We introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed quasi-ϕ-asymptotically nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality problem for a γ-inverse strongly monotone mapping in Banach spaces. Then we study the strong convergence of the algorithm. Our results improve and extend the corresponding results announced by many others.

1. Introduction and Preliminary

Let E be a Banach space with the dual E*. A mapping A:D(A)⊂E→E* is said to be monotone if, for each x,y∈D(A), the following inequality holds:
(1.1)〈Ax-Ay,x-y〉≥0.Ais said to be γ-inverse strongly monotone if there exists a positive real number γ such that
(1.2)〈x-y,Ax-Ay〉≥γ‖Ax-Ay‖2,∀x,y∈D(A).
If A is γ-inverse strongly monotone, then it is Lipschitz continuous with constant 1/γ, that is, ∥Ax-Ay∥≤(1/γ)∥x-y∥, for all x,y∈D(A), and hence uniformly continuous.

Let C be a nonempty closed convex subset of E and f:C×C→ℝ a bifunction, where ℝ is the set of real numbers. The equilibrium problem for f is to find x^∈C such that
(1.3)f(x^,y)≥0
for all y∈C. The set of solutions of (1.3) is denoted by EP(f). Given a mapping T:C→E*, let f(x,y)=〈Tx,y-x〉 for all x,y∈C. Then x^∈EP(f) if and only if 〈Tx^,y-x^〉≥0 for all y∈C; that is, x^ is a solution of the variational inequality. Numerous problems in physics, optimization, engineering, and economics reduce to find a solution of (1.3). Some methods have been proposed to solve the equilibrium problem; see, for example, Blum and Oettli [1] and Moudafi [2]. For solving the equilibrium problem, let us assume that f satisfies the following conditions:

f(x,x)=0 for all x∈C;

f is monotone, that is, f(x,y)+f(y,x)≤0 for all x,y∈C;

for each x,y,z∈C, limt→0f(tz+(1-t)x,y)≤f(x,y);

for each x∈C, the function y↦f(x,y) is convex and lower semicontinuous.

Let E be a Banach space with the dual E*. We denote by J the normalized duality mapping from E to 2E* defined by
(1.4)J(x)={x*∈E*:〈x,x*〉=‖x‖2=‖x*‖2},
where 〈·,·〉 denotes the generalized duality pairing. Let dim E≥2, and the modulus of smoothness of E is the function ρE:[0,∞)→[0,∞) defined by
(1.5)ρE(τ)=sup{‖x+y‖+‖x-y‖2-1:‖x‖=1;‖y‖=τ}.
The space E is said to be smooth if ρE(τ)>0, for all τ>0 and E is called uniformly smooth if and only if limt→0+(ρE(t)/t)=0. A Banach space E is said to be strictly convex if ∥x+y∥/2<1 for x,y∈E with ∥x∥=∥y∥=1 and x≠y. The modulus of convexity of E is the function δE:(0,2]→[0,1] defined by
(1.6)δE(ϵ)=inf{1-‖x+y2‖:‖x‖=‖y‖=1;ϵ=‖x-y‖}.E is called uniformly convex if and only if δE(ϵ)>0 for every ϵ∈(0,2]. Let p>1, then E is said to be p-uniformly convex if there exists a constant c>0 such that δE(ϵ)≥cϵp for all ϵ∈(0,2]. Observe that every p-uniformly convex space is uniformly convex. We know that if E is uniformly smooth, strictly convex, and reflexive, then the normalized duality mapping J is single-valued, one-to-one, onto and uniformly norm-to-norm continuous on each bounded subset of E. Moreover, if E is a reflexive and strictly convex Banach space with a strictly convex dual, then J-1 is single-valued, one-to-one, surjective, and it is the duality mapping from E* into E and thus JJ-1=IE* and J-1J=IE (see, [3]). It is also well known that E is uniformly smooth if and only if E* is uniformly convex.

Let C be a nonempty closed convex subset of a Banach space E and T:C→C a mapping. A point x∈C is said to be a fixed point of T provided Tx=x. A point x∈C is said to be an asymptotic fixed point of T provided C contains a sequence {xn} which converges weakly to x such that limn→∞∥xn-Txn∥=0. In this paper, we use F(T) and F(T)~ to denote the fixed point set and the asymptotic fixed point set of T and use → to denote the strong convergence and weak convergence, respectively. Recall that a mapping T:C→C is called nonexpansive if
(1.7)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.
A mapping T:C→C is called asymptotically nonexpansive if there exists a sequence {kn} of real numbers with kn→1 as n→∞ such that
(1.8)‖Tnx-Tny‖≤kn‖x-y‖,∀x,y∈C,∀n≥1.

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [4] in 1972. They proved that if C is a nonempty bounded closed convex subset of a uniformly convex Banach space E, then every asymptotically nonexpansive self-mapping T of C has a fixed point. Further, the set F(T) is closed and convex. Since 1972, a host of authors have studied the weak and strong convergence problems of the iterative algorithms for such a class of mappings (see, e.g., [4–6] and the references therein).

It is well known that 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 nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [7] recently introduced a generalized projection operator ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces.

Next, we assume that E is a smooth Banach space. Consider the functional defined by
(1.9)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,∀x,y∈E.
Following Alber [7], 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 following minimization problem:
(1.10)ϕ(x-,x)=infy∈Cϕ(y,x).
It follows from the definition of the function ϕ that
(1.11)(‖y‖-‖x‖)2≤ϕ(y,x)≤(‖y‖+‖x‖)2,∀x,y∈E.
If E is a Hilbert space, then ϕ(y,x)=∥y-x∥2 and ΠC=PC is the metric projection of H onto C.

Remark 1.1 (see [<xref ref-type="bibr" rid="B9">8</xref>, <xref ref-type="bibr" rid="B27">9</xref>]).

If E is a reflexive, strictly convex, and smooth Banach space, then for x,y∈E, ϕ(x,y)=0 if and only if x=y.

Let C be a nonempty, closed, and convex subset of a smooth Banach E and T a mapping from C into itself. The mapping T is said to be relatively nonexpansive if F(T)~=F(T)≠∅,ϕ(p,Tx)≤ϕ(p,x),forallx∈C,p∈F(T). The mapping T is said to be ϕ-nonexpansive if ϕ(Tx,Ty)≤ϕ(x,y),forallx,y∈C. The mapping T is said to be quasi-ϕ-nonexpansive if F(T)≠∅,ϕ(p,Tx)≤ϕ(p,x),forallx∈C,p∈F(T). The mapping T is said to be relatively asymptotically nonexpansive if there exists some real sequence {kn} with kn≥1 and kn→1 as n→∞ such that F(T)~=F(T)≠∅,ϕ(p,Tnx)≤knϕ(p,x),forallx∈C,p∈F(T). The mapping T is said to be ϕ-asymptotically nonexpansive if there exists some real sequence {kn} with kn≥1 and kn→1 as n→∞ such that ϕ(Tnx,Tny)≤knϕ(x,y),forallx,y∈C. The mapping T is said to be quasi-ϕ-asymptotically nonexpansive if there exists some real sequence {kn} with kn≥1 and kn→1 as n→∞ such that F(T)≠∅,ϕ(p,Tnx)≤knϕ(p,x),forallx∈C,p∈F(T). The mapping T is said to be asymptotically regular on C if, for any bounded subset K of C, limsupn→∞{∥Tn+1x-Tnx∥:x∈K}=0. The mapping T is said to be closed on C if, for any sequence {xn} such that limn→∞xn=x0 and limn→∞Txn=y0, Tx0=y0.

We remark that a ϕ-asymptotically nonexpansive mapping with a nonempty fixed point set F(T) is a quasi-ϕ-asymptotically nonexpansive mapping, but the converse may be not true. The class of quasi-ϕ-nonexpansive mappings and quasi-ϕ-asymptotically nonexpansive mappings is more general than the class of relatively nonexpansive mappings and relatively asymptotically nonexpansive mappings, respectively.

Recently, many authors studied the problem of finding a common element of the set of fixed points of nonexpansive or relatively nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of variational inequalities in the frame work of Hilbert spaces and Banach spaces, respectively; see, for instance, [10–21] and the references therein.

In 2009, Takahashi and Zembayashi [22] introduced the following iterative process:
(1.12)x0=x∈C,yn=J-1(αnJxn+(1-αn)JSxn),un∈C,f(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Hn={z∈C:ϕ(z,un)≤ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx-Jxn〉≥0},xn+1=ΠHn∩Wnx,∀n≥1,
where f:C×C→ℝ is a bifunction satisfying (A1)–(A4), J is the normalized duality mapping on E and S:C→C is a relatively nonexpansive mapping. They proved the sequence {xn} defined by (1.12) converges strongly to a common point of the set of solutions of the equilibrium problem (1.3) and the set of fixed points of S provided the control sequences {αn} and {rn} satisfy appropriate conditions in Banach spaces.

Qin et al. [8] introduced the following iterative scheme on the equilibrium problem (1.3) and a family of quasi-ϕ-nonexpansive mapping:
(1.13)x0∈E,C1=C,x1=ΠC1x0,yn=J-1(αn,0Jxn+Σi=1Nαn,iJTixn),un∈C,f(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x0,∀n≥1.
Strong convergence theorems of common elements are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.

Very recently, for finding a common element of ∩i=1rF(Ti)∩EP(f,B)∩VI(A,C) Zegeye [23] proposed the following iterative algorithm:
(1.14)x0∈C0=C,zn=ΠCJ-1(Jxn-λnAxn),yn=J-1(α0Jxn+Σi=1rαiJTizn),un∈C,f(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x0,∀n≥1,
where Ti:C→C is closed quasi-ϕ-nonexpansive mapping (i=1,…,r), f:C×C→ℝ is a bifunction satisfying (A1)–(A4) and A is a γ-inverse strongly monotone mapping of C into E*. Strong convergence theorems for iterative scheme (1.14) are obtained under some conditions on parameters in 2-uniformly convex and uniformly smooth real Banach space E.

In this paper, inspired and motivated by the works mentioned above, we introduce an iterative process for finding a common element of the set of common fixed points of a finite family of closed quasi-ϕ-asymptotically nonexpansive mappings, the solution set of equilibrium problem, and the solution set of the variational inequality problem for a γ-inverse strongly monotone mapping in Banach spaces. The results presented in this paper improve and generalize the corresponding results announced by many others.

In order to the main results of this paper, we need the following lemmas.

Lemma 1.2 (see [<xref ref-type="bibr" rid="B20">24</xref>]).

Let E be a 2-uniformly convex and smooth Banach space. Then, for all x,y∈E, one has
(1.15)‖x-y‖≤2c2‖Jx-Jy‖,
where J is the normalized duality mapping of E and 1/c(0<c≤1) is the 2-uniformly convex constant of E.

Lemma 1.3 (see [<xref ref-type="bibr" rid="B1">7</xref>, <xref ref-type="bibr" rid="B7">25</xref>]).

Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Then
(1.16)ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y),∀x∈C,y∈E.

Lemma 1.4 (see [<xref ref-type="bibr" rid="B7">25</xref>]).

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

Lemma 1.5 (see [<xref ref-type="bibr" rid="B1">7</xref>]).

Let C be a nonempty closed convex subset of a smooth Banach space E, let x∈E and let z∈C. Then
(1.17)z=ΠCx⟺〈y-z,Jx-Jz〉≤0,∀y∈C.

We denote by NC(v) the normal cone for C⊂E at a point v∈C, that is, NC(v)={x*∈E*:〈v-y,x*〉≥0,forally∈C}. We shall use the following lemma.

Lemma 1.6 (see [<xref ref-type="bibr" rid="B10">26</xref>]).

Let C be a nonempty closed convex subset of a Banach space E and let A be a monotone and hemicontinuous operator of C into E* with C=D(A). Let S⊂E×E* be an operator defined as follows:
(1.18)Sv={Av+NC(v),v∈C,∅,v∉C.
Then S is maximal monotone and S-1(0)=VI(C,A).

We make use of the function V:E×E*→ℝ defined by
(1.19)V(x,x*)=‖x‖2-2〈x,x*〉+‖x*‖2,
for all x∈E and x*∈E* (see [7]). That is, V(x,x*)=ϕ(x,J-1x*) for all x∈E and x*∈E*.

Lemma 1.7 (see [<xref ref-type="bibr" rid="B1">7</xref>]).

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

Lemma 1.8 (see [<xref ref-type="bibr" rid="B2">1</xref>]).

Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C×C to ℝ satisfying (A1)–(A4), and let r>0 and x∈E. Then, there exists z∈C such that
(1.21)f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.

Lemma 1.9 (see [<xref ref-type="bibr" rid="B19">22</xref>]).

Let C be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). For r>0 and x∈E, define a mapping Tr:E→C as follows:
(1.22)Tr(x)={z∈C:f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}
for all x∈E. Then, the following hold:

Tr is single-valued;

Tr is firmly nonexpansive, that is, for any x,y∈E,
(1.23)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉;

F(Tr)=
EP
(f);

EP
(f) is closed and convex;

ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x),forallq∈F(Tr).

Lemma 1.10 (see [<xref ref-type="bibr" rid="B9">8</xref>, <xref ref-type="bibr" rid="B25">23</xref>]).

Let E be a uniformly convex Banach space, s>0 a positive number and Bs(0) a closed ball of E. Then there exists a strictly increasing, continuous, and convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(1.24)‖∑i=0N(αixi)‖2≤∑i=0Nαi‖xi‖2-αkαlg(‖xk-xl‖)
for any k,l∈{0,1,…,N}, for all x0,x1,…,xN∈Bs(0)={x∈E:∥x∥≤s} and α0,α1,…,αn∈[0,1] such that ∑i=0Nαi=1.

Lemma 1.11 (see [<xref ref-type="bibr" rid="B3">27</xref>]).

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

2. Main Results Theorem 2.1.

Let C be a nonempty, closed, and convex subset of a 2-uniformly convex and uniformly smooth real Banach space E and Ti:C→C a closed quasi-ϕ-asymptotically nonexpansive mapping with sequence {kn,i}⊂[1,∞) such that limn→∞kn,i=1 for each 1≤i≤N. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). Let A be a γ-inverse strongly monotone mapping of C into E* with constant γ>0 such that F=(⋂i=1NF(Ti))⋂
EP
(f)⋂
VI
(C,A)≠∅ and F is bounded. Assume that Ti is asymptotically regular on C for each 1≤i≤N and ∥Ax∥≤∥Ax-Ap∥ for all x∈C and p∈F. Define a sequence {xn} in C in the following manner:
(2.1)x0∈C0=Cchosenarbitrarily,zn=ΠCJ-1(Jxn-λnAxn),yn=J-1(αn,0Jxn+αn,1JT1nzn+⋯+αn,NJTNnzn),un∈C,f(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+∑i=1Nαn,i(kn,i-1)Ln},xn+1=ΠCn+1x0
for every n≥0, where {rn} is a real sequence in [a,∞) for some a>0, J is the normalized duality mapping on E and Ln=sup{ϕ(p,xn):p∈F}<∞. Assume that {αn,0}, {αn,1}, …, {αn,N} are real sequences in (0,1) such that ∑i=0Nαn,i=1 and liminfn→∞αn,0αn,i>0, for all i∈{1,2,…,N}. Let {λn} be a sequence in [s,t] for some 0<s<t<c2γ/2, where 1/c is the 2-uniformly convex constant of E. Then the sequence {xn} converges strongly to ΠFx0.

Proof.

We break the proof into nine steps.

Step 1.

ΠFx0 is well defined for x0∈C.

By Lemma 1.11 we know that F(Ti) is a closed convex subset of C for every 1≤i≤N. Hence F=(⋂i=1NF(Ti))⋂EP(f)⋂VI(C,A)≠∅ is a nonempty closed convex subset of C. Consequently, ΠFx0 is well defined for x0∈C.

Step 2.

Cn is closed and convex for each n≥0.

It is obvious that C0=C is closed and convex. Suppose that Cn is closed and convex for some integer n. Since the defining inequality in Cn+1 is equivalent to the inequality:
(2.2)2〈z,Jxn-Jun〉≤‖xn‖2-‖un‖2+∑i=1Nαn,i(kn,i-1)Ln,
we have that Cn+1 is closed and convex. So Cn is closed and convex for each n≥0. This in turn shows that ΠCn+1x0 is well defined.

Step 3.

F⊂Cn for all n≥0.

We do this by induction. For n=0, we have F⊂C=C0. Suppose that F⊂Cn for some n≥0. Let p∈F⊂C. Putting un=Trnyn for all n≥0, we have that Trn is quasi-ϕ-nonexpansive from Lemma 1.9. Since Ti is quasi-ϕ-asymptotically nonexpansive, we have
(2.3)ϕ(p,un)=ϕ(p,Trnyn)≤ϕ(p,yn)=ϕ(p,J-1(αn,0Jxn+∑i=1Nαn,iJTinzn))=‖p‖2-2〈p,αn,0Jxn+∑i=1Nαn,iJTinzn〉+‖αn,0Jxn+∑i=1Nαn,iJTinzn‖2≤‖p‖2-2αn,0〈p,Jxn〉-2∑i=1Nαn,i〈p,JTinzn〉+αn,0‖xn‖2+∑i=1Nαn,i‖Tinzn‖2=αn,0ϕ(p,xn)+∑i=1Nαn,iϕ(p,Tinzn)≤αn,0ϕ(p,xn)+∑i=1Nαn,ikn,iϕ(p,zn).
Moreover, by Lemmas 1.3 and 1.7, we get that
(2.4)ϕ(p,zn)=ϕ(p,ΠCJ-1(Jxn-λnAxn))≤ϕ(p,J-1(Jxn-λnAxn))=V(p,Jxn-λnAxn)≤V(p,(Jxn-λnAxn)+λnAxn)-2〈J-1(Jxn-λnAxn)-p,λnAxn〉=V(p,Jxn)-2λn〈J-1(Jxn-λnAxn)-p,Axn〉=ϕ(p,xn)-2λn〈xn-p,Axn〉-2λn〈J-1(Jxn-λnAxn)-xn,Axn〉≤ϕ(p,xn)-2λn〈xn-p,Axn-Ap〉-2λn〈xn-p,Ap〉+2〈J-1(Jxn-λnAxn)-xn,-λnAxn〉.
Thus, since p∈VI(C,A) and A is γ-inverse strongly monotone, we have from (2.4) that
(2.5)ϕ(p,zn)≤ϕ(p,xn)-2λnγ‖Axn-Ap‖2+2〈J-1(Jxn-λnAxn)-xn,-λnAxn〉.
Therefore, from (2.5), Lemma 1.2 and the fact that λn<c2γ/2 and ∥Ax∥≤∥Ax-Ap∥ for all x∈C and p∈F, we have
(2.6)ϕ(p,zn)≤ϕ(p,xn)-2λnγ‖Axn-Ap‖2+4c2λn2‖Axn-Ap‖2=ϕ(p,xn)+2λn(2c2λn-γ)‖Axn-Ap‖2≤ϕ(p,xn).
Substituting (2.6) into (2.3), we get
(2.7)ϕ(p,un)≤ϕ(p,xn)+∑i=1Nαn,i(kn,i-1)Ln,
that is, p∈Cn+1. By induction, F⊂Cn and the iteration algorithm generated by (2.1) is well defined.

Step 4.

limn→∞ϕ(xn,x0) exists and {xn} is bounded.

Noticing that xn=ΠCnx0 and Lemma 1.3, we have
(2.8)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(p,x0)-ϕ(p,xn)≤ϕ(p,x0)
for all p∈F and n≥0. This shows that the sequence {ϕ(xn,x0)} is bounded. From xn=ΠCnx0 and xn+1=ΠCn+1x0∈Cn+1⊂Cn, we obtain that
(2.9)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥0,
which implies that {ϕ(xn,x0)} is nondecreasing. Therefore, the limit of {ϕ(xn,x0)} exists and {xn} is bounded.

Step 5.

We have xn→x*∈C.

By Lemma 1.3, we have, for any positive integer m≥n, that
(2.10)ϕ(xm,xn)=ϕ(xm,ΠCnx0)≤ϕ(xm,x0)-ϕ(ΠCnx0,x0)=ϕ(xm,x0)-ϕ(xn,x0).
In view of Step 4 we deduce that ϕ(xm,xn)→0 as m,n→∞. It follows from Lemma 1.4 that ∥xm-xn∥→0 as m,n→∞. Hence {xn} is a Cauchy sequence of C. Since E is a Banach space and C is closed subset of E, there exists a point x*∈C such that xn→x*(n→∞).

Step 6.

We havex*∈⋂i=1NF(Ti).

By taking m=n+1 in (2.10), we have
(2.11)limn→∞ϕ(xn+1,xn)=0.
From Lemma 1.4, it follows that
(2.12)limn→∞‖xn+1-xn‖=0.
Noticing that xn+1∈Cn+1, we obtain
(2.13)ϕ(xn+1,un)≤ϕ(xn+1,xn)+∑i=1Nαn,i(kn,i-1)Ln.
From (2.11), limn→∞kn,i=1 for any 1≤i≤N, and Lemma 1.4, we know
(2.14)limn→∞‖xn+1-un‖=0.
Notice that
(2.15)‖xn-un‖≤‖xn-xn+1‖+‖xn+1-un‖
for all n≥0. It follows from (2.12) and (2.14) that
(2.16)limn→∞‖xn-un‖=0,
which implies that un→x* as n→∞. Since J is uniformly norm-to-norm continuous on bounded sets, from (2.16), we have
(2.17)limn→∞‖Jxn-Jun‖=0.
Let s=sup{∥xn∥,∥T1nxn∥,∥T2nxn∥,…,∥TNnxn∥:n∈ℕ}. Since E is uniformly smooth Banach space, we know that E* is a uniformly convex Banach space. Therefore, from Lemma 1.10 we have, for any p∈F, that
ϕ(p,un)=ϕ(p,Trnyn)≤ϕ(p,yn)=ϕ(p,J-1(αn,0Jxn+∑i=1Nαn,iJTinzn))=‖p‖2-2αn,0〈p,Jxn〉-2∑i=1Nαn,i〈p,JTinzn〉+‖αn,0Jxn+∑i=1Nαn,iJTinzn‖2≤‖p‖2-2αn,0〈p,Jxn〉-2∑i=1Nαn,i〈p,JTinzn〉+αn,0‖xn‖2+∑i=1Nαn,i‖Tinzn‖2-αn,0αn,1g(‖Jxn-JT1nzn‖)=αn,0ϕ(p,xn)+∑i=1Nαn,iϕ(p,Tinzn)-αn,0αn,1g(‖Jxn-JT1nzn‖)≤αn,0ϕ(p,xn)+∑i=1Nαn,ikn,iϕ(p,zn)-αn,0αn,1g(‖Jxn-JT1nzn‖).
Therefore, from (2.6) and (2.18), we have
(2.19)ϕ(p,un)≤ϕ(p,xn)+∑i=1Nαn,i(kn,i-1)ϕ(p,xn)-αn,0αn,1g(‖Jxn-JT1nzn‖)+2λn(2c2λn-γ)‖Axn-Ap‖2∑i=1Nαn,ikn,i.
It follows from λn<c2γ/2 that
(2.20)αn,0αn,1g(‖Jxn-JT1nzn‖)≤ϕ(p,xn)-ϕ(p,un)+∑i=1Nαn,i(kn,i-1)ϕ(p,xn).
On the other hand, we have
(2.21)|ϕ(p,xn)-ϕ(p,un)|=|‖xn‖2-‖un‖2-2〈p,Jxn-Jun〉|≤|‖xn‖-‖un‖|(‖xn‖+‖un‖)+2‖Jxn-Jun‖‖p‖≤‖xn-un‖(‖xn‖+‖un‖)+2‖Jxn-Jun‖‖p‖.
It follows from (2.16) and (2.17) that
(2.22)limn→∞(ϕ(p,xn)-ϕ(p,un))=0.
Since limn→∞kn,i=1 and liminfn→∞αn,0αn,1>0, from (2.20) and (2.22) we have
(2.23)limn→∞g(‖Jxn-JT1nzn‖)=0.
Therefore, from the property of g, we obtain
(2.24)limn→∞‖Jxn-JT1nzn‖=0.
Since J-1 is uniformly norm-to-norm continuous on bounded sets, we have
(2.25)limn→∞‖xn-T1nzn‖=0,
and hence T1nzn→x* as n→∞. Since ∥T1n+1zn-x*∥≤∥T1n+1zn-T1nzn∥+∥T1nzn-x*∥, it follows from the asymptotic regularity of T1 that
(2.26)limn→∞‖T1n+1zn-x*‖=0.
That is, T1(T1nzn)→x* as n→∞. From the closedness of T1, we get T1x*=x*. Similarly, one can obtain that Tix*=x* for i=2,…,N. So, x*∈⋂i=1NF(Ti).

Moreover, from (2.19) we have that
(2.27)2λn(γ-2c2λn)‖Axn-Ap‖2(1-αn,0)≤2λn(γ-2c2λn)‖Axn-Ap‖2∑i=1Nαn,ikn,i≤ϕ(p,xn)-ϕ(p,un)+∑i=1Nαn,i(kn,i-1)ϕ(p,xn),
which implies that
(2.28)limn→∞‖Axn-Ap‖=0.
Now, Lemmas 1.3 and 1.7 imply that
(2.29)ϕ(xn,zn)=ϕ(xn,ΠCJ-1(Jxn-λnAxn))≤ϕ(xn,J-1(Jxn-λnAxn))=V(xn,Jxn-λnAxn)≤V(xn,(Jxn-λnAxn)+λnAxn)-2〈J-1(Jxn-λnAxn)-xn,λnAxn〉=ϕ(xn,xn)+2〈J-1(Jxn-λnAxn)-xn,-λnAxn〉=2〈J-1(Jxn-λnAxn)-xn,-λnAxn〉≤2‖J-1(Jxn-λnAxn)-J-1Jxn‖⋅‖λnAxn‖.
In view of Lemma 1.2 and the fact that ∥Ax∥≤∥Ax-Ap∥ for all x∈C, p∈F, we have
(2.30)ϕ(xn,zn)≤4c2λn2‖Axn-Ap‖2≤4c2t2‖Axn-Ap‖2.
From (2.28) and Lemma 1.4 we get
(2.31)limn→∞‖xn-zn‖=0,
and hence zn→x* as n→∞.

Step 7.

We have x*∈VI(C,A).

Let S⊂E×E* be an operator as follows:
(2.32)Sv={Av+NC(v),v∈C,∅,v∉C.
By Lemma 1.6, S is maximal monotone and S-1(0)=VI(C,A). Let (v,w)∈G(S). Since w∈Sv=Av+NC(v), we have w-Av∈NC(v). It follows from zn∈C that
(2.33)〈v-zn,w-Av〉≥0.
On the other hand, from zn=ΠCJ-1(Jxn-λnAxn) and Lemma 1.5 we obtain that
(2.34)〈v-zn,Jzn-(Jxn-λnAxn)〉≥0,
and hence
(2.35)〈v-zn,Jxn-Jznλn-Axn〉≤0.
Then, from (2.33) and (2.35), we have
(2.36)〈v-zn,w〉≥〈v-zn,Av〉≥〈v-zn,Av〉+〈v-zn,Jxn-Jznλn-Axn〉=〈v-zn,Av-Axn+Jxn-Jznλn〉=〈v-zn,Av-Azn〉+〈v-zn,Azn-Axn〉+〈v-zn,Jxn-Jznλn〉≥-‖v-zn‖⋅‖Azn-Axn‖-‖v-zn‖⋅‖Jxn-Jzn‖s.
Hence we have 〈v-x*,w〉≥0 as n→∞, since the uniform continuity of J and A imply that the right side of (2.36) goes to 0 as n→∞. Thus, since S is maximal monotone, we have x*∈S-1(0) and hence x*∈VI(C,A).

Step 8.

We have x*∈EP(f)=F(Tr).

Let p∈F. From un=Trnyn, (2.3), (2.6) and Lemma 1.9 we obtain that
(2.37)ϕ(un,yn)=ϕ(Trnyn,yn)≤ϕ(p,yn)-ϕ(p,Trnyn)≤ϕ(p,xn)+∑i=1Nαn,i(kn,i-1)ϕ(p,xn)-ϕ(p,un).
It follows from (2.22) and kn,i→1 that ϕ(un,yn)→0 as n→∞. Now, by Lemma 1.4 we have that ∥un-yn∥→0 as n→∞. Consequently, we obtain that ∥Jun-Jyn∥→0 and yn→x* from un→x* as n→∞. From the assumption rn>a, we get
(2.38)limn→∞‖Jun-Jyn‖rn=0.
Noting that un=Trnyn, we obtain
(2.39)f(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C.
From (A2), we have
(2.40)〈y-un,Jun-Jynrn〉≥-f(un,y)≥f(y,un),∀y∈C.
Letting n→∞, we have from un→x*, (2.38) and (A4) that f(y,x*)≤0(forally∈C). For t with 0<t≤1 and y∈C, let yt=ty+(1-t)x*. Since y∈C and x*∈C, we have yt∈C and hence f(yt,x*)≤0. Now, from (A1) and (A4) we have
(2.41)0=f(yt,yt)≤tf(yt,y)+(1-t)f(yt,x*)≤tf(yt,y)
and hence f(yt,y)≥0. Letting t→0, from (A3), we have f(x*,y)≥0. This implies that x*∈EP(f). Therefore, in view of Steps 6, 7, and 8 we have x*∈F.

Step 9.

We have x*=ΠFx0.

From xn=ΠCnx0, we get
(2.42)〈xn-z,Jx0-Jxn〉≥0,∀z∈Cn.
Since F⊂Cn for all n≥1, we arrive at
(2.43)〈xn-p,Jx0-Jxn〉≥0,∀p∈F.
Letting n→∞, we have
(2.44)〈x*-p,Jx0-Jx*〉≥0,∀p∈F,
and hence x*=ΠFx0 by Lemma 1.5. This completes the proof.

Strong convergence theorem for approximating a common element of the set of solutions of the equilibrium problem and the set of fixed points of a finite family of closed quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces may not require that E be 2-uniformly convex. In fact, we have the following theorem.

Theorem 2.2.

Let C be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E and Ti:C→C a closed quasi-ϕ-asymptotically nonexpansive mapping with sequence {kn,i}⊂[1,∞) such that limn→∞kn,i=1 for each 1≤i≤N. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) such that F=(⋂i=1NF(Ti))⋂EP(f)≠∅ and F is bounded. Assume that Ti is asymptotically regular on C for each 1≤i≤N. Define a sequence {xn} in C in the following manner:
(2.45)x0∈C0=Cchosenarbitrarily,yn=J-1(αn,0Jxn+αn,1JT1nxn+⋯+αn,NJTNnxn),un∈C,f(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+∑i=1Nαn,i(kn,i-1)Ln},xn+1=ΠCn+1x0
for every n≥0, where {rn} is a real sequence in [a,∞) for some a>0, J is the normalized duality mapping on E and Ln=sup{ϕ(p,xn):p∈F}<∞. Assume that {αn,0}, {αn,1}, …, {αn,N} are real sequences in (0,1) such that ∑i=0Nαn,i=1 and liminfn→∞αn,0αn,i>0, for all i∈{1,2,…,N}. Then the sequence {xn} converges strongly to ΠFx0.

Proof.

Put A≡0 in Theorem 2.1. We have zn=xn. Thus, the method of proof of Theorem 2.1 gives the required assertion without the requirement that E is 2-uniformly convex.

As some corollaries of Theorems 2.1 and 2.2, we have the following results immediately.

Corollary 2.3.

Let C be a nonempty, closed, and convex subset of a Hilbert space H and Ti:C→C a closed quasi-ϕ-asymptotically nonexpansive mapping with sequence {kn,i}⊂[1,∞) such that limn→∞kn,i=1 for each 1≤i≤N. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). Let A be a γ-inverse strongly monotone mapping of C into H with constant γ>0 such that F=(⋂i=1NF(Ti))⋂EP(f)⋂
VI
(C,A)≠∅ and F is bounded. Assume that Ti is asymptotically regular on C for each 1≤i≤N and ∥Ax∥≤∥Ax-Ap∥ for all x∈C and p∈F. Define a sequence {xn} in C in the following manner:
(2.46)x0∈C0=Cchosenarbitrarily,zn=PC(xn-λnAxn),yn=αn,0xn+αn,1T1nzn+⋯+αn,NTNnzn,un∈C,f(un,y)+1rn〈y-un,un-yn〉≥0,∀y∈C,Cn+1={z∈Cn:‖z-un‖2≤‖z-xn‖2+∑i=1Nαn,i(kn,i-1)Ln},xn+1=PCn+1x0
for every n≥0, where {rn} is a real sequence in [a,∞) for some a>0 and Ln=sup{∥xn-p∥2:p∈F}<∞. Assume that {αn,0}, {αn,1}, …, {αn,N} are real sequences in (0,1) such that ∑i=0Nαn,i=1 and liminfn→∞αn,0αn,i>0, for all i∈{1,2,…,N}. Let {λn} be a sequence in [s,t] for some 0<s<t<γ/2. Then the sequence {xn} converges strongly to PFx0.

Corollary 2.4.

Let C be a nonempty, closed, and convex subset of a Hilbert space H and Ti:C→C a closed quasi-ϕ-asymptotically nonexpansive mapping with sequence {kn,i}⊂[1,∞) such that limn→∞kn,i=1 for each 1≤i≤N. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) such that F=(⋂i=1NF(Ti))⋂EP(f)≠∅ and F is bounded. Assume that Ti is asymptotically regular on C for each 1≤i≤N. Define a sequence {xn} in C in the following manner:
(2.47)x0∈C0=Cchosenarbitrarily,yn=αn,0xn+αn,1T1nxn+⋯+αn,NTNnxn,un∈C,f(un,y)+1rn〈y-un,un-yn〉≥0,∀y∈C,Cn+1={z∈Cn:‖z-un‖2≤‖z-xn‖2+∑i=1Nαn,i(kn,i-1)Ln},xn+1=PCn+1x0
for every n≥0, where {rn} is a real sequence in [a,∞) for some a>0 and Ln=sup{∥xn-p∥2:p∈F}<∞. Assume that {αn,0}, {αn,1}, …, {αn,N} are real sequences in (0,1) such that ∑i=0Nαn,i=1 and liminfn→∞αn,0αn,i>0, for all i∈{1,2,…,N}. Let {λn} be a sequence in [s,t] for some 0<s<t<γ/2. Then the sequence {xn} converges strongly to PFx0.

Remark 2.5.

Theorems 2.1 and 2.2 extend the main results of [23] from quasi-ϕ-nonexpansive mappings to more general quasi-ϕ-asymptotically nonexpansive mappings.

Acknowledgments

The research was supported by Fundamental Research Funds for the Central Universities (Program no. ZXH2012K001), supported in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing and it was also supported by the Science Research Foundation Program in Civil Aviation University of China (2012KYM04).

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