# Weak Convergence Theorems for Strictly Pseudocontractive Mappings and Generalized Mixed Equilibrium Problems

Authors: Jong Soo Jung
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce a new iterative method for finding a common element of the set of fixed points of a strictly pseudocontractive mapping, the set of solutions of a generalized mixed equilibrium problem, and the set of solutions of a variational inequality problem for an inverse-strongly-monotone mapping in Hilbert spaces and then show that the sequence generated by the proposed iterative scheme converges weakly to a common element of the above three sets under suitable control conditions. The results in this paper substantially improve, develop, and complement the previous well-known results in this area.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and induced norm ∥·∥. Let C be a nonempty closed convex subset of H. Let B:C→H be a nonlinear mapping and φ:C→ℝ be a function, and Θ be a bifunction of C×C into ℝ, where ℝ is the set of real numbers.Then, we consider the following generalized mixed equilibrium problem of findingx∈C such that(1.1)Θ(x,y)+〈Bx,y-x〉+φ(y)-φ(x)≥0,∀y∈C,
which was introduced by Peng and Yao [1] recently. The set of solutions of the problem (1.1) is denoted by GMEP(Θ,φ,B). Here some special cases of the problem (1.1) are stated as follows.Ifφ=0, then the problem (1.1) reduced the following generalized equilibrium problem (GEP) of finding x∈C such that(1.2)Θ(x,y)+〈Bx,y-x〉≥0,∀y∈C,
which was studied by S. Takahashi and M. Takahashi [2]. The set of solutions of the problem (1.2) is denoted by GEP(Θ,B).IfB=0, then the problem (1.1) reduces the following mixed equilibrium problem of finding x∈C such that(1.3)Θ(x,y)+φ(y)-φ(x)≥0,∀y∈C,
which was studied by Ceng and Yao [3] (see also [4]). The set of solutions of the problem (1.3) is denoted by MEP(Θ,φ).Ifφ=0 and B=0, then the problem (1.1) reduces the following equilibrium problem of finding x∈C such that(1.4)Θ(x,y)≥0,∀y∈C.
The set of solutions of the problem (1.4) is denoted by EP(Θ).Ifφ=0 and Θ(x,y)=0 for all x,y∈C, the problem (1.1) reduces the following variational inequality problem of finding x∈C such that(1.5)〈Bx,y-x〉≥0,∀y∈C.
The set of solutions of the problem (1.5) is denoted by VI(C,B).The problem (1.1) is very general in the sense that it includes, as special cases, fixed point problems, optimization problems, variational inequality problems, minimax problems, Nash equilibrium problems in noncooperative games, and others; see, for example, [3, 5–7].The class of pseudocontractive mappings is one of the most important classes of mappings among nonlinear mappings. We recall that a mappingS:C→H is said to be k-strictly pseudocontractive if there exists a constant k∈[0,1) such that(1.6)‖Sx-Sy‖2≤‖x-y‖2+k‖(I-S)x-(I-S)y‖2,∀x,y∈C.
Note that the class of k-strictly pseudocontractive mappings includes the class of nonexpansive mappings as a subclass. That is, S is nonexpansive (i.e., ∥Sx-Sy∥≤∥x-y∥,∀x,y∈C) if and only if S is 0-strictly pseudocontractive. The mapping S is also said to be pseudocontractive if k=1 and S is said to be strongly pseudocontractive if there exists a constant λ∈(0,1) such that S-λI is pseudocontractive. Clearly, the class of k-strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudocontractive mappings. Also we remark that the class of strongly pseudocontractive mappings is independent of the class of k-strictly pseudocontractive mappings (see [8, 9]). Recently, many authors have been devoting the studies on the problems of finding fixed points to the class of pseudocontractive mappings; see, for example, [10–15] and the references therein.Recently, in order to study the problems (1.1)–(1.5) coupled with the fixed point problem, many authors have introduced some iterative schemes for finding a common element of the set of the solutions of the problem (1.1)–(1.5) and the set of fixed points of a countable family of nonexpansive mappings and have studied strong convergence of the sequences generated by the proposed schemes; see [1–4, 16–18] and the references therein. Also we refer to [19–21] for the problems (1.1), (1.3), and (1.5) combined to the fixed point problem for nonexpansive semigroups and strictly pseudocontractrive mappings.In this paper, inspired and motivated by [18, 22–27], we introduce a new iterative method for finding a common element of the set of fixed points of a k-strictly pseudocontractive mapping, the set of solutions of a generalized mixed equilibrium problem (1.1), and the set of solutions of the variational inequality problem (1.5) for an inverse-strongly monotone mapping in a Hilbert space. We show that, under suitable conditions, the sequence generated by the proposed iterative scheme converges weakly to a common element of the above three sets. The results in this paper can be viewed as an improvement and complement of the recent results in this direction.


## 2. Preliminaries and Lemmas

LetH be a real Hilbert space and let C be a nonempty closed convex subset of H. In the following, we write xn⇀x to indicate that the sequence {xn} converges weakly to x. xn→x implies that {xn} converges strongly to x. We denote by F(T) the set of fixed points of the mapping T.In a real Hilbert spaceH, we have(2.1)‖λx+(1-λ)y‖2=λ‖x‖2+(1-λ)‖y‖2-λ(1-λ)‖x-y‖2
for all x,y∈H and λ∈ℝ. For every point x∈H, there exists a unique nearest point in C, denoted by PC(x), such that(2.2)‖x-PC(x)‖≤‖x-y‖
for all y∈C. PC is called the metric projection of H onto C. It is well known that PC is nonexpansive and PC satisfies(2.3)〈x-y,PC(x)-PC(y)〉≥‖PC(x)-PC(y)‖2
for every x,y∈H. Moreover, PC(x) is characterized by the properties:(2.4)‖x-y‖2≥‖x-PC(x)‖2+‖y-PC(x)‖2,u=PC(x)⟺〈x-u,u-y〉≥0∀x∈H,y∈C.
In the context of the variational inequality problem for a nonlinear mapping F, this implies that(2.5)u∈VI⁡(C,F)⟺u=PC(u-λFu)foranyλ>0.
It is also well known that H satisfies the Opial condition, that is, for any sequence {xn} with xn⇀x, the inequality(2.6)liminf⁡n→∞‖xn-x‖<liminf⁡n→∞‖xn-y‖
holds for every y∈H with y≠x.A mappingF of C into H is called α-inverse-strongly monotone if there exists a constant α>0 such that(2.7)〈x-y,Fx-Fy〉≥α‖Fx-Fy‖2,∀x,y∈C.
We know that if F=I-T, where T is a nonexpansive mapping of C into itself and I is the identity mapping of H, then F is 1/2-inverse-strongly monotone and VI(C,F)=F(T). A mapping F of C into H is called strongly monotone if there exists a positive real number η such that(2.8)〈x-y,Fx-Fy〉≥η‖x-y‖2,∀x,y∈C.
In such a case, we say F is η-strongly monotone. If F is η-strongly monotone and κ-Lipschitzian, continuous, that is, ∥Fx-Fy∥≤L∥x-y∥ for all x,y∈C, then F is η/κ2-inverse-strongly monotone. If F is an α-inverse-strongly monotone mapping of C into H, then it is obvious that F is 1/α-Lipschitzian. We also have that for all x,y∈C and λ>0,(2.9)‖(I-λF)x-(I-λF)y‖2=‖(x-y)-λ(Fx-Fy)‖2=‖x-y‖2-2λ〈x-y,Fx-Fy〉+λ2‖Fx-Fy‖2≤‖x-y‖2+λ(λ-2α)‖Fx-Fy‖2.
So, if λ≤2α, then I-λF is a nonexpansive mapping of C into H. The following result for the existence of solutions of the variational inequality problem for inverse-strongly monotone mappings was given in Takahashi and Toyoda [27].Proposition 2.1.
LetC be a bounded closed convex subset of a real Hilbert space and let F be an α-inverse-strongly monotone mapping of C into H. Then, VI(C,F) is nonempty.A set-valued mappingQ:H→2H is called monotone if for all x,y∈H, f∈Qx and g∈Qy imply 〈x-y,f-g〉≥0. A monotone mapping Q:H→2H is maximal if the graph G(Q) of Q is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping Q is maximal if and only if for (x,f)∈H×H, 〈x-y,f-g〉≥0 for every (y,g)∈G(Q) implies f∈Qx. Let F be an inverse-strongly monotone mapping of C into H and let NCv be the normal cone to C at v, that is, NCv={w∈H:〈v-u,w〉≥0,forallu∈C}, and define(2.10)Qv={Fv+NCv,v∈C,∅,v∉C.
Then Q is maximal monotone and 0∈Qv if and only if v∈VI(C,F); see [28, 29].For solving the equilibrium problem for a bifunctionΘ:C×C→ℝ, let us assume that Θ and φ satisfy 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,
(2.11)lim⁡t↓0Θ(tz+(1-t)x,y)≤Θ(x,y),(A4) for eachx∈C,y↦Θ(x,y) is convex and lower semicontinuous,(A5) for eachy∈C, x↦Θ(x,y) is weakly upper semicontiunuous,(B1) for eachx∈H and r>0, there exist a bounded subset Dx⊆C and yx∈C such that for any z∈C∖Dx,
(2.12)Θ(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0,(B2)C is a bounded set.The following lemmas were given in [1, 5].Lemma 2.2 (see [5]).
LetC be a nonempty closed convex subset of H and Θ a bifunction of C×C into ℝ satisfying (A1)–(A4). Let r>0 and x∈H. Then, there exists z∈C such that
(2.13)Θ(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.Lemma 2.3 (see [1]).
LetC be a nonempty closed convex subset of H. Let Θ be a bifunction form C×C to ℝ which satisfies (A1)–(A5) and φ:C→ℝ a proper lower semicontinuous and convex function. For r>0 and x∈H, define a mapping Tr:H→C as follows:
(2.14)Tr(x)={z∈C:Θ(z,y)+φ(y)-φ(z)+1r〈y-z,z-x〉≥0,∀y∈C}
for all z∈H. Assume that either (B1) or (B2) holds. Then, the following hold:
(1)  for eachx∈H,Tr(x)≠∅,
(2)Tr is single-valued,
(3)Tr is firmly nonexpansive, that is, for any x,y∈H,
(2.15)‖Trx-Try‖2≤〈Trx-Try,x-y〉,
(4)F(Tr)=MEP(Θ,φ),
(5)MEP(Θ,φ) is closed and convex.We also need the following lemmas for the proof of our main results.Lemma 2.4 (see [30]).
LetH be a real Hilbert space, let {αn} be a sequence of real numbers such that 0<a≤αn≤b<1 for all n≥1 and let {vn} and {wn} be sequences in H such that, for some c(2.16)limsup⁡n→∞‖vn‖≤c,limsup⁡n→∞‖wn‖≤c,limsup⁡n→∞‖αnvn+(1-αn)wn‖=c.
Then limn→∞∥vn-wn∥=0.Lemma 2.5 (see [27]).
LetC be a nonempty closed convex subset of a real Hilbert spaces H and let {xn} be a sequence in H. If
(2.17)‖xn+1-x‖≤‖xn-x‖,∀x∈C,∀n≥1,
then {PCxn} converges strongly to some z∈C, where PC stands for the metric projection of H onto C.Lemma 2.6 (see [31]).
LetH be a Hilbert space, C a closed convex subset of H. If T is a k-strictly pseudocontractive mapping on C, then the fixed point set F(T) is closed convex, so that the projection PF(T) is well defined.Lemma 2.7 (see [31]).
LetH be a Hilbert space, C a closed convex subset of H, and T:C→H a k-strictly pseudocontractive mapping. Define a mapping S:C→H by Sx=λx+(1-λ)Tx for all x∈C. Then, as λ∈[k,1), S is a nonexpansive mapping such that F(S)=F(T).


## 3. Main Results

In this section, we introduce a new iterative scheme for finding a common point of the set of fixed points of ak-strictly pseudocontractive mapping, the set of solutions of the problem (1.1), and the set of solutions of the problem (1.5) for an inverse-strongly monotone mapping.Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from C×C to ℝ satisfying (A1)–(A5) and φ:C→ℝ a lower semicontinuous and convex function. Let F,B be two α, β-inverse-strongly monotone mappings of C into H, respectively. Let T be a k-strictly pseudocontractive mapping of C into itself for some k∈[0,1) such that Ω1:=F(T)⋂GMEP(Θ,φ,B)⋂VI(C,F)≠∅. Assume that either (B1) or (B2) holds. Let {xn} and {un} be sequences generated by x1∈C and
(3.1)Θ(un,y)+〈Bxn,y-un〉+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=S(αnxn+(1-αn)PC(un-λnFun)),∀n≥1,
where S:C→C is a mapping defined by Sx=kx+(1-k)Tx for x∈C,{αn}⊂[0,1] and {rn}⊂(0,∞). Assume that {αn}⊂[a,b] for some a,b∈(0,1),λn∈[c,d]⊂(0,2α) and rn∈[e,f]⊂(0,2β). Then {xn} and {un} converge weakly to z∈Ω1, where z=limn→∞PΩ1(xn).Proof.
From now, we putzn=PC(un-λnFun).
We divide the proof into several steps.
Step 1.
We show that{xn} is bounded. To this end, let p∈Ω1:=F(T)⋂GMEP(Θ,φ,B)⋂VI(C,F) and {Trn} be a sequence of mappings defined as in Lemma 2.3. Then, since F(S)=F(T) by Lemma 2.7, p=Sp. Also, from (4) in Lemma 2.3 and (2.5), it follows that p=Trn(p-rnBp) and p=PC(p-λnFp). From zn=PC(un-λnFun) and the fact that PC and I-λnF are nonexpansive, it follows that
(3.2)‖zn-p‖≤‖(I-λnF)un-(I-λnF)p‖≤‖un-p‖.
Also, by un=Trn(xn-rnBxn)∈C and the β-inverse-strongly monotonicity of B, we have with rn∈(0,2β),
(3.3)‖un-p‖2≤‖xn-rnBxn-(p-rnBp)‖2≤‖xn-p‖2-2rn〈xn-p,Bxn-Bp〉+rn2‖Bxn-Bp‖2≤‖xn-p‖2+rn(rn-2β)‖Bxn-Bp‖2≤‖xn-p‖2,
that is, ∥un-p∥≤∥xn-p∥, and so
(3.4)‖zn-p‖≤‖xn-p‖.
So, by using the convexity of ∥·∥2, (3.2) and (3.3), we have
(3.5)‖xn+1-p‖2=‖S(αnxn+(1-αn)zn)-Sp‖2≤‖αn(xn-p)+(1-αn)(zn-p)‖2≤αn‖xn-p‖2+(1-αn)‖zn-p‖2≤αn‖xn-p‖2+(1-αn)(‖xn-p‖2+rn(rn-2β)‖Bxn-Bp‖2)≤‖xn-p‖2+(1-b)e(f-2β)‖Bxn-Bp‖2≤‖xn-p‖2.
So, there exists r∈ℝ such that
(3.6)r=lim⁡n→∞‖xn-p‖.
Therefore, {xn} is bounded, and so are {un} and {zn} by (3.2) and (3.4). Moreover, from (3.5), it follows that
(3.7)(1-b)e(2β-f)‖Bxn-Bp‖2≤‖xn-p‖2-‖xn+1-p‖2,
which implies that
(3.8)lim⁡n→∞‖Bxn-Bp‖=0.Step 2.
We show thatlimn→∞∥xn-un∥=0. To this end, let p∈Ω1. Since Trn is firmly nonexpansive and un=Trn(xn-rnBxn), we have
(3.9)‖un-p‖2=‖Trn(xn-rnBxn)-Trn(p-rnBp)‖2≤〈Trn(xn-rnBxn)-Trn(p-rnBp),xn-rnBxn-(p-rnBp)〉=〈xn-rnBxn-(p-rnBp),un-p〉=12{‖un-p‖2+‖xn-rnBxn-(p-rnBp)‖2}-12{‖xn-rnBxn-(p-rnBp)-(un-p)‖2}≤12{‖un-p‖2+‖xn-p‖2-‖xn-un-rn(Bxn-Bp)‖2}≤12{‖un-p‖2+‖xn-p‖2-‖xn-un‖2+2rn〈Bxn-Bp,xn-un〉-rn2‖Bxn-Bp‖2},
and hence
(3.10)‖un-p‖2≤‖xn-p‖2-‖xn-un‖2+2rn〈Bxn-Bp,xn-un〉-rn2‖Bxn-Bp‖2≤‖xn-p‖2-‖xn-un‖2+2rn〈Bxn-Bp,xn-un〉≤‖xn-p‖2-‖xn-un‖2+2rn‖Bxn-Bp‖‖xn-un‖.
On the other hand, by using the convexity of ∥·∥2, (3.2) and (3.10), we obtain
(3.11)‖xn+1-p‖2=‖S(αnxn+(1-αn)zn)-Sp‖2≤αn‖xn-p‖2+(1-αn)‖zn-p‖2≤αn‖xn-p‖2+(1-αn)‖un-p‖2≤αn‖xn-p‖2+(1-αn)[‖xn-p‖2-‖xn-un‖2+2rn‖Bxn-Bp‖‖xn-un‖],
and hence
(3.12)(1-b)‖xn-un‖2≤(1-αn)‖xn-un‖2≤‖xn-p‖2-‖xn+1-p‖2+2rn‖Bxn-Bp‖‖xn-un‖≤‖xn-p‖2-‖xn+1-p‖2+2f‖Bxn-Bp‖M1,
where M1=sup{∥xn∥+∥un∥:n≥1}. Since limn→∞∥xn+1-p∥2=limn→∞∥xn-p∥2 and limn→∞∥Bxn-Bp∥=0 in (3.8), we obtain
(3.13)lim⁡n→∞‖xn-un‖=0
and so is the limit of {Bxn-Bun} since B is Lipschitz.Step 3.
We show thatlimn→∞∥xn-Szn∥=0. Indeed, let p∈Ω1 and set r=limn→∞∥xn-p∥. Being S nonexpansive and F(T)=F(S), from (3.4) we can write
(3.14)‖Szn-p‖≤‖zn-p‖≤‖xn-p‖,
and hence limsupn→∞∥Szn-p∥≤r. By (3.4), we also have
(3.15)limsup⁡n→∞‖αn(xn-p)+(1-αn)(Szn-p)‖=limsup⁡n→∞[αn‖xn-p‖+(1-αn)‖zn-p‖]≤limsup⁡n→∞‖xn-p‖=lim⁡n→∞‖xn-p‖=r.
By Lemma 2.4, we obtain limn→∞∥Szn-xn∥=0.Step 4.
We show thatlimn→∞∥un-zn∥=0. Using zn=PC(un-λnFun), p=PC(p-λnFp), we compute
(3.16)‖zn-p‖2≤‖(un-λnFun)-(p-λnFp)‖2≤‖un-p‖2-2λn〈un-p,Fun-Fp〉+λn2‖Fun-Fp‖2≤‖xn-p‖2+λn(λn-2α)‖Fun-Fp‖2.
So, we get
(3.17)‖xn+1-p‖2=‖S(αnxn+(1-αn)zn)-Sp‖2≤α‖xn-p‖2+(1-αn)‖zn-p‖2≤αn‖xn-p‖2+(1-αn)‖xn-p‖2+(1-αn)λn(λn-2α)‖Fun-Fp‖2=‖xn-p‖2+(1-αn)λn(λn-2α)‖Fun-Fp‖2.
From conditions αn∈[a,b]⊂(0,1) and λn∈[c,d]⊂(0,2α), it follows that
(3.18)(1-b)c(2α-d)‖Fun-Fp‖2≤(1-αn)λn(2α-λn)‖Fun-Fp‖2≤‖xn-p‖2-‖xn+1-p‖2.
By r=limn→∞∥xn-p∥, we obtain
(3.19)lim⁡n→∞‖Fun-Fp‖=0.
On the other hand, using zn=PC(un-λnFun) and (2.3), we observe that
(3.20)‖zn-p‖2=‖PC(un-λnFun)-PC(p-λnFp)‖2≤〈(un-λnFun)-(p-λnFp),zn-p〉≤12{‖un-p‖2+‖zn-p‖2-‖(un-zn)-λn(Fun-Fp)‖2}≤12{‖xn-p‖2+‖zn-p‖2-‖un-zn‖2+2λn〈un-zn,Fun-Fp〉-λn2‖Fun-Fp‖2},
that is,
(3.21)‖zn-p‖2≤‖xn-p‖2-‖un-zn‖2+2λn〈un-zn,Fun-Fp〉-λn2‖Fun-Fp‖2.
Thus, by (3.21), we have
(3.22)‖xn+1-p‖2=‖S(αnxn+(1-αn)zn)-Sp‖2≤αn‖xn-p‖2+(1-αn)‖zn-p‖2≤αn‖xn-p‖2+(1-αn)‖xn-p‖2-(1-αn)‖un-zn‖2+(1-αn)2λn〈un-zn,Fun-Fp〉-λn2‖Fun-Fp‖2,
which implies that
(3.23)(1-b)‖un-zn‖2≤(1-αn)‖un-zn‖2≤‖xn-p‖2-‖xn+1-p‖2+2d(1-a)M2‖Fun-Fp‖,
where M2=sup{∥zn∥+∥un∥:n≥1}. From limn→∞∥Fun-Fp∥=0 in (3.19) and r=limn→∞∥xn-p∥, we conclude that limn→∞∥un-zn∥=0.Step 5.
We show thatlimn→∞∥xn-Sxn∥=0. Indeed, since
(3.24)‖Sxn-xn‖≤‖Sxn-Sun‖+‖Sun-Szn‖+‖Szn-xn‖,
by Step 2, Step 3, and Step 4, we have
(3.25)lim⁡n→∞‖xn-Sxn‖=0.Step 6.
We show that any of its weak cluster pointz of {xn} belongs in Ω1. In this case, there exists a subsequence {xni} which converges weakly to z. By Step 2 and Step 4, without loss of generality, we may assume that {zni} converges weakly to z∈C. Since
(3.26)‖Szn-zn‖≤‖Szn-xn‖+‖xn-un‖+‖un-zn‖,
from Step 2, Step 3, and Step 4, it follows that ∥Szn-zn∥→0 and Szni⇀z.
We will show thatz∈Ω1. First we show that z∈F(S)=F(T). Assume that z∉F(S). Since zni⇀z and Sz≠z, by the Opial condition, we obtain
(3.27)liminf⁡i→∞‖zni-z‖<liminf⁡i→∞‖zni-Sz‖≤liminf⁡i→∞(‖zni-Szni‖+‖Szni-Sz‖)≤liminf⁡i→∞‖zni-z‖,
which is a contradiction. Thus we have z∈F(S)=F(T).
Next we prove thatz∈VI(C,F). Let
(3.28)Qv={Fv+NCv,v∈C,∅,v∉C,
where NCv is normal cone to C at v. We have already known that in this case the mapping Q is maximal monotone, and 0∈Qv if and only if v∈VI(C,F). Let (v,w)∈G(Q). Since w-Fv∈NCv and zn∈C, we have
(3.29)〈v-zn,w-Fv〉≥0.
On the other hand, from zn=PC(un-λnFun), we have
(3.30)〈v-zn,zn-(un-λnFun)〉≥0,
that is,
(3.31)〈v-zn,zn-unλn+Fun〉≥0.
Thus, we obtain
(3.32)〈v-zni,w〉≥〈v-zni,Fv〉≥〈v-zni,Fv〉-〈v-zni,zni-uniλni+Funi〉=〈v-zni,Fv-Fzni〉+〈v-zni,Fzni-Funi〉-〈v-zni,zni-uniλni〉≥〈v-zni,Fzni-Funi〉-〈v-zni,zni-uniλni〉.
Since ∥zn-un∥→0 in Step 4 and F is α-inverse-strongly monotone, it follows from (3.32) that
(3.33)〈v-z,w〉≥0,asi⟶∞.
Since Q is maximal monotone, we have z∈Q-10 and hence z∈VI(C,F).
Finally, we show thatz∈GMEP(Θ,φ,B). By un=Srn(xn-rnBxn), we know that
(3.34)Θ(un,y)+〈Bxn,y-un〉+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C.
It follows from (A2) that
(3.35)〈Bxn,y-un〉+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥Θ(y,un),∀y∈C.
Hence
(3.36)〈Bxni,y-uni〉+φ(y)-φ(uni)+1rni〈y-uni,uni-xni〉≥Θ(y,uni),∀y∈C.
For t with 0<t≤1 and y∈C, let yt=ty+(1-t)z. Since y∈C and z∈C, we have yt∈C and hence Θ(yt,z)≤0. So, from (3.36), we have
(3.37)〈yt-uni,Byt〉≥〈yt-uni,Byt〉-φ(yt)+φ(uni)-〈yt-uni,Bxni〉-〈yt-uni,uni-xnirni〉+Θ(yt,uni)=〈yt-uni,Byt-Buni〉+〈yt-uni,Buni-Bxni〉-φ(yt)+φ(uni)-〈yt-uni,uni-xnirni〉+Θ(yt,uni).
Since ∥uni-xni∥→0 by Step 2, we have ∥Buni-Bxni∥→0 and ∥(uni-xni)/rni∥≤∥(uni-xni)/e∥→0, that is, (uni-xni)/rni→0. Also by ∥un-zn∥→0 in Step 4, we have uni⇀z. Moreover, from the inverse-strongly monotonicity of B, we have 〈yt-uni,Byt-Buni〉≥0. So, from (A4) and the weak lower semicontinuity of φ, if follows that
(3.38)〈yt-z,Byt〉≥-φ(yt)+φ(z)+Θ(yt,z)asi⟶∞.
By (A1), (A4), and (3.38), we also obtain
(3.39)0=Θ(yt,yt)+φ(yt)-φ(yt)≤tΘ(yt,y)+(1-t)Θ(yt,z)+tφ(yt)+(1-t)φ(z)-φ(yt)≤t[Θ(yt,y)+φ(y)-φ(yt)]+(1-t)〈yt-z,Byt〉=t[Θ(yt,y)+φ(y)-φ(yt)]+(1-t)t〈y-z,Byt〉,
and hence
(3.40)0≤Θ(yt,y)+φ(y)-φ(yt)+(1-t)〈y-z,Byt〉.
Letting t→0 in (3.40), we have for each y∈C(3.41)Θ(z,y)+〈Bz,y-z〉+φ(y)-φ(z)≥0.
This implies that z∈GMEP(Θ,φ,B). Therefore, we have z∈Ω1.
Let{xnj} be another subsequence of {xn} such that xnj⇀z′. Then, we have z′∈Ω1. If z≠z′, from the Opial condition, we have
(3.42)lim⁡n→∞‖xn-z‖=liminf⁡i→∞‖xni-z‖<liminf⁡i→∞‖xni-z′‖=lim⁡n→∞‖xn-z′‖=liminf⁡j→∞‖xnj-z′‖<liminf⁡j→∞‖xnj-z‖=lim⁡n→∞‖xn-z‖.
This is a contradiction. So, we have z=z′. This implies that
(3.43)xn⇀z∈Ω1.
Also from Step 2, it follows that un⇀z∈Ω1.
Letwn=PΩ1(xn). Since z∈Ω1, we have
(3.44)〈xn-wn,wn-z〉≥0.
Since ∥xn+1-p∥≤∥xn-p∥ for p∈Ω1, by Lemma 2.5, we have that {wn} converges strongly to some z0∈Ω1. Since {xn} converges weakly to z, we have
(3.45)〈z-z0,z0-z〉≥0.
Therefore, we obtain
(3.46)z=z0=lim⁡n→∞PΩ1(xn).
This completes the proof.As direct consequences of Theorem3.1, we also obtain the following new weak convergence theorems for the problems (1.2) and (1.3) and fixed point problem of a strict pseudocontractive mapping.Corollary 3.2.
LetH,C,Θ,B, and F be as in Theorem 3.1. Let T be a k-strictly pseudocontractive mapping of C into itself for some k∈[0,1) such that Ω2:=F(T)⋂GEP(Θ,B)⋂VI(C,F)≠∅. Assume that either (B1) or (B2) holds. Let {xn} and {un} be sequences generated by x1∈C and
(3.47)Θ(un,y)+〈Bxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=S(αnxn+(1-αn)PC(un-λnFun)),∀n≥1,
where S:C→C is a mapping defined by Sx=kx+(1-k)Tx for x∈C,{αn}⊂[0,1] and {rn}⊂(0,∞). Assume that {αn}⊂[a,b] for some a,b∈(0,1),λn∈[c,d]⊂(0,2α) and rn∈[e,f]⊂(0,2β). Then {xn} and {un} converge weakly to z∈Ω2, where z=limn→∞PΩ2(xn).Proof.
Puttingφ≡0 in Theorem 3.1, we obtain the desired result.Corollary 3.3.
LetH,C,Θ, and B be as in Corollary 3.2. Let T be a k-strictly pseudocontractive mapping of C into itself for some k∈[0,1) such that Ω3:=F(T)⋂GEP(Θ,B)≠∅. Assume that either (B1) or (B2) holds. Let {xn} and {un} be sequences generated by x1∈C and
(3.48)Θ(un,y)+〈Bxn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=S(αnxn+(1-αn)un),∀n≥1,
where S:C→C is a mapping defined by Sx=kx+(1-k)Tx for x∈C,{αn}⊂[0,1] and {rn}⊂(0,∞). Assume that {αn}⊂[a,b] for some a,b∈(0,1) and rn∈[e,f]⊂(0,2β). Then {xn} and {un} converge weakly to z∈Ω3, where z=limn→∞PΩ3(xn).Proof.
PuttingF≡0 in Corollary 3.2, we obtain the desired result.Corollary 3.4.
LetH,C,Θ,φ, and F be as in Theorem 3.1. Let T be a k-strictly pseudocontractive mapping of C into itself for some k∈[0,1) such that Ω4:=F(T)⋂MEP(Θ,φ)⋂VI(C,F)≠∅. Assume that either (B1) or (B2) holds. Let {xn} and {un} be sequences generated by x1∈C and
(3.49)Θ(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=S(αnxn+(1-αn)PC(un-λnFun)),∀n≥1,
where S:C→C is a mapping defined by Sx=kx+(1-k)Tx for x∈C,{αn}⊂[0,1] and {rn}⊂(0,∞). Assume that {αn}⊂[a,b] for some a,b∈(0,1),λn∈[c,d]⊂(0,2α) and rn∈[e,f]⊂(0,∞). Then {xn} and {un} converge weakly to z∈Ω4, where z=limn→∞PΩ4(xn).Proof.
PuttingB≡0 in Theorem 3.1, we obtain the desired result.Corollary 3.5.
LetH,C,Θ and φ be as in Theorem 3.1. Let T be a k-strictly pseudocontractive mapping of C into itself for some k∈[0,1) such that Ω5:=F(T)⋂MEP(Θ,φ)≠∅. Assume that either (B1) or (B2) holds. Let {xn} and {un} be sequences generated by x1∈C and
(3.50)Θ(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,xn+1=S(αnxn+(1-αn)un),∀n≥1,
where S:C→C is a mapping defined by Sx=kx+(1-k)Tx for x∈C,{αn}⊂[0,1] and {rn}⊂(0,∞). Assume that {αn}⊂[a,b] for some a,b∈(0,1) and rn∈[e,f]⊂(0,∞). Then {xn} and {un} converge weakly to z∈Ω5, where z=limn→∞PΩ5(xn).Proof.
PuttingF≡0 in Corollary 3.4, we obtain the desired result.Remark 3.6.
(1)
As a new result for a new iterative scheme, Theorem3.1 develops and complements the corresponding results, which were obtained recently by many authors in references and others; for example, see [22–24, 26]. In particular, even though F≡0 in Theorem 3.1, Theorem 3.1 develops and complements Theorem 3.1 of Ceng et al. [22] in the following aspects:
(a)
the iterative scheme (3.1) in Theorem 3.1 is a new one different from those in Theorem 3.1 of [22].(b)
the equilibrium problem in Theorem 3.1 of [22] is extended to the case of generalized mixed equilibrium problem.(2)
We point out that our iterative schemes in Corollaries3.2, 3.3, 3.4 and 3.5 are new ones different from those in the literature (see [22–24, 26] and others in references).



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