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

Authors: Siwaporn Saewan; Poom Kumam
Journal: Journal of Applied Mathematics
 (2012)

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## 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 generalizedf 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.


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## Body

## 1. Introduction

LetE 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 PC: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 πC:E*→C and ΠE: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 thesystem of mixed variational inequalities (SMVI) which is to find x̂,ŷ,ẑ∈C such that(1.1)〈δ1T1ẑ+Jx̂-Jẑ,y-x̂〉+f1(y)-f1(x̂)≥0,∀y∈C,〈δ2T2x̂+Jŷ-Jx̂,y-ŷ〉+f2(y)-f2(ŷ)≥0,∀y∈C,〈δ3T3ŷ+Jẑ-Jŷ,y-ẑ〉+f3(y)-f3(ẑ)≥0,∀y∈C,
where δj>0,Tj:C→E*,fj: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.Iffj(x)=0 for j=1,2,3, for all x∈C, (1.1) is equivalent to find x̂, ŷ and ẑ∈C such that(1.2)〈δT1ẑ+Jx̂-Jẑ,y-x̂〉≥0,∀y∈C,〈δ2T2x̂+Jŷ-Jx̂,y-ŷ〉≥0,∀y∈C,〈δ3T3ŷ+Jẑ-Jŷ,y-ẑ〉≥0,∀y∈C.
The problem (1.2) is called the system of variational inequalities we denote by (SVI).IfT2=T3,f2(x)=f3(x),  for all x∈C and ŷ=ẑ, then (1.1) is reduced to find x̂,ŷ∈C such that(1.3)〈δ1T1ŷ+Jx̂-Jŷ,y-x̂〉+f1(y)-f1(x̂)≥0,∀y∈C,〈δ2T2x̂+Jŷ-Jx̂,y-ŷ〉+f2(y)-f2(ŷ)≥0,∀y∈C,
which is studied by Zhang et al. [7].IfT=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(1.4)〈Tx̂,y-x̂〉+f1(y)-f1(x̂)≥0,∀y∈C.
This iterative method is studied by Wu and Huang [5].Iff1(x)=0, for all x∈C, (1.4) is reduced to find x̂ such that(1.5)〈Tx̂,y-x̂〉≥0,∀y∈C,
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].IfE=H is a Hilbert space, then (1.1) is reduced to find x̂,ŷ,ẑ∈C such that(1.6)〈δ1T1ẑ+x̂-ẑ,y-x̂〉+f1(y)-f1(x̂)≥0,∀y∈C,〈δ2T2x̂+ŷ-x̂,y-ŷ〉+f2(y)-f2(ŷ)≥0,∀y∈C,〈δ3T3ŷ+ẑ-ŷ,y-ẑ〉+f3(y)-f3(ẑ)≥0,∀y∈C.
If fj(x)=0 for j=1,2,3, for all x∈C, (1.6) reduces to the following (SVI):(1.7)〈δ1T1ẑ+x̂-ẑ,y-x̂〉≥0,∀y∈C,〈δ2T2x̂+ŷ-x̂,y-ŷ〉≥0,∀y∈C,〈δ3T3ŷ+ẑ-ŷ,y-ẑ〉≥0,∀y∈C.
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 spaceE 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 limt→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 ρE:[0,∞)→[0,∞) defined by ρE(t)=sup{((∥x+y∥+∥x-y∥)/2)-1:∥x∥=1,∥y∥≤t}. The modulus of convexity of E is the function ηE:[0,2]→[0,1] defined by ηE(ɛ)=inf{1-∥(x+y)/2∥:x,y∈E,∥x∥=∥y∥=1,∥x-y∥≥ɛ}. The normalized duality mappingJ:E→2E* 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.IfE is a reflexive smooth and strictly convex Banach space and J*:E*→2E is the normalized duality mapping on E*, then J-1=J*, JJ*=IE* and J*J=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*.LetE 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]).
Letq>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
(2.1)‖λx+(1-λ)y‖q≤λ‖x‖q+(1-λ)‖y‖q-ςq(λ)g(‖x-y‖)
for all x,y∈Br={x∈E:∥x∥≤r} and λ∈[0,1], where ςq(λ)=λ(1-λ)q+λq(1-λ).For caseq=2, we have(2.2)‖λx+(1-λ)y‖q≤λ‖x‖2+(1-λ)‖y‖2-λ(1-λ)g(‖x-y‖).Lemma 2.2 (Chang [22]).
LetE be a uniformly convex and uniformly smooth Banach space. The following holds:
(2.3)‖ϕ+Φ‖2≤‖ϕ‖2+2〈Φ,J*(ϕ+Φ)〉,∀ϕ,Φ∈E*.Next we recall the concept of the generalizedf-projection operator. Let G:E*×C→ℝ∪{+∞} be a functional defined as follows:(2.4)G(ξ,x)=‖ξ‖2-2〈ξ,x〉+‖x‖2+2ρf(x),
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.
LetE be a real Banach space with its dual E*. Let C be a nonempty closed convex subset of E. It is said that ΠCf:E*→2C is the generalizedf-projection operator if
(2.5)∏Cfξ={u∈C:G(ξ,u)=inf⁡y∈CG(ξ,y)},∀ξ∈E*.In this paper, we fixedρ=1, we have(2.6)G(ξ,x)=‖ξ‖2-2〈ξ,x〉+‖x‖2+2f(x).For the generalizedf-projection operator, Wu and Hung [5] proved the following basic properties.Lemma 2.4 (Wu and Hung [4]).
LetE be a reflexive Banach space with its dual E* and C is a nonempty closed convex subset of E. The following statements hold:(1)
ΠCfξ is nonempty closed convex subset of C for all ξ∈E*;(2)
ifE is smooth, then for all ξ∈E*, x∈ΠCfξ if and only if
(2.7)〈ξ-Jx,x-y〉+ρf(y)-ρf(x)≥0,∀y∈C;(3)
ifE is smooth, then for any ξ∈E*, ΠCfξ=(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]).
Iff(x)≥0 for all x∈C, then for any ρ>0,
(2.8)G(Jx,y)≤G(ξ,y)+2ρf(y),∀ξ∈E*,y∈C,x∈∏Cfξ.Lemma 2.6 (Fan et al. [6]).
LetE 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)
ΠCf:E*→C is single valued and norm to weak continuous;(2)
ifE has the property (h), that is, for any sequence {xn}⊂E,xn⇀x∈E and ∥xn∥→∥x∥, imply that xn→x, then ΠCf:E*→C is continuous.Defined the functionalG2:E×C→ℝ∪{+∞} by(2.9)G2(x,y)=G(Jx,y),∀x∈E,y∈C.


## 3. Generalized Projection Algorithms

Proposition 3.1.
LetC be a nonempty closed and convex subset of a reflexive strictly convex and smooth Banach space E. If fj:C→ℝ∪{+∞} for j=1,2,3 is proper, convex, and lower semicontinuous, then (x̂,ŷ,ẑ) is a solution of (SMVI) equivalent to finding x̂,ŷ,ẑ such that
(3.1)x̂=∏Cf1(Jẑ-δ1T1ẑ),ŷ=∏Cf2(Jx̂-δ2T1x̂),ẑ=∏Cf3(Jŷ-δ3T1ŷ).Proof.
From Lemma2.4 (2) and E is a reflexive strictly convex and smooth Banach space, we known that J is single valued and ΠCfj 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 pointx0,z0∈C, define the sequences{xn},{yn},and{zn}as follows:(3.2)xn+1=(1-αn)xn+αn∏Cf1(Jzn-δ1T1zn),yn+1=∏Cf2(Jxn+1-δ2T2xn+1),zn+1=∏Cf3(Jyn+1-δ3T3yn+1),
where 0<a≤αn≤b<1.Iffj(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 pointx0,z0∈C, define the sequences{xn},{yn},and{zn}as follows:(3.3)xn+1=(1-αn)xn+αn∏C(Jzn-δ1T1zn),yn+1=∏C(Jxn+1-δ2T2xn+1),zn+1=∏C(Jyn+1-δ3T3yn+1),
where 0<a≤αn≤b<1.For solving the problem (1.6), we defined the algorithm as follows:IfE=H is a Hilbert space, then Algorithm 3.2 reduces to the following.Algorithm 3.4.
For an initial pointx0,z0∈C, define the sequences{xn},{yn},and{zn}as follows:(3.4)xn+1=(1-αn)xn+αn∏Cf1(Jzn-δ1T1zn),yn+1=∏Cf2(Jxn+1-δ2T2xn+1),zn+1=∏Cf3(Jyn+1-δ3T3yn+1),
where 0<a≤αn≤b<1.Iffj(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 pointx0,z0∈C, define the sequences{xn},{yn},
and{zn}as follows:(3.5)xn+1=(1-αn)xn+αnPC(Jzn-δ1T1zn),yn+1=PC(Jxn+1-δ2T2xn+1),zn+1=PC(Jyn+1-δ3T3yn+1),
where 0<a≤αn≤b<1.


## 4. Existence and Convergence Analysis

Theorem 4.1.
LetC be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space E with dual space E*. If the mapping Tj:C→E* and fj:C→ℝ∪{+∞} which is convex lower semicontinuous mappings for j=1,2,3 satisfying the following conditions:(i)
〈Tjx,J*(Jx-δjTjx)〉≥0,  for all x∈C for j=1,2,3;(ii)
(J-δjTj) are compact for j=1,2,3;(iii)
fj(0)=0 and fj(x)≥0,  for all x∈C and j=1,2,3; then the system of mixed variational inequality (1.1) has a solution (x̂,ŷ,ẑ) and sequences {xn},{yn}, and {zn} defined by Algorithm 3.2 have convergent subsequences {xni},{yni}, and {zni} such that
(4.1)xni⟶x̂,i⟶∞,yni⟶ŷ,i⟶∞,zni⟶ẑ,i⟶∞.Proof.
SinceE 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, ΠCfj 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 x∈C and y=0, we have
(4.2)G2(x,0)=G(Jx,0)=‖Jx‖2-2〈Jx,0〉+2f(0)=‖Jx‖2=‖x‖2.
By (4.2) and Lemma 2.5, we have
(4.3)G2(∏Cf1(Jzn-δ1T1zn),0)=G(J(∏Cf1(Jzn-δ1T1zn)),0)≤G(Jzn-δ1T1zn,0)=‖Jzn-δ1T1zn‖2.
From Lemma 2.2, and for all x∈C, 〈T1x,J*(Jx-δ1T1x)〉≥0, so for zn∈C, we obtain
(4.4)‖Jzn-δ1T1zn‖2≤‖Jzn‖2-2〈δ1T1zn,J*(Jzn-δ1T1zn)〉≤‖Jzn‖2≤‖zn‖2.
Again by Lemma 2.2, for all x∈C,〈T2x,J*(Jx-δ2T2x)〉≥0, and for xn+1∈C, we have
(4.5)‖yn+1‖2=G2(yn+1,0)=G(Jyn+1,0)=G(J∏Cf2(Jxn+1-δ2T2xn+1),0)≤G(Jxn+1-δ2T2xn+1,0)≤‖Jxn+1-δ2T2xn+1‖2≤‖Jxn+1‖2-2〈δ2T2xn+1,J*(Jxn+1-δ2T2xn+1)〉≤‖Jxn+1‖2≤‖xn+1‖2.
In similar way, for all x∈C, 〈T3x,J*(Jx-δ3T3x)〉≥0, and zn+1∈C, we also have
(4.6)‖zn+1‖2=G(Jzn+1,0)≤G(Jyn+1-δ3T3yn+1,0)=‖Jyn+1-δ3T3yn+1‖2≤‖Jyn+1‖2-2〈δ3T3yn+1,J*(Jyn+1-δ3T3yn+1)〉≤‖yn+1‖2.
It follows from (4.5) and (4.6) that
(4.7)‖zn+1‖2≤‖xn+1‖2,∀n∈N.
From (4.5) and (4.6), we compute
(4.8)‖xn+1‖2≤(1-αn)‖xn‖+αn‖∏Cf1(Jzn-δ1T1zn)‖≤(1-αn)‖xn‖+αn‖zn‖≤(1-αn)‖xn‖+αn‖yn‖≤(1-αn)‖xn‖+αn‖xn‖=‖xn‖.
This implies that the sequences {xn},{yn},{zn}, and {ΠCf1(Jzn-δ1T1zn)} are bounded. For a positive number r such that {xn},{yn},{zn},{ΠCf1(Jzn-δ1T1zn)}∈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
(4.9)‖xn+1‖2=‖(1-αn)xn+αn∏Cf1(Jzn-δ1T1zn)‖2≤(1-αn)‖xn‖2+αn‖∏Cf1(Jzn-δ1T1zn)‖2-αn(1-αn)g‖xn-∏Cf1(Jzn-δ1T1zn)‖=(1-αn)‖xn‖2+αnG2(∏Cf1(Jzn-δ1T1zn,0))-αn(1-αn)g‖xn-∏Cf1(Jzn-δ1T1zn)‖.
Applying (4.3), (4.4), and (4.7), we have
(4.10)αn(1-αn)g‖xn-∏Cf1(Jzn-δ1T1zn)‖≤(1-αn)‖xn‖2-‖xn+1‖2+αnG2(∏Cf1(Jzn-δ1T1zn),0)≤(1-αn)‖xn‖2-‖xn+1‖2+αn‖xn‖2=‖xn‖2-‖xn+1‖2.
Summing (4.10), for n=0,1,2,3,…,k, we have
(4.11)∑n=0kαn(1-αn)g‖xn-∏Cf1(Jzn-δ1T1zn)‖≤‖x0‖2-‖xk+1‖2≤‖x0‖2,
taking k→∞, we get
(4.12)∑n=0∞αn(1-αn)g‖xn-∏Cf1(Jzn-δ1T1zn)‖≤‖x0‖2.
This shows that series (4.12) is converge, we obtain that
(4.13)lim⁡n→∞αn(1-αn)g‖xn-∏Cf1(Jzn-δ1T1zn)‖=0.
From 0<a≤αn≤b<1 for all n, thus ∑n=0∞αn(1-αn)>0 and (4.13), we have
(4.14)lim⁡n→∞g‖xn-∏Cf1(Jzn-δ1T1zn)‖=0.
By property of functional g, we have
(4.15)lim⁡n→∞‖xn-∏Cf1(Jzn-δ1T1zn)‖=0.
Since {zn} is bounded sequence and (J-δ1T1) is compact on C, then sequence {Jzn-δ1T1zn} has a convergence subsequence such that
(4.16){Jzni-δ1T1zni}⟶w0∈E*asi⟶∞.
By the continuity of the ΠCf1, we have
(4.17)lim⁡i→∞∏Cf1(Jzni-δ1T1zni)=∏Cf1(w0).
Again since {xn},{yn} are bounded and (J-δ2T2),(J-δ3T3) are compact on C, then sequences {Jxn-δ2T2xn} and {Jyn-δ3T3yn} have convergence subsequences such that
(4.18){Jxni-δ2T2xni}⟶u0∈E*asi⟶∞,{Jyni-δ3T3yni}⟶v0∈E*asi⟶∞.
By the continuity of ΠCf2 and ΠCf3, we have
(4.19)lim⁡i→∞∏Cf2(Jxni-δ2T2xni)=∏Cf2(u0),(4.20)lim⁡i→∞∏Cf3(Jyni-δ3T3yni)=∏Cf3(v0).
Let
(4.21)∏Cf1(w0)=x̂,(4.22)∏Cf2(u0)=ŷ,(4.23)∏Cf3(v0)=ẑ.
By using the triangle inequality, we have
(4.24)‖xni-x̂‖≤‖xni-∏Cf1(Jzni-δ1T1zni)‖+‖∏Cf1(Jzni-δ1T1zni)-x̂‖.
From (4.15) and (4.17), we have
(4.25)lim⁡i→∞xni=x̂.
By definition of zni, we get
(4.26)‖zni-ẑ‖≤‖∏Cf3(Jyni-δ3T3yni)-ẑ‖.
It follows by (4.20) and (4.23), we obtain
(4.27)lim⁡i→∞zni=ẑ.
In the same way, we also have
(4.28)lim⁡i→∞yni=ŷ.
By the continuity properties of (J-δ1T1),(J-δ2T2),(J-δ3T3), and ΠCfj for j=1,2,3. We conclude that
(4.29)x̂=∏Cf1(Jẑ-δ1T1ẑ),ŷ=∏Cf2(Jx̂-δT2x̂),ẑ=∏Cf3(Jŷ-δ3T3ŷ).
This completes of proof.Theorem 4.2.
LetC be a nonempty compact and convex subset of a uniformly convex and uniformly smooth Banach space E with dual space E*. If the mapping Tj:C→E* and fj:C→ℝ∪{+∞} which is convex lower semicontinuous mappings for j=1,2,3 satisfy the following conditions:(i)
〈Tjx,J*(Jx-δjTjx)〉≥0, for all x∈C for j=1,2,3;(ii)
fj(0)=0 and fj(x)≥0,  for all x∈C for j=1,2,3; then the system of mixed variational inequality (1.1) has a solution (x̂,ŷ,ẑ) and sequences {xn},{yn}, and {zn} defined by Algorithm 3.2 have a convergent subsequences {xni},{yni}, and {zni} such that
(4.30)xni⟶x̂,i⟶∞,yni⟶ŷ,i⟶∞,zni⟶ẑ,i⟶∞.Proof.
In the same way to the proof in Theorem4.1, we have
(4.31)lim⁡n→∞‖xn-∏Cf1(Jzn-δ1T1zn)‖=0.
Hence, there exist subsequences {xni}⊂{xn} and {zni}⊂{zn} such that
(4.32)lim⁡i→∞‖xni-∏Cf1(Jzni-δ1T1zni)‖=0.
From the compactness of C, we have that
(4.33){xni}⟶x̂asi⟶∞,{zni}⟶ẑasi⟶∞,
where x̂,ẑ are points in C. Also, for a sequence {yn}⊃{yni}→ŷasi→∞, where ŷ is a points in C. By the continuity properties of J,T2,T3ΠCf2, and ΠCf3, we obtain that
(4.34)ŷ=∏Cf2(Jx̂-δ2T2x̂),ẑ=∏Cf3(Jŷ-δ3T3ŷ).
From definition of xn+1, we get
(4.35)‖∏Cf1(Jzni-δ1T1zni)-x̂‖=‖∏Cf1(Jzni-δ1T1zni)-x̂+xni+1-(1-αn)xni-αn∏Cf1(Jzni-δ1T1zni)‖=‖xni+1-x̂+(1-αn)∏Cf1(Jzni-δ1T1zni)-xni‖≤‖xni+1-x̂‖+(1-αn)‖xni-∏Cf1(Jzni-δ1T1zni)‖.
By (4.25) and (4.31), we have
(4.36)x̂=∏Cf1(Jẑ-δ1T1ẑ).
This completes of proof.Corollary 4.3.
LetC be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space E with dual space E*. If the mapping Tj:C→E* for j=1,2,3 satisfy the following conditions:(i)
〈Tjx,J*(Jx-δjTjx)〉≥0,  for all x∈C for j=1,2,3;(ii)
(J-δjTj) are compact for j=1,2,3; then the system of mixed variational inequality (1.2) has a solution (x̂,ŷ,ẑ) and sequences {xn},{yn}, and {zn} defined by Algorithm 3.3 have convergent subsequences {xni},{yni}, and {zni} such that xni→x̂,i→∞,yni→ŷ,i→∞, and zni→ẑ,i→∞.IfE=H is a Hilbert space, then H*=H,J*=J=I, so one obtains the following corollary.Corollary 4.4.
LetC be a nonempty closed and convex subset of a Hilbert space H. If the mapping Tj:C→H and fj:C→ℝ∪{+∞} which is convex lower semicontinuous mappings for j=1,2,3 satisfy the following conditions:(i)
〈Tjx,x-δjTjx〉≥0  for j=1,2,3;(ii)
fj(0)=0 and fj(x)≥0  for all x∈C  for j=1,2,3; then the system of mixed variational inequality (1.6) has a solution (x̂,ŷ,ẑ) and sequences {xn},{yn}, and {zn} defined by Algorithm 3.4 have a convergent subsequences {xni},{yni}, and {zni} such that xni→x̂,i→∞,yni→ŷ,i→∞, and zni→ẑ,i→∞.Corollary 4.5.
LetC be a nonempty closed and convex subset of a Hilbert space H. If the mapping Tj:C→H for j=1,2,3 satisfy the conditions: 〈Tjx,x-δjTjx〉≥0 for j=1,2,3; then the system of mixed variational inequality (1.7) has a solution (x̂,ŷ,ẑ) and sequences {xn},{yn}, and {zn} defined by Algorithm 3.5 have a convergent subsequences {xni},{yni}, and {zni} such that xni→x̂,i→∞,yni→ŷ,i→∞, and zni→ẑ,i→∞.Remark 4.6.
Theorems4.1 and 4.2 and Corollary 4.3 extend and improve the results of Zhang et al. [7] and Wu and Huang [5].



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*Source: 413468-2012-04-09.xml*