# A System of Mixed Equilibrium Problems, a General System of Variational Inequality Problems for Relaxed Cocoercive, and Fixed Point Problems for Nonexpansive Semigroup and Strictly Pseudocontractive Mappings

Authors: Poom Kumam; Phayap Katchang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of solutions of a general system of variational inequalities for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in Hilbert spaces. Furthermore, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems. Our results extend and improve the recent results of Chang et al. (2010) and many others.


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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. Recall that a mapping T:C→C is nonexpansive if
(1.1)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.
We denote the set of fixed points ofT by F(T), that is F(T)={x∈C:x=Tx}. A mapping f:C→C is said to be an α-contraction if there exists a coefficient α∈(0,1) such that
(1.2)‖f(x)-f(y)‖≤α‖x-y‖,∀x,y∈C.
Let B:C→H be a mapping. Then B is called:(1)
monotone if
(1.3)〈Bx-By,x-y〉≥0,∀x,y∈C;(2)
d-strongly monotone if there exists a positive real number d such that
(1.4)〈Bx-By,x-y〉≥d‖x-y‖2,∀x,y∈C,
for constant d>0, this implies that
(1.5)‖Bx-By‖≥d‖x-y‖,
that is, B is d-expansive and when d=1, it is expansive;(3)
L-Lipschitz continuous if there exists a positive real number L such that(1.6)‖Bx-By‖≤L‖x-y‖,∀x,y∈C;(4)
c-cocoercive [1, 2] if there exists a positive real number c such that
(1.7)〈Bx-By,x-y〉≥c‖Bx-By‖2,∀x,y∈C,
Clearly, every c-cocoercive map B is (1/c)-Lipschitz continuous;(5)
relaxedc-cocoercive, if there exists a positive real number c such that
(1.8)〈Bx-By,x-y〉≥(-c)‖Bx-By‖2,∀x,y∈C;(6)
relaxed(c,d)-cocoercive, if there exists a positive real number c,d such that
(1.9)〈Bx-By,x-y〉≥(-c)‖Bx-By‖2+d‖x-y‖2,∀x,y∈C,
for c=0, B is d-strongly monotone. This class of mapping is more general than the class of strongly monotone mapping. It is easy to see that we have the following implication: d-strongly monotonicity implying relaxed (c,d)-cocoercivity,(7)
k-strictly pseudocontractive, if there exists a constant k∈[0,1) such that
(1.10)‖Bx-By‖2≤‖x-y‖2+k‖(I-B)x-(I-B)y‖2,∀x,y∈C.Remark 1.1 (see [3, Remark  1.1 pages 135-136]).
IfB:C→H is a LB-Lipschitz continuous and relaxed (c,d)-cocoercive mapping with d>cLB2 and 0<τ<2(d-cLB2)/LB2, then I-τB satisfies the following:
(1.11)‖(I-τB)x-(I-τB)y‖≤(1-τξ)‖x-y‖,∀x,y∈C,
where ξ=(LB2/2)[2(d-cLB2)/LB2-τ].
Similarly, ifD:C→HisLD-Lipschitz continuous and relaxed (c′,d′)-cocoercive mapping with d′>c′LD2 and 0<δ<2(d′-c′LD2)/LD2, then the mapping I-δD satisfies the following:
(1.12)‖(I-δD)x-(I-δD)y‖≤(1-δξ′)‖x-y‖,
where ξ′=(LD2/2)[2(d′-c′LD2)/LD2-δ].LetA be a strongly positive linear bounded operator on H if there is a constant γ̅>0 with the property
(1.13)〈Ax,x〉≥γ̅‖x‖2,∀x∈H.
We recall optimization problem (for short, OP) as the following
(1.14)min⁡x∈Fμ2〈Ax,x〉+12‖x-u‖2-h(x),
where F=∩n=1∞Cn,C1,C2,… are infinitely closed convex subsets of H such that ∩n=1∞Cn≠∅, u∈H, μ≥0 is a real number, A is a strongly positive linear bounded operator on H, and h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H). This kind of optimization problem has been studied extensively by many authors, see, for example, [4–7] when F=∩n=1∞Cn and h(x)=〈x,b〉, where b is a given point in H.On the other hand, a family𝒮={S(s):0≤s<∞} of mappings of C into itself is called a nonexpansive semigroup on C if it satisfies the following conditions:(i)
S(0)x=x for all x∈C;(ii)
S(s+t)=S(s)S(t) for all s,t≥0;(iii)
∥S(s)x-S(s)y∥≤∥x-y∥ for all x,y∈C and s≥0;(iv)
for allx∈C,s↦S(s)x is continuous.We denote byF(𝒮) the set of all common fixed points of𝒮={S(s):s≥0}, that is, F(𝒮)=∩s≥0F(S(s)). It is known that F(𝒮) is closed and convex.Letϕ:C→ℝ be a real-valued function and let {Θk:C×C→ℝ,k=1,2,…,N} be a finite family of equilibrium functions, that is, Θk(u,u)=0 for each u∈C. The system of mixed equilibrium problems (for short, SMEP) for function (Θ1,Θ2,…,ΘN,ϕ) is to find z∈C such that
(1.15)Θ1(z,y)+ϕ(y)-ϕ(z)≥0,∀y∈C,Θ2(z,y)+ϕ(y)-ϕ(z)≥0,∀y∈C,⋮ΘN(z,y)+ϕ(y)-ϕ(z)≥0,∀y∈C.
The set of solutions of (1.15) is denoted by ∩k=1NMEP⁡(Θk,ϕ), where MEP⁡(Θk,ϕ) is the set of solutions of the mixed equilibrium problem (for short, MEP), which is to find z∈C such that
(1.16)Θk(z,y)+ϕ(y)-ϕ(z)≥0,∀y∈C.
In particular, if ϕ≡0, and N=1, then the problem (1.15) reduces to the equilibrium problem (for short, EP), which is to find z∈C such that
(1.17)Θ(z,y)≥0,∀y∈C.
It is well known that the SMEP includes fixed point problem, optimization problem, variational inequality problem, and Nash equilibrium problem as its special cases (see [8–13] for more details).For solving the solutions of a nonexpansive semigroup and the solutions of the system of mixed equilibrium problems were studied by many authors see [14–23] and reference therein. In 2010, Chang et al. [24] studied the following approximation method:
(1.18)Θ1(un(1),x)+ϕ(x)-ϕ(un(1))+1r1〈K′(un(1))-K′(xn),η(x,un(1))〉≥0,∀x∈C,Θ2(un(2),x)+ϕ(x)-ϕ(un(2))+1r2〈K′(un(2))-K′(xn),η(x,un(2))〉≥0,∀x∈C,⋮ΘN(un(N),x)+ϕ(x)-ϕ(un(N))+1rN〈K′(un(N))-K′(xn),η(x,un(N))〉≥0,∀x∈C,xn+1=αnf(Wnxn)+βnxn+γn1tn∫0tnS(s)Wnun(N)ds,
where
(1.19)un(1)=Jr1Θ1xn,un(k)=JrkΘkun(k-1)=JrkΘkJrk-1Θk-1un(k-2)=JrkΘk⋯Jr2Θ2un(1),=JrkΘk⋯Jr2Θ2Jr1Θ1xn,k=2,3,…,N,JrkΘk:C→C,k=1,2,…,N is the mapping defined by (2.22) below, Wn is the mapping defined by (2.12), and 𝒮={S(s):0≤s<∞} is a nonexpansive semigroup. They proved that {xn} converges strongly to a fixed point of F(𝒮)∩F(W)∩(∩k=1NMEP⁡(Θk,ϕ)) under control conditions on the parameters.LetB,D:C→H be two mappings. The general system of variational inequalities problem (see [25]) is to find (x*,y*)∈C×C such that
(1.20)〈τBy*+x*-y*,x-x*〉≥0,∀x∈C,〈δDx*+y*-x*,x-y*〉≥0,∀x∈C,
where τ and δ are two positive real numbers. The set of solutions of the general system of variational inequalities problem is denoted by SVI(C,B,D). In particular, if B=D, then the problem (1.20) reduces to the following equation:
(1.21)〈τBy*+x*-y*,x-x*〉≥0,∀x∈C,〈δBx*+y*-x*,x-y*〉≥0,∀x∈C,
which is defined by Verma [26] (see also Verma [27]), and is called the new system of variational inequalities. Further, if we set D=0, then problem (1.20) reduces to the classical variational inequality is to find x*∈C such that
(1.22)〈Bx*,x-x*〉≥0,∀x∈C.
We denoted by VI⁡(C,B) the set of solutions of the variational inequality problem. The variational inequality problem has been extensively studied in literature, see, for example, [28–31] and references therein. In order to find the solutions of the general system of variational inequality problem (1.20), Wangkeeree and Kamraksa [32] considered the following iterative algorithm:
(1.23)Θ(un,x)+ϕ(x)-ϕ(un)+1r〈K′(un)-K′(xn),η(x,un)〉≥0,∀x∈C,zn=PC(un-δDun),xn+1=αnγf(xn)+βnxn+[(1-βn)I-αnA)]WnPC(zn-τBzn),
where B,D:C→H is a LB-Lipschitz continuous and relaxed (c,d)-cocoercive mapping and LD-Lipschitz continuous and relaxed (c′,d′)-cocoercive mapping, respectively. They proved that {xn} converges strongly to a fixed point of F(Wn)∩MEP⁡(Θ,ϕ)∩SVI(C,B,D) which is a solution of general system of variational inequality (1.20). Very recently, Jaiboon and Kumam [33] studied a new general iterative method for finding a common element of the set of solution of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of variational inequalities for an inverse-strongly monotone mapping in Hilbert spaces, which solves some optimization problems.Inspired and motivated by Chang et al. [24], Jaiboon and Kumam [33], Kumam and Jaiboon [34] and Wangkeeree and Kamraksa [32], the purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of solutions of (1.15), the set of solutions of (1.20) for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroup, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings. Consequently, we prove the strong convergence theorem in Hilbert spaces under control conditions on the parameters. Furthermore, we can apply our results for solving some optimization problems. Our results extend and improve the corresponding results in Chang et al. [24], Kumam and Jaiboon [34], Wangkeeree and Kamraksa [32], and many others.


## 2. Preliminaries

LetH a real Hilbert space and C a nonempty closed convex subset of H. We denote strong convergence (weak convergence) by notation →(⇀). In a real Hilbert space H, it is well known that
(2.1)‖x-y‖2=‖x‖2-‖y‖2-2〈x-y,y〉,(2.2)‖x+y‖2≤‖x‖2+2〈y,x+y〉,(2.3)‖x+y‖2≥‖x‖2+2〈y,x〉,(2.4)‖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.Recall that for every pointx∈H, there exists a unique nearest point in C, denoted by PCx, such that
(2.5)‖x-PCx‖≤‖x-y‖,∀y∈C.PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive mapping of H onto C and satisfies
(2.6)〈x-y,PCx-PCy〉≥‖PCx-PCy‖2
for every x,y∈H. Obviously, this immediately implies that
(2.7)‖(x-y)-(PCx-PCy)‖2≤‖x-y‖2-‖PCx-PCy‖2,∀x,y∈H.
Moreover, PCx is characterized by the following properties: PCx∈C and
(2.8)〈x-PCx,y-PCx〉≤0,‖x-y‖2≥‖x-PCx‖2+‖y-PCx‖2
for all x∈H,y∈C.In order to prove our main results, we need the following lemmas.Lemma 2.1 (see [35]).
LetV:C→H be a k-strict pseudo-contraction, then(1)
the fixed point setF(V) of V is closed convex so that the projection PF(V) is well defined;(2)
define a mappingT:C→H by
(2.9)Tx=tx+(1-t)Vx,∀x∈C.
If t∈[k,1), then T is a nonexpansive mapping such that F(V)=F(T).A family of mappings{Vi:C→H}i=1∞ is called a family of uniformlyk-strict pseudo-contractions, if there exists a constant k∈[0,1) such that
(2.10)‖Vix-Viy‖2≤‖x-y‖2+k‖(I-Vi)x-(I-Vi)y‖2,∀x,y∈C,∀i≥1.
Let {Vi:C→C}i=1∞ be a countable family of uniformly k-strict pseudo-contractions. Let {Ti:C→C}i=1∞ be the sequence of nonexpansive mappings defined by (2.9), that is,
(2.11)Tix=tx+(1-t)Vix,∀x∈C,∀i≥1,t∈[k,1).Let{Ti} be a sequence of nonexpansive mappings of C into itself defined by (2.11) and let {μi} be a sequence of nonnegative numbers in [0,1]. For each n≥1, define a mapping Wn of C into itself as follows:
(2.12)Un,n+1=I,Un,n=μnTnUn,n+1+(1-μn)I,Un,n-1=μn-1Tn-1Un,n+(1-μn-1)I,⋮Un,k=μkTkUn,k+1+(1-μk)I,Un,k-1=μk-1Tk-1Un,k+(1-μk-1)I,⋮Un,2=μ2T2Un,3+(1-μ2)I,Wn=Un,1=μ1T1Un,2+(1-μ1)I.
Such a mapping Wn is nonexpansive from C to C and it is called the W-mapping generated by T1,T2,…,Tn and μ1,μ2,…,μn.For eachn,k∈ℕ, let the mapping Un,k be defined by (2.12). Then we can have the following crucial conclusions concerning Wn. You can find them in [36]. Now we only need the following similar version in Hilbert spaces.Lemma 2.2 (see [36]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T1,T2,… be nonexpansive mappings of C into itself such that ∩n=1∞F(Tn) is nonempty, let μ1,μ2,… be real numbers such that 0≤μn≤b<1 for every n≥1.  Then,(1)
Wn is nonexpansive and F(Wn)=∩i=1nF(Ti), for all n≥1;(2)
for everyx∈C and k∈ℕ, the limit lim⁡n→∞Un,kx exists;(3)
a mappingW:C→C defined by
(2.13)Wx:=lim⁡n→∞Wnx=lim⁡n→∞Un,1x,∀x∈C
is a nonexpansive mapping satisfying F(W)=∩i=1∞F(Ti) and it is called the W-mapping generated by T1,T2,… and μ1,μ2,….Lemma 2.3 (see [37]).
LetC be a nonempty closed convex subset of a Hilbert space H,  {Ti:C→C} a countable family of nonexpansive mappings with ∩i=1∞F(Ti)≠∅,  {μi} a real sequence such that 0<μi≤b<1,foralli≥1. If D is any bounded subset of C,  then
(2.14)lim⁡n→∞sup⁡x∈D‖Wx-Wnx‖=0.Lemma 2.4 (see [38]).
Each Hilbert spaceH satisfies Opial’s condition, that is, for any sequence {xn}⊂H with xn⇀x, the inequality
(2.15)liminf⁡n→∞‖xn-x‖<liminf⁡n→∞‖xn-y‖
holds for each y∈H with y≠x.Lemma 2.5 (see [39]).
AssumeA is a strongly positive linear bounded operator on H with coefficient γ̅>0 and 0<ρ≤∥A∥-1. Then, ∥I-ρA∥≤1-ργ̅.For solving the system of mixed equilibrium problems (1.15), let us assume that function Θk:H×H→ℝ,k=1,2,…,N satisfies the following conditions:(H1)
Θk is monotone, that is, Θk(x,y)+Θk(y,x)≤0, for all x,y∈H;(H2)
for each fixedy∈H, x↦Θk(x,y) is convex and upper semicontinuous;(H3)
for eachx∈H,y↦Θk(x,y) is convex.Letη:H×H→H and B:H→H be two mappings. B is said to be(1)
monotone if
(2.16)〈Bx-By,η(x,y)〉≥0,∀x,y∈H;(2)
d-strongly monotone if there exists a positive real number d such that
(2.17)〈Bx-By,η(x,y)〉≥d‖x-y‖2,∀x,y∈H;(3)
L-Lipschitz continuous if there exists a constant L>0 such that
(2.18)‖η(x,y)‖≤L‖x-y‖,∀x,y∈H.LetK:H→ℝ be a differentiable functional on H, which is called:(1)
η-convex [40] if
(2.19)K(y)-K(x)≥〈K′(x),η(y,x)〉,∀x,y∈H,
where K′(x) is the Fréchet derivative of K at x;(2)
η-strongly convex [41] if there exists a constant σ>0 such that
(2.20)K(y)-K(x)-〈K′(x),η(y,x)〉≥σ2‖x-y‖2,∀x,y∈H.In particular, ifη(x,y)=x-y for all x,y∈H, then K is said to be strongly convex.Lemma 2.6 (see [42]).
LetH  be a real Hilbert space and let ϕ be a lower semicontinuous and convex functional from H to ℝ. Let Θ be a bifunction from H×H to ℝ satisfying (H1)–(H3). Assume that(i)
η:H×H→H is λ-Lipschitz continuous with constant λ>0 such that
(a)
η(x,y)+η(y,x)=0,forallx,y∈H,(b)
η(·,·) is affine in the first variable,(c)
for each fixedx∈H, y↦η(x,y) is sequentially continuous from the weak topology to the weak topology;(ii)
K:H→ℝ is η-strongly convex with constant σ>0 and its derivative K′ is sequentially continuous from the weak topology to the strong topology;(iii)
for eachx∈H, there exist bounded subsets Ex⊂H and zx∈H such that for any y∈H∖Ex,
(2.21)Θ(y,zx)+ϕ(zx)-ϕ(y)+1r〈K′(y)-K′(x),η(zx,y)〉<0.
For given r>0, let JrΘ:H→H be the mapping defined by
(2.22)JrΘ(x)={y∈H:Θ(y,z)+ϕ(z)-ϕ(y)+1r〈K′(y)-K′(x),η(z,y)〉≥0,∀z∈H}for all x∈H. Then(1)
JrΘ is single-valued.(2)
F(JrΘ)=MEP⁡(Θ,ϕ), where MEP⁡(Θ,ϕ) is the set of solution of the mixed equilibrium problem,
(2.23)Θ(x,y)+ϕ(y)-ϕ(x)≥0,∀y∈H.(3)
MEP⁡(Θ,ϕ) is closed and convex.Lemma 2.7 (see [43]).
Let{xn} and {vn} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with 0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1. Suppose xn+1=(1-βn)vn+βnxn for all integers n≥0 and limsup⁡n→∞(∥vn+1-vn∥-∥xn+1-xn∥)≤0. Then, lim⁡n→∞∥vn-xn∥=0.Lemma 2.8 (see [44]).
Assume{xn} is a sequence of nonnegative real numbers such that
(2.24)xn+1≤(1-an)xn+bn,∀n≥0,
where {an} is a sequence in (0,1) and {bn} is a sequence in ℝ such that(1)
∑n=1∞an=∞,(2)
limsup⁡n→∞(bn/an)≤0 or ∑n=1∞|bn|<∞.Then, lim⁡n→∞xn=0.Lemma 2.9 (see [45]).
LetC be a nonempty closed convex subset of a real Hilbert space H and g:C→ℝ∪{∞} a proper lower-semicontinuous differentiable convex function. If z is a solution to the minimization problem
(2.25)g(z)=inf⁡x∈Cg(x),
then
(2.26)〈g′(x),x-z〉≥0,x∈C.
In particular, if z solves problem OP, then
(2.27)〈u+[γf-(I+μA)]z,x-z〉≤0.Lemma 2.10 (see [46]).
LetC be a nonempty bounded closed convex subset of a Hilbert space H and let 𝒮={S(s):0≤s<∞} be a nonexpansive semigroup on C, then for any h≥0,
(2.28)lim⁡t→∞sup⁡x∈C‖1t∫0tT(s)xds-T(h)(1t∫0tT(s)xds)‖=0.Lemma 2.11 (see [47]).
Let C be a nonempty bounded closed convex subset ofH, {xn} a sequence in C, and 𝒮={S(s):0≤s<∞} a nonexpansive semigroup on C. If the following conditions are satisfied:(i)
xn⇀z;(ii)
limsup⁡s→∞limsup⁡n→∞∥S(s)xn-xn∥=0, then z∈𝒮.Lemma 2.12 (see [25]).
For givenx*,y*∈C and (x*,y*) is a solution of the problem (1.20) if and only if x* is a fixed point of the mapping G:C→C is defined by
(2.29)G(x)=PC[PC(x-δDx)-τBPC(x-δDx)],∀x∈H,
where y*=PC(x-δDx), δ and τ are positive constants and B,D:H→H are two mappings.Throughout this paper, the set of fixed points of the mappingG is denoted by SVI(C,B,D).Lemma 2.13 (see [32]).
LetG:C→C be defined in Lemma 2.12. If B:H→H is a LB-Lipschitzian and relaxed (c,d)-cocoercive mapping and D:H→H is a LD-Lipschitz and relaxed (c′,d′)-cocoercive mapping where τ≤2(d-cLB2)/LB2 and δ≤2(d′-c′LD2)/LD2, then G is nonexpansive.Lemma 2.14 (demiclosedness principle [48]).
Assume thatS is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space H. If S has a fixed point, then I-S is demiclosed; that is, whenever {xn} is a sequence in C converging weakly to some x∈C (for short, xn⇀x∈C), and the sequence {(I-S)xn} converges strongly to some y (for short, (I-S)xn→y), it follows that (I-S)x=y. Here I is the identity operator of H.


## 3. Main Results

In this section, we prove a strong convergence theorem of an iterative algorithm (3.1) for finding the solutions of a common element of the set of solutions of (1.15), the set of solutions of (1.20) for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space.Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H which C+C⊂C and let f be a contraction of C into itself with α∈(0,1). Let ϕ be a lower semicontinuous and convex functional from H to ℝ and let {Θk:H×H→ℝ,k=1,2,…,N} be a finite family of equilibrium functions satisfying conditions (H1)–(H3). Let 𝒮={S(s):0≤s<∞} be a nonexpansive semigroup on C and let {tn} be a positive real divergent sequence. Let {Vi:C→C}i=1∞ be a countable family of uniformly k-strict pseudo-contractions, let {Ti:C→C}i=1∞ be the countable family of nonexpansive mappings defined by Tix=tx+(1-t)Vix,forallx∈C,foralli≥1,t∈[k,1), let  Wn be the W-mapping defined by (2.12), and let W be a mapping defined by (2.13) with F(W)≠∅. Let A be a strongly positive linear bounded operator on H with coefficient γ̅>0 and let 0<γ<(1+μγ̅)/α, B:H→H be a LB-Lipschitz continuous and relaxed (c,d)-cocoercive mapping with d>cLB2, and let  D:H→H be a LD-Lipschitz continuous and relaxed (c′,d′)-cocoercive mapping with d′>c′LD2. Suppose that Ω:=F(𝒮)∩F(W)∩𝔉∩SVI(C,B,D)≠∅, where 𝔉=(∩k=1NMEP⁡(Θk,ϕ)). Let μ>0, γ>0 and rk>0,k=1,2,…,N, which are constants. For given x1∈H arbitrarily and fixed u∈H, suppose {xn}, {yn}, {zn} and {un(k)},k=1,2,…,N are the sequences generated iteratively by
(3.1)Θ1(un(1),x)+ϕ(x)-ϕ(un(1))+1r1〈K′(un(1))-K′(xn),η(x,un(1))〉≥0,∀x∈H,Θ2(un(2),x)+ϕ(x)-ϕ(un(2))+1r2〈K′(un(2))-K′(xn),η(x,un(2))〉≥0,∀x∈H,⋮ΘN(un(N),x)+ϕ(x)-ϕ(un(N))+1rN〈K′(un(N))-K′(xn),η(x,un(N))〉≥0,∀x∈H,zn=PC(un(N)-δDun(N)),yn=PC(zn-τBzn),xn+1=αn[u+γf(Wnxn)]+βnxn+[(1-βn)I-αn(I+μA)]1tn∫0tnS(s)Wnynds,
where
(3.2)un(1)=Jr1Θ1xn,un(k)=JrkΘkun(k-1)=JrkΘkJrk-1Θk-1un(k-2)=JrkΘk⋯Jr2Θ2un(1),=JrkΘk⋯Jr2Θ2Jr1Θ1xn,k=2,3,…,N,JrkΘk:H→H,k=1,2,…,N is the mapping defined by (2.22) and {αn} and {βn} are two sequences in (0,1) for all n∈ℕ. Assume the following conditions are satisfied:(C1)
η:H×H→H is λ-Lipschitz continuous with constant λ>0 such that
(a)
η(x,y)+η(y,x)=0,forallx,y∈H,(b)
x↦η(x,y) is affine,(c)
for each fixedy∈H, y↦η(x,y) is sequentially continuous from the weak topology to the weak topology;(C2)
K:H→ℝ is η-strongly convex with constant σ>0 and its derivative K′ is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant ν>0 such that σ>λν;(C3)
for eachk∈{1,2,…,N} and for all x∈H, there exist bounded subsets Ex⊂H and zx∈H such that for any y∈H∖Ex,
(3.3)Θk(y,zx)+ϕ(zx)-ϕ(y)+1rk〈K′(y)-K′(x),η(zx,y)〉<0;(C4)
lim⁡n→∞αn=0 and ∑n=1∞αn=∞;(C5)
0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1;(C6)
0<τ<2(d-cLB2)/LB2 and 0<δ<2(d′-c′LD2)/LD2.Then, {xn} converges strongly to x*∈Ω, which solves the following optimization problem (OP):
(3.4)min⁡x*∈Ωμ2〈Ax*,x*〉+12‖x*-u‖2-h(x*),
and (x*,y*) is a solution of the general system of variational inequality problem (1.20) such that y*=PC(x*-δDx*).Proof.
By the condition (C4) and (C5), we may assume, without loss of generality, thatαn≤(1-βn)(1+μ∥A∥)-1 for all n∈ℕ. Indeed, A is a strongly positive bounded linear operator on H, we have
(3.5)‖A‖=sup⁡{|〈Ax,x〉|:x∈H,‖x‖=1}.
Observe that
(3.6)〈((1-βn)I-αn(I+μA))x,x〉=1-βn-αn-αnμ〈Ax,x〉≥1-βn-αn-αnμ‖A‖≥0,
so this shows that (1-βn)I-αn(I+μA) is positive. It follows that
(3.7)‖(1-βn)I-αn(I+μA)‖=sup⁡{|〈((1-βn)I-αn(I+μA))x,x〉|:x∈H,‖x‖=1}=sup⁡{1-βn-αn-αnμ〈Ax,x〉:x∈H,‖x‖=1}≤1-βn-αn-αnμγ̅.
We shall divide the proofs into several steps.Step 1.
We show that{xn} is bounded.
Letx*∈Ω:=F(𝒮)∩F(W)∩(∩k=1NMEP⁡(Θk,ϕ))∩SVI(C,B,D). In fact, by the assumption that for each k∈{1,2,…,N}, JrkΘk is nonexpansive. Let 𝒜N:=JrNΘN⋯Jr2Θ2Jr1Θ1 and 𝒜0=I. Then, we have x*=𝒜Nx* and un(N)=𝒜Nxn. Since x*∈SVI(C,B,D), then
(3.8)x*=PC[PC(x*-δDx*)-τBPC(x*-δDx*)]=PC[PC(I-δD)ANx*-τBPC(I-δD)ANx*].
Putting y*=PC(x*-δDx*)=PC(I-δD)𝒜Nx*, we have x*=PC(y*-τBy*). Since x*=S(s)x*,foralls≥0 and x*=Wnx*,foralln≥1, therefore, we have
(3.9)x*=ANx*=PC(y*-τBy*)=WnPC(y*-τBy*)=S(s)WnPC(y*-τBy*).
Because PC and 𝒜N are nonexpansive mappings and from Remark 1.1, we have
(3.10)‖yn-x*‖=‖PC(zn-τBzn)-PC(y*-τBy*)‖≤‖(I-τB)zn-(I-τB)y*‖≤‖zn-y*‖=‖PC(un(N)-δDun(N))-PC(x*-δDx*)‖≤‖(I-δD)un(N)-(I-δD)x*‖≤‖un(N)-x*‖=‖ANxn-ANx*‖≤‖xn-x*‖
which yields that
(3.11)‖xn+1-x*‖=‖(1tn∫0tnS(s)Wnynds-x*)αnu+αn(γf(Wnxn)-(I+μA)x*)+βn(xn-x*)+((1-βn)I-αn(I+μA))(1tn∫0tnS(s)Wnynds-x*)‖≤αn‖u‖+αn‖γf(Wnxn)-(I+μA)x*‖+βn‖xn-x*‖+(1-βn-αn(1+μγ̅))‖xn-x*‖≤αn‖u‖+αn‖γf(Wnxn)-γf(x*)‖+αn‖γf(x*)-(I+μA)x*‖+βn‖xn-x*‖+(1-βn-αn(1+μγ̅))‖xn-x*‖≤αn‖u‖+αnγα‖xn-x*‖+αn‖γf(x*)-(I+μA)x*‖+βn‖xn-x*‖+(1-βn-αn(1+μγ̅))‖xn-x*‖=αn(‖u‖+‖γf(x*)-(I+μA)x*‖)+(1-αn(1+μγ̅)+αnγα)‖xn-x*‖=(1-αn((1+μγ̅)-γα))‖xn-x*‖+αn((1+μγ̅)-γα)‖u‖+‖γf(x*)-(I+μA)x*‖(1+μγ̅)-γα.
It follows from (3.11) and induction that
(3.12)‖xn-x*‖≤max⁡{‖x1-p‖,‖u‖+‖γf(x*)-(I+μA)x*‖(1+μγ̅)-γα},n≥1.
Hence, {xn} is bounded, so are {yn}, {zn}, {Wnxn}, {f(Wnxn)}, {un(k)} for all k=1,2,…,N and {KnWnyn}, where Kn=(1/tn)∫0tnS(s)ds.Step 2.
We prove thatlim⁡n→∞∥xn+1-xn∥=0 and lim⁡n→∞∥un+1(N)-un(N)∥=0.
Again, from Remark1.1, we have the following estimates:
(3.13)‖yn+1-yn‖=‖PC(zn+1-τBzn+1)-PC(zn-τBzn)‖≤‖(zn+1-τBzn+1)-(zn-τBzn)‖≤‖zn+1-zn‖=‖PC(un+1(N)-δDun+1(N))-PC(un(N)-δDun(N))‖≤‖(un+1(N)-δDun+1(N))-(un(N)-δDun(N))‖≤‖un+1(N)-un(N)‖=‖ANxn+1-ANxn‖≤‖xn+1-xn‖.
On the other hand, since Ti and Un,i are nonexpansive, we have
(3.14)‖Wn+1yn-Wnyn‖=‖μ1T1Un+1,2yn-μ1T1Un,2yn‖≤μ1‖Un+1,2yn-Un,2yn‖=μ1‖μ2T2Un+1,3yn-μ2T2Un,3yn‖≤μ1μ2‖Un+1,3yn-Un,3yn‖⋮≤μ1μ2⋯μn‖Un+1,n+1yn-Un,n+1yn‖≤M1∏i=1nμi,
where M1≥0 is a constant such that ∥Un+1,n+1yn-Un,n+1yn∥≤M1 for all n≥0. It follows from (3.13) and (3.14) that we have
(3.15)‖Wn+1yn+1-Wnyn‖≤‖Wn+1yn+1-Wn+1yn‖+‖Wn+1yn-Wnyn‖≤‖yn+1-yn‖+M1∏i=1nμi≤‖xn+1-xn‖+M1∏i=1nμi.
It follows that
(3.16)‖Kn+1Wn+1yn+1-KnWnyn‖=‖1tn+1∫0tn+1S(s)Wn+1yn+1ds-1tn∫0tnS(s)Wnynds‖≤1tn+1∫0tn+1‖S(s)Wn+1yn+1-S(s)Wnyn‖ds+‖1tn+1∫0tn+1S(s)Wnynds-1tn∫0tnS(s)Wnynds‖≤‖Wn+1yn+1-Wnyn‖+‖1tn+1∫tntn+1S(s)Wnynds+1tn+1∫0tnS(s)Wnynds-1tn∫0tnS(s)Wnynds‖≤‖Wn+1yn+1-Wnyn‖+1tn+1∫tntn+1‖S(s)Wnyn‖ds+|1tn+1-1tn|∫0tn‖S(s)Wnyn‖ds≤‖Wn+1yn+1-Wnyn‖+2(1-tntn+1)M2≤‖xn+1-xn‖+M1∏i=1nμi+2(1-tntn+1)M2,
where M2=max⁡{∥S(s)Wnyn∥}.
Settingxn+1=(1-βn)vn+βnxn, for all n≥1, we have
(3.17)vn=xn+1-βnxn1-βn=αn(u+γf(Wnxn))+((1-βn)I-αn(I+μA))KnWnyn1-βn.
Then, we obtain
(3.18)vn+1-vn=αn+1(u+γf(Wn+1xn+1))+((1-βn+1)I-αn+1(I+μA))Kn+1Wn+1yn+11-βn+1-αn(u+γf(Wnxn))+((1-βn)I-αn(I+μA))KnWnyn1-βn=αn+11-βn+1(u+γf(Wn+1xn+1))-αn1-βn(u+γf(Wnxn))+Kn+1Wn+1yn+1-KnWnyn+αn1-βn(I+μA)KnWnyn-αn+11-βn+1(I+μA)Kn+1Wn+1yn+1=αn+11-βn+1((u+γf(Wn+1xn+1))-(I+μA)Kn+1Wn+1yn+1)+αn1-βn((I+μA)KnWnyn-u-γf(Wnxn))+Kn+1Wn+1yn+1-KnWnyn.
It follows from (3.16) and (3.18) that
(3.19)‖vn+1-vn‖-‖xn+1-xn‖≤αn+11-βn+1(‖u‖+‖γf(Wn+1xn+1)‖+‖(I+μA)Kn+1Wn+1yn+1‖)+αn1-βn(‖(I+μA)KnWnyn‖+‖u‖+‖γf(Wnxn)‖)+M1∏i=1nμi+2(1-tntn+1)M2.
By the conditions (C4), (C5) and from tn∈(0,∞), tn→∞ and 0<μi≤b<1,foralli≥1, we have
(3.20)limsup⁡n→∞(‖vn+1-vn‖-‖xn+1-xn‖)≤0.
Hence, by Lemma 2.7, we obtain
(3.21)lim⁡n→∞‖vn-xn‖=0.
It follows that
(3.22)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞(1-βn)‖vn-xn‖=0.
Applying (3.22) into (3.13), we obtain that
(3.23)lim⁡n→∞‖yn+1-yn‖=lim⁡n→∞‖zn+1-zn‖=lim⁡n→∞‖un+1(N)-un(N)‖=0.Step 3.
We show thatlim⁡n→∞∥KnWnyn-yn∥=0, lim⁡n→∞∥yn-S(s)yn∥=0, and lim⁡n→∞∥un(k+1)-un(k)∥=0, where Kn=(1/tn)∫0tnS(s)ds.
Sincexn+1=αn(u+γf(Wnxn))+βnxn+((1-βn)I-αn(I+μA))KnWnyn, we have
(3.24)‖xn-KnWnyn‖≤‖xn-xn+1‖+‖xn+1-KnWnyn‖=‖xn-xn+1‖+‖αn(u+γf(Wnxn))+βnxn+((1-βn)I-αn(I+μA))KnWnyn-KnWnyn‖=‖xn-xn+1‖+‖αn((u+γf(Wnxn))-(I+μA)KnWnyn)+βn(xn-KnWnyn)‖≤‖xn-xn+1‖+αn(‖u‖+‖γf(Wnxn)‖+‖(I+μA)KnWnyn‖)+βn‖xn-KnWnyn‖,
that is
(3.25)‖xn-KnWnyn‖≤11-βn‖xn-xn+1‖+αn1-βn(‖u‖+‖γf(Wnxn)‖+‖(I+μA)KnWnyn‖).
By (C4), (C5), and (3.22) it follows that
(3.26)lim⁡n→∞‖KnWnyn-xn‖=0.
SinceJrNΘN:C→C is firmly nonexpansive, un(N)=𝒜Nxn, where 𝒜N:=JrNΘN⋯Jr2Θ2Jr1Θ1 and x*∈Ω, we have
(3.27)‖un(N)-x*‖2=‖ANxn-ANx*‖2≤〈ANxn-ANx*,xn-x*〉=〈un(N)-x*,xn-x*〉=12(‖un(N)-x*‖2+‖xn-x*‖2-‖xn-un(N)‖2),
and hence
(3.28)‖un(N)-x*‖2≤‖xn-x*‖2-‖xn-un(N)‖2.
Observe that
(3.29)‖xn+1-x*‖2=‖((1-βn)I-αn(I+μA))(KnWnyn-x*)+βn(xn-x*)+αn(u+γf(Wnxn)-(I+μA)x*)‖2=‖((1-βn)I-αn(I+μA))(KnWnyn-x*)+βn(xn-x*)‖2+αn2‖u+γf(Wnxn)-(I+μA)x*‖2+2βnαn〈xn-x*,u+γf(Wnxn)-(I+μA)x*〉+2αn〈((1-βn)I-αn(I+μA))(KnWnyn-x*),u+γf(Wnxn)-(I+μA)x*〉≤[(1-βn-αn-αnμγ̅)‖KnWnyn-x*‖+βn‖xn-x*‖]2+αn2‖u+γf(Wnxn)-(I+μA)x*‖2+2βnαn〈xn-x*,u+γf(Wnxn)-(I+μA)x*〉+2αn〈((1-βn)I-αn(I+μA))(KnWnyn-x*),u+γf(Wnxn)-(I+μA)x*〉=[(1-βn-αn-αnμγ̅)‖KnWnyn-x*‖+βn‖xn-x*‖]2+cn≤(1-βn-αn-αnμγ̅)2‖KnWnyn-x*‖2+βn2‖xn-x*‖2+2(1-βn-αn-αnμγ̅)βn‖KnWnyn-x*‖‖xn-x*‖+cn≤(1-βn-αn-αnμγ̅)2‖KnWnyn-x*‖2+βn2‖xn-x*‖2+(1-βn-αn-αnμγ̅)βn[‖KnWnyn-x*‖2+‖xn-x*‖2]+cn=[(1-αn-αnμγ̅)2-2(1-αn-αnμγ̅)βn+βn2]‖KnWnyn-x*‖2+βn2‖xn-x*‖2+[(1-αn-αnμγ̅)βn-βn2][‖KnWnyn-x*‖2+‖xn-x*‖2]+cn=[(1-αn-αnμγ̅)2-(1-αn-αnμγ̅)βn]‖KnWnyn-x*‖2+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖yn-x*‖2+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn,
where
(3.30)cn=αn2‖u+γf(Wnxn)-(I+μA)x*‖2+2βnαn〈xn-x*,u+γf(Wnxn)-(I+μA)x*〉+2αn〈((1-βn)I-αn(I+μA))(KnWnyn-x*),u+γf(Wnxn)-(I+μA)x*〉.
It follows from condition (C4) that
(3.31)lim⁡n→∞cn=0.
Putting (3.28) into (3.29) and using also (3.10), we have
(3.32)‖xn+1-x*‖2≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖yn-x*‖2+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖un(N)-x*‖2+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅){‖xn-x*‖2-‖xn-un(N)‖2}+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn=(1-αn-αnμγ̅)2‖xn-x*‖2-(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖xn-un(N)‖2+cn≤‖xn-x*‖2-(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖xn-un(N)‖2+cn.
It follows that
(3.33)(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖xn-un(N)‖2≤‖xn-x*‖2-‖xn+1-x*‖2+cn≤‖xn-xn+1‖(‖xn-x*‖+‖xn+1-x*‖)+cn.
Therefore, by (3.22) and (3.31), we get
(3.34)lim⁡n→∞‖xn-un(N)‖=0.
Since
(3.35)‖un(N)-KnWnyn‖≤‖un(N)-xn‖+‖xn-KnWnyn‖,
and by (3.26) and (3.70), we have
(3.36)lim⁡n→∞‖un(N)-KnWnyn‖=0.
Since B is a LB-Lipschitz continuous and relaxed (c,d)-cocoercive mapping on B and 0<τ<2(d-cLB2)/LB2 for any x*∈Ω, we have
(3.37)‖yn-x*‖2=‖PC(zn-τBzn)-PC(y*-τBy*)‖2≤‖(zn-y*)-τ(Bzn-By*)‖2=‖zn-y*‖2-2τ〈zn-y*,Bzn-By*〉+τ2‖Bzn-By*‖2≤‖xn-x*‖2-2τ{-c‖Bzn-By*‖2+d‖zn-y*‖2}+τ2‖Bzn-By*‖2=‖xn-x*‖2+2τc‖Bzn-By*‖2-2τd‖zn-y*‖2+τ2‖Bzn-By*‖2≤‖xn-x*‖2+2τc‖Bzn-By*‖2-2τdLB2‖Bzn-By*‖2+τ2‖Byn-Bp‖2=‖xn-x*‖2+(2τc+τ2-2τdLB2)‖Bzn-By*‖2.
Similarly, since D is a LD-Lipschitz continuous and relaxed (c′,d′)-cocoercive mapping on D and 0<δ<2(d′-c′LD2)/LD2, we also have
(3.38)‖zn-y*‖2≤‖xn-x*‖2+(2δc′+δ2-2δd′LD2)‖Dun(N)-Dx*‖2.
Substituting (3.37) into (3.29), we have
(3.39)‖xn+1-x*‖2≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)×{∥xn-x*∥2+(2τc+τ2-2τdLB2)‖Bzn-By*‖2}+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn=(1-αn-αnμγ̅)2‖xn-x*‖2+(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)(2τc+τ2-2τdLB2)‖Bzn-By*‖2+cn≤‖xn-x*‖2+(2τc+τ2-2τdLB2)‖Bzn-By*‖2+cn.
Again, substituting (3.38) into (3.29) and using also (3.10), we get
(3.40)‖xn+1-x*‖2≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖yn-x*‖2+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖zn-y*‖2+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)×{‖xn-x*‖2+(2δc′+δ2-2δd′LD2)‖Dun(N)-Dx*‖2}+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn=(1-αn-αnμγ̅)2‖xn-x*‖2+(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)×(2δc′+δ2-2δd′LD2)‖Dun(N)-Dx*‖2+cn≤‖xn-x*‖2+(2δc′+δ2-2δd′LD2)‖Dun(N)-Dx*‖2+cn.
Therefore, by (3.39) and (3.40), we have
(3.41)(2τdLB2-2τc-τ2)‖Bzn-By*‖2≤‖xn-x*‖2-‖xn+1-x*‖2+cn≤‖xn-xn+1‖(‖xn-x*‖+‖xn+1-x*‖)+cn,(2δd′LD2-2δc′-δ2)‖Dun(N)-Dx*‖2≤‖xn-x*‖2-‖xn+1-x*‖2+cn≤‖xn-xn+1‖(‖xn-x*‖+‖xn+1-x*‖)+cn.
It follows from (3.22) and (3.31) that  we obtain
(3.42)lim⁡n→∞‖Bzn-By*‖=0,(3.43)lim⁡n→∞‖Dun(N)-Dx*‖=0.
From (2.6), we have
(3.44)‖zn-y*‖2=‖PC(un(N)-δDun(N))-PC(x*-δDx*)‖2≤〈(un(N)-δDun(N))-(x*-δDx*),zn-y*〉=12{‖(un(N)-δDun(N))-(x*-δDx*)‖2+‖zn-y*‖2-‖[(un(N)-δDun(N))-(x*-δDx*)]-(zn-y*)‖2}≤12{‖un(N)-x*‖2+‖zn-y*‖2-‖(un(N)-zn)-(x*-y*)-δ(Dun(N)-Dx*)‖2}=12{‖un(N)-x*‖2+‖zn-y*‖2-‖(un(N)-zn)-(x*-y*)‖2+2δ〈(un(N)-zn)-(x*-y*),Dun(N)-Dx*〉-δ2‖Dun(N)-Dx*‖2}≤12{‖xn-x*‖2+‖zn-y*‖2-‖(un(N)-zn)-(x*-y*)‖2+2δ‖(un(N)-zn)-(x*-y*)‖‖Dun(N)-Dx*‖-δ2‖Dun(N)-Dx*‖2}.
So, we obtain
(3.45)‖zn-y*‖2≤‖xn-x*‖2-‖(un(N)-zn)-(x*-y*)‖2+2δ‖(un(N)-zn)-(x*-y*)‖‖Dun(N)-Dx*‖-δ2‖Dun(N)-Dx*‖2.
By (3.29), we get
(3.46)‖xn+1-x*‖2≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖yn-x*‖2+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖zn-y*‖2+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)×{‖xn-x*‖2-‖(un(N)-zn)-(x*-y*)‖2+2δ‖(un(N)-zn)-(x*-y*)‖‖Dun(N)-Dx*‖-δ2‖Dun(N)-Dx*‖2}+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn=(1-αn-αnμγ̅)2‖xn-x*‖2+(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)×{-‖(un(N)-zn)-(x*-y*)‖2+2δ‖(un(N)-zn)-(x*-y*)‖‖Dun(N)-Dx*‖-δ2‖Dun(N)-Dx*‖2}+cn≤‖xn-x*‖2-(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖(un(N)-zn)-(x*-y*)‖2+2δ‖(un(N)-zn)-(x*-y*)‖‖Dun(N)-Dx*‖-δ2(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖Dun(N)-Dx*‖2+cn
which implies that
(3.47)(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖(un(N)-zn)-(x*-y*)‖2≤‖xn-x*‖2-‖xn+1-x*‖2+2δ‖(un(N)-zn)-(x*-y*)‖‖Dun(N)-Dx*‖-δ2(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖Dun(N)-Dx*‖2+cn≤‖xn-xn+1‖(‖xn-x*‖+‖xn+1-x*‖)+2δ‖(un(N)-zn)-(x*-y*)‖‖Dun(N)-Dx*‖-δ2(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖Dun(N)-Dx*‖2+cn.
From (3.22), (3.31), and (3.43), we have
(3.48)lim⁡n→∞‖(un(N)-zn)-(x*-y*)‖=0.
Now, from (2.2) and (2.7), we observe that
(3.49)‖(zn-yn)+(x*-y*)‖2=‖(zn-τBzn)-(y*-τBy*)-[PC(zn-τBzn)-PC(y*-τBy*)]+τ(Bzn-By*)‖2≤‖(zn-τBzn)-(y*-τBy*)-[PC(zn-τBzn)-PC(y*-τBy*)]‖2+2τ〈Bzn-By*,(zn-yn)+(x*-y*)〉≤‖(zn-τBzn)-(y*-τBy*)‖2-‖PC(zn-τBzn)-PC(y*-τBy*)‖2+2τ‖Bzn-By*‖‖(zn-yn)+(x*-y*)‖≤‖(zn-τBzn)-(y*-τBy*)‖2-‖KnWnPC(zn-τBzn)-KnWnPC(y*-τBy*)‖2+2τ‖Bzn-By*‖‖(zn-yn)+(x*-y*)‖=‖(zn-τBzn)-(y*-τBy*)‖2-‖KnWnyn-KnWnx*‖2+2τ‖Bzn-By*‖‖(zn-yn)+(x*-y*)‖=(‖(zn-τBzn)-(y*-τBy*)‖-‖KnWnyn-x*‖)×(‖(zn-τBzn)-(y*-τBy*)‖+‖KnWnyn-x*‖)+2τ‖Bzn-By*‖‖(zn-yn)+(x*-y*)‖≤‖(zn-τBzn)-(y*-τBy*)-(KnWnyn-x*)‖×(‖(zn-τBzn)-(y*-τBy*)‖+‖KnWnyn-x*‖)+2τ‖Bzn-By*‖‖(zn-yn)+(x*-y*)‖=‖(un(N)-KnWnyn)+(x*-y*)-(un(N)-zn)-τ(Bzn-By*)‖×(‖(zn-τBzn)-(y*-τBy*)‖+‖KnWnyn-x*‖)+2τ‖Bzn-By*‖‖(zn-yn)+(x*-y*)‖.
It follows from (3.36), (3.42), and (3.48) that  we have
(3.50)lim⁡n→∞‖(zn-yn)+(x*-y*)‖=0,
since
(3.51)‖KnWnyn-yn‖≤‖KnWnyn-un(N)‖+‖(un(N)-zn)-(x*-y*)‖+‖(zn-yn)+(x*-y*)‖.
It follows from (3.36), (3.48) and (3.50), we get
(3.52)lim⁡n→∞‖KnWnyn-yn‖=0,
and from (3.26), and (3.52) that  we have
(3.53)lim⁡n→∞‖xn-yn‖=0.
Since {Wnyn} is a bounded sequence in C, from Lemma 2.10 for all s≥0, we have
(3.54)lim⁡n→∞‖KnWnyn-S(s)KnWnyn‖=lim⁡n→∞‖1tn∫0tnS(s)Wnynds-S(s)(1tn∫0tnS(s)Wnynds)‖=0,
and since
(3.55)‖yn-S(s)yn‖≤‖yn-KnWnyn‖+‖KnWnyn-S(s)KnWnyn‖+‖S(s)KnWnyn-S(s)yn‖≤2‖yn-KnWnyn‖+‖KnWnyn-S(s)KnWnyn‖,
it follows from (3.52) and (3.54) that  we get
(3.56)lim⁡n→∞‖yn-S(s)yn‖=0.
On the other hand, sinceJrkΘk:H→H is firmly nonexpansive, 𝒜k:=JrkΘk⋯Jr2Θ2Jr1Θ1,k=1,2,…,N and x*∈Ω, we have
(3.57)‖Ak+1xn-x*‖2=‖Jrk+1Θk+1Akxn-Jrk+1Θk+1x*‖2≤〈Jrk+1Θk+1Akxn-x*,Akxn-x*〉=12(‖Jrk+1Θk+1Akxn-x*‖2+‖Akxn-x*‖2-‖Jrk+1Θk+1Akxn-Akxn‖2),
and hence
(3.58)‖Ak+1xn-x*‖2≤‖xn-x*‖2-‖Ak+1xn-Akxn‖2.
From (3.10), (3.29), and (3.58), for each k=1,2,…,N-1, we have
(3.59)‖xn+1-x*‖2≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖yn-x*‖2+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖Akxn-x*‖2+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖Ak+1xn-x*‖2+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn≤(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅){‖xn-x*‖2-‖Ak+1xn-Akxn‖2}+(1-αn-αnμγ̅)βn‖xn-x*‖2+cn=(1-αn-αnμγ̅)2‖xn-x*‖2-(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖Ak+1xn-Akxn‖2+cn≤‖xn-x*‖2-(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖Ak+1xn-Akxn‖2+cn.
It follows that
(3.60)(1-αn-αnμγ̅)(1-βn-αn-αnμγ̅)‖Ak+1xn-Akxn‖2≤‖xn-x*‖2-‖xn+1-x*‖2+cn≤‖xn-xn+1‖(‖xn-x*‖+‖xn+1-x*‖)+cn.
Therefore, by (3.22) and (3.31), we get
(3.61)lim⁡n→∞‖Ak+1xn-Akxn‖=0thatislim⁡n→∞‖un(k+1)-un(k)‖=0.Step 4.
We prove that(3.62)limsup⁡n→∞〈u+[γf-(I+μA)]x*,xn-x*〉≤0,
where x* is a solution of the optimization problem:
(3.63)min⁡x∈Ωμ2〈Ax*,x*〉+12‖x*-u‖2-h(x*).
To show this inequality, we can choose a subsequence {yni} of {yn} such that
(3.64)lim⁡i→∞〈u+[γf-(I+μA)]x*,yni-x*〉=limsup⁡n→∞〈u+[γf-(I+μA)]x*,yn-x*〉.
Since{yni} is bounded, there exists a subsequence {ynij} of {yni} which converges weakly to z∈C. Without loss of generality, we can assume that yni⇀z. From (3.53), we get xni⇀z.
Next, we show thatz∈Ω:=F(𝒮)∩F(W)∩𝔉∩SVI(C,B,D), where 𝔉=(∩k=1NMEP⁡(Θk,ϕ)).
(1) First, we prove thatz∈F(𝒮). Indeed, from Lemma 2.11 and (3.56), we get z∈F(𝒮), that is, z=S(s)z,foralls≥0.
(2) Next, we show thatz∈F(W)=∩n=1∞F(Wn), where F(Wn)=∩i=1nF(Ti),foralln≥1 and F(Wn+1)⊂F(Wn). Assume that z∉F(W), then there exists a positive integer m such that z∉F(Tm) and so z∉∩i=1mF(Ti). Hence for any n≥m,z∉∩i=1nF(Ti)=F(Wn), that is, z≠Wnz. This together with z=S(s)z,foralls≥0, shows z=S(s)z≠S(s)Wnz,foralls≥0; therefore, we have z≠KnWnz,foralln≥m. It follows from the Opial’s condition and (3.52) that
(3.65)liminf⁡i→∞‖yni-z‖<liminf⁡i→∞‖yni-KniWniz‖≤liminf⁡i→∞(‖yni-KniWniyni‖+‖KniWniyni-KniWniz‖)≤liminf⁡i→∞‖yni-z‖,
which is a contradiction. Thus, we get z∈F(W).
(3) Now, we prove thatz∈𝔉. Since 𝒜k+1=Jrk+1Θk+1𝒜k,k=1,2,…,N-1, and un(k+1)=𝒜k+1xn, we have
(3.66)Θ(Ak+1xn,x)+ϕ(x)-ϕ(Ak+1xn)+1rk+1〈K′(Ak+1xn)-K′(Akxn),η(x,Ak+1xn)〉≥0,∀x∈H.
It follows that
(3.67)1rk+1〈K′(Ak+1xni)-K′(Akxni),η(x,Ak+1xni)〉≥-Θ(Ak+1xni,x)-ϕ(x)+ϕ(Ak+1xni)
for all x∈H. From (3.61) and by conditions (C1)(c) and (C2), we get
(3.68)lim⁡ni→∞1rk+1〈K′(Ak+1xni)-K′(Akxni),η(x,Ak+1xni)〉=0.
By the assumption that ϕ is lower semicontinuous, then it is weakly lower semicontinuous and by the condition (H2) that x↦(-Θi(x,y)) is lower semicontinuous, then it is weakly lower semicontinuous. Since yni⇀z, it follows from (3.36), (3.52), and (3.61) that uni(k)⇀z for each k=1,2,…,N-1. Taking the lower limit ni→∞ in (3.67), we have
(3.69)Θk+1(z,x)+ϕ(x)-ϕ(z)≥0,∀x∈H,∀k=0,1,2,…,N-1.
Therefore, z∈∩k=1NMEP⁡(Θk,ϕ).
(4) Next, we show thatz∈SVI(C,B,D). By (3.36) and (3.52), we have
(3.70)‖un(N)-yn‖≤‖un(N)-KnWnyn‖+‖KnWnyn-yn‖⟶0asn⟶∞.
By Lemma 2.13 that G is a nonexpansive, we obtain
(3.71)‖yn-G(yn)‖=‖PC[PC(un(N)-δDun(N))-τBPC(un(N)-δDun(N))]-G(yn)‖=‖G(un(N))-G(yn)‖≤‖un(N)-yn‖.
Thus,
(3.72)lim⁡n→∞‖yn-G(yn)‖=0.
By Lemma 2.14, we obtain that z∈SVI(C,B,D). Hence z∈Ω is proved.
Now, from Lemma2.9, (3.64), and (3.53), we have
(3.73)limsup⁡n→∞〈u+[γf-(I+μA)]x*,xn-x*〉=limsup⁡n→∞〈u+[γf-(I+μA)]x*,yn-x*〉=lim⁡i→∞〈u+[γf-(I+μA)]x*,yni-x*〉=〈u+[γf-(I+μA)]x*,z-x*〉≤0.
By (3.52), (3.53), and (3.73), we obtain
(3.74)limsup⁡n→∞〈u+[γf-(I+μA)]x*,KnWnyn-x*〉≤0.Step 5.
Finally, we show thatxn→x*. From (3.1), we obtain
(3.75)‖xn+1-x*‖2=‖αn(u+γf(Wnxn))+βnxn+((1-βn)I-αn(I+μA))KnWnyn-x*‖2=‖((1-βn)I-αn(I+μA))(KnWnyn-x*)+βn(xn-x*)+αn(u+γf(Wnxn)-(I+μA)x*)‖2=‖((1-βn)I-αn(I+μA))(KnWnyn-x*)+βn(xn-x*)‖2+2αn〈((1-βn)I-αn(I+μA))(KnWnyn-x*),u+γf(Wnxn)-(I+μA)x*〉+2αnβn〈xn-x*,u+γf(Wnxn)-(I+μA)x*〉+αn2∥u+γf(Wnxn)-(I+μA)x*∥2≤[(1-βn-αn(1+μγ̅))‖KnWnyn-x*‖+βn‖xn-x*‖]2+2αn(1-βn)γ〈KnWnyn-x*,f(Wnxn)-f(x*)〉+2αn(1-βn)〈KnWnyn-x*,u+γf(x*)-(I+μA)x*〉-2αn2γ〈(I+μA)(KnWnyn-x*),f(Wnxn)-f(x*)〉-2αn2〈(I+μA)(KnWnyn-x*),u+γf(x*)-(I+μA)x*〉+2αnβnγ〈xn-x*,f(Wnxn)-f(x*)〉+2αnβn〈xn-x*,u+γf(x*)-(I+μA)x*〉+αn2‖u+γf(Wnxn)-(I+μA)x*‖2≤[(1-βn-αn(1+μγ̅))‖KnWnyn-x*‖+βn‖xn-x*‖]2+2αn(1-βn)γ‖KnWnyn-x*‖‖f(Wnxn)-f(x*)‖+2αn(1-βn)〈KnWnyn-x*,u+γf(x*)-(I+μA)x*〉-2αn2γ‖(I+μA)(KnWnyn-x*)‖‖f(Wnxn)-f(x*)‖-2αn2‖(I+μA)(KnWnyn-x*)‖‖u+γf(x*)-(I+μA)x*‖+2αnβnγ‖xn-x*‖‖f(Wnxn)-f(x*)‖+2αnβn〈xn-x*,u+γf(x*)-(I+μA)x*〉+αn2‖u+γf(Wnxn)-(I+μA)x*‖2≤[(1-βn-αn(1+μγ̅))‖xn-x*‖+βn‖xn-x*‖]2+2αn(1-βn)γα‖xn-x*‖2+2αn(1-βn)〈KnWnyn-x*,u+γf(x*)-(I+μA)x*〉-2αn2γα‖(I+μA)(KnWnyn-x*)‖‖xn-x*‖-2αn2‖(I+μA)(KnWnyn-x*)‖‖u+γf(x*)-(I+μA)x*‖+2αnβnγα‖xn-x*‖2+2αnβn〈xn-x*,u+γf(x*)-(I+μA)x*〉+αn2‖u+γf(Wnxn)-(I+μA)x*‖2=[(1-αn(1+μγ̅))+2αnγα]‖xn-x*‖2+αn{‖u+γf(Wnxn)-(I+μA)x*‖22(1-βn)〈KnWnyn-x*,u+γf(x*)-(I+μA)x*〉-2αnγα‖(I+μA)(KnWnyn-x*)‖‖xn-x*‖-2αn‖(I+μA)(KnWnyn-x*)‖‖u+γf(x*)-(I+μA)x*‖+2βn〈xn-x*,u+γf(x*)-(I+μA)x*〉+αn‖u+γf(Wnxn)-(I+μA)x*‖2}=[1-2αn(1+μγ̅)+αn2(1+μγ̅)2+2αnγα]‖xn-x*‖2+αn{‖u+γf(Wnxn)-(I+μA)x*‖22(1-βn)〈KnWnyn-x*,u+γf(x*)-(I+μA)x*〉-2αnγα‖(I+μA)(KnWnyn-x*)‖‖xn-x*‖-2αn‖(I+μA)(KnWnyn-x*)‖‖u+γf(x*)-(I+μA)x*‖+2βn〈xn-x*,u+γf(x*)-(I+μA)x*〉+αn‖u+γf(Wnxn)-(I+μA)x*‖2}.=[1-2αn(1+μγ̅-γα)]‖xn-x*‖2+αn{‖u+γf(Wnxn)-(I+μA)x*‖22(1-βn)〈KnWnyn-x*,u+γf(x*)-(I+μA)x*〉+2βn〈xn-x*,u+γf(x*)-(I+μA)x*〉+αn[(1+μγ̅)2‖xn-x*‖2-2γα‖(I+μA)(KnWnyn-x*)‖‖xn-x*‖-2‖(I+μA)(KnWnyn-x*)‖‖u+γf(x*)-(I+μA)x*‖+‖u+γf(Wnxn)-(I+μA)x*‖2]}.
Since {xn}, {f(Wnxn)}, and {KnWnyn} are bounded, there exist M>0 such that
(3.76)(1+μγ̅)2‖xn-x*‖2-2γα‖(I+μA)(KnWnyn-x*)‖‖xn-x*‖-2‖(I+μA)(KnWnyn-x*)‖‖u+γf(x*)-(I+μA)x*‖+‖u+γf(Wnxn)-(I+μA)x*‖2≤M
for all n≥0. It follows that
(3.77)‖xn+1-x*‖2≤(1-αnan)‖xn-x*‖2+αnbn,
where
(3.78)an=2(1+μγ̅-γα),bn=2(1-βn)〈KnWnyn-x*,u+γf(x*)-(I+μA)x*〉+2βn〈xn-x*,u+γf(x*)-(I+μA)x*〉+αnM.
Applying Lemma 2.8 to (3.77), we conclude that xn→x*. This completes the proof.Remark 3.2.
For example, of the control conditions (C4)–(C6), we setαn=1/10n,βn=n/(n+1). We set B,D is a 1-Lipschitz continuous and relaxed (0,1)-cocoercive mapping, (i.e., LB=1=LD and c=0=c′,d=1=d′).
Then, we can chooseτ∈(0,2) and δ∈(0,2) which satisfies the condition (C6) in Theorem 3.1.Corollary 3.3.
LetC be a nonempty closed convex subset of a real Hilbert space H which C+C⊂C and let  f be a contraction of C into itself with α∈(0,1). Let ϕ be a lower semicontinuous and convex functional from H to ℝ and let Θ:H×H→ℝ be a finite family of equilibrium functions satisfying conditions (H1)–(H3). Let 𝒮={S(s):0≤s<∞} be a nonexpansive semigroup on C and let {tn} be a positive real divergent sequence. Let {Vi:C→C}i=1∞ be a countable family of uniformly k- strict pseudo-contractions, let {Ti:C→C}i=1∞ be the countable family of nonexpansive mappings defined by Tix=tx+(1-t)Vix,forallx∈C,foralli≥1,t∈[k,1),  let Wn be the W-mapping defined by (2.12), and let W be a mapping defined by (2.13) with F(W)≠∅. Let A be a strongly positive linear bounded operator on H  with coefficient γ̅>0 and let 0<γ<(1+μγ̅)/α,  B:H→H be a LB-Lipschitz continuous and relaxed (c,d)-cocoercive mapping with d>cLB2, and let D:H→H be a LD-Lipschitz continuous and relaxed (c′,d′)-cocoercive mapping with d′>c′LD2. Suppose that Ω:=F(𝒮)∩F(W)∩MEP⁡(Θ,ϕ)∩SVI(C,B,D)≠∅. Let μ>0,  γ>0 and r>0,  which are constants. For given  x1∈H arbitrarily and fixed u∈H, suppose {xn},  {yn},{zn}, and{un} are the sequences generated iteratively by
(3.79)Θ(un,x)+ϕ(x)-ϕ(un)+1r〈K′(un)-K′(xn),η(x,un)〉≥0,∀x∈H,zn=PC(un-δDun),yn=PC(zn-τBzn),xn+1=αn[u+γf(Wnxn)]+βnxn+[(1-βn)I-αn(I+μA)]1tn∫0tnS(s)Wnynds,
where un=JrΘxn such that JrΘ:H→H is the mapping defined by (2.22) and {αn} and {βn} are two sequences in (0,1) for all n∈ℕ. If the functions η:H×H→H and K:H→ℝ satisfy the conditions (C1)–(C6) as given in Theorem 3.1, then {xn} converges strongly to x*∈Ω, which solves the following optimization problem (OP):
(3.80)min⁡x*∈Ωμ2〈Ax*,x*〉+12‖x*-u‖2-h(x*),
and (x*,y*) is a solution of the general system of variational inequality problem (1.20) such that y*=PC(x*-δDx*).Proof.
TakingN=1 in Theorem 3.1. Hence, the conclusion follows. This completes the proof.Corollary 3.4.
LetC be a nonempty closed convex subset of a real Hilbert space H which C+C⊂C and let f be a contraction of C into itself with α∈(0,1). Let 𝒮={S(s):0≤s<∞} be a nonexpansive semigroup on C and let {tn} be a positive real divergent sequence. Let {Vi:C→C}i=1∞ be a countable family of uniformly k-strict pseudo-contractions, let {Ti:C→C}i=1∞ be the countable family of nonexpansive mappings defined by Tix=tx+(1-t)Vix,forallx∈C,foralli≥1,t∈[k,1),  let Wn be the W-mapping defined by (2.12), and  let W be a mapping defined by (2.13) with F(W)≠∅.  Let A be a strongly positive linear bounded operator on H with coefficient γ̅>0 and let 0<γ<(1+μγ̅)/α,  B:H→H be a  LB-Lipschitz continuous and relaxed (c,d)-cocoercive mapping with d>cLB2, and let D:H→H be a  LD-Lipschitz continuous and relaxed (c′,d′)-cocoercive mapping with d′>c′LD2. Suppose that Ω:=F(𝒮)∩F(W)∩SVI(C,B,D)≠∅. Let μ>0 and γ>0, which are constants. For given x1∈H arbitrarily and fixed u∈H,  suppose {xn},  {yn},  and{zn} are the sequences generated iteratively by
(3.81)zn=PC(xn-δDxn),yn=PC(zn-τBzn),xn+1=αn[u+γf(Wnxn)]+βnxn+[(1-βn)I-αn(I+μA)]1tn∫0tnS(s)Wnynds,
where {αn} and {βn} are two sequences in (0,1) for all n∈ℕ. If the sequence {xn} satisfy the conditions (C1)–(C6) as given in Theorem 3.1, then {xn} converges strongly to x*∈Ω, which solves the following optimization problem (OP):
(3.82)min⁡x*∈Ωμ2〈Ax*,x*〉+12‖x*-u‖2-h(x*),
and (x*,y*) is a solution of the general system of variational inequality problem (1.20) such that y*=PC(x*-δDx*).Proof.
PutΘ(x,y)≡ϕ(x)≡0 for all x,y∈H and r=1. Take K(x)=∥x∥2/2 and η(y,x)=y-x, for all x,y∈H. Then, we get un=PCxn=xn in Corollary 3.3. Hence, the conclusion follows. This completes the proof.Corollary 3.5.
LetC be a nonempty closed convex subset of a real Hilbert space H and  let f be a contraction of H into itself with α∈(0,1). Let 𝒮={S(s):0≤s<∞} be a nonexpansive semigroup on C and let {tn} be a positive real divergent sequence. Let A be a strongly positive linear bounded operator on H with coefficient γ̅>0 and let 0<γ<(1+μγ¯)/α,  B:H→H be a LB-Lipschitz continuous and relaxed (c,d)-cocoercive mapping with d>cLB2. Suppose that Ω:=F(𝒮)∩B-10≠∅. Let μ>0 and γ>0, which are constants. For given x1∈H arbitrarily and fixed u∈H, suppose the {xn},  {yn}, and {zn} are the sequences generated iteratively by
(3.83)zn=xn-τBxn,yn=zn-τBzn,xn+1=αn[u+γf(xn)]+βnxn+[(1-βn)I-αn(I+μA)]1tn∫0tnS(s)ynds,
where {αn} and {βn} are two sequences in (0,1) for all n∈ℕ. If the sequence {xn} satisfy the conditions (C1)–(C6) as given in Theorem 3.1, then {xn} converges strongly to x*∈Ω.Proof.
Settingτ=δ, C≡H, D≡B and Wn≡PH≡I in Corollary 3.4, it follows from the proof of Theorem  4.1 in [25] that B-10=VI⁡(H,B). Hence, the conclusion follows. This completes the proof.



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*Source: 414831-2012-05-06.xml*