# A New Hybrid Iterative Scheme for Countable Families of Relatively Quasi-Nonexpansive Mappings and System of Equilibrium Problems

Authors: Yekini Shehu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We construct a new iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly smooth
and strictly convex real Banach space with Kadec-Klee property using the properties of generalizedf-projection operator. Using this result, we discuss strong convergence theorem concerning variational inequality and convex minimization problems in Banach spaces. Our results extend many known recent results in the literature.


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## Body

## 1. Introduction

LetE be a real Banach space with dual E* and C a nonempty, closed,  and convex subset of E. A mapping T:C→C is called nonexpansive if(1.1)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.
A point x∈C is called a fixed point of T if Tx=x. The set of fixed points of T is denoted by F(T):={x∈C:Tx=x}.We denote byJ the normalized duality mapping from E to 2E* defined by (1.2)J(x)={f∈E*:〈x,f〉=‖x‖2=‖f‖2}.
The following properties of J are well known (the reader can consult [1–3] for more details).  (1)
IfE is uniformly smooth, then J is norm-to-norm uniformly continuous on each bounded subset of E.(2)
J(x)≠∅,x∈E.(3)
IfE is reflexive, then J is a mapping from E onto E*.(4)
IfE is smooth, then J is single valued.Throughout this paper, we denote byϕ the functional on E×E defined by(1.3)ϕ(x,y)=‖x‖2-2〈x,J(y)〉+‖y‖2,∀x,y∈E.
It is obvious from (1.3) that(1.4)(‖x‖-‖y‖)2≤ϕ(x,y)≤(‖x‖+‖y‖)2,∀x,y∈E.Definition 1.1.
LetC be a nonempty subset of E, and let T be a mapping from C into E. A point p∈C is said to be an asymptotic fixed point of T if C contains a sequence {xn}n=0∞ which converges weakly to p and limn→∞∥xn-Txn∥=0. The set of asymptotic fixed points of T is denoted by F̂(T). We say that a mapping T is relatively nonexpansive (see, e.g., [4–9]) if the following conditions are satisfied:(R1)
F(T)≠∅,(R2)
ϕ(p,Tx)≤ϕ(p,x),for allx∈C,p∈F(T),(R3)
F(T)=F̂(T).IfT satisfies (R1) and (R2), then T is said to be relatively quasi-nonexpansive. It is easy to see that the class of relatively quasi-nonexpansive mappings contains the class of relatively nonexpansive mappings. Many authors have studied the methods of approximating the fixed points of relatively quasi-nonexpansive mappings (see, e.g., [10–12] and the references cited therein). Clearly, in Hilbert space H, relatively quasi-nonexpansive mappings and quasi-nonexpansive mappings are the same, for ϕ(x,y)=∥x-y∥2,for allx,y∈H, and this implies that (1.5)ϕ(p,Tx)≤ϕ(p,x)⟺‖Tx-p‖≤‖x-p‖,∀x∈C,p∈F(T).
The examples of relatively quasi-nonexpansive mappings are given in [11].LetF be a bifunction of C×C into ℝ. The equilibrium problem (see, e.g., [13–25]) is to find x*∈C such that(1.6)F(x*,y)≥0,
for all y∈C. We will denote the solutions set of (1.6) by EP(F). Numerous problems in physics, optimization, and economics reduce to find a solution of problem (1.6). The equilibrium problems include fixed point problems, optimization problems, and variational inequality problems as special cases (see, e.g., [26]).In [7], Matsushita and Takahashi introduced a hybrid iterative scheme for approximation of fixed points of relatively nonexpansive mapping in a uniformly convex real Banach space which is also uniformly smooth: x0∈C,(1.7)yn=J-1(αnJxn+(1-αn)JTxn),Hn={w∈C:ϕ(w,yn)≤ϕ(w,xn)},Wn={w∈C:〈xn-w,Jx0-Jxn〉≥0},xn+1=ΠHn∩Wnx0,n≥0.
They proved that {xn}n=0∞ converges strongly to ΠF(T)x0, where F(T)≠∅.In [27], Plubtieng and Ungchittrakool introduced the following hybrid projection algorithm for a pair of relatively nonexpansive mappings: x0∈C,(1.8)zn=J-1(βn(1)Jxn+βn(2)JTxn+βn(3)JSxn),yn=J-1(αnJx0+(1-αn)Jzn),Cn={z∈C:ϕ(z,yn)≤ϕ(z,xn)+αn(‖x0‖2+2〈w,Jxn-Jx0〉)},Qn={z∈C:〈xn-z,Jxn-Jx0〉≤0},xn+1=PCn∩Qnx0,
where {αn}, {βn(1)}, {βn(2)}, and {βn(3)} are sequences in (0,1) satisfying βn(1)+βn(2)+βn(3)=1 and T and S are relatively nonexpansive mappings and J is the single-valued duality mapping on E. They proved under the appropriate conditions on the parameters that the sequence {xn} generated by (1.8) converges strongly to a common fixed point of T and S.In [9], Takahashi and Zembayashi introduced the following hybrid iterative scheme for approximation of fixed point of relatively nonexpansive mapping which is also a solution to an equilibrium problem in a uniformly convex real Banach space which is also uniformly smooth: x0∈C, C1=C, x1=ΠC1x0, (1.9)yn=J-1(αnJxn+(1-αn)JTxn),F(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={w∈Cn:ϕ(w,un)≤ϕ(w,xn)},xn+1=ΠCn+1x0,n≥1,
where J is the duality mapping on E. Then, they proved that {xn}n=0∞ converges strongly to ΠFx0, where F=EP(F)∩F(T)≠∅.Furthermore, in [28], Qin et al. introduced the following hybrid iterative algorithm for approximation of common fixed point of two countable families of closed relatively quasi-nonexpansive mappings in a uniformly convex and uniform smooth real Banach space: (1.10)zi,n=J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSixn),yi,n=J-1(αn,iJx0+(1-αn,i)Jzi,n),Cn,i={z∈C:ϕ(z,yi,n)≤ϕ(z,xn)+αn,i(‖x0‖2+2〈z,Jxn-Jx0〉},Cn=⋂i∈ICn,i,Q0=C,Qn={z∈Qn-1:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,n≥0.
They proved that the sequence {xn} converges strongly to a common fixed point of the countable families {Ti} and {Si} of closed relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space under some appropriate conditions on {βn,i(1)}, {βn,i(2)}, {βn,i(3)}, and {αn,i}.Recently, Li et al. [29] introduced the following hybrid iterative scheme for approximation of fixed points of a relatively nonexpansive mapping using the properties of generalized f-projection operator in a uniformly smooth real Banach space which is also uniformly convex: x0∈C, C0=C, (1.11)yn=J-1(αnJxn+(1-αn)JTxn),Cn+1={w∈Cn:G(w,Jyn)≤G(w,Jxn)},xn+1=ΠCn+1fx0,n≥0.
They proved a strong convergence theorem for finding an element in the fixed points set of T. We remark here that the results of Li  et al. [29] extended and improved on the results of Matsushita and Takahashi [7].Quite recently, motivated by the results of Takahashi and Zembayashi [9], Cholamjiak and Suantai [30] proved the following strong convergence theorem by hybrid iterative scheme for approximation of common fixed point of a countable family of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in uniformly convex and uniformly smooth Banach space.Theorem 1.2.
LetE be a uniformly convex real Banach space which is also uniformly smooth, and let C be a nonempty, closed, and convex subset of E. For each k=1,2,…,m, let Fk be a bifunction from C×C satisfying (A1)–(A4). Suppose {Ti}i=1∞ is an infinitely countable family of closed and relatively quasi-nonexpansive mappings of C into itself such that Ω:=⋂k=1mEP(Fk)∩(⋂i=1∞F(Ti))≠∅. Suppose {xn}n=0∞ is iteratively generated by x0∈C,C0=C,
(1.12)yi,n=J-1(αnJxn+(1-αn)JTixn),ui,n=Trm,nFmTrm-1,nFm-1⋯Tr2,nF2Tr1,nF1yi,n,Cn+1={z∈Cn:sup⁡i≥1⁡ϕ(z,ui,n)≤ϕ(z,xn)},xn+1=ΠCn+1x0,n≥0.
Assume that {αn}n=1∞ and {rk,n}n=1∞(k=1,2,…,m) are sequences which satisfy the following conditions:(i)
limsupn→∞αn<1,(ii)
liminfn→∞rk,n>0(k=1,2,…,m).Then, {xn}n=0∞ converges strongly to ΠΩx0.Motivated by the above-mentioned results and the on-going research, it is our purpose in this paper to prove a strong convergence theorem for two countable families of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly smooth and strictly convex real Banach space with Kadec-Klee property using the properties of generalizedf-projection operator. Our results extend the results of Matsushita and Takahashi [7], Takahashi and Zembayashi [9], Qin et al. [28], Cholamjiak and Suantai [30], Li et al. [29], and many other recent known results in the literature.


## 2. Preliminaries

LetE be a real Banach space. The modulus of smoothness of E is the function ρE:[0,∞)→[0,∞) defined by (2.1)ρE(t)≔sup⁡{12(‖x+y‖+‖x-y‖)-1:‖x‖≤1,∥y∥≤t}.E is uniformly smooth if and only if (2.2)lim⁡t→0⁡ρE(t)t=0.
Let dim E≥2. The modulus of convexity of E is the function δE:(0,2]→[0,1] defined by (2.3)δE(ϵ):=inf⁡{1-‖x+y2‖:‖x‖=‖y‖=1;ϵ=‖x-y‖}.E is uniformly convex if, for any ϵ∈(0,2], there exists a δ=δ(ϵ)>0 such that if x,y∈E with ∥x∥≤1,∥y∥≤1, and ∥x-y∥≥ϵ, then ∥(1/2)(x+y)∥≤1-δ. Equivalently, E is uniformly convex if and only if δE(ϵ)>0 for all ϵ∈(0,2]. A normed space E is called strictly convex if for all x,y∈E,x≠y, ∥x∥=∥y∥=1, we have ∥λx+(1-λ)y∥<1,for allλ∈(0,1).LetE be a smooth, strictly convex, and reflexive real Banach space, and let C be a nonempty, closed, and convex subset of E. Following Alber [31], the generalized projection ΠC from E onto C is defined by(2.4)ΠC(x)≔argmin⁡y∈C⁡ϕ(y,x),∀x∈E.
The existence and uniqueness of ΠC follows from the property of the functional ϕ(x,y) and strict monotonicity of the mapping J (see, e.g., [3, 31–34]). If E is a Hilbert space, then ΠC is the metric projection of H onto C.Next, we recall the concept of generalizedf-projector operator, together with its properties. Let G:C×E*→ℝ∪{+∞} be a functional defined as follows:(2.5)G(ξ,φ)=‖ξ‖2-2〈ξ,φ〉+‖φ‖2+2ρf(ξ),
where ξ∈C, φ∈E*, ρ is a positive number, and f:C→ℝ∪{+∞} is proper, convex, and lower semicontinuous. From the definitions of G and f, it is easy to see the following properties:(i)
G(ξ,φ) is convex and continuous with respect to φ when ξ is fixed,(ii)
G(ξ,φ) is convex and lower semicontinuous with respect to ξ when φ is fixed.Definition 2.1 (see Wu and Huang [35]).
LetE be a real Banach space with its dual E*. Let C be a nonempty, closed, and convex subset of E. We say that ΠCf:E*→2C is a generalized f-projection operator if
(2.6)ΠCfφ={u∈C:G(u,φ)=inf⁡ξ∈C⁡G(ξ,φ)},∀φ∈E*.For the generalizedf-projection operator, Wu and Huang [35] proved the following theorem basic properties.Lemma 2.2 (see Wu and Huang [35]).
LetE be a real reflexive Banach space with its dual E*. Let C be a nonempty, closed, and convex subset of E. Then, the following statements hold:(i)
ΠCf is a nonempty closed convex subset of C for all φ∈E*,(ii)
ifE is smooth, then, for all φ∈E*,x∈ΠCf if and only if
(2.7)〈x-y,φ-Jx〉+ρf(y)-ρf(x)≥0,∀y∈C,(iii)
ifE is strictly convex and f:C→ℝ∪{+∞} is positive homogeneous (i.e., f(tx)=tf(x) for all t>0 such that tx∈C where x∈C), then ΠCf is a single-valued mapping.Fan et al. [36] showed that the condition f is positive homogeneous which appeared in Lemma 2.2 can be removed.Lemma 2.3 (see Fan et al. [36]).
LetE be a real reflexive Banach space with its dual E* and C a nonempty, closed, and convex subset of E. Then, if E is strictly convex, then ΠCf is a single-valued mapping.Recall thatJ is a single-valued mapping when E is a smooth Banach space. There exists a unique element φ∈E* such that φ=Jx for each x∈E. This substitution in (2.5) gives(2.8)G(ξ,Jx)=‖ξ‖2-2〈ξ,Jx〉+‖x‖2+2ρf(ξ).
Now, we consider the second generalized f-projection operator in a Banach space.Definition 2.4.
LetE be a real Banach space and C a nonempty, closed, and convex subset of E. We say that ΠCf:E→2C is a generalized f-projection operator if
(2.9)ΠCfx={u∈C:G(u,Jx)=inf⁡ξ∈C⁡G(ξ,Jx)},∀x∈E.Obviously, the definition ofT:C→C is a relatively quasi-nonexpansive mapping and is equivalent to(R′1)
F(T)≠∅,(R′2)
G(p,JTx)≤G(p,Jx),for allx∈C,p∈F(T).Lemma 2.5 (see Li et al. [29]).
LetE be a Banach space, and let f:E→ℝ∪{+∞} be a lower semicontinuous convex functional. Then, there exists x*∈E* and α∈ℝ such that
(2.10)f(x)≥〈x,x*〉+α,∀x∈E.We know that the following lemmas hold for operatorΠCf.Lemma 2.6 (see Li et al. [29]).
LetC be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. Then, the following statements hold:(i)
ΠCfx is a nonempty closed and convex subset of C for all x∈E,(ii)
for allx∈E,x̂∈ΠCfx if and only if
(2.11)〈x̂-y,Jx-Jx̂〉+ρf(y)-ρf(x)≥0,∀y∈C,(iii)
ifE is strictly convex, then ΠCfx is a single-valued mapping.Lemma 2.7 (see Li et al. [29]).
LetC be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. Let x∈E and x̂∈ΠCfx. Then,
(2.12)ϕ(y,x̂)+G(x̂,Jx)≤G(y,Jx),∀y∈C.The fixed points setF(T) of a relatively quasi-nonexpansive mapping is closed and convex as given in the following lemma.Lemma 2.8 (see Chang et al. [37]).
LetC be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex real Banach space E which also has Kadec-Klee property. Let T be a closed relatively quasi-nonexpansive mapping of C into itself. Then, F(T) is closed and convex.Also, this following lemma will be used in the sequel.Lemma 2.9 (see Cho et al. [38]).
LetE be a uniformly convex real Banach space. For arbitrary r>0, let Br(0):={x∈E:∥x∥≤r} and λ,μ,γ∈[0,1] such that λ+μ+γ=1. Then, there exists a continuous strictly increasing convex function
(2.13)g:[0,2r]⟶R,g(0)=0,
such that, for every x,y,z∈Br(0), the following inequality holds:
(2.14)‖λx+μy+γz‖2≤λ‖x‖2+μ‖y‖2-λμg(‖x-y‖).For solving the equilibrium problem for a bifunctionF:C×C→ℝ, let us assume that F satisfies the following conditions:(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 eachx,y∈C,limt→0F(tz+(1-t)x,y)≤F(x,y),(A4)
for eachx∈C,y↦F(x,y) is convex and lower semicontinuous.Lemma 2.10 (see Blum and Oettli [26]).
LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, and let F be a bifunction of C×C into ℝ satisfying (A1)–(A4). Let r>0 and x∈E. Then, there exists z∈C such that
(2.15)F(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈K.Lemma 2.11 (see Takahashi and Zembayashi [39]).
LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Assume that F:C×C→ℝ satisfies (A1)–(A4). For r>0 and x∈E, define a mapping TrF:E→C as follows:
(2.16)TrF(x)={z∈C:F(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}
for all  z∈E. Then, the following hold: (1)
TrF is singlevalued,(2)
TrF is firmly nonexpansive-type mapping, that is, for any  x,y∈E,
(2.17)〈TrFx-TrFy,JTrFx-JTrFy〉≤〈TrFx-TrFy,Jx-Jy〉,(3)
F(TrF)=EP(F),(4)
EP(F) is closed and convex.Lemma 2.12 (see Takahashi and Zembayashi [39]).
LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Assume that F:C×C→ℝ satisfies (A1)–(A4), and let r>0. Then, for each x∈E and q∈F(TrF),
(2.18)ϕ(q,TrFx)+ϕ(TrFx,x)≤ϕ(q,x).For the rest of this paper, the sequence{xn}n=0∞ converges strongly to p and will be denoted by xn→p as n→∞, {xn}n=0∞ converges weakly to p and will be denoted by xn⇀p and we will assume that βn,i(1),βn,i(2),βn,i(3)∈[0,1],for alli=1,2,3,… such that βn,i(1)+βn,i(2)+βn,i(3)=1,for alln≥0.We recall that a Banach spaceE has Kadec-Klee property if, for any sequence {xn}n=0∞⊂E and x∈E with xn⇀x and ∥xn∥→∥x∥, xn→x as n→∞. We note that every uniformly convex Banach space has the Kadec-Klee property. For more details on Kadec-Klee property, the reader is referred to [2, 33].Lemma 2.13 (see Li et al. [29]).
LetE be a Banach space and y∈E. Let f:E→ℝ∪{+∞} be a proper, convex, and lower semicontinuous mapping with convex domain D(f). If {xn} is a sequence in D(f) such that xn⇀x∈int(D(f)) and limn→∞G(xn,Jy)=G(x,Jy), then limn→∞∥xn∥=∥x∥.


## 3. Main Results

Theorem 3.1.
LetE be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. For each k=1,2,…,m, let Fk be a bifunction from C×C satisfying (A1)–(A4). Suppose {Ti}i=1∞ and {Si}i=1∞ are two countable families of closed relatively quasi-nonexpansive mappings of C into itself such that Ω≔⋂k=1mEP(Fk)∩(⋂i=1∞F(Ti))∩(⋂i=1∞F(Si))≠∅. Let f:E→ℝ be a convex and lower semicontinuous mapping with C⊂int(D(f)), and suppose {xn}n=0∞ is iteratively generated by x0∈C,C1,i=C,C1=∩i=1∞C1,i,x1=ΠC1fx0,
(3.1)zn,i=J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSixn),yn,i=J-1(αn,iJxn+(1-αn,i)Jzn,i),un,i=Trm,nFmTrm-1,nFm-1⋯Tr2,nF2Tr1,nF1yn,i,Cn+1,i={z∈Cn,i:G(z,Jun,i)≤G(z,Jxn)},Cn+1=⋂i=1∞Cn+1,i,xn+1=ΠCn+1fx0,n≥1,
with the conditions(i)
liminfn→∞βn,i(1)βn,i(2)>0,(ii)
liminfn→∞βn,i(1)βn,i(3)>0,(iii)
0≤αn,i≤α<1 for some α∈(0,1),(iv)
{rk,n}n=1∞⊂(0,∞)(k=1,2,…,m) satisfying liminfn→∞rk,n>0(k=1,2,…,m).Then, {xn}n=0∞ converges strongly to ΠΩfx0.Proof.
We first show thatCn,for alln≥1 is closed and convex. It is obvious that C1,i=C is closed and convex. Suppose Ck,i is closed and convex for some k>1. For each z∈Ck,i, we see that G(z,Juk,i)≤G(z,Jxk) is equivalent to
(3.2)2(〈z,Jxk〉-〈z,Juk,i〉)≤‖xk‖2-‖uk,i‖2.
By the construction of the set Ck+1,i, we see that Ck+1,i is closed and convex. Therefore, Ck+1=⋂i=1∞Ck+1,i is also closed and convex. Hence, Cn,for alln≥1 is closed and convex.
By takingθnk=Trk,nFkTrk-1,nFk-1⋯Tr2,nF2Tr1,nF1,k=1,2,…,m and θn0=I for all n≥1, we obtain un,i=θnmyn,i.
We next show thatΩ⊂Cn,for alln≥1. For n=1, we have Ω⊂C=C1. Then, for each x*∈Ω, we obtain
(3.3)G(x*,Jun,i)=G(x*,Jθnmyn,i)≤G(x*,Jyn,i)=G(x*,(αn,iJxn+(1-αn,i)Jzn,i))=‖x*‖2-2αn,i〈x*,Jxn〉-2(1-αn,i)〈x*,Jzn,i〉+‖αn,iJxn+(1-αn,i)Jzn,i‖2+2ρf(x*)≤‖x*‖2-2αn,i〈x*,Jxn〉-2(1-αn,i)〈x*,Jzn,i〉+αn,i‖xn‖2+(1-αn,i)‖zn,i‖2+2ρf(x*)=αn,iG(x*,Jxn)+(1-αn,i)G(x*,Jzn,i)=αn,iG(x*,Jxn)+(1-αn,i)G(x*,(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSixn))≤αn,iG(x*,Jxn)+(1-αn,i)(‖x*‖2-2βn,i(1)〈x*,Jxn〉-2βn,i(2)〈x*,JTixn〉-2βn,i(3)〈x*,JSixn〉+βn,i(1)‖xn‖2+βn,i(2)‖Tixn‖2+βn,i(3)‖Sixn‖2+2ρf(x*))=αn,iG(x*,Jxn)+(1-αn,i)(βn,i(1)G(x*,Jxn)+βn,i(2)G(x*,JTixn)+βn,i(3)G(x*,JSixn))≤G(x*,Jxn).
So, x*∈Cn. This implies that Ω⊂Cn,for alln≥1. Therefore, {xn} is well defined.
We now show thatlimn→∞G(xn,Jx0) exists. Since f:E→ℝ is convex and lower semicontinuous, applying Lemma 2.5, we see that there exists u*∈E* and α∈ℝ such that
(3.4)f(y)≥〈y,u*〉+α,∀y∈E.
It follows that
(3.5)G(xn,Jx0)=‖xn‖2-2〈xn,Jx0〉+‖x0‖2+2ρf(xn)≥‖xn‖2-2〈xn,Jx0〉+‖x0‖2+2ρ〈xn,u*〉+2ρα=‖xn‖2-2〈xn,Jx0-ρu*〉+‖x0‖2+2ρα≥‖xn‖2-2‖xn‖‖Jx0-ρu*‖+‖x0‖2+2ρα=(‖xn‖-‖Jx0-ρu*‖)2+‖x0‖2-‖Jx0-ρu*‖2+2ρα.
Since xn=ΠCnfx0, it follows from (3.5) that
(3.6)G(x*,Jx0)≥G(xn,Jx0)≥(‖xn‖-‖Jx0-ρu*‖)2+‖x0‖2-‖Jx0-ρu*‖2+2ρα
for each x*∈F. This implies that {xn}n=0∞ is bounded and so is {G(xn,Jx0)}n=0∞. By the construction of Cn, we have that Cn+1⊂Cn and xn+1=ΠCn+1fx0∈Cn. It then follows from Lemma 2.7 that
(3.7)ϕ(xn+1,xn)+G(xn,Jx0)≤G(xn+1,Jx0).
It is obvious that
(3.8)ϕ(xn+1,xn)≥(‖xn+1‖-‖xn‖)2≥0,
and so {G(xn,Jx0)}n=0∞ is nondecreasing. It follows that the limit of {G(xn,Jx0)}n=0∞ exists.
Now since{xn}n=0∞ is bounded in C and E is reflexive, we may assume that xn⇀p, and since Cn is closed and convex for each n≥1, it is easy to see that p∈Cn for each n≥1. Again since xn=ΠCnfx0, from the definition of ΠCnf, we obtain
(3.9)G(xn,Jx0)≤G(p,Jx0),∀n≥1.
Since
(3.10)lim⁡⁡inf⁡n→∞⁡G(xn,Jx0)=lim⁡⁡inf⁡n→∞⁡{‖xn‖2-2〈xn,Jx0〉+‖x0‖2+2ρf(xn)}≥‖p‖2-2〈p,Jx0〉+‖x0‖2+2ρf(p)=G(p,Jx0),
then we obtain
(3.11)G(p,Jx0)≤lim⁡⁡inf⁡n→∞⁡G(xn,Jx0)≤limsup⁡n→∞⁡G(xn,Jx0)≤G(p,Jx0).
This implies that limn→∞G(xn,Jx0)=G(p,Jx0). By Lemma 2.13, we obtain limn→∞∥xn∥=∥p∥. In view of Kadec-Klee property of E, we have that limn→∞xn=p.
We next show thatp∈⋂k=1mEP(Fk)∩(⋂i=1∞F(Ti))∩(⋂i=1∞F(Si)). We first show that p∈⋂i=1∞F(Ti))∩(⋂i=1∞F(Si). By the fact that Cn+1⊂Cn and xn+1=ΠCn+1fx0∈Cn, we obtain
(3.12)ϕ(xn+1,un,i)≤ϕ(xn+1,xn).
Now, (3.7) implies that
(3.13)ϕ(xn+1,un,i)≤ϕ(xn+1,xn)≤G(xn+1,Jx0)-G(xn,Jx0).
Taking the limit as n→∞ in (3.13), we obtain
(3.14)lim⁡n→∞⁡ϕ(xn+1,xn)=0.
Therefore,
(3.15)lim⁡n→∞⁡ϕ(xn+1,un,i)=0,∀i≥1.
It then yields that limn→∞(∥xn+1∥-∥un,i∥)=0,for alli≥1. Since limn→∞∥xn+1∥=∥p∥, we have
(3.16)lim⁡n→∞⁡‖un,i‖=‖p‖,∀i≥1.
Hence,
(3.17)lim⁡n→∞⁡‖Jun,i‖=‖Jp‖,∀i≥1.
This implies that {∥Jun,i∥}n=0∞,i≥1 is bounded in E*. Since E is reflexive, and so E* is reflexive, we can then assume that Jun,i⇀f0∈E*,for alli≥1. In view of reflexivity of E, we see that J(E)=E*. Hence, there exists x∈E such that Jx=f0. Since
(3.18)ϕ(xn+1,un,i)=‖xn+1‖2-2〈xn+1,Jun,i〉+‖un,i‖2=‖xn+1‖2-2〈xn+1,Jun,i〉+‖Jun,i‖2,
taking the limit inferior of both sides of (3.18) and in view of weak lower semicontinuity of ∥·∥, we have
(3.19)0≥‖p‖2-2〈p,f0〉+‖f0‖2=‖p‖2-2〈p,Jx〉+‖Jx‖2=‖p‖2-2〈p,Jx〉+‖x‖2=ϕ(p,x),
that is, p=x. This implies that f0=Jp and so Jun,i⇀Jp,for alli≥1. It follows from limn→∞∥Jun,i∥=∥Jp∥,for alli≥1 and Kadec-Klee property of E* that Jun,i→Jp,for alli≥1. Note that J-1:E*→E is hemicontinuous; it yields that un,i⇀p,for alli≥1. It then follows from limn→∞∥un,i∥=∥p∥,for alli≥1 and Kadec-Klee property of E that limn→∞un,i=p,for alli≥1. Hence,
(3.20)lim⁡n→∞⁡‖xn-un,i‖=0,∀i≥1.
Since J is uniformly norm-to-norm continuous on bounded sets and limn→∞∥xn-un,i∥=0,for alli≥1, we obtain
(3.21)lim⁡n→∞⁡‖Jxn-Jun,i‖=0,∀i≥1.
Since {xn} is bounded, so are {zn,i}, {JTixn}, and {JSixn}. Also, since E is uniformly smooth, E* is uniformly convex. Then, from Lemma 2.9, we have
(3.22)G(x*,Jun,i)=G(x*,Jθnmyn,i)≤G(x*,Jyn,i)=G(x*,(αn,iJxn+(1-αn,i)Jzn,i))=‖x*‖2-2αn,i〈x*,Jxn〉-2(1-αn,i)〈x*,Jzn,i〉+‖αn,iJxn+(1-αn,i)Jzn,i‖2+2ρf(x*)≤‖x*‖2-2αn,i〈x*,Jxn〉-2(1-αn,i)〈x*,Jzn,i〉+αn,i‖xn‖2+(1-αn,i)‖zn,i‖2+2ρf(x*)=αn,iG(x*,Jxn)+(1-αn,i)G(x*,Jzn,i)=αn,iG(x*,Jxn)+(1-αn,i)G(x*,(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSixn))≤αn,iG(x*,Jxn)+(1-αn,i)(‖x*‖2-2βn,i(1)〈x*,Jxn〉-2βn,i(2)〈x*,JTixn〉-2βn,i(3)〈x*,JSixn〉+βn,i(1)‖xn‖2+βn,i(2)‖Tixn‖2+βn,i(3)‖Sixn‖2-βn,i(1)βn,i(2)g(‖Jxn-JTixn‖)+2ρf(x*))=αn,iG(x*,Jxn)+(1-αn,i)(βn,i(1)G(x*,Jxn)+βn,i(2)G(x*,JTixn)+βn,i(3)G(x*,JSixn)-βn,i(1)βn,i(2)g(‖Jxn-JTixn‖))≤αn,iG(x*,Jxn)+(1-αn,i)(βn,i(1)G(x*,Jxn)+βn,i(2)G(x*,Jxn)+βn,i(3)G(x*,Jxn)-βn,i(1)βn,i(2)g(‖Jxn-JTixn‖))=αn,iG(x*,xn)+(1-αn,i)(G(x*,Jxn)-βn,i(1)βn,i(2)g(‖Jxn-JTixn‖))≤G(x*,Jxn)-(1-αn,i)βn,i(1)βn,i(2)g(‖Jxn-JTixn‖).
It then follows that
(3.23)(1-α)βn,i(1)βn,i(2)g(‖Jxn-JTixn‖)≤(1-αn,i)βn,i(1)βn,i(2)g(‖Jxn-JTixn‖)≤G(x*,Jxn)-G(x*,Jun,i).
But
(3.24)G(x*,Jxn)-G(x*,Jun,i)=‖xn‖2-‖un,i‖2-2〈x*,Jxn-Jun,i〉≤|‖xn‖2-‖un,i‖2|+2|〈x*,Jxn-Jun,i〉‖un,i‖2|≤|‖xn‖-‖un,i‖|(‖xn‖+‖un,i‖)+2‖x*‖‖Jxn-Jun,i‖≤‖xn-un,i‖(‖xn‖+‖un,i‖)+2‖x*‖‖Jxn-Jun,i‖.
From limn→∞∥xn-un,i∥=0 and limn→∞∥Jxn-Jun,i∥=0, we obtain
(3.25)G(x*,Jxn)-G(x*,Jun,i)⟶0,n⟶∞.
Using the condition liminfn→∞βn,i(1)βn,i(2)>0, we have
(3.26)lim⁡n→∞⁡g(‖Jxn-JTixn‖)=0,∀i≥1.
By property of g, we have limn→∞∥Jxn-JTixn∥=0,for alli≥1. Since J-1 is also uniformly norm-to-norm continuous on bounded sets, we have
(3.27)lim⁡n→∞⁡‖xn-Tixn‖=0,∀i≥1.
Similarly, we can show that
(3.28)lim⁡n→∞⁡‖xn-Sixn‖=0,∀i≥1.
Since xn→p and Ti,Si are closed, we have p∈(⋂i=1∞F(Ti))∩(⋂i=1∞F(Si)).
Next, we show thatp∈⋂k=1mEP(Fk). Now, by Lemma 2.12, we obtain
(3.29)ϕ(un,i,yn,i)=ϕ(θnmyn,i,yn,i)≤ϕ(x*,yn,i)-ϕ(x*,θnmyn,i)≤ϕ(x*,xn)-ϕ(x*,un,i)⟶0,n⟶∞.
It then yields that limn→∞(∥un,i∥-∥yn,i∥)=0. Since limn→∞∥un,i∥=∥p∥,i≥1, we have
(3.30)lim⁡n→∞⁡‖yn,i‖=‖p‖,i≥1.
Hence,
(3.31)lim⁡n→∞⁡‖Jyn,i‖=‖Jp‖,i≥1.
This implies that {∥Jyn,i∥}n=0∞ is bounded in E*. Since E is reflexive, and so E* is reflexive, we can then assume that Jyn,i⇀f1∈E*. In view of reflexivity of E, we see that J(E)=E*. Hence, there exists z∈E such that Jz=f1. Since
(3.32)ϕ(un,i,yn,i)=‖un,i‖2-2〈un,i,Jyn,i〉+‖yn,i‖2=‖un,i‖2-2〈un,i,Jyn,i〉+‖Jyn,i‖2,
taking the limit inferior of both sides of (3.32) and in view of weak lower semicontinuity of ∥·∥, we have
(3.33)0≥‖p‖2-2〈p,f1〉+‖f1‖2=‖p‖2-2〈p,Jz〉+‖Jz‖2=‖p‖2-2〈p,Jz〉+‖z‖2=ϕ(p,z),
that is, p=z. This implies that f1=Jp and so Jyn,i⇀Jp. It follows from limn→∞∥Jyn,i∥=∥Jp∥ and Kadec-Klee property of E* that Jyn,i→Jp. Note that J-1:E*→E is hemicontinuous; it yields that yn,i⇀p. It then follows from limn→∞∥yn,i∥=∥p∥ and Kadec-Klee property of E that limn→∞yn,i=p,i≥1.  By the fact that θnk,k=1,2,…,m is relatively nonexpansive and using Lemma 2.12 again, we have that
(3.34)ϕ(θnkyn,i,yn,i)≤ϕ(x*,yn,i)-ϕ(x*,θnkyn,i)≤ϕ(x*,xn)-ϕ(x*,θnkyn,i).
Observe that
(3.35)ϕ(x*,un,i)=ϕ(x*,θnmyn,i)=ϕ(x*,Trm,nFmTrm-1,nFm-1⋯Trk,nFkTrk-1,nFk-1⋯Tr2,nF2Tr1,nF1yn,i)=ϕ(x*,Trm,nFmTrm-1,nFm-1⋯θnkyn,i)≤ϕ(x*,θnkyn,i).
Using (3.35) in (3.34), we obtain
(3.36)ϕ(θnkyn,i,yn,i)≤ϕ(x*,xn)-ϕ(x*,un,i)⟶0,n⟶∞.
It then yields that limn→∞(∥θnkyn,i∥-∥yn,i∥)=0. Since limn→∞∥yn,i∥=∥p∥, we have
(3.37)lim⁡n→∞⁡‖θnkyn,i‖=‖p‖,k=1,2,…,m.
This implies that {∥θnkyn,i∥}n=0∞ is bounded in E. Since E is reflexive, we can then assume that θnkyn,i⇀w∈E. Since
(3.38)ϕ(θnkyn,i,yn,i)=‖θnkyn,i‖2-2〈θnkyn,i,Jyn,i〉+‖yn,i‖2=‖θnkyn,i‖2-2〈θnkyn,i,Jyn,i〉+‖Jyn,i‖2,
taking the limit inferior of both sides of (3.38) and in view of weak lower semicontinuity of ∥·∥, we have
(3.39)0≥‖w‖2-2〈w,Jp〉+‖p‖2=‖w‖2-2〈w,Jp〉+‖Jp‖2=ϕ(w,p),
that is, p=w. This implies that θnkyn,i⇀p. It follows from limn→∞∥θnkyn,i∥=∥p∥ and Kadec-Klee property of E that
(3.40)θnkyn,i⟶p,n⟶∞,k=1,2,…,m.
Similarly, limn→∞∥p-θnk-1yn,i∥=0,k=1,2,…,m. This further implies that
(3.41)lim⁡n→∞⁡‖θnkyn,i-θnk-1yn,i‖=0,i≥1.
Also, since J is uniformly norm-to-norm continuous on bounded sets and using (3.41), we obtain
(3.42)lim⁡n→∞⁡‖Jθnkyn,i-Jθnk-1yn,i‖=0,i≥1.
Since liminfn→∞rk,n>0(k=1,2,…,m),
(3.43)lim⁡n→∞⁡‖Jθnkyn,i-Jθnk-1yn,i‖rk,n=0.
By Lemma 2.11, we have that for each k=1,2,…,m(3.44)Fk(θnkyn,i,y)+1rk,n〈y-θnkyn,i,Jθnkyn,i-Jθnk-1yn,i〉≥0,∀y∈C.
Furthermore, using (A2), we obtain
(3.45)1rk,n〈y-θnkyn,i,Jθnkyn,i-Jθnk-1yn,i〉≥Fk(y,θnkyn,i).
By (A4), (3.43), and θnkyn,i→p, we have for each k=1,2,…,m(3.46)Fk(y,p)≤0,∀y∈C.
For fixed y∈C, let zt,y≔ty+(1-t)p for all t∈(0,1]. This implies that zt,y∈C. This yields that Fk(zt,y,p)≤0. It follows from (A1) and (A4) that
(3.47)0=Fk(zt,y,zt,y)≤tFk(zt,y,y)+(1-t)Fk(zt,y,p)≤tFk(zt,y,y),
and hence
(3.48)0≤Fk(zt,y,y).
From condition (A3), we obtain
(3.49)Fk(p,y)≥0,∀y∈C.
This implies that p∈EP(Fk),k=1,2,…,m. Thus, p∈⋂k=1mEP(Fk). Hence, we have p∈Ω=⋂k=1mEP(Fk)∩(⋂n=0∞F(Ti))∩(⋂i=1∞F(Si)).
Finally, we show thatp=ΠΩfx0. Since Ω=⋂k=1mEP(Fk)∩(⋂n=0∞F(Ti))∩(⋂i=1∞F(Si)) is a closed and convex set, from Lemma 2.6, we know that ΠFfx0 is single valued and denote w=ΠΩfx0. Since xn=ΠCnfx0 and w∈Ω⊂Cn, we have
(3.50)G(xn,Jx0)≤G(w,Jx0),∀n≥1.
We know that G(ξ,Jφ) is convex and lower semicontinuous with respect to ξ when φ is fixed. This implies that
(3.51)G(p,Jx0)≤lim⁡⁡inf⁡n→∞⁡G(xn,Jx0)≤lim⁡⁡sup⁡n→∞⁡G(xn,Jx0)≤G(w,Jx0).
From the definition of ΠΩfx0 and p∈Ω, we see that p=w. This completes the proof.Takef(x)=0 for all x∈E in Theorem 3.1, then G(ξ,Jx)=ϕ(ξ,x) and ΠCfx0=ΠCx0. Then we obtain the following corollary.Corollary 3.2.
LetE be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. For each k=1,2,…,m, let Fk be a bifunction from C×C satisfying (A1)–(A4). Suppose {Ti}i=1∞ and {Si}i=1∞ are two countable families of closed relatively quasi-nonexpansive mappings of C into itself such that Ω≔⋂k=1mEP(Fk)∩(⋂n=1∞F(Ti))∩(⋂n=1∞F(Si))≠∅. Suppose {xn}n=0∞ is iteratively generated by x0∈C,C1,i=C,C1=∩i=1∞C1,i,x1=ΠC1x0,
(3.52)zn,i=J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSixn),yn,i=J-1(αn,iJxn+(1-αn,i)Jzn,i),un,i=Trm,nFmTrm-1,nFm-1⋯Tr2,nF2Tr1,nF1yn,i,Cn+1,i={z∈Cn,i:ϕ(z,un,i)≤ϕ(z,xn)},Cn+1=⋂i=1∞Cn+1,i,xn+1=ΠCn+1x0,n≥1,
with the conditions(i)
liminfn→∞βn,i(1)βn,i(2)>0,(ii)
liminfn→∞βn,i(1)βn,i(3)>0,(iii)
0≤αn,i≤α<1 for some α∈(0,1),(iv)
{rk,n}n=1∞⊂(0,∞)(k=1,2,…,m) satisfying liminfn→∞rk,n>0(k=1,2,…,m).Then, {xn}n=0∞ converges strongly to ΠΩx0.Corollary 3.3 (see Li et al. [29]).
LetE be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty, closed, and convex subset of E. Suppose T is a relatively nonexpansive mapping of C into itself such that Ω:=F(T)≠∅. Let f:E→ℝ be a convex and lower semicontinuous mapping with C⊂int(D(f)), and suppose {xn}n=0∞ is iteratively generated by x0∈C, C0=C,
(3.53)yn=J-1(αnJxn+(1-αn)JTxn),Cn+1={w∈Cn:G(w,Jyn)≤G(w,Jxn)},xn+1=ΠCn+1fx0,n≥0.
Suppose {αn}n=1∞ is a sequence in (0,1) such that limsupn→∞αn<1. Then, {xn}n=0∞ converges strongly to ΠΩx0.Corollary 3.4 (see Takahashi and Zembayashi [9]).
LetE be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty, closed, and convex subset of E. Let F be a bifunction from C×C satisfying (A1)–(A4). Suppose T is a relatively nonexpansive mapping of C into itself such that Ω≔EP(F)∩F(T)≠∅. Let {xn}n=0∞ be iteratively generated by x0∈C, C1=C, x1=ΠC1x0,
(3.54)yn=J-1(αn,iJxn+(1-αn,i)JTxn),F(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={w∈Cn:ϕ(w,un)≤ϕ(w,xn)},xn+1=ΠCn+1x0,n≥1,
where J is the duality mapping on E. Suppose {αn,i}n=1∞ is a sequence in (0,1) such that liminfn→∞αn,i(1-αn,i)>0 and {rn}n=1∞⊂(0,∞) satisfying liminfn→∞rn>0. Then, {xn}n=0∞ converges strongly to ΠΩx0.


## 4. Applications

LetA be a monotone operator from C into E*, the classical variational inequality is to find x*∈C such that(4.1)〈y-x,Ax*〉≥0,∀y∈C.
The set of solutions of (4.1) is denoted by VI(C,A).Letφ:C→ℝ be a real-valued function. The convex minimization problem is to find x*∈C such that(4.2)φ(x*)≤φ(y),∀y∈C.
The set of solutions of (4.2) is denoted by CMP(φ).The following lemmas are special cases of Lemmas  2.8 and Lemma  2.9 of [39].Lemma 4.1.
LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Assume that A:C→E* is a continuous and monotone operator. For r>0 and x∈E, define a mapping TrA:E→C as follows:
(4.3)TrA(x)={z∈C:〈Az,y-z〉+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.
Then, the following hold:(1)
TrA is singlevalued,(2)
F(TrA)=VI(C,A),(3)
VI(C,A) is closed and convex,(4)
ϕ(q,TrAx)+ϕ(TrAx,x)≤ϕ(q,x),for allq∈F(TrA).Lemma 4.2.
LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Assume that φ:C→ℝ is lower semicontinuous and convex. For r>0 and x∈E, define a mapping Trφ:E→C as follows:
(4.4)Trφ(x)={z∈C:φ(y)+1r〈y-z,Jz-Jx〉≥φ(z),∀y∈C}.
Then, the following hold:(1)
Trφ is single valued,(2)
F(Trφ)=CMP(φ),(3)
CMP(φ) is closed and convex,(4)
ϕ(q,Trφx)+ϕ(Trφx,x)≤ϕ(q,x),for allq∈F(Trφ).Then we obtain the following theorems from Theorem3.1.Theorem 4.3.
LetE be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. For each k=1,2,…,m, let Ak be a continuous and monotone operator from C into E*. Suppose {Ti}i=1∞ and {Si}i=1∞ are two countable families of closed relatively quasi-nonexpansive mappings of C into itself such that Ω≔⋂k=1mVI(C,Ak)∩(⋂i=1∞F(Ti))∩(⋂i=1∞F(Si))≠∅. Let f:E→ℝ be a convex and lower semicontinuous mapping with C⊂int(D(f)), and suppose {xn}n=0∞ is iteratively generated by x0∈C,C1,i=C,C1=∩i=1∞C1,i,x1=ΠC1fx0,
(4.5)zn,i=J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSixn),yn,i=J-1(αn,iJxn+(1-αn,i)Jzn,i),un,i=Trm,nAmTrm-1,nAm-1⋯Tr2,nA2Tr1,nA1yn,i,Cn+1,i={z∈Cn,i:G(z,Jun,i)≤G(z,Jxn)},Cn+1=⋂i=1∞Cn+1,i,xn+1=ΠCn+1fx0,n≥1,
with the conditions(i)
liminfn→∞βn,i(1)βn,i(2)>0,(ii)
liminfn→∞βn,i(1)βn,i(3)>0,(iii)
0≤αn,i≤α<1 for some α∈(0,1),(iv)
{rk,n}n=1∞⊂(0,∞)(k=1,2,…,m) satisfying liminfn→∞rk,n>0(k=1,2,…,m).Then, {xn}n=0∞ converges strongly to ΠΩfx0.Theorem 4.4.
LetE be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property.  Let C be a nonempty, closed, and convex subset of E. For each k=1,2,…,m, let φk:C→ℝ be lower semicontinuous and convex. Suppose {Ti}i=1∞ and {Si}i=1∞ are two countable families of closed relatively quasi-nonexpansive mappings of C into itself such that Ω:=⋂k=1mCMP(φk)∩(⋂i=1∞F(Ti))∩(⋂i=1∞F(Si))≠∅. Let f:E→ℝ be a convex and lower semicontinuous mapping with C⊂int(D(f)), and suppose {xn}n=0∞ is iteratively generated by x0∈C,C1,i=C,C1=⋂i=1∞C1,i,x1=ΠC1fx0,
(4.6)zn,i=J-1(βn,i(1)Jxn+βn,i(2)JTixn+βn,i(3)JSixn),yn,i=J-1(αn,iJxn+(1-αn,i)Jzn,i),un,i=Trm,nφmTrm-1,nφm-1⋯Tr2,nφ2Tr1,nφ1yn,i,Cn+1,i={z∈Cn,i:G(z,Jun,i)≤G(z,Jxn)},Cn+1=⋂i=1∞Cn+1,i,xn+1=ΠCn+1fx0,n≥1,
with the conditions(i)
liminfn→∞βn,i(1)βn,i(2)>0,(ii)
liminfn→∞βn,i(1)βn,i(3)>0,(iii)
0≤αn,i≤α<1 for some α∈(0,1),(iv)
{rk,n}n=1∞⊂(0,∞)(k=1,2,…,m) satisfying liminfn→∞rk,n>0(k=1,2,…,m).Then, {xn}n=0∞ converges strongly to ΠΩfx0.



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*Source: 131890-2011-06-09.xml*