# An Iterative Method for Solving a System of Mixed Equilibrium Problems, System of Quasivariational Inclusions, and Fixed Point Problems of Nonexpansive Semigroups with Application to Optimization Problems

Authors: Pongsakorn Sunthrayuth; Poom Kumam
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce a general implicit iterative scheme base on viscosity approximation method with aϕ-strongly pseudocontractive mapping for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed point for a nonexpansive semigroup, and the set of solutions of system of variational inclusions with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mappings in Hilbert spaces. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator.


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## Body

## 1. Introduction

Throughout this paper we denoted byℕ and ℝ+ the set of all positive integers and all positive real numbers, respectively. We always assume that H be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively, C is a nonempty closed convex subset of H. Let φ:C→ℝ be a real-valued function and Θ:C×C→ℝ be an equilibrium bifunction. The mixed equilibrium problem (for short, MEP) is to find x*∈C such that(1.1)Θ(x*,y)+φ(y)-φ(x*)≥0,∀y∈C.
The set of solutions of (1.1) is denoted by MEP(Θ,φ), that is,(1.2)MEP⁡(Θ,φ)={x*∈C:Θ(x*,y)+φ(y)-φ(x*)≥0,∀y∈C}.In particular, ifφ=0, this problem reduces to the equilibrium problem, that is, to find x*∈C such that(1.3)Θ(x*,y)≥0,∀y∈C.
The set of solution of (1.3) is denoted by EP(Θ).Mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [1–6]). Some methods have been proposed to solve the equilibrium problem, see, for instance, [7–21].LetA be a strongly positive bounded linear operator on H, that is, there exists a constant γ̅>0 such that(1.4)〈Ax,x〉≥γ̅‖x‖2,∀x∈H.Recall that, a mappingf:H→H is said to be contractive if there exists a constant α∈(0,1) such that(1.5)‖f(x)-f(y)‖≤α‖x-y‖,∀x,y∈H.
A mapping T:H→H is said to be (i)
nonexpansive if
(1.6)‖Tx-Ty‖≤‖x-y‖,∀x,y∈H,(ii)
pseudocontractive if
(1.7)〈Tx-Ty,x-y〉≤‖x-y‖2,∀x,y∈H,(iii)
ϕ-strongly pseudocontractive if there exists a continuous and strictly increasing function ϕ:ℝ+→ℝ+ with ϕ(0)=0 such that
(1.8)〈Tx-Ty,x-y〉≤‖x-y‖2-ϕ(‖x-y‖)‖x-y‖,∀x,y∈H.It is obvious that pseudocontractive mapping is more general thanϕ-strongly pseudocontractive mapping. If ϕ(t)=αt with 0<α<1, then ϕ-strongly pseudocontractive mapping reduces to β-strongly pseudocontractive mapping with 1-α=β∈(0,1), which is more general than contractive mapping.Definition 1.1.
A one-parameter family mapping𝒮={T(t):t∈ℝ+} from C into itself is said to be a nonexpansive semigroup on C if it satisfies the following conditions: (i)
T(0)x=xforallx∈C,(ii)
T(s+t)=T(s)∘T(t) for all s,t∈ℝ+,(iii)
for eachx∈C the mapping t↦T(t)x is continuous,(iv)
∥T(t)x-T(t)y∥≤∥x-y∥ for all x,y∈C and t∈ℝ+.Remark 1.2.
We denote byF(𝒮) the set of all common fixed points of 𝒮, that is, F(𝒮):=⋂t∈ℝ+F(T(t))={x∈C:T(t)x=x}.Recall the following definitions of a nonlinear mappingB:C→H, the following are mentioned.Definition 1.3.
The nonlinear mappingB:C→H is said to be (i)
monotone if
(1.9)〈Bx-By,x-y〉≥0,∀x,y∈C,(ii)
β-strongly monotone if there exists a constant β>0 such that
(1.10)〈Bx-By,x-y〉≥β‖x-y‖2,∀x,y∈C,(iii)
L-Lipschitz continuous if there exists a constant L>0 such that
(1.11)‖Bx-By‖≤L‖x-y‖,∀x,y∈C,(iv)
ν-inverse-strongly monotone if there exists a constant ν>0 such that
(1.12)〈Bx-By,x-y〉≥ν‖Bx-By‖2,∀x,y∈C,(v)
relaxed(c,d)-cocoercive if there exists a constants c,d>0 such that
(1.13)〈Bx-By,x-y〉≥(-c)‖Bx-By‖2+d‖dx-y‖2,∀x,y∈C.
The resolvent operator technique for solving variational inequalities and variational inclusions is interesting and important. The resolvent equation technique is used to develop powerful and efficient numerical techniques for solving various classes of variational inequalities, inclusions, and related optimization problems.Definition 1.4.
LetM:H→2H be a multivalued maximal monotone mapping. The single-valued mapping J(M,ρ):H→H,defined by
(1.14)J(M,ρ)(u)=(I+ρM)-1(u),∀u∈H,
is called resolvent operator associated withM, where ρ is any positive number and I is the identity mapping.
Next, we consider a system of quasivariational inclusions problem is to find(x*,y*)∈H×H such that
(1.15)0∈x*-y*+ρ1(B1y*+M1x*),0∈y*-x*+ρ2(B2x*+M2y*),
where Bi:H→H and Mi:H→2H are nonlinear mappings for each i=1,2.
As special cases of the problem (1.15), we have the following results. (1)
IfB1=B2=B and M1=M2=M, then the problem (1.15) is reduces to the following. Find (x*,y*)∈H×H such that
(1.16)0∈x*-y*+ρ1(By*+Mx*),0∈y*-x*+ρ2(Bx*+My*).(2)
Further, ifx*=y* in problem (1.16), then the problem (1.16) is reduces to the following. Find x*∈H such that
(1.17)0∈Bx*+Mx*.The problem (1.17) is called variational inclusion problem. We denote by VI(H,B,M) the set of solutions of the variational inclusion problem (1.17). Next, we consider two special cases of the problem (1.17). (1)
M=∂ϕ:H→2H, where ϕ:H→ℝ∪{+∞} is a proper convex lower semicontinuous function and ∂ϕ is the subdifferential of ϕ then the quasivariational inclusion problem (1.17) is equivalent to finding x*∈H such that 〈Bx*,x-x*〉+ϕ(x)-ϕ(x*)≥0, for all x∈H, which is said to be the mixed quasivariational inequality.(2)
IfM=∂δC, where C is a nonempty closed convex subset of H, and δC:H→[0,∞) is the indicator function of C, that is,
(1.18)δC(x)={0,x∈C,+∞,x∉C,
then the quasivariational inclusion problem (1.17) is equivalent to the classical variational inequality problem denoted by VI(C,B) which is to find x*∈C such that
(1.19)〈Bx*,x-x*〉≥0,∀x∈C.This problem is called Hartman-Stampacchia variational inequality problem (see e.g., [22–24]).
It is known that problem (1.17) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity, variational inequalities, optimal control, mathematical economics, equilibria, and game theory. Also various types of variational inclusions problems have been extended and generalized (see [25–40] and the references therein).
On the other hand, the following optimization problem has been studied extensively by many authors:(1.20)min⁡x∈Ω¯μ2〈Ax,x〉+12‖x-u‖2-h(x),
where Ω¯=⋂n=1∞Cn,C1,C2,… are infinitely many 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 all x∈H). This kind of optimization problem has been studied extensively by many authors (see, e.g. [41–44]) when Ω¯=⋂n=1∞Cn and h(x)=〈x,b〉, where b is a given point in H.
Li et al. [45] introduced two steps of iterative procedures for the approximation of common fixed point of a nonexpansive semigroup 𝒮={T(t):t∈ℝ+} on a nonempty closed convex subset C in a Hilbert space. Recently, Liu et al. [46] introduced a hybrid iterative scheme for finding a common element of the set of solutions of system of mixed equilibrium problems, the set of common fixed points for nonexpansive semigroup and the set of solution of quasivariational inclusions with multivalued maximal monotone mappings and inverse-strongly monotone mappings. Very recently, Hao [47] introduced a general iterative method for finding a common element of solution set of quasivariational inclusion problems and the set of common fixed points of an infinite family of nonexpansive mappings.
In this paper, motivated and inspired by Li et al. [45], Liu et al. [46], and Hao [47], we introduce a general implicit iterative algorithm base on viscosity approximation methods with a ϕ-strongly pseudocontractive mapping which is more general than a contraction mapping for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed point for a nonexpansive semigroup, and the set of solutions of system of variational inclusions (1.15) with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mappings in Hilbert spaces. We prove that the proposed iterative algorithm converges strongly to a common element of the above three sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results obtained in this paper extend and improve several recent results in this area.


## 2. Preliminaries

In the sequel, we usexn⇀x and xn→x to denote the weak convergence and strong convergence of the sequence {xn} in H, respectively.This collects some results that will be used in the proofs of our main results.Proposition 2.1 (see [21]).
(i) The resolvent operator  J(M,ρ) associated with  M is single-valued and nonexpansive for all  ρ>0, that is,
(2.1)‖J(M,ρ)(x)-J(M,ρ)(y)‖≤‖x-y‖,∀x,y∈H,∀ρ>0.(ii) The resolvent operator J(M,ρ) is 1-inverse-strongly monotone, that is,
(2.2)‖J(M,ρ)(x)-J(M,ρ)(y)‖2≤〈x-y,J(M,ρ)(x)-J(M,ρ)(y)〉,∀x,y∈H.
Obviously, this immediately implies that
(2.3)‖(x-y)-(J(M,ρ)(x)-J(M,ρ)(y))‖2≤‖x-y‖2-‖J(M,ρ)(x)-J(M,ρ)(y)‖2,∀x,y∈H.For solving the equilibrium problem for bifunctionΘ:H×H→ℝ, let us assume that satisfies the following conditions:(H1)
Θ(x,x)=0 for all x∈H;(H2)
Θ is monotone, that is, Θ(x,y)+Θ(y,x)≤0 for all x,y∈H;(H3)
for eachy∈H, x↦Θ(x,y) is concave and upper semicontinuous;(H4)
for eachy∈H, x↦Θ(x,y) is convex;(H5)
for eachy∈H, x↦Θ(x,y) is lower semicontinuous.Definition 2.2.
A mapη:H×H→Hiscalled Lipschitz continuous, if there exists a constant L>0 such that
(2.4)‖η(x,y)‖≤L‖x-y‖,∀x,y∈H.
A differentiable function K:H→ℝ on a convex set H is called (i)
η—convex[7] if(2.5)K(y)-K(x)≥〈K′(x),η(y,x)〉,∀x,y∈H,
where K′(x) is the Fréchet differentiable of K at x,(i)
η—strongly convex [7] if there exists a constant ν>0 such that(2.6)K(y)-K(x)-〈K′(x),η(x,y)〉≥(ν2)‖x-y‖2,∀x,y∈H.
Let Θ:H×H→ℝ be an equilibrium bifunction satisfying the conditions (H1)–(H5). Let r be any given positive number. For a given point x∈H, consider the following auxiliary problem for MEP (for short, MEP (x,y)) to find y∈H such that
(2.7)Θ(y,z)+φ(z)-φ(y)+1r〈K′(y)-K′(x),η(z,y)〉≥0,∀z∈H,
where η:H×H→H is a mapping, and K′(x) is the Fréchet derivative of a functional K:H→ℝ at x. Let Vr(Θ,φ):H→H be the mapping such that for each x∈H, Vr(Θ,φ)(x) is the set of solutions of MEP (x,y), that is,
(2.8)Vr(Θ,φ)(x)={1ry∈H:Θ(y,z)+φ(z)-φ(y)+1r〈K′(y)-K′(x),η(z,y)〉≥0,∀z∈H},∀x∈H.
Then the following conclusion holds.Proposition 2.3 (see [7]).
LetH be a real Hilbert space,  φ:H→ℝ be a a lower semicontinuous and convex functional. Let  Θ:H×H→ℝ be an equilibrium bifunction satisfying conditions  (H1)–(H5). Assume that  (i)
η:H×H→ℝ is Lipschitz continuous with constant σ>0 such that
(a)
η(x,y)+η(y,x)=0 for all x,y∈H;(b)
η(·,·) is affine in the first variable;(c)
for each fixedy∈H, x↦η(y,x) is continuous from the weak topology to the weak topology;(ii)
K:H→ℝ is η-strongly convex with constant μ>0, and its derivative K′ is continuous from the weak topology to the strong topology;(iii)
for eachx∈H, there exists a bounded subset Dx⊂H and zx∈H such that for all y∉Dx,(2.9)Θ(y,zx)+φ(zx)-φ(y)+1r〈K′(y)-K′(x),η(zx,y)〉<0.
Then the following hold: (i)
Vr(Θ,φ) is single valued;(ii)
F(Vr(Θ,φ))=MEP(Θ,φ);(iii)
MEP(Θ,φ) is closed and convex.Lemma 2.4 (see [48]).
LetC be a nonempty bounded closed and convex subset of a real Hilbert space  H. Let  𝒮={T(t):t∈ℝ+} be a nonexpansive semigroup on  C, then for all  h>0,
(2.10)lim⁡t→∞sup⁡x∈C⁡‖1t∫0tT(s)xds-T(h)(1t∫0tT(s)xds)‖=0.Lemma 2.5 (see [49]).
LetX be a uniformly convex Banach space,  C be a nonempty closed and convex subset of  X, and  T:C→X be a nonexpansive mapping. Then  I-T is demiclosed at zero.Lemma 2.6 (see [50]).
Assume thatA is a strongly positive linear bounded operator on  H with coefficient  γ̅>0 and  0<ρ≤∥A∥-1. Then  ∥I-ρA∥≤1-ργ̅.Lemma 2.7 (see [51]).
LetX be a Banach space and  f:X→X be a  ϕ-strongly pseudocontractive and continuous mapping. Then  f has a unique fixed point in  X.Lemma 2.8.
In a real Hilbert spaceH, the following inequality holds:
(2.11)‖x+y‖2≤‖x‖2+2〈y,x+y〉,∀x,y∈H.The following lemma can be found in [52, 53] (see also Lemma  2.2 in [54]).Lemma 2.9.
LetC be a nonempty closed and convex subset of a real Hilbert space H and g:C→ℝ∪{+∞} be a proper lower semicontinuous differentiable convex function differentiable convex function. If x* is a solution to the minimization problem
(2.12)g(x*)=inf⁡x∈Cg(x),
then
(2.13)〈g′(x),x-x*〉≥0,x∈C.
In particular, if x* solves the optimization problem
(2.14)min⁡x∈Ωμ2〈Ax,x〉+12‖x-u‖2-h(x),
then
(2.15)〈u+(γf-(I+μA))x*,x-x*〉≤0,x∈C,
where h is a potential function for γf.The following lemmas can be found in ([55, 56]). For the sake of the completeness, one includes its proof in a Hilbert space version. Without loss of generality, one assumes that c,d∈(0,1) and LB∈[1,∞).Lemma 2.10.
LetH be a real Hilbert space,  B:H→H be an  LB-Lipschitzian and relaxed  (c,d)-cocoercive mapping. Then, one has
(2.16)‖(I-ρB)x-(I-ρB)y‖2≤(1+2ρcLB2-2ρd+ρ2LB2)‖x-y‖2,
where ρ>0. In particular, if 0<ρ≤2(d-cLB2)/LB2, then I-ρB is nonexpansive.Proof.
For allx,y∈H, we have
(2.17)‖(I-ρB)x-(I-ρB)y‖2=‖(x-y)-(ρBx-ρBy)‖2=‖x-y‖2-2ρ〈Bx-By,x-y〉+ρ2‖Bx-By‖2≤‖x-y‖2-2ρ[-c‖Bx-By‖2+d‖x-y‖2]+ρ2‖Bx-By‖2=‖x-y‖2-2ρd‖x-y‖2+2ρc‖Bx-By‖2+ρ2‖Bx-By‖2≤(1+2ρcLB2-2ρd+ρ2LB2)‖x-y‖2.
It is clear that, if 0<ρ≤2(d-cLB2)/LB2, then I-ρB is nonexpansive. This completes the proof.Lemma 2.11.
LetH be a real Hilbert space,  Mi:H→2H be the a maximal monotone mapping and  Bi:H→H be an  Li-Lipschitzian and relaxed  (ci,di)-cocoercive mapping for all  i=1,2. Let  Q:H→H be a mapping defined by
(2.18)Qx:=J(M1,ρ1)[J(M2,ρ2)(x-ρ2B2x)-ρ1B1J(M2,ρ2)(x-ρ2B2x)],∀x∈H.
If 0<ρi≤2(di-ciLi2)/Li2 for all i=1,2, then Q:H→H is nonexpansive.Proof.
By Lemma2.10, we know that (I-ρ2B2) and (I-ρ1B1) are nonexpansive, for all x,y∈H, we have
(2.19)‖Qx-Qy‖=‖J(M1,ρ1)[J(M2,ρ2)(x-ρ2B2x)-ρ1B1J(M2,ρ2)(x-ρ2B2x)]-J(M1,ρ1)[J(M2,ρ2)(y-ρ2B2y)-ρ1B1J(M2,ρ2)(y-ρ2B2y)]‖≤‖[J(M2,ρ2)(x-ρ2B2x)-ρ1B1J(M2,ρ2)(x-ρ2B2x)]-[J(M2,ρ2)(y-ρ2B2y)-ρ1B1J(M2,ρ2)(y-ρ2B2y)]‖=‖J(M2,ρ2)(I-ρ2B2)(I-ρ1B1)x-J(M2,ρ2)(I-ρ2B2)(I-ρ1B1)y‖≤‖(I-ρ2B2)(I-ρ1B1)x-(I-ρ2B2)(I-ρ1B1)y‖≤‖(I-ρ2B2)x-(I-ρ2B2)y‖≤‖x-y‖,
which implies that Q is nonexpansive. This completes the proof.Lemma 2.12.
For all(x*,y*)∈H×H, where  y*=J(M2,ρ2)(x*-ρ2B2x*),  (x*,y*) is a solution of the problem (1.15) if and only if  x* is a fixed point of the mapping  Q:H→H defined as in Lemma 2.11.Proof.
Let(x*,y*)∈H×H be a solution of the problem (1.15). Then, we have
(2.20)y*-ρ1B1y*∈(I+ρ1M1)x*,x*-ρ2B2x*∈(I+ρ2M2)y*,
which implies that
(2.21)x*=J(M1,ρ1)(y*-ρ1B1y*),y*=J(M2,ρ2)(x*-ρ2B2x*).
We can deduce that (2.21) is equivalent to
(2.22)x*=J(M1,ρ1)[J(M2,ρ2)(x*-ρ2B2x*)-ρ1B1J(M2,ρ2)(x*-ρ2B2x*)].
This completes the proof.


## 3. Main Results

Now, in this section, we prove our main results of this article. Before proving the main result we need the following lemma.Lemma 3.1.
LetH be a real Hilbert space. Let  𝒮={T(t):t∈ℝ+} be a nonexpansive semigroup from  H into itself. Then  I-σt(·) is monotone, where  σt(x):=(1/t)∫0tT(s)xds for all  x∈H and  t≥0.Proof.
For allx,y∈H, we have
(3.1)〈x-y,(I-σt(⋅))x-(I-σt(⋅))y〉=〈x-y,(I-1t∫0tT(s)ds)x-(I-1t∫0tT(s)ds)y〉=‖x-y‖2-〈x-y,1t∫0tT(s)xds-1t∫0tT(s)yds〉≥‖x-y‖2-‖x-y‖1t∫0t‖T(s)x-T(s)y‖ds≥‖x-y‖2-‖x-y‖2=0,
which implies that I-σt(·) is monotone. This completes the proof.Theorem 3.2.
LetH be a real Hilbert space. Let  φi:H→ℝ(i=1,2,…,N) be a finite family of lower semicontinuous and convex function,  Θi:H×H→ℝ(i=1,2,…,N) be a finite family of bifunctions satisfying  (H1)–(H5), and   ηi:H×H→H be a finite family of Lipschitz continuous mappings with a constant  σi(i=1,2,…,N). Let  𝒮={T(t):t∈ℝ+} be a nonexpansive semigroup from  H into itself,  Bi:H→H(i=1,2) be an  Li-Lipschitzian and relaxed  (ci,di)-cocoercive mapping with  ρi∈(0,2(di-ciLi2)/Li2] for all  i=1,2 and  Mi:H→2H(i=1,2) be a maximal monotone mapping. Assume that  Ω:=F(𝒮)∩⋂k=1NMEP  (Θk,φk)∩F(Q)≠∅, where  Q is defined as in Lemma 2.11. Let  f:H→H be a  ϕ-strongly pseudocontractive mapping with  limt→+∞ϕ(t)=+∞ and  A be a strongly positive linear bounded operator on  H with a coefficient  γ̅>0. Let  μ>0 and  γ>0 be two constants such that  0<γ<1+μγ̅. Let  {ri,n}(i=1,2,…,N) be a finite family of positive real sequence such that  liminfn→∞ri,n>0,  {αn} and  {βn} be two sequences in  [0,1], and  {tn} be a positive real divergent sequence. For any fixed  u∈H, let  {xn} be the sequence defined by
(3.2)Θ1(un(1),x)+φ1(x)-φ1(un(1))+1r1,n〈K1′(un(1))-K1′(xn),η1(x,un(1))〉≥0,∀x∈H,Θ2(un(2),x)+φ2(x)-φ2(un(2))+1r2,n〈K2′(un(2))-K2′(un(1)),η1(x,un(2))〉≥0,∀x∈H,⋮ΘN(un(N),x)+φN(x)-φN(un(N))+1rN,n〈KN′(un(N))-KN′(un(N-1)),ηN(x,un(N))〉≥0,∀x∈H,yn=J(M2,ρ2)(un(N)-ρ2B2un(N)),xn=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μA))1tn∫0tnT(s)J(M1,ρ1)(yn-ρ1B1yn)ds,
where
(3.3)un(1)=Vr1,n(Θ1,φ1)xn,un(i)=Vri,n(Θi,φi)un(i-1)=Vri,n(Θi,φi)Vri-1,n(Θi-1,φi-1)un(i-2)=Vri,n(Θi,φi)⋅⋅⋅Vr2,n(Θ2,φ2)un(1)=Vri,n(Θi,φi)⋯Vr2,n(Θ2,φ2)Vr1,n(Θ1,φ1)xn,i=2,3,…,N,
and Vri,n(Θi,φi):H→H, i=1,2,…,N is the mapping defined by (2.8). Assume the following. (i)
ηi:H×H→H is Lipschitz continuous with constant σi>0(i=1,2,…,N) such that
(a)
ηi(x,y)+ηi(y,x)=0 for all x,y∈H,(b)
ηi(·,·) is affine in the first variable,(c)
for each fixedy∈H, x↦ηi(y,x) is sequentially continuous from the weak topology to the weak topology.(ii)
Ki:H→ℝ is ηi-strongly convex with constant μi>0, and its derivative Ki′ is not only continuous from the weak topology to the strong topology but also Lipschitz continuous with constant νi such that μi≥σiνi.(iii)
For alli=1,2,...,N and for all x∈H, there exists a bounded subset Dx⊂H and zx∈H such that for all y∉Dx,(3.4)Θi(y,zn)+φi(zx)-φi(y)+1ri,n〈Ki′(y)-Ki′(x),ηi(zx,y)〉<0.
If the following conditions are satisfied: (C1)
limn→∞αn=0,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1, then the sequence {xn} defined by (3.2) converges strongly to x*∈Ω:=F(𝒮)∩⋂k=1NMEP(Θk,φk)∩F(Q), provided Vri,n(Θi,φi) is firmly nonexpansive, where x* is the unique solution of the variational inequality
(3.5)〈u+(γf-(I+μA))x*,z-x*〉≤0,∀z∈Ω,
or, equivalently, x* is the unique solution of the optimization problem
(3.6)min⁡x∈Ωμ2〈Ax,x〉+12‖x-u‖2-h(x),
where h is a potential function for γf and (x*,y*) is the solution of the problem (1.15), where y*=J(M2,ρ2)(x*-ρ2B2x*).Proof.
By the conditions(C1) and (C2), we may assume, without loss of generality, that αn≤(1-βn)(1+μ∥A∥)-1 for all n∈ℕ. Since A is a linear bounded self-adjoint operator on H, by (1.4), we have
(3.7)‖A‖=sup⁡{|〈Au,u〉|:u∈H,‖u‖=1}.
Observe that
(3.8)〈((1-βn)I-αn(I+μA))u,u〉=1-βn-αn-αnμ〈Au,u〉≥1-βn-αn-αnμ‖A‖≥0.
This shows that (1-βn)I-αn(I+μA) is positive. It follows that
(3.9)‖(1-βn)I-αn(I+μA)‖=sup⁡{|〈((1-βn)I-αn(I+μA))u,u〉|:u∈H,‖u‖=1}=sup⁡{1-βn-αn-αnμ〈Au,u〉:u∈H,‖u‖=1}≤1-βn-αn-αnμγ̅.
In fact, by the assumption that for all k∈{1,2,…,N}, Vrk,n(Θk,φk) is nonexpansive. Setting Vnk:=Vrk,n(Θk,φk)···Vr2,n(Θ2,φ2)Vr1,n(Θ1,φ1) for k∈{1,2,…,N} and Vn0:=I. Define a mapping Wn:H→H by
(3.10)Wnx:=1tn∫0tnT(s)QVnNxds,∀x∈H.
Hence, by Lemma 2.11 and nonexpansiveness (semigroup) of T(s) and Vnk, for all x,y∈C, we have
(3.11)‖Wnx-Wny‖=‖1tn∫0tnT(s)QVnNxds-1tn∫0tnT(s)QVnNyds‖≤‖QVnNx-QVnNy‖≤‖VnNx-VnNy‖≤‖x-y‖,
which implies that Wn is nonexpansive.
First, we show that{xn} defined by (3.2) is well defined. Define a mapping Tnf:H→H by
(3.12)Tnfx:=αn(u+γf(x))+βnx+((1-βn)I-αn(I+μA))Wnx,∀x∈H.
Indeed, by Lemma2.6, and from (3.11), for all x,y∈H, we have
(3.13)〈Tnfx-Tnfy,x-y〉=αnγ〈f(x)-f(y),x-y〉+βn〈x-y,x-y〉+〈((1-βn)I-αn(I+μA))(Wx-Wy),x-y〉≤αnγ(‖x-y‖2-ϕ(∥x-y∥)‖x-y‖)+βn‖x-y‖2+(1-βn-αn-αnμγ̅)‖x-y‖2=[1+αn(γ-(1+μγ̅))]∥x-y∥2-αnγϕ(∥x-y∥)∥x-y∥≤∥x-y∥2-αnγϕ(∥x-y∥)∥x-y∥.
This shows that Tnf is a ϕ-strongly pseudocontractive and strongly continuous. It follows from Lemma 2.7 that Tnf has a unique fixed point xn∈H, that is, {xn} defined by (3.2) is well defined.
Next, we show that uniqueness of the solution of the variational inequality (3.5). Suppose that x̃,x*∈Ω satisfy (3.5), then
(3.14)〈u+(γf-(I+μA))x*,x̃-x*〉≤0,〈u+(γf-(I+μA))x̃,x*-x̃〉≤0.
Adding up (3.14), we have
(3.15)0≥〈(I+μA)x*-(I+μA)x̃-γ(f(x*)-f(x̃)),x*-x̃〉=〈(I+μA)x*-(I+μA)x̃,x*-x̃〉-γ〈f(x*)-f(x̃),x*-x̃〉≥(1+μγ̅)‖x*-x̃‖2-γ‖x*-x̃‖2+ϕ(‖x*-x̃‖)(‖x*-x̃‖)=(1+μγ̅-γ)‖x*-x̃‖2+ϕ(‖x*-x̃‖)(‖x*-x̃‖).
It follows that
(3.16)(1+μγ̅-γ)‖x*-x̃‖+ϕ(‖x*-x̃‖)≤0,
which is a contradiction. Hence, x̃=x* and the uniqueness is proved.
Next, we show that{xn} is bounded. Taking x¯∈Ω, it follows from Lemma 2.11 that
(3.17)x¯=J(M1,ρ1)[J(M2,ρ2)(x¯-ρ2B2x¯)-ρ1B1J(M2,ρ2)(x¯-ρ2B2x¯)].
Putting y¯=J(M2,ρ2)(x¯-ρ2B2x¯), we have x¯=J(M1,ρ1)(y¯-ρ1B1y¯). Setting zn:=VnNxn and vn:=J(M1,ρ1)(yn-ρ1B1yn), then
(3.18)xn=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μA))1tn∫0tnT(s)vnds.
Since for all k∈{1,2,…,N}, Vrk,n(Θk,φk) is nonexpansive, we also have that  VnN is nonexpansive and x¯=VnNx¯, then
(3.19)‖zn-x¯‖=‖VnNxn-VnNx¯‖≤‖xn-x¯‖,∀n∈N.
By nonexpansiveness of J(Mi,ρi) and I-ρiBi  (i=1,2), we have
(3.20)‖vn-x¯‖=‖J(M1,ρ1)(yn-ρ1B1yn)-J(M1,ρ1)(y¯-ρ1B1y¯)‖≤‖(yn-ρ1B1yn)-(y¯-ρ1B1y¯)‖≤‖yn-y¯‖=‖J(M2,ρ2)(zn-ρ2B2zn)-J(M2,ρ2)(x¯-ρ2B2x¯)‖≤‖(zn-ρ2B2zn)-(x¯-ρ2B2x¯)‖≤‖zn-x¯‖≤‖xn-x¯‖.
It follows from (3.20) that
(3.21)‖xn-x¯‖2=〈xn-x¯,xn-x¯〉=αn〈u+γf(xn)-(I+μA)x¯,xn-x¯〉+βn〈xn-x¯,xn-x¯〉+〈((1-βn)I-αn(I+μA))(1tn∫0tn(T(s)vn-x¯)ds),xn-x¯〉=αnγ〈f(xn)-f(x¯),xn-x¯〉+αn〈u+γf(x¯)-(I+μA)x¯,xn-x¯〉+βn〈xn-x¯,xn-x¯〉+〈((1-βn)I-αn(I+μA))(1tn∫0tn(T(s)vn-x¯)ds),xn-x¯〉≤αnγ(‖xn-x¯‖2-ϕ(‖xn-x¯‖)‖xn-x¯‖)+αn〈u+γf(x¯)-(I+μA)x¯,xn-x¯〉βn‖xn-x¯‖2+(1-βn-αn-αnμγ̅)‖xn-x¯‖2,
and, so
(3.22)γϕ(‖xn-x¯‖)(‖xn-x¯‖)+(1-γ+μγ̅)‖xn-x¯‖2≤〈u+γf(x¯)-(I+μA)x¯,xn-x¯〉.
It follows that
(3.23)γϕ(‖xn-x¯‖)‖xn-x¯‖≤〈u+γf(x¯)-(I+μA)x¯,xn-x¯〉≤‖u+γf(x¯)-(I+μA)x¯‖‖xn-x¯‖.
Hence
(3.24)‖xn-x¯‖≤ϕ-1(‖u+γf(x¯)-(I+μA)x¯‖γ),
which implies that {xn} is bounded, so are {zn}, {yn}, and {vn}. Since f is ϕ-strongly pseudocontractive, we have
(3.25)〈f(xn)-f(x¯),xn-x¯〉≤‖xn-x¯‖2-ϕ(‖xn-x¯‖)‖xn-x¯‖≤‖xn-x¯‖2.
Thus {f(xn)} is bounded.
Next, we show thatlimn→∞∥xn-T(h)xn∥=0, for all h≥0. From (3.18), we observe that
(3.26)‖xn-1tn∫0tnT(s)vnds‖≤αn‖u+γf(xn)-(I+μA)1tn∫0tnT(s)vnds‖+βn‖xn-1tn∫0tnT(s)vnds‖.
It follows that
(3.27)‖xn-1tn∫0tnT(s)vnds‖≤αn1-βn‖u+γf(xn)-(I+μA)1tn∫0tnT(s)vnds‖.
By the conditions (C1) and (C2), we obtain
(3.28)lim⁡n→∞‖xn-1tn∫0tnT(s)vnds‖=0.
Let B={ω∈H:∥ω-x¯∥≤ϕ-1(∥u+γf(x¯)-(I+μA)x¯∥/γ)}, then B is nonempty bounded closed and convex subset of H, which is T(h)-invariant for all h≥0 and contains {xn}. It follows from Lemma 2.4 that
(3.29)lim⁡n→∞‖1tn∫0tnT(s)vnds-T(h)(1tn∫0tnT(s)vnds)‖=0.
On the other hand, we note that
(3.30)‖xn-T(h)xn‖≤‖xn-1tn∫0tnT(s)vnds‖+‖1tn∫0tnT(s)vnds-T(h)(1tn∫0tnT(s)vnds)‖+‖T(h)(1tn∫0tnT(s)vnds)-T(h)xn‖≤2‖xn-1tn∫0tnT(s)vnds‖+‖1tn∫0tnT(s)vnds-T(h)(1tn∫0tnT(s)vnds)‖.
From (3.28) and (3.29), for all h≥0, we have
(3.31)lim⁡n→∞‖xn-T(h)xn‖=0.
Next, we show thatlimn→∞∥Vnkxn-Vnk-1xn∥=0 for all k∈{1,2,…,N}. Since Vrk,n(Θk,φk) is firmly nonexpansive for all k∈{1,2,…,N} and Vnk=Vrk,n(Θk,φk)Vrk-1,n(Θk-1,φk-1)···Vr1,n(Θ1,φ1) for k∈{1,2,…,N}, hence for x¯∈Ω, we have
(3.32)‖Vnkxn-x¯‖2=‖Vrk,n(Θk,φk)Vnk-1xn-Vrk,n(Θk,φk)x¯‖2≤〈Vrk,n(Θk,φk)Vnk-1xn-Vrk,n(Θk,φk)x¯,Vnk-1xn-x¯〉=〈Vnkxn-x¯,Vnk-1xn-x¯〉=12{‖Vnkxn-x¯‖2‖Vnkxn-x¯‖2+‖Vnk-1xn-x¯‖2-‖Vnkxn-Vnk-1xn‖2}.
It follows that
(3.33)‖Vnkxn-x¯‖2≤‖xn-x¯‖2-‖Vnkxn-Vnk-1xn‖2.
Now, by Lemma 2.8, we have
(3.34)‖xn-x¯‖2=‖αnu+αn(γf(xn)-(I+μA)x¯)+βn(xn-1tn∫0tnT(s)vnds)+(I-αn(I+μA))(1tn∫0tnT(s)vnds-x¯)‖2≤‖(I-αn(I+μA))(1tn∫0tnT(s)vnds-x¯)+βn(xn-1tn∫0tnT(s)vnds)‖2+2αn〈u+γf(xn)-(I+μA)x¯,xn-x¯〉≤(‖(I-αn(I+μA))(1tn∫0tnT(s)vnds-x¯)‖+βn‖xn-1tn∫0tnT(s)vnds‖)2+2αn〈u+γf(xn)-(I+μA)x¯,xn-x¯〉≤((1-αn-αnμγ̅)‖vn-x¯‖+βn‖xn-1tn∫0tnT(s)vnds‖)2+2αn‖u+γf(xn)-(I+μA)x¯‖‖xn-x¯‖=(1-αn-αnμγ̅)2‖vn-x¯‖2+cn,
where
(3.35)cn:=βn2‖xn-1tn∫0tnT(s)vnds‖2+2(1-αn-αnμγ̅)βn‖vn-x¯‖‖xn-1tn∫0tnT(s)vnds‖+2αn‖u+γf(xn)-(I+μA)x¯‖‖xn-x¯‖.
From the condition (C1) and (3.28), we have
(3.36)lim⁡n→∞cn=0.
From (3.20), we observe that
(3.37)‖vn-x¯‖≤‖yn-x¯‖≤‖VnNxn-x¯‖≤‖Vnkxn-x¯‖,∀k∈{1,2,…,N}.
Substituting (3.33) into (3.34), we have
(3.38)‖xn-x¯‖2≤(1-αn-αnμγ̅)2{‖xn-x¯‖2-‖Vnkxn-Vnk-1xn‖2}+cn={1-2αn(1+μγ̅)+αn2(1+μγ̅)2}‖xn-x¯‖2-(1-αn-αnμγ̅)2‖Vnkxn-Vnk-1xn‖2+cn,
which in turn implies that
(3.39)(1-αn-αnμγ̅)2‖Vnkxn-Vnk-1xn‖2≤{1+αn2(1+μγ̅)2}‖xn-x¯‖2-‖xn-x¯‖2+cn.
From the condition (C1) and from (3.36), we obtain that
(3.40)lim⁡n→∞‖Vnkxn-Vnk-1xn‖=0.
On the other hand, we observe that
(3.41)‖xn-zn‖=‖xn-VnNxn‖=‖∑k=1N(Vnkxn-Vnk-1xn)‖≤∑k=1N‖Vnkxn-Vnk-1xn‖.
Then, we have
(3.42)lim⁡n→∞‖xn-zn‖=0.
Moreover, we observe that
(3.43)‖zn-1tn∫0tnT(s)vnds‖≤‖zn-xn‖+‖xn-1tn∫0tnT(s)vnds‖,
and hence
(3.44)lim⁡n→∞‖zn-1tn∫0tnT(s)vnds‖=0.
Next, we show that for all x¯,y¯∈Ω, limn→∞∥B1yn-B1y¯∥=0 and limn→∞∥B2zn-B2x¯∥=0. By the cocoercivity of the mapping B1, we have
(3.45)‖vn-x¯‖2=‖J(M1,ρ1)(yn-ρ1B1yn)-J(M1,ρ1)(y¯-ρ1B1y¯)‖2≤‖(yn-ρ1B1yn)-(y¯-ρ1B1y¯)‖2=‖(yn-y¯)-ρ1(B1yn-B1y¯)‖2=‖yn-y¯‖2-2ρ1〈B1yn-B1y¯,yn-y¯〉+ρ12‖B1yn-B1y¯‖2≤‖xn-x¯‖2-2ρ1[-c1‖B1yn-B1y¯‖2+d1‖yn-y¯‖2]+ρ12‖B1yn-B1y¯‖2≤‖xn-x¯‖2+(2ρ1c1+ρ12-2ρ1d1L12)‖B1yn-B1y¯‖2.
Similarly, we have
(3.46)‖yn-y¯‖2=‖J(M2,ρ2)(zn-ρ2B2zn)-J(M2,ρ2)(x¯-ρ2B2x¯)‖2≤‖(zn-ρ2B2zn)-(x¯-ρ2B2x¯)‖2=‖(zn-x¯)-ρ2(B2zn-B2x¯)‖2=‖zn-x¯‖2-2ρ2〈B2zn-B2x¯,zn-x¯〉+ρ22‖B2zn-B2x¯‖2≤‖xn-x¯‖2-2ρ2[-c2‖B2zn-B2x¯‖2+d2‖zn-x¯‖2]+ρ22‖B2zn-B2x¯‖2≤‖xn-x¯‖2+(2ρ2c2+ρ22-2ρ2d2L22)‖B2zn-B2x¯‖2.
Substituting (3.45) into (3.34), we have
(3.47)‖xn-x¯‖2≤(1-αn-αnμγ̅)2{‖xn-x¯‖2+(2ρ1c1+ρ12-2ρ1d1L12)‖B1yn-B1y¯‖2}+cn.
Again, from (3.34), we obtain
(3.48)‖xn-x¯‖2≤(1-αn-αnμγ̅)2‖vn-x¯‖2+cn≤(1-αn-αnμγ̅)2‖yn-y¯‖2+cn≤(1-αn-αnμγ̅)2{‖xn-x¯‖2+(2ρ2c2+ρ22-2ρ2d2L22)‖B2zn-B2x¯‖2}+cn.
Therefore, from (3.47) and (3.48), we obtain
(3.49)(1-αn-αnμγ̅)2(-2ρ1c1-ρ12+2ρ1d1L12)‖B1yn-B1y¯‖2≤(1-αn-αnμγ̅)2‖xn-x¯‖2-‖xn-x¯‖2+cn≤{1+αn2(1+μγ̅)2}‖xn-x¯‖2-‖xn-x¯‖2+cn,(1-αn-αnμγ̅)2(-2ρ2c2-ρ22+2ρ2d2L22)‖B2zn-B2x¯‖2≤(1-αn-αnμγ̅)2‖xn-x¯‖2-‖xn-x¯‖2+cn≤{1+αn2(1+μγ̅)2}‖xn-x¯‖2-‖xn-x¯‖2+cn.
From the condition (C1) and from (3.36), we obtain that
(3.50)lim⁡n→∞‖B1yn-B1y¯‖=0,lim⁡n→∞‖B2zn-B2x¯‖=0.
Next, we show that limn→∞∥zn-vn∥=0 and limn→∞∥xn-vn∥=0. By the nonexpansiveness of I-ρ2B2, we have
(3.51)‖yn-y¯‖2=‖J(M2,ρ2)(zn-ρ2B2zn)-J(M2,ρ2)(x¯-ρ2B2x¯)‖2≤〈(zn-ρ2B2zn)-(x¯-ρ2B2x¯),yn-y¯〉=12{‖(zn-ρ2B2zn)-(x¯-ρ2B2x¯)‖2+‖yn-y¯‖2-‖(zn-ρ2B2zn)-(x¯-ρ2B2x¯)-(yn-y¯)‖2}≤12{‖zn-x¯‖2+‖yn-y¯‖2-‖(zn-yn)-ρ2(B2zn-B2x¯)-(x¯-y¯)‖2}=12{‖zn-x¯‖2+‖yn-y¯‖2-‖(zn-yn)-(x¯-y¯)‖2+2ρ2〈(zn-yn)-(x¯-y¯),B2zn-B2x¯〉-ρ22‖B2zn-B2x¯‖2‖B2zn-B2x¯‖2}.
So, we obtain
(3.52)‖yn-y¯‖2≤‖zn-x¯‖2-‖(zn-yn)-(x¯-y¯)‖2+2ρ2〈(zn-yn)-(x¯-y¯),B2zn-B2x¯〉.
Substituting (3.52) into (3.34), we have
(3.53)‖xn-x¯‖2≤(1-αn-αnμγ̅)2{‖zn-x¯‖2-‖(zn-yn)-(x¯-y¯)‖2+2ρ2〈(zn-yn)-(x¯-y¯),B2zn-B2x¯〉‖zn-x¯‖2}+cn,
which in turn implies that
(3.54)(1-αn-αnμγ̅)2‖(zn-yn)-(x¯-y¯)‖2≤(1-αn-αnμγ̅)2‖xn-x¯‖2-‖xn-x¯‖2+(1-αn-αnμγ̅)22ρ2‖(zn-yn)-(x¯-y¯)‖×‖B2zn-B2x¯‖+cn≤{1+αn2(1+μγ̅)2}‖xn-x¯‖2-‖xn-x¯‖2+(1-αn-αnμγ̅)22ρ2‖(zn-yn)-(x¯-y¯)‖×‖B2zn-B2x¯‖+cn.
From the condition (C1) and from (3.36), (3.50), we obtain that
(3.55)lim⁡n→∞‖(zn-yn)-(x¯-y¯)‖=0.
On the other hand, by Lemma 2.8 and from (2.3), we have
(3.56)‖(yn-vn)+(x¯-y¯)‖2=‖(yn-ρ1B1yn)-(y¯-ρ1B1y¯)-[J(M1,ρ1)(yn-ρ1B1yn)-J(M1,ρ1)(y¯-ρ1B1y¯)]+ρ1(B1yn-B1y¯)‖2≤‖(yn-ρ1B1yn)-(y¯-ρ1B1y¯)-[J(M1,ρ1)(yn-ρ1B1yn)-J(M1,ρ1)(y¯-ρ1B1y¯)]‖2+2ρ1〈B1yn-B1y¯,(yn-vn)+(x¯-y¯)〉≤‖(yn-ρ1B1yn)-(y¯-ρ1B1y¯)‖2-‖J(M1,ρ1)(yn-ρ1B1yn)-J(M1,ρ1)(y¯-ρ1B1y¯)‖2+2ρ1‖B1yn-B1y¯‖‖(yn-vn)+(x¯-y¯)‖≤‖(yn-ρ1B1yn)-(y¯-ρ1B1y¯)‖2-‖1tn∫0tnT(s)J(M1,ρ1)(yn-ρ1B1yn)ds-1tn∫0tnT(s)J(M1,ρ1)(y¯-ρ1B1y¯)ds‖2+2ρ1‖B1yn-B1y¯‖‖(yn-vn)+(x¯-y¯)‖=‖(yn-ρ1B1yn)-(y¯-ρ1B1y¯)‖2-‖1tn∫0tnT(s)vnds-1tn∫0tnT(s)x¯ds‖2+2ρ1‖B1yn-B1y¯‖‖(yn-vn)+(x¯-y¯)‖≤‖(yn-ρ1B1yn)-(y¯-ρ1B1y¯)-(1tn∫0tnT(s)vnds-x¯)‖×{‖(yn-ρ1B1yn)-(y¯-ρ1B1y¯)‖+‖1tn∫0tnT(s)vnds-x¯‖}+2ρ1‖B1yn-B1y¯‖‖(yn-vn)+(x¯-y¯)‖=‖(zn-1tn∫0tnT(s)vnds)+(x¯-y¯)-(zn-yn)-ρ1(B1yn-B1y¯)‖×{‖(yn-ρ1B1yn)-(y¯-ρ1B1y¯)‖+‖1tn∫0tnT(s)vnds-x¯‖}+2ρ1‖B1yn-B1y¯‖‖(yn-vn)+(x¯-y¯)‖.
From (3.44), (3.50), and (3.55), we obtain that
(3.57)lim⁡n→∞‖(yn-vn)+(x¯-y¯)‖=0.
In fact, since
(3.58)‖zn-vn‖≤‖(zn-yn)-(x¯-y¯)‖+‖(yn-vn)+(x¯-y¯)‖,
so
(3.59)lim⁡n→∞‖zn-vn‖=0.
And since
(3.60)‖xn-vn‖≤‖xn-zn‖+‖zn-vn‖,
and so
(3.61)lim⁡n→∞‖xn-vn‖=0.
Next, we show thatx̃∈Ω:=F(𝒮)∩⋂k=1NMEP(Θk,φk)∩F(Q).(i)
We first show thatx̃∈F(𝒮). Since {xn} is bounded, there exists a subsequence {xnj} of {xn} such that xnj⇀x̃∈H as j→∞. From (3.31) and Lemma 2.5, we obtain that x̃∈F(𝒮).(ii)
Now, we show thatx̃∈⋂k=1NMEP (Θk,φk). Since Vnk=Vrk,n(Θk,φk)Vnk-1 for k∈{1,2,…,N}. Hence, for all x∈H and for all k∈{1,2,…,N}, we obtain(3.62)Θk(Vnkxn,x)+φk(x)-φk(Vnkxn)+1rk,n〈Kk′(Vnkxn)-Kk′(Vnk-1xn),ηk(x,Vnkxn)〉≥0.
And hence
(3.63)〈Kk′(Vnkxnj)-Kk′(Vnk-1xnj)rk,n,ηk(x,Vnkxnj)〉≥-Θk(Vnkxnj,x)-φk(x)+φk(Vnkxnj),∀x∈H.
By the assumptions thatφk is lower semicontinuous and by conditions (H4), (H5), the mapping x↦(-Θk(x,y)) is lower semicontinuous. So, they are weakly lower semicontinuous. Since xnj⇀x̃, we have that Vnkxn⇀x̃ for all k∈{1,2,…,N} and from (i)(c), (ii), (3.40). Now, taking lower limit as j→∞ in (3.63), we obtain that
(3.64)Θk(x̃,x)+φk(x)-φk(x̃)≥0,∀x∈H,∀k∈{1,2,…,N}.
Therefore x̃∈⋂k=1NMEP(Θk,φk).
(iii) Now, we show thatx̃∈F(Q), where Q is defined as in Lemma 2.11. Since Q is nonexpansive. Then, we have
(3.65)‖vn-Qvn‖=‖J(M1,ρ1)(yn-ρ1B1yn)-Qvn‖=‖J(M1,ρ1)[J(M2,ρ2)(zn-ρ2B2zn)-ρ1B1J(M2,ρ2)(zn-ρ2B2zn)]-Qvn‖=‖Qzn-Qvn‖≤‖zn-vn‖.
From (3.59), we have limn→∞∥vn-Qvn∥=0. Since xnj⇀x̃ and from (3.61), we also have vnj⇀x̃. Hence, we obtain by Lemma 2.5 that x̃∈F(Q).
Next, we show that{xn} is sequentially compact, namely, there is a subsequence {xnj}⊂{xn} that converges strongly to x̃∈Ω as j→∞. From (3.23), replacing x¯ by x̃ to obtain
(3.66)γϕ(‖xn-x̃‖)(‖xn-x̃‖)≤〈u+γf(x̃)-(I+μA)x̃,xn-x̃〉.
Now, replacing n with nj in (3.66) and letting j→∞, since xnj⇀x̃, we obtain that xnj→x̃ as j→∞.
Next, we show thatx̃ is the unique solution of the variational inequality (3.5). Since
(3.67)xn=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μA))1tn∫0tnT(s)vnds,
we derive that
(3.68)(-u+(I+μA-γf)xn)=-(1-βn)αn(xn-1tn∫0tnT(s)vnds)+(I+μA)(xn-1tn∫0tnT(s)vnds).
For all z∈Ω, it follows that
(3.69)〈-u+(I+μA-γf)xn,xn-z〉=-(1-βn)αn〈1tn(I-1tn∫0tnT(s)QVnNds)xn-(I-1tn∫0tnT(s)QVnNds)z,xn-z〉+〈(I+μA)(xn-1tn∫0tnT(s)vnds),xn-z〉.
By Lemma2.10, we obtain that I-σt(·) is monotone, where σt(·):=(1/t)∫0tT(s)ds. Then, from (3.69), we have
(3.70)〈-u+(I+μA-γf)xn,xn-z〉≤〈(I+μA)(xn-σt(vn)),xn-z〉.
Now, replacing n by nj in (3.70) and letting j→∞ and xnj→x̃, we notice that
(3.71)xnj-1tnj∫0tnjT(s)vnjds→0.
Then, we have
(3.72)〈u+(γf-(I+μA))x̃,z-x̃〉≤0,∀z∈Ω.
That is, x̃ is the solution of variational inequality (3.5).
Finally, we show that{xn} converges strongly to x̃∈Ω. Suppose that there exists another subsequence xnk→x̂ as k→∞. We note Lemma 2.5 that x̂∈Ω is the solution of the variational inequality (3.5). Hence x̃=x̂=x* by uniqueness. In summary, we have shown that {xn} is sequentially compact and each cluster point of the sequence {xn} is equal to x*. Then, we conclude that xn→x* as n→∞. This completes the proof.From Theorem3.2, we can deduce the following result.Theorem 3.3.
LetC be a nonempty closed and convex subset of a real Hilbert space  H. Let  φi:C→ℝ(i=1,2,…,N) be a finite family of lower semicontinuous and convex function,  Θi:C×C→ℝ(i=1,2,…,N) be a finite family of bifunctions satisfying  (H1)–(H5),  ηi:C×H→H be a finite family of Lipschitz continuous mappings with a constant  σi(i=1,2,…,N). Let  𝒮={T(t):t∈ℝ+} be a nonexpansive semigroup from  C into  H and  Bi:H→H(i=1,2) be an  Li-Lipschitzian and relaxed  (ci,di)-cocoercive mapping with  ρi∈(0,2(di-ciLi2)/Li2] for all  i=1,2. Let  Q¯:C→C be a mapping defined by
(3.73)Q¯x:=PC[PC(x-ρ2B2x)-ρ1B1PC(x-ρ2B2x)].
Assume that Ω:=F(𝒮)∩⋂k=1NMEP(Θk,φk)∩F(Q¯)≠∅. Let f:H→H be a ϕ-strongly pseudocontractive mapping with limt→+∞ϕ(t)=+∞ and A be a strongly positive linear bounded operator on H with a coefficient γ̅>0. Let μ>0 and γ>0 be two constants such that 0<γ<1+μγ̅. Let {ri,n}(i=1,2,…,N) be a finite family of positive real sequence such that liminfn→∞ri,n>0, {αn} and {βn} be two sequences in [0,1], and {tn} be a positive real divergent sequence. For any fixed u∈H, let {xn} be the sequence defined by
(3.74)Θ1(un(1),x)+φ1(x)-φ1(un(1))+1r1,n〈K1′(un(1))-K1′(xn),η1(x,un(1))〉≥0,∀x∈C,Θ2(un(2),x)+φ2(x)-φ2(un(2))+1r2,n〈K2′(un(2))-K2′(un(1)),η1(x,un(2))〉≥0,∀x∈C,⋮ΘN(un(N),x)+φN(x)-φN(un(N))+1rN,n〈KN′(un(N))-KN′(un(N-1)),ηN(x,un(N))〉≥0,∀x∈C,yn=PC(un(N)-ρ2B2un(N)),xn=αn(u+γf(xn))+βnxn+((1-βn)I-αn(I+μA))1tn∫0tnT(s)PC(yn-ρ1B1yn)ds,
where
(3.75)un(1)=Vr1,n(Θ1,φ1)xn,un(i)=Vri,n(Θi,φi)un(i-1)=Vri,n(Θi,φi)Vri-1,n(Θi-1,φi-1)un(i-2)=Vri,n(Θi,φi)⋯Vr2,n(Θ2,φ2)un(1)=Vri,n(Θi,φi)⋯Vr2,n(Θ2,φ2)Vr1,n(Θ1,φ1)xn,i=2,3,…,N,
and Vri,n(Θi,φi):C→C, i=1,2,…,N is the mapping defined by (2.8). Assume the following.(i)
ηi:C×C→ℝ is Lipschitz continuous with constant σi>0(i=1,2,…,N) such that
(a)
ηi(x,y)+ηi(y,x)=0 for all x,y∈C,(b)
ηi(·,·) is affine in the first variable,(c)
for each fixedy∈C, x↦ηi(y,x) is sequentially continuous from the weak topology to the weak topology.(ii)
Ki:C→ℝ is ηi—strongly convex with constant μi>0, and its derivative Ki′ is not only continuous from the weak topology to the strong topology but also Lipschitz continuous with constant νi such that μi≥σiνi.(iii)
For alli=1,2,…,N and for all x∈C, there exists a bounded subset Dx⊂C and zx∈C such that for all y∉Dx,(3.76)Θi(y,zn)+φi(zx)-φi(y)+1ri,n〈Ki′(y)-Ki′(x),ηi(zx,y)〉<0.
If the following conditions are satisfied:(C1)
limn→∞αn=0,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1, then the sequence {xn} defined by (3.74) converges strongly to x*∈Ω:=F(𝒮)∩⋂k=1NMEP(Θk,φk)∩F(Q¯), provided Vri,n(Θi,φi) is firmly nonexpansive, where x* is the unique solution of the variational inequality
(3.77)〈u+(γf-(I+μA))x*,z-x*〉≤0,∀z∈Ω,
or, equivalently, x* is the unique solution of the optimization problem
(3.78)min⁡x∈Ωμ2〈Ax,x〉+12‖x-u‖2-h(x),
where h is a potential function for γf and (x*,y*) is the solution of general system of variational inequality problem
(3.79)〈ρ1B1y*+x*-y*,x-x*〉≥0,∀x∈C,〈ρ2B2x*+y*-x*,x-y*〉≥0,∀x∈C.Proof.
From Theorem3.2, taking M1=M2=∂δC, where C is a nonempty closed convex subset of H and δC:H→[0,+∞) is the indicator function of C, then, we have J(M1,ρ1)=J(M2,ρ2)=PC and the quasivariational inclusion problem (1.17) is equivalent to the classical variational inequality (1.19). Thus, we can get the desired conclusion immediately.



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*Source: 979870-2012-02-06.xml*