# A New General Iterative Method for Solution of a New General System of Variational Inclusions for Nonexpansive Semigroups in Banach Spaces

Authors: Pongsakorn Sunthrayuth; Poom Kumam
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

We introduce a new general system of variational inclusions in Banach spaces and propose a
new iterative scheme for finding common element of the set of solutions of the variational inclusion with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point
of nonexpansive semigroups in a uniformly convex and 2-uniformly smooth Banach space. Furthermore,
strong convergence theorems are established under some certain control conditions. As applications, finding
a common solution for a system of variational inequality problems and minimization problems is given.


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## Body

## 1. Introduction

In the theory of variational inequalities and variational inclusions, the development of an efficient and implementable iterative algorithm is interesting and important. The important generalization of variational inequalities calledvariational inclusions, have been extensively studied and generalized in different directions to study a wide class of problems arising in optimization, nonlinear programming, finance, economics, and applied sciences.Variational inequalities are being used as a mathematical programming tool in modeling a wide class of problems arising in several branches of pure and applied mathematics. Several numerical techniques for solving variational inequalities and the related optimization problem have been considered by many authors.Throughout this paper, we denoted byℕ and ℝ+ the set of all positive integers and all positive real numbers, respectively. Let X be a real Banach space and X* be its dual space. Let U={x∈X:∥x∥=1} denote the unit sphere of X. X is said to be uniformly convex if for each ϵ∈(0,2], there exists a constant δ>0 such that for all x,y∈U,(1.1)‖x-y‖≥ϵimplies‖x+y2‖≤1-δ.
The norm on X is said to be Gâteaux differentiable if the limit(1.2)lim⁡t→0‖x+ty‖-‖x‖t
exists for each x,y∈U, and in this case X is smooth. Moreover, we say that the norm X is said to have a uniformly Gâteaux differentiable if the above limit is attained uniformly for all x,y∈U and in this case X is said to be uniformly smooth. We define a function ρ:[0,∞)→[0,∞), called the modulus of smoothness of X, as follows:(1.3)ρ(τ)=sup⁡{12(‖x+y‖+‖x-y‖)-1:x,y∈X,‖x‖=1,‖y‖=τ}.
It is know that X is uniformly smooth if and only if limτ→0ρ(τ)/τ=0. Let q be a fixed real number 1<q≤2. A Banach space X is said to be q-uniformly smooth if there exists a constant c>0 such that ρ(τ)≤cτq for all τ>0. From [1], we know the following property.Letq be a real number with 1<q≤2 and let X be a Banach space. Then, X is q-uniformly smooth if and only if there exists a constant K≥1 such that(1.4)‖x+y‖q+‖x-y‖q≤2(‖x‖q+‖Ky‖q),∀x,y∈X.
The best constant K in the above inequality is called the q-uniformly smoothness constant of X (see [1] for more details).LetX be a real Banach space and X* the dual space of X. Let 〈·,·〉 denote the pairing between X and X*. For q>1, the generalized duality mapping(1.5)Jq(x)={f∈X*:〈x,f〉=‖x‖q,‖f‖=‖x‖q-1},∀x∈X.
In particular, if q=2, the mapping J2 is called the normalized duality mapping and usually, we write J2=J. If X is a Hilbert space, then J=I is the identity. Further, we have the following properties of the generalized duality mapping Jq: (1)
Jq(x)=∥x∥q-2J2(x) for all x∈X with x≠0,(2)
Jq(tx)=tq-1Jq(x) for all x∈X and t∈[0,∞),(3)
Jq(-x)=-Jq(x) for all x∈X.It is know that ifX is smooth, then J is single-valued, which is denoted by j.Definition 1.1.
LetC be a nonempty closed convex subset of X. A mapping T:C→X is said to be (i)
nonexpansive if
(1.6)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C,(ii)
Lipschitzian if there exists a constant L>0 such that
(1.7)‖Tx-Ty‖≤L‖x-y‖,∀x,y∈C,(iii)
contraction if there exists a constant α∈(0,1) such that
(1.8)‖Tx-Ty‖≤α‖x-y‖,∀x,y∈C.Remark 1.2.
We denoteF(T) as the set of fixed points of T. We know that F(T) is nonempty if C is bounded; for more detail see [2].Definition 1.3.
A one-parameter family𝒮={T(t):t∈ℝ+} from C of X into itself is said to be a nonexpansive semigroup on C if it satisfies the following conditions: (i)
T(0)x=x for all x∈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 1.4.
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}. We know that F(𝒮) is nonempty if C is bounded, see [3].LetC be a nonempty closed convex subset of a smooth Banach space X. Recall the following definitions of a nonlinear mapping B:C→X, the following are mentioned.Definition 1.5.
Given a mappingB:C→X, (i)
B is said to be accretive(1.9)〈Bx-By,J(x-y)〉≥0,∀x,y∈C,(ii)
B is said to be α-strongly accretive if there exists a constant α>0 such that
(1.10)〈Bx-By,J(x-y)〉≥α‖x-y‖2,∀x,y∈C,(iii)
B is said to be α-inverse-strongly accretive or α-cocoerciveif there exists a constant α>0 such that
(1.11)〈Bx-By,J(x-y)〉≥α‖Bx-By‖2,∀x,y∈C,(iv)
B is said to be α-relaxed cocoercive if there exists a constant α>0 such that
(1.12)〈Bx-By,J(x-y)〉≥-α‖Bx-By‖2,∀x,y∈C,(v)
B is said to be (α,β)-relaxed cocoercive if there exist positive constants α>0 and β>0 such that
(1.13)〈Bx-By,J(x-y)〉≥-α‖Bx-By‖2+β‖x-y‖2,∀x,y∈C.Remark 1.6.
(1) Everyα-strongly accretive mapping is an accretive mapping.
(2) Everyα-strongly accretive mapping is a (β,α)-relaxed cocoercive mapping for any positive constant β but the converse is not true in general. Then, the class of relaxed cocoercive operators is more general than the class of strongly accretive operators.
(3) Evidently, the definition of the inverse-strongly accretive operator is based on that of the inverse-strongly monotone operator in real Hilbert spaces (see, e.g., [4]).
(4) The notion of the cocoercivity is applied in several directions, especially for solving variational inequality problems using the auxiliary problem principle and projection methods [5]. Several classes of relaxed cocoercive variational inequalities have been studied in [6, 7].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 1.7.
LetM:X→2X be a multivalued maximal accretive mapping. The single-valued mapping J(M,ρ):X→X, defined by
(1.14)J(M,ρ)(u)=(I+ρM)-1(u),∀u∈X,
is called resolvent operator associated with M, where ρ is any positive number and I is the identity mapping.In 2010, Qin et al. [8] introduced a system of quasivariational inclusions as follows. Find (x*,y*)∈X×X such that(1.15)0∈x*-y*+ρ1(B1y*+M1x*),0∈y*-x*+ρ2(B2x*+M2y*),
where Bi:X→X and Mi:X→2X are nonlinear mappings for all i=1,2. As special cases of problem (1.15), we have the following. (1)
IfB1=B2=B and M1=M2=M, then problem (1.15) is reduced to the following. Find (x*,y*)∈X×X 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 problem (1.16) is reduced to the following. Find x*∈X such that
(1.17)0∈Bx*+Mx*.The problem (1.17) is called variational inclusion problem denoted by V(X,B,M).Here we have examples of the variational inclusion (1.17). If M=∂δC, where C is a nonempty closed convex subset of X, and δC:X→[0,∞)(1.18)δC(x)={0,x∈C,+∞,x∉C,
then the variational inclusion problem (1.17) is equivalent (see [9]) to finding u∈C such that(1.19)〈Bu,v-u〉≥0,∀x∈C.
This problem is called Hartman-Stampacchia variational inequality problem denoted by VI(C,B).LetD be a subset of C, and let P be a mapping of C into D. Then, P is said to be sunny if(1.20)P(Px+t(x-Px))=Px,
whenever Px+t(x-Px)∈C for x∈C and t≥0. A mapping P of C into itself is called a retraction if P2=P. If a mapping P of C into itself is a retraction, then Pz=z for all z∈R(P), where R(P) is the range of P. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D.In 2006, Aoyama et al. [10] considered the following problem: find u∈C such that(1.21)〈Au,J(v-u)〉≥0,∀v∈C.
They proved that the variational inequality (1.21) is equivalent to a fixed point problem. The element u∈C is a solution of the variational inequality (1.21) if and only if u∈C satisfies the following equation:(1.22)u=PC(u-λAu),
where λ>0 is a constant and PC is a sunny nonexpansive retraction from X onto C.The following results describe a characterization of sunny nonexpansive retractions on a smooth Banach space.Proposition 1.8 (see [11]).
LetX be a smooth Banach space and C a nonempty subset of X. Let P:X→C be a retraction and J the normalized duality mapping on X. Then the following are equivalent: (1)
P is sunny and nonexpansive,(2)
〈x-Px,J(y-Px)〉≤0, for all x∈X, y∈C.Proposition 1.9 (see [12]).
LetC be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and T a nonexpansive mapping of C into itself with F(T)≠∅. Then the set F(T) is a sunny nonexpansive retract of C.For the class of nonexpansive mappings, one classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping [13, 14]. More precisely, take t∈(0,1), and define a contraction Tt:C→C by(1.23)Ttx=tu+(1-t)Ttx,∀x∈C,
where u∈C is a fixed point. Banachs contraction mapping principle guarantees that Tt has a unique fixed point xt∈C, that is,(1.24)xt=tu+(1-t)Ttxt.
It is unclear, in general, what the behavior of xt is as t→0, even if T has a fixed point. However, in the case of T having a fixed point, Ceng et al. [15] proved that, if X is a Hilbert space, then xt converges strongly to a fixed point of T. Reich [14] extended Browders result to the setting of Banach spaces and proved that, if X is a uniformly smooth Banach space, then xt converges strongly to a fixed point of T, and the limit defines the (unique) sunny nonexpansive retraction from C onto F(T).Reich [14] showed that, if X is uniformly smooth and D is the fixed point set of a nonexpansive mapping from C into itself, then there is a unique sunny nonexpansive retraction from C onto D and it can be constructed as follows.Proposition 1.10 (see [14]).
LetX be a uniformly smooth Banach space and T:C→C a nonexpansive mapping such that F(T)≠∅. For each fixed u∈C and every t∈(0,1), the unique fixed point xt∈C of the contraction C∋x↦tu+(1-t)Tx converges strongly as t→0 to a fixed point of T. Define P:C→D by Pu=s-limt→0xt. Then P is the unique sunny nonexpansive retract from C onto D; that is, P satisfies the property,
(1.25)〈x-Px,J(y-Px)〉≤0,∀x∈C,y∈D.Many authors have studied the problems of finding a common element of the set of fixed points of a nonexpansive mapping and one of the sets of solutions to the variational inclusion and variational inequalities (1.15)–(1.17) and (1.21) by using different iterative methods (see, e.g., [10, 16–27]).Recently, Qin et al. [8] considered the problem of finding the solutions of a general system of variational inclusion (1.15) with α-inverse strongly accretive mappings. To be more precise, they obtained the following results.Lemma 1.11.
For any(x*,y*)∈X×X, 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 defined by
(1.26)Qx:=J(M1,ρ1)[J(M2,ρ2)(x-ρ2B2x)-ρ1B1J(M2,ρ2)(x-ρ2B2x)].Theorem QCCK (see [8]).
LetX be a uniformly convex and 2-uniformly smooth Banach space with the smoothness constant K. Let Mi:X→2X be a maximal monotone mapping and Bi:X→X be a ρi-inverse-strongly accretive mapping, respectively, for all i=1,2. Let T:X→X be a ϵ-strict pseudocontraction such that F(T)≠∅. Define a mapping S by Sx=(1-ϵ/K2)x+(ϵ/K2)Tx, for all x∈X. Assume that Ω=F(T)∩F(Q), where Q is defined as in Lemma 1.11. Let x1=u∈C and let {xn} be a sequence defined by
(1.27)zn=J(M2,ρ2)(xn-ρ2B2xn),yn=J(M1,ρ1)(zn-ρ1B1zn),xn+1=αnu+βnxn+(1-βn-αn)[μSxn+(1-μ)yn],∀n≥1,
where μ∈(0,1), ρ1∈(0,γ1/K2], ρ2∈(0,γ2/K2], and {αn}, {βn} are sequences in (0,1). If the control consequences {αn} and {βn} satisfy the following restrictions: (C1)
0<liminfn→∞βn≤limsupn→∞βn<1,(C2)
limn→∞αn=0 and ∑n=1∞αn=∞,then {xn} converges strongly to x*=PΩu, where PΩ is the sunny nonexpansive retraction from X onto Ω and (x*,y*) is a solution of the problem (1.15), where y*=J(M2,ρ2)(x*-ρ2B2x*).Iterative methods for nonexpansive mappings have recently been applied to solve minimization problems; see, for example, [28–32]. Let H be a real Hilbert space, whose inner product and norm are denoted by ∥·,·∥ and 〈·〉, respectively. Let A be a strongly positive bounded linear operator on H: that is, there is a constant γ>0 with property(1.28)〈Ax,x〉≥γ¯‖x‖2,∀x∈H.
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H(1.29)min⁡x∈F12〈Ax,x〉-〈x,u〉,
where F is the fixed point set of a nonexpansive mapping T on H and u is a given point in H.In [33], Moudafi introduced the viscosity approximation method and proved that if H is a real Hilbert space, the sequence {xn} generated by the following algorithm:(1.30)x0=u∈Hchosenarbitrarily,xn+1=αnf(xn)+(1-αn)Txn,∀n≥0,
where f:C→C is a contraction mapping with a constant α∈(0,1) and {αn}⊂(0,1) satisfies certain conditions, converges strongly to a fixed point of T in C which is the unique solution of the following variational inequality:(1.31)〈(f-I)x*,x-x*〉≤0,∀x∈F(T).
In 2006, Marino and Xu [34] introduced the following general iterative method:(1.32)x0=u∈Hchosenarbitrarily,xn+1=αnγf(xn)+(I-αnA)Txn,∀n≥0,
where A is a strongly positive bounded linear operator on a Hilbert space H. They proved that, if the sequence {αn} of parameters satisfies appropriate conditions, then the sequence {xn} generated by (1.32) converges strongly to the unique solution of the variational inequality:(1.33)〈(γf-A)x*,x-x*〉≤0,∀x∈F(T),
which is the optimality condition for the minimization problem:(1.34)min⁡x∈C12〈Ax,x〉-h(x),
where C is the fixed point set of a nonexpansive mapping T and h is a potential function for γf (i.e., h′(x)=γf(x) for all x∈H).In a smooth Banach space, we always assume thatA is strongly positive (see [35]), that is, a constant γ¯>0 with the property(1.35)〈Ax,J(x)〉≥γ¯‖x‖2,‖aI-bA‖=sup⁡‖x‖≤1|〈(aI-bA)x,J(x)〉|,a∈[0,1],b∈[-1,1],
where I is the identity mapping and J is the normalized duality mapping.Recently, Sunthrayuth and Kumam [36] introduced the following iterative method for nonexpansive semigroup 𝒮={T(t):t∈ℝ+} in Banach spaces,(1.36)x0=u∈Cchosenarbitrarily,xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)1tn∫0tnT(s)xnds,∀n≥0.
They proved strong convergence theorem of the iterative scheme {xn} defined by (1.36) converges strongly to the common fixed point of x*∈F(𝒮) solving the variational inequality(1.37)〈(γf-A)x*,J(x-x*)〉≤0,∀x∈F(S).
In 2010, Kamraksa and Wangkeeree [37] introduced a general iterative approximation method for finding common elements of the set of solutions to a general system of variational inclusions with Lipschitzian and relaxed cocoercive mappings and the set common fixed points of a countable family of strict pseudocontractions. They proved the strong convergence theorems of such iterative scheme for finding a common element of such two sets which is a unique solution of some variational inequality and is also the optimality condition for some minimization problems in a strictly convex and 2-uniformly smooth Banach space.In this paper, we are motivated and inspire by idea of Qin et al. [8] and Sunthrayuth and Kumam [36].First, we introduce a new general system of variational inclusions in Banach spaces as follows.LetX be Banach spaces. We consider a system of quasivariational inclusions as follows. Finding (x*,y*,z*)∈X×X×X such that(1.38)0∈x*-y*+ρ1(B1y*+M1x*),0∈y*-z*+ρ2(B2z*+M2y*),0∈z*-x*+ρ3(B3x*+M3z*),
which is called a new general system of variational inclusions in Banach spaces, Bi:X→X and Mi:X→2X are nonlinear mappings for all i=1,2,3. As special cases of problem (1.38), we have the following. (1)
IfB1=B2=B3=B and M1=M2=M3=M, then problem (1.38) is reduced to the following. Finding (x*,y*,z*)∈X×X×X such that
(1.39)0∈x*-y*+ρ1(By*+Mx*),0∈y*-z*+ρ2(Bz*+My*),0∈z*-x*+ρ3(Bx*+Mz*).(2)
Further, ifB3=M3=0, z*=x*, then problem (1.38) is reduced to problem (1.15).Second, we study a general iterative approximation method (3.1) below, for finding common elements of the set of solutions of a new general system of variational inclusions (1.38) with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mappings and the set common fixed points of nonexpansive semigroup in the framework of Banach spaces. Moreover, we prove the strong convergence of the proposed iterative method under some certain control conditions. The results presented in this paper extend and improve the results of Qin et al. [8] and Sunthrayuth and Kumam [36], and many authors.


## 2. Preliminaries

This section collects some results that will be used in the proofs of our main results.Lemma 2.1 (see [38]).
The resolvent operatorJ(M,ρ) associated with M is single valued and nonexpansive for all ρ>0.Lemma 2.2 (see [39]).
LetX be a real 2-uniformly smooth Banach space with the best smoothness constant K. Then, the following inequality holds:
(2.1)‖x+y‖2≤‖x‖2+2〈y,Jx〉+2‖Ky‖2,∀x,y∈X.Lemma 2.3 (see [40]).
In a real Banach spaceX, the following inequality holds:
(2.2)‖x+y‖2≤‖x‖2+2〈y,J(x+y)〉,∀x,y∈X.Now, we present the concept of a uniformly asymptotically regular semigroup (see [41–43]).Definition 2.4.
LetC be a nonempty closed convex subset of a Banach space X, 𝒮={T(t):t∈ℝ+} be a continuous operator semigroup on C. Then 𝒮 is said to be uniformly asymptotically regular (in short, u.a.r.) on C if for all h≥0 and any bounded subset B of C such that
(2.3)lim⁡t→∞sup⁡x∈B‖T(h)T(t)x-T(t)x‖=0.Lemma 2.5 (see [44]).
LetC be a nonempty closed convex subset of a uniformly Banach space X, B be a bounded closed convex subset of C. If we denote 𝒮={T(t):t∈ℝ+} a nonexpansive semigroup on C such that F(𝒮):=⋂t∈ℝ+F(T(t))≠∅. For all h≥0, the set σt(x)=(1/t)∫0tT(s)xds, then
(2.4)lim⁡t→∞sup⁡x∈B‖σt(x)-T(h)σt(x)‖=0.Remark 2.6.
It is easy to check that the set{σt:t∈ℝ+} defined by Lemma 2.5 is a u.a.r. nonexpansive semigroup on C (see [45] for more detail).Lemma 2.7 (see [46]).
LetC be a nonempty closed convex subset of X and let 𝒮={T(t):t∈ℝ+} be a u.a.r. nonexpansive semigroup on C such F(𝒮):=⋂t∈ℝ+F(T(t))≠∅ and at least there exists a T(t) which is demicompact. Then, for each x∈C, there exists a sequence {T(tn):tn∈ℝ+,n∈ℕ}⊂T(t), such that {T(tn)x} converges strongly to some point in F(𝒮), where limn→∞tn=∞.Remark 2.8.
By Lemma2.7, we can see that, for each x∈C, there is a corresponding unique point y∈F(𝒮), thus we can define a mapping T such that Tx=y and it is easy to see that F(T)=F(𝒮).Remark 2.9.
From the definition ofT, we can see that T is a nonexpansive mapping. Actually, by Lemma 2.7, let x1∈C; then there exists a sequence {T(tn)} of {T(h)} such that Tx1=limn→∞T(tn)x1=y1∈F(𝒮). Further, for any other point x2∈C, by the definition of a u.a.r., we can get a subsequence {T(tnk)} of {T(tn)} such that T(tnk)→y2∈F(𝒮), then
(2.5)‖Tx1-Tx2‖=lim⁡k→∞‖T(tnk)x1-T(tnk)x2‖≤‖x1-x2‖.Lemma 2.10 (see [35]).
LetC be a nonempty closed convex subset of a reflexive, smooth Banach space X which admits a weakly sequentially continuous duality mapping J from X into X*, T be a nonexpansive mapping such that F(T)≠∅, f:C→C be a contraction mapping with a coefficient α∈(0,1) and A be a strongly positive bounded linear operator with a coefficient γ¯>0. Let t∈(0,1) such that t≤∥A∥-1 and 0<γ<γ¯/α which satisfies t→0. Then the sequence {xt} defined by
(2.6)xt=tγf(xt)+(I-tA)Txt,
converges strongly to the common fixed point x* as t→0, where x* is a unique solution in F(𝒮) of the variational inequality
(2.7)〈(γf-A)x*,J(z-x*)〉≤0,∀z∈F(S).Lemma 2.11 (see [47]).
LetC be a closed convex subset of a strictly convex Banach space X. Let {Tn:n∈ℕ} be a sequence of nonexpansive mappings on C. Suppose ⋂n=1∞F(Tn) is nonempty. Let {μn} be a sequence of positive numbers with ∑n=1∞μn=1. Then a mapping S on C defined by Sx=∑n=1∞μnTnx for all x∈C is well defined, nonexpansive and F(S)=⋂n=1∞F(Tn) holds.Lemma 2.12 (see [48]).
Let{xn} and {vn} 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)vn+βnxn for all integers n≥0 and limsupn→∞(∥vn+1-vn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥vn-xn∥=0.Lemma 2.13 (see [35]).
Assume thatA is a strongly positive linear bounded operator on a smooth Banach space X with coefficient γ¯>0 and 0<ρ≤∥A∥-1. Then ∥I-ρA∥≤1-ργ¯.If a Banach spaceX admits a sequentially continuous duality mapping J from weak topology to weak star topology, then by Lemma 1 of [49], we have that duality mapping J is a single value. In this case, the duality mapping J is said to be a weakly sequentially continuous duality mapping, that is, for each {xn}⊂X with xn⇀x, we have J(xn)⇀*J(x) (see [49–51] for more details).A Banach spaceX is said to be satisfying Opial's condition if for any sequence xn⇀x for all x∈X implies (2.8)limsup⁡n→∞‖xn-x‖<limsup⁡n→∞‖xn-y‖,∀y∈X,withx≠y.By Theorem 1 in [49], it is well known that if X admits a weakly sequentially continuous duality mapping, then X satisfies Opial's condition, and X is smooth.Lemma 2.14 (see [50], Demiclosed principle).
LetC be a nonempty closed convex subset of a reflexive Banach space X which satisfies Opial's condition and suppose T:C→X is nonexpansive. Then the mapping I-T is demiclosed at zero, that is, xn⇀x and xn-Txn→0 implies x=Tx.Lemma 2.15 (see [30]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.9)an+1≤(1-τn)an+σn,
where {τn} is a sequence in (0,1) and {σn} is a sequence in ℝ such that (i)
∑n=0∞τn=∞,(ii)
limsupn→∞(σn/τn)≤0 or ∑n=0∞|σn|<∞.Then, limn→∞an=0.LetX be a real 2-uniformly smooth Banach space X with the smoothness constant K. Let B:X→X be an LB-Lipschitzian and relaxed (c,d)-cocoercive mapping, we defined a function p:(0,+∞)→(-∞,+∞) by (2.10)p(ρ):=1+2ρcLB2-2ρd+2ρ2K2LB2,∀ρ∈(0,+∞).
Consequence, we put
(2.11)θρ={p(ρ),ifp(ρ)>0,11+K,ifp(ρ)≤0.In order to prove our main result, the following lemmas are needed.Lemma 2.16.
LetX be a real 2-uniformly smooth Banach space X with the smoothness constant K. Let B:X→X be an LB-Lipschitzian and relaxed (c,d)-cocoercive mapping. Then
(2.12)‖(I-ρB)x-(I-ρB)y‖2≤θρ2‖x-y‖2.
In particular, if 0<ρ≤(d-cLB2)/K2LB2, then I-ρB is nonexpansive.Proof.
For allx,y∈X, from Lemma 2.2 and by the cocoercivity of the mapping B, we have
(2.13)‖(I-ρB)x-(I-ρB)y‖2=‖(x-y)-(ρBx-ρBy)‖2≤‖x-y‖2-2ρ〈Bx-By,J(x-y)〉+2ρ2K2‖Bx-By‖2≤‖x-y‖2-2ρ[-c‖Bx-By‖2+d‖x-y‖2]+2ρ2K2‖Bx-By‖2=‖x-y‖2-2ρd‖x-y‖2+2ρc‖Bx-By‖2+2ρ2K2‖Bx-By‖2≤(1+2ρcLB2-2ρd+2ρ2K2LB2)‖x-y‖2=θρ2‖x-y‖2.
It follows that
(2.14)‖(I-ρB)x-(I-ρB)y‖≤θρ‖x-y‖.
It is clear that, if 0<ρ≤(d-cLB2)/K2LB2. Thus, we have I-ρB is nonexpansive.Lemma 2.17.
LetX be a 2-uniformly smooth Banach space X. Let Mi:X→2X be the a maximal monotone mapping and Bi:X→X be an Li-Lipschitzian and relaxed (ci,di)-cocoercive mapping for i=1,2,3. Let Q:X→X be a mapping defined by
(2.15)Qx:=J(M1,ρ1)(J(M2,ρ2)(J(M3,ρ3)(x-ρ3B3x)-ρ2B2J(M3,ρ3)(x-ρ3B3x)-ρ1B1J(M2,ρ2)J(M3,ρ3)(x-ρ3B3x)-ρ2B2J(M3,ρ3)(x-ρ3B3x))),∀x∈X.
If 0<ρi≤(di-ciLi2)/K2Li2 and θρ=max{θρi} for all i=1,2,3, then the mapping Q is nonexpansive.Proof.
For allx,y∈X, we have
(2.16)‖Qx-Qy‖≤‖J(M1,ρ1)[J(M2,ρ2)(J(M3,ρ3)(x-ρ3B3x)-ρ2B2J(M3,ρ3)(x-ρ3B3x))-ρ1B1J(M2,ρ2)(J(M3,ρ3)(x-ρ3B3x)-ρ2B2J(M3,ρ3)(x-ρ3B3x))]-J(M1,ρ1)[J(M2,ρ2)(J(M3,ρ3)(y-ρ3B3y)-ρ2B2J(M3,ρ3)(y-ρ3B3y))-ρ1B1J(M2,ρ2)(J(M3,ρ3)(y-ρ3B3y)-ρ2B2J(M3,ρ3)(y-ρ3B3y))]‖≤‖J(M2,ρ2)(J(M3,ρ3)(x-ρ3B3x)-ρ2B2J(M3,ρ3)(x-ρ3B3x))-ρ1B1J(M2,ρ2)(J(M3,ρ3)(x-ρ3B3x)-ρ2B2J(M3,ρ3)(x-ρ3B3x))-J(M2,ρ2)(J(M3,ρ3)(y-ρ3B3y)-ρ2B2J(M3,ρ3)(y-ρ3B3y))-ρ1B1J(M2,ρ2)(J(M3,ρ3)(y-ρ3B3y)-ρ2B2J(M3,ρ3)(y-ρ3B3y))‖=‖J(M2,ρ2)(I-ρ2B2)J(M3,ρ3)(I-ρ3B3)(I-ρ1B1)x-J(M2,ρ2)(I-ρ2B2)J(M3,ρ3)(I-ρ3B3)(I-ρ1B1)y‖≤‖(I-ρ2B2)J(M3,ρ3)(I-ρ3B3)(I-ρ1B1)x-(I-ρ2B2)J(M3,ρ3)(I-ρ3B3)(I-ρ1B1)y‖.
From Lemma 2.16 and by the nonexpansiveness of J(Mi,ρi) for all i=1,2,3, we have (I-ρ2B2)J(M3,ρ3)(I-ρ3B3)(I-ρ1B1) is a nonexpansive mapping, which implies that the mapping Q is nonexpansive.Lemma 2.18.
For all(x*,y*,z*)∈X×X×X, where y*=J(M2,ρ2)(z*-ρ2B2z*) and z*=J(M3,ρ3)(x*-ρ3B3x*), (x*,y*,z*) is a solution of the problem (1.38) if and only if x* is a fixed point of the mapping Q defined as in Lemma 2.17.Proof.
Let(x*,y*,z*)∈X×X×X be a solution of the problem (1.38). Then, we have
(2.17)y*-ρ1B1y*∈(I+ρ1M1)x*,z*-ρ2B2z*∈(I+ρ2M2)y*,x*-ρ3B3x*∈(I+ρ3M3)z*,
which implies that
(2.18)x*=J(M1,ρ1)(y*-ρ1B1y*),y*=J(M2,ρ2)(z*-ρ2B2z*),z*=J(M3,ρ3)(x*-ρ3B3x*).
We can deduce that (2.18) is equivalent to
(2.19)x*=J(M1,ρ1)[J(M2,ρ2)(J(M3,ρ3)(x*-ρ3B3x*)-ρ2B2J(M3,ρ3)(x*-ρ3B3x*)-ρ1B1J(M2,ρ2)J(M3,ρ3)(x*-ρ3B3x*)-ρ2B2J(M3,ρ3)(x*-ρ3B3x*))].
This completes the proof.


## 3. Main Results

In this section, we prove that the iterative scheme (3.1) below converges strongly to common element of the set of solutions of the variational inclusion with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point of a family of nonexpansive semigroup in a uniformly convex and 2-uniformly smooth Banach space under some certain control conditions.Now, we prove the strong convergence theorem of the sequence (3.1) for solving the problem (1.38).Theorem 3.1.
LetX be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant K. Let Mi:X→2X be a maximal monotone mapping and Bi:X→X be a Li-Lipschitzian and relaxed (ci,di)-cocoercive mapping with ρi∈(0,(di-ciLi2)/K2Li2] and θρ=max{θρi} for all i=1,2,3. Let 𝒮={T(t):t∈ℝ+} be a nonexpansive semigroup from X into itself and at least there exists a T(t) which is demicompact. Assume that Ω:=F(𝒮)∩F(Q)≠∅, where Q is defined as in Lemma 2.17. Let f:X→X be a contraction mapping with a coefficient α∈(0,1) and A be a strongly positive linear bounded self adjoint operator with a coefficient γ¯∈(0,1) such that ∥A∥≤1 and 0<γ<γ¯/α. Let {xn} be a sequence defined by
(3.1)x1∈Xchosenarbitrarily,zn=J(M3,ρ3)(xn-ρ3B3xn),yn=J(M2,ρ2)(zn-ρ2B2zn),vn=J(M1,ρ1)(yn-ρ1B1yn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)[μ1tn∫0tnT(s)xnds+(1-μ)vn],∀n≥1,
where μ∈(0,1), {αn}n=1∞, {βn}n=1∞ are the sequences in (0,1) which satisfies αn+βn≤1 and {tn}n=1∞ is a positive real divergent sequence satisfy the following restrictions: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1,(C3)
limn→∞|tn+1-tn|/tn+1=0.Then the sequence {xn} defined by (3.1) converges strongly to x*, which x* solves the variational inequality
(3.2)〈(γf-A)x*,J(z-x*)〉≤0,∀z∈Ω,
and (x*,y*,z*) is a solution of general system of variational inequality problem (1.38), where y*=J(M2,ρ2)(z*-ρ2B2z*) and z*=J(M3,ρ3)(x*-ρ3B3x*).Proof.
First, we show{xn} is bounded. By the condition (C1), we may assume, with no loss of generality, that αn≤(1-βn)∥A∥-1. Since A is a linear bounded operator on X, by (1.35), we have ∥A∥=sup{|〈Au,J(u)〉|:u∈X, ∥u∥=1}. Observe that
(3.3)〈((1-βn)I-αnA)u,J(u)〉=1-βn-αn〈Au,J(u)〉≥1-βn-αn‖A‖≥0,
that is, (1-βn)I-αnA is positive. It follows that
(3.4)‖(1-βn)I-αnA‖=sup⁡{〈((1-βn)I-αnA)u,J(u)〉:u∈X,‖u‖=1}=sup⁡{1-βn-αn〈Au,J(u)〉:u∈X,‖u‖=1}≤1-βn-αnγ¯.
Taking x¯∈Ω, it follows from Lemma 2.17 that
(3.5)x¯=J(M1,ρ1)[J(M2,ρ2)(J(M3,ρ3)(x¯-ρ3B3x¯)-ρ2B2J(M3,ρ3)(x¯-ρ3B3x¯)-ρ1B1J(M2,ρ2)J(M3,ρ3)(x¯-ρ3B3x¯)-ρ2B2J(M3,ρ3)(x¯-ρ3B3x¯))].
Putting y¯=J(M2,ρ2)(z¯-ρ2B2z¯) and z¯=J(M1,ρ1)(x¯-ρ3B3x¯). Then x¯=J(M1,ρ1)(y¯-ρ1B1y¯). It follows from Lemmas 2.1 and 2.16 that
(3.6)‖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)(z¯-ρ2B2z¯)‖≤‖(zn-ρ2B2zn)-(z¯-ρ2B2z¯)‖≤‖zn-z¯‖=‖J(M3,ρ3)(xn-ρ3B3xn)-J(M3,ρ3)(x¯-ρ3B3x¯)‖≤‖(xn-ρ3B3xn)-(x¯-ρ3B3x¯)‖≤‖xn-x¯‖
and setting ϵn=μ(1/tn)∫0tnT(s)xnds+(1-μ)vn. From (3.6), we obtain
(3.7)‖ϵn-x¯‖=‖μ(1tn∫0tnT(s)xnds-x¯)+(1-μ)(vn-x¯)‖≤μ‖1tn∫0tnT(s)xnds-x¯‖+(1-μ)‖vn-x¯‖≤μ‖xn-x¯‖+(1-μ)‖xn-x¯‖=‖xn-x¯‖.
It follows from (3.7) that
(3.8)‖xn+1-x¯‖=‖αn(γf(xn)-Ax¯)+βn(xn-x¯)+((1-βn)I-αnA)(ϵn-x¯)‖≤αn‖γf(xn)-Ax¯‖+βn‖xn-x¯‖+(1-βn-αnγ¯)‖ϵn-x¯‖≤αnγ‖f(xn)-f(x¯)‖+αn‖γf(x¯)-Ax¯‖+βn‖xn-x¯‖+(1-βn-αnγ¯)‖xn-x¯‖≤(1-αnγ¯)‖xn-x¯‖+αnγα‖xn-x¯‖+αn‖γf(x¯)-Ax¯‖=(1-αn(γ¯-γα))‖xn-x¯‖+αn‖γf(x¯)-Ax¯‖≤max⁡{‖xn-x¯‖,‖γf(x¯)-Ax¯‖γ¯-γα}.
By induction, we have
(3.9)‖xn-x¯‖≤max⁡{‖x1-x¯‖,‖γf(x¯)-Ax¯‖γ¯-γα},∀n≥1.
Hence, {xn} is bounded, so are {zn}, {yn}, and {vn}. On the other hand, by nonexpansiveness of J(Mi,ρi) and I-ρiBi for all i=1,2,3, we have
(3.10)‖vn+1-vn‖=‖J(M1,ρ1)(yn+1-ρ1B1yn+1)-J(M1,ρ1)(yn-ρ1B1yn)‖≤‖(yn+1-ρ1B1yn+1)-(yn-ρ1B1yn)‖≤‖yn+1-yn‖=‖J(M2,ρ2)(zn+1-ρ2B2zn+1)-J(M2,ρ2)(zn-ρ2B2zn)‖≤‖zn+1-zn‖=‖J(M3,ρ3)(xn+1-ρ3B3xn+1)-J(M3,ρ3)(xn-ρ3B3xn)‖≤‖xn+1-xn‖.
Now, we estimate
(3.11)‖1tn+1∫0tn+1T(s)xn+1ds-1tn∫0tnT(s)xnds‖=‖1tn+1∫0tn+1[T(s)xn+1-T(s)xn]ds+1tn+1∫0tn+1T(s)xnds-1tn∫0tnT(s)xnds‖=‖1tn+1∫0tn+1[T(s)xn+1-T(s)xn]ds+1tn+1∫0tnT(s)xnds+1tn+1∫tntn+1T(s)xnds-1tn∫0tnT(s)xnds‖=‖1tn+1∫0tn+1[T(s)xn+1-T(s)xn]ds+(1tn+1-1tn)∫0tnT(s)xnds+1tn+1∫tntn+1T(s)xnds‖,
for x¯∈Ω, it follows that
(3.12)‖1tn+1∫0tn+1T(s)xn+1ds-1tn∫0tnT(s)xnds‖=‖1tn+1∫0tn+1[T(s)xn+1-T(s)xn]ds+(1tn+1-1tn)∫0tn[T(s)xn-x¯]ds+1tn+1∫tntn+1[T(s)xn-x¯]ds‖≤1tn+1∫0tn+1‖T(s)xn+1-T(s)xn‖ds+|1tn+1-1tn|∫0tn‖T(s)xn-x¯‖ds+1tn+1∫tntn+1‖T(s)xn-x¯‖ds≤‖xn+1-xn‖+[|1tn+1-1tn|tn+1tn+1|tn+1-tn|]‖xn-x¯‖≤‖xn+1-xn‖+(2|tn+1-tn|tn+1)‖xn-x¯‖.
It follows from (3.10) and (3.12) that
(3.13)‖ϵn+1-ϵn‖=‖(μ1tn+1∫0tn+1T(s)xn+1ds+(1-μ)vn+1)-(μ1tn∫0tnT(s)xnds+(1-μ)vn)‖≤μ‖1tn+1∫0tn+1T(s)xn+1ds-1tn∫0tnT(s)xnds‖+(1-μ)‖vn+1-vn‖≤‖xn+1-xn‖+(2|tn+1-tn|tn+1)‖xn-x¯‖+(1-μ)‖xn+1-xn‖≤‖xn+1-xn‖+(2|tn+1-tn|tn+1)M1,
where M1>0 is an appropriate constant such that M1=supn≥1{∥xn-x¯∥}.
Settingln=(xn+1-βnxn)/(1-βn), for all n∈ℕ. Then xn+1=βnxn+(1-βn)ln, for all n∈ℕ, we have (3.14)ln+1-ln=xn+2-βn+1xn+11-βn+1-xn+1-βnxn1-βn=αn+1γf(xn+1)+((1-βn+1)I-αn+1A)ϵn+11-βn+1-αnγf(xn)+((1-βn)I-αnA)ϵn1-βn=αn+11-βn+1(γf(xn+1)-Aϵn+1)+αn1-βn(Aϵn-γf(xn))+ϵn+1-ϵn
and, hence,
(3.15)‖ln+1-ln‖≤αn+11-βn+1‖γf(xn+1)-Aϵn+1‖+αn1-βn‖Aϵn-γf(xn)‖+‖ϵn+1-ϵn‖,
which, combined with (3.13) yields that
(3.16)‖ln+1-ln‖≤αn+11-βn+1‖γf(xn+1)-Aϵn+1‖+αn1-βn‖Aϵn-γf(xn)‖+‖xn+1-xn‖+(2|tn+1-tn|tn+1)M1.
It follows that
(3.17)‖ln+1-ln‖-‖xn+1-xn‖≤αn+11-βn+1‖γf(xn+1)-Aϵn+1‖+αn1-βn‖Aϵn-γf(xn)‖+(2|tn+1-tn|tn+1)M1.
By conditions (C1)–(C3), we have
(3.18)limsup⁡n→∞(‖ln+1-ln‖-‖xn+1-xn‖)≤0.
Then, from Lemma 2.12, we obtain
(3.19)lim⁡n→∞‖ln-xn‖=0,
observing that
(3.20)xn+1-xn=(1-βn)(ln-xn),
and, hence,
(3.21)lim⁡n→∞‖xn+1-xn‖=0.
Since xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)ϵn, then we have
(3.22)xn+1-xn=αn(γf(xn)-Axn)+((1-βn)I-αnA)(ϵn-xn).
It follows that
(3.23)(1-βn-αnγ¯)‖ϵn-xn‖≤‖xn-xn+1‖+αn‖γf(xn)-Axn‖.
From (3.21) and by the conditions (C1), (C2), we obtain that
(3.24)lim⁡n→∞‖ϵn-xn‖=0.
Define the mapping Gn by
(3.25)Gnx=μ1tn∫0tnT(s)xds+(1-μ)Qx,∀x∈X,
where Q is defined as in Lemma 2.17. By Lemma 2.11, we see that Gn is nonexpansive such that
(3.26)F(Gn)=F(S)∩F(Q).
From (3.24), it follows that
(3.27)lim⁡n→∞‖Gnxn-xn‖=0.
Since {T(t):t∈ℝ+} is a u.a.r. nonexpansive semigroup by Remark 2.6, and at least there exists a T(t) which is demicompact, then by Lemma 2.7, we can define the mapping T:X→X such that Ty=limn→∞(1/tn)∫0tnT(s)yds for all y∈X. Furthermore, let G:X→X be the mapping define by Gy=limn→∞Gny, for all y∈X. It follows from the nonexpansivity of G and
(3.28)Gy:=μTy+(1-μ)Qy,∀y∈X,
such that
(3.29)F(G)=F(T)∩F(Q)=F(S)∩F(Q)=F(Gn).
On the other hand, we observe that for any bounded subset C of X containing {xn},
(3.30)‖xn-Gxn‖≤‖xn-Gnxn‖+‖Gnxn-Gxn‖≤‖xn-Gnxn‖+sup⁡ω∈C‖Gnω-Gω‖.
By Lemma 2.7 and (3.27), we obtain that
(3.31)lim⁡n→∞‖xn-Gxn‖=0.
Since X is a uniformly convex, hence it is reflexive and {xn} is bounded, then there exists a subsequence {xnj} of {xn} which converges weakly to some z∈X as j→∞. Again, since Banach space X has a weakly sequentially continuous duality mapping satisfying Opial's condition, then by (3.32), we have by Lemma 2.14, that z∈F(G)=Ω.
Next, we show thatlimsupn→∞〈γf(x*)-Ax*,J(xn-x*)〉≤0, where x*=limt→0xt with xt be the fixed point of the contraction x↦tγf(x)+(I-tA)Gx. Since G is a nonexpansive mapping, it follows from Lemma 2.10 that x*∈F(G)=Ω, which solves the variational inequality(3.32)〈(γf-A)x*,J(z-x*)〉≤0,∀z∈F(G)
and (x*,y*,z*) is a solution of general system of variational inequality problem (1.38) such that y*=J(M2,ρ2)(z*-ρ2B2z*) and z*=J(M3,ρ3)(x*-ρ3B3x*).
Let{xnj} be a subsequence of {xn} such that(3.33)limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)〉=lim⁡j→∞〈γf(x*)-Ax*,J(xnj-x*)〉.
Since the duality mapping J is single-valued and weakly sequentially continuous from X to X*, we obtain that
(3.34)limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)〉=lim⁡j→∞〈γf(x*)-Ax*,J(xnj-x*)〉=〈γf(x*)-Ax*,J(z-x*)〉≤0,
as required.
Finally, we show thatlimn→∞∥xn-x*∥=0. Now, from Lemma 2.3, we have (3.35)‖xn+1-x*‖2=‖αn(γf(xn)-Ax*)+βn(xn-x*)+((1-βn)I-αnA)(ϵn-x*)‖2≤‖((1-βn)I-αnA)(ϵn-x*)+βn(xn-x*)‖2+2αn〈γf(xn)-Ax*,J(xn+1-x*)〉≤((1-βn-αnγ¯)‖ϵn-x*‖+βn‖xn-x*‖)2+2αn〈γf(xn)-Ax*,J(xn+1-x*)〉≤(1-αnγ¯)2‖xn-x*‖2+2αnγ〈f(xn)-f(x*),J(xn+1-x*)〉+2αn〈γf(x*)-Ax*,J(xn+1-x*)〉≤(1-αnγ¯)2‖xn-x*‖2+2αnγα‖xn-x*‖‖xn+1-x*‖+2αn〈γf(x*)-Ax*,J(xn+1-x*)〉≤(1-αnγ¯)2‖xn-x*‖2+αnγα(‖xn-x*‖2+‖xn+1-x*‖2)+2αn〈γf(x*)-Ax*,J(xn+1-x*)〉,
which implies that
(3.36)‖xn+1-x*‖2≤(1-αnγ¯)2+αnγα1-αnγα‖xn-x*‖2+2αn1-αnγα〈γf(x*)-Ax*,J(xn+1-x*)〉=1-2αnγ¯+αnγα1-αnγα‖xn-x*‖2+αn2γ¯21-αnγα‖xn-x*‖2+2αn1-αnγα〈γf(x*)-Ax*,J(xn+1-x*)〉≤[1-2αn(γ¯-γα)1-αnγα]‖xn-x*‖2+2αn(γ¯-γα)1-αnγα[1γ¯-γα〈γf(x*)-Ax*,J(xn+1-x*)〉+αnγ¯22(γ¯-γα)M2],
where M2>0 is an appropriate constant such that M2=supn≥1{∥xn-x*∥}. Put τn=2αn(γ¯-γα)/(1-αnγα) and δn=(1/(γ¯-γα))〈γf(x*)-Ax*,J(xn+1-x*)〉+(αnγ¯2/2(γ¯-γα))M2. Then (3.32) reduces to formula
(3.37)‖xn+1-x*‖2≤(1-τn)‖xn-x*‖2+τnδn.
By the condition (C1) and (3.31), it easy to see that
(3.38)lim⁡n→∞τn=0,∑n=1∞τn=∞,limsup⁡n→∞δn≤0.
Applying Lemma 2.15 to (3.37), we obtain xn→x* as n→∞, that is limn→∞∥xn-x*∥=0. This completes the proof.Remark 3.2.
Theorem3.1 mainly improves Theorem 2.1 of Qin et al. [8], in the following respects: (1)
from a system variational inclusion to a general system of variational inclusions,(2)
from the class of inverse-strongly accretive mappings to the class of Lipchitzian and relaxed cocoercive mappings.TakingA≡I and γ=1 in Theorem 3.1, we can obtain the following result.Corollary 3.3.
LetX be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant K. Let Mi:X→2X be a maximal monotone mapping and Bi:X→X be a Li-Lipschitzian and relaxed (ci,di)-cocoercive mapping with ρi∈(0,(di-ciLi2)/K2Li2] and θρ=max{θρi} for all i=1,2,3. Let 𝒮={T(t):t∈ℝ+} be a nonexpansive semigroup from X into itself and at least there exists a T(t) which is demicompact. Assume that Ω:=F(𝒮)∩F(Q)≠∅, where Q is defined as in Lemma 2.17. Let f:X→X be a contraction mapping with a coefficient α∈(0,1). Let {xn} be a sequence defined by
(3.39)x1∈Xchosenarbitrarily,zn=J(M3,ρ3)(xn-ρ3B3xn),yn=J(M2,ρ2)(zn-ρ2B2zn),vn=J(M1,ρ1)(yn-ρ1B1yn),xn+1=αnf(xn)+βnxn+(1-βn-αn)[μ1tn∫0tnT(s)xnds+(1-μ)vn],∀n≥1,
where μ∈(0,1), {αn}n=1∞,{βn}n=1∞ are the sequences in (0,1) which satisfies αn+βn≤1 and {tn}n=1∞ is a positive real divergent sequence satisfy the following restrictions: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1,(C3)
limn→∞(|tn+1-tn|/tn+1)=0. Then the sequence {xn} defined by (3.39) converges strongly to x*, which x* solves the variational inequality
(3.40)〈(f-I)x*,J(z-x*)〉≤0,∀z∈Ω,
and (x*,y*,z*) is a solution of general system of variational inequality problem (1.38), where y*=J(M2,ρ2)(z*-ρ2B2z*) and z*=J(M3,ρ3)(x*-ρ3B3x*).Takingf(x)=u in Corollary 3.3, we can obtain the following result.Corollary 3.4.
LetX be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant K. Let Mi:X→2X be a maximal monotone mapping and Bi:X→X be a Li-Lipschitzian and relaxed (ci,di)-cocoercive mapping with ρi∈(0,(di-ciLi2)/K2Li2] and θρ=max{θρi} for all i=1,2,3. Let 𝒮={T(t):t∈ℝ+} be a nonexpansive semigroup from X into itself and at least there exists a T(t) which is demicompact. Assume that Ω:=F(𝒮)∩F(Q)≠∅, where Q is defined as in Lemma 2.17. Let {xn} be a sequence defined by
(3.41)x1∈Xchosenarbitrarily,zn=J(M,ρ3)(xn-ρ3B3xn),yn=J(M,ρ2)(zn-ρ2B2zn),vn=J(M,ρ1)(yn-ρ1B1yn),xn+1=αnu+βnxn+(1-βn-αn)[μ1tn∫0tnT(s)xnds+(1-μ)vn],∀n≥1,
where μ∈(0,1), {αn}n=1∞, {βn}n=1∞ are the sequences in (0,1) which satisfies αn+βn≤1 and {tn}n=1∞ is a positive real divergent sequence satisfy the following restrictions: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1,(C3)
limn→∞(|tn+1-tn|/tn+1)=0.Then the sequence {xn} defined by (3.41) converges strongly to x*, which x* solves the variational inequality
(3.42)〈u-x*,J(z-x*)〉≤0,∀z∈Ω,
and (x*,y*,z*) is a solution of general system of variational inequality problem (1.38), where y*=J(M2,ρ2)(z*-ρ2B2z*) and z*=J(M3,ρ3)(x*-ρ3B3x*).


## 4. Some Applications

As some applications of Theorem3.1, we obtain the following results.Lemma 4.1.
For all(x*,y*,z*)∈X×X×X, where y*=J(M,ρ2)(z*-ρ2Bz*) and z*=J(M,ρ3)(x*-ρ3Bx*), (x*,y*,z*) is a solution of the problem (1.39) if and only if x* is a fixed point of the mapping Q′ defined by
(4.1)Q′x:=J(M,ρ1)(J(M,ρ2)(J(M,ρ3)(x-ρ3Bx)-ρ2BJ(M,ρ3)(x-ρ3Bx)-ρ1BJ(M,ρ2)J(M,ρ3)(x-ρ3Bx)-ρ2BJ(M,ρ3)(x-ρ3Bx))),∀x∈X.Corollary 4.2.
LetX be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant K. Let M:X→2X be a maximal monotone mapping and B:X→X be a L-Lipschitzian and relaxed (c,d)-cocoercive mapping with ρi∈(0,(d-cL2)/K2L2] and θρ=max{θρi} for all i=1,2,3. Let 𝒮={T(t):t∈ℝ+} be a nonexpansive semigroup from X into itself and at least there exists a T(t) which is demicompact. Assume that Ω:=F(𝒮)∩F(Q′)≠∅, where Q′ is defined as in Lemma 4.1. Let f:X→X be a contraction mapping with a coefficient α∈(0,1) and A be a strongly positive linear bounded self adjoint operator with a coefficient γ¯∈(0,1) such that ∥A∥≤1 and 0<γ<γ¯/α. Let {xn} be a sequence defined by
(4.2)x1∈Xchosenarbitrarily,zn=J(M,ρ3)(xn-ρ3Bxn),yn=J(M,ρ2)(zn-ρ2Bzn),vn=J(M,ρ1)(yn-ρ1Byn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)[μ1tn∫0tnT(s)xnds+(1-μ)vn],∀n≥1,
where μ∈(0,1), {αn}n=1∞,{βn}n=1∞ are the sequences in (0,1) which satisfies αn+βn≤1 and {tn}n=1∞ is a positive real divergent sequence satisfy the following restrictions: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1,(C3)
limn→∞(|tn+1-tn|/tn+1)=0.Then the sequence {xn} defined by (4.2) converges strongly to x*, which x* solves the variational inequality
(4.3)〈(γf-A)x*,J(z-x*)〉≤0,∀z∈Ω,
and (x*,y*,z*) is a solution of system of variational inequality problem (1.39), where y*=J(M,ρ2)(z*-ρ2Bz*) and z*=J(M,ρ3)(x*-ρ3Bx*).Lemma 4.3 (see [8]).
For all(x*,y*)∈X×X, 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′ defined by
(4.4)Q′x:=J(M1,ρ1)[J(M2,ρ2)(x-ρ2B2x)-ρ1B1J(M2,ρ2)(x-ρ2B2x)].Corollary 4.4.
LetX be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant K. Let Mi:X→2X be a maximal monotone mapping and Bi:X→X be a Li-Lipschitzian and relaxed (ci,di)-cocoercive mapping with ρi∈(0,(di-ciLi2)/K2Li2] and θρ=max{θρi} for all i=1,2. Let 𝒮={T(t):t∈ℝ+} be a nonexpansive semigroup from X into itself and at least there exists a T(t) which is demicompact. Assume that Ω:=F(𝒮)∩F(Q′)≠∅, where Q′ is defined as in Lemma 4.3. Let f:X→X be a contraction mapping with a coefficient α∈(0,1) and A be a strongly positive linear bounded self adjoint operator with a coefficient γ¯∈(0,1) such that ∥A∥≤1 and 0<γ<γ¯/α. Let {xn} be a sequence defined by
(4.5)x1∈Xchosenarbitrarily,yn=J(M,ρ2)(xn-ρ2B2xn),vn=J(M,ρ1)(yn-ρ1B1yn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)[μ1tn∫0tnT(s)xnds+(1-μ)vn],∀n≥1,
where μ∈(0,1), {αn}n=1∞,{βn}n=1∞ are the sequences in (0,1) which satisfies αn+βn≤1 and {tn}n=1∞ is a positive real divergent sequence satisfy the following restrictions: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1,(C3)
limn→∞(|tn+1-tn|/tn+1)=0.Then the sequence {xn} defined by (4.5) converges strongly to x*, which x* solves the variational inequality
(4.6)〈(γf-A)x*,J(z-x*)〉≤0,∀z∈Ω,
and (x*,y*) is a solution of system of variational inequality problem (1.15), where y*=J(M2,ρ2)(x*-ρ2B2x*).If we setB1=B2=B and M1=M2=M, we obtain the following result.Lemma 4.5 (see [8]).
For all(x*,y*)∈X×X, where y*=J(M,ρ2)(x*-ρ2Bx*), (x*,y*) is a solution of the problem (1.16) if and only if x* is a fixed point of the mapping Q′′ defined by
(4.7)Q′′x:=J(M,ρ1)[J(M,ρ2)(x-ρ2Bx)-ρ1BJ(M,ρ2)(x-ρ2Bx)].Corollary 4.6.
LetX be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant K. Let M:X→2X be a maximal monotone mapping and B:X→X be a L-Lipschitzian and relaxed (c,d)-cocoercive mapping with ρi∈(0,(d-cL2)/K2L2] and θρ=max{θρi} for all i=1,2. Let 𝒮={T(t):t∈ℝ+} be a nonexpansive semigroup from X into itself and at least there exists a T(t) which is demicompact. Assume that Ω:=F(𝒮)∩F(Q′)≠∅, where Q′′ is defined as in Lemma 4.5. Let f:X→X be a contraction mapping with a coefficient α∈(0,1) and A be a strongly positive linear bounded self adjoint operator with a coefficient γ¯∈(0,1) such that ∥A∥≤1 and 0<γ<γ¯/α. Let {xn} be a sequence defined by
(4.8)x1∈Xchosenarbitrarily,yn=J(M,ρ2)(xn-ρ2Bxn),vn=J(M,ρ1)(yn-ρ1Byn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)[μ1tn∫0tnT(s)xnds+(1-μ)vn],∀n≥1,
where μ∈(0,1), {αn}n=1∞, {βn}n=1∞ are the sequences in (0,1) which satisfies αn+βn≤1 and {tn}n=1∞ is a positive real divergent sequence satisfy the following restrictions: (C1)
limn→∞αn=0 and ∑n=1∞αn=∞,(C2)
0<liminfn→∞βn≤limsupn→∞βn<1,(C3)
limn→∞(|tn+1-tn|/tn+1)=0.Then the sequence {xn} defined by (4.8) converges strongly to x*, which x* solves the variational inequality
(4.9)〈(γf-A)x*,J(z-x*)〉≤0,∀z∈Ω,
and (x*,y*) is a solution of system of variational inequality problem (1.16), where y*=J(M,ρ2)(x*-ρ2Bx*).



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*Source: 187052-2011-07-07.xml*