# A Hybrid Iterative Scheme for Mixed Equilibrium Problems, General System of Variational Inequality Problems, and Fixed Point Problems in Hilbert Spaces

Authors: S. Imnang; S. Suantai
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

We introduce a new iterative scheme for finding a common element of the set of solutions of a general system of variational inequalities, the set of solutions of a mixed equilibrium problem, and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Using the demiclosedness principle for nonexpansive mappings, we prove that the iterative sequence converges strongly to a common element of the above three sets under some control conditions, and we also give some examples for mappings which satisfy conditions of the main result.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and C a nonempty closed convex subset of H. A mapping T:C→C is said to be nonexpansive mapping if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C. The fixed point set of T is denoted by F(T):={x∈C:Tx=x}.Halpern [1] studied the following iteration formula for approximating a fixed point of T. For an arbitrary v∈C, let the sequence {xn} be defined by x1∈C,(1.1)xn+1=αnv+(1-αn)Txn,n≥1,
where {αn} is a sequence of real numbers in [0,1] that satisfies the following conditions: C1:limn→∞αn=0 and C2:∑n=1∞αn=∞.Actually, Halpern studied the special case of (1.1) in which αn=n-σ, σ∈(0,1), and v=0 and proved that {xn} converges strongly to a fixed point of T. Under a different restriction on the parameter {αn}, Lions [2] improved the result of Halpern, still in Hilbert spaces. He proved strong convergence of {xn} to a fixed point of T, where the real sequence {αn} satisfies the following conditions: (1.2)C1:lim⁡n→∞αn=0,C2:∑n=1∞αn=∞,C3:lim⁡n→∞|αn+1-αn|αn+12=0.
Reich [3] proved that the result of Halpern remains true when H is uniformly smooth. It was observed that both Halpern's and Lion's conditions on the real sequence {αn} excluded the canonical choice αn=1/(n+1). This was overcome by Wittmann [4] who proved, still in Hilbert spaces, the strong convergence of {xn} to a fixed point of T if {αn} satisfies the following conditions: (1.3)C1:lim⁡n→∞αn=0,C2:∑n=1∞αn=∞,C3:∑n=1∞|αn+1-αn|<∞.Recall that the classical variational inequality, denoted byVI(C,A), is to find an x*∈C such that (1.4)〈Ax*,y-x*〉≥0,∀y∈C.
The variational inequality, nonconvex variational inequality, and mixed variational inequality have been widely studied in the literature; see, for example, Ceng et al. [5–14], Chang et al. [15], Noor [16–18], Peng and Yao [19–21], Plubtieng and Punpaeng [22], Yao et al. [23], Zeng and Yao [24], Zhao and He [25], and the references therein.For solving the variational inequality problem in the finite-dimensional Euclidean spaceℝn under the assumption that a set C⊂ℝn is closed and convex, a mapping A of C into ℝn is monotone and k-Lipschitz continuous, and VI(C,A) is nonempty, Korpelevich [26] introduced the following so-called extragradient method: (1.5)x0=x∈C,yn=PC(xn-λAxn),xn+1=PC(xn-λAyn),
for every n=0,1,2,…, where λ∈(0,1/k) and PC is the projection of ℝn onto C. He showed that the sequences {xn} and {yn} generated by this iterative process converge to the same point z∈VI(C,A). In 2007, Y. Yao and J. C. Yao [27] introduced a new iterative scheme for finding the common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings in a real Hilbert space.Let C be a nonempty closed convex subset of a real Hilbert spaceH.LetA,B:C→H be two mappings. In 2008, Ceng et al. [12] considered the following problem of finding (x*,y*)∈C×C such that(1.6)〈λAy*+x*-y*,x-x*〉≥0,∀x∈C,〈μBx*+y*-x*,x-y*〉≥0,∀x∈C,
which is called a general system of variational inequalities, where λ>0 and μ>0 are two constants. In particular, if A=B, then the problem (1.6) reduces to finding (x*,y*)∈C×C such that(1.7)〈λAy*+x*-y*,x-x*〉≥0,∀x∈C,〈μAx*+y*-x*,x-y*〉≥0,∀x∈C,
which was defined by Verma [28] and is called the new system of variational inequalities. Further, if we add up the requirement that x*=y*, then the problem (1.7) reduces to the classical variational inequality VI(C,A). Ceng et al. [12] introduced and studied a relaxed extragradient method for finding a common element of the set of solutions of the problem (1.6) for the α- and β-inverse-strongly monotone mappings and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Let x1=v∈C, and {xn}, {yn} are given by (1.8)yn=PC(xn-μBxn),xn+1=anv+bnxn+(1-an-bn)SPC(yn-λAyn),n≥1,
where λ∈(0,2α), μ∈(0,2β), and {an},{bn}⊂[0,1]. Then, they proved that the sequence {xn} converges strongly to a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the problem (1.6) under some control conditions.Letφ:C→ℝ∪{+∞} be a proper extended real-valued function and F a bifunction from C×C to ℝ, where ℝ is the set of real numbers. Ceng and Yao [13] considered the following mixed equilibrium problem:(1.9)findx∈Csuch thatF(x,y)+φ(y)≥φ(x),∀y∈C.The set of solutions of (1.9) is denoted by MEP(F,φ). It is easy to see that x is a solution of the problem (1.9) impling that x∈domφ={x∈C∣φ(x)<+∞}.Ifφ=0, then the mixed equilibrium problem (1.9) becomes the following equilibrium problem:(1.10)findx∈Csuch thatF(x,y)≥0,∀y∈C.
If F=0, then the mixed equilibrium problem (1.9) reduces to the convex minimization problem(1.11)findx∈Csuch thatφ(y)≥φ(x),∀y∈C.The set of solutions of (1.10) is denoted by EP(F).Ifφ=0 and F(x,y)=〈Ax,y-x〉 for all x,y∈C, where A is a mapping from C into H, then (1.9) reduces to the classical variational inequality and EP(F)=VI(C,A).The equilibrium problems and variational inequality problems can be applied for the problems in science and technology, economics, optimizations, and control theory. The problems of finding a solution of the systems of variational inequalities which is also a solution of the mixed equilibrium problems and the problem of finding a solution of the mixed equilibrium problems under the constraints that it is also a solution of the system of variational inequalities are very useful and important for studying problems in science and applied science.Inspired and motivated by these facts, we introduce a new iteration process for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a nonexpansive mapping, and the set of solutions of a general system of variational inequalities in a real Hilbert space. Start with an arbitraryv∈C, and let x1∈C, {xn},{un}, and {yn} be the sequences generated by(1.12)F(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=PC(un-μBun),xn+1=anv+(1-an)TPC(yn-λAyn),n≥1,
where λ>0 and μ>0 are two constants, {rn}⊂(0,∞) and {an}⊂[0,1]. Using the demiclosedness principle for nonexpansive mappings, we will show that the sequence {xn} converges strongly to a common element of the above three sets under some control conditions.


## 2. Preliminaries

In this section, we recall the well-known results and give some useful lemmas that will be used in the next section.LetH be a real Hilbert space with inner product 〈·,·〉 and C a nonempty closed convex subset of H.A mappingA:C→H is called monotone if (2.1)〈Ax-Ay,x-y〉≥0,∀x,y∈C.A is called α-strongly monotone if there exists a positive real number α>0 such that (2.2)〈Ax-Ay,x-y〉≥α‖x-y‖2,∀x,y∈C.
This implies that (2.3)‖Ax-Ay‖≥α‖x-y‖,
that is, A is α-expansive and, when α=1, it is expansive.We can see easily that the following implications in monotonicity, strong monotonicity, and expansiveness hold:(2.4)strongmonotonicity⇒monotonicity⇓expansiveness.The mappingA is called L-Lipschitz continuous (or Lipschitzian) if there exists a constant L≥0 such that (2.5)‖Ax-Ay‖≤L‖x-y‖,∀x,y∈C.A is called α-inverse-strongly monotone (or α-cocoercive) if there exists a positive real number α>0 such that (2.6)〈Ax-Ay,x-y〉≥α‖Ax-Ay‖2,∀x,y∈C.
It is obvious that every α-inverse-strongly monotone mapping A is monotone and Lipschitz continuous. Also, if A is α-strongly and L-Lipschitz continuous, then A is (α/L2)-inverse-strongly monotone, but inverse-strongly monotone need not be strongly monotone.A mappingA is called relaxed c-cocoercive, if there exists a constant c>0 such that (2.7)〈Ax-Ay,x-y〉≥(-c)‖Ax-Ay‖2,∀x,y∈C.
A mapping A is called relaxed (c,d)-cocoercive, if there exist two constants c,d>0 such that (2.8)〈Ax-Ay,x-y〉≥(-c)‖Ax-Ay‖2+d‖x-y‖2,∀x,y∈C.
For c=0, A is d-strongly monotone. This class of mappings is more general than the class of strongly monotone mappings. As a result, we have the following implication: d-strong monotonicity ⇒ relaxed (c,d)-cocoercivity.It is known that if the operator is Lipschitz continuous, then the relaxed cocoercivity is strongly monotone, but the strongly monotone does not imply the cocoercivity as shown in the following example.Example 2.1.
LetH=R, C=[1,∞), and A:C→H be defined by Ax=x2, x∈C. For x,y∈C, we have
(2.9)〈Ax-Ay,x-y〉=(x2-y2)(x-y)=(x+y)|x-y|2≥2|x-y|2.
Hence, A is 2-strongly monotone. If A is μ-cocoercive for some μ>0, then 〈Ax-Ay,x-y〉=(x+y)|x-y|2≥μ|x2-y2|2. This implies x+y≤1/μ for all x,y∈[1,∞) which is a contradiction. Hence, A is not μ-cocoercive for any μ>0.For every pointx∈H, there exists a unique nearest point in C, denoted by PCx, such that (2.10)‖x-PCx‖≤‖x-y‖,∀y∈C.PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive mapping of H onto C and satisfies(2.11)〈x-y,PCx-PCy〉≥‖PCx-PCy‖2,∀x,y∈H.
Obviously, this immediately implies that(2.12)‖(x-y)-(PCx-PCy)‖2≤‖x-y‖2-‖PCx-PCy‖2,∀x,y∈H.
Recall that PCx is characterized by the following properties: PCx∈C and(2.13)〈x-PCx,y-PCx〉≤0,‖x-y‖2≥‖x-PCx‖2+‖PCx-y‖2,
for all x∈H and y∈C; see Goebel and Kirk [29] for more details.For solving the mixed equilibrium problem, let us assume the following assumptions for the bifunctionF,φ and the set C:(A1)
F(x,x)=0 for all x∈C;(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0 for all x,y∈C;(A3)
for eachy∈C, x↦F(x,y) is weakly upper semicontinuous;(A4)
for eachx∈C, y↦F(x,y) is convex;(A5)
for eachx∈C, y↦F(x,y) is lower semicontinuous;(B1)
for eachx∈H and r>0, there exist a bounded subset Dx⊆C and yx∈C such that for any z∈C∖Dx(2.14)F(z,yx)+φ(yx)+1r〈yx-z,z-x〉<φ(z);(B2)
C is a bounded set.In the sequel we will need to use the following lemma.Lemma 2.2 (see [30]).
LetC be a nonempty closed convex subset of H. Let F be a bifunction from C×C to ℝ satisfying (A1)–(A5), and let φ:C→ℝ∪{+∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r>0 and x∈H, define a mapping Tr:H→C as follows
(2.15)Tr(x)={z∈C:F(z,y)+φ(y)+1r〈y-z,z-x〉≥φ(z),∀y∈C}
for all x∈H. Then the following conclusions hold:(1)
for eachx∈H, Tr(x)≠∅;(2)
Tr is single valued;(3)
Tr is firmly nonexpansive, that is, for any x,y∈H,
(2.16)‖Tr(x)-Tr(y)‖2≤〈Trx-Try,x-y〉;(4)
F(Tr)=MEP(F,φ);(5)
MEP(F,φ) is closed and convex.We also need the following lemmas.Lemma 2.3 (see [31]).
Let{an} be a sequence of nonnegative real numbers satisfying the property
(2.17)an+1≤(1-tn)an+bn+tncn,∀n≥1,
where {tn}, {bn}, and {cn} satisfy the restrictions
(2.18)(i)∑n=1∞tn=∞,(ii)∑n=1∞bn<∞,(iii)limsup⁡n→∞cn≤0.
Then limn→∞an=0.Lemma 2.4 (see [32]).
Let(H,〈·,·〉) be an inner product space. Then, for all x,y,z∈H and α,β,γ∈[0,1] with α+β+γ=1, one has
(2.19)‖αx+βy+γz‖2=α‖x‖2+β‖y‖2+γ‖z‖2-αβ‖x-y‖2-αγ‖x-z‖2-βγ‖y-z‖2.Lemma 2.5 (see [29] (demiclosedness principle)).
Assume thatT is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space H. If T has a fixed point, then I-T is demiclosed: that is, whenever {xn} is a sequence in C converging weakly to some x∈C (for short, xn⇀x∈C) and the sequence {(I-T)xn} converges strongly to some y (for short, (I-T)xn→y), it follows that (I-T)x=y. Here I is the identity operator of H.The following lemma is an immediate consequence of an inner product.Lemma 2.6.
In a real Hilbert spaceH, there holds the inequality
(2.20)‖x+y‖2≤‖x‖2+2〈y,x+y〉,∀x,y∈H.In 2009, Kangtunyakarn and Suantai [33] introduced a new mapping called the S-mapping as follows.Let{Ti}i=1N be a finite family of nonexpansive mappings of C into itself. For each n∈ℕ, and j=1,2,…,N, let αj(n)=(α1n,j,α2n,j,α3n,j) be such that α1n,j,α2n,j,α3n,j∈[0,1] with α1n,j+α2n,j+α3n,j=1. Very recently, in 2009, Kangtunyakarn and Suantai [33] introduced the new mapping Sn:C→C as follows: (2.21)Un,0=I,Un,1=α1n,1T1Un,0+α2n,1Un,0+α3n,1I,Un,2=α1n,2T2Un,1+α2n,2Un,1+α3n,2I,Un,3=α1n,3T3Un,2+α2n,3Un,2+α3n,3I,⋮Un,N-1=α1n,N-1TN-1Un,N-2+α2n,N-1Un,N-2+α3n,N-1I,Sn=Un,N=α1n,NTNUn,N-1+α2n,NUn,N-1+α3n,NI.
The mapping Sn is called the S-mapping generated by T1,T2,…,TN and α1(n),α2(n),…,αN(n). Nonexpansivity of each Ti ensures the nonexpansivity of Sn. They also showed the following useful fact.Lemma 2.7 (see [33]).
LetC be a nonempty closed convex subset of a strictly convex Banach space X. Let {Ti}i=1N be a finite family of nonexpansive mappings of C into itself with ∩i=1NF(Ti)≠∅, and let αj=(α1j,α2j,α3j), j=1,2,…,N, where α1j,α2j,α3j∈[0,1], α1j+α2j+α3j=1, α1j∈(0,1) for all j=1,2,…,N-1, α1N∈(0,1] and α2j,α3j∈[0,1) for all j=1,2,…,N. Let S be the S-mapping generated by T1,T2,…,TN and α1,α2,…,αN. Then F(S)=∩i=1NF(Ti).Lemma 2.8 (see [12]).
For givenx*,y*∈C, (x*,y*) is a solution of the problem (1.6) if and only if x* is a fixed point of the mapping G:C→C defined by
(2.22)G(x)=PC[PC(x-μBx)-λAPC(x-μBx)],∀x∈C,
where y*=PC(x*-μBx*).Throughout this paper, the set of fixed points of the mappingG is denoted by GVI(C,A,B).Next, we prove a lemma which is very useful for our consideration.Lemma 2.9.
LetG:C→C be defined by
(2.23)G(x)=PC[PC(x-μBx)-λAPC(x-μBx)],∀x∈C,
where λ,μ>0 and A,B:C→H are two mappings. If I-λA and I-μB are nonexpansive mappings, then G is nonexpansive.Proof.
For anyx,y∈C, we have
(2.24)‖G(x)-G(y)‖=‖PC[PC(x-μBx)-λAPC(x-μBx)]-PC[PC(y-μBy)-λAPC(y-μBy)]‖2≤‖PC(x-μBx)-λAPC(x-μBx)-(PC(y-μBy)-λAPC(y-μBy))‖=‖(I-λA)PC(I-μB)x-(I-λA)PC(I-μB)y‖≤‖x-y‖.
This shows that G is a nonexpansive mapping.


## 3. Main Results

In this section, we prove strong convergence theorems of the iterative scheme (1.12) to a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a nonexpansive mapping, and the set of solutions of a general system of variational inequality in a real Hilbert space.Theorem 3.1.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let F be a function from C×C to ℝ satisfying (A1)–(A5) and φ:C→ℝ∪{+∞} a proper lower semicontinuous and convex function. Let λ,μ>0, and let A,B:C→H be such that I-λA and I-μB are nonexpansive mappings. Let T be a nonexpansive self-mapping of C such that Ω=F(T)∩GVI(C,A,B)∩MEP(F,φ)≠∅. Assume that either (B1) or (B2) holds and that v is an arbitrary point in C. Let the sequences {xn}, {yn}, and {un} be defined by (1.12), {rn}⊂(0,∞) and {an}⊂[0,1] such that(C1)
limn→∞an=0, limn→∞(an+1/an)=1, and ∑n=1∞an=∞,(C2)
liminfn→∞rn>0 and ∑n=1∞|rn+1-rn|<∞.If limn→∞∥Ayn-Ay*∥=0 and limn→∞∥Bun-Bx*∥=0 for all x*∈Ω and y*=PC(x*-μBx*), then {xn} converges strongly to x¯=PΩv and (x¯,y¯) is a solution of the problem (1.6), where y¯=PC(x¯-μBx¯).Proof.
Letx*∈Ω and {Trn} a sequence of mappings defined as in Lemma 2.2. It follows from Lemma 2.8 that
(3.1)x*=PC[PC(x*-μBx*)-λAPC(x*-μBx*)].
Put y*=PC(x*-μBx*) and tn=PC(yn-λAyn), then x*=PC(y*-λAy*) and
(3.2)xn+1=anv+(1-an)Ttn.
Since I-λA, I-μB, Trn, and PC are nonexpansive mappings, we have
(3.3)‖tn-x*‖2=‖PC(yn-λAyn)-PC(y*-λAy*)‖2≤‖yn-y*‖2=‖PC(un-μBun)-PC(x*-μBx*)‖2≤‖un-x*‖2=‖Trnxn-Trnx*‖2≤‖xn-x*‖2,
which implies that
(3.4)‖xn+1-x*‖=‖anv+(1-an)Ttn-x*‖≤an‖v-x*‖+(1-an)‖tn-x*‖≤an‖v-x*‖+(1-an)‖xn-x*‖≤max⁡{‖v-x*‖,‖x1-x*‖}.
Thus, {xn} is bounded. Consequently, the sequences {un}, {yn}, {tn}, {Ayn}, {Bun}, and {Ttn} are also bounded. Also, observe that
(3.5)‖tn+1-tn‖=‖PC(yn+1-λAyn+1)-PC(yn-λAyn)‖≤‖yn+1-yn‖=‖PC(un+1-μBun+1)-PC(un-μBun)‖≤‖un+1-un‖.
On the other hand, from un=Trnxn∈domφ and un+1=Trn+1xn+1∈domφ, we have
(3.6)F(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,(3.7)F(un+1,y)+φ(y)-φ(un+1)+1rn+1〈y-un+1,un+1-xn+1〉≥0,∀y∈C.
Putting y=un+1 in (3.6) and y=un in (3.7), we have
(3.8)F(un,un+1)+φ(un+1)-φ(un)+1rn〈un+1-un,un-xn〉≥0,F(un+1,un)+φ(un)-φ(un+1)+1rn+1〈un-un+1,un+1-xn+1〉≥0.
From the monotonicity of F, we obtain that
(3.9)〈un+1-un,un-xnrn-un+1-xn+1rn+1〉≥0,
and hence
(3.10)〈un+1-un,un-un+1+un+1-xn-rnrn+1(un+1-xn+1)〉≥0.
Without loss of generality, let us assume that there exists a real number c such that rn>c>0 for all n∈ℕ. Then, we have
(3.11)‖un+1-un‖2≤〈un+1-un,xn+1-xn+(1-rnrn+1)(un+1-xn+1)〉≤‖un+1-un‖(‖xn+1-xn‖+|1-rnrn+1|‖un+1-xn+1‖),
and hence
(3.12)‖un+1-un‖≤‖xn+1-xn‖+1rn+1|rn+1-rn|‖un+1-xn+1‖≤‖xn+1-xn‖+1c|rn+1-rn|‖un+1-xn+1‖.
It follows from (3.5) and (3.12) that
(3.13)‖tn+1-tn‖≤‖xn+1-xn‖+1c|rn+1-rn|‖un+1-xn+1‖.
Next, we show that ∥xn+1-xn∥→0 as n→∞. From (3.13), we have
(3.14)‖xn+2-xn+1‖=‖an+1v+(1-an+1)Ttn+1-anv-(1-an)Ttn‖=‖(an+1-an)v+(an-an+1)Ttn+1+(1-an)(Ttn+1-Ttn)‖≤|an+1-an|(‖v‖+‖Ttn+1‖)+(1-an)‖tn+1-tn‖≤|an+1-an|(‖v‖+‖Ttn+1‖)+(1-an)‖xn+1-xn‖+1c|rn+1-rn|‖un+1-xn+1‖≤(1-an)‖xn+1-xn‖+1c|rn+1-rn|M+2M|an+1-an|,
where M=max{∥v∥,supn∥Ttn+1∥,supn∥un+1-xn+1∥}. By the assumptions on {an} and {rn} and Lemma 2.3, we conclude that
(3.15)lim⁡n→∞‖xn+1-xn‖=0.
From (C1), (C2), (3.5), (3.12), and (3.15), we also have ∥un+1-un∥→0, ∥tn+1-tn∥→0, and ∥yn+1-yn∥→0, as n→∞.
Since(3.16)xn+1-xn=an(v-xn)+(1-an)(Ttn-xn),
we have that
(3.17)‖Ttn-xn‖⟶0asn⟶∞.
Next, we prove that limn→∞∥xn-un∥=0. From Lemma 2.2(3), we have
(3.18)‖un-x*‖2=‖Trnxn-Trnx*‖2≤〈Trnxn-Trnx*,xn-x*〉=〈un-x*,xn-x*〉=12(‖un-x*‖2+‖xn-x*‖2-‖xn-un‖2).
Hence,
(3.19)‖un-x*‖2≤‖xn-x*‖2-‖xn-un‖2.
From Lemma 2.4, (3.3), and (3.19), we have
(3.20)‖xn+1-x*‖2≤an‖v-x*‖2+(1-an)‖tn-x*‖2≤an‖v-x*‖2+(1-an)‖un-x*‖2≤an‖v-x*‖2+(1-an)[‖xn-x*‖2-‖xn-un‖2]≤an‖v-x*‖2+‖xn-x*‖2-(1-an)‖xn-un‖2.
It follows that
(3.21)(1-an)‖xn-un‖2≤an‖v-x*‖2+‖xn-x*‖2-‖xn+1-x*‖2≤an‖v-x*‖2+(‖xn-x*‖+‖xn+1-x*‖)‖xn+1-xn‖.
From conditions (C1) and (3.15), we obtain
(3.22)lim⁡n→∞‖xn-un‖=0.
By (3.17) and (3.22), we have
(3.23)‖Ttn-un‖≤‖Ttn-xn‖+‖xn-un‖⟶0,asn⟶∞.
Next, we prove that ∥Ttn-tn∥→0 as n→∞. From (2.11) and nonexpansiveness of I-μB, we get
(3.24)‖yn-y*‖2=‖PC(un-μBun)-PC(x*-μBx*)‖2≤〈(un-μBun)-(x*-μBx*),yn-y*〉=12[‖(un-μBun)-(x*-μBx*)‖2+‖yn-y*‖2-‖(un-μBun)-(x*-μBx*)-(yn-y*)‖2]≤12[‖un-x*‖2+‖yn-y*‖2-‖(un-x*)-(yn-y*)‖2+2μ〈(un-x*)-(yn-y*),Bun-Bx*〉-μ2∥Bun-Bx*∥2].
By (3.3), we obtain
(3.25)‖yn-y*‖2≤‖un-x*‖2-‖(un-x*)-(yn-y*)‖2+2μ〈(un-x*)-(yn-y*),Bun-Bx*〉-μ2‖Bun-Bx*‖2≤‖xn-x*‖2-‖(un-x*)-(yn-y*)‖2+2μ〈(un-x*)-(yn-y*),Bun-Bx*〉-μ2‖Bun-Bx*‖2.
Hence,
(3.26)‖xn+1-x*‖2≤an‖v-x*‖2+(1-an)‖yn-y*‖2≤an‖v-x*‖2+(1-an)×[‖xn-x*‖2-‖(un-x*)-(yn-y*)‖2+2μ〈(un-x*)-(yn-y*),Bun-Bx*〉-μ2‖Bun-Bx*‖2]≤an‖v-x*‖2+‖xn-x*‖2-(1-an)‖(un-x*)-(yn-y*)‖2+(1-an)2μ‖(un-x*)-(yn-y*)‖‖Bun-Bx*‖,
which implies that
(3.27)(1-an)‖(un-x*)-(yn-y*)‖2≤an‖v-x*‖2+(1-an)2μ‖(un-x*)-(yn-y*)‖‖Bun-Bx*‖+(‖xn-x*‖+‖xn+1-x*‖)‖xn+1-xn‖.
By this together with (C1), (3.15), and limn→∞∥Bun-Bx*∥=0, we obtain
(3.28)‖(un-x*)-(yn-y*)‖⟶0asn⟶∞.
From Lemma 2.6 and (2.12), it follows that
(3.29)‖(yn-tn)+(x*-y*)‖2=‖(yn-λAyn)-(y*-λAy*)-[PC(yn-λAyn)-PC(y*-λAy*)]+λ(Ayn-Ay*)‖2≤‖(yn-λAyn)-(y*-λAy*)-[PC(yn-λAyn)-PC(y*-λAy*)]‖2+2λ〈Ayn-Ay*,(yn-tn)+(x*-y*)〉≤‖(yn-λAyn)-(y*-λAy*)‖2-‖PC(yn-λAyn)-PC(y*-λAy*)‖2+2λ‖Ayn-Ay*‖‖(yn-tn)+(x*-y*)‖≤‖(yn-λAyn)-(y*-λAy*)‖2-‖TPC(yn-λAyn)-TPC(y*-λAy*)‖2+2λ‖Ayn-Ay*‖‖(yn-tn)+(x*-y*)‖≤‖(yn-λAyn)-(y*-λAy*)-(Ttn-x*)‖×[‖(yn-λAyn)-(y*-λAy*)‖+‖Ttn-x*‖]+2λ‖Ayn-Ay*‖‖(yn-tn)+(x*-y*)‖=‖un-Ttn+x*-y*-(un-yn)-λ(Ayn-Ay*)‖×[‖(yn-λAyn)-(y*-λAy*)‖+‖Ttn-x*‖]+2λ‖Ayn-Ay*‖‖(yn-tn)+(x*-y*)‖.
By this together with (3.23), (3.28), and limn→∞∥Ayn-Ay*∥=0, we obtain ∥(yn-tn)+(x*-y*)∥→0 as n→∞. This together with (3.17), (3.22), and (3.28), we obtain that
(3.30)‖Ttn-tn‖≤‖Ttn-xn‖+‖xn-un‖+‖(un-yn)-(x*-y*)‖+‖(yn-tn)+(x*-y*)‖⟶0,asn⟶∞.
Next, we show that
(3.31)limsup⁡n→∞〈v-x¯,xn-x¯〉≤0,
where x¯=PΩv.
Indeed, since{tn} and {Ttn} are two bounded sequences in C, we can choose a subsequence {tni} of {tn} such that tni⇀z∈C and
(3.32)limsup⁡n→∞〈v-x¯,Ttn-x¯〉=lim⁡i→∞〈v-x¯,Ttni-x¯〉.
Since limn→∞∥Ttn-tn∥=0, we obtain that Ttni⇀z as i→∞.
Next, we show thatz∈Ω.
(a) We first showz∈F(T).
Sincetni⇀z and ∥Ttn-tn∥→0, we obtain by Lemma 2.5 that z∈F(T).
(b) Now, we show thatz∈GVI(C,A,B).
From (3.30) and (3.17), we have
(3.33)‖tn-xn‖≤‖Ttn-tn‖+‖Ttn-xn‖⟶0asn⟶∞.
Furthermore, by Lemma 2.9, we have that G:C→C is nonexpansive. Then, we have
(3.34)‖tn-G(tn)‖=‖PC(yn-λAyn)-G(tn)‖=‖PC[P(un-μBun)-λAP(un-μBun)]-G(tn)‖=‖G(un)-G(tn)‖≤‖un-tn‖≤‖un-xn‖+‖xn-tn‖⟶0asn⟶∞.
Again by Lemma 2.5, we have z∈GVI(C,A,B).
(c) We show thatz∈MEP(F,φ). Since tni⇀z and ∥xn-tn∥→0, we obtain that xni⇀z. From ∥un-xn∥→0, we also obtain that uni⇀z. By using the same argument as that in the proof of [30, Theorem  3.1, page 1825], we can show that z∈MEP(F,φ). Therefore, there holds z∈Ω.
On the other hand, it follows from (2.13), (3.17), and Ttni⇀z as i→∞ that
(3.35)limsup⁡n→∞〈v-x¯,xn-x¯〉=limsup⁡n→∞〈v-x¯,Ttn-x¯〉=lim⁡i→∞〈v-x¯,Ttni-x¯〉=〈v-x¯,z-x¯〉≤0.
Hence, we have
(3.36)‖xn+1-x¯‖2=〈anv+(1-an)Ttn-x¯,xn+1-x¯〉=an〈v-x¯,xn+1-x¯〉+(1-an)〈Ttn-x¯,xn+1-x¯〉≤an〈v-x¯,xn+1-x¯〉+12(1-an)(‖tn-x¯‖2+‖xn+1-x¯‖2)≤an〈v-x¯,xn+1-x¯〉+12(1-an)(‖xn-x¯‖2+‖xn+1-x¯‖2),
which implies that
(3.37)‖xn+1-x¯‖2≤(1-an)‖xn-x¯‖2+2an〈v-x¯,xn+1-x¯〉.
By this together with (C1) and (3.35), we have by Lemma 2.3 that {xn} converges strongly to x¯. This completes the proof.The following examples provide mappingsA and B which satisfy those conditions in Theorem 3.1.Example 3.2.
LetA,B:C→H be α-inverse-strongly monotone and β-inverse-strongly monotone, respectively. If λ∈(0,2α) and μ∈(0,2β), then we have(1)
I-λA and I-μB are nonexpansive,(2)
∥Ayn-Ay*∥→0 and ∥Bun-Bx*∥→0 as n→∞ for all x*∈Ω and y*=PC(x*-μBx*), where {yn} and {un} are sequences defined as in Theorem 3.1.Proof.
(1) For anyx,y∈C, we have
(3.38)‖(I-λA)x-(I-λA)y‖2=‖(x-y)-λ(Ax-Ay)‖2=‖x-y‖2+λ2‖Ax-Ay‖2-2λ〈x-y,Ax-Ay〉≤‖x-y‖2+λ2‖Ax-Ay‖2-2λα‖Ax-Ay‖2=‖x-y‖2+λ(λ-2α)‖Ax-Ay‖2≤∥x-y∥2,
hence, I-λA is nonexpansive. Similarly, we can show that I-μB is nonexpansive.
(2) Let{xn}, {yn}, and {un} be the sequences defined as in Theorem 3.1. From (3.3), we have
(3.39)‖xn+1-x*‖2≤an‖v-x*‖2+(1-an)‖tn-x*‖2=an‖v-x*‖2+(1-an)‖PC(yn-λAyn)-PC(y*-λAy*)‖2≤an‖v-x*‖2+(1-an)‖(yn-λAyn)-(y*-λAy*)‖2≤an‖v-x*‖2+(1-an)[‖yn-y*‖2+λ(λ-2α)‖Ayn-Ay*‖2]≤an‖v-x*‖2+‖xn-x*‖2+(1-an)λ(λ-2α)‖Ayn-Ay*‖2,‖xn+1-x*‖2≤an‖v-x*‖2+(1-an)‖tn-x*‖2≤an‖v-x*‖2+(1-an)‖yn-y*‖2≤an‖v-x*‖2+(1-an)‖(un-μBun)-(x*-μBx*)‖2≤an‖v-x*‖2+(1-an)[‖un-x*‖2+μ(μ-2β)‖Bun-Bx*‖2]≤an‖v-x*‖2+‖xn-x*‖2+(1-an)μ(μ-2β)‖Bun-Bx*‖2.
Therefore, we have
(3.40)-(1-an)λ(λ-2α)‖Ayn-Ay*‖2≤an‖v-x*‖2+(‖xn-x*‖+‖xn+1-x*‖)‖xn+1-xn‖,-(1-an)μ(μ-2β)‖Bun-Bx*‖2≤an‖v-x*‖2+(‖xn-x*‖+‖xn+1-x*‖)‖xn+1-xn‖.
From (3.15) and (C1), we obtain
(3.41)‖Ayn-Ay*‖⟶0,‖Bun-Bx*‖⟶0asn⟶∞.Example 3.3.
LetA:C→H be an LA-Lipschitzian and relaxed (c,d)-cocoercive mapping and B:C→H an LB-Lipschitzian and relaxed (c′,d′)-cocoercive mapping. If 0<λ<(2(d-cLA2))/LA2 and 0<μ<(2(d′-c′LB2))/LB2, then we have(1)
I-λA and I-μB are nonexpansive,(2)
∥Ayn-Ay*∥→0 and ∥Bun-Bx*∥→0 as n→∞ for all x*∈Ω and y*=PC(x*-μBx*), where {yn} and {un} are sequences defined as in Theorem 3.1.Proof.
(1) For anyx,y∈C, we have
(3.42)‖(I-μB)x-(I-μB)y‖2=‖(x-y)-μ(Bx-By)‖2=‖x-y‖2+μ2‖Bx-By‖2-2μ〈x-y,Bx-By〉≤‖x-y‖2+μ2‖Bx-By‖2-2μ[-c′‖Bx-By‖2+d′‖x-y‖2]=‖x-y‖2+μ2‖Bx-By‖2+2μc′‖Bx-By‖2-2μd′‖x-y‖2≤‖x-y‖2+μ2LB2‖x-y‖2+2μc′LB2‖x-y‖2-2μd′‖x-y‖2=(1+2μc′LB2-2μd′+μ2LB2)‖x-y‖2≤‖x-y‖2,
hence, I-μB is nonexpansive. Similarly, we can show that I-λA is nonexpansive.
(2) Let{xn}, {yn}, and {un} be the sequences defined as in Theorem 3.1. From (3.3), we have
(3.43)‖xn+1-x*‖2≤an‖v-x*‖2+(1-an)‖tn-x*‖2≤an‖v-x*‖2+(1-an)‖yn-y*‖2≤an‖v-x*‖2+(1-an)‖(un-μBun)-(x*-μBx*)‖2≤an‖v-x*‖2+(1-an)×[‖un-x*‖2-2μ〈un-x*,Bun-Bx*〉+μ2‖Bun-Bx*‖2]≤an‖v-x*‖2+(1-an)×[‖un-x*‖2+2μc′‖Bun-Bx*‖2-2μd′‖un-x*‖2+μ2‖Bun-Bx*‖2]≤an‖v-x*‖2+(1-an)[‖xn-x*‖2+(2μc′+μ2-2μd′LB2)‖Bun-Bx*‖2]≤an‖v-x*‖2+‖xn-x*‖2+(1-an)(2μc′+μ2-2μd′LB2)‖Bun-Bx*‖2,‖xn+1-x*‖2≤an‖v-x*‖2+(1-an)‖tn-x*‖2=an‖v-x*‖2+(1-an)‖PC(yn-λAyn)-PC(y*-λAy*)‖2≤an‖v-x*‖2+(1-an)‖(yn-λAyn)-(y*-λAy*)‖2≤an‖v-x*‖2+(1-an)[‖xn-x*‖2+(2λc+λ2-2λdLA2)‖Ayn-Ay*‖2]≤an‖v-x*‖2+‖xn-x*‖2+(1-an)(2λc+λ2-2λdLA2)‖Ayn-Ay*‖2.
Therefore, we have
(3.44)-(1-an)(2μc′+μ2-2μd′LB2)‖Bun-Bx*‖2≤an‖v-x*‖2+(‖xn-x*‖+‖xn+1-x*‖)‖xn+1-xn‖,-(1-an)(2λc+λ2-2λdLA2)‖Ayn-Ay*‖2≤an‖v-x*‖2+(‖xn-x*‖+‖xn+1-x*‖)‖xn+1-xn‖.
From (3.15) and (C1), we obtain
(3.45)‖Ayn-Ay*‖⟶0,‖Bun-Bx*‖⟶0asn⟶∞.By using the same proof as in Examples3.2 and 3.3, we can obtain the following example.Example 3.4.
LetA be an α-inverse-strongly monotone mapping of C into H and B be a L-Lipschitzian and relaxed (c,d)-cocoercive mapping of C into H. If λ∈(0,2α) and 0<μ<(2(d-cL2))/L2, then we have(1)
I-λA and I-μB are nonexpansive,(2)
∥Ayn-Ay*∥→0 and ∥Bun-Bx*∥→0 as n→∞ for all x*∈Ω and y*=PC(x*-μBx*), where {yn} and {un} are sequences defined as in Theorem 3.1.Let𝒜 be the class of all α-inverse-strongly monotone mappings from C into H, ℬ the class of all β-inverse-strongly monotone mappings from C into H, 𝒞 the class of all L-Lipschitzian and relaxed (c,d)-cocoercive mappings from C into H, and 𝒟 the class of all L1-Lipschitzian and relaxed (c′,d′)-cocoercive mappings from C into H.Theorem 3.5.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let F be a function from C×C to ℝ satisfying (A1)–(A5) and φ:C→ℝ∪{+∞} a proper lower semicontinuous and convex function. Let {Ti}i=1N be a finite family of nonexpansive self-mappings of C such that Ω=∩i=1NF(Ti)∩GVI(C,A,B)∩MEP(F,φ)≠∅. Let αj=(α1j,α2j,α3j), j=1,2,…,N, where α1j,α2j,α3j∈[0,1], α1j+α2j+α3j=1, α1j∈(0,1) for all j=1,2,…,N-1, α1N∈(0,1] and α2j,α3j∈[0,1) for all j=1,2,…,N. Let S be the S-mapping generated by T1,T2,…,TN and α1,α2,…,αN. Assume that either (B1) or (B2) holds and that v is an arbitrary point in C. Let x1∈C and {xn},{un},{yn} the sequences generated by
(3.46)F(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=PC(un-μBun),xn+1=anv+(1-an)SPC(yn-λAyn),n≥1.
If one of the following conditions is satisfied:(1)
A∈𝒜, B∈ℬ, λ∈(0,2α), and μ∈(0,2β),(2)
A∈𝒞, B∈𝒟, 0<λ<(2(d-cL2))/L2, and 0<μ<(2(d′-c′L12))/L12,(3)
A∈𝒜, B∈𝒞, λ∈(0,2α), and 0<λ<(2(d-cL2))/L2,and the sequences {rn} and {an} are as in Theorem 3.1, then {xn} converges strongly to x¯=PΩv and (x¯,y¯) is a solution of the problem (1.6), where y¯=PC(x¯-μBx¯).Proof.
By Lemma2.7, we obtain that S is nonexpansive and F(S)=∩i=1NF(Ti). Hence, the result is obtained directly from Theorem 3.1 and Examples 3.2–3.4.From Theorem3.1 and Examples 3.2–3.4, we obtain the following result.Corollary 3.6.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let F be a function from C×C to ℝ satisfying (A1)–(A5) and φ:C→ℝ∪{+∞} a proper lower semicontinuous and convex function. Let T be a nonexpansive self-mapping of C such that Ω=F(T)∩GVI(C,A,B)∩MEP(F,φ)≠∅. Assume that either (B1) or (B2) holds and that v is an arbitrary point in C. If one of the following conditions is satisfied:(1)
A∈𝒜, B∈ℬ, λ∈(0,2α), and μ∈(0,2β),(2)
A∈𝒞, B∈𝒟, 0<λ<(2(d-cL2))/L2, and 0<μ<(2(d′-c′L12))/L12,(3)
A∈𝒜, B∈𝒞, λ∈(0,2α), and 0<λ<(2(d-cL2))/L2,and the sequences {rn}, {an} are as in Theorem 3.1, then the sequence {xn} generated by (1.12) converges strongly to x¯=PΩv and (x¯,y¯) is a solution of the problem (1.6), where y¯=PC(x¯-μBx¯).Letφ=0 in Theorem 3.1. From Theorem 3.1 and Examples 3.2–3.4, we obtain the following result.Corollary 3.7.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let F be a function from C×C to ℝ satisfying (A1)–(A5). Let T be a nonexpansive self-mapping of C such that Ω=F(T)∩GVI(C,A,B)∩EP(F)≠∅. Let x1,v∈C and {xn},{un},{yn} be the sequences generated by
(3.47)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=PC(un-μBun),xn+1=anv+(1-an)TPC(yn-λAyn),n≥1.
If one of the following conditions is satisfied:(1)
A∈𝒜, B∈ℬ, λ∈(0,2α), and μ∈(0,2β),(2)
A∈𝒞, B∈𝒟, 0<λ<(2(d-cL2))/L2, and 0<μ<(2(d′-c′L12))/L12,(3)
A∈𝒜, B∈𝒞, λ∈(0,2α), and 0<λ<(2(d-cL2))/L2,and the sequences {rn}, {an} are as in Theorem 3.1, then {xn} converges strongly to x¯=PΩv and (x¯,y¯) is a solution of the problem (1.6), where y¯=PC(x¯-μBx¯).Remark 3.8.
In Theorem3.5, if F≡0, then the sequence {xn} generated by (3.46) converges strongly to a solution of the minimization problem which is also a solution of a system of variational inequalities.



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