# Hybrid Gradient-Projection Algorithm for Solving Constrained Convex Minimization Problems with Generalized Mixed Equilibrium Problems

Authors: Lu-Chuan Ceng; Ching-Feng Wen
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

It is well known that the gradient-projection algorithm (GPA) for solving constrained convex minimization problems has been proven to have only weak convergence unless the underlying Hilbert space is finite dimensional. In this paper, we introduce a new hybrid gradient-projection algorithm for solving constrained convex minimization problems with generalized mixed equilibrium problems in a real Hilbert space. It is proven that three sequences generated by this algorithm converge strongly to the unique solution of some variational inequality, which is also a common element of the set of solutions of a constrained convex minimization problem, the set of solutions of a generalized mixed equilibrium problem, and the set of fixed points of a strict pseudocontraction in a real Hilbert space.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let C be a nonempty closed convex subset of H and let PC be the metric projection of H onto C. Recall that a ρ-Lipschitz continuous mapping T:C→H is a mapping on C such that
(1.1)∥Tx-Ty∥≤ρ∥x-y∥,∀x,y∈C,
where ρ≥0 is a constant. In particular, if ρ∈[0,1) then T is called a contraction on C; if ρ=1 then T is called a nonexpansive mapping on C. A mapping A:C→H is called monotone if
(1.2)〈Ax-Ay,x-y〉≥0,∀x,y∈C.
A mapping A:C→H is called α-inverse strongly monotone if there exists a constant α>0 such that
(1.3)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C;
see, for example, [1]. A self-mapping S:C→C is called k-strictly pseudocontractive if there exists a constant k∈[0,1) such that
(1.4)∥Sx-Sy∥2≤∥x-y∥2+k∥(I-S)x-(I-S)y∥2,∀x,y∈C;
see, for example, [2]. In particular, if k=0, then S reduces to a nonexpansive self-mapping on C.Consider the following constrained convex minimization problem:(1.5)minimize{f(x):x∈C},
where f:C→R is a real-valued convex function. If f is (Frechet) differentiable, then the gradient-projection method (for short, GPM) generates a sequence {xn} via the recursive formula
(1.6)xn+1=PC(xn-λ∇f(xn)),∀n≥0,
or more generally,
(1.7)xn+1=PC(xn-λn∇f(xn)),∀n≥0,
where in both (1.6) and (1.7), the initial guess x0 is taken from C arbitrarily, the parameters, λ or λn, are positive real numbers, and PC is the metric projection from H onto C. The convergence of the algorithms (1.6) and (1.7) depends on the behavior of the gradient ∇f. As a matter of fact, it is known that if ∇f is strongly monotone and Lipschitzian; namely, there are constants η,L>0 satisfying the properties
(1.8)〈∇f(x)-∇f(y),x-y〉≥η∥x-y∥2,(1.9)∥∇f(x)-∇f(y)∥≤L∥x-y∥
for all x,y∈C, then, for 0<λ<2η/L2, the operator
(1.10)T:=PC(I-λ∇f)
is a contraction; hence, the sequence {xn} defined by algorithm (1.6) converges in norm to the unique solution of the minimization (1.5). More generally, if the sequence {λn} is chosen to satisfy the property
(1.11)0<liminfn→∞λn≤limsupn→∞λn<2ηL2
then the sequence {xn} defined by algorithm (1.7) converges in norm to the unique minimizer of (1.5). However, if the gradient ∇f fails to be strongly monotone, the operator T defined in (1.10) would fail to be contractive; consequently, the sequence {xn} generated by algorithm (1.6) may fail to converge strongly (see Section 4 in Xu [3]). The following theorem states that if the Lipschitz condition (1.9) holds, then the algorithms (1.6) and (1.7) can still converge in the weak topology.Theorem 1.1 (see [3, Theorem 3.2]).
Assume the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇f satisfies the Lipschitz condition (1.9). Let the sequence of parameters, {λn}, satisfy the condition
(1.12)0<liminfn→∞λn≤limsupn→∞λn<2L.
Then the sequence {xn} generated by the gradient-projection algorithm (1.7) converges weakly to a minimizer of (1.5).From the above theorem, it is known that the gradient-projection algorithm has weak convergence, in general, unless the underlying Hilbert space is finite dimensional. This gives naturally rise to a question how to appropriately modify the gradient-projection algorithm so as to have strong convergence. Xu [3] gave two such modifications, one of which is simply a convex combination of a contraction with the point generated by the projected gradient algorithm.Theorem 1.2 (see [3, Theorem 5.2]).
Assume the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇f satisfies the Lipschitz condition (1.9). Let Q:C→C be a ρ-contraction with ρ∈[0,1). Let a sequence {xn} be generated by the following hybrid gradient-projection algorithm:
(1.13)xn+1=αnQxn+(1-αn)PC(xn-λn∇f(xn)),∀n≥0.
Assume the sequence {λn} satisfies the condition (1.12) and, in addition, the following conditions are satisfied for {λn} and {αn}⊂[0,1]:(i)
αn→0;(ii)
∑n=0∞‍αn=∞;(iii)
∑n=0∞‍|αn+1-αn|<∞;(iv)
∑n=0∞‍|λn+1-λn|<∞.
Then the sequence{xn} converges in norm to a minimizer of (1.5) which is also the unique solution of the variational inequality of finding x*∈Ω such that
(1.14)〈(I-Q)x*,x-x*〉≥0,∀x∈Ω.
In other words, x* is the unique fixed point of the contraction PΩQ, x*=PΩQx*.On the other hand, Peng and Yao [4] recently introduced the following generalized mixed equilibrium problem of finding x¯∈C such that
(1.15)Θ(x¯,y)+φ(y)-φ(x¯)+〈Fx¯,y-x¯〉≥0,∀y∈C,
where F:C→H is a nonlinear mapping and φ:C→R is a function and Θ:C×C→R is a bifunction. The set of solutions of problem (1.15) is denoted by GMEP. Subsequently, Yao et al. [5] and Ceng and Yao [6] also considered problem (1.15).The problem (1.15) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games, and others; see, for example, [7–15]. Here some special cases of problem (1.15) are stated as follows.IfF=0, then problem (1.15) reduces to the following mixed equilibrium problem of finding x¯∈C such that
(1.16)Θ(x¯,y)+φ(y)-φ(x¯)≥0,∀y∈C,
which was considered by Ceng and Yao [7] and Bigi et al. [16]. Very recently, Peng [10] further discussed this problem. The set of solutions of this problem is denoted by MEP.Ifφ=0, then problem (1.15) reduces to the following generalized equilibrium problem of finding x¯∈C such that
(1.17)Θ(x¯,y)+〈Fx¯,y-x¯〉≥0,∀y∈C,
which was studied by S. Takahashi and W. Takahashi [8].Ifφ=0 and F=0, then problem (1.15) reduces to the following equilibrium problem of finding x¯∈C such that
(1.18)Θ(x¯,y)≥0,∀y∈C.IfΘ=0,  φ=0 and F=A, then problem (1.15) reduces to the following classical variational inequality of finding x¯∈C such that
(1.19)〈Ax¯,y-x¯〉≥0,∀y∈C,
whose solution set is denoted by VI(C,A).The variational inequalities have been extensively studied in the literature; see [14, 17–27] and the references therein. In 2006, Nadezhkina and Takahashi [22, 25] and Zeng and Yao [18] proposed some variants of Korpelevič's extragradient method [17] for finding an element of Fix(S)∩VI(C,A), where S is a nonexpansive self-mapping on C.Very recently, Peng [10] also introduced a variant of Korpelevič's extragradient method [17] for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a strict pseudocontraction, and the set of solutions of a variational inequality for a monotone, Lipschitz continuous mapping.Theorem 1.3 (see [10, Theorem 3.1]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ:C×C→R be a bifunction satisfying conditions (H1)–(H4) and φ:C→R a lower semicontinuous and convex function with assumptions (A1) or (A2), where(H1)
Θ(x,x)=0, for all x∈C;(H2)
Θ is monotone, that is, Θ(x,y)+Θ(y,x)≤0, for all x,y∈C;(H3)
for eachy∈C,  x↦Θ(x,y) is weakly upper semicontinuous;(H4)
for eachx∈C,  y↦Θ(x,y) is convex and lower semicontinuous;(A1)
for eachx∈H and r>0, there exists a bounded subset Dx⊂C and yx∈C such that for any z∈C∖Dx,
(1.20)Θ(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0;(A2)
C is a bounded set.
LetA:C→H be a monotone and L-Lipschitz-continuous mapping and S:C→C be a k-strict pseudocontraction for some 0≤k<1 such that Fix(S)∩VI(C,A)∩MEP≠∅. For given x0∈H arbitrarily, let {xn}, {tn}, {yn}, {un}, {zn} be sequences generated by
(1.21)Θ(tn,y)+φ(y)-φ(tn)+1rn〈y-tn,tn-xn〉≥0,∀y∈C,yn=PC(tn-λnAtn),un=PC(tn-λnAyn),zn=αnun+(1-αn)Sun,Cn={z∈C:∥zn-z∥2≤∥xn-z∥2-(1-αn)(αn-ɛ)∥tn-Stn∥2},Qn={z∈H:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,∀n≥0.
Assume that {λn}⊂[a,b] for some a,b∈(0,1/L), {αn}⊂[c,d] for some c,d∈(k,1) and let {rn}⊂(0,∞) satisfy liminfn→∞rn>0. Then, {xn}, {tn}, {yn}, {un}, {zn} converge strongly to w=PFix(S)∩VI(C,A)∩MEPx.Furthermore, related iterative methods for solving fixed point problems, variational inequalities, equilibrium problems, and optimization problems can be found in [1, 2, 6, 11, 13–16, 19, 20, 24, 26–35].In this paper, letC be a nonempty closed convex subset of a real Hilbert space H. Let Θ:C×C→R be a bifunction satisfying conditions (H1)–(H4) and φ:C→R a lower semicontinuous and convex function with assumptions (A1) or (A2). Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Let the gradient ∇f be L-Lipschitzian with constant L>0 and F:C→H be an α-inverse strongly monotone mapping. Let S:C→C be a k-strictly pseudocontractive mapping such that Fix(S)∩Ω∩GMEP≠∅. Let Q:C→C be a ρ-contraction with ρ∈[0,1/2). For given x0∈C arbitrarily, let the sequences {xn}, {yn} and {zn} be generated iteratively by
(1.22)Θ(zn,y)+φ(y)-φ(zn)+〈Fxn,y-zn〉+1rn〈y-zn,zn-xn〉≥0,∀y∈C,yn=αnQxn+(1-αn)PC(zn-λn∇f(zn)),xn+1=βnxn+γnPC(zn-λn∇f(zn))+δnSyn,∀n≥0,
where {λn}⊂(0,2/L],{rn}⊂(0,2α] and {αn}, {βn}, {γn}, {δn} are four sequences in [0,1] such that βn+γn+δn=1 for all n≥0. It is proven that under very mild conditions, the sequences {xn}, {yn} and {zn} converge strongly to the unique solution of the variational inequality of finding x*∈Fix(S)∩Ω∩GMEP such that
(1.23)〈(I-Q)x*,x-x*〉≥0,∀x∈Fix(S)∩Ω∩GMEP.
In other words, x* is the unique fixed point of the contraction PFix(S)∩Ω∩GMEPQ,  x*=PFix(S)∩Ω∩GMEPQx*. The result presented in this paper generalizes and improves some well-known results in the literature. Indeed, compared with some well-known results in the literature, our result improves and extends them in the following aspects.(i)
Compared with Xu [3, Theorem 3.2], a weak convergence result, our result is a strong convergence result.(ii)
Our problem of finding an element ofFix(S)∩Ω∩GMEP is more general than the problem of finding an element of Fix(S)∩VI(C,A) in [14, 18, 22, 23, 25].(iii)
In our algorithm (1.22), Xu's modified gradient-projection algorithm in [3, Theorem 5.2] is rewritten as the second iteration step
(1.24)yn=αnQxn+(1-αn)PC(zn-λn∇f(zn)).Here the main purpose of the reason why we use such an iteration step is to play a convenience and efficiency role in the computation of an element ofΩ. Therefore, Xu's algorithm (1.13) is extended to develop our algorithm (1.22).(iv)
Our problem of finding an element ofFix(S)∩Ω∩GMEP is more general than the problem of finding an element of Ω in Xu [3]. In addition, it is worth pointing out that Xu's conditions ∑n=0∞‍|αn+1-αn|<∞ and ∑n=0∞‍|λn+1-λn|<∞ in the above Theorem 1.2 are replaced by the weaker conditions limn→∞(αn-αn+1)=0 and limn→∞(λn-λn+1)=0 in our result (see Theorem 3.2 in Section 3).


## 2. Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥ and C a nonempty closed convex subset of H. We write → to indicate that the sequence {xn} converges strongly to x and ⇀ to indicate that the sequence {xn} converges weakly to x. Moreover, we use ωw(xn) to denote the weak ω-limit set of the sequence {xn}, that is,
(2.1)ωw(xn):={x:xni⇀xforsomesubsequence{xni}of{xn}}.For every pointx∈H, there exists a unique nearest point in C, denoted by PCx, such that
(2.2)∥x-PCx∥≤∥x-y∥,∀x∈C.PC is called the metric projection of H onto C. We know that PC is a firmly nonexpansive mapping of H onto C; that is, there holds the following relation:
(2.3)〈PCx-PCy,x-y〉≥∥PCx-PCy∥2,∀x,y∈H.
Consequently, PC is nonexpansive and monotone. It is also known that PC is characterized by the following properties: PCx∈C and
(2.4)〈x-PCx,PCx-y〉≥0,(2.5)∥x-y∥2≥∥x-PCx∥2+∥y-PCx∥2,
for all x∈H,  y∈C; see [36] for more details. Let A:C→H be a monotone mapping. In the context of the variational inequality, this implies that
(2.6)x∈VI(C,A)⟺x=PC(x-λAx)∀λ>0.A set-valued mappingT:H→2H is called monotone if for all x,y∈H,  f∈Tx and g∈Ty imply 〈f-g,x-y〉≥0. A monotone mapping T:H→2H is called maximal if its graph G(T) is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x,f)∈H×H, 〈f-g,x-y〉≥0 for every (y,g)∈G(T) implies f∈Tx.LetA:C→H be a monotone, k-Lipschitz-continuous mapping and let NCv be the normal cone to C at v∈C, that is, NCv={w∈H:〈v-u,w〉≥0, for all u∈C}. Define
(2.7)Tv={Av+NCv,ifv∈C,∅,ifv∉C.
Then, T is maximal monotone and 0∈Tv if and only if v∈VI(C,A); see [37].Recall that a mappingS:C→C is called a strict pseudocontraction if there exists a constant 0≤k<1 such that
(2.8)∥Sx-Sy∥2≤∥x-y∥2+k∥(I-S)x-(I-S)y∥2,∀x,y∈C.
In this case, we also say that S is a k-strict pseudocontraction. A mapping A:C→H is called α-inverse strongly monotone if there exists a constant α>0 such that
(2.9)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.
It is obvious that any α-inverse strongly monotone mapping is Lipschitz continuous. Meantime, observe that (2.8) is equivalent to
(2.10)〈Sx-Sy,x-y〉≤∥x-y∥2-1-k2∥(I-S)x-(I-S)y∥2,∀x,y∈C.
It is easy to see that if S is a k-strictly pseudocontractive mapping, then I-S is ((1-k)/2)-inverse strongly monotone and hence (2/(1-k))-Lipschitz continuous. Thus, S is Lipschitz continuous with constant (1+k)/(1-k). We denote by Fix(S) the set of fixed points of S. It is clear that the class of strict pseudocontractions strictly includes the one of nonexpansive mappings which are mappings S:C→C such that ∥Sx-Sy∥≤∥x-y∥ for all x,y∈C.In order to prove our main result in the next section, we need the following lemmas and propositions.Lemma 2.1 (see [7]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Θ:C×C→R be a bifunction satisfying conditions (H1)–(H4) and let φ:C→R be a lower semicontinuous and convex function. For r>0 and x∈H, define a mapping Tr(Θ,φ):H→C as follows:
(2.11)Tr(Θ,φ)(x)={z∈C:Θ(z,y)+φ(y)-φ(z)+1r〈y-z,z-x〉≥0,∀y∈C}
for all x∈H. Assume that either (A1) or (A2) holds. Then the following conclusions hold:(i)
Tr(Θ,φ)(x)≠∅ for each x∈H and Tr(Θ,φ) is single-valued;(ii)
Tr(Θ,φ) is firmly nonexpansive, that is, for any x,y∈H,
(2.12)∥Tr(Θ,φ)x-Tr(Θ,φ)y∥2≤〈Tr(Θ,φ)x-Tr(Θ,φ)y,x-y〉;(iii)
Fix(Tr(Θ,φ))=MEP(Θ,φ);(iv)
MEP(Θ,φ) is closed and convex.Remark 2.2.
Ifφ=0, then Tr(Θ,φ) is rewritten as TrΘ.The following lemma is an immediate consequence of an inner product.Lemma 2.3.
In a real Hilbert spaceH, there holds the inequality
(2.13)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.Proposition 2.4 (see [6, Proposition 2.1]).
LetC, H, Θ, φ, and Tr(Θ,φ) be as in Lemma 2.1. Then the following relation holds:
(2.14)∥Ts(Θ,φ)x-Tt(Θ,φ)x∥2≤s-ts〈Ts(Θ,φ)x-Tt(Θ,φ)x,Ts(Θ,φ)x-x〉
for all s,t>0 and x∈H.Recall thatS:C→C is called a quasi-strict pseudocontraction if the fixed point set of S, Fix(S), is nonempty and if there exists a constant 0≤k<1 such that
(2.15)∥Sx-p∥2≤∥x-p∥2+k∥x-Sx∥2∀x∈C,p∈Fix(S).
We also say that S is a k-quasi-strict pseudocontraction if condition (2.15) holds.Proposition 2.5 (see [2, Proposition 2.1]).
AssumeC is a nonempty closed convex subset of a real Hilbert space H and let S:C→C be a self-mapping on C.(i)
IfS is a k-strict pseudocontraction, then S satisfies the Lipschitz condition
(2.16)∥Sx-Sy∥≤1+k1-k∥x-y∥,∀x,y∈C.(ii)
IfS is a k-strict pseudocontraction, then the mapping I-S is demiclosed (at 0). That is, if {xn} is a sequence in C such that xn⇀x~ and (I-S)xn→0, then (I-S)x~=0, that is, x~∈Fix(S).(iii)
IfS is a k-quasi-strict pseudocontraction, then the fixed point set Fix(S) of S is closed and convex so that the projection PFix(S) is well defined.The following lemma was proved by Suzuki [30].Lemma 2.6 (see [30]).
Let{xn} and {yn} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose xn+1=(1-βn)yn+βnxn for all integers n≥0 and limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥yn-xn∥=0.Lemma 2.7 (see [34]).
Let{an} be a sequence of nonnegative numbers satisfying the condition
(2.17)an+1≤(1-δn)an+δnσn,∀n≥0,
where {δn}, {σn} are sequences of real numbers such that(i)
{δn}⊂[0,1] and ∑n=0∞‍δn=∞, or equivalently,
(2.18)∏n=0∞‍(1-δn):=limn→∞∏j=0n‍(1-δj)=0;(ii)
limsupn→∞σn≤0, or,(iii)
∑n=0∞‍δnσn is convergent.
Thenlimn→∞an=0.


## 3. Strong Convergence Theorem

In order to prove our main result, we shall need the following lemma given in [21].Lemma 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let S:C→C be a k-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers. Assume (γ+δ)k≤γ. Then
(3.1)∥γ(x-y)+δ(Sx-Sy)∥≤(γ+δ)∥x-y∥,∀x·y∈C.We are now in a position to state and prove our main result.Theorem 3.2.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H. Let Θ:C×C→R be a bifunction satisfying conditions (H1)–(H4) and φ:C→R a lower semicontinuous and convex function with assumptions (A1) or (A2). Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇f is L-Lipschitzian with constant L>0 and F:C→H is an α-inverse strongly monotone mapping. Let S:C→C be a k-strictly pseudocontractive mapping such that Fix(S)∩Ω∩GMEP≠∅. Let Q:C→C be a ρ-contraction with ρ∈[0,1/2). For given x0∈C arbitrarily, let the sequences {xn}, {yn}, and {zn} be generated iteratively by
(3.2)Θ(zn,y)+φ(y)-φ(zn)+〈Fxn,y-zn〉+1rn〈y-zn,zn-xn〉≥0,∀y∈C,yn=αnQxn+(1-αn)PC(zn-λn∇f(zn)),xn+1=βnxn+γnPC(zn-λn∇f(zn))+δnSyn,∀n≥0,
where {λn}⊂(0,2/L],  {rn}⊂(0,2α], and {αn}, {βn}, {γn}, {δn} are four sequences in [0,1] such that(i)
0<liminfn→∞λn≤limsupn→∞λn<2/L and limn→∞(λn-λn+1)=0;(ii)
0<liminfn→∞rn≤limsupn→∞rn<2α and limn→∞(rn-rn+1)=0;(iii)
βn+γn+δn=1 and (γn+δn)k≤γn<(1-2ρ)δn for all n≥0;(iv)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(v)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞δn>0;(vi)
limn→∞(γn+1/(1-βn+1)-γn/(1-βn))=0.
Then the sequences{xn},  {yn}, and {zn} converge strongly to the unique solution of the variational inequality of finding x*∈Fix(S)∩Ω∩GMEP such that
(3.3)〈(I-Q)x*,x-x*〉≥0,∀x∈Fix(S)∩Ω∩GMEP.
In other words, x* is the unique fixed point of the contraction PFix(S)∩Ω∩GMEPQ, x*=PFix(S)∩Ω∩GMEPQx*.Proof.
First it is obvious that there hold the following assertions:(a)
x*∈C solves the minimization (1.5);(b)
x* solves the fixed point equation
(3.4)x*=PC(I-λ∇f)x*,whereλ>0 is any fixed positive number;(c)
x* solves the variational inequality of finding x*∈C such that
(3.5)〈∇f(x*),x-x*〉≥0,∀x∈C,
where its solution set is denoted byVI(C,∇f).
We divide the proof into several steps.
Step  1. We claim that limn→∞∥xn+1-xn∥=0.
Indeed, first, we can write (3.2) as xn+1=βnxn+(1-βn)un, for all n≥0, where un=(xn+1-βnxn)/(1-βn). It follows that
(3.6)un+1-un=xn+2-βn+1xn+11-βn+1-xn+1-βnxn1-βn=γn+1PC(zn+1-λn+1∇f(zn+1))+δn+1Syn+11-βn+1-γnPC(zn-λn∇f(zn))+δnSyn1-βn=γn+1[PC(zn+1-λn+1∇f(zn+1))-PC(zn-λn∇f(zn))]+δn+1(Syn+1-Syn)1-βn+1+(γn+11-βn+1-γn1-βn)PC(zn-λn∇f(zn))+(δn+11-βn+1-δn1-βn)Syn.
From Lemma 3.1 and (3.2), we get
(3.7)∥γn+1[PC(zn+1-λn+1∇f(zn+1))-PC(zn-λn∇f(zn))]+δn+1(Syn+1-Syn)∥≤∥γn+1(yn+1-yn)+δn+1(Syn+1-Syn)∥+γn+1∥[PC(zn+1-λn+1∇f(zn+1))-yn+1]+[yn-PC(zn-λn∇f(zn))]∥≤(γn+1+δn+1)∥yn+1-yn∥+γn+1αn+1∥Qxn+1-PC(zn+1-λn+1∇f(zn+1))∥+γn+1αn∥Qxn-PC(zn-λn∇f(zn))∥.
Let {Trn(Θ,φ)} be a sequence of mappings defined as in Lemma 2.1. Note that the L-Lipschitz continuity of ∇f implies that the gradient ∇f is (1/L)-ism [31]. Since ∇f and F are (1/L)-inverse strongly monotone mapping and α-inverse strongly monotone mapping, respectively, then we have
(3.8)∥(I-λ∇f)x-(I-λ∇f)y∥2=∥x-y∥2-2λ〈∇f(x)-∇f(y),x-y〉+λ2∥∇f(x)-∇f(y)∥2≤∥x-y∥2+λ(λ-2L)∥∇f(x)-∇f(y)∥2,∥(I-μF)x-(I-μF)y∥2≤∥x-y∥2+μ(μ-2α)∥Fx-Fy∥2.
It is clear that if 0≤λ≤2/L and 0≤μ≤2α, then (I-λ∇f) and (I-μF) are nonexpansive. It follows from that
(3.9)∥PC(zn+1-λn+1∇f(zn+1))-PC(zn-λn∇f(zn))∥≤∥zn+1-λn+1∇f(zn+1)-(zn-λn∇f(zn))∥≤∥zn+1-λn+1∇f(zn+1)-(zn-λn+1∇f(zn))∥+|λn+1-λn|∥∇f(zn)∥≤∥zn+1-zn∥+|λn+1-λn|∥∇f(zn)∥=∥Trn+1(Θ,φ)(xn+1-rn+1Fxn+1)-Trn(Θ,φ)(xn-rnFxn)∥+|λn+1-λn|∥∇f(zn)∥≤∥Trn+1(Θ,φ)(xn+1-rn+1Fxn+1)-Trn+1(Θ,φ)(xn-rnFxn)∥+∥Trn+1(Θ,φ)(xn-rnFxn)-Trn(Θ,φ)(xn-rnFxn)∥+|λn+1-λn|∥∇f(zn)∥≤∥(xn+1-rn+1Fxn+1)-(xn-rnFxn)∥+∥Trn+1(Θ,φ)(xn-rnFxn)-Trn(Θ,φ)(xn-rnFxn)∥+|λn+1-λn|∥∇f(zn)∥≤∥(xn+1-rn+1Fxn+1)-(xn-rn+1Fxn)∥+|rn+1-rn|∥Fxn∥+∥Trn+1(Θ,φ)(xn-rnFxn)-Trn(Θ,φ)(xn-rnFxn)∥+|λn+1-λn|∥∇f(zn)∥≤∥xn+1-xn∥+∥Trn+1(Θ,φ)(xn-rnFxn)-Trn(Θ,φ)(xn-rnFxn)∥+|rn+1-rn|∥Fxn∥+|λn+1-λn|∥∇f(zn)∥.
Then,
(3.10)∥yn+1-yn∥≤∥PC(zn+1-λn+1∇f(zn+1))-PC(zn-λn∇f(zn))∥+αn+1∥Qxn+1-PC(zn+1-λn+1∇f(zn+1))∥+αn∥Qxn-PC(zn-λn∇f(zn))∥≤∥xn+1-xn∥+∥Trn+1(Θ,φ)(xn-rnFxn)-Trn(Θ,φ)(xn-rnFxn)∥+|rn+1-rn|∥Fxn∥+|λn+1-λn|∥∇f(zn)∥+αn∥Qxn-PC(zn-λn∇f(zn))∥+αn+1∥Qxn+1-PC(zn+1-λn+1∇f(zn+1))∥.
So, from (3.6), (3.7), and (3.10), we have
(3.11)∥un+1-un∥≤∥xn+1-xn∥+(1+γn+11-βn+1)αn∥Qxn-PC(zn-λn∇f(zn))∥+(1+γn+11-βn+1)αn+1∥Qxn+1-PC(zn+1-λn+1∇f(zn+1))∥+|γn+11-βn+1-γn1-βn|(∥PC(zn-λn∇f(zn))∥+∥Syn∥)+∥Trn+1(Θ,φ)(xn-rnFxn)-Trn(Θ,φ)(xn-rnFxn)∥+|rn+1-rn|∥Fxn∥+|λn+1-λn|∥∇f(zn)∥.
Utilizing Proposition 2.4 and condition (ii), we have
(3.12)limn→∞∥Trn+1(Θ,φ)(xn-rnFxn)-Trn(Θ,φ)(xn-rnFxn)∥=0.
This implies that
(3.13)limsupn→∞(∥un+1-un∥-∥xn+1-xn∥)≤0.
Hence by Lemma 2.6, we get limn→∞∥un-xn∥=0. Consequently,
(3.14)limn→∞∥xn+1-xn∥=limn→∞(1-βn)∥un-xn∥=0.Step  2. We claim that limn→∞∥∇f(zn)-∇f(x*)∥=0 and limn→∞∥Fxn-Fx*∥=0.
Indeed, letx*∈Fix(S)∩Ω∩GMEP. Then we have x*=Sx*,  x*=PC(x*-λn∇f(x*)) and
(3.15)x*=Trn(Θ,φ)(x*-rnFx*).
Hence from (3.8), we have
(3.16)∥PC(zn-λn∇f(zn))-PC(x*-λn∇f(x*))∥2≤∥(zn-λn∇f(zn))-(x*-λn∇f(x*))∥2≤∥zn-x*∥2+λn(λn-2L)∥∇f(zn)-∇f(x*)∥2,(3.17)∥zn-x*∥2=∥Trn(Θ,φ)(xn-rnFxn)-Trn(Θ,φ)(x*-rnFx*)∥2≤∥(xn-rnFxn)-(x*-rnFx*)∥2≤∥xn-x*∥2+rn(rn-2α)∥Fxn-Fx*∥2.
It follows from (3.2), (3.16), and (3.17) that
(3.18)∥yn-x*∥2≤(1-αn)∥PC(zn-λn∇f(zn))-PC(x*-λn∇f(x*))∥2+αn∥Qxn-x*∥2≤αn∥Qxn-x*∥2+∥zn-x*∥2+λn(λn-2L)∥∇f(zn)-∇f(x*)∥2≤αn∥Qxn-x*∥2+∥xn-x*∥2+rn(rn-2α)∥Fxn-Fx*∥2+λn(λn-2L)∥∇f(zn)-∇f(x*)∥2.
Utilizing the convexity of ∥·∥, we have
(3.19)∥xn+1-x*∥2=∥βn(xn-x*)+(1-βn)11-βn[γn(PC(zn-λn∇f(zn))-x*)+δn(Syn-x*)]∥2≤βn∥xn-x*∥2+(1-βn)∥γn1-βn(PC(zn-λn∇f(zn))-x*)+δn1-βn(Syn-x*)∥2=βn∥xn-x*∥2+(1-βn)∥γn(yn-x*)+δn(Syn-x*)1-βn+αnγn1-βn(PC(zn-λn∇f(zn))-Qxn)∥2≤βn∥xn-x*∥2+(1-βn)∥γn(yn-x*)+δn(Syn-x*)1-βn∥2+Mαn≤βn∥xn-x*∥2+(1-βn)∥yn-x*∥2+Mαn,
where M>0 is some appropriate constant. So, from (3.18) and (3.19), it follows that
(3.20)∥xn+1-x*∥2≤∥xn-x*∥2+rn(rn-2α)(1-βn)∥Fxn-Fx*∥2+λn(λn-2L)(1-βn)∥∇f(zn)-∇f(x*)∥2+(M+∥Qxn-x*∥2)αn.
Therefore,
(3.21)λn(2L-λn)(1-βn)∥∇f(zn)-∇f(x*)∥2+rn(2α-rn)(1-βn)∥Fxn-Fx*∥2≤∥xn-x*∥2-∥xn+1-x*∥2+(M+∥Qxn-x*∥2)αn≤(∥xn-x*∥+∥xn+1-x*∥)∥xn-xn+1∥+(M+∥Qxn-x*∥2)αn.
Since liminfn→∞λn(2/L-λn)(1-βn)>0,  liminfn→∞rn(2α-rn)(1-βn)>0,  ∥xn-xn+1∥→0 and αn→0, we have
(3.22)limn→∞∥∇f(zn)-∇f(x*)∥=0,limn→∞∥Fxn-Fx*∥=0.Step  3. We claim that limn→∞∥Syn-yn∥=0.
Indeed, setvn=PC(zn-λn∇f(zn)). Noticing the firm nonexpansivity of Trn(Θ,φ), we have
(3.23)∥zn-x*∥2=∥Trn(Θ,φ)(xn-rnFxn)-Trn(Θ,φ)(x*-rnFx*)∥2≤〈(xn-rnFxn)-(x*-rnFx*),zn-x*〉=12(∥xn-x*-rn(Fxn-Fx*)∥2+∥zn-x*∥2-∥(xn-x*)-rn(Fxn-Fx*)-(zn-x*)∥2)≤12(∥xn-x*∥2+∥zn-x*∥2-∥(xn-zn)-rn(Fxn-Fx*)∥2)=12(∥xn-x*∥2+∥zn-x*∥2-∥xn-zn∥2+2rn〈xn-zn,Fxn-Fx*〉-rn2∥Fxn-Fx*∥2),(3.24)∥vn-x*∥2=∥PC(zn-λn∇f(zn))-PC(x*-λn∇f(x*))∥2≤〈zn-λn∇f(zn)-(x*-λn∇f(x*)),vn-x*〉=12(∥zn-λn∇f(zn)-(x*-λn∇f(x*))∥2+∥vn-x*∥2-∥zn-λn∇f(zn)-(x*-λn∇f(x*))-(vn-x*)∥2)≤12(∥zn-x*∥2+∥vn-x*∥2-∥zn-vn∥2+2λn〈∇f(zn)-∇f(x*),zn-vn〉-λn2∥∇f(zn)-∇f(x*)∥2)≤12(∥xn-x*∥2+∥vn-x*∥2-∥zn-vn∥2+2λn〈∇f(zn)-∇f(x*),zn-vn〉).
Thus, we have
(3.25)∥zn-x*∥2≤∥xn-x*∥2-∥xn-zn∥2+2rn〈xn-zn,Fxn-Fx*〉-rn2∥Fxn-Fx*∥2,(3.26)∥vn-x*∥2≤∥xn-x*∥2-∥zn-vn∥2+2λn∥∇f(zn)-∇f(x*)∥∥zn-vn∥.
It follows that
(3.27)∥yn-x*∥2≤αn∥Qxn-x*∥2+(1-αn)∥vn-x*∥2≤αn∥Qxn-x*∥2+∥vn-x*∥2≤αn∥Qxn-x*∥2+∥xn-x*∥2-∥zn-vn∥2+2λn∥∇f(zn)-∇f(x*)∥∥zn-vn∥.
From (3.18), (3.19), and (3.25), we have
(3.28)∥xn+1-x*∥2≤βn∥xn-x*∥2+(1-βn)αn∥Qxn-x*∥2+(1-βn)∥zn-x*∥2+Mαn≤∥xn-x*∥2-(1-βn)∥xn-zn∥2+2(1-βn)rn∥xn-zn∥∥Fxn-Fx*∥+(M+∥Qxn-x*∥2)αn.
It follows that
(3.29)(1-βn)∥xn-zn∥2≤(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥+(M+∥Qxn-x*∥2)αn+2(1-βn)rn∥xn-zn∥∥Fxn-Fx*∥.
Note that ∥xn+1-xn∥→0,  αn→0 and ∥Fxn-Fx*∥→0. Then we immediately deduce that
(3.30)limn→∞∥xn-zn∥=0.
From (3.19) and (3.27), we have
(3.31)∥xn+1-x*∥2≤∥xn-x*∥2-(1-βn)∥zn-vn∥2+2λn(1-βn)∥∇f(zn)-∇f(x*)∥∥zn-vn∥+(M+∥Qxn-x*∥2)αn.
So, we obtain
(3.32)(1-βn)∥zn-vn∥2≤(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥+(M+∥Qxn-x*∥2)αn+2λn(1-βn)∥∇f(zn)-∇f(x*)∥∥zn-vn∥.
Note that ∥xn+1-xn∥→0,  αn→0 and ∥∇f(zn)-∇f(x*)∥→0. Then we immediately conclude that
(3.33)limn→∞∥zn-vn∥=0.
This together with ∥yn-vn∥≤αn∥Qxn-vn∥→0, implies that
(3.34)limn→∞∥zn-yn∥=0.
Thus, from (3.30) and (3.34), we deduce that
(3.35)limn→∞∥xn-yn∥=0.
Since
(3.36)∥δn(Syn-xn)∥≤∥xn+1-xn∥+γn∥PC(zn-λn∇f(zn))-xn∥≤∥xn+1-xn∥+γn∥yn-xn∥+γnαn∥Qxn-PC(zn-λn∇f(zn))∥.
Therefore,
(3.37)limn→∞∥Syn-xn∥=0,limn→∞∥Syn-yn∥=0.Step  4. We claim that limsupn→∞〈Qx*-x*,xn-x*〉≤0 where x*=PFix(S)∩Ω∩GMEPQx*.
Indeed, since{xn} is bounded, there exists a subsequence {xni} of {xn} such that xni⇀u and
(3.38)limsupn→∞〈Qx*-x*,xn-x*〉=limi→∞〈Qx*-x*,xni-x*〉=〈Qx*-x*,u-x*〉.
We can obtain that u∈Fix(S)∩Ω∩GMEP. First, we show u∈Ω(=VI(C,∇f)). Since xn-zn→0 and vn-zn→0, we conclude that zni⇀u and vni⇀u. Let
(3.39)Tv={∇f(v)+NCv,ifv∈C,∅,ifv∉C,
where NCv is the normal cone to C at v∈C. We have already mentioned that in this case, the mapping T is maximal monotone, and 0∈Tv if and only if v∈VI(C,∇f)(=Ω); see [37]. Let G(T) be the graph of T and let (v,w)∈G(T). Then, we have w∈Tv=∇f(v)+NCv and hence w-∇f(v)∈NCv. So, we have 〈v-t,w-∇f(v)〉≥0 for all t∈C. On the other hand, from vn=PC(zn-λn∇f(zn)) and v∈C, we have
(3.40)〈zn-λn∇f(zn)-vn,vn-v〉≥0
and hence
(3.41)〈v-vn,vn-znλn+∇f(zn)〉≥0.
From 〈v-t,w-∇f(v)〉≥0 for all t∈C and vni∈C, we have
(3.42)〈v-vni,w〉≥〈v-vni,∇f(v)〉≥〈v-vni,∇f(v)〉-〈v-vni,vni-zniλni+∇f(zni)〉=〈v-vni,∇f(v)-∇f(vni)〉+〈v-vni,∇f(vni)-∇f(zni)〉-〈v-vni,vni-zniλni〉≥〈v-vni,∇f(vni)-∇f(zni)〉-〈v-vni,vni-zniλni〉.
Hence, we obtain 〈v-u,w〉≥0 as i→∞. Since T is maximal monotone, we have u∈T-10 and hence u∈VI(C,∇f)(=Ω).
Secondly, let us showu∈Fix(S). Since xn-yn→0 and xni⇀u, we have yni⇀u. Also, since yn-Syn→0, it follows that yni-Syni→0 as i→∞. So, in terms of Proposition 2.5(ii) we obtain u∈Fix(S).
Next, let us showu∈GMEP. From zn=Trn(Θ,φ)(xn-rnFxn), we know that
(3.43)Θ(zn,y)+φ(y)-φ(zn)+〈Fxn,y-zn〉+1rn〈y-zn,zn-xn〉≥0,∀y∈C.
From (H2), it follows that
(3.44)φ(y)-φ(zn)+〈Fxn,y-zn〉+1rn〈y-zn,zn-xn〉≥Θ(y,zn),∀y∈C.
Replacing n by ni, we have
(3.45)φ(y)-φ(zni)+〈Fxni,y-zni〉+〈y-zni,zni-xnirni〉≥Θ(y,zni),∀y∈C.
Put zs=sy+(1-s)u for all s∈(0,1] and y∈C. Then, we have zs∈C. So, from (3.45), we have
(3.46)〈zs-zni,Fzs〉≥〈zs-zni,Fzs〉-φ(zs)+φ(zni)-〈zs-zni,Fxni〉-〈zs-zni,zni-xnirni〉+Θ(zs,zni)=〈zs-zni,Fzs-Fzni〉+〈zs-zni,Fzni-Fxni〉-φ(zs)+φ(zni)-〈zs-zni,zni-xnirni〉+Θ(zs,zni).
Since ∥zni-xni∥→0, we have ∥Fzni-Fxni∥→0. Further, from the monotonicity of F, we have 〈zs-zni,Fzs-Fzni〉≥0. So, from (H4), the weakly lower semicontinuity of φ,(zni-xni)/rni→0 and zni⇀u, we have
(3.47)〈zs-zni,Fzs〉≥-φ(zs)+φ(u)+Θ(zs,u),
as i→∞. From (H1), (H4), and (3.47), we also have
(3.48)0=Θ(zs,zs)+φ(zs)+φ(zs)≤sΘ(zs,y)+(1-s)Θ(zs,u)+sφ(y)+(1-s)φ(u)-φ(zs)=s[Θ(zs,y)+φ(y)-φ(zs)]+(1-s)[Θ(zs,u)+φ(u)-φ(zs)]≤s[Θ(zs,y)+φ(y)-φ(zs)]+(1-s)〈zs-u,Fzs〉=s[Θ(zs,y)+φ(y)-φ(zs)]+(1-s)s〈y-u,Fzs〉,
and hence
(3.49)0≤Θ(zs,y)+φ(y)-φ(zs)+(1-s)〈y-u,Fzs〉.
Letting s→0, we have, for each y∈C,
(3.50)0≤Θ(u,y)+φ(y)-φ(u)+〈y-u,Fu〉.
This shows that u∈GMEP. Therefore, u∈Fix(S)∩Ω∩GMEP. Hence, it follows from (2.4) that
(3.51)limsupn→∞〈Qx*-x*,xn-x*〉=limi→∞〈Qx*-x*,xni-x*〉=〈Qx*-x*,u-x*〉≤0.Step  5. We claim that limn→∞∥xn-x*∥=0.
Indeed, from (3.2) and the convexity of ∥·∥, we have
(3.52)∥xn+1-x*∥2=∥βn(xn-x*)+γn(yn-x*)+δn(Syn-x*)+γnαn(PC(zn-λn∇f(zn))-Qxn)∥2≤∥βn(xn-x*)+γn(yn-x*)+δn(Syn-x*)∥2+2γnαn〈PC(zn-λn∇f(zn))-Qxn,xn+1-x*〉≤βn∥xn-x*∥2+(1-βn)∥11-βn[γn(yn-x*)+δn(Syn-x*)]∥2+2γnαn〈PC(zn-λn∇f(zn))-x*,xn+1-x*〉+2γnαn〈x*-Qxn,xn+1-x*〉.
Utilizing Lemma 3.1, we get from (3.52)
(3.53)∥xn+1-x*∥2≤βn∥xn-x*∥2+(1-βn)∥yn-x*∥2+2γnαn∥PC(zn-λn∇f(zn))-x*∥∥xn+1-x*∥+2γnαn〈x*-Qxn,xn+1-x*〉≤βn∥xn-x*∥2+(1-βn)[(1-αn)∥zn-x*∥2+2αn〈Qxn-x*,yn-x*〉]+2γnαn∥zn-x*∥∥xn+1-x*∥+2γnαn〈x*-Qxn,xn+1-x*〉.
From (3.17), we note that ∥zn-x*∥≤∥xn-x*∥. Hence we have
(3.54)∥xn+1-x*∥2≤βn∥xn-x*∥2+(1-βn)(1-αn)∥xn-x*∥2+2αn(1-βn)〈Qxn-x*,yn-x*〉+2γnαn∥xn-x*∥∥xn+1-x*∥+2γnαn〈x*-Qxn,xn+1-x*〉≤[1-(1-βn)αn]∥xn-x*∥2+2αnγn〈Qxn-x*,yn-xn+1〉+2αnδn〈Qxn-x*,yn-x*〉+2αnγn∥xn-x*∥∥xn+1-x*∥≤[1-(1-βn)αn]∥xn-x*∥2+2αnγn∥Qxn-x*∥∥yn-xn+1∥+2αnδn〈Qxn-x*,xn-x*〉+2αnδn〈Qxn-x*,yn-xn〉+2αnγn∥xn-x*∥∥xn+1-x*∥≤[1-(1-βn)αn]∥xn-x*∥2+2αnγn∥Qxn-x*∥∥yn-xn+1∥+2αnδnρ∥xn-x*∥2+2αnδn〈Qx*-x*,xn-x*〉+2αnδn∥Qxn-x*∥∥yn-xn∥+2αnγn∥xn-x*∥∥xn+1-x*∥≤[1-(1-βn)αn]∥xn-x*∥2+2αnγn∥Qxn-x*∥∥yn-xn+1∥+2αnδnρ∥xn-x*∥2+2αnδn〈Qx*-x*,xn-x*〉+2αnδn∥Qxn-x*∥∥yn-xn∥+αnγn(∥xn-x*∥2+∥xn+1-x*∥2),
that is,
(3.55)∥xn+1-x*∥2≤[1-(1-2ρ)δn-γn1-αnγnαn]∥xn-x*∥2+[(1-2ρ)δn-γn]αn1-αnγn×{2γn(1-2ρ)δn-γn∥Qxn-x*∥∥yn-xn+1∥+2δn(1-2ρ)δn-γn∥Qxn-x*∥∥yn-xn∥+2δn(1-2ρ)δn-γn〈Qx*-x*,xn-x*〉}.
Note that liminfn→∞((1-2ρ)δn-γn)/(1-αnγn)>0. It follows that (∑n=0∞‍((1-2ρ)δn-γn)/(1-αnγn))αn=∞. It is clear that
(3.56)limsupn→∞{2γn(1-2ρ)δn-γn∥Qxn-x*∥∥yn-xn+1∥+2δn(1-2ρ)δn-γn∥Qxn-x*∥∥yn-xn∥+2δn(1-2ρ)δn-γn〈Qx*-x*,xn-x*〉}≤0.
Therefore, all conditions of Lemma 2.7 are satisfied. This immediately implies that xn→x*. It is readily seen that both {yn} and {zn} converge strongly to the same point x*. The proof is complete.Utilizing Theorem3.2, we establish the following corollaries.Corollary 3.3.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H. Let Θ:C×C→R be a bifunction satisfying conditions (H1)–(H4) and φ:C→R be a lower semicontinuous and convex function with assumptions (A1) or (A2). Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇f is L-Lipschitzian with constant L>0 and F:C→H is an α-inverse strongly monotone mapping. Let S:C→C be a k-strictly pseudocontractive mapping such that Fix(S)∩Ω∩GMEP≠∅. For fixed u∈C and given x0∈C arbitrarily, let the sequences {xn}, {yn} and {zn} be generated iteratively by
(3.57)Θ(zn,y)+φ(y)-φ(zn)+〈Fxn,y-zn〉+1rn〈y-zn,zn-xn〉≥0,∀y∈C,yn=αnu+(1-αn)PC(zn-λn∇f(zn)),xn+1=βnxn+γnPC(zn-λn∇f(zn))+δnSyn,∀n≥0,
where {λn}⊂(0,2/L],  {rn}⊂(0,2α], and {αn}, {βn}, {γn}, {δn} are four sequences in [0,1] such that:(i)
0<liminfn→∞λn≤limsupn→∞λn<2/L and limn→∞(λn-λn+1)=0;(ii)
0<liminfn→∞rn≤limsupn→∞rn<2α and limn→∞(rn-rn+1)=0;(iii)
βn+γn+δn=1 and (γn+δn)k≤γn<δn for all n≥0;(iv)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(v)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞δn>0;(vi)
limn→∞(γn+1/(1-βn+1)-γn/(1-βn))=0.
Then,{xn}, {yn} and {zn} converge strongly to the same point x*=PFix(S)∩Ω∩GMEPu.Corollary 3.4.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H. Let Θ:C×C→R be a bifunction satisfying conditions (H1)–(H4) and φ:C→R be a lower semicontinuous and convex function with assumptions (A1) or (A2). Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇f is L-Lipschitzian with constant L>0 and F:C→H is an α-inverse strongly monotone mapping. Let S:C→C be a nonexpansive mapping such that Fix(S)∩Ω∩GMEP≠∅. Let Q:C→C be a ρ-contraction with ρ∈[0,1/2). For given x0∈C arbitrarily, let the sequences {xn}, {yn} and {zn} be generated iteratively by
(3.58)Θ(zn,y)+φ(y)-φ(zn)+〈Fxn,y-zn〉+1rn〈y-zn,zn-xn〉≥0,∀y∈C,yn=αnQxn+(1-αn)PC(zn-λn∇f(zn)),xn+1=βnxn+γnPC(zn-λn∇f(zn))+δnSyn,∀n≥0,
where {λn}⊂(0,2/L], {rn}⊂(0,2α] and {αn}, {βn}, {γn}, {δn} are four sequences in [0,1] such that(i)
0<liminfn→∞λn≤limsupn→∞λn<2/L and limn→∞(λn-λn+1)=0;(ii)
0<liminfn→∞rn≤limsupn→∞rn<2α and limn→∞(rn-rn+1)=0;(iii)
βn+γn+δn=1 and γn<(1-2ρ)δn for all n≥0;(iv)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(v)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞γn>0;(vi)
limn→∞(γn+1/(1-βn+1)-γn/(1-βn))=0.
Then{xn}, {yn}, and {zn} converge strongly to the same point x*=PFix(S)∩Ω∩GMEPQx*.Corollary 3.5.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H. Let Θ:C×C→R be a bifunction satisfying conditions (H1)–(H4) and φ:C→R a lower semicontinuous and convex function with assumptions (A1) or (A2). Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇f is L-Lipschitzian with constant L>0 and F:C→H is an α-inverse strongly monotone mapping. Let S:C→C be a nonexpansive mapping such that Fix(S)∩Ω∩GMEP≠∅. For fixed u∈C and given x0∈C arbitrarily, let the sequences {xn}, {yn}, and {zn} be generated iteratively by
(3.59)Θ(zn,y)+φ(y)-φ(zn)+〈Fxn,y-zn〉+1rn〈y-zn,zn-xn〉≥0,∀y∈C,yn=αnu+(1-αn)PC(zn-λn∇f(zn)),xn+1=βnxn+γnPC(zn-λn∇f(zn))+δnSyn,∀n≥0,
where {λn}⊂(0,2/L], {rn}⊂(0,2α], and {αn}, {βn}, {γn}, {δn} are four sequences in [0,1] such that:(i)
0<liminfn→∞λn≤limsupn→∞λn<2/L and limn→∞(λn-λn+1)=0;(ii)
0<liminfn→∞rn≤limsupn→∞rn<2α and limn→∞(rn-rn+1)=0;(iii)
βn+γn+δn=1 and γn<δn for all n≥0;(iv)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(v)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞γn>0;(vi)
limn→∞(γn+1/(1-βn+1)-γn/(1-βn))=0.
Then,{xn}, {yn}, and {zn} converge strongly to the same point x*=PFix(S)∩Ω∩GMEPu.Corollary 3.6.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H. Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇f is L-Lipschitzian with constant L>0 and A:C→H is an α-inverse strongly monotone mapping. Let S:C→C be a k-strictly pseudocontractive mapping such that Fix(S)∩Ω∩VI(C,A)≠∅. For fixed u∈C and given x0∈C arbitrarily, let the sequences {xn}, {yn}, and {zn} be generated iteratively by
(3.60)zn=PC(xn-rnAxn),yn=αnu+(1-αn)PC(zn-λn∇f(zn)),xn+1=βnxn+γnPC(zn-λn∇f(zn))+δnSyn,∀n≥0,
where {λn}⊂(0,2/L],  {rn}⊂(0,2α], and {αn}, {βn}, {γn}, {δn} are four sequences in [0,1] such that:(i)
0<liminfn→∞λn≤limsupn→∞λn<2/L and limn→∞(λn-λn+1)=0;(ii)
0<liminfn→∞rn≤limsupn→∞rn<2α and limn→∞(rn-rn+1)=0;(iii)
βn+γn+δn=1 and (γn+δn)k≤γn<δn for all n≥0;(iv)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(v)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞δn>0;(vi)
limn→∞(γn+1/(1-βn+1)-γn/(1-βn))=0.
Then,{xn}, {yn} and {zn} converge strongly to the same point x*=PFix(S)∩Ω∩VI(C,A)u.Proof.
In Theorem3.2, putting Θ=0,  φ=0 and F=A, the following relation
(3.61)Θ(zn,y)+φ(y)-φ(zn)+〈Fxn,y-zn〉+1rn〈y-zn,zn-xn〉≥0,∀y∈C,
is reduced to
(3.62)〈Axn,y-zn〉+1rn〈y-zn,zn-xn〉≥0,∀y∈C.
This implies that
(3.63)〈y-zn,xn-rnAxn-zn〉≤0,∀y∈C.
So, it follows that zn=PC(xn-rnAxn) for all n≥0. Then, by Theorem 3.2, we obtain the desired result.LetT:C→C be a k~-strictly pseudocontractive mapping. For recent convergence result for strictly pseudocontractive mappings, we refer to Zeng et al. [38]. Putting F=I-T, we know that
(3.64)∥(I-F)x-(I-F)y∥2≤∥x-y∥2+k~∥Fx-Fy∥2,∀x,y∈C.
Note that
(3.65)∥(I-F)x-(I-F)y∥2=∥x-y∥2+∥Fx-Fy∥2-2〈x-y,Fx-Fy〉.
Hence
(3.66)〈x-y,Fx-Fy〉≥1-k~2∥Fx-Fy∥2,∀x,y∈C.
This implies that the mapping F=I-T is ((1-k~)/2)-inverse-strongly monotone.Corollary 3.7.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H. Let Θ:C×C→R be a bifunction satisfying conditions (H1)–(H4) and φ:C→R a lower semicontinuous and convex function with assumptions (A1) or (A2). Suppose the minimization (1.5) is consistent and let Ω denote its solution set. Assume the gradient ∇f is L-Lipschitzian with constant L>0 and T:C→C is a k~-strictly pseudocontractive mapping. Let S:C→C be a k-strictly pseudocontractive mapping such that Fix(S)∩Ω∩GMEP≠∅, where F=I-T. For fixed u∈C and given x0∈C arbitrarily, let the sequences {xn}, {yn}, and {zn} be generated iteratively by
(3.67)zn=Trn(Θ,φ)((1-rn)xn+rnTxn),yn=αnu+(1-αn)PC(zn-λn∇f(zn)),xn+1=βnxn+γnPC(zn-λn∇f(zn))+δnSyn,∀n≥0,
where {λn}⊂(0,2/L],  {rn}⊂(0,1-k~] and {αn}, {βn}, {γn}, {δn} are four sequences in [0,1] such that(i)
0<liminfn→∞λn≤limsupn→∞λn<2/L and limn→∞(λn-λn+1)=0;(ii)
0<liminfn→∞rn≤limsupn→∞rn<1-k~ and limn→∞(rn-rn+1)=0;(iii)
βn+γn+δn=1 and (γn+δn)k≤γn<δn for all n≥0;(iv)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(v)
0<liminfn→∞βn≤limsupn→∞βn<1 and liminfn→∞δn>0;(vi)
limn→∞(γn+1/(1-βn+1)-γn/(1-βn))=0.
Then,{xn}, {yn} and {zn} converge strongly to the same point x*=PFix(S)∩Ω∩GMEPu.Proof.
SinceT is a k~-strictly pseudocontractive mapping, the mapping F=I-T is (1-k~)/2-inverse-strongly monotone. In this case, put α=(1-k~)/2. Then, we conclude that
(3.68)zn=Trn(Θ,φ)(xn-rnFxn)=Trn(Θ,φ)(xn-rn(I-T)xn)=Trn(Θ,φ)((1-rn)xn+rnTxn).
So, by Theorem 3.2, we obtain the desired result.



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*Source: 678353-2012-10-30.xml*