝛼𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛2𝛼𝑛1𝛼𝑛𝑔𝑥𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛=1𝛼𝑛𝑥𝑛2+𝛼𝑛𝐺2𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛,0𝛼𝑛1𝛼𝑛𝑔𝑥𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛.(4.9) Applying (4.3), (4.4), and (4.7), we have 𝛼𝑛1𝛼𝑛𝑔𝑥𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛1𝛼𝑛𝑥𝑛2𝑥𝑛+12+𝛼𝑛𝐺2𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛,01𝛼𝑛𝑥𝑛2𝑥𝑛+12+𝛼𝑛𝑥𝑛2=𝑥𝑛2𝑥𝑛+12.(4.10) Summing (4.10), for 𝑛=0,1,2,3,,𝑘, we have 𝑘𝑛=0𝛼𝑛1𝛼𝑛𝑔𝑥𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛𝑥02𝑥𝑘+12𝑥02,(4.11) taking 𝑘, we get 𝑛=0𝛼𝑛1𝛼𝑛𝑔𝑥𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛𝑥02.(4.12) This shows that series (4.12) is converge, we obtain that lim𝑛𝛼𝑛1𝛼𝑛𝑔𝑥𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛=0.(4.13) From 0<𝑎𝛼𝑛𝑏<1 for all 𝑛, thus 𝑛=0𝛼𝑛(1𝛼𝑛)>0 and (4.13), we have lim𝑛𝑔𝑥𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛=0.(4.14) By property of functional 𝑔, we have lim𝑛𝑥𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛=0.(4.15) Since {𝑧𝑛} is bounded sequence and (𝐽𝛿1𝑇1) is compact on 𝐶, then sequence {𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛} has a convergence subsequence such that 𝐽𝑧𝑛𝑖𝛿1𝑇1𝑧𝑛𝑖𝑤0𝐸as𝑖.(4.16) By the continuity of the Π𝑓1𝐶, we have lim𝑓𝑖1𝐶𝐽𝑧𝑛𝑖𝛿1𝑇1𝑧𝑛𝑖=𝑓1𝐶𝑤0.(4.17) Again since {𝑥𝑛},{𝑦𝑛} are bounded and (𝐽𝛿2𝑇2),(𝐽𝛿3𝑇3) are compact on 𝐶, then sequences {𝐽𝑥𝑛𝛿2𝑇2𝑥𝑛} and {𝐽𝑦𝑛𝛿3𝑇3𝑦𝑛} have convergence subsequences such that 𝐽𝑥𝑛𝑖𝛿2𝑇2𝑥𝑛𝑖𝑢0𝐸as𝑖,𝐽𝑦𝑛𝑖𝛿3𝑇3𝑦𝑛𝑖𝑣0𝐸as𝑖.(4.18) By the continuity of Π𝑓2𝐶 and Π𝑓3𝐶, we have lim𝑓𝑖2𝐶𝐽𝑥𝑛𝑖𝛿2𝑇2𝑥𝑛𝑖=𝑓2𝐶𝑢0,(4.19)lim𝑓𝑖3𝐶𝐽𝑦𝑛𝑖𝛿3𝑇3𝑦𝑛𝑖=𝑓3𝐶𝑣0.(4.20) Let 𝑓1𝐶𝑤0=̂𝑥,(4.21)𝑓2𝐶𝑢0=̂𝑦,(4.22)𝑓3𝐶𝑣0=̂𝑧.(4.23) By using the triangle inequality, we have 𝑥𝑛𝑖𝑥̂𝑥𝑛𝑖𝑓1𝐶𝐽𝑧𝑛𝑖𝛿1𝑇1𝑧𝑛𝑖+𝑓1𝐶𝐽𝑧𝑛𝑖𝛿1𝑇1𝑧𝑛𝑖.̂𝑥(4.24) From (4.15) and (4.17), we have lim𝑖𝑥𝑛𝑖=̂𝑥.(4.25) By definition of 𝑧𝑛𝑖, we get 𝑧𝑛𝑖̂𝑧𝑓3𝐶𝐽𝑦𝑛𝑖𝛿3𝑇3𝑦𝑛𝑖.̂𝑧(4.26) It follows by (4.20) and (4.23), we obtain lim𝑖𝑧𝑛i=̂𝑧.(4.27) In the same way, we also have lim𝑖𝑦𝑛𝑖=̂𝑦.(4.28) By the continuity properties of (𝐽𝛿1𝑇1),(𝐽𝛿2𝑇2),(𝐽𝛿3𝑇3), and Π𝑓𝑗𝐶 for 𝑗=1,2,3. We conclude that ̂𝑥=𝑓1𝐶𝐽̂𝑧𝛿1𝑇1,̂𝑧̂𝑦=𝑓2𝐶𝐽̂𝑥𝛿𝑇2,̂𝑥̂𝑧=𝑓3𝐶𝐽̂𝑦𝛿3𝑇3.̂𝑦(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 𝑇𝑗𝐶𝐸 and 𝑓𝑗𝐶{+} which is convex lower semicontinuous mappings for 𝑗=1,2,3 satisfy the following conditions:(i)𝑇𝑗𝑥,𝐽(𝐽𝑥𝛿𝑗𝑇𝑗𝑥)0, for all 𝑥𝐶 for 𝑗=1,2,3;(ii)𝑓𝑗(0)=0 and 𝑓𝑗(𝑥)0, for all 𝑥𝐶 for 𝑗=1,2,3; then the system of mixed variational inequality (1.1) has a solution (̂𝑥,̂𝑦,̂𝑧) and sequences {𝑥𝑛},{𝑦𝑛}, and {𝑧𝑛} defined by Algorithm 3.2 have a convergent subsequences {𝑥𝑛𝑖},{𝑦𝑛𝑖}, and {𝑧𝑛𝑖} such that 𝑥𝑛𝑖𝑦̂𝑥,𝑖,𝑛𝑖𝑧̂𝑦,𝑖,𝑛𝑖̂𝑧,𝑖.(4.30)

Proof. In the same way to the proof in Theorem 4.1, we have lim𝑛𝑥𝑛𝑓1𝐶𝐽𝑧𝑛𝛿1𝑇1𝑧𝑛=0.(4.31) Hence, there exist subsequences {𝑥𝑛𝑖}{𝑥𝑛} and {𝑧𝑛𝑖}{𝑧𝑛} such that lim𝑖𝑥𝑛𝑖𝑓1𝐶𝐽𝑧n𝑖𝛿1𝑇1𝑧𝑛𝑖=0.(4.32) From the compactness of 𝐶, we have that 𝑥𝑛𝑖𝑧̂𝑥as𝑖,𝑛𝑖̂𝑧as𝑖,(4.33) where ̂𝑥,̂𝑧 are points in 𝐶. Also, for a sequence {𝑦𝑛}{𝑦𝑛𝑖}̂𝑦as𝑖, where ̂𝑦 is a points in 𝐶. By the continuity properties of 𝐽,𝑇2,𝑇3Π𝑓2𝐶, and Π𝑓3𝐶, we obtain that ̂𝑦=𝑓2𝐶𝐽̂𝑥𝛿2𝑇2,̂𝑥̂𝑧=𝑓3𝐶𝐽̂𝑦𝛿3𝑇3.̂𝑦(4.34) From definition of 𝑥𝑛+1, we get 𝑓1𝐶𝐽𝑧𝑛𝑖𝛿1𝑇1𝑧𝑛𝑖=̂𝑥𝑓1𝐶𝐽𝑧𝑛i𝛿1𝑇1𝑧𝑛𝑖̂𝑥+𝑥𝑛𝑖+11𝛼𝑛𝑥𝑛𝑖𝛼𝑛𝑓1𝐶𝐽𝑧𝑛𝑖𝛿1𝑇1𝑧𝑛𝑖=𝑥𝑛𝑖+1̂𝑥+1𝛼𝑛𝑓1𝐶𝐽𝑧𝑛𝑖𝛿1𝑇1𝑧𝑛𝑖𝑥𝑛𝑖𝑥𝑛𝑖+1+̂𝑥1𝛼𝑛𝑥𝑛𝑖𝑓1𝐶𝐽𝑧𝑛𝑖𝛿1𝑇1𝑧𝑛𝑖.(4.35) By (4.25) and (4.31), we have ̂𝑥=𝑓1𝐶𝐽̂𝑧𝛿1𝑇1.̂𝑧(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 𝑇𝑗𝐶𝐸 for 𝑗=1,2,3 satisfy the following conditions:(i)𝑇𝑗𝑥,𝐽(𝐽𝑥𝛿𝑗𝑇𝑗𝑥)0, for all 𝑥𝐶 for 𝑗=1,2,3;(ii)(𝐽𝛿𝑗𝑇𝑗) are compact for 𝑗=1,2,3; then the system of mixed variational inequality (1.2) has a solution (̂𝑥,̂𝑦,̂𝑧) and sequences {𝑥𝑛},{𝑦𝑛}, and {z𝑛} defined by Algorithm 3.3 have convergent subsequences {𝑥𝑛𝑖},{𝑦𝑛𝑖}, and {𝑧𝑛𝑖} such that 𝑥𝑛𝑖̂𝑥,𝑖,𝑦𝑛𝑖̂𝑦,𝑖, and 𝑧𝑛𝑖̂𝑧,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 𝑇𝑗𝐶𝐻 and 𝑓𝑗𝐶{+} which is convex lower semicontinuous mappings for 𝑗=1,2,3 satisfy the following conditions:(i)𝑇𝑗𝑥,𝑥𝛿𝑗𝑇𝑗𝑥0 for 𝑗=1,2,3;(ii)𝑓𝑗(0)=0 and 𝑓𝑗(𝑥)0 for all 𝑥𝐶 for 𝑗=1,2,3; then the system of mixed variational inequality (1.6) has a solution (̂𝑥,̂𝑦,̂𝑧) and sequences {𝑥𝑛},{𝑦𝑛}, and {𝑧𝑛} defined by Algorithm 3.4 have a convergent subsequences {𝑥𝑛𝑖},{𝑦𝑛𝑖}, and {𝑧𝑛𝑖} such that 𝑥𝑛𝑖̂𝑥,𝑖,𝑦𝑛𝑖̂𝑦,𝑖, and 𝑧𝑛𝑖̂𝑧,𝑖.

Corollary 4.5. Let 𝐶 be a nonempty closed and convex subset of a Hilbert space 𝐻. If the mapping 𝑇𝑗𝐶𝐻 for 𝑗=1,2,3 satisfy the conditions: 𝑇𝑗𝑥,𝑥𝛿𝑗𝑇𝑗𝑥0 for 𝑗=1,2,3; then the system of mixed variational inequality (1.7) has a solution (̂𝑥,̂𝑦,̂𝑧) and sequences {𝑥𝑛},{𝑦𝑛}, and {𝑧𝑛} defined by Algorithm 3.5 have a convergent subsequences {𝑥𝑛𝑖},{𝑦𝑛𝑖}, and {𝑧𝑛𝑖} such that 𝑥𝑛𝑖̂𝑥,𝑖,𝑦𝑛𝑖̂𝑦,𝑖, and 𝑧𝑛𝑖̂𝑧,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).

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