# An Iterative Method for Solving the Generalized System of Relaxed Cocoercive Quasivariational Inclusions and Fixed Point Problems of an Infinite Family of Strictly Pseudocontractive Mappings

Authors: Pattanapong Tianchai; Rabian Wangkeeree
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

We introduce an iterative scheme by the viscosity approximation to find the set of solutions of the generalized system of relaxed cocoercive quasivariational inclusions and the set of common fixed points of an infinite family of strictly pseudocontractive mappings problems in Hilbert spaces. We suggest and analyze an iterative scheme under some appropriate conditions imposed on the parameters;  we prove that another strong convergence theorem for the above two sets is obtained. The results presented in this paper improve and extend the main results of Li and Wu (2010) and many others.


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## Body

## 1. Introduction and Preliminaries

LetH be a real Hilbert space with inner product and norm being denoted by 〈·,·〉 and ∥·∥, respectively, and let C be a nonempty closed convex subset of H. Recall that PC is the metric projection of H onto C; that is, for each x∈H there exists the unique point in PCx∈C such that (1.1)∥x-PCx∥=miny∈C∥x-y∥.
A mapping T:C→C is called nonexpansive if (1.2)∥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 (1.3)∥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 mappings of C into itself, then F(T) is nonempty (see [1]). Recall that a mapping A:C→H is said to be (i)
monotone if
(1.4)〈Ax-Ay,x-y〉≥0,∀x,y∈C,(ii)
k-Lipschitz continuous if there exists a constant k>0 such that
(1.5)∥Ax-Ay∥≤k∥x-y∥,∀x,y∈C,
if k=1, then A is a nonexpansive,(iii)
pseudocontractive if
(1.6)∥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
(1.7)∥Ax-Ay∥2≤∥x-y∥2+k∥(I-A)x-(I-A)y∥2,∀x,y∈C,
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
(1.8)〈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
(1.9)〈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
(1.10)〈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
(1.11)〈Ax-Ay,x-y〉≥(-α)∥Ax-Ay∥2+r∥x-y∥2,∀x,y∈C,
it is obvious that any the r-strongly monotonicity implies to the relaxed (α,r)-cocoercivity.Recall that a set-valued mappingM:H→2H is called monotone if for all x,y∈H,f∈Mx and g∈My imply 〈x-y,f-g〉≥0. A monotone mapping M:H→2H is maximal if the graph of G(M) of M is not properly contained in the graph of any other monotone mappings.The existence common fixed points for a finite family of nonexpansive mappings has been considered by many authers (see [2–5] and the references therein).In this paper, we study the mappingWn defined by(1.12)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,… form a family of infinitely nonexpansive mappings of C into itself. It is obvious that Wn is nonexpansive from 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: (1.13)minx∈C{12〈Ax,x〉-〈x,b〉},
where A is a bounded linear operator on H, C is the fixed point set of a nonexpansive mapping S on H and b is a given point in H. Recall that A be a strongly positive bounded linear operator on H if there exists a constant γ¯>0 such that (1.14)〈Ax,x〉≥γ¯∥x∥2,∀x∈H.Marino and Xu [6] introduced the following general iterative scheme based on the viscosity approximation method introduced by Moudafi [7]:(1.15)xn+1=αnγf(xn)+(I-αnA)Sxn,∀n∈ℕ,
where 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 those conditions are corrected, if F(S)≠∅, then the sequence {xn} generated by (1.15) converges strongly to the unique solution z of the variational inequality (1.16)〈(A-γf)z,x-z〉≥0,∀x∈F(S),
which is the optimality condition for the minimization problem (1.17)minx∈F(S){12〈Ax,x〉-h(x)},
where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H).The so-called the system of generalized quasivariational inclusions problem is to find(x*,y*)∈H×H such that(1.18)0∈x*-y*+λ1((B1+C1)y*+M1x*),0∈y*-x*+λ2((B2+C2)x*+M2y*),
where Bi,Ci:H→H, Mi:H→2H are nonlinear mappings and λi>0 for each i=1,2. As special cases of problem (1.18), we have the following. (1)
IfB1=B2=B, C1=C2=C and M1=M2=M, then problem (1.18) is reduced to find (x*,y*)∈H×H such that
(1.19)0∈x*-y*+λ1((B+C)y*+Mx*),0∈y*-x*+λ2((B+C)x*+My*).(2)
IfC1=C2=0, then problem (1.18) is reduced to find (x*,y*)∈H×H such that
(1.20)0∈x*-y*+λ1(B1y*+M1x*),0∈y*-x*+λ2(B2x*+M2y*),
which called that the system of quasivariational inclusions problem.(3)
IfB1=B2=B and M1=M2=M, then problem (1.20) is reduced to find (x*,y*)∈H×H such that
(1.21)0∈x*-y*+λ1(By*+Mx*),0∈y*-x*+λ2(Bx*+My*).(4)
Ifx*=y*, then problem (1.21) is reduced to find x*∈H such that
(1.22)0∈Bx*+Mx*.
We denote by VI(H,B,M) the set of solutions of variational inclusion of the problem (1.22).(5)
IfM=∂ϕ:H→2H, where ϕ:H→ℝ∪{+∞} is a proper convex lower semicontinuous function and ∂ϕ is the subdifferential of ϕ, then problem (1.22) is equivalent to find x*∈H such that
(1.23)〈Bx*,v-x*〉+ϕ(v)-ϕ(x*)≥0,∀v∈H,
which is said to be the mixed quasivariational inequality (see, e.g., [8, 9] for more details).(6)
Ifϕ is the indicator function of C, then problem (1.23) is equivalent to the classical variational inequality problem, denoted by VI(C,B), to find x*∈C such that
(1.24)〈Bx*,v-x*〉≥0,∀v∈C.Iiduka and Takahashi [10] introduced 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 (1.24) as the following theorem.Theorem IT.
LetC be a 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
(1.25)xn+1=αnx+(1-αn)SPC(xn-λnBxn),∀n∈ℕ,
where {αn}⊂[0,1) and {λn}⊂[a,b] for some a,b∈(0,2α) 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 [11]).
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 with M, where λ is any positive number and I is the identity mapping.Recently, Zhang et al. [11] considered the problem (1.22). To be more precise, they proved the following theorem.Theorem ZLC.
LetH be a real Hilbert space, B:H→H be an α-inverse-strongly monotone mapping, M:H→2H be a maximal monotone mapping, and 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 variational inclusion (1.22). Suppose that x1=x∈H and {xn} is the sequence defined by
(1.26)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.Very recently, Li and Wu [12] introduced an iterative scheme:(1.27)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 and Sk is a mapping on H defined by Skx=kx+(1-k)Sx such that S is a k-strictly pseudocontractive mapping on H with a fixed point. They proved that under missing condition of μ, it should be 0<μ<1 by those Lemma 1.6, others are corrected, if Ω=F(S)∩VI(H,B,M)≠∅, then the sequence {xn} generated by (1.27) converges strongly to z=PΩ(I-A+γf)z of the variational inequality (1.28)〈(A-γf)z,x-z〉≥0,∀x∈Ω,
which is the optimality condition for the minimization problem: (1.29)minx∈Ω{12〈Ax,x〉-h(x)},
where h is a potential function for γf.Inspired and motivated by the works mentioned above, in this paper, we introduce an iterative scheme (2.2) below by the viscosity approximation to find the set of solutions of the generalized system of relaxed cocoercive quasivariational inclusions and the set of common fixed points of an infinite family of strictly pseudocontractive mappings problems in Hilbert spaces. We suggest and analyze an iterative scheme under some appropriate conditions imposed on the parameters, we prove that another strong convergence theorem for the above two sets is obtained. The results presented in this paper improve and extend the main results of Li and Wu [12] and many others.We collect the following lemmas which be used in the proof for the main results in the next section.Lemma 1.2 (see [6]).
LetH be a Hilbert space, C be a nonempty closed convex subset of H, f:H→H be a contraction with coefficient 0<α<1, and A be a strongly positive linear bounded operator with coefficient γ¯>0. Then, (1)
if0<γ<γ¯/α, then 〈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 1.3 (see [13]).
Let{xn} and {zn} be bounded sequences in a Banach space E and {βn} be a sequence in [0,1] which satisfies the following condition:
(1.30)0<lim infn→∞βn≤lim supn→∞βn<1.
Suppose that
(1.31)xn+1=(1-βn)xn+βnzn,n≥1
and lim supn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0. Then limn→∞∥zn-xn∥=0.Lemma 1.4 (see [14]).
Assume{an} is a sequence of nonnegative real numbers such that
(1.32)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)
∑n=1∞ηn=∞;(2)
lim supn→∞(δn/ηn)≤0 or ∑n=1∞|δn|<∞.Then limn→∞an=0.Lemma 1.5 (see [15]).
LetC be a nonempty closed convex subset of a Hilbert space H, define mapping Wn as (1.12), 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=1∞F(Si) for each n≥1;(2)
for eachx∈C and for each positive integer k, limn→∞Un,kx exists;(3)
the mappingW:C→C define by
(1.33)Wx:=limn→∞Wnx=limn→∞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 1.6 (see [11]).
The resolvent operatorJM,λ associated with M is single-valued and nonexpansive for all λ>0.Lemma 1.7 (see [11]).
u∈H is a solution of variational inclusion (1.22) if and only if u=JM,λ(u-λBu), for all λ>0, that is,
(1.34)VI(H,B,M)=F(JM,λ(I-λB)),∀λ>0.Lemma 1.8.
For any(x*,y*)∈H×H, where y*=JM2,λ2(I-λ2(B2+C2))x*, we have (x*,y*) is a solution of problem (1.18) if and only if x* is a fixed point of the mapping D defined by
(1.35)Dx=JM1,λ1((I-λ1(B1+C1))JM2,λ2(I-λ2(B2+C2))x).Proof.
Observe from (1.18) that
(1.36)0∈x*-y*+λ1((B1+C1)y*+M1x*),0∈y*-x*+λ2((B2+C2)x*+M2y*)⇔x*=JM1,λ1(I-λ1(B1+C1))y*,y*=JM2,λ2(I-λ2(B2+C2))x*⇔Dx*=JM1,λ1((I-λ1(B1+C1))JM2,λ2(I-λ2(B2+C2))x*)=x*.Lemma 1.9 (see [16]).
LetC be a closed convex subset of a strictly convex Banach space E. Let S and T be two nonexpansive mappings on C. Suppose that F(S)∩F(T) is nonempty. Then a mapping R on C defined by Rx=aSx+(1-a)Tx, where a∈(0,1), for x∈C is well defined and nonexpansive and F(R)=F(S)∩F(T) holds.Lemma 1.10 (see [17]).
LetH be a real Hilbert space, let C be a nonempty closed convex subset of H, and let S:C→C be a nonexpansive mapping. Then I-S is demiclosed at zero.Lemma 1.11 (see [18]).
LetC be a nonempty closed convex subset of a real Hilbert space H and T:C→C be a k-strict pseudocontraction. Define S:C→H 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).


## 2. Main Results

LetH be a real Hilbert space, Mi:H→2H be a maximal monotone mapping, Bi:H→H be a relaxed (αi,ri)-cocoercive and ξi-Lipschitz continuous mappings, respectively, Ci:H→H be a relaxed (α¯i,r¯i)-cocoercive and ξ¯i-Lipschitz continuous mappings, respectively, for each i=1,2. Let A:H→H be a strongly positive linear bounded self-adjoint operator mapping with coefficient δ¯∈(0,1] such that ∥A∥≤1 and f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Tn:H→H} be a family of kn-strictly pseudocontractive mappings with a fixed point such that kn∈[0,1) for all n∈ℕ. Define Snx=δnx+(1-δn)Tnx, where δn∈[kn,1), for all n∈ℕ, and let Wn:H→H be a W-mapping generated by {Sn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1).Define sequence of mappings{Pn:H→H} and mapping Q:H→H as follows: (2.1)Pnx=αnγf(Wnx)+βnx+((1-βn)I-αnA)(γnWnx+(1-γn)Qx),Qx=JM1,λ1((I-λ1(B1+C1))JM2,λ2(I-λ2(B2+C2))x),
for all n∈ℕ, where {αn},{βn},{γn}⊂(0,∥A∥-1), λi∈(0,ϵi], ϵi=min{2(ri-αiξi2)/ξi(ξi+ξ¯i),2(r¯i-α¯iξ¯i2)/ξ¯i(ξi+ξ¯i)}, ri>αiξi2, r¯i>α¯iξ¯i2, for each i=1,2 and 0<γ<δ¯/δ.Under some appropriate imposed on the parameters{αn} and {βn}, we also know that ∥(1-βn)I-αnA∥≤1-βn-αnδ¯, and, we also have that Wn,JMi,λi and I-λi(Bi+Ci) are nonexpansive for each i=1,2 (see argument in the proof of Theorem 2.1 below). Observe that Q is a nonexpansive, and so Pn is a contraction with coefficient 1-(δ¯-γδ)αn. Therefore, by Banach contraction principle guarantees that Pn has a unique fixed point in H.By the idea above, we obtain an iteration scheme by the viscosity approximation for solving the generalized system of relaxed cocoercive quasivariational inclusions and fixed point problems of an infinite family of strictly pseudocontractive mappings as the following theorem.Theorem 2.1.
LetH be a real Hilbert space, Mi:H→2H be a maximal monotone mapping, Bi:H→H be a relaxed (αi,ri)-cocoercive and ξi-Lipschitz continuous mappings, respectively, Ci:H→H be a relaxed (α¯i,r¯i)-cocoercive and ξ¯i-Lipschitz continuous mappings, respectively, for each i=1,2. Let A:H→H be a strongly positive linear bounded self-adjoint operator mapping with coefficient δ¯∈(0,1] such that ∥A∥≤1 and f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Tn:H→H} be a family of kn-strictly pseudocontractive mappings with a fixed point such that kn∈[0,1) for all n∈ℕ. Define Snx=δnx+(1-δn)Tnx, where δn∈[kn,1), for all n∈ℕ, and let Wn:H→H be a W-mapping generated by {Sn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω:=⋂n=1∞F(Tn)∩F(D)≠∅ and 0<γ<δ¯/δ where D:H→H defined by Dx=JM1,λ1((I-λ1(B1+C1))JM2,λ2(I-λ2(B2+C2))x). For x1∈H, suppose that {xn} be generated iteratively by
(2.2)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 {αn},{βn},{γn}⊂(0,1), λi∈(0,ϵi], ϵi=min{2(ri-αiξi2)/ξi(ξi+ξ¯i), 2(r¯i-α¯iξ¯i2)/ξ¯i(ξi+ξ¯i)}, ri>αiξi2, r¯i>α¯iξ¯i2, for each i=1,2, satisfying the following conditions: (C1)
limn→∞αn=limn→∞|γn+1-γn|=0;(C2)
0<lim infn→∞βn≤lim supn→∞βn<1;(C3)
∑n=1∞αn=∞ and 0<limn→∞γn<1.Then the sequence {xn} converges strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(2.3)〈(A-γf)w,y-w〉≥0,∀y∈Ω,
which is the optimality condition for the minimization problem
(2.4)minx∈Ω{12〈Ax,x〉-h(x)},
where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H) and (w,JM2,λ2(I-λ2(B2+C2))w) is a solution of problem (1.18).Proof.
From (C1) and (C2), we haveαn→0 and lim supn→∞βn<1. Thus, we may assume without loss of generality that αn≤(1-βn)∥A∥-1 for all n∈ℕ. For each i=1,2, since
(2.5)0<λi≤ϵi≤2(ri-αiξi2)ξi(ξi+ξ¯i),0<λi≤ϵi≤2(r¯i-α¯iξ¯i2)ξ¯i(ξi+ξ¯i),
where ri>αiξi2 and r¯i>α¯iξ¯i2, we have
(2.6)λi2ξi(ξi+ξ¯i)+2λiαiξi2-2λiri≤0,λi2ξ¯i(ξi+ξ¯i)+2λiα¯iξ¯i2-2λir¯i≤0.
For any x,y∈H, it follows by the relaxed (αi,ri)-cocoercivity and ξi-Lipschitz continuity of Bi and the relaxed (α¯i,r¯i)-cocoercivity and ξ¯i-Lipschitz continuity of Ci that
(2.7)∥(I-λi(Bi+Ci))x-(I-λi(Bi+Ci))y∥2=∥(x-y)-λi((Bi+Ci)x-(Bi+Ci)y)∥2=∥x-y∥2-2λi〈x-y,(Bi+Ci)x-(Bi+Ci)y〉+λi2∥(Bi+Ci)x-(Bi+Ci)y∥2=∥x-y∥2-2λi〈x-y,Bix-Biy〉-2λi〈x-y,Cix-Ciy〉+λi2∥(Bix-Biy)+(Cix-Ciy)∥2≤∥x-y∥2-2λi(-αi∥Bix-Biy∥2+ri∥x-y∥2)-2λi(-α¯i∥Cix-Ciy∥2+r¯i∥x-y∥2)+λi2(∥Bix-Biy∥+∥Cix-Ciy∥)2≤∥x-y∥2-2λi(-αiξi2∥x-y∥2+ri∥x-y∥2)-2λi(-α¯iξ¯i2∥x-y∥2+r¯i∥x-y∥2)+λi2(ξi∥x-y∥+ξ¯i∥x-y∥)2=(1+(λi2ξi(ξi+ξ¯i)+2λiαiξi2-2λiri)+(λi2ξ¯i(ξi+ξ¯i)+2λiα¯iξ¯i2-2λir¯i))∥x-y∥2≤∥x-y∥2,
which implies that I-λi(Bi+Ci) is a nonexpansive. Since, A is a linear bounded self-adjoint operator, we have
(2.8)∥A∥=sup{|〈Ax,x〉|:x∈E,∥x∥=1}.
Observe that
(2.9)〈((1-βn)I-αnA)x,x〉=(1-βn)〈x,x〉-αn〈Ax,x〉≥(1-βn)-αn∥A∥≥0.
Therefore, we obtain (1-βn)I-αnA is positive. Thus, by the strong positively of A, we get
(2.10)∥(1-βn)I-αnA∥=sup{〈((1-βn)I-αnA)x,x〉:x∈E,∥x∥=1}=sup{(1-βn)〈x,x〉-αn〈Ax,x〉:x∈E,∥x∥=1}≤1-βn-αnδ¯.
Pickx*∈Ω. Then, we have(2.11)x*=Dx*=JM1,λ1((I-λ1(B1+C1))JM2,λ2(I-λ2(B2+C2))x*).
And, let y*=JM2,λ2(x*-λ2(B2+C2)x*). Therefore, from (2.11), we have
(2.12)x*=JM1,λ1(y*-λ1(B1+C1)y*).
By the nonexpansivity of JMi,λi and I-λi(Bi+Ci), we have
(2.13)∥zn-y*∥=∥JM2,λ2(xn-λ2(B2+C2)xn)-JM2,λ2(x*-λ2(B2+C2)x*)∥≤∥(xn-λ2(B2+C2)xn)-(x*-λ2(B2+C2)x*)∥≤∥xn-x*∥.
And, we have
(2.14)∥yn-x*∥=∥JM1,λ1(zn-λ1(B1+C1)zn)-JM1,λ1(y*-λ1(B1+C1)y*)∥≤∥(zn-λ1(B1+C1)zn)-(y*-λ1(B1+C1)y*)∥≤∥zn-y*∥.
Therefore, by (2.13) and (2.14), we have
(2.15)∥yn-x*∥≤∥zn-y*∥≤∥xn-x*∥.
Lettn=γnWnxn+(1-γn)yn. Since Snx=δnx+(1-δn)Tnx, where δn∈[kn,1) and {Tn} be a family of kn-strict pseudocontraction. By Lemma 1.11, we have Sn is a nonexpansive and F(Sn)=F(Tn). Therefore, by Lemma 1.5(1), we get F(Wn)=⋂i=1∞F(Si)=⋂i=1∞F(Ti), which implies that Wnx*=x*. It follows by (2.15) and the nonexpansivity of Wn that(2.16)∥tn-x*∥=∥γnWnxn+(1-γn)yn-x*∥=∥γn(Wnxn-x*)+(1-γn)(yn-x*)∥≤γn∥xn-x*∥+(1-γn)∥yn-x*∥≤∥xn-x*∥.
From (2.16), by the contraction of f and the nonexpansivity of Wn, we have
(2.17)∥xn+1-x*∥=∥αnγf(Wnxn)+βnxn+((1-βn)I-αnA)tn-x*∥=∥αn(γf(Wnxn)-Ax*)+βn(xn-x*)+((1-βn)I-αnA)(tn-x*)∥≤αn∥γf(Wnxn)-Ax*∥+βn∥xn-x*∥+∥(1-βn)I-αnA∥∥tn-x*∥≤αnγ∥f(Wnxn)-f(x*)∥+αn∥γf(x*)-Ax*∥+βn∥xn-x*∥+(1-βn-αnδ¯)∥xn-x*∥≤αnγδ∥Wnxn-x*∥+αn∥γf(x*)-Ax*∥+(1-αnδ¯)∥xn-x*∥≤(1-(δ¯-γδ)αn)∥xn-x*∥+αn∥γf(x*)-Ax*∥≤max{∥xn-x*∥,∥γf(x*)-Ax*∥δ¯-γδ}.
It follows from induction that
(2.18)∥xn+1-x*∥≤max{∥x1-x*∥,∥γf(x*)-Ax*∥δ¯-γδ},
for all n∈ℕ. Hence, {xn} is bounded, and so are {yn},{zn},{tn}, {Wnxn}, {f(Wnxn)} and {Atn}.
Next, we prove that∥vn-xn∥→0,∥xn+1-xn∥→0 and ∥tn-xn∥→0 as n→∞. By the nonexpansivity of JMi,λi and I-λi(Bi+Ci), we have(2.19)∥yn+1-yn∥=∥JM1,λ1(zn+1-λ1(B1+C1)zn+1)-JM1,λ1(zn-λ1(B1+C1)zn)∥≤∥(zn+1-λ1(B1+C1)zn+1)-(zn-λ1(B1+C1)zn)∥≤∥zn+1-zn∥.
Similarly, we have
(2.20)∥zn+1-zn∥≤∥xn+1-xn∥.
Therefore, from (2.19) and (2.20), we have
(2.21)∥yn+1-yn∥≤∥zn+1-zn∥≤∥xn+1-xn∥.
By the nonexpansivity of Si and Un,i, we have
(2.22)∥Wn+1xn-Wnxn∥=∥Un+1,1xn-Un,1xn∥=∥μ1S1Un+1,2xn+(1-μ1)xn-(μ1S1Un,2xn+(1-μ1)xn)∥≤μ1∥Un+1,2xn-Un,2xn∥=μ1∥μ2S2Un+1,3xn+(1-μ2)xn-(μ2S2Un,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 (2.22), by the nonexpansivity of Wn+1, we have
(2.23)∥Wn+1xn+1-Wnxn∥≤∥Wn+1xn+1-Wn+1xn∥+∥Wn+1xn-Wnxn∥≤∥xn+1-xn∥+M∏i=1nμi.
Since
(2.24)tn+1-tn=γn+1Wn+1xn+1+(1-γn+1)yn+1-(γnWnxn+(1-γn)yn)=γn+1(Wn+1xn+1-Wnxn)+(γn+1-γn)(Wnxn-yn)+(1-γn+1)(yn+1-yn),
combining (2.21), (2.23), and (2.24), we have
(2.25)∥tn+1-tn∥≤γn+1∥Wn+1xn+1-Wnxn∥+|γn+1-γn|∥Wnxn-yn∥+(1-γn+1)∥yn+1-yn∥≤γn+1(∥xn+1-xn∥+M∏i=1nμi)+|γn+1-γn|∥Wnxn-yn∥+(1-γn+1)∥yn+1-yn∥≤M∏i=1nμi+|γn+1-γn|∥Wnxn-yn∥+∥xn+1-xn∥.
Letvn=(xn+1-βnxn)/(1-βn). Then we have(2.26)xn+1=(1-βn)vn+βnxn.
Since
(2.27)vn+1-vn=αn+1γf(Wn+1xn+1)+((1-βn+1)I-αn+1A)tn+11-βn+1-αnγf(Wnxn)+((1-βn)I-αnA)tn1-βn=αn+11-βn+1(γf(Wn+1xn+1)-Atn+1)-αn1-βn(γf(Wnxn)-Atn)+(tn+1-tn),
combining (2.25) and (2.27), we have
(2.28)∥vn+1-vn∥≤αn+11-βn+1∥γf(Wn+1xn+1)-Atn+1∥+αn1-βn∥γf(Wnxn)-Atn∥+∥tn+1-tn∥≤αn+11-βn+1∥γf(Wn+1xn+1)-Atn+1∥+αn1-βn∥γf(Wnxn)-Atn∥+M∏i=1nμi+|γn+1-γn|∥Wnxn-yn∥+∥xn+1-xn∥.
Therefore, by (C1), (C2) and limn→∞∏i=1nμi=0, we get
(2.29)lim supn→∞(∥vn+1-vn∥-∥xn+1-xn∥)≤0.
From (2.26) and (2.29), by (C2) and Lemma 1.3, we obtain
(2.30)∥vn-xn∥→0asn→∞.
From (2.26), by (2.30), we obtain
(2.31)∥xn+1-xn∥=(1-βn)∥vn-xn∥→0asn→∞.
Since
(2.32)xn+1-xn=αnγf(Wnxn)+βnxn+((1-βn)I-αnA)tn-xn=αn(γf(Wnxn)-Atn)+(1-βn)(tn-xn),
therefore,
(2.33)(1-βn)∥tn-xn∥≤αn∥γf(Wnxn)-Atn∥+∥xn+1-xn∥.
From (2.31), by (C1) and (C2), we obtain
(2.34)∥tn-xn∥→0asn→∞.
For allx,y∈H, by Lemma 1.2(2), the nonexpansivity of PΩ and the contraction of f, we have(2.35)∥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), by 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.
Next, we claim that(2.36)lim supn→∞〈γf(w)-Aw,xn-w〉≤0.
To show this inequality, we choose a subsequence {xni} of {xn} such that
(2.37)lim supn→∞〈γf(w)-Aw,xn-w〉=limi→∞〈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 sequence of mappings {Rn:H→H} and mapping R:H→H by(2.38)Rnx=γnWnx+(1-γn)Dx,∀n∈ℕ,Rx=limn→∞Rnx.
Thus, by Lemma 1.5(3) and (C3), we have
(2.39)Rx=aWx+(1-a)Dx,
where 0<a=limn→∞γn<1. Since, W and D are nonexpansive and by Lemma 1.5(3), F(W)=⋂i=1∞F(Si). Therefore, by Lemma 1.9, we get R is a nonexpansive and F(R)=F(W)∩F(D)=⋂i=1∞F(Si)∩F(D). Since, Snx=δnx+(1-δn)Tnx, where δn∈[kn,1). Thus, by Lemma 1.11, we obtain
(2.40)F(R)=⋂i=1∞F(Si)∩F(D)=⋂i=1∞F(Ti)∩F(D)=Ω.
From (2.34), we have ∥tni-xni∥→0 as i→∞. Thus, from (2.38), we get ∥Rxni-xni∥→0 as i→∞. It follows from xni⇀w¯ and by Lemma 1.10 that w¯∈F(R), that is w¯∈Ω. Therefore, from (2.37), we obtain
(2.41)lim supn→∞〈γf(w)-Aw,xn-w〉=limi→∞〈γf(w)-Aw,xni-w〉=〈(γf-A)w,w¯-w〉≤0.
Next, we prove thatxn→w as n→∞. Since w∈Ω, we have Wnw=w, and the same as in (2.16), we have (2.42)∥tn-w∥≤∥xn-w∥.
It follows by the contraction of f and the nonexpansivity of Wn that
(2.43)∥xn+1-w∥2=∥αnγf(Wnxn)+βnxn+((1-βn)I-αnA)tn-w∥2=∥((1-βn)I-αnA)(tn-w)+βn(xn-w)+αn(γf(Wnxn)-Aw)∥=∥((1-βn)I-αnA)(tn-w)+βn(xn-w)∥2+αn2∥γf(Wnxn)-Aw∥2+2αnβn〈xn-w,γf(Wnxn)-Aw〉+2αn〈((1-βn)I-αnA)(tn-w),γf(Wnxn)-Aw〉≤((1-βn-αnδ¯)∥tn-w∥+βn∥xn-w∥)2+αn2∥γf(Wnxn)-Aw∥2+2αnβnγ〈xn-w,f(Wnxn)-f(w)〉+2αnβn〈xn-w,γf(w)-Aw〉+2αn(1-βn)γ〈tn-w,f(Wnxn)-f(w)〉+2αn(1-βn)〈tn-w,γf(w)-Aw〉-2αn2〈A(tn-w),γf(Wnxn)-Aw〉≤(1-αnδ¯)2∥xn-w∥2+2αnβnγ∥xn-w∥∥f(Wnxn)-f(w)∥+2αn(1-βn)γ∥tn-w∥∥f(Wnxn)-f(w)∥+θn≤(1-αnδ¯)2∥xn-w∥2+2αnβnγδ∥xn-w∥2+2αn(1-βn)γδ∥xn-w∥2+θn=(1-αnδ¯)2∥xn-w∥2+2αnγδ∥xn-w∥2+θn=(1-2(δ¯-γδ)αn)∥xn-w∥2+αn2δ¯2∥xn-w∥2+θn≤(1-ηn)∥xn-w∥2+δn′,
where ηn:=(δ¯-γδ)αn∈(0,1), δn′:=αn2δ¯2∥xn-w∥2+θn and
(2.44)θn:=αn2∥γf(Wnxn)-Aw∥2+2αnβn〈xn-w,γf(w)-Aw〉+2αn(1-βn)〈tn-w,γf(w)-Aw〉-2αn2〈A(tn-w),γf(Wnxn)-Aw〉.
By (2.34), (2.41), (C1) and (C3), we can found that ∑n=1∞ηn=∞ and lim supn→∞(δn′/ηn)≤0. By Lemma 1.4, we obtain {xn} converges strongly to w. This proof is completed.Remarks 2.2.
Theorem2.1 improve and extend to the main results of Li and Wu [12] for solving the generalized system of relaxed cocoercive quasivariational inclusions and fixed points problems of an infinite family of strictly pseudocontractive mappings.


## 3. Applications

Theorem 3.1.
LetH be a real Hilbert space, M:H→2H be a maximal monotone mapping, B:H→H be a relaxed (α,r)-cocoercive and ξ-Lipschitz continuous mappings, respectively, C:H→H be a relaxed (α¯,r¯)-cocoercive and ξ¯-Lipschitz continuous mappings, respectively. Let A:H→H be a strongly positive linear bounded self-adjoint operator mapping with coefficient δ¯∈(0,1] such that ∥A∥≤1 and f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Tn:H→H} be a family of kn-strictly pseudocontractive mappings with a fixed point such that kn∈[0,1) for all n∈ℕ. Define Snx=δnx+(1-δn)Tnx, where δn∈[kn,1), for all n∈ℕ, and let Wn:H→H be a W-mapping generated by {Sn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω:=⋂n=1∞F(Tn)∩F(D)≠∅ and 0<γ<δ¯/δ where D:H→H defined by Dx=JM,λ1((I-λ1(B+C))JM,λ2(I-λ2(B+C))x). For x1∈H, suppose that {xn} be generated iteratively by
(3.1)zn=JM,λ2(xn-λ2(B+C)xn),yn=JM,λ1(zn-λ1(B+C)zn),xn+1=αnγf(Wnxn)+βnxn+((1-βn)I-αnA)(γnWnxn+(1-γn)yn),
for all n∈ℕ, where {αn},{βn},{γn}⊂(0,1), λi∈(0,ϵ],ϵ=min{2(r-αξ2)/ξ(ξ+ξ¯),2(r¯-α¯ξ¯2)/ξ¯(ξ+ξ¯)}, r>αξ2, r¯>α¯ξ¯2, for each i=1,2, satisfying the following conditions: (C1)
limn→∞αn=limn→∞|γn+1-γn|=0;(C2)
0<lim infn→∞βn≤lim supn→∞βn<1;(C3)
∑n=1∞αn=∞ and 0<limn→∞γn<1.Then the sequence {xn} converges strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(3.2)〈(A-γf)w,y-w〉≥0,∀y∈Ω,
which is the optimality condition for the minimization problem
(3.3)minx∈Ω{12〈Ax,x〉-h(x)},
where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H) and (w,JM,λ2(I-λ2(B+C))w) is a solution of problem (1.19).Proof.
It is concluded obviously, from Theorem2.1 by putting M1=M2=M,B1=B2=B and C1=C2=C.Theorem 3.2.
LetH be a real Hilbert space, M:H→2H be a maximal monotone mapping, B:H→H be a relaxed (α,r)-cocoercive and ξ-Lipschitz continuous mappings, respectively, C:H→H be a relaxed (α¯,r¯)-cocoercive and ξ¯-Lipschitz continuous mappings, respectively. Let A:H→H be a strongly positive linear bounded self-adjoint operator mapping with coefficient δ¯∈(0,1] such that ∥A∥≤1 and f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Tn:H→H} be a family of nonexpansive mappings. Define Snx=δnx+(1-δn)Tnx, where δn∈[0,1), for all n∈ℕ, and let Wn:H→H be a W-mapping generated by {Sn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω:=⋂n=1∞F(Tn)∩F(D)≠∅ and 0<γ<δ¯/δ where D:H→H defined by Dx=JM,λ1((I-λ1(B+C))JM,λ2(I-λ2(B+C))x). For x1∈H, suppose that {xn} be generated iteratively by
(3.4)zn=JM,λ2(xn-λ2(B+C)xn),yn=JM,λ1(zn-λ1(B+C)zn),xn+1=αnγf(Wnxn)+βnxn+((1-βn)I-αnA)(γnWnxn+(1-γn)yn),
for all n∈ℕ, where {αn}, {βn}, {γn}⊂(0,1), λi∈(0,ϵ],ϵ=min{2(r-αξ2)/ξ(ξ+ξ¯),2(r¯-α¯ξ¯2)/ξ¯(ξ+ξ¯)}, r>αξ2, r¯>α¯ξ¯2, for each i=1,2, satisfying the following conditions: (C1)
limn→∞αn=limn→∞|γn+1-γn|=0;(C2)
0<lim infn→∞βn≤lim supn→∞βn<1;(C3)
∑n=1∞αn=∞ and 0<limn→∞γn<1.Then the sequence {xn} converges strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(3.5)〈(A-γf)w,y-w〉≥0,∀y∈Ω,
which is the optimality condition for the minimization problem
(3.6)minx∈Ω{12〈Ax,x〉-h(x)},
where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H) and (w,JM,λ2(I-λ2(B+C))w) is a solution of problem (1.19).Proof.
It is concluded obviously, from Theorem3.1 by putting kn=0 for all n∈ℕ.Theorem 3.3.
LetH be a real Hilbert space, Mi:H→2H be a maximal monotone mapping, Bi:H→H be a relaxed (αi,ri)-cocoercive and ξi-Lipschitz continuous mappings, respectively, for each i=1,2. Let A:H→H be a strongly positive linear bounded self-adjoint operator mapping with coefficient δ¯∈(0,1] such that ∥A∥≤1 and f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Tn:H→H} be a family of kn-strictly pseudocontractive mappings with a fixed point such that kn∈[0,1) for all n∈ℕ. Define Snx=δnx+(1-δn)Tnx, where δn∈[kn,1), for all n∈ℕ, and let Wn:H→H be a W-mapping generated by {Sn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω:=⋂n=1∞F(Tn)∩F(D)≠∅ and 0<γ<δ¯/δ where D:H→H defined by Dx=JM1,λ1((I-ρ1B1)JM2,λ2(I-ρ2B2)x) such that λi=ρi/2 for each i=1,2. For x1∈H, suppose that {xn} be generated iteratively by
(3.7)zn=JM2,λ2(xn-ρ2B2xn),yn=JM1,λ1(zn-ρ1B1zn),xn+1=αnγf(Wnxn)+βnxn+((1-βn)I-αnA)(γnWnxn+(1-γn)yn),
for all n∈ℕ, where {αn},{βn},{γn}⊂(0,1), ρi∈(0,2(ri-αiξi2)/ξi2], ri>αiξi2, for each i=1,2, satisfying the following conditions: (C1)
limn→∞αn=limn→∞|γn+1-γn|=0;(C2)
0<lim infn→∞βn≤lim supn→∞βn<1;(C3)
∑n=1∞αn=∞ and 0<limn→∞γn<1.Then the sequence {xn} converges strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(3.8)〈(A-γf)w,y-w〉≥0,∀y∈Ω,
which is the optimality condition for the minimization problem
(3.9)minx∈Ω{12〈Ax,x〉-h(x)},
where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H) and (w,JM2,λ2(I-ρ2B2)w) is a solution of problem
(3.10)0∈x*-y*+λ1(2B1y*+M1x*),0∈y*-x*+λ2(2B2x*+M2y*).Proof.
It is concluded obviously, from Theorem2.1 by putting Bi=Ci for each i=1,2.Theorem 3.4.
LetH be a real Hilbert space, M:H→2H be a maximal monotone mapping, B:H→H be a relaxed (α,r)-cocoercive and ξ-Lipschitz continuous mappings, respectively. Let A:H→H be a strongly positive linear bounded self-adjoint operator mapping with coefficient δ¯∈(0,1] such that ∥A∥≤1 and f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Tn:H→H} be a family of kn-strictly pseudocontractive mappings with a fixed point such that kn∈[0,1) for all n∈ℕ. Define Snx=δnx+(1-δn)Tnx, where δn∈[kn,1), for all n∈ℕ, and let Wn:H→H be a W-mapping generated by {Sn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω:=⋂n=1∞F(Tn)∩F(D)≠∅ and 0<γ<δ¯/δ where D:H→H defined by Dx=JM,λ1((I-ρ1B)JM,λ2(I-ρ2B)x) such that λi=ρi/2 for each i=1,2. For x1∈H, suppose that {xn} be generated iteratively by
(3.11)zn=JM,λ2(xn-ρ2Bxn),yn=JM,λ1(zn-ρ1Bzn),xn+1=αnγf(Wnxn)+βnxn+((1-βn)I-αnA)(γnWnxn+(1-γn)yn),
for all n∈ℕ, where {αn},{βn},{γn}⊂(0,1), ρi∈(0,2(r-αξ2)/ξ2], r>αξ2, for each i=1,2, satisfying the following conditions: (C1)
limn→∞αn=limn→∞|γn+1-γn|=0;(C2)
0<lim infn→∞βn≤lim supn→∞βn<1;(C3)
∑n=1∞αn=∞ and 0<limn→∞γn<1.Then the sequence {xn} converges strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(3.12)〈(A-γf)w,y-w〉≥0,∀y∈Ω,
which is the optimality condition for the minimization problem
(3.13)minx∈Ω{12〈Ax,x〉-h(x)},
where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H) and (w,JM,λ2(I-ρ2B)w) is a solution of problem
(3.14)0∈x*-y*+λ1(2By*+Mx*),0∈y*-x*+λ2(2Bx*+My*).Proof.
It is concluded obviously, from Theorem3.3 by putting B1=B2=B and M1=M2=M.Theorem 3.5.
LetH be a real Hilbert space, M:H→2H be a maximal monotone mapping, B:H→H be a relaxed (α,r)-cocoercive and ξ-Lipschitz continuous mappings, respectively. Let A:H→H be a strongly positive linear bounded self-adjoint operator mapping with coefficient δ¯∈(0,1] such that ∥A∥≤1 and f:H→H be a contraction mapping with coefficient δ∈(0,1). Let {Tn:H→H} be a family of nonexpansive mappings. Define Snx=δnx+(1-δn)Tnx, where δn∈[0,1), for all n∈ℕ, and let Wn:H→H be a W-mapping generated by {Sn} and {μn} such that {μn}⊂(0,μ], for some μ∈(0,1). Assume that Ω:=⋂n=1∞F(Tn)∩F(D)≠∅ and 0<γ<δ¯/δ where D:H→H defined by Dx=JM,λ1((I-ρ1B)JM,λ2(I-ρ2B)x) such that λi=ρi/2 for each i=1,2. For x1∈H, suppose that {xn} be generated iteratively by
(3.15)zn=JM,λ2(xn-ρ2Bxn),yn=JM,λ1(zn-ρ1Bzn),xn+1=αnγf(Wnxn)+βnxn+((1-βn)I-αnA)(γnWnxn+(1-γn)yn),
for all n∈ℕ, where {αn},{βn},{γn}⊂(0,1), ρi∈(0,2(r-αξ2)/ξ2], r>αξ2, for each i=1,2, satisfying the following conditions: (C1)
limn→∞αn=limn→∞|γn+1-γn|=0;(C2)
0<lim infn→∞βn≤lim supn→∞βn<1;(C3)
∑n=1∞αn=∞ and 0<limn→∞γn<1.Then the sequence {xn} converges strongly to w∈Ω where w=PΩ(I-A+γf)w is a unique solution of the variational inequality
(3.16)〈(A-γf)w,y-w〉≥0,∀y∈Ω,
which is the optimality condition for the minimization problem
(3.17)minx∈Ω{12〈Ax,x〉-h(x)},
where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H) and (w,JM,λ2(I-ρ2B)w) is a solution of problem
(3.18)0∈x*-y*+λ1(2By*+Mx*),0∈y*-x*+λ2(2Bx*+My*).Proof.
It is concluded obviously, from Theorem3.4 by putting kn=0 for all n∈ℕ.



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*Source: 683584-2010-09-29.xml*