We consider a hybrid projection algorithm based on the shrinking projection method for two families of quasi-ϕ-nonexpansive mappings. We establish strong convergence theorems for approximating the common element of the set of the common fixed points of such two families and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. As applications, at the end of the paper we first apply our results to consider the problem of finding a zero point of an inverse-strongly monotone operator and we finally utilize our results to study the problem of finding a solution of the complementarity problem. Our results improve and extend the corresponding results announced by recent results.

1. Introduction

Let E be a Banach space and let C be a nonempty, closed, and convex subset of E. Let A:C→E* be an operator. The classical variational inequality problem [1] for A is to find x*∈C such that

〈Ax*,y-x*〉≥0,∀y∈C,
where E* denotes the dual space of E and 〈·,·〉 the generalized duality pairing between E and E*. The set of all solutions of (1.1) is denoted by VI(A,C). Such a problem is connected with the convex minimization problem, the complementarity, the problem of finding a point x*∈E satisfying 0=Ax*, and so on. First, we recall that a mapping A:C→E* is said to be

monotone if 〈Ax-Ay,x-y〉≥0,forallx,y∈C,

α-inverse-strongly monotone if there exists a positive real number α such that

〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.
In this paper, we assume that the operator A satisfies the following conditions:

A is α-inverse-strongly monotone,

∥Ay∥≤∥Ay-Au∥ for all y∈C and u∈VI(A,C).

Let J be the normalized duality mapping from E into 2E* given by

Jx={x*∈E*:〈x,x*〉=∥x∥∥x*∥,∥x∥=∥x*∥},∀x∈E.
It is well known that if E* is uniformly convex, then J is uniformly continuous on bounded subsets of E. Some properties of the duality mapping are given in [2–4].

Recall that a mapping T:C→C is said to be nonexpansive if

∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.
If C is a nonempty closed convex subset of a Hilbert space H and PC:H→C is the metric projection of H onto C, then PC is a nonexpansive mapping. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [5] recently introduced a generalized projection operator C in a Banach space E which is an analogue of the metric projection in Hilbert spaces.

Consider the functional ϕ:E×E→ℝ defined by

ϕ(y,x)=∥y∥2-2〈y,Jx〉+∥x∥2
for all x,y∈E, where J is the normalized duality mapping from E to E*. Observe that, in a Hilbert space H, (1.5) reduces to ϕ(y,x)=∥x-y∥2 for all x,y∈H. The generalized projection ΠC:E→C is a mapping that assigns to an arbitrary point x∈E the minimum point of the functional ϕ(y,x), that is, ΠCx=x*, where x* is the solution to the minimization problem:

ϕ(x*,x)=infy∈Cϕ(y,x).
The existence and uniqueness of the operator ΠC follows from the properties of the functional ϕ(y,x) and strict monotonicity of the mapping J (see, e.g., [2, 5–7]). In Hilbert spaces, ΠC=PC, where PC is the metric projection. It is obvious from the definition of the function ϕ that

(∥y∥-∥x∥)2⩽ϕ(y,x)⩽(∥y∥+∥x∥)2 for all x,y∈E,

ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2〈x-z,Jz-Jy〉 for all x,y,z∈E,

ϕ(x,y)=〈x,Jx-Jy〉+〈y-x,Jy〉⩽∥x∥∥Jx-Jy∥+∥y-x∥∥y∥ for all x,y∈E,

if E is a reflexive, strictly convex, and smooth Banach space, then for all x,y∈E,

ϕ(x,y)=0iffx=y.
For more details see [2, 3]. Let C be a closed convex subset of E, and let T be a mapping from C into itself. We denote by F(T) the set of fixed point of T. A point p in C is said to be an asymptotic fixed point of T [8] if C contains a sequence {xn} which converges weakly to p such that limn→∞∥xn-Txn∥=0. The set of asymptotic fixed points of T will be denoted by F̂(T). A mapping T from C into itself is called nonexpansive if ∥Tx-Ty∥⩽∥x-y∥ for all x,y∈C and relatively nonexpansive [9–11] if F̂(T)=F(T) and ϕ(p,Tx)⩽ϕ(p,x) for all x∈C and p∈F(T). The asymptotic behavior of relatively nonexpansive mappings which was studied in [9–11] is of special interest in the convergence analysis of feasibility, optimization, and equilibrium methods for solving the problems of image processing, rational resource allocation, and optimal control. The most typical examples in this regard are the Bregman projections and the Yosida type operators which are the cornerstones of the common fixed point and optimization algorithms discussed in [12] (see also the references therein).

The mapping T is said to be ϕ-nonexpansive if ϕ(Tx,Ty)≤ϕ(x,y) for all x,y∈C. T is said to be quasi-ϕ-nonexpansive if F(T)≠∅ and ϕ(p,Tx)≤ϕ(p,x) for all x∈C and p∈F(T).

Remark 1.1.

The class of quasi-ϕ-nonexpansive is more general than the class of relatively nonexpansive mappings [9, 10, 13–15] which requires the strong restriction F̂(T)=F(T).

Next, we give some examples which are closed quasi-ϕ-nonexpansive [16].

Example 1.2.

(1) Let E be a uniformly smooth and strictly convex Banach space and let A be a maximal monotone mapping from E to E such that its zero set A-10 is nonempty. The resolvent Jr=(J+rA)-1J is a closed quasi-ϕ-nonexpansive mapping from E onto D(A) and F(Jr)=A-10.

(2) Let ΠC be the generalized projection from a smooth, strictly convex, and reflexive Banach space E onto a nonempty closed convex subset C of E. Then ΠC is a closed and quasi-ϕ-nonexpansive mapping from E onto C with F(ΠC)=C.

Iiduka and Takahashi [17] introduced the following algorithm for finding a solution of the variational inequality for an operator A that satisfies conditions (C1)-(C2) in a 2 uniformly convex and uniformly smooth Banach space E. For an initial point x0=x∈C, define a sequence {xn} by

xn+1=ΠCJ-1(Jxn-λnxn),∀n≥0.
where J is the duality mapping on E, and ΠC is the generalized projection of E onto C. Assume that λn∈[a,b] for some a,b with 0<a<b<c2α/2 where 1/c is the 2 uniformly convexity constant of E. They proved that if J is weakly sequentially continuous, then the sequence {xn} converges weakly to some element z in VI(A,C) where z=limn→∞ΠVI(A,C)(xn).

The problem of finding a common element of the set of the variational inequalities for monotone mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many authors; see, for instance, [18–20] and the references cited therein.

On the other hand, one classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping (see [21]). More precisely, let t∈(0,1) and define a contraction Gt:C→C by Gtx=tx0+(1-t)Tx for all x∈C, where x0∈C is a fixed point in C. Applying Banach's Contraction Principle, there exists a unique fixed point xt of Gt in C. It is unclear, in general, what is the behavior of xt as t→0 even if T has a fixed point. However, in the case of T having a fixed point, Browder [21] proved that the net {xt} defined by xt=tx0+(1-t)Txt for all t∈(0,1) converges strongly to an element of F(T) which is nearest to x0 in a real Hilbert space. Motivated by Browder [21], Halpern [22] proposed the following iteration process:

x0∈C,xn+1=αnx0+(1-αn)Txn,n⩾0
and proved the following theorem.

Theorem 1 H.

Let C be a bounded closed convex subset of a Hilbert space H and let T be a nonexpansive mapping on C. Define a real sequence {αn} in [0,1] by αn=n-θ,0<θ<1. Define a sequence {xn} by (1.9). Then {xn} converges strongly to the element of F(T) which is the nearest to u.

Recently, Martinez-Yanes and Xu [23] have adapted Nakajo and Takahashi's [24] idea to modify the process (1.9) for a single nonexpansive mapping T in a Hilbert space H:

x0=x∈Cchosenarbitrary,yn=αnx0+(1-αn)Txn,Cn={v∈C:∥yn-v∥2⩽∥xn-v∥2+αn(∥x0∥2+2〈xn-x0,v〉)},Qn={v∈C:〈xn-v,x0-xn〉⩾0},xn+1=PCn∩Qnx0,
where PC denotes the metric projection of H onto a closed convex subset C of H. They proved that if {αn}⊂(0,1) and limn→∞αn=0, then the sequence {xn} generated by (1.10) converges strongly to PF(T)x.

In [15] (see also [13]), Qin and Su improved the result of Martinez-Yanes and Xu [23] from Hilbert spaces to Banach spaces. To be more precise, they proved the following theorem.

Theorem 1 QS.

Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, and let T:C→C be a relatively nonexpansive mapping. Assume that {αn} is a sequence in (0,1) such that limn→∞αn=0. Define a sequence {xn} in C by the following algorithm:
x0=x∈Cchosenarbitrary,yn=J-1(αnJx0+(1-αn)JTxn),Cn={v∈C:ϕ(v,yn)≤αnϕ(v,yn)+(1-αn)ϕ(v,xn)},Qn={v∈C:〈xn-v,Jx0-Jxn〉⩾0},xn+1=ΠCn∩Qnx0,
where J is the single-valued duality mapping on E. If F(T) is nonempty, then {xn} converges to ΠF(T)x0.

In [14], Plubtieng and Ungchittrakool introduced the following hybrid projection algorithm for a pair of relatively nonexpansive mappings:

x0=x∈Cchosenarbitrary,zn=J-1(βn(1)Jxn+βn(2)JTxn+βn(3)JSxn),yn=J-1(αnJx0+(1-αn)Jzn),Hn={z∈C:ϕ(z,yn)⩽ϕ(z,xn)+αn(∥x0∥2+2〈z,Jxn-Jx〉)},Wn={z∈C:〈xn-z,Jx-Jxn〉⩾0},xn+1=PHn∩Wnx,n=0,1,2,…,
where {αn}, {βn(1)}, {βn(2)}, and {βn(3)} are sequences in [0,1] satisfying βn(1)+βn(2)+βn(3)=1 for all n∈ℕ∪{0} and T,S are relatively nonexpansive mappings and J is the single-valued duality mapping on E. They proved, under appropriate conditions on the parameters, that the sequence {xn} generated by (1.12) converges strongly to a common fixed point of T and S.

Very recently, Qin et al. [25] introduced a new hybrid projection algorithm for two families of quasi-ϕ-nonexpansive mappings which are more general than relatively nonexpansive mappings to have strong convergence theorems in the framework of Banach spaces. To be more precise, they proved the following theorem.

Theorem 1 QCKZ.

Let E be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let {Si}i∈I and {Ti}i∈I be two families of closed quasi-ϕ-nonexpansive mappings of C into itself with F:=⋂i∈IF(Ti)∩⋂i∈IF(Si) being nonempty, where I is an index set. Let the sequence {xn} be generated by the following manner:
x0=x∈Cchosenarbitrary,zn,i=J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSixn),yn,i=J-1(αn,iJx0+(1-αn,i)Jzn,i),Cn,i={u∈C:ϕ(u,yn,i)⩽ϕ(u,xn)+αn,i(∥x0∥2+2〈u,Jxn-Jxn〉)},Cn=⋂i∈ICn,i,Q0=C,Qn={u∈Qn-1:〈xn-u,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,n=0,1,2,…,
where J is the duality mapping on E, and {αn,i} and {βn,i(i)}(i=1,2,3,…) are sequences in (0,1) satisfying

βn,i(1)+βn,i(2)+βn,i(3)=1 for all i∈I,

limn→∞αn,i=0 for all i∈I,

lim infn→∞βn,i(2)βn,i(3)>0 and limn→∞βn,i(1)=0 for all i∈I.

Then the sequence {xn} converges strongly to ΠFx0.

On the other hand, recently, Takahashi et al. [26] introduced the following hybrid method (1.14) which is different from Nakajo and Takahashi's [24] hybrid method. It is called the shrinking projection method. They obtained the following result.

Theorem 1 NT.

Let C be a nonempty closed convex subset of a Hilbert space H. Let T be a nonexpansive mapping of C into H such that F(T)≠∅ and let x0∈H. For C1=C and x1=PC1x0, define a sequence {xn} of C as follows:
yn=αnxn+(1-αn)Txn,Cn+1={z∈Cn:∥yn-z∥≤∥xn-z∥},xn+1=PCn+1x0,∀n≥0,
where 0≤αn<a<1 for all n∈ℕ. Then {xn} converges strongly to z0=PF(T)(x0).

Motivated and inspired by Iiduka and Takahashi [17], Martinez-Yanes and Xu [23], Matsushita and Takahashi [13], Plubtieng and Ungchittrakool [14], Qin and Su [15], Qin et al. [25], and Takahashi et al. [26], we introduce a new hybrid projection algorithm basing on the shrinking projection method for two families of quasi-ϕ-nonexpansive mappings which are more general than relatively nonexpansive mappings to have strong convergence theorems for approximating the common element of the set of common fixed points of two families of quasi-ϕ-nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. As applications, the problem of finding a zero point of an inverse-strongly monotone operator and the problem of finding a solution of the complementarity problem are studied. Our results improve and extend the corresponding results announced by recent results.

2. Preliminaries

Let E be a real Banach space with duality mapping J. We denote strong convergence of {xn} to x by xn→x and weak convergence by xn⇀x. A multivalued operator T:E→2E* with domain D(T) and range R(T) is said to be monotone if 〈x1-x2,y1-y2〉≥0 for each xi∈D(T) and yi∈Txi,i=1,2. A monotone operator T is said to be maximal if its graph G(T)={(x,y):y∈Tx} is not properly contained in the graph of any other monotone operators.

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. It is said to be uniformly convex if limn→∞∥xn-yn∥=0 for any two sequences {xn},{yn} in E such that ∥xn∥=∥yn∥=1 and limn→∞∥(xn+yn)/2∥=1. Let U={x∈E:∥x∥=1} be the unit sphere of E. Then the Banach space E is said to be smooth provided that

limt→0∥x+ty∥-∥x∥t
exists for each x,y∈U. It is also said to be uniformly smooth if the limit is attained uniformly for x,y∈U. It is well know that if E is smooth, then the duality mapping J is single valued. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded subset of E. Some properties of the duality mapping are given in [2, 3, 27–29]. We define the function δ:[0,2]→[0,1] which is called the modulus of convexity of E as follows:

δ(ε)=inf{1-∥x+y2∥:x,y∈C,∥x∥=∥y∥=1,∥x-y∥≥ε}.
Then E is said to be 2 uniformly convex if there exists a constant c>0 such that constant δ(ε)>cε2 for all ε∈(0,2]. Constant 1/c is called the 2 uniformly convexity constant of E. A 2 uniformly convex Banach space is uniformly convex; see [30, 31] for more details. We know the following lemma of 2 uniformly convex Banach spaces.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B4">32</xref>, <xref ref-type="bibr" rid="B7">33</xref>]).

Let E be a 2 uniformly convex Banach, then for all x,y from any bounded set of E and jx∈Jx,jy∈Jy,
〈x-y,jx-jy〉≥c22∥x-y∥2,
where 1/c is the 2 uniformly convexity constant of E.

Now we present some definitions and lemmas which will be applied in the proof of the main result in the next section.

Lemma 2.2 (Kamimura and Takahashi [<xref ref-type="bibr" rid="B18">7</xref>]).

Let E be a uniformly convex and smooth Banach space and let {yn}, {zn} be two sequences of E such that either {yn} or {zn} is bounded. If limn→∞ϕ(yn,zn)=0, then limn→∞∥yn-zn∥=0.

Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed convex subset of E, and let x∈E. Then
ϕ(y,ΠCx)+ϕ(ΠCx,x)⩽ϕ(y,x)
for all y∈C.

Lemma 2.5 (Qin et al. [<xref ref-type="bibr" rid="B26">25</xref>]).

Let E be a uniformly convex and smooth Banach space, let C be a closed convex subset of E, and let T be a closed quasi-ϕ-nonexpansive mapping of C into itself. Then F(T) is a closed convex subset of C.

Let E be a reflexive strictly convex, smooth, and uniformly Banach space and the duality mapping from E to E*. Then J-1 is also single valued, one to one, and surjective, and it is the duality mapping from E* to E. We need the following mapping V which is studied in Alber [5]:

V(x,x*)=∥x∥2-2〈x,x*〉+∥x∥2
for all x∈E and x*∈E*. Obviously, V(x,x*)=ϕ(x,J-1(x*)). We know the following lemma.

Lemma 2.6 (Kamimura and Takahashi [<xref ref-type="bibr" rid="B18">7</xref>]).

Let E be a reflexive, strictly convex, and smooth Banach space, and let V be as in (2.5). Then
V(x,x*)+2〈J-1(x*)-x,y*〉≤V(x,x*+y*)
for all x∈E and x*,y*∈E*.

Lemma 2.7 (see [<xref ref-type="bibr" rid="B13">34</xref>,Lemma <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M359"><mml:mrow><mml:mn>1.4</mml:mn></mml:mrow></mml:math></inline-formula>]).

Let E be a uniformly convex Banach space and Br(0)={x∈E:∥x∥⩽r} be a closed ball of E. Then there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
∥λx+μy+γz∥2⩽λ∥x∥2+μ∥y∥2+γ∥z∥2-λμg(∥x-y∥)
for all x,y,z∈Br(0) and λ,μ,γ∈[0,1] with λ+μ+γ=1.

An operator A of C into E* is said to be hemicontinuous if, for all x,y∈C, the mapping F of [0,1) into E* defined by F(t)=A(tx+(1-t)y) is continuous with respect to the weak* topology of E*. We denote by NC(v) the normal cone for C at a point v∈C, that is,

NC(v)={x*∈E*:〈v-y,x*〉≥0,∀y∈C}.

Lemma 2.8 (see [<xref ref-type="bibr" rid="B30">35</xref>]).

Let C be a nonempty closed convex subset of a Banach space E and A a monotone, hemicontinuous operator of C into E*. Let T⊂E×E* be an operator defined as follows:
Tv={Av+NC(v),v∈C,∅,v∉C.
Then T is maximal monotone and T-10=VI(A,C).

3. Main Results

In this section, we prove strong convergence theorem which is our main result.

Theorem 3.1.

Let E be a 2 uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let A be an operator of C into E* satisfying (C1) and (C2), and let {Si}i∈I and {Ti}i∈I be two families of closed quasi-ϕ-nonexpansive mappings of C into itself with F:=⋂i∈IF(Ti)∩⋂i∈IF(Si)∩VI(A,C) being nonempty, where I is an index set. Let {xn} be a sequence generated by the following manner:
x0∈Cchosenarbitrary,C1,i=C,C1=⋂i=1∞C1,i,x1=ΠC1(x0)∀i∈I,wn,i=ΠCJ-1(Jxn-λn,iAxn),zn,i=J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i),yn,i=J-1(αn,iJx0+(1-αn,i)Jzn,i),Cn+1,i={u∈Cn,i:ϕ(u,yn,i)⩽ϕ(u,xn)+αn,i(∥x0∥2+2〈u,Jxn-Jx0〉)},Cn+1=⋂i∈ICn+1,i,xn+1=ΠCn+1x0,∀n≥0,
where J is the duality mapping on E, and {λn,i},{αn,i}, and {βn,i(j)}(j=1,2,3) are sequences in (0,1) satisfying the following conditions:

limn→∞αn,i=0 for all i∈I;

for all i∈I, {λn,i}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2 uniformly convexity constant of E;

βn,i(1)+βn,i(2)+βn,i(3)=1 for all i∈I and if one of the following conditions is satisfied:

lim infn→∞βn,i(1)βn,i(l)>0 for all l=2,3 and for all i∈I,

lim infn→∞βn,i(2)βn,i(3)>0 and limn→∞βn,i(1)=0 for all i∈I.

Then the sequence {xn} converges strongly to ΠFx0, where ΠF is the generalized projection from C onto F.Proof.

We divide the proof into six steps.

Step 1.

Show that ΠFx0 and ΠCn+1x0 are well defined.

To this end, we prove first that F is closed and convex. It is obvious that VI(A,C) is a closed convex subset of C. By Lemma 2.5, we know that ⋂i∈IF(Ti)∩⋂i∈IF(Si) is closed and convex. Hence F:=⋂i∈IF(Ti)∩⋂i∈IF(Si)∩VI(A,C) is a nonempty, closed, and convex subset of C. Consequently, ΠFx0 is well defined.

We next show that Cn+1 is convex for each n≥0. From the definition of Cn, it is obvious that Cn is closed for each n≥0. Notice that
Cn+1,i={u∈Cn,i:ϕ(u,yn,i)≤ϕ(u,xn)+αn,i(∥x0∥2+2〈u,Jxn-Jx0〉)}
is equivalent to
Cn+1,i′={u∈Cn,i:2〈u,Jxn-Jyn,i〉-2αn,i〈u,Jxn-Jx0〉≤∥xn∥2-∥yn,i∥2+αn,i∥x0∥2}.
It is easy to see that Cn+1,i′ is closed and convex for all n≥0 and i∈I. Therefore, Cn+1=⋂i∈ICn+1,i=⋂i∈ICn+1,i′ is closed and convex for every n≥0. This shows that ΠCn+1x0 is well defined.

Step 2.

Show that F:=⋂i∈IF(Ti)∩⋂i∈IF(Si)∩VI(A,C)⊂Cn for all n≥0.

Put vn,i=J-1(Jxn-λn,iAxn). We have to show that F⊂Cn for all n≥0. For all u∈F, we know from Lemmas 2.4 and 2.6 that
ϕ(u,wn,i)=ϕ(u,ΠCvn,i)≤ϕ(u,vn,i)=ϕ(u,J-1(Jxn-λn,iAxn))=V(u,Jxn-λn,iAxn)≤V(u,(Jxn-λn,iAxn)+λn,iAxn)-2〈J-1(Jxn-λn,iAxn)-u,λn,iAxn〉=V(u,Jxn)-2λn,i〈vn,i-u,Axn〉=ϕ(u,xn)-2λn,i〈xn-u,Axn〉+2〈vn,i-xn,-λn,iAxn〉.
Since u∈VI(A,C) and from condition (C1), we have
-2λn,i〈xn-u,Axn〉=-2λn,i〈xn-u,Axn-Au〉-2λn,i〈xn-u,Au〉≤-2αλn,i∥Axn-Au∥2.
From Lemma 2.1, and condition (C2), we also have
2〈vn,i-xn,-λn,iAxn〉=2〈J-1(Jxn-λn,iAxn)-J-1(Jxn),-λn,iAxn〉≤2∥J-1(Jxn-λn,iAxn)-J-1(Jxn)∥∥λn,iAxn∥≤4c2∥JJ-1(Jxn-λn,iAxn)-JJ-1(Jxn)∥∥λn,iAxn∥=4c2∥(Jxn-λn,iAxn)-(Jxn)∥∥λn,iAxn∥≤4c2λn,i2∥Axn∥2≤4c2λn,i2∥Axn-Au∥2.
Subtituting (3.6) and (3.5) into (3.4) and using the assumption (ii), we obtain
ϕ(u,wn,i)≤ϕ(u,xn)-2αλn,i∥Axn-Au∥2+4c2λn,i2∥Axn-Au∥2.≤ϕ(u,xn)+2λn,i(2c2λn,i-α)∥Axn-Au∥2≤ϕ(u,xn).
It follows from the convexity of ∥·∥2 and (3.7) that
ϕ(u,zn,i)=ϕ(u,J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i))=∥u∥2-2〈u,βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i〉+∥βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i∥2≤∥u∥2-2βn,i(1)〈u,Jxn〉-2βn,i(2)〈u,JTixn〉-2βn,i(2)〈u,JSiwn,i〉+βn,i(1)∥Jxn∥2+βn,i(2)∥JTixn∥2+βn,i(3)∥JSiwn,i∥2=βn,i(1)ϕ(u,xn)+βn,i(2)ϕ(u,Tixn)+βn,i(3)ϕ(u,Siwn,i)≤βn,i(1)ϕ(u,xn)+βn,i(2)ϕ(u,xn)+βn,i(3)ϕ(u,wn,i)≤ϕ(u,xn),
and hence
ϕ(u,yn,i)=ϕ(u,J-1(αn,iJx0+(1-αn,i)Jzn,i))=∥u∥2-2〈u,αn,iJx0+(1-αn,i)Jzn,i〉+∥αn,iJx0+(1-αn,i)Jzn,i∥2≤∥u∥2-2αn,i〈u,Jx0〉-2(1-αn,i)〈u,Jzn,i〉+αn,i∥x0∥2+(1-αn,i)∥zn,i∥2≤αn,iϕ(u,x0)+(1-αn,i)ϕ(u,zn,i)≤αn,iϕ(u,x0)+(1-αn,i)ϕ(u,xn)=ϕ(u,xn)+αn,i[ϕ(u,x0)-ϕ(u,xn)]≤ϕ(u,xn)+αn,i(∥x0∥2+2〈u,Jxn-Jx0〉).
This show that u∈Cn+1,i for each i∈I. That is, u∈Cn=⋂i∈ICn,i for all n≥0. This show that
F:=⋂i∈IF(Ti)∩⋂i∈IF(Si)∩VI(A,C)⊂Cn,∀n≥0.

Step 3.

Show that limn→∞ϕ(xn,x0) exists.

We note that Cn+1,i⊂Cn,i for all n≥0 and for all i∈I. Hence
Cn+1=⋂i∈ICn+1,i⊂Cn=⋂i∈ICn,i.
From xn+1=ΠCn+1x0∈Cn+1⊂Cn and xn=ΠCnx0∈Cn, we have
ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥1.
This shows that {ϕ(xn,x0)} is nondecreasing. On the other hand, from Lemma 2.4, we have
ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(w,x0)-ϕ(w,xn)≤ϕ(w,x0)
for each w∈F⊂Cn. This show that {ϕ(xn,x0)} is bounded. Consequently, limn→∞ϕ(xn,x0) exists.

Step 4.

Show that {xn} is a convergent sequence in C.

Since xm=ΠCmx0∈Cn for any m≥n. It follows that
ϕ(xm,xn)=ϕ(xm,ΠCnx0)≤ϕ(xm,x0)-ϕ(ΠCnx0,x0)=ϕ(xm,x0)-ϕ(xn,x0).
Letting m,n→∞ in (3.14), we have ϕ(xm,xn)→0. It follows from Lemma 2.2 that
limm,n→∞∥xm-xn∥=0.
Hence {xn} is a Cauchy sequence in C. By the completeness of E and the closedness of C, we can assume that
xn→p∈Casn→∞.

Step 5.

We show that p∈F:=⋂i∈IF(Ti)∩⋂i∈IF(Si)∩VI(A,C).

(I) We first show that p∈⋂i∈IF(Ti)∩⋂i∈IF(Si). Taking m=n+1 in (3.14), one arrives that
limn→∞ϕ(xn+1,xn)=0.
From Lemma 2.2, we obtain
limn→∞∥xn+1-xn∥=0.
Noticing that xn+1=ΠCn+1x0, from the definition of Cn,i for every i∈I, we obtain
ϕ(xn+1,yn,i)⩽ϕ(xn+1,xn)+αn,i(∥x0∥2+2〈u,Jxn-Jx0〉).
It follows from (3.17) and limn→∞αn,i=0 and the fact that {Jxn} is bounded that
limn→∞ϕ(xn+1,yn,i)=0,∀i∈I.
From Lemma 2.2, we obtain
limn→∞∥xn+1-yn,i∥=0,∀i∈I.
It follows from (3.18) that
limn→∞∥xn-yn,i∥=0,∀i∈I.
Since J is uniformly norm-to-norm continuity on bounded sets, for every i∈I, one has
limn→∞∥Jxn-Jyn,i∥=limn→∞∥Jxn+1-Jxn∥=0,∀i∈I.
For every i∈I, we obtain from the properties of ϕ that
ϕ(zn,i,xn)=ϕ(zn,i,yn,i)+ϕ(yn,i,xn)+2〈zn,i-yn,i,Jyn,i-Jxn〉≤ϕ(zn,i,yn,i)+ϕ(yn,i,xn)+2∥zn,i-yn,i∥∥Jyn,i-Jxn∥.
On the other hand, for all i∈I, we have
ϕ(zn,i,yn,i)=∥zn,i∥2-2〈zn,i,αn,iJx0+(1-αn,i)Jzn,i〉+∥αn,iJx0+(1-αn,i)Jzn,i∥2≤∥zn,i∥2-2αn,i〈zn,i,Jx0〉-2(1-αn,i)〈zn,i,Jzn,i〉+αn,i∥x0∥2+(1-αn,i)∥zn,i∥2=αn,i(∥zn,i∥2-2〈zn,i,Jx0〉+∥x0∥2)=αn,iϕ(zn,i,x0).
It follows form (ii) that
limn→∞ϕ(zn,i,yn,i)=0,∀i∈I.
Notice that
ϕ(yn,i,xn)=∥yn,i∥2-2〈yn,i,Jxn〉+∥xn∥2=∥yn,i∥2-2〈yn,i,Jxn〉+∥xn∥2+∥xn+1∥2-∥xn+1∥2-2〈xn+1,Jyn,i〉+2〈xn+1,Jyn,i〉=ϕ(xn+1,yn,i)-2〈yn,i,Jxn〉+∥xn∥2-∥xn+1∥2+2〈xn+1,Jyn,i〉=ϕ(xn+1,yn,i)+(∥xn-xn+1∥)(∥xn∥+∥xn+1∥)-2〈yn,i,Jxn-Jyn,i〉-2〈yn,i,Jyn,i〉+2〈xn+1,Jyn,i〉=ϕ(xn+1,yn,i)+(∥xn-xn+1∥)(∥xn∥+∥xn+1∥)+2〈yn,i,Jyn,i-Jxn〉+2〈xn+1-yn,i,Jyn,i〉≤ϕ(xn+1,yn,i)+(∥xn-xn+1∥)(∥xn∥+∥xn+1∥)+2∥yn,i∥∥Jyn,i-Jxn∥+2∥xn+1-yn,i∥∥Jyn,i∥.
Applying (3.18), (3.20), (3.21), and (3.23) to the last inequality, we obtain
limn→∞ϕ(yn,i,xn)=0,∀i∈I.
Combining (3.26) with (3.28) in (3.24), we have
limn→∞ϕ(zn,i,xn)=0,∀i∈I.
From Lemma 2.2, we have
limn→∞∥zn,i-xn∥=0,∀i∈I.
Since J is uniformly norm-to-norm continuity on bounded sets, for every i∈I, one has
limn→∞∥Jzn,i-Jxn∥=0,∀i∈I.
Let r=supn≥1{∥xn∥,∥Tixn∥,∥Sixn∥} for every i∈I. Therefore Lemma 2.7 implies that there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) satisfying g(0)=0 and (2.7).

Case I.

Assume that (a) holds. Applying (2.7), we can calculate
ϕ(u,zn,i)=ϕ(u,J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i))=∥u∥2-2〈u,βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i〉+∥βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i∥2≤∥u∥2-2βn,i(1)〈u,Jxn〉-2βn,i(2)〈u,JTixn〉-2βn,i(2)〈u,JSiwn,i〉+βn,i(1)∥xn∥2+βn,i(2)∥Tixn∥2+βn,i(3)∥Siwn,i∥2-βn,i(1)βn,i(2)g(∥Jxn-JTixn∥)=βn,i(1)ϕ(u,xn)+βn,i(2)ϕ(u,Tixn)+βn,i(3)ϕ(u,Siwn,i)-βn,i(1)βn,i(2)g(∥Jxn-JTixn∥)≤βn,i(1)ϕ(u,xn)+βn,i(2)ϕ(u,xn)+βn,i(3)ϕ(u,wn,i)-βn,i(1)βn,i(2)g(∥Jxn-JTixn∥)≤βn,i(1)ϕ(u,xn)+βn,i(2)ϕ(u,xn)+βn,i(3)ϕ(u,xn)-βn,i(1)βn,i(2)g(∥Jxn-JTixn∥)=ϕ(u,xn)-βn,i(1)βn,i(2)g(∥Jxn-JTixn∥).
This implies that
βn,i(1)βn,i(2)g(∥Jxn-JTixn∥)≤ϕ(u,xn)-ϕ(u,zn,i),∀i∈I.
On the other hand, for every i∈I, one has
ϕ(u,xn)-ϕ(u,zn,i)=∥xn∥2-∥zn,i∥2-2〈u,Jxn-Jzn,i〉≤∥xn-zn,i∥(∥xn∥+∥zn,i∥)+2∥u∥∥Jxn-Jzn,i∥.
It follows from (3.30) and (3.31) that
ϕ(u,xn)-ϕ(u,zn,i)→0asn→∞,∀i∈I.
Applying lim infn→∞βn,i(1)βn,i(2)>0 and (3.35) in (3.33) we get
g(∥Jxn-JTixn∥)→0asn→∞,∀i∈I.
It follows from the property of g that
∥Jxn-JTixn∥→0asn→∞,∀i∈I.
Since J-1 is also uniformly norm-to-norm continuity on bounded sets, for every i∈I, one has
limn→∞∥xn-Tixn∥=0,∀i∈I.
In a similar way, one has
limn→∞∥xn-Siwn,i∥=0,∀i∈I.
On the other hand, we observe from (3.7) that
ϕ(u,zn,i)=ϕ(u,J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i))=∥u∥2-2〈u,βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i〉+∥βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i∥2≤∥u∥2-2βn,i(1)〈u,Jxn〉-2βn,i(2)〈u,JTixn〉-2βn,i(2)〈u,JSiwn,i〉+βn,i(1)∥Jxn∥2+βn,i(2)∥JTixn∥2+βn,i(3)∥JSiwn,i∥2-βn,i(1)βn,i(2)g(∥Jxn-JTixn∥)=βn,i(1)ϕ(u,xn)+βn,i(2)ϕ(u,Tixn)+βn,i(3)ϕ(u,Siwn,i)-βn,i(1)βn,i(2)g(∥Jxn-JTixn∥)≤βn,i(1)ϕ(u,xn)+βn,i(2)ϕ(u,xn)+βn,i(3)ϕ(u,wn,i)-βn,i(1)βn,i(2)g(∥Jxn-JTixn∥)≤βn,i(1)ϕ(u,xn)+βn,i(2)ϕ(u,xn)+βn,i(3)[ϕ(u,xn)+2λn,i(2c2λn,i-α)∥Axn-Au∥2]=ϕ(u,xn)+2βn,i(3)λn,i(2c2λn,i-α)∥Axn-Au∥2.
Hence
2a(α-2c2b)∥Axn-Au∥2≤ϕ(u,xn)-ϕ(u,zn,i).
Using (3.35), we can conclude that
limn→∞∥Axn-Au∥=0,∀i∈I.
From (3.6), we can calculate
ϕ(xn,wn,i)=ϕ(xn,ΠCvn,i)≤ϕ(xn,vn,i)=ϕ(xn,J-1(Jxn-λn,iAxn))=V(xn,Jxn-λn,iAxn)≤V(xn,(Jxn-λn,iAxn)+λn,iAxn)-2〈J-1(Jxn-λn,iAxn)-u,λn,iAxn〉=V(xn,Jxn)+2〈vn,i-xn,-λn,iAxn〉=ϕ(xn,xn)+2〈vn,i-xn,-λn,iAxn〉=2〈vn,i-xn,-λn,iAxn〉≤4c2λn,i2∥Axn-Au∥.
It follows from (3.42) and the fact that {λn,i} is bounded that
limn→∞ϕ(xn,wn,i)=0,∀i∈I.
From Lemma 2.2, we have
limn→∞∥xn-wn,i∥=0,∀i∈I.
Hence wn,i→p as n→∞ for each i∈I. From (3.39) and (3.45), we have
limn→∞∥wn,i-Siwn,i∥=0,∀i∈I.
The closedness of Ti and Si implies that p∈⋂i∈IF(Ti)∩⋂i∈IF(Si).

Case II.

Assume that (b) holds. We observe that
ϕ(u,zn,i)=ϕ(u,J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i))=∥u∥2-2〈u,βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i〉+∥βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i∥2≤∥u∥2-2βn,i(1)〈u,Jxn〉-2βn,i(2)〈u,JTixn〉-2βn,i(2)〈u,JSiwn,i〉+βn,i(1)∥Jxn∥2+βn,i(2)∥JTixn∥2+βn,i(3)∥JSiwn,i∥2-βn,i(2)βn,i(3)g(∥JSiwn,i-JTixn∥)=βn,i(1)ϕ(u,xn)+βn,i(2)ϕ(u,Tixn)+βn,i(3)ϕ(u,Siwn,i)-βn,i(2)βn,i(3)g(∥JSiwn,i-JTixn∥)≤βn,i(1)ϕ(u,xn)+βn,i(2)ϕ(u,xn)+βn,i(3)ϕ(u,wn,i)-βn,i(2)βn,i(3)g(∥JSiwn,i-JTixn∥)≤βn,i(1)ϕ(u,xn)+βn,i(2)ϕ(u,xn)+βn,i(3)ϕ(u,xn)-βn,i(2)βn,i(3)g(∥JSiwn,i-JTixn∥)=ϕ(u,xn)-βn,i(2)βn,i(3)g(∥JSiwn,i-JTixn∥).
This implies that
βn,i(2)βn,i(3)g(∥JSiwn,i-JTixn∥)≤ϕ(u,xn)-ϕ(u,zn,i),∀i∈I.
On the other hand, for every i∈I, one has
ϕ(u,xn)-ϕ(u,zn,i)=∥xn∥2-∥zn,i∥2-2〈u,Jxn-Jzn,i〉≤∥xn-zn,i∥(∥xn∥+∥zn,i∥)+2∥u∥∥Jxn-Jzn,i∥.
It follows from (3.30) and (3.31) that
ϕ(u,xn)-ϕ(u,zn,i)→0asn→∞,∀i∈I.
Applying lim infn→∞βn,i(2)βn,i(3)>0 and (3.50) we get
g(∥JSiwn,i-JTixn∥)→0asn→∞,∀i∈I.
It follows from the property of g that
∥JSiwn,i-JTixn∥→0asn→∞,∀i∈I.
Since J-1 is also uniformly norm-to-norm continuity on bounded sets, for every i∈I, one has
limn→∞∥Tixn-Siwn,i∥=0,∀i∈I.
On the other hand, we can calculate
ϕ(Tixn,zn,i)=ϕ(Tixn,J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i))=∥Tixn∥2-2〈Tixn,βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i〉+∥βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i)∥2≤∥Tixn∥2-2βn,i(1)〈Tixn,Jxn〉-2βn,i(2)〈Tixn,JTixn〉-2βn,i(3)〈Tixn,JSiwn,i〉+βn,i(1)∥xn∥2+βn,i(2)∥Tixn∥2+βn,i(3)∥Siwn,i∥2≤βn,i(1)ϕ(Tixn,xn)+βn,i(3)ϕ(Tixn,Siwn,i).
Observe that
ϕ(Tixn,Siwn,i)=∥Tixn∥2-2〈Tixn,JSiwn,i〉+∥Siwn,i∥2=∥Tixn∥2-2〈Tixn,JTixn〉+2〈Tixn,JTixn-JSiwn,i〉+∥Siwn,i∥2≤∥Siwn,i∥2-∥Tixn∥2+2∥Tixn∥∥JTixn-JSiwn,i∥≤∥Siwn,i-Tixn∥(∥Siwn,i∥+∥Tixn∥)+2∥Tixn∥∥JTixn-JSiwn,i∥.
It follows from (3.52) and (3.53) that
limn→∞ϕ(Tixn,Siwn,i)=0,∀i∈I.
Applying limn→∞βn,i(1)=0 and (3.56) and the fact that {ϕ(Tixn,xn)} is bounded to (3.54), we obtain
limn→∞ϕ(Tixn,zn,i)=0,∀i∈I.
From Lemma 2.2, one obtains
limn→∞∥Tixn-zn,i∥=0,∀i∈I.
We observe that
∥Tixn-xn∥≤∥Tixn-zn,i∥+∥zn,i-xn∥.
It follows from (3.30) and (3.58) that
limn→∞∥Tixn-xn∥=0,∀i∈I.
By the same proof as in Case I, we obtain that
limn→∞∥xn-wn,i∥=0,∀i∈I.
Hence wn,i→p as n→∞ for each i∈I and
limn→∞∥Jxn-Jwn,i∥=0,∀i∈I.
Combining (3.53), (3.60), and (3.61), we also have
limn→∞∥Siwn,i-wn,i∥=0,∀i∈I.
It follows from the closedness of Ti and Si that p∈⋂i∈IF(Ti)∩⋂i∈IF(Si).

(II) Now, we show that p∈VI(A,C).

Let T⊂E×E* be an operator defined by
Tv={Av+NC(v),v∈C,∅,v∉C.
By Lemma 2.8, we have that T is maximal monotone and T-10=VI(A,C). Let (v,w)∈G(T). Since w∈Tv=Av+NC(v), we obtain that w-Av∈NC(v). From xn=ΠCnx0⊂Cn⊂C, we have
〈v-xn,w-Av〉≥0.
Since A is α-inverse strongly monotone, we can calculate
〈v-xn,w〉≥〈v-xn,Av〉=〈v-xn,Av-Axn〉+〈v-xn,Axn〉≥〈v-xn,Axn〉.
From wn,i=ΠCJ-1(Jxn-λn,iAxn) and by Lemma 2.3, we have
〈v-wn,i,Jwn,i-Jxn-λn,iAxn〉≥0.
This implies that
〈v-wn,i,Jxn-Jwn,iλn,i-Axn〉≤0.
Since A is α-inverse strongly monotone, we have also that A is 1/α-Lipschitzian. Hence
〈v-xn,w〉≥〈v-xn,Axn〉+〈v-wn,i,Jxn-Jwn,iλn,i-Axn〉=〈v-wn,i,Axn〉+〈wn,i-xn,Axn〉-〈v-wn,i,Axn〉+〈v-wn,i,Jxn-Jwn,iλn,i〉=〈wn,i-xn,Axn〉+〈v-wn,i,Jxn-Jwn,iλn,i〉≥-∥wn,i-xn∥∥Axn∥-∥v-wn,i∥∥Jxn-Jwn,ia∥
for all n≥0. By Taking the limit as n→∞ and by (3.61) and (3.62), we obtain 〈v-p,w〉≥0. By the maximality of T we obtain p∈T-10 and hence p∈VI(A,C). Hence p∈F.

Step 6.

Finally, we show that p=ΠFx0.

From xn=ΠCnx0, we have
〈Jx0-Jxn,xn-z〉≥0,∀z∈Cn.
Since F⊂Cn, we also have
〈Jx0-Jxn,xn-u〉≥0,∀u∈F.
By taking limit in (3.71), we obtain that
〈Jx0-Jp,p-u〉≥0,∀u∈F.
By Lemma 2.3, we can conclude that p=ΠFx0. This completes the proof.

Remark 3.2.

Theorem 3.1 improves and extends main results of Iiduka and Takahashi [17], Martinez-Yanes and Xu [23], Matsushita and Takahashi [13], Plubtieng and Ungchittrakool [14], Qin and Su [15], and Qin et al. [25] because it can be applied to solving the problem of finding the common element of the set of common fixed points of two families of quasi-ϕ-nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly monotone operator.

4. Applications

From Theorem 3.1 we can obtain some new and interesting strong convergence theorems. Now we give some examples as follows.

If βn,i(1)=0 for all n≥0, Ti=Si for all i∈I and A=0 in Theorem 3.1, then we have the following result.

Corollary 4.1.

Let E be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let {Ti}i∈I be a family of closed quasi-ϕ-nonexpansive mappings of C into itself with F:=⋂i∈IF(Ti) being nonempty, where I is an index set. Let {xn} be a sequence generated by the following manner:
x0∈Cchosenarbitrary,C1,i=C,C1=⋂i=1∞C1,i,x1=ΠC1(x0)∀i∈I,yn,i=J-1(αn,iJx0+(1-αn,i)JTixn),Cn+1,i={u∈Cn,i:ϕ(u,yn,i)⩽ϕ(u,xn)+αn,i(∥x0∥2+2〈u,Jxn-Jx0〉)},Cn+1=⋂i∈ICn+1,i,xn+1=ΠCn+1x0,∀n≥0,
where J is the duality mapping on E, and {αn,i} is a sequence in (0,1) such that lim supn→∞αn,i=0,foralli∈I. Then the sequence {xn} converges strongly to ΠFx0, where ΠF is the generalized projection from C onto F.

Now we consider the problem of finding a zero point of an inverse-strongly monotone operator of E into E*. Assume that A satisfies the following conditions:

A is α-inverse-strongly monotone,

A-10={u∈E:Au=0}≠∅.

Corollary 4.2.

Let E be a 2 uniformly convex and uniformly smooth Banach space. Let A be an operator of E into E* satisfying (C1) and (C2), and let {Si}i∈I and {Ti}i∈I be two families of closed quasi-ϕ-nonexpansive mappings of E into itself with F:=⋂i∈IF(Ti)∩⋂i∈IF(Si)∩A-10 being nonempty, where I is an index set. Let {xn} be a sequence generated by the following manner:
x0∈Echosenarbitrary,C1,i=E,C1=⋂i=1∞C1,i,x1=ΠC1(x0)∀i∈I,wn,i=J-1(Jxn-λn,iAxn),zn,i=J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i),yn,i=J-1(αn,iJx0+(1-αn,i)Jzn,i),Cn+1,i={u∈Cn,i:ϕ(u,yn,i)⩽ϕ(u,xn)+αn,i(∥x0∥2+2〈u,Jxn-Jx0〉)},Cn+1=⋂i∈ICn+1,i,xn+1=ΠCn+1x0,∀n≥0,
where J is the duality mapping on E, and {λn,i},{αn,i}, and {βn,i(j)}(j=1,2,3) are sequences in (0,1) such that

limn→∞αn,i=0 for all i∈I;

for all i∈I, {λn,i}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2 uniformly convexity constant of E;

βn,i(1)+βn,i(2)+βn,i(3)=1 for all i∈I and if one of the following conditions is satisfied:

lim infn→∞βn,i(1)βn,i(l)>0 for all l=2,3 and for all i∈I,

lim infn→∞βn,i(2)βn,i(3)>0 and limn→∞βn,i(1)=0 for all i∈I.

Then the sequence {xn} converges strongly to ΠFx0, where ΠF is the generalized projection from C onto F.Proof.

Setting C=E in Theorem 3.1, we get that ΠE is the identity mapping, that is, ΠEx=x for all x∈E. We also have VI(A,E)=A-10. From Theorem 3.1, we can obtain the desired conclusion easily.

Let X be a nonempty closed convex cone in E, and let A be an operator from X into E*. We define its polar in E* to be the set

X*={y*∈E*:〈x,y*〉≥0∀x∈X}.
Then an element x in X is called a solution of the complementarity problem if

Ax∈X*,〈x,Ax〉=0.
The set of all solutions of the complementarity problem is denoted by CP(A,X). Several problems arising in different fields, such as mathematical programming, game theory, mechanics, and geometry, are to find solutions of the complementarity problems.

Corollary 4.3.

Let E be a 2 uniformly convex and uniformly smooth Banach space, and let X be a nonempty closed convex subset of E. Let A be an operator of X into E* satisfying (C1) and (C2), and let {Si}i∈I and {Ti}i∈I be two families of closed quasi-ϕ-nonexpansive mappings of X into itself with F:=⋂i∈IF(Ti)∩⋂i∈IF(Si)∩CP(A,X) being nonempty, where I is an index set. Let {xn} be a sequence generated by the following manner:
x0∈Xchosenarbitrary,C1,i=X,C1=⋂i=1∞C1,i,x1=ΠC1(x0)∀i∈I,wn,i=ΠXJ-1(Jxn-λn,iAxn),zn,i=J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSiwn,i),yn,i=J-1(αn,iJx0+(1-αn,i)Jzn,i),Cn+1,i={u∈Cn,i:ϕ(u,yn,i)⩽ϕ(u,xn)+αn,i(∥x0∥2+2〈u,Jxn-Jx0〉)},Cn+1=⋂i∈ICn+1,i,xn+1=ΠCn+1x0,∀n≥0,
where J is the duality mapping on E, and {λn,i},{αn,i},and{βn,i(j)}(j=1,2,3) are sequences in (0,1) such that

limn→∞αn,i=0 for all i∈I;

for all i∈I,{λn,i}⊂[a,b] for some a,b with0<a<b<c2α/2, where1/c is the2 uniformly convexity constant of E;

βn,i(1)+βn,i(2)+βn,i(3)=1 for all i∈I and if one of the following conditions is satisfied:

lim infn→∞βn,i(1)βn,i(l)>0for all l=2,3and for all i∈I,

lim infn→∞βn,i(2)βn,i(3)>0 and limn→∞βn,i(1)=0 for all i∈I.

Then the sequence {xn} converges strongly to ΠFx0, where ΠF is the generalized projection from C onto F.Proof.

From [29, Lemma 7.1.1], we have VI(A,X)=CP(A,X). From Theorem 3.1, we can obtain the desired conclusion easily.

Acknowledgments

The first author would like to thank The Thailand Research Fund, Grant TRG5280011 for financial support. The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments and pointing out a major error in the original version of this paper.

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