# An Iterative Shrinking Projection Method for Solving Fixed Point Problems of Closed andϕ-Quasi-Strict Pseudocontractions along with Generalized Mixed Equilibrium Problems in Banach Spaces

Authors: Kasamsuk Ungchittrakool
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We provide some new type of mappings associated with pseudocontractions by introducing some actual examples in smooth and strictly convex Banach spaces. Moreover, we also find the significant inequality related to the mappings mentioned in the paper and the mappings defined from generalized mixed equilibrium problems on Banach spaces. We propose an iterative shrinking projection method for finding a common solution of generalized mixed equilibrium problems and fixed point problems of closed andϕ-quasi-strict pseudo-contractions. Our results hold in reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this paper improve and extend the corresponding results of Zhou and Gao (2010) and many others.


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## Body

## 1. Introduction

It is well known that, in an infinite-dimensional Hilbert space, the normal Mann’s iterative algorithm [1] has only weak convergence, in general, even for nonexpansive mappings [2, 3]. Consequently, in order to obtain strong convergence, Nakajo and Takahashi [4] modified the normal Mann’s iteration algorithm, later well known as a hybrid projection iteration method. Since then, the hybrid method has undergone rapid developments. For the details, the readers are referred to papers [5–9] and the references therein.LetE be a smooth Banach space, and let E* be the dual of E. The function ϕ:E×E→ℝ is defined by
(1.1)ϕ(y,x)=‖y‖2-2〈y,Jx〉+‖x‖2
for all x,y∈E, which was studied by Alber [10], Kamimura and Takahashi [11], and Reich [12], where J is the normalized duality mapping from E to 2E* defined by
(1.2)J(x)={f∈E*:〈x,f〉=‖x‖2=‖f‖2},
where 〈·,·〉 denotes the duality paring. It is well known that if E is smooth, then J is single valued and if E is strictly convex, then J is injective (one to one). Let C be a closed convex subset of E, and let T be a mapping from C into itself. We denote by F(T) the set of fixed points of T. A point p in C is said to be an asymptotic fixed point of T [12] if C contains a sequence {xn}, which converges weakly to p such that the strong limn→∞(xn-Txn)=0. The set of asymptotic fixed points of  T will be denoted by F^(T). A mapping T from C into itself is called nonexpansive if ∥Tx-Ty∥⩽∥x-y∥ for all x,y∈C and relatively nonexpansive [13–16] if F^(T)=F(T) and ϕ(p,Tx)⩽ϕ(p,x) for all x∈C and p∈F(T). The asymptotic behavior of relatively nonexpansive mapping was studied in [13–16].In 2005, Matsushita and Takahashi [16] proposed the hybrid iteration method with generalized projection for relatively nonexpansive mapping T in the framework of uniformly smooth and uniformly convex Banach spaces E as follows:
(1.3)x0∈Cchosen  arbitrarily,yn=J-1(αnJxn+(1-αn)JTxn),Cn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qn(x0),
where J is the duality mapping on E and ΠC(·) is the generalized projection from E onto a nonempty closed convex subset C. Based on the guidelines of Matsushita and Takahashi [16], Plubtieng and Ungchittrakool [17, 18] studied and developed (1.3) to the case of two relatively nonexpansive mappings and finite family of relatively nonexpansive mappings, respectively.On the other hand, for a real Banach spaceE and the dual E*, let C be a nonempty closed convex subset of E. Let Θ:C×C→ℝ a bifunction, φ:C→ℝ a real-valued function, and A:C→E* be a nonlinear mapping. The generalized mixed equilibrium problem is to find x∈C such that
(1.4)Θ(x,y)+〈Ax,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
The solution set of (1.4) is denoted by GMEP(Θ,A,φ), that is,
(1.5)GMEP⁡(Θ,A,φ)={x∈C:Θ(x,y)+〈Ax,y-x〉+φ(y)-φ(x)≥0,∀y∈C}.
If A=0, the problem (1.4) reduces to the mixed equilibrium problem for Θ, denoted by MEP(Θ,φ), which is to find x∈C such that
(1.6)Θ(x,y)+φ(y)-φ(x)≥0,∀y∈C.
If Θ=0, the problem (1.4) reduces to the mixed variational inequality of Browder type, denoted by VI(C,A,φ), which is to find x∈C such that
(1.7)〈Ax,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
If A=0 and φ=0, the problem (1.4) reduces to the equilibrium problem for Θ (for short, EP), denoted by EP(Θ), which is to find x∈C such that
(1.8)Θ(x,y)≥0,∀y∈C.
If Θ=0 and A=0, the problem (1.4) reduces to the minimization problem for φ, denoted by Argmin(φ), which is to find x∈C such that
(1.9)φ(x)≤φ(y),∀y∈C.The above formulation (1.7) was shown in [19] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of (1.8). However, (1.4) is very general, it covers the problems mentioned above as special cases.In 2007, S. Takahashi and W. Takahashi [20] and Tada and Takahashi [21, 22] proved weak and strong convergence theorems for finding a common element of the set of solution of an equilibrium problem (1.8) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Takahashi et al. [9] studied a strong convergence theorem by the hybrid method for a family of nonexpansive mappings in Hilbert spaces as follows: x0∈H, C1=C and x1=PC1x0, and let
(1.10)yn=αnxn+(1-αn)Tnxn,Cn+1={z∈Cn:‖yn-z‖≤‖xn-z‖},xn+1=PCn+1x0,n∈N,
where 0⩽αn⩽a<1 for all n∈ℕ and {Tn} is a sequence of nonexpansive mappings of C into itself such that ⋂n=1∞‍F(Tn)≠∅. They proved that if {Tn} satisfies some appropriate conditions, then {xn} converges strongly to P⋂n=1∞‍F(Tn)x0.Motivated by Takahashi et al. [9], Takahashi and Zembayashi [23] (see also [24]) introduced and proved a hybrid projection algorithm for solving equilibrium problems and fixed point problems of a relatively nonexpansive mapping S in the framework of uniformly smooth and uniformly convex Banach space as follows:
(1.11)x0=x,C0=C,yn=J-1(αnJxn+(1-αn)JSxn),un∈CsuchthatΘ(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x,
where ΠCn+1(·) is the generalized projection from E onto Cn+1. Under some appropriate assumptions on Θ, {αn}, and {rn}, they proved that the sequence {xn} converges strongly to ΠF(S)∩EP(Θ)(x0).In 2010, Zhou and Gao [25] introduced the definition of a quasi-strict pseudocontraction related to the function ϕ and proved a hybrid projection algorithm for finding a fixed point of a closed and quasi-strict pseudocontraction in more general framework than uniformly smooth and uniformly convex Banach spaces as follows:(1.12)x0∈E,chosenarbitrarily,C1=C,x1=ΠC1(x0),un∈CsuchthatCn+1={z∈Cn:ϕ(xn,Txn)≤21-k〈xn-z,Jxn-JTxn〉},xn+1=ΠCn+1(x0),
where ΠCn+1 is the generalized projection from E onto Cn+1. They proved that the sequence {xn} converges strongly to ΠF(T)(x0).Motivated and inspired by the above research work, in this paper, we provide some new type of mappings associated with pseudocontractions by introducing some actual examples in smooth and strictly convex Banach spaces. Furthermore, by employing the inequality that appeared in Lemma2.10 together with (1.12) and some facts of Zhou and Gao [25], we create an iterative shrinking projection method for finding a common solution of generalized mixed equilibrium problems and fixed point problems of closed and ϕ-quasi-strict pseudocontractions in the framework of reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this research improve and extend the corresponding results of Zhou and Gao [25] and many others.


## 2. Preliminaries

In this paper, we denote byE and E* a real Banach space and the dual space of E, respectively. Let C be a nonempty closed convex subset of E. We denote by J the normalized duality mapping from E to 2E* defined by (1.1). Let S(E):={x∈E:∥x∥=1} be the unit sphere of E. Then, a Banach space E is said to be strictly convex∥(x+y)/2∥<1 for all x,y∈S(E) and x≠y. It is also said to be uniformly convex if limn→∞∥xn-yn∥=0 for any two sequences {xn}and{yn} in S(E) such that limn→∞∥(xn+yn)/2∥=1. The Banach space E is said to be smooth provided
(2.1)lim⁡t→0‖x+ty‖-‖x‖t
exists for each x,y∈S(E). In this case, the norm of E is said to be Gâteaux differentiable. The norm of E is said to be Fréchet differentiable if for each x∈S(E), the limit (2.1) 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 (2.1) is attained uniformly for x,y∈S(E).A Banach spaceE is said to have the property (K) (or Kadec-Klee property) if for any sequence {xn}⊂E such that xn→x weakly and ∥xn∥→∥x∥, then ∥xn-x∥→0. For more information concerning property (K) the reader is referred to [26] and references cited there.A mappingT:C→C is said to be closed if for any sequence {xn}⊂C with xn→x and Txn→y, x=y.We also know the following properties (see [27–29] for details):(i)
ifE is smooth (⇔E* is strictly convex), then J is single-valued,(ii)
ifE is strictly convex (⇔E* is smooth), then J is one-to-one (i.e., J(x)∩J(y)=∅ for all x≠y),(iii)
ifE is reflexive (⇔E* is reflexive), then J is surjective,(iv)
ifE* is smooth and reflexive, then J-1:E*→2E is single-valued and demicontinuous (i.e., if {xn*}⊂E* such that xn*→x*, then J-1(xn*)→J-1(x*)),(v)
E is uniformly smooth if and only if E* is uniformly convex,(vi)
ifE is uniformly convex, then
(a)
it is strictly convex,(b)
it is reflexive,(c)
it satisfies the property(K),(vii)
ifE is a Hilbert space, then J is the identity operator.It is also very well known that ifC is a nonempty closed convex subset of a Hilbert space H and PC:H→C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces, and, consequently, it is not available in more general Banach spaces. In this connection, Alber [10] recently introduced a generalized projection operator ΠC in a Banach space E that is an analogue of the metric projection in Hilbert spaces.LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, for any x∈E. 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 to the minimization problem
(2.2)ϕ(x¯,x)=min⁡y∈Cϕ(y,x).
Existence and uniqueness of the operator ΠC follow from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J (see, e.g., [10, 11, 27, 29, 30]). In Hilbert spaces, ϕ(y,x)=∥y-x∥2 and ΠC coincides with the metric projection PC.It is obvious from the definition of functionϕ that
(2.3)(‖y‖-‖x‖)2≤ϕ(y,x)≤(‖y‖+‖x‖)2,∀x,y∈E,(2.4)ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2〈x-z,Jz-Jy〉,∀x,y,z∈E.Remark 2.1.
IfE is a reflexive strictly convex and smooth Banach space, then, for x,y∈E,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 (2.3), we have ∥x∥=∥y∥. This implies 〈x,Jy〉=∥x∥2=∥Jy∥2. From the definitions of J, we have Jx=Jy. That is, x=y; one may consult [27, 29] for the details.
T is said to be a ϕ-quasi-strict pseudocontraction [25, page 230] if there exists a constant k∈[0,1) and F(T)≠∅ such that ϕ(p,Tx)≤ϕ(p,x)+kϕ(x,Tx) for all x∈C and p∈F(T). In particular, T is said to be ϕ-quasi-nonexpansive if k=0 and T is said to be ϕ-quasi-pseudocontractive if k=1.
LetC be a nonempty closed convex subset of a real Banach space E with the dual E*. A mapping A:D(A)⊂E→E* is said to be monotone if, for each x,y∈D(A), the following inequality holds:
(2.5)〈x-y,Ax-Ay〉⩾0.Ais said to be r-inverse strongly monotone if there exists a positive real number r such that
(2.6)〈x-y,Ax-Ay〉≥r‖Ax-Ay‖E*2,∀x,y∈D(A).A is said to be r-quasi inverse strongly monotone if A-1(0)={z∈D(A):Az=0}≠∅ and there exists a positive real number r such that for each u∈A-1(0), the following inequality holds:
(2.7)〈x-u,Ax〉⩾r‖Ax‖E*2,∀x∈D(A).
Without loss of generality we can assume that the constantr∈(0,1/2] since if A is r^-quasi inverse strongly monotone (or r^-inverse strongly monotone), then we can find r∈(0,1/2] such that r^≥r and then r^∥Ax∥E*2≥r∥Ax∥E*2 (or r^∥Ax-Ay∥E*2≥r∥Ax-Ay∥E*2) for all r^>0 (i.e., A is r-quasi inverse strongly monotone).
The following is the example ofϕ-quasi-strict pseudocontractions in the framework of a smooth and strictly convex Banach space E.Example 2.2.
LetE be a smooth and strictly convex Banach space E with dual E* and J a normalized duality mapping. Let α∈[0,1], and define the mapping Aα:E→E by
(2.8)Aα(x)=αx,∀x∈E.
It follows that the mapping JAα:E→E* with the set of zero points is given as
(2.9)(JAα)-1(0)={z∈E:0=JAα(z)=J(αz)=αJz}={{0},if0<α≤1,E,ifα=0.
Thus for each u∈(JAα)-1(0) and r∈(0,1/2] if 0<α≤1, we obtain
(2.10)〈x-u,JAαx〉=〈x-0,JAαx〉=〈x,J(αx)〉=〈x,αJ(x)〉=α‖x‖2≥α2‖x‖2=〈αx,J(αx)〉=〈Aαx,J(Aαx)〉=‖Aαx‖2=‖JAαx‖E*2≥r‖JAαx‖E*2,∀x∈E.
On the other hand, if α=0, we have 〈x-u,JAαx〉=0=r∥0∥E*2=r∥JAαx∥E*2,for  allx∈E. Therefore, JAα is r-quasi inverse strongly monotone. So, we can define the following set:
(2.11)Q={JA~:E→E*∣〈x-u,JA~x〉≥r‖JA~x‖E*2,∀x∈E,u∈(JA~)-1(0)},
where A~:E→E and r∈(0,1/2] (note that JAα∈𝒬≠∅). In the same way, let A^:E→E, and define
(2.12)L={JA^∈Q∣J(x-A^x)=Jx-JA^x,〈x,JA^x〉=〈A^x,Jx〉,∀x∈E}.
We claim thatJAα∈ℒ. Note that JAα∈𝒬, and let us consider
(2.13)J(x-Aαx)=J(x-αx)=(1-α)Jx=Jx-αJx=Jx-J(αx)=Jx-JAαx.
Furthermore, it is also found that
(2.14)〈x,JAαx〉=〈x,J(αx)〉=〈αx,Jx〉=〈Aαx,Jx〉.
This means that JAα∈ℒ≠∅. So, we have the claim.
Now, we pick arbitrarilyJA^∈ℒ and claim that (I-A^):E→E is a ϕ-quasi-strict pseudocontraction (i.e., ∃k∈[0,1) s.t. ϕ(q,(I-A^)x)⩽ϕ(q,x)+kϕ(x,(I-A^)x) for all x∈E and q∈F(I-A^)).
SinceJA^∈ℒ, we have
(2.15)‖A^x‖2=〈A^x,JA^x〉=〈x-(I-A^)x,Jx-(Jx-JA^x)〉=〈x-(I-A^)x,Jx〉-〈x-(I-A^)x,J(I-A^)x〉=‖x‖2-〈(I-A^)x,Jx〉-〈x,J(I-A^)x〉+‖(I-A^)x‖2=(‖x‖2-2〈x,J(I-A^)x〉+‖(I-A^)x‖2)+〈x,J(I-A^)x〉-〈(I-A^)x,Jx〉=ϕ(x,(I-A^)x)+〈x,Jx〉-〈x,JA^x〉-〈x,Jx〉+〈A^x,Jx〉=ϕ(x,(I-A^)x).
Note thatℒ⊂𝒬, and thus (JA^)-1(0)≠∅. It is not difficult to check that (JA^)-1(0)=F(I-A^). Therefore, for each q∈F(I-A^), we have
(2.16)ϕ(q,(I-A^)x)=‖q‖2-2〈q,J(I-A^)x〉+‖(I-A^)x‖2=‖q‖2-2〈q,Jx-JA^x〉+‖(I-A^)x‖2=(‖q‖2-2〈q,Jx〉+‖x‖2)-‖x‖2+2〈q,JA^x〉+‖(I-A^)x‖2=ϕ(q,x)-‖x‖2-2〈x-q,JA^x〉+2〈x,JA^x〉+‖(I-A^)x‖2≤ϕ(q,x)-‖x‖2-2r‖JA^x‖2+2〈x,Jx〉-2〈x,Jx-JA^x〉+‖(I-A^)x‖2=ϕ(q,x)-2r‖A^x‖2+(‖x‖2-2〈x,J(I-A^)x〉+‖(I-A^)x‖2)=ϕ(q,x)-2rϕ(x,(I-A^)x)+ϕ(x,(I-A^)x)=ϕ(q,x)+(1-2r)ϕ(x,(I-A^)x)=ϕ(q,x)+kϕ(x,(I-A^)x),
where k:=(1-2r). So, we obtain the desired result.
Next, we claim that the mapping(I-Aα) as mentioned previously is closed. Let {xn}⊂E such that xn→x; then we have
(2.17)(I-Aα)xn=xn-Aαxn=xn-αxn=(1-α)xn⟶(1-α)xasn⟶∞.
Moreover, (I-Aα)x=x-Aαx=x-αx=(1-α)x. This shows that (I-Aα) is a closed mapping.Remark 2.3.
A relatively nonexpansive mapping is aϕ-quasi-strict pseudocontraction, but the converse may be not true.Example 2.4.
LetE be a reflexive, strictly convex, and smooth Banach space. Let A⊂E×E* be a maximal monotone mapping such that A-1(0) is nonempty. Then, Jr=(J+rA)-1J is a closed and ϕ-quasi-strict pseudocontraction from E onto D(A) with constant k=0 and F(Jr)=A-1(0).Example 2.5.
LetΠC be the generalized projection from a smooth, strictly convex, and reflexive Banach space E onto a nonempty closed convex subset C of E. Then, ΠC is a closed and ϕ-quasi-strict pseudocontraction from E onto C with constant k=0 and F(ΠC)=C.
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,
(2.18)lim⁡t↓0Θ(tz+(1-t)x,y)≤Θ(x,y),(A4)
for eachx∈C, y↦Θ(x,y) is convex and lower semicontinuous.Lemma 2.6 (Blum and Oettli [19]).
Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaceE, and let Θ be a bifunction of C×C into ℝ satisfying (A1)–(A4). Let  r>0  and x∈E. Then, there exists z∈C such that(2.19)Θ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 2.7 (Takahashi and Zembayashi [24]).
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 (A1)–(A4). For r>0 and x∈E, define a mapping Tr:E→C as follows:
(2.20)Trx={z∈C:Θ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}
for all x∈C. Then, the following hold:(i)
Tr is single valued,(ii)
Tr is firmly nonexpansive-type mapping, that is, for any x,y∈H,
(2.21)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉,(iii)
F(Tr)=EP(Θ),(iv)
EP(Θ) is closed and convexLemma 2.8 (Takahashi and Zembayashi [24]).
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(Tr),
(2.22)ϕ(p,Trx)+ϕ(Trx,x)≤ϕ(p,x).Lemma 2.9 (Zhang [31]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Let A:C→E* be a continuous and monotone mapping, φ:C→ℝ a lower semi-continuous and convex function, and Θ a bifunction from C×C to ℝ satisfying (A1)–(A4). For r>0 and x∈E, there exists u∈C such that
(2.23)Θ(u,y)+〈Au,y-u〉+φ(y)-φ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C.
Define a mapping Kr:C→C as follows:
(2.24)Kr(x)={u∈C:Θ(u,y)+〈Au,y-u〉+φ(y)-φ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C}
for all x∈C. Then, the following conclusions hold:(1)
Kr is single valued,(2)
Kr is firmly nonexpansive type, that is, for all x,y∈E,
(2.25)〈Krx-Kry,JKrx-JKry〉≤〈Krx-Kry,Jx-Jy〉,(3)
F(Kr)=GMEP(Θ,A,φ),(4)
GMEP(Θ,A,φ) is closed and convex,(5)
ϕ(p,Krx)+ϕ(Krx,x)≤ϕ(p,x)∀p∈F(Kr),x∈E.The following lemmas are crucial for the proofs of the main results in this paper.Lemma 2.10.
LetE be a reflexive, strictly convex, and smooth Banach space. Assume that C is a nonempty closed convex subset of E. Let T:C→C be a ϕ-quasi-strict pseudocontraction, and let Kr:C→C be as in Lemma 2.9 such that Ω:=F(T)∩GMEP(Θ,A,φ)≠∅. Then
(2.26)ϕ(x,KrTx)+ϕ(KrTx,Tx)≤21-k〈x-p,Jx-JTx〉+2〈x-p,JTx-JKrTx〉
for all x∈C and p∈Ω.Proof.
Letx∈C and p∈Ω. By the ϕ-quasi-strict pseudocontractility of T and (2.4) we have
(2.27)ϕ(p,Tx)≤ϕ(p,x)+kϕ(x,Tx)⇔ϕ(p,x)+ϕ(x,Tx)+2〈p-x,Jx-JTx〉≤ϕ(p,x)+kϕ(x,Tx)⇔ϕ(x,Tx)≤21-k〈x-p,Jx-JTx〉.
It follows from (2.4), Lemma 2.9 (5), and (2.27) that
(2.28)ϕ(p,x)+ϕ(x,KrTx)+2〈p-x,Jx-JKrTx〉=ϕ(p,KrTx)≤ϕ(p,Tx)-ϕ(KrTx,Tx)≤ϕ(p,x)+kϕ(x,Tx)-ϕ(KrTx,Tx)≤ϕ(p,x)+2k1-k〈x-p,Jx-JTx〉-ϕ(KrTx,Tx),
and then
(2.29)ϕ(x,KrTx)+ϕ(KrTx,Tx)≤2k1-k〈x-p,Jx-JTx〉+2〈x-p,Jx-JKrTx〉=2k1-k〈x-p,Jx-JTx〉+2〈x-p,Jx-JTx〉+2〈x-p,JTx-JKrTx〉=(2k1-k+2)〈x-p,Jx-JTx〉+2〈x-p,JTx-JKrTx〉=21-k〈x-p,Jx-JTx〉+2〈x-p,JTx-JKrTx〉.Lemma 2.11 (Alber and Reich [10]).
LetC be a nonempty closed convex subset of a smooth Banach space E, x0∈C, and x∈E. Then, x0=ΠCx if and only if
(2.30)〈x0-y,Jx-Jx0〉≥0,∀y∈C.Lemma 2.12 (Alber [30]).
LetE be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed convex subset of E, and let x∈E. Then
(2.31)ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x),∀y∈C.


## 3. Main Result

Theorem 3.1.
LetE be a reflexive, strictly convex, and smooth Banach space such that E and E* have the property (K). Assume that C is a nonempty closed convex subset of E. Let T:C→C be a closed and ϕ-quasi-strict pseudocontraction, Θ a bifunction from C×C to ℝ satisfying (A1)–(A4), φ:C→ℝ a lower semi-continuous and convex function, and A:C→E* a continuous and monotone mapping such that Ω:=F(T)∩GMEP(Θ,A,φ)≠∅. Define a sequence {xn} in C by the following algorithm:
(3.1)x0∈E,chosenarbitrarily,C1=C,x1=ΠC1(x0),un∈CsuchthatΘ(un,y)+〈Aun,y-un〉+φ(y)-φ(un)+1rn〈y-un,Jun-JTxn〉≥0,∀y∈C,Cn+1={21-kz∈Cn:ϕ(xn,un)+ϕ(un,Txn)≤21-k〈xn-z,Jxn-JTxn〉+2〈xn-z,JTxn-Jun〉},xn+1=ΠCn+1(x0),
where k∈[0,1) and rn>0 for all n∈ℕ with liminfn→∞rn>0. Then, {xn} converges strongly to ΠΩ(x0).Proof.
The proof is divided into seven steps.
Step  1. Show that Ω is closed and convex.
From step 1 of Zhou and Gao [25], F(T) is closed and convex and by Lemma 2.9(4) GMEP(Θ,A,φ) is closed and convex. So, Ω:=F(T)∩GMEP(Θ,A,φ) is closed and convex.
Step  2. Show that Cn is closed and convex for all n≥1.
Forn=1, C1=C is closed and convex. Assume that Ck is closed and convex for some k∈ℕ. For z∈Ck+1, one obtains that
(3.2)ϕ(xk,uk)+ϕ(uk,Txk)≤21-k〈xk-z,Jxk-JTxk〉+2〈xk-z,JTxk-Juk〉.
It is not hard to see that the continuity and linearity of 〈·,Jxk-JTxk〉 and 〈·,JTxk-Juk〉 allow Ck+1 to be closed and convex. Then, for all n≥1, Cn is closed and convex.
Step  3. Show that Ω⊂Cn for all n≥1.
It is obvious thatΩ:=F(T)∩GMEP(Θ,A,φ)⊂C=C1. Suppose that Ω⊂Ck for some k∈ℕ. For any p∈Ω, we have p∈Ck. Notice that xk=ΠCk(x0), and then by Lemma 2.9 we obtain uk=KrkTxk. So, it follows from Lemma 2.10 that
(3.3)ϕ(xk,uk)+ϕ(uk,Txk)≤21-k〈xk-p,Jxk-JTxk〉+2〈xk-p,JTxk-Juk〉.
This means that p∈Ck+1. By mathematical induction, Ω⊂Cn for all n≥1. Therefore, Ω⊂⋂n=1∞‍Cn=:D≠∅.
Step 4. Show that limn→∞ϕ(xn,x0) exists.
Byxn=ΠCnx0 and Lemma 2.12, we have
(3.4)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(w,x0)-ϕ(w,xn)≤ϕ(w,x0),
for each w∈Ω⊂Cn and for all n≥1. Therefore, the sequence {ϕ(xn,x0)} is bounded.
On the other hand, noticing thatxn=ΠCnx0 and xn+1=ΠCn+1x0∈Cn+1⊂Cn, ϕ(xn,x0)=minz∈Cnϕ(z,x0)≤ϕ(xn+1,x0) for all n≥1. Therefore, ϕ(xn,x0) is nondecreasing. It follows that the limit of ϕ(xn,x0) exists.
Step  5. Show that xn→q as n→∞, where q=ΠDx0.
Sincexn+1=ΠCn+1x0∈Cn+1⊂Cn for any positive integer n and by Lemma 2.12, we have
(3.5)ϕ(xn+1,xn)=ϕ(xn+1,ΠCnx0)≤ϕ(xn+1,x0)-ϕ(ΠCnx0,x0)=ϕ(xn+1,x0)-ϕ(xn,x0).
Letting n→∞ in (3.5), one has ϕ(xn+1,xn)→0 as n→∞. Without loss of generality, we can assume that xn→q weakly as n→∞ (passing to a subsequence if necessary). It is easy to show that q∈Cn for all n≥1. Hence q∈⋂n=1∞Cn=D. Noticing that ϕ(xn,x0)≤ϕ(xn+1,x0)≤ϕ(q,x0), we have
(3.6)(q,x0)≤liminf⁡n→∞ϕ(xn,x0)≤limsup⁡n→∞ϕ(xn,x0)≤ϕ(q,x0),
which implies that ϕ(xn,x0)→ϕ(q,x0) as n→∞. Hence ∥xn∥→∥q∥. By the property (K) of E, we have xn→q. From Lemma 2.11, we have
(3.7)〈xn-y,Jx0-Jxn〉≥0,∀y∈D.
Hence
(3.8)〈q-y,Jx0-Jq〉≥0,∀y∈D,
which implies that q=ΠDx0.
Step  6. Show that q∈Ω.
We prove first that{Txn} and {un}={KrnTxn} are bounded. Indeed, taking p∈Ω=F(T)∩GMEP(Θ,A,φ)⊂Cn+1, we have
(3.9)‖p‖2-2‖p‖‖Txn‖+‖Txn‖2=(‖p‖-‖Txn‖)2≤ϕ(p,Txn)≤ϕ(p,xn)+kϕ(xn,Txn)≤ϕ(p,xn)+2k1-k〈xn-p,Jxn-JTxn〉≤ϕ(p,xn)+2k1-k‖xn-p‖‖xn‖+2k1-k‖xn-p‖‖Txn‖.
Then
(3.10)‖Txn‖2≤(ϕ(p,xn)-‖p‖2+2k1-k‖xn-p‖‖xn‖)+(2k1-k‖xn-p‖+2‖p‖)‖Txn‖≤M+K‖Txn‖=M+12(2K‖Txn‖)≤M+12(K2+‖Txn‖2)=M+12K2+12‖Txn‖2,
where M:=sup{ϕ(p,xn)-∥p∥2+(2k/(1-k))∥xn-p∥∥xn∥:n∈ℕ} and K:=sup{(2k/(1-k))∥xn-p∥+2∥p∥:n∈ℕ}. Thus
(3.11)‖Txn‖2≤2M+K2
for all n∈ℕ. Therefore {Txn} is bounded. Note that ϕ(p,KrnTxn)≤ϕ(p,Txn) for all n∈ℕ. Therefore {un}={KrnTxn} is also bounded. From xn+1∈Cn+1, one has
(3.12)ϕ(xn,un)+ϕ(un,Txn)≤21-k〈xn-xn+1,Jxn-JTxn〉+2〈xn-xn+1,JTxn-Jun〉.
By Step  5, we obtain that xn+1-xn→0. Taking limit on both sides of (3.12), we obtain ϕ(xn,un)+ϕ(un,Txn)→0 as n→∞. Noting that 0≤(∥xn∥-∥un∥)2≤ϕ(xn,un), 0≤(∥un∥-∥Txn∥)2≤ϕ(un,Txn), and ∥xn∥→∥q∥, it is implied that
(3.13)‖q‖=lim⁡n→∞‖xn‖=lim⁡n→∞‖un‖=lim⁡n→∞‖Txn‖,
and consequently
(3.14)‖Jq‖=lim⁡n→∞‖Jxn‖=lim⁡n→∞‖Jun‖=lim⁡n→∞‖JTxn‖.
This implies that {J(un)} and {J(Txn)} are bounded. Since E is reflexive, E* is also reflexive. So we can assume that
(3.15)J(un)⟶f∈E*
weakly. On the other hand, in view of the reflexivity of E, one has J(E)=E*, which means that, for f∈E*, there exists xf∈E, such that Jxf=f. It follows that
(3.16)ϕ(xn,un)=‖xn‖2-2〈xn,J(un)〉+‖un‖2=‖xn‖2-2〈xn,J(un)〉+‖J(un)‖2.
Taking liminfn→∞ on both sides of the equality above, we have
(3.17)0≥‖q‖2-2〈q,f〉+‖f‖2=‖q‖2-2〈u0,Jxf〉+‖Jxf‖2=ϕ(q,xf).
Therefore ϕ(q,xf)=0 and consequently q=xf, which implies that f=Jq. Hence
(3.18)J(un)⟶Jq∈E*
weakly. Since ∥J(un)∥→∥Jq∥ and E* has the property (K), we have
(3.19)‖J(un)-Jq‖⟶0.
Noting that J-1:E*→E is demicontinuous, we have
(3.20)un⟶q∈E
weakly. Since ∥un∥→∥q∥ and E has the property (K), we obtain that un→q as n→∞. Similarly, it is not difficult to show that Txn→q as n→∞. From xn→q and the closeness property of T, we have Tq=q.
Next, we want to show thatq∈GMEP(Θ,A,φ). Define G:C×C→ℝ by G(x,y)=Θ(x,y)+〈Ax,y-x〉+φ(y)-φ(x) for all x,y∈C. It is not hard to verify that G satisfies conditions (A1)–(A4). It follows from un=KrnTxn and (A2) that
(3.21)1rn〈y-un,Jun-Jxn〉≥G(y,un),∀y∈C.
By using (A4) and liminfn→∞rn>0, we obtain 0≥G(y,q) for all y∈C. For t∈(0,1] and y∈C, let yt=ty+(1-t)q. So, from (A1) and (A4), we have
(3.22)0=G(yt,yt)=G(yt,ty+(1-t)q)≤tG(yt,y)+(1-t)G(yt,q)≤tG(yt,y).
Dividing by t, we have
(3.23)G(yt,y)≥0,∀y∈C.
From (A3) we have 0≤limt→0G(yt,y)=limt→0G(ty+(1-t)q,y)≤G(q,y) for all y∈C, and hence q∈GMEP(Θ,A,φ). So, q∈F(T)∩GMEP(Θ,A,φ)=Ω.
Step  7. Show that q=ΠΩx0.
It follows from Steps 5 and 6 that(3.24)ϕ(q,x0)≤ϕ(ΠΩx0,x0)≤ϕ(q,x0),
which implies that ϕ(ΠΩx0,x0)=ϕ(q,x0). Hence, q=ΠΩx0. Then {xn} converges strongly to q=ΠΩx0. This completes the proof.IfT is closed ϕ-quasi-nonexpansive, then Theorem 3.1 is reduced to the following corollary.Corollary 3.2.
LetE be a reflexive, strictly convex, and smooth Banach space such that E and E* have the property (K). Assume that C is a nonempty closed convex subset of E. Let T:C→C be a ϕ-closed quasi-nonexpansive mapping, Θ a bifunction from C×C to ℝ satisfying (A1)–(A4), φ:C→ℝ a lower semi-continuous and convex function, and A:C→E* a continuous and monotone mapping such that Ω:=F(T)∩GMEP(Θ,A,φ)≠∅. Define a sequence {xn} in C by the following algorithm:
(3.25)x0∈E,chosenarbitrarily,C1=C,x1=ΠC1(x0),un∈CsuchthatΘ(un,y)+〈Aun,y-un〉+φ(y)-φ(un)+1rn〈y-un,Jun-JTxn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(xn,un)+ϕ(un,Txn)≤2〈xn-z,Jxn-Jun〉},xn+1=ΠCn+1(x0),
where rn>0 for all n∈ℕ with liminfn→∞rn>0. Then {xn} converges strongly to ΠΩ(x0).IfE=H is a Hilbert space, then Theorem 3.1 is reduced to the following corollary.Corollary 3.3.
LetH be a Hilbert space. Assume that C is a nonempty closed convex subset of H. Let T:C→C be a closed and ϕ-quasi-strict pseudocontraction, Θ a bifunction from C×C to ℝ satisfying (A 1)–(A 4), φ:C→ℝ a lower semi-continuous and convex function, and A:C→H a continuous and monotone mapping such that Ω:=F(T)∩GMEP(Θ,A,φ)≠∅. Define a sequence {xn} in C by the following algorithm:
(3.26)x0∈H,chosenarbitrarily,C1=C,x1=ΠC1(x0),un∈Csuchthat,Θ(un,y)+〈Aun,y-un〉+φ(y)-φ(un)+1rn〈y-un,un-Txn〉≥0,∀y∈C,Cn+1={21-kz∈Cn:‖xn-un‖2+‖un-Txn‖2≤21-k〈xn-z,xn-Txn〉+2〈xn-z,Txn-un〉},xn+1=ΠCn+1(x0),
where k∈[0,1) and rn>0 for all n∈ℕ with liminfn→∞rn>0. Then {xn} converges strongly to ΠΩ(x0).IfA=0 and φ=0, then we have the following corollary.Corollary 3.4.
LetE be a reflexive, strictly convex, and smooth Banach space such that E and E* have the property (K). Assume that C is a nonempty closed convex subset of E. Let T:C→C be a closed and ϕ-quasi-strict pseudocontraction and Θ a bifunction from C×C to ℝ satisfying (A 1)–(A 4) such that Λ:=F(T)∩EP(Θ)≠∅. Define a sequence {xn} in C by the following algorithm:
(3.27)x0∈E,chosenarbitrarily,C1=C,x1=ΠC1(x0),un∈CsuchthatΘ(un,y)+1rn〈y-un,Jun-JTxn〉≥0∀y∈C,Cn+1={21-kz∈Cn:ϕ(xn,un)+ϕ(un,Txn)≤21-k〈xn-z,Jxn-JTxn〉+2〈xn-z,JTxn-Jun〉},xn+1=ΠCn+1(x0),
where k∈[0,1) and rn>0 for all n∈ℕ with liminfn→∞rn>0. Then {xn} converges strongly to ΠΩ(x0).Corollary 3.5 ([25, Theorem 3.1]).
LetE be a reflexive, strictly convex and smooth Banach space such that E and E* have the property (K). Assume that C is a nonempty closed convex subset of E. Let T:C→C be a ϕ-closed quasi-strict pseudocontraction. Define a sequence {xn} in C by the following algorithm:
(3.28)x0∈E,chosen  arbitrarily,C1=C,x1=ΠC1(x0),un∈Csuch  thatCn+1={z∈Cn:ϕ(xn,Txn)≤21-k〈xn-z,Jxn-JTxn〉},xn+1=ΠCn+1(x0)
where k∈[0,1). Then {xn} converges strongly to ΠΩ(x0).Proof.
PutΘ=0, A=0, φ=0, and rn=1 for all n≥1 in Theorem 3.1. Then, Krn=ΠC for all n⩾1. So, un=ΠCTxn for all n⩾1 (note that x1=ΠCx0). Since xn=ΠCnx0∈Cn⊂C and then Txn∈C for all n⩾1, we have un=Txn for all n⩾1. Thus ϕ(xn,un)+ϕ(un,Txn)=ϕ(xn,Txn) and JTxn-Jun=0 for all n≥1. For this reason, (1.12) is a special case of (3.1). Applying Theorem 3.1, we have the desired result.Remark 3.6.
It is well known that every uniformly convex and uniformly smooth Banach space satisfies all assumptions of Banach space in Theorem3.1. On the other hand, in general, Musielak-Orlicz space [26] need not be uniformly convex or uniformly smooth, however,  any strictly convex,  reflexive  and smooth  Musielak-Orlicz space [26] satisfies all Banach-space assumptions of Theorem 3.1. It can be written as the following diagram:
For this reason, Theorem3.1 can be viewed as a more general one and can be applied widely in both the fixed point problems and the equilibrium problems.



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*Source: 536283-2012-09-03.xml*