# A New Iterative Scheme for Countable Families of Weak Relatively

Authors: Yekini Shehu
Journal: Abstract and Applied Analysis
 (2010)

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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 weak relatively nonexpansive
mappings which is also a solution to a system of generalized mixed equilibrium problems in a uniformly convex
real Banach space which is also uniformly smooth using the properties of generalizedf-projection operator.
Using this result, we discuss strong convergence theorem concerning general H-monotone mappings and system
of generalized mixed equilibrium 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 let C be 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.
From [4], in uniformly convex and uniformly smooth Banach spaces, we have(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 {Tn}n=0∞ be a countable family of mappings from C into E. A point p∈C is said to be an asymptotic fixed point of {Tn}n=0∞ if C contains a sequence {xn}n=0∞ which converges weakly to p and limn→∞∥xn-Tnxn∥=0. The set of asymptotic fixed points of T is denoted by F̂({Tn}n=0∞). One says that {Tn}n=0∞ is countable family of relatively nonexpansive mappings (see, e.g., [5]) if the following conditions are satisfied:(R1)
F({Tn}n=0∞)≠∅;(R2)
ϕ(p,Tnx)≤ϕ(p,x), for all x∈C,  p∈F(Tn),  n≥0;(R3)
∩n=0∞F(Tn)=F̂({Tn}n=0∞).Definition 1.2.
A pointp∈C is said to be a strong asymptotic fixed point of {Tn}n=0∞ if C contains a sequence {xn}n=0∞ which converges strongly to p and limn→∞∥xn-Tnxn∥=0. The set of strong asymptotic fixed points of T is denoted by F̃({Tn}n=0∞). One says that a mapping {Tn}n=0∞ is countable family of weak relatively nonexpansive mappings (see, e.g., [5]) if the following conditions are satisfied:(R1)
F({Tn}n=0∞)≠∅;(R2)
ϕ(p,Tnx)≤ϕ(p,x), for all x∈C,  p∈F(Tn),  n≥0;(R3)
∩n=0∞F(Tn)=F̃({Tn}n=0∞).Definition 1.3.
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., [6–11]) if the following conditions are satisfied:(R1)
F(T)≠∅;(R2)
ϕ(p,Tx)≤ϕ(p,x), for all x∈C,  p∈F(T);(R3)
F(T)=F̂(T).Definition 1.4.
A pointp∈C is said to be an strong asymptotic fixed point of T if C contains a sequence {xn}n=0∞ which converges strongly to p and limn→∞∥xn-Txn∥=0. The set of strong asymptotic fixed points of T is denoted by F̃(T). We say that a mapping T is weak relatively nonexpansive (see, e.g., [12, 13]) if the following conditions are satisfied:(R1)
F(T)≠∅;(R2)
ϕ(p,Tx)≤ϕ(p,x), for all x∈C,  p∈F(T);(R3)
F(T)=F̃(T).Definition1.3 (Definition 1.4) is a special form of Definition 1.1 (Definition 1.2) as Tn≡T, for all n≥0. Furthermore, Su  et al. [5] gave an example which is a countable family of weak relatively nonexpansive mappings but not a countable family of relatively nonexpansive mappings. It is obvious that relatively nonexpansive mapping is weak relatively nonexpansive mapping. In fact, for any mapping T:C→C, we have F(T)⊂F̃(T)⊂F̂(T). Therefore, if T is relatively nonexpansive mapping, then F(T)=F̃(T)=F̂(T). Kang et al. [12] gave an example of a weak relatively nonexpansive mapping which is not relatively nonexpansive.LetF:C×C→ℝ be a bifunction, A:C→E* a mapping and φ:C→ℝ a real-valued function. The generalized mixed equilibrium problem is to find x∈C (see, e.g., [14–19]) such that(1.5)F(x,y)+〈Ax,y-x〉+φ(y)-φ(x)≥0,
for all y∈C. We will denote the solutions set of (1.5) by GMEP(F,φ). Thus (1.6)GMEP(F,A,φ)∶={x*∈C:F(x*,y)+〈Ax*,y-x*〉+φ(y)-φ(x*)≥0,∀y∈C}.
If φ=0 and A=0, then problem (1.5) reduces to an equilibrium problem studied by many authors (see, e.g., [20–28]), which is to find x*∈C such that(1.7)F(x*,y)≥0
for all y∈C. We shall denote the solutions set of (1.7) by EP(F).Ifφ=0 and E=H (a real Hilbert space), then problem (1.5) reduces to a generalized equilibrium problem studied by many authors (see, e.g., [29–31]), which is to find x*∈C such that(1.8)F(x*,y)+〈Ax*,y-x*〉≥0
for all y∈C.IfA=0 and E=H, then problem (1.5) reduces to mixed equilibrium problem considered by many authors (see, e.g., [32–34]), which is to find x*∈C such that(1.9)F(x*,y)+φ(y)-φ(x*)≥0
for all y∈C.The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [35]). Some methods have been proposed to solve the mixed equilibrium problem (see, e.g., [33, 34, 36]). Numerous problems in physics, optimization and economics reduce to find a solution of problem (1.8).In [9], 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,(1.10)yn=J-1(αnJxn+(1-αn)JTxn),Hn={w∈C:ϕ(w,yn)≤ϕ(w,xn)},Wn={w∈C:〈xn-w,Jx0-Jxn〉},xn+1=ΠHn∩Wnx0,n≥0
They proved that {xn}n=0∞ converges strongly to ΠF(T)x0, where F(T)≠∅.In [37], Plubtieng and Ungchittrakool introduced the following hybrid projection algorithm for a pair of relatively nonexpansive mappings: x0∈C,(1.11)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 (1.11) converges strongly to a common fixed point of T and S.In [10], 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,(1.12)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 ΠΩx0, where Ω=EP(F)∩F(T)≠∅.Recently, Li et al. [38] 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,(1.13)yn=J-1(αnJxn+(1-αn)JTxn),Cn+1={w∈Cn:G(w,Jyn)≤G(w,Jxn)},xn+1=ΠCn+1fx0,n≥1.
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. [38] extended and improved on the results of Matsushita and Takahashi [9].Quite recently, motivated by the results of Matsushita and Takahashi [9] and Plubtieng and Ungchittrakool [37], Su et al. [5] proved the following strong convergence theorem by hybrid iterative scheme for approximation of common fixed point of two countable families of weak relatively nonexpansive mappings in uniformly convex and uniformly smooth Banach space.Theorem 1.5.
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 {Tn}n=1∞ and {Sn}n=1∞ are two countable families of weak relatively nonexpansive mappings of C into itself such that Ω:=(∩n=1∞F(Tn))∩(∩n=1∞F(Sn))≠∅. Suppose that {xn}n=0∞ is iteratively generated by x0∈C,
(1.14)zn=J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn),yn=J-1(αnJxn+(1-αn)Jzn),Cn={w∈Cn-1∩Qn-1:ϕ(w,yn)≤ϕ(w,xn)},C0={w∈C:ϕ(w,y0)≤ϕ(w,x0)},Qn={w∈Cn-1∩Qn-1:〈xn-w,Jx0-Jxn〉≥0},Q0=C,xn+1=ΠCn∩Qnx0,n≥1,
with the conditions(i)
liminfn→∞βn(1)βn(2)>0;(ii)
liminfn→∞βn(1)βn(3)>0;(iii)
0≤αn≤α<1 for some α∈(0,1).
Then,{xn}n=0∞ converges strongly to ΠΩx0.Motivated by the above-mentioned results and the ongoing research, it is our purpose in this paper to prove a strong convergence theorem for two countable families of weak relatively nonexpansive mappings in a uniformly convex real Banach space which is also uniformly smooth using the properties of generalizedf-projection operator. Our results extend the results of Li et al. [38], Su et al. [5] 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→0ρE(t)t=0.
Let dimE≥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 all λ∈(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 [39], the generalized projection ΠC from E onto C is defined by (2.4)ΠCx∶=argmin⁡y∈Cϕ(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, 4, 39–41]).  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,ψ)=‖ξ‖2-2〈ξ,ψ〉+‖ψ‖2+2ρf(ξ),
where ξ∈C,  ψ∈E*,  ρ is a positive number and f:C→ℝ∪{+∞} is proper, convex and lower semi-continuous. From the definitions of G and f, it is easy to see the following properties:(i)
G(ξ,ψ) is convex and continuous with respect to ψ when ξ is fixed;(ii)
G(ξ,ψ) is convex and lower semi-continuous with respect to ξ when ψ is fixed.Definition 2.1 (Wu and Huang [42]).
LetE be a real Banach space with its dual E*. Let C be a nonempty, closed and convex subset of E. One says that ΠCf:E*→2C is a generalized f-projection operator if
(2.6)ΠCfψ={u∈C:G(u,ψ)=inf⁡ξ∈CG(ξ,ψ)},∀ψ∈E*.For the generalizedf-projection operator, Wu and Huang [42] proved the following theorem basic properties.Lemma 2.2 (Wu and Huang [42]).
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 ψ∈E*;(ii)
ifE is smooth, then for all ψ∈E*,  x∈ΠCf if and only if
(2.7)〈x-y,ψ-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. [43] showed that the condition f is positive homogeneous which appeared in Lemma 2.2 and can be removed.Lemma 2.3 (Fan et al. [43]).
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 ψ∈E* such that ψ=Jx for each x∈E. This substitution in (4.3) 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. One says  that ΠCf:E→2C is a generalized f-projection operator if
(2.9)ΠCfx={u∈C:G(u,Jx)=inf⁡ξ∈CG(ξ,Jx)},∀x∈E.Obviously, the definition of{Tn}n=0∞ is a countably family of weak relatively nonexpansive mappings is equivalent to(R′1)
F({Tn}n=0∞)≠∅;(R′2)
G(p,JTnx)≤G(p,Jx), for all x∈C,  p∈F(Tn),  n≥0;(R′3)
∩n=0∞F(Tn)=F̃({Tn}n=0∞).Lemma 2.5 (Li et al. [38]).
LetE be a Banach space and f:E→ℝ∪{+∞} a lower semi-continuous 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 (Li et al. [38]).
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 (Li et al. [38]).
LetC be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. Let x∈E and x̂∈ΠCf. Then
(2.12)ϕ(y,x̂)+G(x̂,Jx)≤G(y,Jx),∀y∈C.Lemma 2.8 (Su et al. [5]).
LetC be a nonempty, closed, and convex subset of a smooth, strictly convex Banach space E. Let T be a weak relatively 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 (Kamimura and Takahashi [4]).
LetC be a nonempty, closed, and convex subset of a smooth, uniformly convex Banach space E. Let {xn}n=1∞ and {yn}n=1∞ be sequences in E such that either {xn}n=1∞ or {yn}n=1∞ is bounded. If limn→∞ϕ(xn,yn)=0, then limn→∞∥xn-yn∥=0.Lemma 2.10 (Cho et al. [44]).
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,z∈C, limsupt↓0F(tz+(1-t)x,y)≤F(x,y);(A4)
for eachx∈C, y↦F(x,y) is convex and lower semicontinuous.Lemma 2.11 (Liu et al. [14] and Zhang [19]).
LetC be a nonempty, closed, and convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that F:C×C→ℝ satisfies (A1)–(A4), A:C→E* a continuous and monotone mapping, and φ:C→ℝ a lower semicontinuous and convex functional. For r>0 and x∈E, there exists z∈C such that
(2.15)Q(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C,
where Q(z,y)=F(z,y)+〈Az,y-z〉+φ(y)-φ(z),z,y∈C. Furthermore, define a mapping Tr:E→C as follows:
(2.16)Tr(x)={z∈C:Q(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.
Then, the following hold:(i)
Tr is single-valued;(ii)
for anyx,y∈E,
(2.17)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉;(iii)
F(Tr)=GMEP(F,A,φ);(iv)
GMEP(F,A,φ) is closed and convex.Lemma 2.12 (Zhang [19]).
LetC be a nonempty, closed, and 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(Tr),
(2.18)ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x).For the rest of this paper, the sequence{xn}n=0∞ converges strongly to p shall be denoted by xn→p as n→∞, and we shall assume that βn(1),βn(2),βn(1)∈[0,1] such that βn(1)+βn(2)+βn(1)=1, for all n≥0.


## 3. Main Results

Theorem 3.1.
LetE be a uniformly convex real Banach space which is also uniformly smooth. 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), Ak:C→E* a continuous and monotone mapping and φk:C→ℝ a lower semicontinuous and convex functional. Suppose that {Tn}n=0∞ and {Sn}n=0∞ are two countable families of weak relatively nonexpansive mappings of C into itself such that Ω∶=∩k=1mGMEP(Fk,Ak,φk)∩(∩n=0∞F(Tn))∩(∩n=0∞F(Sn))≠∅. Let f:E→ℝ be a convex and lower semicontinuous mapping with C⊂int(D(f)), and suppose that {xn}n=0∞ is iteratively generated by x0∈C,  C0=C,
(3.1)zn=J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn),yn=J-1(αnJxn+(1-αn)Jzn),un=Trm,nQmTrm-1,nQm-1⋯Tr2,nQ2Tr1,nQ1yn,Cn+1={w∈Cn:G(w,Jun)≤G(w,Jxn)},xn+1=ΠCn+1fx0,n≥0,
with the conditions(i)
liminfn→∞βn(1)βn(2)>0;(ii)
liminfn→∞βn(1)βn(3)>0;(iii)
0≤αn≤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.
By Lemma2.8, we know that (∩n=0∞F(Tn))∩(∩n=0∞F(Sn) is closed and convex. We also know from Lemma 2.11(iv) that ∩k=1mGMEP(Fk,Ak,φk) is closed and convex. Hence, Ω=∩k=1mGMEP(Fk,Ak,φk)∩(∩n=0∞F(Tn))∩(∩n=0∞F(Sn)) is a nonempty, closed and convex subset of C. Consequently, ΠΩfx0 is well defined. We first show that Cn, for all n≥0 is closed and convex. It is obvious that C0=C is closed and convex. Thus, we only need to show that Cn is closed and convex for each n≥1. Since G(z,Jun)≤G(z,Jxn) is equivalent to
(3.2)2(〈z,Jxn〉-〈z,Jun〉)≤‖xn‖2-‖un‖2,
this implies that Cn+1 is closed and convex  for all n≥0. This shows that ΠCn+1fx0 is well defined for all n≥0. By taking θnk=Trk,nQkTrk-1,nQk-1⋯Tr2,nQ2Tr1,nQ1, k=1,2,…,m and θn0=I for all n≥1, we obtain un=θnmyn. We next show that Ω⊂Cn, for all n≥0. For n=0, we have Ω⊂C=C0. Then for each x*∈Ω, we obtain
(3.3)G(x*,Jun)≤G(x*,Jθnmyn)≤G(x*,Jyn)=G(x*,(αnJxn+(1-αn)Jzn))=‖x*‖2-2αn〈x*,Jxn〉-2(1-αn)〈x*,Jzn〉+‖αnJxn+(1-αn)Jzn‖2+2ρf(x*)≤‖x*‖2-2αn〈x*,Jxn〉-2(1-αn)〈x*,Jzn〉+αn‖Jxn‖2+(1-αn)‖Jzn‖2+2ρf(x*)=αnG(x*,Jxn)+(1-αn)G(x*,Jzn)=αnG(x*,Jxn)+(1-αn)G(x*,βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn)≤αnG(x*,Jxn)+(1-αn)(‖x*‖2-2βn(1)〈x*,Jxn〉-2βn(2)〈x*,JTnxn〉-2βn(3)〈x*,JSnxn〉+βn(1)‖xn‖2+βn(2)‖Tnxn‖2+βn(3)‖Snxn‖2+2ρf(x*))=αnG(x*,Jxn)+(1-αn)(βn(1)G(x*,Jxn)+βn(2)G(x*,JTnxn)+βn(3)G(x*,JSnxn))≤G(x*,Jxn).
So, x*∈Cn. This implies that ∅≠Ω⊂Cn, for all n≥0. It follows that {xn}n=0∞ is well defined for all n≥0.
We now show thatlimn→∞G(xn,Jx0) exists. Since f:E→ℝ is a convex and lower semi-continuous, 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*∈Ω. This implies that {xn}n=0∞ is bounded and so is {G(xn,Jx0)}n=0∞. By the construction of Cn, we have that Cm⊂Cn and xm=ΠCmfx0∈Cn for any positive integer m≥n. It then follows from Lemma 2.7 that
(3.7)ϕ(xm,xn)+G(xn,Jx0)≤G(xm,Jx0).
It is obvious that
(3.8)ϕ(xm,xn)≥(‖xm‖-‖xn‖)2≥0.
In particular,
(3.9)ϕ(xn+1,xn)+G(xn,Jx0)≤G(xn+1,Jx0),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. By the fact that Cm⊂Cn and xm=ΠCmfx0∈Cn for any positive integer m≥n, we obtain
(3.10)ϕ(xm,un)≤ϕ(xm,xn).
Now, (3.7) implies that
(3.11)ϕ(xm,un)≤ϕ(xm,xn)≤G(xm,Jx0)-G(xn,Jx0).
Taking the limit as m,n→∞ in (3.11), we obtain
(3.12)lim⁡n→∞ϕ(xm,xn)=0.
It then follows from Lemma 2.9 that ∥xm-xn∥→0 as m,n→∞. Hence, {xn}n=0∞ is Cauchy. Since E is a Banach space and C is closed and convex, then there exists p∈C such that xn→p as n→∞.
Now sinceϕ(xm,xn)→0 as m,n→∞ we have in particular that ϕ(xn+1,xn)→0 as n→∞ and this further implies that limn→∞∥xn+1-xn∥=0. Since xn+1=ΠCn+1x0∈Cn+1, we have
(3.13)ϕ(xn+1,un)≤ϕ(xn+1,xn),∀n≥0.
Then, we obtain
(3.14)lim⁡n→∞ϕ(xn+1,un)=0.
Since E is uniformly convex and smooth, we have from Lemma 2.9 that
(3.15)lim⁡n→∞‖xn+1-xn‖=0=lim⁡n→∞‖xn+1-un‖.
So,
(3.16)‖xn-un‖≤‖xn+1-xn‖+‖xn+1-un‖.
Hence,
(3.17)lim⁡n→∞‖xn-un‖=0.
Since J is uniformly norm-to-norm continuous on bounded sets and limn→∞∥xn-un∥=0, we obtain
(3.18)lim⁡n→∞‖Jxn-un‖=0.
Since {xn} is bounded, so are {zn}, {JTnxn}, and {JSnxn}. Let r∶=supn≥0{∥xn∥,∥Tnxn∥,∥Snxn∥}. Then from Lemma 2.10, we have
(3.19)G(x*,Jun)≤G(x*,Jθnmyn)≤G(x*,Jyn)=G(x*,(αnJxn+(1-αn)Jzn))=‖x*‖2-2αn〈x*,Jxn〉-2(1-αn)〈x*,Jzn〉+‖αnJxn+(1-αn)Jzn‖2+2ρf(x*)≤‖x*‖2-2αn〈x*,Jxn〉-2(1-αn)〈x*,Jzn〉+αn‖Jxn‖2+(1-αn)‖Jzn‖2+2ρf(x*)=αnG(x*,Jxn)+(1-αn)G(x*,Jzn)=αnG(x*,Jxn)+(1-αn)G(x*,(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn))≤αnG(x*,Jxn)+(1-αn)(‖x*‖2-2βn(1)〈x*,Jxn〉-2βn(2)〈x*,JTnxn〉-2βn(3)〈x*,JSnxn〉+βn(1)‖xn‖2+βn(2)‖Tnxn‖2+βn(3)‖Snxn‖2-βn(1)βn(2)g(‖Jxn-JTnxn‖)+2ρf(x*))=αnG(x*,Jxn)+(1-αn)(βn(1)G(x*,Jxn)+βn(2)G(x*,JTnxn)+βn(3)G(x*,JSnxn)-βn(1)βn(2)g(‖Jxn-JTnxn‖))≤αnG(x*,Jxn)+(1-αn)(βn(1)G(x*,Jxn)+βn(2)G(x*,Jxn)+βn(3)G(x*,Jxn)-βn(1)βn(2)g(‖Jxn-JTnxn‖))=αnG(x*,Jxn)+(1-αn)(G(x*,Jxn)-βn(1)βn(2)g(‖Jxn-JTnxn‖))≤G(x*,Jxn)-(1-αn)βn(1)βn(2)g(‖Jxn-JTnxn‖).
It then follows that
(3.20)(1-α)βn(1)βn(2)g(‖Jxn-JTnxn‖)≤(1-αn)βn(1)βn(2)g(‖Jxn-JTnxn‖)≤G(x*,Jxn)-G(x*,Jun).
But
(3.21)G(x*,Jxn)-G(x*,Jun)=‖xn‖2-‖un‖2-2〈x*,Jxn-Jun〉≤|‖xn‖2-‖un‖2|+2|〈x*,Jxn-Jun〉|≤|‖xn‖-‖un‖|(‖xn‖-‖un‖)+2‖x*‖‖Jxn-Jun‖≤‖xn-un‖(‖xn‖+‖un‖)+2‖x*‖‖Jxn-Jun‖.
From limn→∞∥xn-un∥=0 and limn→∞∥Jxn-Jun∥=0, we obtain
(3.22)G(x*,Jxn)-G(x*,Jun)⟶0,n⟶∞.
Using the condition liminfn→∞βn(1)βn(2)>0, we have
(3.23)lim⁡n→∞g(‖Jxn-JTnxn‖)=0.
By property of g, we have limn→∞∥Jxn-JTnxn∥=0. Since J-1 is also uniformly norm-to-norm continuous on bounded sets, we have
(3.24)lim⁡n→∞‖xn-Tnxn‖=0.
Similarly, we can show that
(3.25)lim⁡n→∞‖xn-Snxn‖=0.
Since xn→p and {Tn}, {Sn} are uniformly closed, we have p∈(∩n=0∞F(Tn))∩(∩n=0∞F(Sn)).
Next, we show thatp∈∩k=1mGMEP(Fk,Ak,φk). Now, by Lemma 2.12, we obtain
(3.26)ϕ(un,yn)=ϕ(θnmyn,yn)≤ϕ(x*,yn)-ϕ(x*,θnmyn)≤ϕ(x*,xn)-ϕ(x*,un)⟶0,n⟶∞.
Using Lemma 2.9, we have limn→∞∥un-yn∥=0. Furthermore,
(3.27)‖xn-yn‖≤‖xn-un‖+‖un-yn‖⟶0,n⟶∞.
Since xn→p as n→∞ and ∥xn-yn∥→0 as n→∞, then yn→p as n→∞. By the fact that θnk,  k=1,2,…,m is relatively nonexpansive and using Lemma 2.12 again, we have that
(3.28)ϕ(θnkyn,yn)≤ϕ(x*,yn)-ϕ(x*,θnkyn)≤ϕ(x*,xn)-ϕ(x*,θnkyn).
Observe that
(3.29)ϕ(x*,un)=ϕ(x*,θnmyn)=ϕ(x*,Trm,nQmTrm-1,nQm-1⋯Trk,nQkTrk-1,nQk-1⋯Tr2,nQ2Tr1,nQ1yn)=ϕ(x*,Trm,nQmTrm-1,nQm-1⋯θnkyn)≤ϕ(x*,θnkyn).
Using (3.29) in (3.28), we obtain
(3.30)ϕ(θnkyn,yn)≤ϕ(x*,xn)-ϕ(x*,un)⟶0,n⟶∞.
Then Lemma 2.9 implies that limn→∞∥yn-θnkyn∥=0,  k=1,2,…,m. Now
(3.31)‖p-θnkyn‖≤‖yn-θnkyn‖+‖yn-p‖⟶0,n⟶∞,k=1,2,…,m.
Similarly, limn→∞∥p-θnk-1yn∥=0,  k=1,2,…,m. This further implies that
(3.32)lim⁡n→∞‖θnkyn-θnk-1yn‖=0.
Also, since J is uniformly norm-to-norm continuous on bounded sets and using (3.32), we obtain
(3.33)lim⁡n→∞‖Jθnkyn-Jθnk-1yn‖=0.
Since liminfn→∞rk,n>0,  (k=1,2,…,m),
(3.34)lim⁡n→∞‖Jθnkyn-Jθnk-1yn‖rk,n=0.
By Lemma 2.11, we have that for each k=1,2,…,m(3.35)Qk(θnkyn,y)+1rk,n〈y-θnkyn,Jθnkyn-Jθnk-1yn〉≥0,∀y∈C.
Furthermore, using (A2) we obtain
(3.36)1rk,n〈y-θnkyn,Jθnkyn-Jθnk-1yn〉≥Qk(y,θnkyn).
By (A4), (3.34), and θnkyn→p, we have for each k=1,2,…,m(3.37)Qk(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∈C. This yields that Qk(zt,p)≤0. It follows from (A1) and (A4) that
(3.38)0=Qk(zt,zt)≤tQk(zt,y)+(1-t)Qk(zt,p)≤tQk(zt,y)
and hence
(3.39)0≤Qk(zt,y).
From condition (A3), we obtain
(3.40)Qk(p,y)≥0,∀y∈C.
This implies that p∈GMEP(Fk,Ak,φk),  k=1,2,…,m. Thus, p∈∩k=1mGMEP(Fk,Ak,φk). Hence, we have p∈Ω=∩k=1mGMEP(Fk,Ak,φk)∩(∩n=0∞F(Tn))∩(∩n=0∞F(Sn)).
Finally, we show thatp=ΠΩfx0. Since Ω=∩k=1mGMEP(Fk,Ak,φk)∩(∩n=0∞F(Tn))∩(∩n=0∞F(Sn)) is a closed and convex set, from Lemma 2.6, we know that ΠΩfx0 is single valued and denote w=ΠΩfx0. Since xn=ΠCnfx0 and w∈Ω⊂Cn, we have
(3.41)G(xn,Jx0)≤G(w,Jx0),∀n≥0.
We know that G(ξ,Jφ) is convex and lower semi-continuous with respect to ξ when φ is fixed. This implies that
(3.42)G(p,Jx0)≤liminf⁡n→∞G(xn,Jx0)≤limsup⁡n→∞G(xn,Jx0)≤G(w,Jx0).
From the definition of ΠΩfx0 and p∈Ω, we see that p=w. This completes the proof.Corollary 3.2 (Li et al. [38]).
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 that T:C→C is a relatively nonexpansive mapping of C into itself such that F(T)≠∅ and f:E→ℝ is a convex and lower semicontinuous mapping with C⊂int(D(f)). Suppose that {xn}n=0∞ is iteratively generated by x0∈C,  C0=C(3.43)yn=J-1(αnJxn+(1-αn)JTxn),Cn+1={w∈Cn:G(w,Jyn)≤G(w,Jxn)},xn+1=ΠCn+1fx0,n≥0,
where J is the duality mapping on E. Suppose that {αn}n=0∞ is a sequence in (0,1) such that limsupn→∞αn<1. Then, {xn}n=0∞ converges strongly to ΠF(T)fx0.Corollary 3.3.
LetE be a uniformly convex real Banach space which is also uniformly smooth. 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), Ak:C→E*  a continuous and monotone mapping and φk:C→ℝ  a lower semicontinuous and convex functional. Suppose {Tn}n=0∞ and {Sn}n=0∞ are two countable families of weak relatively nonexpansive mappings of C into itself such that Ω:=∩k=1mGMEP(Fk,Ak,φk)∩(∩n=0∞F(Tn))∩(∩n=0∞F(Sn))≠∅. Suppose that {xn}n=0∞ is iteratively generated by x0∈C,  C0=C(3.44)zn=J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn),yn=J-1(αnJxn+(1-αn)Jzn),un=Trm,nQmTrm-1,nQm-1⋯Tr2,nQ2Tr1,nQ1yn,Cn+1={w∈Cn:ϕ(w,un)≤ϕ(w,xn)},xn+1=ΠCn+1x0,n≥0,
with the conditions(i)
liminfn→∞βn(1)βn(2)>0;(ii)
liminfn→∞βn(1)βn(3)>0;(iii)
0≤αn≤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.Proof.
Takef(x)=0 for all x∈E in Theorem 3.1, G(ξ,Jx)=ϕ(ξ,x) and ΠCfx0=ΠCx0. Then, the desired conclusion follows.Remark 3.4.
Corollary3.3 extends and improves on Theorem 1.5. In fact, the iterative procedure (3.44) is simpler than (1.14) in the following two aspects: (a) the process of computing Qn={w∈Cn-1∩Qn-1:〈xn-w,Jx0-Jxn〉≥0} is removed; (b) the process of computing ΠCn∩Qn is replaced by computing ΠCn.


## 4. Applications

A mappingH from E to E* is said to be (i)
monotone if〈Hx-Hy,x-y〉≥0, for all x,y∈E;(ii)
strictly monotone ifH is monotone and 〈Hx-Hy,x-y〉=0 if and only if x=y;(iii)
β-Lipschitz continuous if there exists a constant β≥0 such that ∥Hx-Hy∥≤β∥x-y∥, for all x,y∈E.LetM be a set-valued mapping from E to E* with domain D(M)={z∈E:Mz≠∅} and range R(M)=∪{Mz:z∈D(M)}. A set-valued mapping M is said to be (i)
monotone if〈x1-x2,y1-y2〉≥0 for each xi∈D(M) and yi∈Mxi,  i=1,2;(ii)
r-strongly monotone if 〈x1-x2,y1-y2〉≥r∥x1-x2∥2 for each xi∈D(M) and yi∈Mxi,  i=1,2;(iii)
maximal monotone ifM is monotone and its graph G(M):={(x,y):y∈Mx} is not properly contained in the graph of any other monotone operator;(iv)
a generalH-monotone if M is monotone and (H+λM)E=E* holds for every λ>0, where H is a mapping from E to E*.We denote the set{x∈E:0∈Mx} by M-10. From Li et al. [38], we know that if H:E→E* is strictly monotone and M:E→2E* is general H-monotone mapping, then M-10 is closed and convex. Furthermore, for every λ>0 and x*∈E*, there exists a unique x∈D(M) such that x=(H+λM)-1x*. Thus, we can define a single-value mapping Tλ:E→D(M) by Tλx=(H+λM)-1Hx. It is obvious that M-10=F(Tλ) for all λ>0.Lemma 4.1 (Alber, [39]).
IfE is a uniformly convex and uniformly smooth Banach space, δE(ϵ) is the modulus of convexity of E, and ρE(t) is the modulus of smoothness of E, then the inequalities
(4.1)8d2δE(‖x-ξ‖4d)≤ϕ(x,ξ)≤4d2ρE(4‖x-ξ‖d)
hold for all x and ξ in E, where d=(∥x∥2+∥ξ∥2)/2.Lemma 4.2 (Xia and Huang [45]).
LetE be a Banach space with dual space E*,H:E→E* a strictly monotone mapping, and M:E→2E* a general H-monotone mapping. Then(i)
(H+λM)-1 is a single-valued mapping;(ii)
ifE is reflexive and M:E→2E* is r-strongly monotone, (H+λM)-1 is Lipschitz continuous with constant 1/λr, where r>0.Theorem 4.3.
LetE be a uniformly convex real Banach space which is also uniformly smooth with δE(ϵ)≥kϵ2 and ρE(t)≤ct2 for some k,c>0. For each k=1,2,…,m, let Fk be a bifunction from E×E satisfying (A1)–(A4), Ak:E→E* a continuous and monotone mapping, and φk:E→ℝ a lower semicontinuous and convex functional. Suppose that H:E→E* is a strictly monotone and β-Lipschitz continuous mapping and Mi:E→2E* is a general H-monotone mapping and ri-strongly monotone mapping with ri>0,  i=1,2 such that Ω:=∩k=1mGMEP(Fk,Ak,φk)∩M1-10∩M2-10≠∅. Let TλMi=(H+λMi)-1H,  i=1,2, let  f:E→ℝ   a convex and lower semicontinuous mapping with D(f)=E and suppose for each n≥0 that there exists a λn>0 such that 64cβ2≤min{(1/2)kλn2r12,(1/2)kλn2r22}. Let {xn}n=0∞ be iteratively generated by x0∈E,  C0=E,
(4.2)zn=J-1(βn(1)Jxn+βn(2)JTλnM1xn+βn(3)JTλnM2xn),yn=J-1(αnJxn+(1-αn)Jzn),un=Trm,nQmTrm-1,nQm-1⋯Tr2,nQ2Tr1,nQ1yn,Cn+1={w∈Cn:G(w,Jyn)≤G(w,Jxn)},xn+1=ΠCn+1fx0,n≥0,
with the conditions(i)
liminfn→∞βn(1)βn(2)>0;(ii)
liminfn→∞βn(1)βn(3)>0;(iii)
0≤αn≤α<1;(iv)
liminfn→∞λn>0;(v)
{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 only need to prove that{TλnM1} and {TλnM2} are countable families of weak relatively nonexpansive mappings with common fixed points sets ∩n=0∞F(TλnM1)=M1-10 and ∩n=0∞F(TλnM2)=M2-10, respectively. Firstly, we have ∩n=0∞F(TλnM1)=M1-10≠∅. Secondly, we show that ϕ(p,TλnM1w)≤ϕ(p,w), for all w∈E,  p∈F(TλnM1),  n≥0. Now, by Lemma 4.2 and the Lipschitz continuity of H, we have
(4.3)‖TλnM1p-TλnM1w‖=‖(H+λnM1)-1Hp-(H+λnM1)-1Hw‖≤1λnr1‖Hp-Hw‖≤βλnr1‖p-w‖.
By (4.3) and Lemma 4.1,
(4.4)ϕ(p,TλnM1w)=ϕ(TλnM1p,TλnM1w)≤4d2ρE(4‖TλnM1p-TλnM1w‖d)≤64c‖TλnM1p-TλnM1w‖2≤64cβ2λn2r12‖p-w‖2,ϕ(p,w)≥8d2δE(‖p-w‖4d)≥12k‖p-w‖2.
Since 64cβ2≤(1/2)kλn2r12, it follows from (4.4) that ϕ(p,TλnM1w)≤ϕ(p,w), for all w∈E, p∈F(TλnM1), n≥0. Thirdly, we show that F̃({TλnM1})=∩n=0∞F(TλnM1)=M1-10. We first show that F̃({TλnM1})⊂M1-10. Let p∈F̃({TλnM1}), then there exists {xn}⊂E such that xn→p and limn→∞∥xn-TλnM1xn∥=0. Since H is β-Lipschitz continuous,
(4.5)‖Hxn-HTλnM1xn‖≤β‖xn-TλnM1xn‖.
Letting n→∞, we obtain
(4.6)1λn(Hxn-HTλnM1xn)⟶0.
It follows from (1/λn)(Hxn-HTλnM1xn)∈M1TλnM1 and the monotonicity of M1 that
(4.7)〈x-TλnM1,x*-1λn(Hxn-HTλnM1xn)〉≥0
for all ∈D(M1) and x*∈M1x.Taking the limit as n→∞, we obtain
(4.8)〈x-p,x*〉≥0
for all ∈D(M1) and x*∈M1x. By the maximality of M1, we know that p∈M1-10. On the other hand, we know that F(TλnM1)=M1-10, F(TλnM1)⊂F̃(TλnM1) for all n≥0, therefore, M1-10=∩n=0∞F(TλnM1)=F̃(∩n=0∞TλnM1). Thus, we have proved that {TλnM1} is a countable family of weak relatively nonexpansive mappings with common fixed points sets ∩n=0∞F(TλnM1)=M1-10. By following the same arguments, we can show that {TλnM2} is a countable family of weak relatively nonexpansive mappings with common fixed points sets ∩n=0∞F(TλnM2)=M2-10.LetE be a uniformly convex and uniformly smooth Banach space, H=J and M a maximal monotone mapping. Then, we can define Jλ=(J+λM)-1J for all λ>0. We know that Jλ is relatively nonexpansive and therefore weak relatively nonexpansive and M-10=F(Jλ) for all λ>0 (see, e.g., [2]), where F(Jλ) denotes the fixed points set of Jλ. By Corollary 3.3, we obtain the following theorem.Theorem 4.4.
LetE be a uniformly convex real Banach space which is also uniformly smooth. For each k=1,2,…,m, let Fk be a bifunction from E×E satisfying (A1)–(A4), Ak:E→E* a continuous and monotone mapping, and φk:E→ℝ a lower semicontinuous and convex functional. For each i=1,2, let Mi⊂E×E* a maximal monotone operator, and let JλMi=(J+λMi)-1J for all λ>0, and suppose C is a nonempty closed and convex subset of E such that D(Mi)⊂C⊂J-1(∩λ>0R(J+λMi)),  i=1,2. Assume that Ω:=∩k=1mGMEP(Fk,Ak,φk)∩M1-10∩M2-10≠∅ and let {xn}n=0∞ be iteratively generated by x0∈E,  C0=E,
(4.9)zn=J-1(βn(1)Jxn+βn(2)JJTλnM1xn+βn(3)JJTλnM2xn),yn=J-1(αnJxn+(1-αn)Jzn),un=Trm,nQmTrm-1,nQm-1⋯Tr2,nQ2Tr1,nQ1yn,Cn+1={w∈Cn:ϕ(w,yn)≤ϕ(w,xn)},xn+1=ΠCn+1x0,n≥0,
with the conditions(i)
liminfn→∞βn(1)βn(2)>0;(ii)
liminfn→∞βn(1)βn(3)>0;(iii)
0≤αn≤α<1;(iv)
liminfn→∞λn>0;(v)
{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.



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*Source: 861031-2011-01-24.xml*