# A Modified Halpern-TypeIterative Method of a System of Equilibrium Problems and a Fixed Point for a Totally Quasi-ϕ-Asymptotically Nonexpansive Mapping in a Banach Space

Authors: Preedaporn Kanjanasamranwong; Poom Kumam; Siwaporn Saewan
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The purpose of this paper is to introduce the modified Halpern-type iterative method by the generalizedf-projection operator for finding a common solution of fixed-point problem of a totally quasi-ϕ-asymptotically nonexpansive mapping and a system of equilibrium problems in a uniform smooth and
strictly convex Banach space with the Kadec-Klee property. Consequently, we prove the strong convergence for a common solution of above two sets. Our result presented in this paper generalize and improve
the result of Chang et al., (2012), and some others.


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## Body

## 1.   Introduction

In 1953, Mann [1] introduced the following iteration process which is now known as Mann's iteration:
(1.1)xn+1=αnxn+(1-αn)Txn,
where T is nonexpansive, the initial guess element x1∈C is arbitrary, and {αn} is a sequence in [0,1]. Mann iteration has been extensively investigated for nonexpansive mappings. In an infinite-dimensional Hilbert space, Mann iteration can conclude only weak conviergence (see [2, 3]).Later, in 1967, Halpern [4] considered the following algorithm:
(1.2)x1∈C,xn+1=αnx1+(1-αn)Txn,∀n≥0,
where T is nonexpansive. He proved the strong convergence theorem of {xn} to a fixed point of T under some control condition {αn}. Many authors improved and studied the result of Halpern [4] such as Qin et al. [5], Wang et al. [6], and reference therein.In 2008-2009, Takahashi and Zembayashi [7, 8] studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of the Banach spaces.On the other hand, Li et al. [9] introduced the following hybrid iterative scheme for approximation fixed points of relatively nonexpansive mapping using the generalized f-projection operator in a uniformly smooth real Banach space which is also uniformly convex. They obtained strong convergence theorem for finding an element in the fixed point set of T.Recently, Ofoedu and Shehu [10] extended algorithm of Li et al. [9] to prove a strong convergence theorem for a common solution of a system of equilibrium problem and the set of common fixed points of a pair of relatively quasi-nonexpansive mappings in the Banach spaces by using generalized f-projection operator. Chang et al. [11] extended and improved Qin and Su [12] to obtain a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of relatively nonexpansive mappings in a Banach space.Very recently, Chang et al. [13] extended the results of Qin et al. [5] and Wang et al. [6] to consider a modification to the Halpern-type iteration algorithm for a total quasi-ϕ-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces.The purpose of this paper is to be motivated and inspired by the works mentioned above, we introduce a modified Halpern-type iterative method by using the new hybrid projection algorithm of the generalizedf-projection operator for solving the common solution of fixed point for totally quasi-ϕ-asymptoically nonexpansive mappings and the system of equilibrium problems in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. The results presented in this paper improve and extend the corresponding ones announced by many others.


## 2.   Preliminaries and Definitions

LetE be a real Banach space with dual E*, and let C be a nonempty closed and convex subset of E. Let {θi}i∈Γ:C×C→ℝ be a bifunction, where Γ is an arbitrary index set. The system of equilibrium problems is to find x∈C such that
(2.1)θi(x,y)≥0,i∈Γ,∀y∈C.
If Γ is a singleton, then problem (2.1) reduces to the equilibrium problem, which is to find x∈C such that
(2.2)θ(x,y)≥0,∀y∈C.
A mapping T from C into itself is said to be nonexpansive if
(2.3)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.T is said to be asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) with kn→1 as n→∞ such that
(2.4)‖Tnx-Tny‖≤kn‖x-y‖,∀x,y∈C.T is said to be total asymptotically nonexpansive if there exist nonnegative real sequences νn, μn with νn→0, μn→0 as n→∞ and a strictly increasing continuous function φ:ℝ+→ℝ+ with φ(0)=0 such that
(2.5)‖Tnx-Tny‖≤‖x-y‖+νnψ(‖x-y‖)+μn,∀x,y∈C,∀n≥1.
A point x∈C is a fixed pointof T provided Tx=x. Denote by F(T) the fixed point set of T; that is, F(T)={x∈C:Tx=x}. A point p in C is called an asymptotic fixed point of T if C contains a sequence {xn} which converges weakly to p such that limn→∞∥xn-Txn∥=0. The asymptotic fixed point set of T is denoted by F^(T).Thenormalized duality mappingJ:E→2E* is defined by J(x)={x*∈E*:〈x,x*〉=∥x∥2,∥x*∥=∥x∥}. If E is a Hilbert space, then J=I, where I is the identity mapping. Consider the functional defined by
(2.6)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,
where J is the normalized duality mapping and 〈·,·〉 denote the duality pairing of E and E*.IfE is a Hilbert space, then ϕ(y,x)=∥y-x∥2. It is obvious from the definition of ϕ that
(2.7)(‖y‖-‖x‖)2≤ϕ(y,x)≤(‖y‖+‖x‖)2,∀x,y∈E.
A mapping T from C into itself is said to be ϕ-nonexpansive [14, 15] if
(2.8)ϕ(Tx,Ty)≤ϕ(x,y),∀x,y∈C.T is said to be quasi-ϕ-nonexpansive [14, 15] if F(T)≠∅ and
(2.9)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).T is said to be asymptoticallyϕ-nonexpansive [15] if there exists a sequence {kn}⊂[0,∞) with kn→1 as n→∞ such that
(2.10)ϕ(Tnx,Tny)≤knϕ(x,y),∀x,y∈C.T is said to be quasi-ϕ-asymptotically nonexpansive [15] if F(T)≠∅ and there exists a sequence {kn}⊂[0,∞) with kn→1 as n→∞ such that
(2.11)ϕ(p,Tnx)≤knϕ(p,x),∀x∈C,p∈F(T),∀n≥1.T is said to be totally quasi-ϕ-asymptotically nonexpansive, if F(T)≠∅ and there exist nonnegative real sequences νn, μn with νn→0, μn→0 as n→∞ and a strictly increasing continuous function φ:ℝ+→ℝ+ with φ(0)=0 such that
(2.12)ϕ(p,Tnx)≤ϕ(p,x)+νnφ(ϕ(p,x))+μn,∀n≥1,∀x∈C,p∈F(T).
A mapping T from C into itself is said to be closed if for any sequence {xn}⊂C such that limn→∞xn=x0 and limn→∞Txn=y0, then Tx0=y0.Alber [16] introduced the generalized projectionΠC:E→C is a map that assigns to an arbitrary point x∈E the minimum point of the functional ϕ(x,y); that is, ΠCx=x¯, where x¯ is the solution of the minimization problem:
(2.13)ϕ(x¯,x)=inf⁡y∈Cϕ(y,x).
The existence and uniqueness of the operator ΠC follows from the properties of the functional ϕ(y,x) and the strict monotonicity of the mapping J (see, e.g., [16–20]). If E is a Hilbert space, then ϕ(x,y)=∥x-y∥2 and ΠC becomes the metric projection PC:H→C. If C is a nonempty, closed, and convex subset of a Hilbert space H, then PC is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. Later, Wu and Huang [21] introduced a new generalized f-projection operator in the Banach space. They extended the definition of the generalized projection operators and proved some properties of the generalized f-projection operator. Next, we recall the concept of the generalized f-projection operator. Let G:C×E*→ℝ∪{+∞} be a functional defined by
(2.14)G(y,ϖ)=‖y‖2-2〈y,ϖ〉+‖ϖ‖2+2ρf(y),
where y∈C,  ϖ∈E*,ρ is positive number, and f:C→ℝ∪{+∞} is proper, convex, and lower semicontinuous. From the definition of G, Wu and Huang [21] proved the following properties:(1)
G(y,ϖ) is convex and continuous with respect to ϖ when y is fixed;(2)
G(y,ϖ) is convex and lower semicontinuous with respect to y when ϖ is fixed.Definition 2.1.
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 generalizedf-projection operator if
(2.15)πCfϖ={u∈C:G(u,ϖ)=inf⁡y∈CG(y,ϖ),∀ϖ∈E*}.A Banach spaceE with norm ∥·∥ is called strictly convex if ∥(x+y)/2∥<1 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. Let U={x∈E:∥x∥=1} be the unit sphere of E. A Banach space E is called smooth if the limit limt→0((∥x+ty∥-∥x∥)/t) exists for each x,y∈U. It is also called uniformly smooth if the limit exists uniformly for all x,y∈U. The modulus of smoothness of E is the function ρE:[0,∞)→[0,∞) defined by ρE(t)=sup{(∥x+y∥+∥x-y∥)/2-1:∥x∥=1,∥y∥≤t}. The modulus of convexity of E (see [22]) is the function δE:[0,2]→[0,1] defined by δE(ɛ)=inf{1-∥(x+y)/2∥:x,y∈E,∥x∥=∥y∥=1,∥x-y∥≥ɛ}. In this paper we denote the strong convergence and weak convergence of a sequence {xn} by xn→x and xn⇀x, respectively.Remark 2.2.
The basic properties ofE, E*, J, and J-1 (see [18]) are as follows.(i)
IfE is an arbitrary Banach space, then J is monotone and bounded.(ii)
IfE is a strictly convex, then J is strictly monotone.(iii)
IfE is a smooth, then J is single valued and semicontinuous.(iv)
IfE is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.(v)
IfE is reflexive smooth and strictly convex, then the normalized duality mapping J is single valued, one-to-one, and onto.(vi)
IfE is a reflexive strictly convex and smooth Banach space and J is the duality mapping from E into E*, then J-1 is also single valued, bijective, and is also the duality mapping from E* into E, and thus JJ-1=IE* and J-1J=IE.(vii)
IfE is uniformly smooth, then E is smooth and reflexive.(viii)
E is uniformly smooth if and only if E* is uniformly convex.(ix)
IfE is a reflexive and strictly convex Banach space, then J-1 is norm-weak*-continuous.Remark 2.3.
IfE is a reflexive, strictly convex, and smooth Banach space, then ϕ(x,y)=0, if and only if x=y. It is sufficient to show that if ϕ(x,y)=0 then x=y. From (2.6), we have ∥x∥=∥y∥. This implies that 〈x,Jy〉=∥x∥2=∥Jy∥2. From the definition of J, one has Jx=Jy. Therefore, we have x=y (see [18, 20, 23] for more details).Recall that a Banach spaceE has the Kadec-Klee property [18, 20, 24], if for any sequence {xn}⊂E and x∈E with xn⇀x and ∥xn∥→∥x∥, then ∥xn-x∥→0 as n→∞. It is well known that if E is a uniformly convex Banach space, then E has the Kadec-Klee property.We also need the following lemmas for the proof of our main results.Lemma 2.4 (see Change et al. [25]).
LetC be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let T:C→C be a closed and total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequence νn and μn with νn→0, μn→0 as n→∞ and a strictly increasing continuous function ζ:ℝ+→ℝ+ with ζ(0)=0. If μ1=0, then the fixed point set F(T) is a closed convex subset of C.Lemma 2.5 (see Wu and Hung [21]).
LetE be a real reflexive Banach space with its dual E* and C a nonempty, closed, and convex subset of E. The following statement hold:(1)
πCfϖ is a nonempty, closed and convex subset of C for all ϖ∈E*;(2)
ifE is smooth, then for all ϖ∈E*, x∈πCfϖ if and only if
(2.16)〈x-y,ϖ-Jx〉+ρf(y)-ρf(x)≥0,∀y∈C;(3)
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 πCfϖ is single-valued mapping.Lemma 2.6 (see Fan et al. [26]).
LetE be a real reflexive Banach space with its dual E* and C be a nonempty, closed and convex subset of E. If E is strictly convex, then πCfϖ is single valued.Recall thatJ is single-valued mapping when E is a smooth Banach space. There exists a unique element ϖ∈E* such that ϖ=Jx where x∈E. This substitution in (2.14) gives
(2.17)G(y,Jx)=‖y‖2-2〈y,Jx〉+‖x‖2+2ρf(y).Now we consider the second generalizedf projection operator in Banach space (see [9]).Definition 2.7.
LetE be a real smooth Banach space, and let C be a nonempty, closed, and convex subset of E. We say that ΠCf:E→2C is generalized f-projection operator if
(2.18)ΠCfx={u∈C:G(u,Jx)=inf⁡y∈CG(y,Jx),∀x∈E}.Lemma 2.8 (see Deimling [27]).
LetE be a Banach space, and let f:E→ℝ∪{+∞} be a lower semicontinuous convex function. Then there exist x*∈E* and α∈ℝ such that
(2.19)f(x)≥〈x,x*〉+α,∀x∈E.Lemma 2.9 (see Li et al. [9]).
LetE be a reflexive smooth Banach space, and let C be a nonempty, closed, and convex subset of E. The following statements hold:  (1)
ΠCfx is nonempty, closed and convex subset of C for all x∈E;(2)
for allx∈E, x^∈ΠCfx if and only if
(2.20)〈x^-y,Jx-Jx^〉+ρf(y)-ρf(x^)≥0,∀y∈C;(3)
ifE is strictly convex, then ΠCf is single-valued mapping.Lemma 2.10 (see Li et al. [9]).
LetE be a real reflexive smooth Banach space, let C be a nonempty, closed, and convex subset of E, x∈E, and let x^∈ΠCfx. Then
(2.21)ϕ(y,x^)+G(x^,Jx)≤G(y,Jx),∀y∈C.Remark 2.11.
LetE be a uniformly convex and uniformly smooth Banach space and f(x)=0 for all x∈E, then Lemma 2.10 reduces to the property of the generalized projection operator considered by Alber [16].Iff(y)≥0 for all y∈C and f(0)=0, then the definition of totally quasi-ϕ-asymptotically nonexpansive T is equivalent to if F(T)≠∅, and there exist nonnegative real sequences νn, μn with νn→0, μn→0 as n→∞ and a strictly increasing continuous function ζ:ℝ+→ℝ+ with ζ(0)=0 such that
(2.22)G(p,Tnx)≤G(p,x)+νnζG(p,x)+μn,∀n≥1,∀x∈C,p∈F(T).For solving the equilibrium problem for a bifunctionθ:C×C→ℝ, let us assume that θ satisfies the following conditions:  (A1)
θ(x,x)=0 for all x∈C;(A2)
θ is monotone; that is, θ(x,y)+θ(y,x)≤0 for all x,y∈C;(A3)
for eachx,y,z∈C,
(2.23)lim⁡t↓0θ(tz+(1-t)x,y)≤θ(x,y);(A4)
for eachx∈C, y↦θ(x,y) is convex and lower semicontinuous.For example, letA be a continuous and monotone operator of C into E* and define
(2.24)θ(x,y)=〈Ax,y-x〉,∀x,y∈C.
Then, θ satisfies (A1)–(A4). The following result is in Blum and Oettli [28].Lemma 2.12 (see Blum and Oettli [28]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let θ be a bifunction from C×C to ℝ satisfying (A1)–(A4), and let r>0 and x∈E. Then, there exists z∈C such that
(2.25)θ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 2.13 (see Takahashi and Zembayashi [8]).
LetC be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E, and let θ be a bifunction from C×C to ℝ satisfying conditions (A1)–(A4). For all r>0 and x∈E, define a mapping Trθ:E→C as follows:
(2.26)Trθx={z∈C:θ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.
Then the following hold:  (1)
Trθ is single-valued;(2)
Trθ is a firmly nonexpansive-type mapping [29]; that is, for all x,y∈E,
(2.27)〈Trθx-Trθy,JTrθx-JTrθy〉≤〈Trθx-Trθy,Jx-Jy〉;(3)
F(Trθ)=EP(θ);(4)
EP(θ) is closed and convex.Lemma 2.14 (see Takahashi and Zembayashi [8]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let θ be a bifunction from C×C to ℝ satisfying (A1)–(A4), and let r>0. Then, for x∈E and q∈F(Trθ),
(2.28)ϕ(q,Trθx)+ϕ(Trθx,x)≤ϕ(q,x).


## 3.   Main Result

Theorem 3.1.
LetC be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. For each j=1,2,…,m, let θj be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A4). Let S:C→C be a closed totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences νn, μn with νn→0, μn→0 as n→∞, and a strictly increasing continuous function ψ:ℝ+→ℝ+ with ψ(0)=0. Let f:E→ℝ be a convex and lower semicontinuous function with C⊂int(D(f)) such that f(x)≥0 for all x∈C and f(0)=0. Assume that ℱ:=F(S)∩(∩j=1mEP(θj))≠∅. For an initial point x1∈E and C1=C, one define the sequence {xn} by
(3.1)un=Trm,nθmTrm-1,nθm-1Trm-2,nθm-2⋯Tr1,nθ1xn,zn=J-1(αnJx1+(1-αn)JSnun),Cn+1={v∈Cn:G(v,Jzn)≤G(v,Jun)≤G(v,Jx1)+(1-αn)G(v,Jxn)+ζn},xn+1=ΠCn+1fx1,n∈ℕ,
where {αn} is a sequence in [0,1], ζn=νnsupq∈ℱψ(G(q,xn))+μn and {rj,n}⊂[d,∞) for some d>0. If limn→∞αn=0, then {xn} converges strongly to Πℱfx0.Proof.
We split the proof into four steps.Step 1.
First, we show thatCn is closed and convex for all n∈ℕ.
ClearlyC1=C is closed and convex. Suppose that Cn is closed and convex for all n∈ℕ. For any υ∈Cn, we know that G(υ,Jzn)≤G(υ,Jxn)+ζn is equivalent to
(3.2)2〈υ,Jxn-Jzn〉≤‖xn‖2-‖zn‖2+ζn.
So, Cn+1 is closed and convex. Hence by induction Cn is closed and convex for all n≥1.Step 2.
We will show that the sequence{xn} is well defined.
We will show by induction thatℱ⊂Cn for all n∈ℕ. It is obvious that ℱ⊂C1=C. Suppose that ℱ⊂Cn for some n∈ℕ. Let q∈ℱ, put un=Knmxn,Knj=Trj,nθjTrj-1,nθj-1…Tr1,nθ1 for all j=1,2,3,…,m, Kn0=I, we have that
(3.3)G(q,Jun)=&G(q,JKnmxn)≤&G(q,Jxn).
From (3.3) and S which is a totally quasi-ϕ asymptotically nonexpansive mappings, it follows that
(3.4)G(q,Jzn)=G(q,(αnJx1+(1-αn)JSnun))=‖q‖2-2αn〈q,Jx1〉-2(1-αn)〈q,JSnun〉+‖αnJx1+(1-αn)JSnun‖2+2ρf(q)≤‖q‖2-2αn〈q,Jx1〉-2(1-αn)〈q,JSnun〉+αn‖Jx1‖2+(1-αn)‖JSnun‖2+2ρf(q)=αnG(q,Jx1)+(1-αn)G(q,JSnun)≤αnG(q,Jx1)+(1-αn)(G(q,Jun)+νnψ(G(q,Jun))+μn)≤αnG(q,Jx1)+(1-αn)G(q,Jxn)+νnsup⁡q∈Fψ(G(q,Jxn))+μn=αnG(q,Jx1)+(1-αn)G(q,Jxn)+ζn.
This shows thatq∈Cn+1 which implies that ℱ⊂Cn+1, and hence, ℱ⊂Cn for all n∈ℕ. and the sequence {xn} is well defined. From xn=ΠCnfx1, we see that
(3.5)〈xn-q,Jx1-Jxn〉+ρf(q)-ρf(xn)≥0,∀q∈Cn.
Since ℱ⊂Cn for each n∈ℕ, we arrive at
(3.6)〈xn-q,Jx1-Jxn〉+ρf(q)-ρf(xn)≥0,∀q∈F.
Hence, the sequence {xn} is well defined.Step 3.
We will show thatxn→p∈ℱ:=F(S)∩(∩j=1mEP(θj)).
Letf:E→ℝ is convex and lower semicontinuous function, follows from Lemma 2.8, there exist x*∈E* and α∈ℝ such that
(3.7)f(y)≥〈y,x*〉+α,∀y∈E.
Since xn∈Cn⊂E, it follows that
(3.8)G(xn,Jx1)=‖xn‖2-2〈xn,Jx1〉+‖x1‖2+2ρf(xn)≥‖xn‖2-2〈xn,Jx1〉+‖x1‖2+2ρ〈xn,x*〉+2ρα=‖xn‖2-2〈xn,Jx1-ρx*〉+‖x1‖2+2ρα≥‖xn‖2-2‖xn‖‖Jx1-ρx*‖+‖x1‖2+2ρα=(‖xn‖-‖Jx1-ρx*‖)2+‖x1‖2-‖Jx1-ρx*‖2+2ρα.
For q∈ℱ and xn=ΠCnfx1, we have
(3.9)G(q,Jx1)≥G(xn,Jx1)≥(‖xn‖-‖Jx1-ρx*‖)2+‖x1‖2-‖Jx1-ρx*‖2+2ρα.
This shows that {xn} is bounded and so is {G(xn,Jx1)}. From the fact that xn+1=ΠCn+1fx1∈Cn+1⊂Cn and xn=ΠCnfx1, it follows from Lemma 2.10 that
(3.10)0≤(‖xn+1-‖xn∥)2≤ϕ(xn+1,xn)≤G(xn+1,Jx1)-G(xn,Jx1).
That is, {G(xn,Jx1)} is nondecreasing. Hence, we obtain that limn→∞G(xn,Jx1) exists. Taking n→∞, we obtain
(3.11)lim⁡n→∞ϕ(xn+1,xn)=0.
Since E is reflexive, {xn} is bounded, and Cn is closed and convex for all n∈ℕ. Without loss of generality, we can assume that xn⇀p∈Cn. From the fact that xn=ΠCnfx1, we get that
(3.12)G(xn,Jx1)≤G(p,Jx1),∀n∈N.
Since f is convex and lower semicontinuous, we have
(3.13)liminf⁡n→∞G(xn,Jx1)=liminf⁡n→∞{‖xn‖2-2〈xn,Jx1〉+‖x1‖2+2ρf(xn)}≥‖p‖2-2〈p,Jx1〉+‖x1‖2+2ρf(p)=G(xn,Jx1).
By (3.12) and (3.13), we get
(3.14)G(p,Jx1)≤liminf⁡n→∞G(xn,Jx1)≤limsup⁡n→∞G(xn,Jx1)≤G(p,Jx1).
That is, limn→∞G(xn,Jx1)=G(p,Jx1); this implies that ∥xn∥→∥p∥; by virtue of the Kadec-Klee property of E, we obtain that
(3.15)lim⁡n→∞xn=p.
We also have
(3.16)lim⁡n→∞xn+1=p.
From (3.15), we get that
(3.17)lim⁡n→∞ζn=lim⁡n→∞(νnsup⁡q∈Fψ(G(q,xn))+μn)=0.
(
a
) We show that p∈∩j=1mEP(θj).
Sincexn+1=ΠCn+1fx1∈Cn+1⊂Cn and the definition of Cn+1, we have
(3.18)G(xn+1,Jun)≤αnG(xn+1,Jx1)+(1-αn)G(xn+1,Jxn)+ζn
is equivalent to
(3.19)ϕ(xn+1,un)≤αnϕ(xn+1,x1)+(1-αn)ϕ(xn+1,xn)+ζn.
From (3.11), (3.15), and (3.17), it follows that
(3.20)lim⁡n→∞ϕ(xn+1,un)=0.
From (2.7), we have
(3.21)(‖xn+1‖-‖un‖)2⟶0.
Since ∥xn+1∥→∥p∥, we have
(3.22)‖un‖⟶‖p‖asn⟶∞.
It follow that
(3.23)‖Jun‖⟶‖Jp‖asn⟶∞.
That is, {∥Jun∥} is bounded in E* and E* is reflexive; we assume that Jun⇀u*∈E*. In view of J(E)=E*, there exists u∈E such that Ju=u*. It follows that
(3.24)ϕ(xn+1,un)=‖xn+1‖2-2〈xn+1,Jyn〉+‖un‖2=‖xn+1‖2-2〈xn+1,Jun〉+‖Jun‖2.
Taking liminfn→∞ on both sides of the equality above and ∥·∥ is the weak lower semicontinuous, it yields that
(3.25)0≥‖p‖2-2〈p,u*〉+‖u*‖2=‖p‖2-2〈p,Ju〉+‖Ju‖2=‖p‖2-2〈p,Ju〉+‖u‖2=ϕ(p,u).
That is, p=u, which implies that u*=Jp. It follows that Jun⇀Jp∈E*. From (3.23) and the Kadec-Klee property of E* we have Jun→Jp as n→∞. Note that J-1:E*→E is norm-weak*-continuous; that is, un⇀p. From (3.22) and the Kadec-Klee property of E, we have
(3.26)lim⁡n→∞un=p.
For q∈F⊂Cn, by nonexpansiveness, we observe that
(3.27)ϕ(q,un)=ϕ(q,Knmxn)≤ϕ(q,Knm-1xn)≤ϕ(q,Knm-2xn)⋮≤ϕ(q,Knjxn).
By Lemma 2.14, we have for j=1,2,3,…,m(3.28)ϕ(Knjxn,xn)&≤ϕ(q,xn)-ϕ(q,Knjxn)≤ϕ(q,xn)-ϕ(q,un).
Since xn,un→p as n→∞, we get ϕ(Knjxn,xn)→0 as n→∞, for j=1,2,3,…,m. From (2.7), it follow that
(3.29)(‖Knjxn‖-‖xn‖)2⟶0.
Since ∥xn∥→∥p∥, we also have
(3.30)‖Knjxn‖⟶‖p‖asn⟶∞.
Since {Knjxn} is bounded and E is reflexive, without loss of generality we assume that Knjyn⇀h. We know that Cn is closed and convex for each n≥1 it is obvious that h∈Cn. Again since
(3.31)ϕ(Knjxn,xn)=‖Knjxn‖2-2〈Knjxn,Jxn〉+‖xn‖2,
taking liminfn→∞ on the both sides of equality above, we have
(3.32)0&≥‖h‖2-2〈h,Jp〉+‖p‖2=ϕ(h,p).
That is, h=p,for all j=1,2,3,…,m; it follow that
(3.33)Knjxn⇀p;
from (3.30), (3.33), and the Kadec-Klee property, it follows that
(3.34)lim⁡n→∞Knjxn=p,∀j=1,2,3,…,m.
By using triangle inequality, we have
(3.35)‖xn-Knjxn‖≤‖xn-p‖+‖p-Knjun‖.
Since xn,Knjxn→p as n→∞, we have
(3.36)lim⁡n→∞‖xn-Knjxn‖=0,∀j=1,2,3,…,m.
Again by using triangle inequality, we have
(3.37)‖Knjxn-Knj-1xn‖≤‖Knjxn-xn‖+‖xn-Knj-1xn‖.
From (3.36), we also have
(3.38)lim⁡n→∞‖Knjxn-Knj-1xn‖=0,∀j=1,2,3,…,m.
Since J is uniformly norm-to-norm continuous, we obtain
(3.39)lim⁡n→∞‖JKnjxn-JKnj-1xn‖=0,∀j=1,2,3,…,m.
From rj,n>0, we have ∥JKnjxn-JKnj-1xn∥/rj,n→0 as n→∞for all j=1,2,3,…,m, and
(3.40)θj(Knjyn,y)+1rj,n〈y-Knjxn,JKnjxn-JKnj-1xn〉≥0,∀y∈C.
By (A2), that
(3.41)‖y-Knjyn‖‖JKnjyn-JKnj-1xn‖rn≥1rj,n〈y-Knjxn,JKnjyn-JKnj-1xn〉≥-θj(Knjxn,y)≥θj(y,Knjxn),∀y∈C,
and Knjxn→p as n→∞, we get θj(y,p)≤0, for all y∈C. For 0<t<1, define yt=ty+(1-t)p, then yt∈C which imply that θj(yt,p)≤0. From (A1), we obtain that
(3.42)0=θj(yt,yt)≤tθj(yt,y)+(1-t)θj(yt,p)≤tθj(yt,y).
We have that θj(yt,y)≥0. From (A3), we have θj(p,y)≥0, for all y∈C and j=1,2,3,…,m. That is, p∈EP(θj), for allj=1,2,3,…,m. This imply that p∈∩j=1mEP(θj).
(b) We show thatp∈F(S).
Sincexn+1=ΠCn+1fx1∈Cn+1⊂Cn and the definition of Cn+1, we have
(3.43)G(xn+1,Jzn)≤αnG(xn+1,Jx1)+(1-αn)G(xn+1,Jxn)+ζn
is equivalent to
(3.44)ϕ(xn+1,zn)≤αnϕ(xn+1,x1)+(1-αn)ϕ(xn+1,xn)+ζn.
Following (3.11), (3.15), and (3.17), we get that
(3.45)lim⁡n→∞ϕ(xn+1,zn)=0.
From (2.7), we also have
(3.46)‖zn‖⟶‖p‖asn⟶∞.
It follows that
(3.47)‖Jzn‖⟶‖Jp‖asn⟶∞.
This implies that {∥Jzn∥} is bounded in E*. Since E is reflexive and E* is also reflexive, we can assume that Jzn⇀z*∈E*. In view of the reflexive of E, we see that J(E)=E*. There exists z∈E such that Jz=z*. It follows that
(3.48)ϕ(xn+1,zn)=‖xn+1‖2-2〈xn+1,Jzn〉+‖zn‖2=‖xn+1‖2-2〈xn+1,Jzn〉+‖Jzn‖2.
Taking liminfn→∞ on both sides of the equality above and in view of the weak lower semicontinuity of norm ∥·∥, it yields that
(3.49)0≥‖p‖2-2〈p,z*〉+‖z*‖2=‖p‖2-2〈p,Jz〉+‖Jz‖2=‖p‖2-2〈p,Jz〉+‖z‖2=ϕ(p,z);
That is p=z, which implies that z*=Jp. It follows that Jzn⇀Jp∈E*.From (3.47) and the Kadec-Klee property of E* we have Jzn→Jp as n→∞. Since J-1:E*→E is norm-weak*-continuous,zn⇀p as n→∞. From (3.46) and the Kadec-Klee property of E, we have
(3.50)lim⁡n→∞zn=p.
Since {xn} is bounded, then a mapping S is also bounded. From the condition limn→∞αn=0, we have that
(3.51)‖Jzn-JSnun‖=lim⁡n→∞αn‖Jx1-JSnun‖=0.
From (3.47), we get
(3.52)‖JSnun‖⟶‖Jp‖asn⟶∞.
Since J-1:E*→E is norm-weak*-continuous,
(3.53)Snun⇀pasn⟶∞.
On the other hand, we observe that
(3.54)|‖Snun‖-‖p‖|=‖J(Snun)‖-‖Jp‖≤‖J(Snun)-Jp‖.
In view of (3.52), we obtain ∥Snun∥→∥p∥. Since E has the Kadee-Klee property, we get
(3.55)Snun⟶pforeachn∈N.
From Snun→p, we get Sn+1un→p; that is, SSnun→p. In view of closeness of S, we have Sp=p. This implies that p∈F(S). From (a) and (b), it follows that p∈∩j=1mEP(θj)∩F(S).Step 4.
We will show thatp=Πℱfx1.
Sinceℱ is closed and convex set from Lemma 2.9, we have Πℱfx1 which is single valued, denoted by υ. By definition xn=ΠCnfx1 and v∈ℱ⊂Cn, we also have
(3.56)G(xn,Jx1)≤G(υ,Jx1),∀n≥1.
By the definition of G and f, we know that, for each given x,G(ξ,Jx) is convex and lower semicontinuous with respect to ξ. So
(3.57)G(p,Jx1)≤liminf⁡n→∞G(xn,Jx1)≤limsup⁡n→∞G(xn,Jx1)≤G(υ,Jx1).
From the definition of Πℱfx1 and since p∈ℱ, we conclude that υ=p=Πℱfx1 and xn→p as n→∞. The proof is completed.Settingνn≡0 and μn≡0 in Theorem 3.1, then we have the following corollary.Corollary 3.2.
LetC be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. For each j=1,2,…,m, let θj be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A4). Let S:C→C be a closed and quasi-ϕ-asymptotically nonexpansive mappings, and let f:E→ℝ be a convex and lower semicontinuous function with C⊂int(D(f)) such that f(x)≥0 for all x∈C and f(0)=0. Assume that ℱ=F(S)∩(∩j=1mEP(θj))≠∅. For an initial point x1∈E and C1=C, we define the sequence {xn} by
(3.58)un=Trm,nθmTrm-1,nθm-1Trm-2,nθm-2⋯Tr1,nθ1xn,zn=J-1(αnJx1+(1-αn)JSnun),Cn+1={v∈Cn:G(v,Jzn)≤G(v,Jun)≤G(v,Jx1)+(1-αn)G(v,Jxn)+ζn},xn+1=ΠCn+1fx1,n∈ℕ,
where {αn} is a sequence in [0,1], ζn=νnsupq∈ℱψ(G(q,xn))+μn, and {rj,n}⊂[d,∞) for some d>0. If limn→∞αn=0, then {xn} converges strongly to Πℱfx1.LetE be a real Banach space, and let C be a nonempty closed convex subset of E. Given a mapping A:C→E*, let θ(x,y)=〈Ax,y-x〉 for all x,y∈C. Then x*∈EP(θ) if and only if 〈Ax*,y-x*〉≥0 for all y∈C; that is, x* is a solution of the classical variational inequality problem. The set of this solution is denoted by VI(A,C). For each r>0 and x∈E, we define the mapping Trθx by
(3.59)Trθx={z∈C:〈Az,y-z〉+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.
Hence, we obtain the following corollary.Corollary 3.3.
LetC be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. For each j=1,2,…,m, let{Aj} be a continuous monotone mapping of C into E*. Let S:C→C be a closed totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences νn, μn with νn→0, μn→0 as n→∞ and a strictly increasing continuous function ψ:ℝ+→ℝ+ with ψ(0)=0, and let f:E→ℝ be a convex and lower semicontinuous function with C⊂int(D(f)) such that f(x)≥0 for all x∈C and f(0)=0. Assume that ℱ=F(S)∩(∩j=1mVI(Aj,C))≠∅. For an initial point x1∈E and C1=C, one defines the sequence {xn} by
(3.60)un=Trm,nθmTrm-1,nθm-1Trm-2,nθm-2⋯Tr1,nθ1xn,zn=J-1(αnJx1+(1-αn)JSnun),Cn+1={v∈Cn:G(v,Jzn)≤G(v,Jun)≤G(v,Jx1)+(1-αn)G(v,Jxn)+ζn},xn+1=ΠCn+1fx1,n∈ℕ,
where ζn=νnsupq∈ℱψ(G(q,xn))+μn, {αn} is a sequence in [0,1], and {rj,n}⊂[d,∞) for some d>0. If limn→∞αn=0, then {xn} converges strongly to Πℱfx1.Iff(x)=0 for all x∈E, we have G(ξ,Jx)=ϕ(ξ,x) and ΠCfx=ΠCx. From Theorem 3.1, we obtain the following corollary.Corollary 3.4.
LetC be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. For each j=1,2,…,m, let θj be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A4). Let S:C→C be a closed totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences νn, μn with νn→0, μn→0 as n→∞ and a strictly increasing continuous function ψ:ℝ+→ℝ+ with ψ(0)=0. Assume that ℱ=F(S)∩(∩j=1mEP(θj))≠∅. For an initial point x1∈E and C1=C, we define the sequence {xn} by
(3.61)un=Trm,nθmTrm-1,nθm-1Trm-2,nθm-2⋯Tr1,nθ1xn,zn=J-1(αnJx1+(1-αn)JSnun),Cn+1={v∈Cn:G(v,Jzn)≤G(v,Jun)≤G(v,Jx1)+(1-αn)G(v,Jxn)+ζn},xn+1=ΠCn+1x1,n∈ℕ,
where {αn} is a sequence in [0,1], ζn=νnsupq∈ℱψ(G(q,xn))+μn, and {rj,n}⊂[d,∞) for some d>0. If limn→∞αn=0, then {xn} converges strongly to Πℱx1.Remark 3.5.
Our main result extends and improves the result of Chang et al. [13] in the following sense. (i)
From the algorithm we used new method replace by the generalizedf-projection method which is more general than generalized projection.(ii)
For the problem, we extend the result to a common problem of fixed point problems and equilibrium problems.



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*Source: 750732-2012-06-27.xml*