Block methods for a convex feasibility problem in a Banach space

Research Article

Block methods for a convex feasibility problem in a Banach space

Mingliang Zhang^a,∗, Ravi P. Agarwal^b

aSchool of Mathematics and Statistics, Henan University, Kaifeng, China.

bDepartment of Mathematics, Texas A&M University, Kingsville, USA.

Communicated by X. Qin

Abstract

In this paper, a convex feasibility problem is investigated based on a block method. Strong conver- gence theorems for common solutions of equilibrium problems and generalized asymptotically quasi-φ- nonexpansive mappings are established in a strictly convex and uniformly smooth Banach space which also has the Kadec-Klee property. The results obtained in this paper unify and improve many corresponding results announced recently. c2016 All rights reserved.

Keywords: Banach space, block method, equilibrium problem, convex feasibility problem, variational inequality.

2010 MSC: 47H05, 90C33.

1. Introduction

The theory of fixed points of nonlinear operators is an important branch of modern mathematics as a bridge between nonlinear functional analysis and convex optimization theory. Many nonlinear problems arising in economics, medicine, engineering, and physics can be studied based on fixed point methods (see [8, 12, 27], and the references therein).

Recently, convex feasibility problems have been intensively investigated based on iterative techniques.

The so called convex feasibility problems which capture lots of applications in various disciplines such as image restoration, and radiation therapy treatment planning are to find a special point in the intersection of common fixed point sets, which is a convex set of a (countable or uncountable) family of nonlinear operators.

∗Corresponding author

Email addresses: [email protected](Mingliang Zhang),[email protected](Ravi P. Agarwal) Received 2016-07-09

Mean-valued iterative methods are efficient for studying the fixed points of Lipschitz continuous nonlinear operators. However, in the framework of infinite-dimensional Hilbert spaces, they are only weakly conver- gent (see [13] and the references therein). In many modern disciplines, including image recovery, economics, control theory, and quantum physics, problems arise in the framework of infinite dimension spaces. In such nonlinear problems, the strong convergence is often much more desirable than the weak convergence. To guarantee the strong convergence of mean-valued iteration algorithms, many authors use different regular- ization methods (see [4, 5, 7, 9–11, 20, 29]). The projection method which was first introduced by Haugazeau [15] has been considered for the approximation of fixed points of nonexpansive mappings. The advantage of projection methods is that the strong convergence of iterative sequences can be guaranteed without any compact assumptions imposed on mappings or spaces.

In this paper, we study on the generalized asymptotically-φ-nonexpansive mappings and equilibrium problems in the terminology of Blum and Oettli [6] based on a block method. The equilibrium problem includes many important problems in nonlinear functional analysis and convex optimization such as the Nash equilibrium problem, variational inequalities, complementarity problems, saddle point problems and game theory. Strong convergence theorems of common solutions are established with the aid of a generalized projection in a Banach space. The results obtained in this paper mainly unify and improve the corresponding results in [14, 16–19, 22, 23, 25].

2. Preliminaries

LetE be a real Banach space andE^∗ the dual space ofE. LetB_E be the unit sphere ofE. Recall that E is said to be a strictly convex space iffkx+yk<2 for all x, y∈BE and x6=y.

Recall thatEis said to have a Gˆateaux differentiable norm iff limt→0kx+tyk−kxk

t exists for eachx, y∈B_E. In this case, we also say that E is smooth. E is said to have a uniformly Gˆateaux differentiable norm if for eachy ∈B_E, the limit is attained uniformly for allx ∈B_E. E is also said to have a uniformly Fr´echet differentiable norm iff the above limit is attained uniformly for x, y ∈ B_E. In this case, we say that E is uniformly smooth. It is known that if E is uniformly smooth, then duality mapping J is uniformly norm- to-norm continuous on every bounded subset ofE. It is also known thatE^∗ is uniformly convex if and only ifE is uniformly smooth.

In this paper, we use→ and * to denote the strong convergence and weak convergence, respectively.

Recall that E has the Kadec-Klee (KK) property if limn→∞kx_n−xk= 0 as n→ ∞, for any sequence {x_n} ⊂E, and x∈E with x_n * x, and kx_nk → kxk asn→ ∞. It is known that every uniformly convex Banach space has the KK property.

Let C be a nonempty closed and convex subset of E and let G:C×C→R be a function. Recall that in equilibrium problem, the aim is to find ¯x∈C such thatG(¯x y)≥0 for ally∈C.We useS(G) to denote the solution set of the equilibrium problem. That is,S(G) ={x∈C:G(x, y)≥0 ∀y∈C}.

In order to study the equilibrium problem, we assume that Gsatisfies the following conditions:

(A1) G(a, a)≡0, ∀a∈C;

(A2) G(b, a) +G(a, b)≤0,∀a, b∈C;

(A3) G(a, b)≥lim sup_t↓0G(tc+ (1−t)a, b),∀a, b, c∈C;

(A4) b7→G(a, b) is convex and weakly lower semi-continuous for all a∈C.

Let T :C →C be a mapping. In this paper, we use F(T) to denote the fixed point set of mapping T.

T is said to be closed if for any sequence {x_n} ⊂C such that limn→∞x_n =x⁰ and limn→∞T x_n=y⁰, then T x⁰ =y⁰.

Recall that the normalized duality mapping J fromE to 2^E∗ is defined by J x={f^∗ ∈E^∗ :kxk²=hx, f^∗i=kf^∗k²}.

Next, we assume that E is a smooth Banach space which means mapping J is single-valued and study the functional

φ(x, y) :=kxk²+kyk²−2hx, J yi ∀x, y∈E.

In [3], Alber studied a new mapping Π_C in a Banach space E which is an analogue of P_C, the metric projection, in Hilbert spaces. Recall that the generalized projection Π_C :E →C is a mapping that assigns to an arbitrary point x ∈ E the minimum point of φ(x, y), that is, ΠCx = ¯x, where ¯x is the solution to the minimization problem φ(¯x, x) = miny∈Cφ(y, x). It is obvious from the definition of function φ that (kxk − kyk)² ≤φ(x, y) for all x, y∈E.

Recall that a pointpis said to be an asymptotic fixed point of mappingT if and only if subsetCcontains a sequence{x_n}which converges weakly topsuch that limn→∞kx_n−T xnk= 0. In this paper, we useFe(T) to stand for the asymptotic fixed point set. Let K be a bounded subset of C. Recall that T is said to be uniformly asymptotically regular onC if and only if lim sup_n→∞sup_x∈K{kTⁿx−Tⁿ⁺¹xk}= 0.

T is said to be relatively nonexpansive iff

φ(p, T x)≤φ(p, x) ∀x∈C, ∀p∈Fe(T) =F(T)6=∅.

T is said to be relatively asymptotically nonexpansive iff

φ(p, Tⁿx)≤(µ_n+ 1)φ(p, x) ∀x∈C, ∀p∈Fe(T) =F(T)6=∅ ∀n≥1, where{µ_n} ⊂[0,∞) is a sequence such thatµn→0 as n→ ∞.

T is said to be quasi-φ-nonexpansive iff

φ(p, T x)≤φ(p, x) ∀x∈C, ∀p∈F(T)6=∅.

T is said to be asymptotically quasi-φ-nonexpansive iff there exists a sequence{µ_n} ⊂[0,∞) withµn→0 asn→ ∞ such that

φ(p, Tⁿx)≤(µn+ 1)φ(p, x) ∀x∈C, ∀p∈F(T)6=∅, ∀n≥1.

Remark 2.1. The class of relatively asymptotically nonexpansive mappings, which was considered in [1] cov- ers the class of relatively nonexpansive mappings. The class of quasi-φ-nonexpansive mappings and the class of asymptotically quasi-φ-nonexpansive mappings cover the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings, respectively. Quasi-φ-nonexpansive mappings and asymptotically quasi-φ-nonexpansive mappings do not require the strong restriction that the fixed point set equals the asymptotic fixed point set, respectively (see [23, 24] and the references therein).

Remark 2.2. We note that the class of quasi-φ-nonexpansive mappings and the class of asymptotically quasi- φ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces, respectively, since thep

φ(x, y) =kx−yk.

T is said to be a generalized asymptotically quasi-φ-nonexpansive mapping if there exist two nonnegative sequences{µ_n} ⊂[0,∞) withµn→0, and{ξ_n} ⊂[0,∞) withξn→0 as n→ ∞ such that

φ(p, Tⁿx)≤(µ_n+ 1)φ(p, x) +ξ_n, ∀x∈C,∀p∈F(T)6=∅, ∀n≥1.

Remark 2.3. The class of generalized asymptotically quasi-φ-nonexpansive mappings, which was introduced in [21] is a generalization of the class of generalized asymptotically quasi-nonexpansive mappings [2] in the framework of Banach spaces. Common fixed points of generalized asymptotically quasi-nonexpansive mappings were investigated via the implicit iterations in [2].

The following lemmas play an important role in this paper.

Lemma 2.4 ([3]). Let E be a strictly convex, reflexive, and smooth Banach space and letC be a nonempty, closed, and convex subset of E. Let x∈E.Then

φ(y,Π_Cx)≤φ(y, x)−φ(Π_Cx, x), ∀y∈C, and hy−x₀, J x−J x₀i ≤0 for all y∈C, if and only if x₀= Π_Cx.

Lemma 2.5 ([27]). Let r be a positive real number and let E be uniformly convex. Then there exists a strictly increasing, continuous, and convex functiong: [0,2r]→R such thatg(0) = 0 and

k(1−t)y+tak²+t(1−t)g(kb−ak)≤tkak²+ (1−t)kbk² for alla, b∈B^r:={a∈E:kak ≤r} and t∈[0,1].

Lemma 2.6([23, 28]). LetE be a strictly convex, smooth, and reflexive Banach space and letC be a closed convex subset of E. Let G be a function, which satisfies (A1)-(A4), from C×C to R. Let x ∈ E and let r >0. Define a mappingW_G,r:E →C by

W^G,rx={z∈C :rG(z, y) +hy−z, J z−J xi ≥0, ∀y∈C}.

Then, there existsz∈C such thatrG(z, y)+hz−y, J z−J xi ≤0 for ally∈C, and the following conclusions hold:

(1) W^G,r is single-valued quasi-φ-nonexpansive and

hW^G,rx−W^G,ry, J W^G,rx−J W^G,ryi ≤ hW^G,rx−W^G,ry, J x−J yi for allx, y∈E;

(2) F(W^G,r) =S(G) is closed and convex;

(3) φ(q, W^G,rx) +φ(W^G,rx, x)≤φ(q, x), ∀q ∈F(W^G,r).

3. Main results

Theorem 3.1. Let E be a strictly convex and uniformly smooth Banach space which also has the KK property. LetC be a nonempty closed and convex subset of E. Let G_i be a bifunction satisfying conditions (A1)-(A4) and let T_i be a generalized asymptotically quasi-φ-nonexpansive mapping on C for every i∈ Λ, where Λ is an arbitrary index set. Assume that Ti is closed and uniformly asymptotically regular on C for every i∈Λ and∩_i∈ΛF(T_i)T∩_i∈ΛS(G_i) is nonempty and bounded. Let {x_n} be a sequence generated in the following process. x₀∈E chosen arbitrarily and



















C_(1,i)=C, ∀i∈Λ,

C₁ =∩_i∈ΛC_(1,i), x₁= Π_C1x₀,

J y_(n,i)= (1−α_(n,i))J T_iⁿx_n+α_(n,i)J x_n,

C_(n+1,i)={z∈C_(n,i) :µ_(n,i)M_(n,i)+φ(z, xn) +ξ_(n,i) ≥φ(z, u_(n,i))}, C_n+1 =∩_i∈ΛC_(n+1,i), x_n+1= Π_Cn+1x₁,

where u_(n,i) ∈ Cn such that r_(n,i)Gi(u_(n,i), y) +hu_(n,i)−y, J u_(n,i)−J y_(n,i)i ≤ 0 for all y ∈ Cn, M_(n,i) = sup{φ(p, x_n) : p∈ ∩_i∈ΛF(T_i)T

∩_i∈ΛS(G_i)}, {α_(n,i)} is a real sequence in (0,1) such that lim infn→∞α_(n,i) (α_(n,i)−1) < 0, and {r_(n,i)} is a real sequence in [r_i,∞), where {r_i} is a positive real number sequence.

Then{x_n} converges strongly to Π∩_i∈ΛF(Ti)T∩_i∈ΛS(Gi)x1.

Proof. Since T_i is closed, we find that ∩_i∈ΛF(T_i) is also closed. Let p_(1,i), p_(2,i) ∈ F(T_i), and p_i = (1− ti)p_(2,i)+tip_(1,i), whereti ∈(0,1), for everyi∈Λ. Note thatξ_(n,i)+ (1 +µ_(n,i))φ(p_(1,i), pi)≥φ(p_(1,i), T_iⁿpi), and ξ_(n,i)+ (1 +µ_(n,i))φ(p_(2,i), pi)≥φ(p_(2,i), T_iⁿpi).It follows that

φ(p_(1,i), T_iⁿpi)−2hp_i−p_(1,i), J pi−J T_iⁿpii=φ(p_(1,i), pi) +φ(pi, T_iⁿpi), and

φ(p_(2,i), T_iⁿpi)−2hp_i−p_(2,i), J pi−J T_iⁿpii=φ(p_(2,i), pi) +φ(pi, T_iⁿpi).

Hence, we have

φ(pi, T_iⁿpi)≤2hp_i−p_(1,i), J pi−J(T_iⁿpi)i+ξ_(n,i)+µ_(n,i)φ(p_(1,i), pi) (3.1) and

φ(pi, T_iⁿpi)≤2hp_i−p_(2,i), J pi−J(T_iⁿpi)i+ξ_(n,i)+µ_(n,i)φ(p_(2,i), pi). (3.2) Multiplyingti and (1−ti) on the both sides of (3.1) and (3.2), respectively, yields that

φ(p_i, T_iⁿp_i)≤(1−t_i)µ_(n,i)φ(p_(2,i), p_i) +ξ_(n,i)+t_iµ_(n,i)φ(p_(1,i), p_i).

It follows that limn→∞φ(p_i, T_iⁿp_i) = 0.This implies that kp_ik= lim

n→∞kT_iⁿpik. (3.3)

It follows that

kJ p_ik= lim

n→∞kJ(T_iⁿp_i)k. (3.4)

SinceE^∗ is reflexive, we may assume that J(T_iⁿpi) converges weakly tog^∗,i.In view of the reflexivity of E, we have E^∗ = J(E). This shows that there exists an element gⁱ ∈E such that g^∗,i =J gⁱ. It follows that φ(p_i, T_iⁿp_i) + 2hp_i, J(T_iⁿp_i)i=kp_ik²+kJ(T_iⁿp_i)k².Taking limit infimum asn→ ∞ on the both sides of the equality above, we obtain that

0≥ kp_ik²+kg^∗,ik²−2hp_i, g^∗,ii=kp_ik²+kgⁱk²−2hp_i, J gⁱi=φ(p_i, gⁱ)≥0.

This implies that gⁱ =p_i,that is, J p_i = g^∗,i. It follows that J(T_iⁿp_i) * J p_i ∈ E^∗. In view of the KK property ofE^∗, we obtain from (3.4) that limn→∞kJ p_i−J(T_iⁿp_i)k= 0.It follows thatT_iⁿp_i * p_i. Using the KK property of E, one finds from (3.3) thatT_iⁿpi → pi as n→ ∞.It follows that TiT_iⁿpi =T_iⁿ⁺¹pi →pi, asn → ∞.Since T_i is closed, one sees that p_i ∈ F(T_i),for every i∈Λ. This proves, for every i∈Λ, that F(T_i) is convex. This in turn implies that∩_i∈ΛF(T_i) is a convex set. Using Lemma 2.6, one obtains that

∩_i∈ΛS(Gi) is closed and convex. This proves that Π∩_i∈ΛS(Gi)T∩_i∈ΛF(Ti)x is well-defined, for any element in E.

Next, we prove that C_n is convex and closed. It is obvious thatC_(1,i)=C is convex and closed. Assume thatC_(m,i) is convex and closed for somem≥1.Letz1, z2∈C_(m+1,i), we see thatz1, z2 ∈C_(m,i). It follows thatz=tz1+ (1−t)z₂ ∈C_(m,i), wheret∈(0,1). Notice thatφ(z1, u_(m,i))−φ(z1, xm)≤ξ_(m,i)+µ_(m,i)M_(m,i), and φ(z₂, u_(m,i))−φ(z₂, x_m)≤ξ_(m,i)+µ_(m,i)M_(m,i),Hence, one has

2hz₁, J x_m−J u_(m,i)i − kx_mk²+ku_(m,i)k² ≤ξ_(m,i)+µ_(m,i)M_(m,i) and

2hz₂, J x_k−J u_(m,i)i − kx_mk²+ku_(m,i)k² ≤ξ_(m,i)+µ_(m,i)M_(m,i).

This findsφ(z, x_m) +ξ_(m,i)+µ_(m,i)M_(m,i)≥φ(z, u_(m,i)),wherez∈C_(m,i).This shows thatC_(m+1,i)is closed and convex. Hence,C_n=∩i∈ΛC_(n,i) is a convex and closed set. This proves that Π_Cn+1x₁ is well defined.

Now, we are in a position to prove that ∩_i∈ΛF(Ti)T

∩_i∈ΛS(Gi)⊂Cn.

Here, ∩_i∈ΛF(T_i)T∩_i∈ΛS(G_i) ⊂ C₁ = C is clear. Suppose that ∩_i∈ΛF(T_i)T∩_i∈ΛS(G_i) ⊂ C_(m,i) for some positive integerm. For anyw∈ ∩_i∈ΛF(Ti)T

∩_i∈ΛS(Gi)⊂C_(m,i), we see that φ(w, u_(m,i))≤φ(w, y_(m,i))

=kwk²+kα_(m,i)J xm+ (1−α_(m,i))J T_i^mxmk²−2hw, α_(m,i)J xm+ (1−α_(m,i))J T_i^mxmi

≤ kwk²−2(1−α_(m,i))hw, J T_i^mxmi −2α_(m,i)hw, J x_mi + (1−α_(m,i))kT_i^mx_mk²+α_(m,i)kx_mk²

≤(1−α_(m,i))φ(w, xm) +α_(m,i)φ(w, xm) + (1−α_(m,i))µ_(m,i)φ(w, xm) + (1−α_(m,i))ξ_(m,i)

≤φ(w, x_m) +ξ_(m,i)+µ_(m,i)φ(w, x_m),

which shows that w ∈ C_(m+1,i). This implies that ∩_i∈ΛS(Gi)T

∩_i∈ΛF(Ti) ⊂ C_(n,i). Hence, we find that

∩_i∈ΛS(G_i)T∩_i∈ΛF(T_i)⊂ ∩_i∈ΛC_(n,i). Using Lemma 2.4, one hashz−x_n, J x₁−J x_ni ≤0 for anyz∈C_n.It follows that

hw−xn, J x1−J xni ≤0 ∀w∈ ∩_i∈ΛS(Gi)\

∩_i∈ΛF(Ti). (3.5)

Using Lemma 2.4 yields that φ(x_n, x₁) ≤ φ(Π_{∩i∈ΛF(Ti)}^T_{∩i∈ΛS(Gi)}x₁, x₁), which shows that {φ(x_n, x₁)} is bounded. Hence, {x_n} is also bounded. Since E is reflexive, we may assume that xn * x. Since¯ Cn is closed and convex, this yields ¯x ∈ C_n. Therefore, φ(x_n, x₁) ≤ φ(¯x, x₁). Since the norm is weakly lower semicontinuous, we have

φ(¯x, x₁)≤lim inf

n→∞ (kx_nk²+kx₁k²−2hx_n, J x₁i) = lim inf

n→∞ φ(x_n, x₁)≤φ(¯x, x₁).

It follows that limn→∞φ(xn, x1) = φ(¯x, x1). Hence, limn→∞kx_nk = k¯xk. Using the KK property of the space, one obtains thatx_nconverges strongly to ¯xasn→ ∞.On the other hand, we find thatφ(x_n+1, x₁)≥ φ(x_n, x₁), which shows that{φ(x_n, x₁)}is nondecreasing. Therefore, limn→∞φ(x_n, x₁) exists. It follows that φ(xn+1, x1)−φ(xn, x1)≥φ(xn+1, xn).Therefore, we have limn→∞φ(xn+1, xn) = 0.Since xn+1 ∈Cn+1, we haveφ(x_n+1, x_n) +ξ_(n,i)+µ_(n,i)M_(n,i)≥φ(x_n+1, u_(n,i))≥0.It follows that limn→∞φ(x_n+1, u_(n,i)) = 0.This yields that limn→∞(ku_(n,i)k − kx_n+1k) = 0. Therefore, limn→∞ku_(n,i)k=kxk.¯ That is, limn→∞kJ u_(n,i)k= limn→∞ku_(n,i)k = kxk¯ = kJxk.¯ This implies that {J u_(n,i)} is bounded. Assume that J u_(n,i) converges weakly tou^(∗,i)∈E^∗. In view of the reflexivity ofE, we see thatJ(E) =E^∗.This shows that there exists an elementuⁱ ∈E such thatJ uⁱ=u^(∗,i).It follows thatφ(xn+1, u_(n,i))+2hx_n+1, J u_(n,i)i=kx_n+1k²+kJ u_(n,i)k². Taking limit infimum as n→ ∞ yields that

0≥ kxk¯ ²−2hx, u¯ ^(∗,i)i+ku^(∗,i)k²=k¯xk²+kJ uⁱk²−2h¯x, J uⁱi=φ(¯x, uⁱ).

That is, ¯x = uⁱ, which in turn implies that Jx¯ = u^(∗,i). Hence, J u_(n,i) * Jx¯ ∈ E^∗. Since E^∗ is uniformly convex, then it has the KK property and we obtain that limn→∞J u_(n,i) = Jx.¯ Since J⁻¹ : E^∗ → E is demicontinuous and E has the KK property, one gets that u_(n,i) →x, as¯ n→ ∞.This implies limn→∞kx_n−u_(n,i)k= 0.Hence, we have limn→∞(φ(w, xn)−φ(w, u_(n,i))) = 0.SinceE^∗is uniformly convex, we find from Lemma 2.5 that

φ(w, u_(n,i))≤ kwk²+kα_(n,i)J x_n+ (1−α_(n,i))J T_iⁿx_nk²−2hw,(1−α_(n,i))J T_iⁿx_n+α_(n,i)J x_ni

≤ kwk²−2(1−α_(n,i))hw, J T_iⁿx_ni −2α_(n,i)hw, J x_ni

−α_(n,i)(1−α_(n,i))g(kJ x_n−J T_iⁿxnk) +α_(n,i)kx_nk²+ (1−α_(n,i))kT_iⁿxnk²

≤φ(w, x_n) + (1−α_(n,i))µ_(n,i)φ(w, x_n)−α_(n,i)(1−α_(n,i))g(kJ x_n−J T_iⁿx_nk) +ξ_(n,i)

≤φ(w, x_n) +µ_(n,i)M_(n,i)−α_(n,i)(1−α_(n,i))g(kJ x_n−J T_iⁿx_nk) +ξ_(n,i).

(3.6)

It follows that limn→∞kJ x_n−J T_iⁿxnk= 0.This implies that limn→∞kJ T_iⁿxn−Jxk¯ = 0.SinceJ⁻¹:E^∗→E is demicontinuous, one has T_iⁿx_n * x.¯ Hence, one has limn→∞kT_iⁿx_nk = k¯xk. Since E has the KK

property, we obtain limn→∞kx¯−T_iⁿx_nk = 0. Since each T_i is uniformly asymptotically regular, one has limn→∞k¯x−T_iⁿ⁺¹xnk = 0. That is, Ti(T_iⁿxn) → x.¯ Using the closedness of Ti, we find ¯x = Tix¯ for each i∈Λ.This proves ¯x∈ ∩_i∈ΛF(Ti).

Next, we show that ¯x∈ ∩_i∈ΛS(G_i).It follows from (3.6) that limn→∞φ(u_(n,i), y_(n,i)) = 0.Hence, we have limn→∞(ku_(n,i)k − ky_(n,i)k) = 0. Since u_(n,i) → x¯ as n → ∞, we find that ky_(n,i)k → kxk¯ as n → ∞, that is, limn→∞kJ y_(n,i)k=kJxk.¯ This shows that {J y_(n,i)} is bounded. Since E is uniformly smooth, one sees thatE^∗ is reflexive. We may assume that J y_(n,i)* y^(∗,i)∈E^∗.There exists yⁱ ∈E such that J yⁱ =y^(∗,i). It follows that φ(u_(n,i), y_(n,i)) + 2hu_(n,i), J y_(n,i)i=ku_(n,i)k²+kJ y_(n,i)k².Therefore, we have

0≥ kxk¯ ²+ky^(∗,i)k²−2h¯x, y^(∗,i)i=k¯xk²+kyⁱk²−2hx, J y¯ ⁱi=φ(¯x, yⁱ).

That is, ¯x =yⁱ.Hence, we have y^(∗,i) =Jx.¯ It follows that J y_(n,i) * Jx¯ ∈E^∗. Since E^∗ is uniformly convex, it has the KK property, we obtain thatJ y_(n,i)−Jx¯→0 asn→ ∞.Using the fact thatJ⁻¹:E^∗→E is demicontinuous, we see that y_(n,i) * x. Using the KK property, we obtain that¯ y_(n,i) → x¯ as n → ∞.

Since E is uniformly smooth, limn→∞kJ y_(n,i)−J u_(n,i)k= 0.Since Gi is monotone, we find that r_(n,i)G_i(y, u_(n,i))≤ ky−u_(n,i)kkJ u_(n,i)−J y_(n,i)k, ∀y∈C_n.

Therefore, one seesGi(y,x)¯ ≤0 for all y∈C.Fory∈C and 0< ti<1, define y_(t,i)= (1−ti)¯x+tiy. This implies that 0≥G_i(y_(t,i),x). Hence, we have 0 =¯ G_i(y_(t,i), y_(t,i))≤t_iG_i(y_(t,i), y).It follows thatG_i(¯x, y)≥0 for all y∈C.This implies that ¯x∈S(G_i) for everyi∈Λ.

Finally, we prove ¯x= Π∩_i∈ΛF(Ti)T∩_i∈ΛS(Gi)x1. Lettingn→ ∞in (3.5), we arrive athx−w, J x¯ ₁−Jxi ≥¯ 0 forw∈ ∩_i∈ΛF(T_i)T

∩_i∈ΛS(G_i).Using Lemma 2.4, we find that ¯x= Π_{∩i∈ΛF(Ti)}^T_{∩i∈ΛS(Gi)}x₁. This completes the proof.

Remark 3.2. Theorem 3.1 mainly improves the corresponding results in Hao [14], Qin et al. [22] and unifies the corresponding results in [14, 16–19, 22, 23, 25]. The framework of the space is applicable to L^p, where p >1.

For the class of asymptotically quasi-φ-nonexpansive mappings, we have the following result.

Corollary 3.3. Let E be a strictly convex and uniformly smooth Banach space which also has the KK property. LetC be a nonempty closed and convex subset of E. Let Gi be a bifunction satisfying conditions (A1)-(A4) and let T_i : C → C be asymptotically quasi-φ-nonexpansive for every i ∈ Λ, where Λ is an arbitrary index set. Assume that T_i is closed and uniformly asymptotically regular on C for every i ∈ Λ.

Assume that ∩_i∈ΛS(Gi)T

∩_i∈ΛF(Ti) is nonempty and bounded. Let {x_n} be a sequence generated in the following process. x₀∈E chosen arbitrarily and



















C_(1,i)=C, ∀i∈Λ,

C₁ =∩i∈ΛC_(1,i), x₁ = Π_C1x₀,

J y_(n,i)=α_(n,i)J xn+ (1−α_(n,i))J T_iⁿxn,

C_(n+1,i)={z∈C_(n,i):φ(z, x_n) +µ_(n,i)M_(n,i)≥φ(z, u_(n,i))}, Cn+1=∩_i∈ΛC_(n+1,i), xn+1 = ΠCn+1x1,

where u_(n,i) ∈ C_n such that r_(n,i)G_i(u_(n,i), y) +hy−u_(n,i), J u_(n,i)−J y_(n,i)i ≥ 0 for all y ∈ C_n, M_(n,i) = sup{φ(p, x_n) : p∈ ∩_i∈ΛS(G_i)T∩_i∈ΛF(T_i)}, {α_(n,i)} is a real sequence in (0,1) such that lim infn→∞α_(n,i) (α_(n,i)−1) < 0, and {r_(n,i)} is a real sequence in [ri,∞), where {r_i} is a positive real number sequence.

Then the sequence {x_n} converges strongly to Π_{∩i∈ΛS(Gi)}^T_{∩i∈ΛF(Ti)}x₁.

Remark 3.4. Corollary 3.3 improves the corresponding results in Kim [17]. To be more clear, we have the following: a single mapping was extended to a family of mappings and a single bifunction was extended to a family of bifunctions, respectively.

For the class of quasi-φ-nonexpansive mappings, the boundedness of the common solution set is not required. Indeed, we have the following result.

Corollary 3.5. Let E be a strictly convex and uniformly smooth Banach space which also has the KK property. LetC be a nonempty closed and convex subset of E. LetGi be a bifunction satisfying (A1)-(A4) and let T_i :C → C be a quasi-φ-nonexpansive for every i∈Λ, where Λ is an arbitrary index set. Assume thatT_i is closed for every i∈Λ. Assume that ∩i∈ΛS(G_i)T

∩i∈ΛF(T_i) is nonempty. Let{x_n} be a sequence generated in the following process. x0 ∈E chosen arbitrarily and



















C_(1,i) =C, ∀i∈Λ,

C₁ =∩_i∈ΛC_(1,i), x₁ = Π_C1x₀,

J y_(n,i)=α_(n,i)J xn+ (1−α_(n,i))J Tixn,

C_(n+1,i)={z∈C_(n,i):φ(z, x_n)≥φ(z, u_(n,i))}, C_n+1=∩i∈ΛC_(n+1,i), x_n+1 = Π_Cn+1x₁,

where u_(n,i) ∈ Cn such that r_(n,i)Gi(u_(n,i), y) +hy −u_(n,i), J u_(n,i) −J y_(n,i)i ≥ 0 for all y ∈ Cn, {α_(n,i)} is a real sequence in (0,1) such that lim infn→∞α_(n,i)(α_(n,i) − 1) < 0, and {r_(n,i)} is a real sequence in [ri,∞), where {r_i} is a positive real number sequence. Then the sequence {x_n} converges strongly to Π_{∩i∈ΛS(Gi)}^T_{∩i∈ΛF(Ti)}x1.

Remark 3.6. Corollary 3.5 improves the corresponding results in Qin et al. [25]. A single equilibrium problem was extended to a family of equilibrium problems and a pair of mappings was extended to a family of mappings. Both bifunctions and mappings are uncountable infinite families. The projection studied in this paper is also different from [25], which is only valid for a countable infinite family mappings. Since every uniformly convex Banach space is a strictly convex Banach space which also has the KK property, we see that Corollary 3.5 is still valid in uniformly smooth and uniformly convex Banach spaces. Corollary 3.5 improves the corresponding results in Qin et al. [24].

4. Applications

In this section, we give some deduced results in the framework of Hilbert spaces. We also consider solutions of a family of variational inequalities and common minimizers of a family of proper, lower semi- continuous, and convex functionals.

Theorem 4.1. Let E be a Hilbert space and let C be a nonempty closed and convex subset of E. Let G_i be a bifunction satisfying(A1)-(A4) and let T_i :C →C be a generalized asymptotically quasi-nonexpansive mapping for every i ∈ Λ, where Λ is an arbitrary index set. Assume that Ti is closed and uniformly asymptotically regular onC for everyi∈Λ. Assume that∩_i∈ΛS(Gi)T

∩_i∈ΛF(Ti) is nonempty and bounded.

Let {x_n} be a sequence generated in the following process. x₀∈E chosen arbitrarily and



















C_(1,i)=C, ∀i∈Λ,

C1=∩_i∈ΛC_(1,i), x1 =P rojC1x0, y_(n,i)=α_(n,i)x_n+ (1−α_(n,i))T_iⁿx_n,

C_(n+1,i) ={z∈C_(n,i):kz−x_nk²+ξ_(n,i)+µ_(n,i)M_(n,i) ≥ kz−u_(n,i)k²}, Cn+1 =∩_i∈ΛC_(n+1,i), xn+1=P rojCn+1x1,

whereu_(n,i)∈C_n such thathy−u_(n,i), u_(n,i)−y_(n,i)i+r_(n,i)G_i(u_(n,i), y)≥0 for ally∈C_n, M_(n,i)= sup{kp− xnk² :p ∈ ∩_i∈ΛS(Gi)T

∩_i∈ΛF(Ti)}, {α_(n,i)} is a real sequence in(0,1) such that lim infn→∞α_(n,i)(α_(n,i)− 1)<0, {r_(n,i)} is a real sequence in [ri,∞), where{r_i} is a positive real number sequence and P roj is the metric projection. Then the sequence{x_n} converges strongly to P roj_{∩i∈ΛS(Gi)}^T_{∩i∈ΛF(Ti)}x₁.

Proof. In the framework of Hilbert spaces, we see thatp

φ(x, y) =kx−ykfor allx, y∈E. The generalized projection is reduced to the metric projection and the generalized asymptotically-φ-nonexpansive mapping is reduced to the generalized asymptotically quasi-nonexpansive mapping. Using Theorem 3.1, we find the desired conclusion immediately.

For the class of quasi-nonexpansive mappings, we have the following result.

Corollary 4.2. Let E be a Hilbert space. Let C be a nonempty closed and convex subset ofE. LetG_i be a bifunction satisfying(A1)-(A4)and letT_i :C→C be a quasi-nonexpansive mapping for everyi∈Λ, where Λ is an arbitrary index set. Assume that Ti is closed for every i∈Λ. Assume that ∩_i∈ΛS(Gi)T

∩_i∈ΛF(Ti) is nonempty. Let {x_n} be a sequence generated in the following process. x₀ ∈E chosen arbitrarily and



















C_(1,i)=C, ∀i∈Λ,

C1=∩_i∈ΛC_(1,i), x1=P rojC1x0, y_(n,i) =α_(n,i)x_n+ (1−α_(n,i))T_ix_n,

C_(n+1,i) ={z∈C_(n,i) :kz−xnk ≥ kz−u_(n,i)k}, Cn+1 =∩_i∈ΛC_(n+1,i), xn+1=P roj_Cn+1x1,

whereu_(n,i)∈Cn such thatr_(n,i)Gi(u_(n,i), y) +hy−u_(n,i), u_(n,i)−y_(n,i)i ≥0 for ally∈Cn, M_(n,i)= sup{kp− x_nk² :p ∈ ∩_i∈ΛS(G_i)T∩_i∈ΛF(T_i)}, {α_(n,i)} is a real sequence in(0,1) such that lim infn→∞α_(n,i)(α_(n,i)− 1)<0, {r_(n,i)} is a real sequence in [r_i,∞), where{r_i} is a positive real number sequence and P roj is the metric projection. Then the sequence{x_n} converges strongly to P roj∩_i∈ΛS(Gi)T∩_i∈ΛF(Ti)x1.

Let A :C → E^∗ be a single valued monotone operator which is continuous along each line segment in C with respect to the weak^∗ topology of E^∗ (hemicontinuous). Recall the following variational inequality.

Finding a point x ∈C such that hx−y, Axi ≤0 for all y ∈ C.The symbol NC(x) stands for the normal cone for C at a pointx ∈C; that is, NC(x) ={x^∗ ∈E^∗ :hx−y, x^∗i ≥ 0, ∀y ∈ C}.From now on, we use V I(C, A) to denote the solution set of the variational inequality.

Theorem 4.3. Let E be a strictly convex and uniformly smooth Banach space which also has the KK property. Let C be a nonempty closed and convex subset of E. Let Λ be an index set and let Ai : C → E^∗ be a single valued, monotone, and hemicontinuous operator. Let G_i be a bifunction satisfying (A1)- (A4). Assume that ∩_i∈ΛS(G_i)T∩_i∈ΛV I(C, A_i) is not empty. Let {x_n} be a sequence generated in the following process. x0∈E chosen arbitrarily and























C1,i=C, ∀i∈Λ,

C1 =∩_i∈∆C_(1,i), x1= Π_C1x0, z_(n,i) =V I(C, A_i+_r¹

i(J−J x_n)),

J y_(n,i) = (1−α_(n,i))J zn,i+α_(n,i)J xn, n≥1, C_(n+1,i)={w∈C_(n,i):φ(w, x_n)≥φ(w, u_n,i)}, Cn+1=∩_i∈∆C_(n+1,i), xn+1= ΠCn+1x0 ∀n≥1,

where u_(n,i) ∈ Cn such that r_(n,i)Gi(u_(n,i), y) +hy−u_(n,i), J u_(n,i)−J y_(n,i)i ≥ 0 for all y ∈ Cn, M_(n,i) = sup{φ(p, x_n) :p∈ ∩_i∈ΛS(G_i)T∩_i∈ΛV I(C, A_i)},{α_(n,i)}is a real sequence in(0,1)such thatlim infn→∞α_(n,i) (α_(n,i)−1) < 0, and {r_(n,i)} is a real sequence in [ri,∞), where {r_i} is a positive real number sequence.

Then{x_n} converges strongly to Π_{∩i∈ΛS(Gi)}^T_{∩i∈ΛV I(C,Ai)}x1. Proof. Define a mapping T_i ⊂E×E^∗ by

T_ix=

(∅, x /∈C, A_ix+N_Cx, x∈C.

Hence,T is maximal monotone andT_i⁻¹(0) =V I(C, A_i) (see [26]). For eachr_i >0, and x∈E,we see that there exists a unique xri in the domain of Ti such that J x∈J xri+riTi(xri), where xri = (J +riTi)⁻¹J x.

Notice that

z_n,i=V I(C, 1

r_i(J −J x_n) +A_i), which is equivalent to

hz_n,i−y, A_iz_n,i+ 1 ri

(J z_n,i−J x_n)i ≤0, ∀y∈C, that is,

1

r_i J xn−J zn,i

∈NC(zn,i) +Aizn,i.

This implies thatz_n,i= (J+r_iT_i)⁻¹J x_n.From [23], we find that (J+r_iT_i)⁻¹Jis closed quasi-φ-nonexpansive withF((J+riTi)⁻¹J) =T_i⁻¹(0). Using Theorem 3.1, we find the desired conclusion immediately.

Example 4.4. Let E=R andC = [0,1]. Define a mappingQ by Qx=

(0, x∈(¹₃,1],

1

3x, x∈[0,¹₃].

Then Q is a generalized asymptotically-φ-nonexpansive mapping with a unique fixed pint 0. We also have the following:

φ(Qⁿx, Qⁿy) =|Qⁿx−Qⁿy|² = 1

3²ⁿ|x−y|²≤ |x−y|²=φ(x, y)x, y∈[0,1 3], φ(Qⁿx, Qⁿy) =|Qⁿx−Qⁿy|² = 0≤ |x−y|² =φ(x, y)∀x, y∈(1

3,1], φ(Qⁿx, Qⁿy) =|Qⁿx−Qⁿy|²

=| 1

3ⁿx−0|²

≤(|1 3ⁿx− 1

3ⁿy|+| 1 3ⁿy|)²

≤( 1

3ⁿ|x−y|+ 1 3ⁿ)²

≤ |x−y|²+ξn, whereξn= ₃¹2n +₃²n for any x∈[0,¹₃], y ∈(¹₃,1], and

φ(Qⁿx, Qⁿy) =|Qⁿx−Qⁿy|²

=|0− 1 3ⁿy|²

≤(| 1 3ⁿy− 1

3ⁿx|+|1 3ⁿx|)²

≤( 1

3ⁿ|y−x|+ 1 3ⁿ)²

≤ |y−x|²+ξn,

where ξn = ₃2n¹ + ₃²n for any x ∈ (¹₃,1], y ∈ [0,¹₃]. This implies that φ(Qⁿx, Qⁿy) ≤ (1 +µn)φ(x, y) +ξn. This shows thatQis a generalized asymptoticallyφ-nonexpansive mapping with a unique fixed point 0. Let G: [0,1]×[0,1]→Rbe defined as

G(x, y) :=−3 4x²−

√3

2 sinx+3 4y²+

√3 2 siny.

It is easy to check that

(1) G(b, a) +G(a, b)≤0, ∀a, b∈[0,1];

(2) G(a, a)≡0, ∀a∈[0,1];

(3) b7→G(a, b) is convex and weakly lower semi-continuous for each a∈[0,1];

(4) G(a, b)≥lim sup_t↓0G(tc+ (1−t)a, b), ∀a, b, c∈[0,1];

and the common solution set of the equilibrium and the fixed point problems is not empty.

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