# On Common Solutions for Fixed-Point Problems of Two Infinite Families of Strictly Pseudocontractive Mappings and the System of Cocoercive Quasivariational Inclusions Problems in Hilbert Spaces

Authors: Pattanapong Tianchai
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

This paper is concerned with a common element of the set of common fixed points for two infinite families of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a novel general iterative scheme based on the viscosity approximation method, and applicability of the results has shown difference with the results of many others existing in the current literature.


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## Body

## 1. Introduction

Throughout this paper, we always assume thatC is a nonempty closed-convex subset of a real Hilbert space H with inner product and norm denoted by 〈·,·〉 and ∥·∥, respectively, and 2H denotes the family of all the nonempty subsets of H.LetB:H→H be a single-valued nonlinear mapping and M:H→2H a set-valued mapping. We consider the following quasivariational inclusion problem, which is to find a point x∈H such that(1.1)θ∈Bx+Mx,
where θ is the zero vector in H. The set of solutions of the problem (1.1) is denoted by VI(H,B,M). As special cases of the problem (1.1), we have the following.  (i)
IfM=∂ϕ:H→2H, where ϕ:H→ℝ∪{+∞} is a proper convex lower semicontinuous function such that ℝ is the set of real numbers, and ∂ϕ is the subdifferential of ϕ, then the quasivariational inclusion problem (1.1) is equivalent to find x∈H such that
(1.2)〈Bx,v-x〉+ϕ(y)-ϕ(x)≥0,∀v,y∈H,
which is called the mixed quasivariational inequality problem (see [1]).(ii)
IfM=∂δC, where δC:H→{0,+∞} is the indicator function of C, that is,
(1.3)δC(x)={0,x∈C,+∞,x∉C,
then the quasivariational inclusion (1.1) is equivalent to find x∈C such that
(1.4)〈Bx,v-x〉≥0,∀v∈C,
which is called Hartman-Stampacchia variational inequality problem (see [2–4]).Recall thatPC is the metric projection of H onto C, that is, for each x∈H, there exists the unique point in PCx∈C such that (1.5)‖x-PCx‖=min⁡y∈C‖x-y‖.
A mapping T:C→C is called nonexpansive if (1.6)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C,
and the mapping f:C→C is called a contraction if there exists a constant α∈(0,1) such that (1.7)‖f(x)-f(y)‖≤α‖x-y‖,∀x,y∈C.
A point x∈C is a fixed point of T provided Tx=x. We denote by F(T) the set of fixed points of T, that is, F(T)={x∈C:Tx=x}. If C⊂H is bounded, closed, and convex and T is a nonexpansive mapping of C into itself, then F(T) is nonempty (see [5]). Recall that a mapping A:C→C is said to be  (i)
monotone if
(1.8)〈Ax-Ay,x-y〉≥0,∀x,y∈C,(ii)
k-Lipschitz continuous if there exists a constant k>0 such that
(1.9)‖Ax-Ay‖≤k‖x-y‖,∀x,y∈C,
if k=1, then A is a nonexpansive,(iii)
pseudocontractive if
(1.10)‖Ax-Ay‖2≤‖x-y‖2+‖(I-A)x-(I-A)y‖2,∀x,y∈C,(iv)
k-strictly pseudocontractive if there exists a constant k∈[0,1) such that
(1.11)‖Ax-Ay‖2≤‖x-y‖2+k‖(I-A)x-(I-A)y‖2,∀x,y∈C,
and it is obvious that A is a nonexpansive if and only if A is a 0-strictly pseudocontractive,(v)
α-strongly monotone if there exists a constant α>0 such that
(1.12)〈Ax-Ay,x-y〉≥α‖x-y‖2,∀x,y∈C,(vi)
α-inverse-strongly monotone (orα-cocoercive) if there exists a constant α>0 such that
(1.13)〈Ax-Ay,x-y〉≥α‖Ax-Ay‖2,∀x,y∈C,
if α=1, then A is called that firmly nonexpansive; it is obvious that any α-inverse-strongly monotone mapping A is monotone and (1/α)-Lipschitz continuous,(vii)
relaxedα-cocoercive if there exists a constant α>0 such that
(1.14)〈Ax-Ay,x-y〉≥(-α)‖Ax-Ay‖2,∀x,y∈C,(viii)
relaxed(α,r)-cocoercive if there exists two constants α,r>0 such that
(1.15)〈Ax-Ay,x-y〉≥(-α)‖Ax-Ay‖2+r‖x-y‖2,∀x,y∈C,
and it is obvious that any r-strongly monotonicity implies to the relaxed (α,r)-cocoercivity.The existence common fixed points for a finite family of nonexpansive mappings have been considered by many authors (see [6–9] and the references therein).In this paper, we study the mappingWn defined by(1.16)Un,n+1=I,Un,n=μnSnUn,n+1+(1-μn)I,Un,n-1=μn-1Sn-1Un,n+(1-μn-1)I,⋮Un,k=μkSkUn,k+1+(1-μk)I,Un,k-1=μk-1Sk-1Un,k+(1-μk-1)I,⋮Un,2=μ2S2Un,3+(1-μ2)I,Wn=Un,1=μ1S1Un,2+(1-μ1)I,
where {μi} is nonnegative real sequence in (0,1), for all i∈ℕ, S1,S2,… from a family of infinitely nonexpansive mappings of C into itself. It is obvious that Wn is a nonexpansive of C into itself, such a mapping Wn is called a  W-mapping generated by S1,S2,…,Sn and μ1,μ2,…,μn.A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping in a real Hilbert spaceH,(1.17)min⁡x∈Ω{12〈Ax,x〉-〈x,b〉},
where A is a bounded linear operator on H, Ω is the fixed-point set of a nonexpansive mapping S on H, and b is a given point in H. Recall that A is a strongly positive bounded linear operator on H if there exists a constant γ¯>0 such that (1.18)〈Ax,x〉≥γ¯‖x‖2,∀x∈H.Marino and Xu [10] introduced the following iterative scheme based on the viscosity approximation method introduced by Moudafi [11]:(1.19)xn+1=αnγf(xn)+(I-αnA)Sxn,∀n∈N,
where x1∈H, A is a strongly positive bounded linear operator on H, f is a contraction on H, and S is a nonexpansive on H. They proved that under some appropriateness conditions imposed on the parameters, if F(S)≠∅, then the sequence {xn} generated by (1.19) converges strongly to the unique solution z=PF(S)(I-A+γf)z of the variational inequality (1.20)〈(A-γf)z,x-z〉≥0,∀x∈F(S),
which is the optimality condition for the minimization problem (1.21)min⁡x∈F(S){12〈Ax,x〉-h(x)},
where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H).Iiduka and Takahashi [12] introduced an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality (1.4) as in the following theorem.Theorem IT.
LetC be a nonempty closed-convex subset of a real Hilbert space  H. Let  B be an  α-inverse-strongly monotone mapping of  C into  H, and let  S be a nonexpansive mapping of  C into itself such that  F(S)∩VI(C,B)≠∅. Suppose that  x1=x∈C and  {xn} is the sequence defined by
(1.22)xn+1=αnx+(1-αn)SPC(xn-λnBxn),
for all n∈ℕ, where {αn}⊂(0,1) and {λn}⊂[a,b] such that 0<a<b<2α satisfying the following conditions:
(C1)
limn→∞αn=0 and ∑n=1∞αn=∞,(C2)
∑n=1∞|αn+1-αn|<∞ and ∑n=1∞|λn+1-λn|<∞,
then{xn} converges strongly to PF(S)∩VI(C,B)x.Definition 1.1 (see [13]).
LetM:H→2H be a multivalued maximal monotone mapping, then the single-valued mapping JM,λ:H→H defined by JM,λ(u)=(I+λM)-1(u), for all u∈H, is called the resolvent operator associated withM, where λ is any positive number, and I is the identity mapping.Recently, Zhang et al. [13] considered the problem (1.1). To be more precise, they proved the following theorem.Theorem ZLC.
LetH be a real Hilbert space, let  B:H→H be an  α-inverse-strongly monotone mapping, let  M:H→2H be a maximal monotone mapping, and let  S:H→H be a nonexpansive mapping. Suppose that the set  F(S)∩VI(H,B,M)≠∅, where  VI(H,B,M) is the set of solutions of quasivariational inclusion (1.1). Suppose that  x1=x∈H and  {xn} is the sequence defined by
(1.23)yn=JM,λ(xn-λBxn),xn+1=αnx+(1-αn)Syn,
for all n∈ℕ, where λ∈(0,2α) and {αn}⊂(0,1) satisfying the following conditions:
(C1)
limn→∞αn=0 and ∑n=1∞αn=∞,(C2)
∑n=1∞|αn+1-αn|<∞,
then{xn} converges strongly to PF(S)∩VI(H,B,M)x.Peng et al. [14] introduced an iterative scheme(1.24)Φ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈H,yn=JM,λ(un-λBun),xn+1=αnf(xn)+(1-αn)Syn,
for all n∈ℕ, where x1∈H, B is an α-cocoercive mapping on H, f is a contraction on H, S is a nonexpansive on H, M is a maximal monotone mapping of H into 2H, and Φ is a bifunction from H×H into ℝ.We note that their iteration is well defined if we letC=H, and the appropriateness of the control conditions αn and λ of their iteration should be {αn}⊂(0,1) and λ∈(0,2α) (see Theorem  3.1 in [14]). They proved that under some appropriateness imposed on the other parameters, if Ω=F(S)∩VI(H,B,M)∩EP(Φ)≠∅, then the sequences {xn}, {yn}, and {un} generated by (1.24) converge strongly to z=PΩf(z) of the variational inequality (1.25)〈z-f(z),x-z〉≥0,∀x∈Ω,
where EP(Φ) is the set of solutions of equilibrium problem defined by (1.26)EP⁡(Φ)={x∈H:Φ(x,y)≥0,∀y∈H}.Moreover, Plubtieng and Sriprad [15] introduced an iterative scheme(1.27)Φ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈H,yn=JM,λ(un-λBun),xn+1=αnγf(xn)+(I-αnA)Snyn,
for all n∈ℕ, where x1∈H, A is a strongly bounded linear operator on H, B is an α-cocoercive mapping on H, f is a contraction on H, Sn is a nonexpansive on H, M is a maximal monotone mapping of H into 2H, and Φ is a bifunction from H×H into ℝ.We note that the appropriateness of the control conditionsαn and λ of their iteration should be {αn}⊂(0,1) and λ∈(0,2α) (see Theorem 3.2 in [15]). They proved that under some appropriateness imposed on the other parameters, if Ω=⋂n=1∞F(Sn)∩VI(H,B,M)∩EP(Φ)≠∅, then the sequences {xn}, {yn}, and {un} generated by (1.27) converge strongly to z=PΩ(I-A+γf)z.On the other hand, Li and Wu [16] introduced an iterative scheme for finding a common element of the set of fixed points of a k-strictly pseudocontractive mapping with a fixed point and the set of solutions of relaxed cocoercive quasivariational inclusions as follows:(1.28)yn=JM,λ(xn-λBxn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)(μSkxn+(1-μ)yn),
for all n∈ℕ, where x1∈H, A is a strongly positive bounded linear operator on H, f is a contraction on H, Sk is a mapping on H defined by Skx=kx+(1-k)Sx for all x∈H, such that S is a k-strictly pseudocontractive mapping on H with a fixed point, B is relaxed cocoercive and Lipschitz continuous mappings on H, and M is a maximal monotone mapping of H into 2H.They proved that under the missing condition ofμ, which should be 0<μ<1 (see Theorem  2.1 in [16]) and some appropriateness imposed on the other parameters, if Ω=F(S)∩VI(H,B,M)≠∅, then the sequence {xn} generated by (1.28) converges strongly to z=PΩ(I-A+γf)z.Very recently, Tianchai and Wangkeeree [17] introduced an implicit iterative scheme for finding a common element of the set of common fixed points of an infinite family of a kn-strictly pseudocontractive mapping and the set of solutions of the system of generalized relaxed cocoercive quasivariational inclusions as follows:(1.29)zn=JM2,λ2(xn-λ2(B2+C2)xn),yn=JM1,λ1(zn-λ1(B1+C1)zn),xn+1=αnγf(Wnxn)+βnxn+((1-βn)I-αnA)(γnWnxn+(1-γn)yn),
for all n∈ℕ, where x1∈H, A is a strongly positive bounded linear operator on H, f is a contraction on H, Wn is a W-mapping on H generated by {Sn} and {μn} such that Snx=δnx+(1-δn)Tnx for all x∈H, Tn is a kn-strictly pseudocontractive mapping on H with a fixed point, Mi is a maximal monotone mapping of H into 2H, and Bi, Ci are two mappings of relaxed cocoercive and Lipschitz continuous mappings on H for each i=1,2.They proved that under some appropriateness imposed on the parameters, ifΩ=⋂n=1∞F(Tn)∩F(D)≠∅ such that the mapping D:H→H defined by (1.30)Dx=JM1,λ1((I-λ1(B1+C1))JM2,λ2(I-λ2(B2+C2))x),∀x∈H,
then the sequence {xn} generated by (1.29) converges strongly to z=PΩ(I-A+γf)z.In this paper, we introduce a novel general iterative scheme (1.32) below by the viscosity approximation method to find a common element of the set of common fixed points for two infinite families of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. Firstly, we introduce a mapping Wn, where Wn is a W-mapping generated by {Rn} and {μn} for solving a common fixed point for two infinite families of strictly pseudocontractive mappings by iteration such that the mapping Rn:H→H defined by(1.31)Rnx=αx+(1-α)(αSnx+(1-α)Tnx),∀x∈H,
for all n∈ℕ, where {Sn:H→H} and {Tn:H→H} are two infinite families of k1 and k2-strictly pseudocontractive mappings with a fixed point, respectively, and {μn}⊂(0,μ] for some μ∈(0,1). It follows that a linear general iterative scheme of the mappings Wn and JMi,λi(I-λiCi) is obtained as follows:(1.32)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+βnBxn+((1-ϵn)I-βnB-αnA)yn,
for all n∈ℕ, where x1=u∈H, Mi:H→2H is a maximal monotone mapping, Ci:H→H is a cocoercive mapping for each i=1,2,…,N, f:H→H is a contraction mapping, and A,B:H→H are two mappings of the strongly positive linear bounded self-adjoint operator mappings.As special cases of the iterative scheme (1.32), we have the following.  (i)
Ifϵn=0 for all n∈ℕ, then (1.32) is reduced to the iterative scheme
(1.33)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+βnBxn+(I-βnB-αnA)yn,∀n∈N.(ii)
IfB≡I, then (1.32) is reduced to the iterative scheme
(1.34)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+βnxn+((1-ϵn-βn)I-αnA)yn,∀n∈N.(iii)
Ifϵn=0 for all n∈ℕ, then (1.34) is reduced to the iterative scheme
(1.35)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)yn,∀n∈N.(iv)
Ifβn=0 for all n∈ℕ, then (1.34) is reduced to the iterative scheme
(1.36)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+((1-ϵn)I-αnA)yn,∀n∈N.(v)
Ifϵn=0 for all n∈ℕ, then (1.36) is reduced to the iterative scheme
(1.37)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+(I-αnA)yn,∀n∈N.(vi)
Ifγ=1 and A≡I, then (1.37) is reduced to the iterative scheme
(1.38)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnf(xn)+(1-αn)yn,∀n∈N.(vii)
IfMi≡Ci≡0 for each i=1,2,…,N and ∑i=1Nρi=1, then (1.32) is reduced to the iterative scheme
(1.39)yn=γnWnxn+(1-γn)xn,xn+1=αnγf(xn)+βnBxn+((1-ϵn)I-βnB-αnA)yn,∀n∈N.Furthermore, ifSn≡Tn for all n∈ℕ, then the mapping Rn:H→H in (1.31) is reduced to(1.40)Rnx=αx+(1-α)Tnx,∀x∈H,
for all n∈ℕ. It follows that the iterative scheme (1.32) is reduced to find a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces.It is well known that the class of strictly pseudocontractive mappings contains the class of nonexpansive mappings; it follows that if the mappingRn is defined as (1.31) and k1=k2=0, then the iterative scheme (1.32) is reduced to find a common element of the set of common fixed points for two infinite families of nonexpansive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces, and if the mapping Rn is defined as (1.40) and k1=k2=0, then the iterative scheme (1.32) is reduced to find a common element of the set of common fixed points for an infinite family of nonexpansive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces.We suggest and analyze the iterative scheme (1.32) above under some appropriateness conditions imposed on the parameters, the strong convergence theorem for the above two sets is obtained, and applicability of the results has shown difference with the results of many others existing in the current literature.


## 2. Preliminaries

We collect the following lemmas which are used in the proof for the main results in the next section.Lemma 2.1.
LetC be a nonempty closed-convex subset of a Hilbert space H then the following inequalities hold:
(1)
〈x-PCx,PCx-y〉≥0,∀x∈H,y∈C,(2)
∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.Lemma 2.2 (see [10]).
LetH be a Hilbert space, let f:H→H be a contraction with coefficient 0<α<1, and let A:H→H be a strongly positive linear bounded operator with coefficient γ¯>0, then
(1)
if0<γ<γ¯/α, then
(2.1)〈x-y,(A-γf)x-(A-γf)y〉≥(γ¯-γα)‖x-y‖2,∀x,y∈H,(2)
if0<ρ≤∥A∥-1, then ∥I-ρA∥≤1-ργ¯.Lemma 2.3 (see [18]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.2)an+1≤(1-ηn)an+δn,n≥1,
where {ηn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that
(1)
limn→∞ηn=0 and ∑n=1∞ηn=∞,(2)
limsupn→∞(δn/ηn)≤0 or ∑n=1∞|δn|<∞,then limn→∞an=0.Lemma 2.4 (see [9]).
LetC be a nonempty closed-convex subset of a Hilbert space H, define mapping Wn as (1.16), let Si:C→C be a family of infinitely nonexpansive mappings with ⋂i=1∞F(Si)≠∅, and let {μi} be a sequence such that 0<μi≤μ<1, for all i≥1, then
(1)
Wn is nonexpansive and F(Wn)=⋂i=1nF(Si) for each n≥1,(2)
for eachx∈C and for each positive integer k, limn→∞Un,kx exists,(3)
the mappingW:C→C defined by
(2.3)Wx∶=lim⁡n→∞Wnx=lim⁡n→∞Un,1x,x∈C,
is a nonexpansive mapping satisfying F(W)=⋂i=1∞F(Si), and it is called the W-mapping generated by S1,S2,… and μ1,μ2,….Lemma 2.5 (see [13]).
The resolvent operatorJM,λ associated with M is single-valued and nonexpansive for all λ>0.Lemma 2.6 (see [13]).
u∈H is a solution of quasivariational inclusion (1.1) if and only if u=JM,λ(u-λBu), for all λ>0, that is,
(2.4)VI⁡(H,B,M)=F(JM,λ(I-λB)),∀λ>0.Lemma 2.7 (see [19]).
LetC be a nonempty closed-convex subset of a strictly convex Banach space X. Let {Tn:n∈ℕ} be a sequence of nonexpansive mappings on C. Suppose that ⋂n=1∞F(Tn)≠∅. Let {αn} be a sequence of positive real numbers such that ∑n=1∞αn=1, then a mapping S on C defined by
(2.5)Sx=∑n=1∞αnTnx,
for x∈C, is well defined, nonexpansive, and F(S)=⋂n=1∞F(Tn) holds.Lemma 2.8 (see [2]).
LetC be a nonempty closed-convex subset of a Hilbert space H and S:C→C a nonexpansive mapping, then I-S is demiclosed at zero. That is, whenever {xn} is a sequence in C weakly converging to some x∈C and the sequence {(I-S)xn} strongly converges to some y, it follows that (I-S)x=y.Lemma 2.9 (see [20]).
LetC be a nonempty closed-convex subset of a real Hilbert space H and T:C→C a k-strict pseudocontraction. Define S:C→C by Sx=αx+(1-α)Tx for each x∈C, then, as α∈[k,1), S is a nonexpansive such that F(S)=F(T).


## 3. Main Results

Lemma 3.1.
LetC be a nonempty closed-convex subset of a real Hilbert space H, and let S,T:C→C be two mappings of k1 and k2-strictly pseudocontractive mappings with a fixed point, respectively. Suppose that F(S)∩F(T)≠∅ and define a mapping R:C→C by
(3.1)Rx=αx+(1-α)(αSx+(1-α)Tx),∀x∈C,
where α∈[k,1)∖{0} such that k=max{k1,k2}, then R is well defined, nonexpansive, and F(R)=F(S)∩F(T).Proof.
Define the mappingsS1,T1:C→C as follows:
(3.2)S1x=αx+(1-α)Sx,T1x=αx+(1-α)Tx,
for all x∈C. By Lemma 2.9, we have S1 and T1 as nonexpansive such that F(S1)=F(S) and F(T1)=F(T). Therefore, for all x∈C, we have
(3.3)Rx=αx+(1-α)(αSx+(1-α)Tx)=αx+α(1-α)Sx+(1-α)2Tx=α2x+α(1-α)Sx+(1-α)αx+(1-α)2Tx=α(αx+(1-α)Sx)+(1-α)(αx+(1-α)Tx)=αS1x+(1-α)T1x.
It follows from Lemma 2.7 that R is well defined, nonexpansive, and F(R)=F(S1)∩F(T1)=F(S)∩F(T).Theorem 3.2.
LetH be a real Hilbert space, let Mi:H→2H be a maximal monotone mapping, and let Ci:H→H be a ξi-cocoercive mapping for each i=1,2,…,N. Let A,B:H→H be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients δ¯,β¯∈(0,1] such that δ¯≤∥A∥≤1 and ∥B∥=β¯, respectively, and let f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Sn:H→H} and {Tn:H→H} be two infinite families of k1 and k2-strictly pseudocontractive mappings with a fixed point such that k1,k2∈[0,1), respectively. Define a mapping Rn:H→H by
(3.4)Rnx=αx+(1-α)(αSnx+(1-α)Tnx),∀x∈H,
for all n∈ℕ, where α∈[k,1)∖{0} such that k=max{k1,k2}. Let Wn:H→H be a W-mapping generated by {Rn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω∶=(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn))∩(⋂i=1NVI(H,Ci,Mi))≠∅ and 0<γ<δ¯/δ. For x1=u∈H, suppose that {xn} is generated iteratively by
(3.5)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+βnBxn+((1-ϵn)I-βnB-αnA)yn,
for all n∈ℕ, where {αn},{γn}⊂(0,1), {βn},{ϵn}⊂[0,1) such that ϵn≤αn, ρi∈(0,1), and λi∈(0,2ξi] for each i=1,2,…,N satisfying the following conditions:
(C1)
limn→∞αn=limn→∞(ϵn/αn)=0,(C2)
0<limn→∞γn<1 and limsupn→∞βn<1,(C3)
∑n=1∞αn=∞ and ∑i=1Nρi=1,(C4)
∑n=1∞|αn+1-αn|<∞, ∑n=1∞|βn+1-βn|<∞, and ∑n=1∞|ϵn+1-ϵn|<∞,(C5)
∑n=1∞|γn+1-γn|<∞ and ∑n=1∞∏i=1nμi<∞,
then the sequences{xn} and {yn} converge strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(3.6)〈(A-γf)w,y-w〉≥0,∀y∈Ω.Proof.
From∥B∥=β¯∈(0,1], ϵn≤αn for all n∈ℕ, (C1) and (C2), we have αn→0, ϵn→0 as n→∞ and limsupn→∞βn<1. Thus, we may assume without loss of generality that αn<(1-ϵn-βn∥B∥)∥A∥-1 for all n∈ℕ. For any x,y∈H and for each i=1,2,…,N, by the ξi-cocoercivity of Ci, we have
(3.7)‖(I-λiCi)x-(I-λiCi)y‖2=‖(x-y)-λi(Cix-Ciy)‖2=‖x-y‖2-2λi〈x-y,Cix-Ciy〉+λi2‖Cix-Ciy‖2≤‖x-y‖2-(2ξi-λi)λi‖Cix-Ciy‖2≤‖x-y‖2,
which implies that I-λiCi is a nonexpansive. Since A and B are two mappings of the linear bounded self-adjoint operators, we have
(3.8)‖A‖=sup⁡{|〈Ax,x〉|:x∈H,‖x‖=1},‖B‖=sup⁡{|〈Bx,x〉|:x∈H,‖x‖=1}.
Observe that
(3.9)〈((1-ϵn)I-βnB-αnA)x,x〉=(1-ϵn)〈x,x〉-βn〈Bx,x〉-αn〈Ax,x〉≥1-ϵn-βn‖B‖-αn‖A‖>0.
Therefore, we obtain that (1-ϵn)I-βnB-αnA is positive. Thus, by the strong positivity of A and B, we get
(3.10)‖(1-ϵn)I-βnB-αnA‖=sup⁡{〈((1-ϵn)I-βnB-αnA)x,x〉:x∈H,‖x‖=1}=sup⁡{(1-ϵn)〈x,x〉-βn〈Bx,x〉-αn〈Ax,x〉:x∈H,‖x‖=1}≤1-ϵn-βnβ¯-αnδ¯≤1-βnβ¯-αnδ¯.
Define the sequences of mappings{Pn:H→H} and {Qn:H→H} as follows:
(3.11)Pnx=αnγf(x)+βnBx+((1-ϵn)I-βnB-αnA)Qnx,∀x∈H,Qnx=γnWnx+(1-γn)∑i=1NρiJMi,λi(I-λiCi)x,∀x∈H,
for all n∈ℕ. Firstly, we prove that Pn has a unique fixed point in H. Note that for all x,y∈H, by (3.11), (C3), the nonexpansiveness of Wn,JMi,λi, and I-λiCi, we have
(3.12)‖Qnx-Qny‖≤γn‖Wnx-Wny‖+(1-γn)∑i=1Nρi‖JMi,λi(I-λiCi)x-JMi,λi(I-λiCi)y‖≤γn‖x-y‖+(1-γn)∑i=1Nρi‖(I-λiCi)x-(I-λiCi)y‖≤γn‖x-y‖+(1-γn)(∑i=1Nρi)‖x-y‖=‖x-y‖.
Therefore, Qn is a nonexpansive. It follows from (3.10), (3.11), (3.12), the contraction of f, and the linearity of A and B that
(3.13)‖Pnx-Pny‖≤αnγ‖f(x)-f(y)‖+βn‖B‖‖x-y‖+‖(1-ϵn)I-βnB-αnA‖‖Qnx-Qny‖≤αnγδ‖x-y‖+βnβ¯‖x-y‖+(1-βnβ¯-αnδ¯)‖x-y‖=(1-(δ¯-γδ)αn)‖x-y‖.
Hence, Pn is a contraction with coefficient 1-(δ¯-γδ)αn∈(0,1). Therefore, Banach contraction principle guarantees that Pn has a unique fixed point in H, and so the iteration (3.5) is well defined.
Next, we prove that{xn} is bounded. Pick p∈Ω. Therefore, by Lemma 2.6, we have
(3.14)p=JMi,λi(I-λiCi)p,
for each i=1,2,…,N. By (3.14), the nonexpansiveness of JMi,λi, and I-λiCi, we have
(3.15)‖JMi,λi(xn-λiCixn)-p‖=‖JMi,λi(xn-λiCixn)-JMi,λi(p-λiCip)‖≤‖(xn-λiCixn)-(p-λiCip)‖≤‖xn-p‖.
Lettn=∑i=1NρiJMi,λi(xn-λiCixn). By (3.14), (C3), the nonexpansiveness of JMi,λi, and I-λiCi, we have
(3.16)‖tn-p‖=‖∑i=1NρiJMi,λi(xn-λiCixn)-∑i=1Nρip‖=‖∑i=1NρiJMi,λi(xn-λiCixn)-∑i=1NρiJMi,λi(p-λiCip)‖≤∑i=1Nρi‖JMi,λi(xn-λiCixn)-JMi,λi(p-λiCip)‖≤∑i=1Nρi‖(xn-λiCixn)-(p-λiCip)‖≤(∑i=1Nρi)‖xn-p‖=‖xn-p‖.
Since Rnx=αx+(1-α)(αSnx+(1-α)Tnx), where α∈[k,1)∖{0}, {Sn} and {Tn} are two infinite families of k1 and k2-strict pseudocontractions with a fixed point, respectively, such that k=max{k1,k2}; therefore, by Lemma 3.1, we have that Rn is a nonexpansive and F(Rn)=F(Sn)∩F(Tn) for all n∈ℕ. It follows from Lemma 2.4(1) that we get F(Wn)=⋂i=1nF(Ri)=(⋂i=1nF(Si))∩(⋂i=1nF(Ti)), which implies that Wnp=p. Hence, by (3.16) and the nonexpansiveness of Wn, we have
(3.17)‖yn-p‖=‖γnWnxn+(1-γn)tn-p‖=‖γn(Wnxn-p)+(1-γn)(tn-p)‖≤γn‖Wnxn-Wnp‖+(1-γn)‖tn-p‖≤γn‖xn-p‖+(1-γn)‖xn-p‖=‖xn-p‖.
By (3.10), (3.17), the contraction of f, and the linearity of A and B, we have
(3.18)‖xn+1-p‖=‖αnγf(xn)+βnBxn+((1-ϵn)I-βnB-αnA)yn-p‖=‖αn(γf(xn)-Ap)+βnB(xn-p)+((1-ϵn)I-βnB-αnA)(yn-p)-ϵnp‖≤αn‖γf(xn)-Ap‖+βn‖B‖‖xn-p‖+‖(1-ϵn)I-βnB-αnA‖‖yn-p‖+ϵn‖p‖≤αnγ‖f(xn)-f(p)‖+αn‖γf(p)-Ap‖+βnβ¯‖xn-p‖+(1-βnβ¯-αnδ¯)‖xn-p‖+αn‖p‖≤(1-(δ¯-γδ)αn)‖xn-p‖+αn(‖γf(p)-Ap‖+‖p‖)≤max⁡{‖xn-p‖,‖γf(p)-Ap‖+‖p‖δ¯-γδ}.
It follows from induction that
(3.19)‖xn+1-p‖≤max⁡{‖x1-p‖,‖γf(p)-Ap‖+‖p‖δ¯-γδ},
for all n∈ℕ. Hence, {xn} is bounded, and so are {yn}, {Wnxn}, {tn}, {f(xn)}, {Ayn}, {Bxn}, and {Byn}.
Next, we prove that∥xn+1-xn∥→0 as n→∞. By (C3), the nonexpansiveness of JMi,λi, and I-λiCi, we have
(3.20)‖tn+1-tn‖=‖∑i=1NρiJMi,λi(xn+1-λiCixn+1)-∑i=1NρiJMi,λi(xn-λiCixn)‖≤∑i=1Nρi‖JMi,λi(xn+1-λiCixn+1)-JMi,λi(xn-λiCixn)‖≤∑i=1Nρi‖(xn+1-λiCixn+1)-(xn-λiCixn)‖≤(∑i=1Nρi)‖xn+1-xn‖=‖xn+1-xn‖.
By the nonexpansiveness of Ri and Un,i, we have
(3.21)‖Wn+1xn-Wnxn‖=‖Un+1,1xn-Un,1xn‖=‖μ1R1Un+1,2xn+(1-μ1)xn-(μ1R1Un,2xn+(1-μ1)xn)‖≤μ1‖Un+1,2xn-Un,2xn‖=μ1‖μ2R2Un+1,3xn+(1-μ2)xn-(μ2R2Un,3xn+(1-μ2)xn)‖≤μ1μ2‖Un+1,3xn-Un,3xn‖⋮≤(∏i=1nμi)‖Un+1,n+1xn-Un,n+1xn‖≤M∏i=1nμi,
for some constant M such that M≥∥Un+1,n+1xn-Un,n+1xn∥≥0. Therefore, from (3.21), by the nonexpansiveness of Wn+1, we have
(3.22)‖Wn+1xn+1-Wnxn‖≤‖Wn+1xn+1-Wn+1xn‖+‖Wn+1xn-Wnxn‖≤‖xn+1-xn‖+M∏i=1nμi.
Since
(3.23)yn+1-yn=(γn+1Wn+1xn+1+(1-γn+1)tn+1)-(γnWnxn+(1-γn)tn)=γn+1(Wn+1xn+1-Wnxn)+(γn+1-γn)(Wnxn-tn)+(1-γn+1)(tn+1-tn),
combining (3.20), (3.22), and (3.23), we have
(3.24)‖yn+1-yn‖≤γn+1‖Wn+1xn+1-Wnxn‖+|γn+1-γn|‖Wnxn-tn‖+(1-γn+1)‖tn+1-tn‖≤γn+1(‖xn+1-xn‖+M∏i=1nμi)+|γn+1-γn|‖Wnxn-tn‖+(1-γn+1)‖xn+1-xn‖≤‖xn+1-xn‖+M∏i=1nμi+|γn+1-γn|‖Wnxn-tn‖.
By the linearity of A and B, we have
(3.25)xn+2-xn+1=(αn+1γf(xn+1)+βn+1Bxn+1+((1-ϵn+1)I-βn+1B-αn+1A)yn+1)-(αnγf(xn)+βnBxn+((1-ϵn)I-βnB-αnA)yn)=((1-ϵn+1)I-βn+1B-αn+1A)(yn+1-yn)+(βn-βn+1)Byn+(αn-αn+1)Ayn+(ϵn-ϵn+1)yn+αn+1γ(f(xn+1)-f(xn))+γ(αn+1-αn)f(xn)+βn+1B(xn+1-xn)+(βn+1-βn)Bxn.
Therefore, by (3.10), (3.24), (3.25), and the contraction of f, we have
(3.26)‖xn+2-xn+1‖≤‖(1-ϵn+1)I-βn+1B-αn+1A‖‖yn+1-yn‖+|βn-βn+1|‖Byn‖+|αn-αn+1|‖Ayn‖+|ϵn-ϵn+1|‖yn‖+αn+1γ‖f(xn+1)-f(xn)‖+γ|αn+1-αn|‖f(xn)‖+βn+1‖B‖‖xn+1-xn‖+|βn+1-βn|‖Bxn‖≤(1-βn+1β¯-αn+1δ¯)‖yn+1-yn‖+|βn-βn+1|‖Byn‖+|αn-αn+1|‖Ayn‖+|ϵn-ϵn+1|‖yn‖+αn+1γδ‖xn+1-xn‖+γ|αn+1-αn|‖f(xn)‖+βn+1β¯‖xn+1-xn‖+|βn+1-βn|‖Bxn‖≤(1-ηn)‖xn+1-xn‖+δn,
where ηn∶=(δ¯-γδ)αn+1∈(0,1) and
(3.27)δn∶=M∏i=1nμi+N(|γn-γn+1|+|ϵn-ϵn+1|+|βn-βn+1|+|αn-αn+1|),
such that
(3.28)N=max⁡{sup⁡n≥1‖Wnxn-tn‖,sup⁡n≥1(‖Byn‖+‖Bxn‖),sup⁡n≥1‖yn‖,sup⁡n≥1(‖Ayn‖+γ‖f(xn)‖)}.
By (C1), (C3), (C4), and (C5), we can find that limn→∞ηn=0, ∑n=1∞ηn=∞, and ∑n=1∞δn<∞; therefore, by (3.26) and Lemma 2.3, we obtain
(3.29)‖xn+1-xn‖⟶0asn⟶∞.
Next, we prove that∥xn-yn∥→0 as n→∞. By the linearity of B, we have
(3.30)‖xn+1-yn‖=‖αnγf(xn)+βnBxn+((1-ϵn)I-βnB-αnA)yn-yn‖=‖αn(γf(xn)-Ayn)+βnB(xn-xn+1)+βnB(xn+1-yn)-ϵnyn‖≤αn‖γf(xn)-Ayn‖+βn‖B‖‖xn-xn+1‖+βn‖B‖‖xn+1-yn‖+ϵn‖yn‖≤αn(‖γf(xn)-Ayn‖+‖yn‖)+βnβ¯‖xn-xn+1‖+βnβ¯‖xn+1-yn‖.
It follows that
(3.31)(1-βnβ¯)‖xn+1-yn‖≤αn(‖γf(xn)-Ayn‖+‖yn‖)+βnβ¯‖xn-xn+1‖.
Hence, by (C1), (C2), (3.29), and (3.31), we have
(3.32)‖xn+1-yn‖⟶0asn⟶∞.
Since
(3.33)‖xn-yn‖≤‖xn-xn+1‖+‖xn+1-yn‖,
therefore, by (3.29) and (3.32), we obtain
(3.34)‖xn-yn‖⟶0asn⟶∞.
For allx,y∈H, by Lemma 2.2(2), the nonexpansiveness of PΩ, the contraction of f, and the linearity of A, we have
(3.35)‖PΩ(I-A+γf)x-PΩ(I-A+γf)y‖≤‖(I-A+γf)x-(I-A+γf)y‖≤γ‖f(x)-f(y)‖+‖I-A‖‖x-y‖≤γδ‖x-y‖+(1-δ¯)‖x-y‖=(1-(δ¯-γδ))‖x-y‖.
Therefore, PΩ(I-A+γf) is a contraction with coefficient 1-(δ¯-γδ)∈(0,1); Banach contraction principle guarantees that PΩ(I-A+γf) has a unique fixed point, say w∈H, that is, w=PΩ(I-A+γf)w. Hence, by Lemma 2.1(1), we obtain
(3.36)〈(A-γf)w,y-w〉≥0,∀y∈Ω.
Next, we claim that(3.37)limsup⁡n→∞〈γf(w)-Aw,xn-w〉≤0.
To show this inequality, we choose a subsequence {xni} of {xn} such that
(3.38)limsup⁡n→∞〈γf(w)-Aw,xn-w〉=lim⁡i→∞〈γf(w)-Aw,xni-w〉.
Since {xni} is bounded, there exists a subsequence {xnij} of {xni} which converges weakly to w¯. Without loss of generality, we can assume that xni⇀w¯ as i→∞.
Next, we prove thatw¯∈Ω. Define the sequence of mappings {Qn:H→H} and the mapping Q:H→H by
(3.39)Qnx=γnWnx+(1-γn)∑i=1NρiJMi,λi(I-λiCi)x,∀x∈H,Qx=lim⁡n→∞Qnx,
for all n∈ℕ. Therefore, by (C2) and Lemma 2.4(3), we have
(3.40)Qx=aWx+(1-a)∑i=1NρiJMi,λi(I-λiCi)x,∀x∈H,
where 0<a=limn→∞γn<1. From (C3), Lemma 2.4(3), we have that W and ∑i=1NρiJMi,λi(I-λiCi) are nonexpansive. Therefore, by (C3), Lemmas 2.4(3), 2.6, 2.7, and 3.1, we have
(3.41)F(Q)=F(W)∩F(∑i=1NρiJMi,λi(I-λiCi))=(⋂i=1∞F(Ri))∩(⋂i=1NF(JMi,λi(I-λiCi)))=(⋂i=1∞F(Si))∩(⋂i=1∞F(Ti))∩(⋂i=1NVI⁡(H,Ci,Mi)),
that is, F(Q)=Ω. From (3.34), we have ∥yni-xni∥→0asi→∞. Thus, from (3.5) and (3.39), we get ∥Qxni-xni∥→0asi→∞. It follows from xni⇀w¯ and by Lemma 2.8 that w¯∈F(Q), that is, w¯∈Ω. Therefore, from (3.36) and (3.38), we obtain
(3.42)limsup⁡n→∞〈γf(w)-Aw,xn-w〉=lim⁡i→∞〈γf(w)-Aw,xni-w〉=〈(γf-A)w,w¯-w〉≤0.
Next, we prove thatxn→w as n→∞. Since w∈Ω, the same as in (3.17), we have
(3.43)‖yn-w‖≤‖xn-w‖.
Therefore, by (3.10), (3.43), Lemma 2.1(2), the contraction of f, and the linearity of A and B, we have
(3.44)‖xn+1-w‖2=‖αnγf(xn)+βnBxn+((1-ϵn)I-βnB-αnA)yn-w‖2=‖αn(γf(xn)-Aw)+βnB(xn-w)+((1-ϵn)I-βnB-αnA)(yn-w)-ϵnw‖2≤‖βnB(xn-w)+((1-ϵn)I-βnB-αnA)(yn-w)‖2+2〈αn(γf(xn)-Aw)-ϵnw,xn+1-w〉≤(βn‖B‖‖xn-w‖+‖(1-ϵn)I-βnB-αnA‖‖yn-w‖)2+2αnγ〈f(xn)-f(w),xn+1-w〉+2αn〈γf(w)-Aw,xn+1-w〉-2ϵn〈w,xn+1-w〉≤(βnβ¯‖xn-w‖+(1-βnβ¯-αnδ¯)‖xn-w‖)2+2αnγδ‖xn-w‖‖xn+1-w‖+2αn〈γf(w)-Aw,xn+1-w〉-2ϵn〈w,xn+1-w〉≤(1-αnδ¯)2‖xn-w‖2+αnγδ(‖xn-w‖2+‖xn+1-w‖2)+2αn〈γf(w)-Aw,xn+1-w〉-2ϵn〈w,xn+1-w〉.
If follows that
(3.45)‖xn+1-w‖2≤1-2αnδ¯+αnγδ1-αnγδ‖xn-w‖2+δn′=(1-2(δ¯-γδ)αn1-αnγδ)‖xn-w‖2+δn′≤(1-ηn′)‖xn-w‖2+δn′,
where ηn′∶=(δ¯-γδ)αn/(1-αnγδ)∈(0,1) and
(3.46)δn′∶=11-αnγδ(αn2δ¯2‖xn-w‖2+2αn〈γf(w)-Aw,xn+1-w〉-2ϵn〈w,xn+1-w〉).
By (3.29), (3.42), (C1), and (C3), we can found that limn→∞ηn′=0, ∑n=1∞ηn′=∞, and limsupn→∞(δn′/ηn′)≤0. Therefore, by Lemma 2.3, we obtain that {xn} converges strongly to w, and so is {yn}. This completes the proof.Remark 3.3.
The iteration (3.5) is the difference with many others as follows.
(1)
Two mappingsA and B of the strongly positive linear bounded self-adjoint operator mappings are used in the iteration of {xn}, which used only one mapping A by many others.(2)
Three parametersαn,βn, and ϵn are used in the iteration of {xn}, which used only two parameters αn and βn by many others.(3)
The parameterβn can be chosen to be βn=0 for all n∈ℕ, because the condition liminfn→∞βn>0 of Suzuki's Lemma (see [21]) is ignored in the control conditions of the iteration, which is used by many others.(4)
A solving of a common fixed point for two infinite families of strictly pseudocontractive mappings by iteration is obtained by the mappingWn, where Wn is a W-mapping generated by {Rn} and {μn} such that Rn is defined as in Theorem 3.2.


## 4. Applications

Theorem 4.1.
LetH be a real Hilbert space, let Mi:H→2H be a maximal monotone mapping, and let Ci:H→H be a ξi-cocoercive mapping for each i=1,2,…,N. Let A,B:H→H be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients δ¯,β¯∈(0,1] such that δ¯≤∥A∥≤1 and ∥B∥=β¯, respectively, and let f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Sn:H→H} and {Tn:H→H} be two infinite families of k1 and k2-strictly pseudocontractive mappings with a fixed point such that k1,k2∈[0,1), respectively. Define a mapping Rn:H→H by
(4.1)Rnx=αx+(1-α)(αSnx+(1-α)Tnx),∀x∈H,
for all n∈ℕ, where α∈[k,1)∖{0} such that k=max{k1,k2}. Let Wn:H→H be a W-mapping generated by {Rn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω∶=(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn))∩(⋂i=1NVI(H,Ci,Mi))≠∅ and  0<γ<δ¯/δ.For x1=u∈H, suppose that {xn} is generated iteratively by
(4.2)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+βnBxn+(I-βnB-αnA)yn,
for all n∈ℕ, where {αn},{γn}⊂(0,1), {βn}⊂[0,1), ρi∈(0,1), and λi∈(0,2ξi] for each i=1,2,…,N satisfying the following conditions:
(C1)
limn→∞αn=0,(C2)
0<limn→∞γn<1 and limsupn→∞βn<1,(C3)
∑n=1∞αn=∞ and ∑i=1Nρi=1,(C4)
∑n=1∞|αn+1-αn|<∞ and ∑n=1∞|βn+1-βn|<∞,(C5)
∑n=1∞|γn+1-γn|<∞ and ∑n=1∞∏i=1nμi<∞,
then the sequences{xn} and {yn} converge strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(4.3)〈(A-γf)w,y-w〉≥0,∀y∈Ω.Proof.
It is concluded from Theorem3.2 immediately, by putting ϵn=0 for all n∈ℕ.Theorem 4.2.
LetH be a real Hilbert space, let Mi:H→2H be a maximal monotone mapping, and let Ci:H→H be a ξi-cocoercive mapping for each i=1,2,…,N. Let A:H→H be a strongly positive linear bounded self-adjoint operator mapping with coefficient δ¯∈(0,1] such that δ¯≤∥A∥≤1, and let f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Sn:H→H} and {Tn:H→H} be two infinite families of k1 and k2-strictly pseudocontractive mappings with a fixed point such that k1,k2∈[0,1), respectively. Define a mapping Rn:H→H by
(4.4)Rnx=αx+(1-α)(αSnx+(1-α)Tnx),∀x∈H,
for all n∈ℕ, where α∈[k,1)∖{0} such that k=max{k1,k2}. Let Wn:H→H be a W-mapping generated by {Rn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω∶=(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn))∩(⋂i=1NVI(H,Ci,Mi))≠∅ and 0<γ<δ¯/δ. For x1=u∈H, suppose that {xn} is generated iteratively by
(4.5)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+βnxn+((1-ϵn-βn)I-αnA)yn,
for all n∈ℕ, where {αn},{γn}⊂(0,1), {βn},{ϵn}⊂[0,1) such that ϵn≤αn, ρi∈(0,1), and λi∈(0,2ξi] for each i=1,2,…,N satisfying the following conditions:
(C1)
limn→∞αn=limn→∞(ϵn/αn)=0,(C2)
0<limn→∞γn<1 and limsupn→∞βn<1,(C3)
∑n=1∞αn=∞ and ∑i=1Nρi=1,(C4)
∑n=1∞|αn+1-αn|<∞, ∑n=1∞|βn+1-βn|<∞, and ∑n=1∞|ϵn+1-ϵn|<∞,(C5)
∑n=1∞|γn+1-γn|<∞ and ∑n=1∞∏i=1nμi<∞,
then the sequences{xn} and {yn} converge strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(4.6)〈(A-γf)w,y-w〉≥0,∀y∈Ω.Proof.
It is concluded from Theorem3.2 immediately, by putting B≡I.Theorem 4.3.
LetH be a real Hilbert space, let Mi:H→2H be a maximal monotone mapping, and let Ci:H→H be a ξi-cocoercive mapping for each i=1,2,…,N. Let A:H→H be a strongly positive linear bounded self-adjoint operator mapping with coefficient δ¯∈(0,1] such that δ¯≤∥A∥≤1, and let f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Sn:H→H} and {Tn:H→H} be two infinite families of k1 and k2-strictly pseudocontractive mappings with a fixed point such that k1,k2∈[0,1), respectively. Define a mapping Rn:H→H by
(4.7)Rnx=αx+(1-α)(αSnx+(1-α)Tnx),∀x∈H,
for all n∈ℕ, where α∈[k,1)∖{0} such that k=max{k1,k2}. Let Wn:H→H be a W-mapping generated by {Rn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω∶=(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn))∩(⋂i=1NVI(H,Ci,Mi))≠∅ and 0<γ<δ¯/δ. For x1=u∈H, suppose that {xn} is generated iteratively by
(4.8)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)yn,
for all n∈ℕ, where {αn},{γn}⊂(0,1), {βn}⊂[0,1), ρi∈(0,1), and λi∈(0,2ξi] for each i=1,2,…,N satisfying the following conditions:
(C1)
limn→∞αn=0,(C2)
0<limn→∞γn<1 and limsupn→∞βn<1,(C3)
∑n=1∞αn=∞ and ∑i=1Nρi=1,(C4)
∑n=1∞|αn+1-αn|<∞ and ∑n=1∞|βn+1-βn|<∞,(C5)
∑n=1∞|γn+1-γn|<∞ and ∑n=1∞∏i=1nμi<∞,
then the sequences{xn} and {yn} converge strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(4.9)〈(A-γf)w,y-w〉≥0,∀y∈Ω.Proof.
It is concluded from Theorem4.2 immediately, by putting ϵn=0 for all n∈ℕ.Theorem 4.4.
LetH be a real Hilbert space, let Mi:H→2H be a maximal monotone mapping, and let Ci:H→H be a ξi-cocoercive mapping for each i=1,2,…,N. Let A:H→H be a strongly positive linear bounded self-adjoint operator mapping with coefficient δ¯∈(0,1] such that δ¯≤∥A∥≤1, and let f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Sn:H→H} and {Tn:H→H} be two infinite families of k1 and k2-strictly pseudocontractive mappings with a fixed point such that k1,k2∈[0,1), respectively. Define a mapping Rn:H→H by
(4.10)Rnx=αx+(1-α)(αSnx+(1-α)Tnx),∀x∈H,
for all n∈ℕ, where α∈[k,1)∖{0} such that k=max{k1,k2}. Let Wn:H→H be a W-mapping generated by {Rn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω∶=(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn))∩(⋂i=1NVI(H,Ci,Mi))≠∅ and 0<γ<δ¯/δ. For x1=u∈H, suppose that {xn} is generated iteratively by
(4.11)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+((1-ϵn)I-αnA)yn,
for all n∈ℕ, where {αn},{γn}⊂(0,1), {ϵn}⊂[0,1) such that ϵn≤αn, ρi∈(0,1), and λi∈(0,2ξi] for each i=1,2,…,N satisfying the following conditions:
(C1)
limn→∞αn=limn→∞(ϵn/αn)=0,(C2)
0<limn→∞γn<1,(C3)
∑n=1∞αn=∞ and ∑i=1Nρi=1,(C4)
∑n=1∞|αn+1-αn|<∞ and ∑n=1∞|ϵn+1-ϵn|<∞,(C5)
∑n=1∞|γn+1-γn|<∞ and ∑n=1∞∏i=1nμi<∞,
then the sequences{xn} and {yn} converge strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(4.12)〈(A-γf)w,y-w〉≥0,∀y∈Ω.Proof.
It is concluded from Theorem4.2 immediately, by putting βn=0 for all n∈ℕ.Theorem 4.5.
LetH be a real Hilbert space, let Mi:H→2H be a maximal monotone mapping, and let Ci:H→H be a ξi-cocoercive mapping for each i=1,2,…,N. Let A:H→H be a strongly positive linear bounded self-adjoint operator mapping with coefficient δ¯∈(0,1] such that δ¯≤∥A∥≤1, and let f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Sn:H→H} and {Tn:H→H} be two infinite families of k1 and k2-strictly pseudocontractive mappings with a fixed point such that k1,k2∈[0,1), respectively. Define a mapping Rn:H→H by
(4.13)Rnx=αx+(1-α)(αSnx+(1-α)Tnx),∀x∈H,
for all n∈ℕ, where α∈[k,1)∖{0} such that k=max{k1,k2}. Let Wn:H→H be a W-mapping generated by {Rn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω∶=(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn))∩(⋂i=1NVI(H,Ci,Mi))≠∅ and 0<γ<δ¯/δ. For x1=u∈H,  suppose that {xn} is generated iteratively by
(4.14)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+(I-αnA)yn,
for all n∈ℕ, where {αn},{γn}⊂(0,1), ρi∈(0,1), and λi∈(0,2ξi] for each i=1,2,…,N satisfying the following conditions:
(C1)
limn→∞αn=0,(C2)
0<limn→∞γn<1,(C3)
∑n=1∞αn=∞ and ∑i=1Nρi=1,(C4)
∑n=1∞|αn+1-αn|<∞,(C5)
∑n=1∞|γn+1-γn|<∞ and ∑n=1∞∏i=1nμi<∞,
then the sequences{xn} and {yn} converge strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(4.15)〈(A-γf)w,y-w〉≥0,∀y∈Ω.Proof.
It is concluded from Theorem4.4 immediately, by putting ϵn=0 for all n∈ℕ.Theorem 4.6.
LetH be a real Hilbert space, let Mi:H→2H be a maximal monotone mapping, and let Ci:H→H be a ξi-cocoercive mapping for each i=1,2,…,N. Let f:H→H be a contraction mapping with coefficient δ∈(0,1), and let {Sn:H→H} and {Tn:H→H} be two infinite families of k1 and k2-strictly pseudocontractive mappings with a fixed point such that k1,k2∈[0,1), respectively. Define a mapping Rn:H→H by
(4.16)Rnx=αx+(1-α)(αSnx+(1-α)Tnx),∀x∈H,
for all n∈ℕ, where α∈[k,1)∖{0} such that k=max{k1,k2}. Let Wn:H→H be a W-mapping generated by {Rn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω∶=(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn))∩(⋂i=1NVI(H,Ci,Mi))≠∅. For x1=u∈H, suppose that {xn} is generated iteratively by
(4.17)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnf(xn)+(1-αn)yn,
for all n∈ℕ, where {αn},{γn}⊂(0,1), ρi∈(0,1), and λi∈(0,2ξi] for each i=1,2,…,N satisfying the following conditions:
(C1)
limn→∞αn=0,(C2)
0<limn→∞γn<1,(C3)
∑n=1∞αn=∞ and ∑i=1Nρi=1,(C4)
∑n=1∞|αn+1-αn|<∞,(C5)
∑n=1∞|γn+1-γn|<∞ and ∑n=1∞∏i=1nμi<∞,
then the sequences{xn} and {yn} converge strongly to w∈Ω where w=PΩf(w) is a unique solution of the variational inequality
(4.18)〈(I-f)w,y-w〉≥0,∀y∈Ω.Proof.
It is concluded from Theorem4.5 immediately, by putting γ=δ¯=1 and A≡I.Theorem 4.7.
LetH be a real Hilbert space. Let A,B:H→H be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients δ¯,β¯∈(0,1] such that δ¯≤∥A∥≤1 and ∥B∥=β¯, respectively, and let f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Sn:H→H} and {Tn:H→H} be two infinite families of k1 and k2-strictly pseudocontractive mappings with a fixed point such that k1,k2∈[0,1), respectively. Define a mapping Rn:H→H by
(4.19)Rnx=αx+(1-α)(αSnx+(1-α)Tnx),∀x∈H,
for all n∈ℕ, where α∈[k,1)∖{0} such that k=max{k1,k2}. Let Wn:H→H be a W-mapping generated by {Rn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω∶=(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn))≠∅ and 0 <γ<δ¯/δ. For x1=u∈H, suppose that {xn} is generated iteratively by
(4.20)yn=γnWnxn+(1-γn)xn,xn+1=αnγf(xn)+βnBxn+((1-ϵn)I-βnB-αnA)yn,
for all n∈ℕ, where {αn},{γn}⊂(0,1) and {βn},{ϵn}⊂[0,1) such that ϵn≤αn satisfying the following conditions:
(C1)
limn→∞αn=limn→∞(ϵn/αn)=0,(C2)
0<limn→∞γn<1 and limsupn→∞βn<1,(C3)
∑n=1∞αn=∞,(C4)
∑n=1∞|αn+1-αn|<∞, ∑n=1∞|βn+1-βn|<∞, and ∑n=1∞|ϵn+1-ϵn|<∞,(C5)
∑n=1∞|γn+1-γn|<∞ and ∑n=1∞∏i=1nμi<∞,
then the sequences{xn} and {yn} converge strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(4.21)〈(A-γf)w,y-w〉≥0,∀y∈Ω.Proof.
It is concluded from Theorem3.2 immediately, by putting Mi≡Ci≡0 for each i=1,2,…,N.Theorem 4.8.
LetH be a real Hilbert space, let Mi:H→2H be a maximal monotone mapping, and let Ci:H→H be a ξi-cocoercive mapping for each i=1,2,…,N. Let A,B:H→H be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients δ¯,β¯∈(0,1] such that δ¯≤∥A∥≤1 and ∥B∥=β¯, respectively, and let f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Tn:H→H} be an infinite family of k-strictly pseudocontractive mappings with a fixed point such that k∈[0,1). Define a mapping Rn:H→H by
(4.22)Rnx=αx+(1-α)Tnx,∀x∈H,
for all n∈ℕ, where α∈[k,1). Let Wn:H→H be a W-mapping generated by {Rn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω∶=(⋂n=1∞F(Tn))∩(⋂i=1NVI(H,Ci,Mi))≠∅ and 0<γ<δ¯/δ. For x1=u∈H, suppose that {xn} is generated iteratively by
(4.23)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+βnBxn+((1-ϵn)I-βnB-αnA)yn,
for all n∈ℕ, where {αn},{γn}⊂(0,1) and {βn},{ϵn}⊂[0,1) such that ϵn≤αn, ρi∈(0,1), and λi∈(0,2ξi] for each i=1,2,…,N satisfying the following conditions:
(C1)
limn→∞αn=limn→∞(ϵn/αn)=0,(C2)
0<limn→∞γn<1 and limsupn→∞βn<1,(C3)
∑n=1∞αn=∞ and ∑i=1Nρi=1,(C4)
∑n=1∞|αn+1-αn|<∞, ∑n=1∞|βn+1-βn|<∞, and ∑n=1∞|ϵn+1-ϵn|<∞,(C5)
∑n=1∞|γn+1-γn|<∞ and ∑n=1∞∏i=1nμi<∞,
then the sequences{xn} and {yn} converge strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(4.24)〈(A-γf)w,y-w〉≥0,∀y∈Ω.Proof.
It is concluded from Theorem3.2 immediately, by putting Sn≡Tn for all n∈ℕ, and note that α∈[k,1) by Lemma 2.9.Theorem 4.9.
LetH be a real Hilbert space, let Mi:H→2H be a maximal monotone mapping, and let Ci:H→H be a ξi-cocoercive mapping for each i=1,2,…,N. Let A,B:H→H be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients δ¯,β¯∈(0,1] such that δ¯≤∥A∥≤1 and ∥B∥=β¯, respectively, and let f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Sn:H→H} and {Tn:H→H} be two infinite families of nonexpansive mappings. Define a mapping Rn:H→H by
(4.25)Rnx=αx+(1-α)(αSnx+(1-α)Tnx),∀x∈H,
for all n∈ℕ, where α∈(0,1). Let Wn:H→H be a W-mapping generated by {Rn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω∶=(⋂n=1∞F(Sn))∩(⋂n=1∞F(Tn))∩(⋂i=1NVI(H,Ci,Mi))≠∅ and 0<γ<δ¯/δ. For x1=u∈H, suppose that {xn} is generated iteratively by
(4.26)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+βnBxn+((1-ϵn)I-βnB-αnA)yn,
for all n∈ℕ, where {αn},{γn}⊂(0,1) and {βn},{ϵn}⊂[0,1) such that ϵn≤αn, ρi∈(0,1), and λi∈(0,2ξi] for each i=1,2,…,N satisfying the following conditions:
(C1)
limn→∞αn=limn→∞(ϵn/αn)=0,(C2)
0<limn→∞γn<1 and limsupn→∞βn<1,(C3)
∑n=1∞αn=∞ and ∑i=1Nρi=1,(C4)
∑n=1∞|αn+1-αn|<∞, ∑n=1∞|βn+1-βn|<∞, and ∑n=1∞|ϵn+1-ϵn|<∞,(C5)
∑n=1∞|γn+1-γn|<∞ and ∑n=1∞∏i=1nμi<∞,
then the sequences{xn} and {yn} converge strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(4.27)〈(A-γf)w,y-w〉≥0,∀y∈Ω.Proof.
It is concluded from Theorem3.2 immediately, by putting k1=k2=0.Theorem 4.10.
LetH be a real Hilbert space, let Mi:H→2H be a maximal monotone mapping, and let Ci:H→H be a ξi-cocoercive mapping for each i=1,2,…,N. Let A,B:H→H be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients δ¯,β¯∈(0,1] such that δ¯≤∥A∥≤1 and ∥B∥=β¯, respectively, and let f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Tn:H→H} be an infinite family of nonexpansive mappings. Define a mapping Rn:H→H by
(4.28)Rnx=αx+(1-α)Tnx,∀x∈H,
for all n∈ℕ, where α∈[0,1). Let Wn:H→H be a W-mapping generated by {Rn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω∶=(⋂n=1∞F(Tn))∩(⋂i=1NVI(H,Ci,Mi))≠∅ and 0<γ<δ¯/δ. For x1=u∈H, suppose that {xn} is generated iteratively by
(4.29)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn),xn+1=αnγf(xn)+βnBxn+((1-ϵn)I-βnB-αnA)yn,
for all n∈ℕ, where {αn},{γn}⊂(0,1) and {βn},{ϵn}⊂[0,1) such that ϵn≤αn, ρi∈(0,1), and λi∈(0,2ξi] for each i=1,2,…,N satisfying the following conditions:
(C1)
limn→∞αn=limn→∞(ϵn/αn)=0,(C2)
0<limn→∞γn<1 and limsupn→∞βn<1,(C3)
∑n=1∞αn=∞ and ∑i=1Nρi=1,(C4)
∑n=1∞|αn+1-αn|<∞, ∑n=1∞|βn+1-βn|<∞, and ∑n=1∞|ϵn+1-ϵn|<∞,(C5)
∑n=1∞|γn+1-γn|<∞ and ∑n=1∞∏i=1nμi<∞,
then the sequences{xn} and {yn} converge strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(4.30)〈(A-γf)w,y-w〉≥0,∀y∈Ω.Proof.
It is concluded from Theorem4.8 immediately, by putting k=0.Theorem 4.11.
LetH be a real Hilbert space. Let A,B:H→H be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients δ¯,β¯∈(0,1] such that δ¯≤∥A∥≤1 and ∥B∥=β¯, respectively, and let f:H→H be a contraction mapping with coefficient δ∈(0,1). Let T:H→H be a nonexpansive mapping. Assume that F(T)≠∅ and 0<γ<δ¯/δ. For x1=u∈H,  suppose that {xn} is generated iteratively by
(4.31)xn+1=αnγf(xn)+βnBxn+((1-ϵn)I-βnB-αnA)(σnTxn+(1-σn)xn),
for all n∈ℕ, where {αn},{σn}⊂(0,1) and {βn},{ϵn}⊂[0,1) such that ϵn≤αn satisfying the following conditions:
(C1)
limn→∞αn=limn→∞(ϵn/αn)=0,(C2)
0<limn→∞σn<1 and limsupn→∞βn<1,(C3)
∑n=1∞αn=∞,(C4)
∑n=1∞|αn+1-αn|<∞ and ∑n=1∞|βn+1-βn|<∞,(C5)
∑n=1∞|ϵn+1-ϵn|<∞ and ∑n=1∞|σn+1-σn|<∞,
then the sequences{xn} and {yn} converge strongly to w∈F(T) where w=PF(T)(I-A+γf)w is a unique solution of the variational inequality
(4.32)〈(A-γf)w,y-w〉≥0,∀y∈F(T).Proof.
From Theorem4.10, putting α=0 and Mi≡Ci≡0 for all i=1,2,…,N. Setting T1≡T, Tn≡I for all n=2,3,…, and let μn⊂(0,μ] for some μ∈(0,1) such that ∑n=1∞∏i=1nμi<∞. Therefore, from the definition of Rn in Theorem 4.10, we have R1=T1=T and Rn=I for all n=2,3,…. Since Wn is a W-mapping generated by {Rn} and {μn}, therefore by the definition of Un,i and Wn in (1.16), we have Un,i=I for all i=2,3,… and Wn=Un,1=μ1R1Un,2+(1-μ1)I=μ1T+(1-μ1)I. Hence, by Theorem 4.10, we obtain
(4.33)yn=γnWnxn+(1-γn)∑i=1NρiJMi,λi(xn-λiCixn)=γn(μ1Txn+(1-μ1)xn)+(1-γn)(∑i=1Nρi)xn=γn(μ1Txn+(1-μ1)xn)+(1-γn)xn=σnTxn+(1-σn)xn,
where σn∶=γnμ1. This completes the proof.



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*Source: 691839-2011-08-11.xml*