# A New Iterative Method for Solving a System of Generalized Mixed Equilibrium Problems for a Countable Family of Generalized Quasi-ϕ-Asymptotically Nonexpansive Mappings

Authors: Wei-Qi Deng
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

By using a specific way of choosing the indexes, we introduce an up-to-date iterative algorithm for approximating common fixed points of a countable family of generalized quasi-ϕ-asymptotically nonexpansive mappings and obtain a strong convergence theorem under some suitable conditions. As application, an iterative solution to a system of generalized mixed equilibrium problems is studied. The results extend those of other authors, in which the involved mappings consist of just finite families.


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## Body

## 1. Introduction

Throughout this paper, we assume thatE is a real Banach space with its dual E*, C is a nonempty closed convex subset of E, and J:E→2E* is the normalized duality mapping defined by
(1)Jx={f∈E*:〈x,f〉=∥x∥2=∥f∥2},∀x∈E.
In the sequel, we use F(T) to denote the set of fixed points of a mapping T.Definition 1.
(1) [1] A mapping T:C→C is said to be generalized quasi-ϕ-asymptotically nonexpansive in the light of [1], if F(T)≠∅, and there exist nonnegative real sequences {νn} and {μn} with νn,μn→0 (as n→∞) such that
(2)ϕ(p,Tnx)≤(1+νn)ϕ(p,x)+μn,∀n≥1,x∈C,p∈F(T),
where ϕ:E×E→ℝ denotes the Lyapunov functional defined by
(3)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,∀x,y∈E.
It is obvious from the definition of ϕ that
(4)(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2.
(2) A mappingT:C→C is said to be uniformly L-Lipschitz continuous, if there exists a constant L>0 such that
(5)∥Tnx-Tny∥≤L∥x-y∥,∀n≥1,x,y∈C.Example 2.
LetC be a unit ball in a real Hilbert space l2, and let T:C→C be a mapping defined by
(6)T(x1,x2,…)=(0,x12,a2x2,a3x3,…),
where {ai} is a sequence in (0,1) satisfying ∏i=2∞‍ai=1/2. It is shown by Goebel and Kirk [2] that
(7)ϕ(p,Tny)≤(1+νn)ϕ(p,y)+μn,∀n≥1,y∈C,p∈F(T),
where ϕ(x,y)=∥x-y∥2, νn=(2∏i=2n‍ai)2-1, for all n≥1, and {μn} is a nonnegative real sequence with μn→0 as n→∞. This shows that the mapping T defined earlier is a generalized quasi-ϕ-asymptotically nonexpansive mapping.
Letθ:C×C→ℝ be a bifunction, ψ:C→ℝ a real valued function, and B:C→E* a nonlinear mapping. The so-called generalized mixed equilibrium problem(GMEP) is to find a u∈C such that
(8)θ(u,y)+〈y-u,Bu〉+ψ(y)-ψ(u)≥0,∀y∈C,
whose set of solutions is denoted by Ω(θ,B,ψ).The equilibrium problem is a unifying model for several problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games, and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems. Many authors have proposed some useful methods to solve the (equilibrium problem) EP, (generalized equilibrium problem) GEP, (mixed equilibrium problem) MEP, and GMEP. Concerning the weak and strong convergence of iterative sequences to approximate a common element of the set of solutions for the GMEP, the set of solutions to variational inequality problems and the set of common fixed points for relatively nonexpansive mappings, quasi-ϕ-nonexpansive mappings, and quasi-ϕ-asymptotically nonexpansive mappings have been studied by many authors in the setting of Hilbert or Banach spaces (e.g., see [3–16] and the references therein).Inspired and motivated by the study mentioned earlier, in this paper, by using a specific way of choosing the indexes, we propose an up-to-date iteration scheme for approximating common fixed points of a countable family of generalized quasi-ϕ-asymptotically nonexpansive mappings and obtain a strong convergence theorem for solving a system of generalized mixed equilibrium problems. The results extend those of the authors, in which the involved mappings consist of just finite families.


## 2. Preliminaries

A Banach spaceE is strictly convex if the following implication holds for x,y∈E:
(9)∥x∥=∥y∥=1,x≠y⟹∥x+y2∥<1.
It is also said to be uniformly convex if for any ϵ>0, there exists a δ>0 such that
(10)∥x∥=∥y∥=1,∥x-y∥≥ϵ⟹∥x+y2∥≤1-δ.
It is known that if E is uniformly convex Banach space, then E is reflexive and strictly convex. A Banach space E is said to be smooth if
(11)limt→0∥x+ty∥-∥x∥t
exists for each x,y∈S(E):={x∈E:∥x∥=1}. In this case, the norm of E is said to be Gâteaux differentiable. The space E is said to have uniformly Gâteaux differentiable norm if for each y∈S(E), the limit (11) is attained uniformly for x∈S(E). The norm of E is said to be Fréchet differentiable if for each x∈S(E), the limit (11) is attained uniformly for y∈S(E). The norm of E is said to be uniformly Fréchet differentiable (and E is said to be uniformly smooth) if the limit (11) is attained uniformly for x,y∈S(E). Note that E (E*, resp.) is uniformly convex ⇔E* (E, resp.) is uniformly smooth.Following Alber [17], the generalized projectionΠC:E→C is defined by
(12)ΠC(x)=arginfy∈Cϕ(y,x),∀x∈E.Lemma 3 (see [17]).
Let E be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Then, the following conclusions hold:(1)
ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y) for all x∈C and y∈E;(2)
ifx∈E and z∈C, then z=ΠCx⇔〈z-y,Jx-Jz〉≥0, for all y∈C;(3)
forx,y∈E, ϕ(x,y)=0 if and only if x=y.Remark 4.
The following basic properties for a Banach spaceE can be found in Cioranescu [18].(i)
IfE is uniformly smooth, then J is uniformly continuous on each bounded subset of E.(ii)
IfE is reflexive and strictly convex, then J-1 is norm-weak continuous.(iii)
IfE is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping J:E→2E* is single valued, one-to-one, and onto.(iv)
A Banach spaceE is uniformly smooth if and only if E* is uniformly convex.(v)
Each uniformly convex Banach spaceE has the Kadec-Klee property; that is, for any sequence {xn}⊂E, if xn⇀x∈E and ∥xn∥→∥x∥, then xn→x, where xn⇀x denotes that {xn} converges weakly to x.Lemma 5 (see [19]).
Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let C be a nonempty closed convex subset of E. Let{xn} and {yn} be two sequences in C such that xn→p and ϕ(xn,yn)→0, where ϕ is the function defined by (3); then, yn→p.Lemma 6 (see [1]).
Let E and C be the same as those in Lemma5. Let T:C→C be a closed and generalized quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences {νn} and {μn}; then, the fixed point set F(T) of T is a closed and convex subset of C.Lemma 7 (see [20]).
LetE be a real uniformly convex Banach space, and let Br(0) be the closed ball of E with center at the origin and radius r>0. Then, there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(13)∥αx+βy∥2≤α∥x∥2+β∥y∥2-αβg(∥x-y∥)
for all x,y,∈Br(0), and α,β∈[0,1] with α+β=1.


## 3. Main Results

Theorem 8.
Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, C a nonempty closed convex subset of E, andTi:C→C, i=1,2,… a countable family of closed and generalized quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences {νn(i)} and {μn(i)} satisfying νn(i)→0 and μn(i)→0 (as n→∞ and for each i≥1), and each Ti is uniformly Li-Lipschitz continuous. Let {αn} be a sequence in [0,ϵ] for some ϵ∈(0,1), and let {βn} be a sequence in [0,1] satisfying 0<liminfn→∞βn(1-βn). Let {xn} be the sequence generated by
(14)x1∈C;C1=C,yn=J-1[αnJxn+(1-αn)Jzn],zn=J-1[βnJxn+(1-βn)JTinmnxn],Cn+1={v∈Cn:ϕ(v,yn)≤ϕ(v,xn)+ξn},xn+1=ΠCn+1x1,∀n≥1,
where ξn:=νmn(in)supp∈Fϕ(p,xn)+μmn(in), ΠCn+1 is the generalized projection of E onto Cn+1, and in and mn satisfy the positive integer equation: n=i+(m-1)m/2, m≥i (m≥i, n=1,2,…); that is, for each n≥1, there exist unique in and mn such that
(15)i1=1,i2=1,i3=2,i4=1,i5=2,i6=3,i7=1,i8=2,…;m1=1,m2=2,m3=2,m4=3,m5=3,m6=3,m7=4,m8=4,….
If F:=⋂i=1∞F(Ti) is nonempty and bounded, then {xn} converges strongly to ΠFx1.Proof.
We divide the proof into several steps.(I)
F and Cn (for all n≥1) both are closed and convex subsets in C.
In fact, it follows from Lemma6 that each F(Ti) is a closed and convex subset of C, so is F. In addition, with C1 (=C) being closed and convex, we may assume that Cn is closed and convex for some n≥2. In view of the definition of ϕ, we have that
(16)Cn+1={v∈C:φ(v)≤a}∩Cn,
where φ(v)=2〈v,Jxn-Jyn〉, and a=∥xn∥2-∥yn∥2+ξn. This shows that Cn+1 is closed and convex.(II)
F is a subset of ⋂n=1∞Cn.
It is obvious thatF⊂C1. Suppose that F⊂Cn for some n≥2. Since E is uniformly smooth, E* is uniformly convex. Then, for any p∈F⊂Cn, we have that
(17)ϕ(p,yn)=ϕ(p,J-1[αnJxn+(1-αn)Jxn])=∥p∥2-2〈p,αnJxn+(1-αn)Jzn〉+∥αnJxn+(1-αn)Jzn∥2≤∥p∥2-2αn〈p,Jxn〉-2(1-αn)〈p,Jzn〉+αn∥xn∥2+(1-αn)∥zn∥2=αnϕ(p,xn)+(1-αn)ϕ(p,zn).
Furthermore, it follows from Lemma 7 that for any p∈F, we have that
(18)ϕ(p,zn)=ϕ(p,J-1[βnJxn+(1-βn)JTinmnxn])=∥p∥2-2〈p,βnJxn+(1-βn)JTinmnxn〉+∥βnJxn+(1-βn)JTinmnxn∥2≤∥p∥2-2βn〈p,Jxn〉-2(1-βn)×〈p,JTinmnxn〉+βn∥xn∥2+(1-βn)∥Tinmnxn∥2-βn(1-βn)g(∥Jxn-JTinmnxn∥)=βnϕ(p,xn)+(1-βn)ϕ(p,Tinmnxn)-βn(1-βn)g(∥Jxn-JTinmnxn∥)≤βnϕ(p,xn)+(1-βn)×[ϕ(p,xn)+νmn(in)supp∈Fϕ(p,xn)+μmn(in)]-βn(1-βn)g(∥Jxn-JTinmnxn∥)≤ϕ(p,xn)+νmn(in)supp∈Fϕ(p,xn)+μmn(in)-βn(1-βn)g(∥Jxn-JTinmnxn∥)=ϕ(p,xn)+ξn-βn(1-βn)g(∥Jxn-JTinmnxn∥).
Substituting (18) into (17) and simplifying it, we have that
(19)ϕ(p,yn)≤ϕ(p,xn)+(1-αn)ξn≤ϕ(p,xn)+ξn.
This implies that p∈Cn+1, and so F⊂Cn+1.(III)
xn→x*∈C as n→∞.
In fact, sincexn=ΠCnx1, from Lemma 3 (2), we have that 〈xn-y,Jx1-Jxn〉≥0, for all y∈Cn. Again, since F⊂⋂n=1∞Cn, we have that 〈xn-p,Jx1-Jxn〉≥0, for all p∈F. It follows from Lemma 3 (1) that for each p∈F and for each n≥1,
(20)ϕ(xn,x1)=ϕ(ΠCnx1,x1)≤ϕ(p,x1)-ϕ(p,xn)≤ϕ(p,x1),
which implies that {ϕ(xn,x1)} is bounded, so is {xn}. Since for all n≥1, xn=ΠCnx1 and xn+1=ΠCn+1x1∈Cn+1⊂Cn, we have ϕ(xn,x1)≤ϕ(xn+1,x1). This implies that {ϕ(xn,x1)} is nondecreasing; hence, the limit
(21)limn→∞ϕ(xn,x1)exists.
Since E is reflexive, there exists a subsequence {xni} of {xn} such that xni⇀x*∈C as i→∞. Since Cn is closed and convex and Cn+1⊂Cn, this implies that Cn is weakly, closed and x*∈Cn for each n≥1. In view of xni=ΠCnix1, we have that
(22)ϕ(xni,x1)≤ϕ(x*,x1),∀i≥1.
Since the norm ∥·∥ is weakly lower semicontinuous, we have that
(23)liminfi→∞ϕ(xni,x1)=liminfi→∞(∥xni∥2-2〈xni,Jx1〉+∥x1∥2)≥∥x*∥2-2〈x*,Jx1〉+∥x1∥2=ϕ(x*,x1),
and so
(24)ϕ(x*,x1)≤liminfi→∞ϕ(xni,x1)≤limsupi→∞ϕ(xni,x1)≤ϕ(x*,x1).
This implies that limi→∞ϕ(xni,x1)=ϕ(x*,x1), and so ∥xni∥→∥x*∥ as i→∞. Since xni⇀x*, by virtue of Kadec-Klee property of E, we obtain that
(25)limi→∞xni=x*.
Since {ϕ(xn,x1)} is convergent, this, together with limi→∞ϕ(xni,x1)=ϕ(x*,x1), shows that limn→∞ϕ(xn,x1)=ϕ(x*,x1). If there exists some subsequence {xnj} of {xn} such that xnj→y as j→∞, then, from Lemma 3 (1), we have that
(26)ϕ(x*,y)=limi,j→∞ϕ(xni,xnj)=limi,j→∞ϕ(xni,ΠCnjx1)≤limi,j→∞(ϕ(xni,x1)-ϕ(ΠCnjx1,x1))=limi,j→∞(ϕ(xni,x1)-ϕ(xnj,x1))=ϕ(x*,x1)-ϕ(x*,x1)=0;
that is, x*=y, and so
(27)limn→∞xn=x*.(IV)
x* is some member of F.
Set𝒦i={k≥1:k=i+(m-1)m/2,m≥i,m∈ℕ} for each i≥1. Note that νmn(ik)=νmk(i) and μmn(ik)=μmk(i) whenever k∈𝒦i for each i≥1. For example, by the definition of 𝒦1, we have that 𝒦1={1,2,4,7,11,16,…}, and i1=i2=i4=i7=i11=i16=···=1. Then, we have that
(28)ξk=νmk(i)supp∈Fϕ(p,xk)+μmk(i),∀k∈𝒦i.
Note that {mk}k∈𝒦i={i,i+1,i+2,…}; that is, mk↑∞ as 𝒦i∋k→∞. It follows from (27) and (28) that
(29)lim𝒦i∋k→∞ξk=0.
Since xn+1∈Cn+1, it follows from (14), (27), and (29) that
(30)ϕ(xk+1,yk)≤ϕ(xk+1,xk)+ξk⟶0
as 𝒦i∋k→∞. Since xk→x* as 𝒦i∋k→∞, it follows from (30) and Lemma 5 that
(31)lim𝒦i∋k→∞yk=x*.
Note that Tikmk=Timk whenever k∈𝒦i for each i≥1. From (17) and (18), for any p∈F, we have that
(32)ϕ(p,yk)≤ϕ(p,xk)+ξk-(1-αk)βk(1-βk)g×(∥Jxk-JTimkxk∥);
that is,
(33)(1-αk)βk(1-βk)g(∥Jxk-JTimkxk∥)≤ϕ(p,xk)+ξk-ϕ(p,yk)⟶0(𝒦i∋k⟶∞).
This, together with assumption conditions imposed on the sequences {αn} and {βn}, shows that lim𝒦i∋k→∞g(∥Jxk-JTimkxk∥)=0. In view of property of g, we have that
(34)lim𝒦i∋k→∞∥Jxk-JTimkxk∥=0.
In addition, Jxk→Jx* implies that lim𝒦i∋k→∞JTimkxk=Jx*. From Remark 4 (ii), it yields that, as 𝒦i∋k→∞,
(35)Timkxk⇀x*,∀i≥1.
Again, since for each i≥1, as 𝒦i∋k→∞,
(36)|∥Timkxk∥-∥x*∥|=|∥JTimkxk∥-∥Jx*∥|≤∥JTimkxk-Jx*∥⟶0.
This, together with (35) and the Kadec-Klee property of E, shows that
(37)lim𝒦i∋k→∞Timkxk=x*,∀i≥1.
On the other hand, by the assumptions that for each i≥1, Ti is uniformly Li-Lipschitz continuous, and noting again that {mk}k∈𝒦i={i,i+1,i+2,…}, that is, mk+1-1=mk for all k∈𝒦i, we then have
(38)∥Timk+1xk-Timkxk∥≤∥Timk+1xk-Timk+1xk+1∥+∥Timk+1xk+1-xk+1∥+∥xk+1-xk∥+∥xk-Timkxk∥≤(Li+1)∥xk+1-xk∥+∥Timk+1xk+1-xk+1∥+∥xk-Timkxk∥.
From (37) and xk→x*, we have that lim𝒦i∋k→∞∥Timk+1xk-Timkxk∥=0, and lim𝒦i∋k→∞Timk+1xk=x*; that is, lim𝒦i∋k→∞Ti(Timk+1-1xk)=x*. It then follows that, for each i≥1,
(39)lim𝒦i∋k→∞Ti(Timkxk)=x*.
In view of the closeness of Ti, it follows from (37) that Tix*=x*, namely, for each i≥1, x*∈F(Ti), and, hence, x*∈F. (V)
x*=ΠFx1, and so xn→ΠFx1 as n→∞.
Putu=ΠFx1. Since u∈F⊂Cn and xn=ΠCnx1, we have that ϕ(xn,x1)≤ϕ(u,x1), for all n≥1. Then,
(40)ϕ(x*,x1)=limn→∞ϕ(xn,x1)≤ϕ(u,x1),
which implies that x*=u since u=ΠFx1, and, hence, xn→x*=ΠFx1.
This completes the proof.


## 4. Applications

LetE be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Let {Bi}i=1∞:C→E* be a sequence of βi-inverse strongly monotone mappings, {ψ}i=1∞:C→ℝ a sequence of lower semicontinuous and convex functions, and {θi}i=1∞:C×C→ℝ a sequence of bifunctions satisfying the following conditions:(A1)
θ(x,x)=0;(A2)
θ is monotone; that is, θ(x,y)+θ(y,x)≤0;(A3)
limsupt↓0θ(x+t(z-x),y)≤θ(x,y);(A4)
the mappingy↦θ(x,y) is convex and lower semicontinuous.A system of generalized mixed equilibrium problems(GMEPs), for {θi}i=1∞, {Bi}i=1∞, and {ψi}i=1∞ is to find an x*∈C such that
(41)θi(x*,y)+〈y-x*,Bix*〉+ψi(y)-ψi(x*)≥0,∀y∈C,i≥1,
whose set of common solutions is denoted by Ω:=∩i=1∞Ωi, where Ωi denotes the set of solutions to generalized mixed equilibrium problem for θi, Bi, and ψi.Define a countable family of mappings{Tr,i}i=1∞:E→C with r>0 as follows:
(42)Tr,i(x)={z∈C:τi(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},∀i≥1,
where τi(x,y)=θi(x,y)+〈y-x,Bix〉+ψi(y)-ψi(x), for all x,y∈C, i≥1. It has been shown by Zhang [15] that (1)
{Tr,i}i=1∞ is a sequence of single-valued mappings;(2)
{Tr,i}i=1∞ is a sequence of closed quasi-ϕ-nonexpansive mappings;(3)
⋂i=1∞F(Tr,i)=Ω.Now, we have the following result.Theorem 9.
LetE be the same as that in Theorem 8, and let C be a nonempty closed convex subset of E. Let {Tr,i}i=1∞:C→C be a sequence of mappings defined by (42) with F:=⋂i=1∞F(Tr,i)≠∅. Let {αn} be a sequence in [0,ϵ] for some ϵ∈(0,1), and let {βn} be a sequence in [0,1] satisfying 0<liminfn→∞βn(1-βn). Let {xn} be the sequence generated by
(43)x1∈C;C1=C,yn=J-1[αnJxn+(1-αn)Jzn],zn=J-1[βnJxn+(1-βn)JTr,inxn],Cn+1={v∈Cn:ϕ(v,yn)≤ϕ(v,xn)},xn+1=ΠCn+1x1,∀n≥1,
where in satisfies the positive integer equation: n=i+(m-1)m/2, and m≥i (m≥i, n=1,2,…). Then, {xn} converges strongly to ΠFx1 which is some solution to the system of generalized mixed equilibrium problems for {Tr,i}i=1∞.Proof.
Note that{Tr,i}i=1∞ are quasi-ϕ-nonexpansive mappings; so, they are obviously generalized quasi-ϕ-asymptotically nonexpansive. Therefore, this conclusion can be obtained immediately from Theorem 8.



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