This paper is concerned with 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. The strong convergence
theorem for the above two sets is obtained by a general iterative scheme based on the shrinking
projection method, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.

1. Introduction

Throughout this paper, we always assume that C is a nonempty closed convex subset of a real Hilbert space H with inner product and norm denoted by 〈·,·〉 and ∥·∥, respectively, 2H denoting the family of all the nonempty subsets of H.

Let B: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,θ∈Bx+Mx,
where θ is the zero vector in H. The set of solutions of problem (1.1) is denoted by VI(H,B,M).

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 ∥x-PCx∥=miny∈C∥x-y∥. A mapping T:C→C is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all 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 is nonempty bounded closed convex subset of H and T is a nonexpansive mapping of C into itself, then F(T) is nonempty (see [1]). Recall that a mapping A:C→C is said to be

monotone if
〈Ax-Ay,x-y〉≥0,∀x,y∈C,

k-Lipschitz continuous if there exists a constant k>0 such that
‖Ax-Ay‖≤k‖x-y‖,∀x,y∈C,
if k=1, then A is a nonexpansive,

pseudocontractive if
‖Ax-Ay‖2≤‖x-y‖2+‖(I-A)x-(I-A)y‖2,∀x,y∈C,

k-strictly pseudocontractive if there exists a constant k∈[0,1) such that
‖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 0-strictly pseudocontractive,

α-strongly monotone if there exists a constant α>0 such that
〈Ax-Ay,x-y〉≥α‖x-y‖2,∀x,y∈C,

α-inverse-strongly monotone (or α-cocoercive) if there exists a constant α>0 such that
〈Ax-Ay,x-y〉≥α‖Ax-Ay‖2,∀x,y∈C,
if α=1, then A is said to be firmly nonexpansive; it is obvious that any α-inverse-strongly monotone mapping A is monotone and (1/α)-Lipschitz continuous.

The existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [2–5] and the references therein).

In this paper, we study the mapping Wn defined byUn,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 mapping of C into itself; such a mapping Wn is called a W-mapping generated by S1,S2,…,Sn and μ1,μ2,…,μn.

Definition 1.1 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

Let M: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. [6] considered the problem (1.1) and the problem of a fixed point of nonexpansive mapping. To be more precise, they proved the following theorem.

Theorem ZLC.

Let H be a real Hilbert space, B:H→H an α-inverse-strongly monotone mapping, M:H→2H a maximal monotone mapping, and T:H→H a nonexpansive mapping. Suppose that the set F(T)∩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
yn=JM,λ(xn-λBxn),xn+1=αnx+(1-αn)Tyn,
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(T)∩VI(H,B,M)(x).

Nakajo and Takahashi [7] introduced an iterative scheme for finding a fixed point of a nonexpansive mapping by a hybrid method which is called that shrinking projection method (or CQ method) as in the following theorem.

Theorem NT.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself such that F(T)≠∅. Suppose that x1=x∈C and {xn} is the sequence defined by
yn=αnxn+(1-αn)Txn,Cn={z∈C:‖yn-z‖≤‖xn-z‖},Qn={z∈C:〈xn-z,x1-xn〉≥0},xn+1=PCn∩Qn(x1),∀n∈N,
where 0≤αn≤α<1. Then, {xn} converges strongly to PF(T)(x1).

In the same way, Kikkawa and Takahashi [8] introduced an iterative scheme for finding a common fixed point of an infinite family of nonexpansive mappings as follows:yn=Wnxn,Cn={z∈C:‖yn-z‖≤‖xn-z‖},Qn={z∈C:〈xn-z,x1-xn〉≥0},xn+1=PCn∩Qn(x1),∀n∈N,
where x1=x∈C and Wn is a W-mapping of C into itself generated by {Tn:C→C} and {μn}. They prove that, if Ω=⋂n=1∞F(Tn)≠∅, then the sequence {xn} generated by (1.11) converges strongly to PΩ(x1).

Recently, Su and Qin [9] modified the shrinking projection method for finding a fixed point of a nonexpansive mapping, for which the convergence rate of the iterative scheme is faster than that of the iterative scheme of Nakajo and Takahashi [7] as follows:yn=αnxn+(1-αn)Txn,Cn={z∈Cn-1∩Qn-1:‖yn-z‖≤‖xn-z‖},n≥1,Qn={z∈Cn-1∩Qn-1:〈xn-z,x0-xn〉≥0},n≥1,C0={z∈C:‖y0-z‖≤‖x0-z‖},Q0=C,xn+1=PCn∩Qn(x0),∀n∈N∪{0},
where x0=x∈C and T is a nonexpansive mapping of C into itself. They prove that, under the parameter 0≤αn≤α<1, if F(T)≠∅, then the sequence {xn} generated by (1.12) converges strongly to PF(T)(x0).

On the other hand, Tada and Takahashi [10] introduced an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of solutions of a fixed point problem of a nonexpansive mapping as follows:un∈CsuchthatF(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=(1-αn)xn+αnTun,Cn={z∈H:‖yn-z‖≤‖xn-z‖},Qn={z∈H:〈xn-z,x1-xn〉≥0},xn+1=PCn∩Qn(x1),∀n∈N,
where x1=x∈H, T is a nonexpansive mapping of C into H and F is a bifunction from C×C into ℝ. They prove that, under the sequences {αn}⊂[α,1] for some α∈(0,1) and {rn}⊂[r,∞) for some r>0, if Ω=F(T)∩EP(F)≠∅, then the sequence {xn} generated by (1.13) converges strongly to PΩ(x1) such that EP(F) is the set of solutions of equilibrium problem defined by EP(F)={x∈C:F(x,y)≥0,∀y∈C}.

In this paper, we introduce an iterative scheme (1.15) for finding 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 by the shrinking projection method in Hilbert spaces as follows:yn=αnWnxn+(1-αn)∑i=1NβiJMi,λi(xn-λiBixn),ϵn=αn(1-αn)‖Wnxn-∑i=1NβiJMi,λi(xn-λiBixn)‖2,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2-ϵn},Qn+1={z∈Cn∩Qn:〈xn-z,x1-xn〉≥0},C1=Q1=H,xn+1=PCn+1∩Qn+1(x1),∀n∈N,
where x1=u∈H chosen arbitrarily, Mi:H→2H is a maximal monotone mapping, Bi:H→H is a ξi-cocoercive mapping for each i=1,2,…,N, and Wn is a W-mapping on H generated by {Sn} and {μn} such that the mapping Sn:H→H defined by Snx=αx+(1-α)Tnx for all x∈H, where {Tn:H→H} is an infinite family of k-strictly pseudocontractive mappings with a fixed point.

It is well known that the class of strictly pseudocontractive mappings contains the class of nonexpansive mappings, and it follows that, if k=0, then the iterative scheme (1.15) 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.

Furthermore, if Mi≡Bi≡0 for all i=1,2,…,N and ∑i=1Nβi=1, then the iterative scheme (1.15) is reduced to extend and improve the results of Kikkawa and Takahashi [8] for finding a common fixed point of an infinite family of k-strictly pseudocontractive mappings as follows:x1=u∈Cchosenarbitrarily,yn=αnWnxn+(1-αn)xn,ϵn=αn(1-αn)‖Wnxn-xn‖2,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2-ϵn},Qn+1={z∈Cn∩Qn:〈xn-z,x1-xn〉≥0},C1=Q1=C,xn+1=PCn+1∩Qn+1(x1),∀n∈N,
and if k=α=0 and setting T1≡T, Tn≡I for all n=2,3,…, then the iterative scheme (1.16) is reduced to find a fixed point of a nonexpansive mapping, for which the convergence rate of the iterative scheme is faster than that of the iterative scheme of Su and Qin [9] as follows:x1=u∈Cchosenarbitrarily,yn=σnTxn+(1-σn)xn,δn=σn(1-σn)‖Txn-xn‖2,Dn+1={z∈Dn∩Qn:‖yn-z‖2≤‖xn-z‖2-δn},Qn+1={z∈Dn∩Qn:〈xn-z,x1-xn〉≥0},D1=Q1=C,xn+1=PDn+1∩Qn+1(x1),∀n∈N.

We suggest and analyze the iterative scheme (1.15) under some appropriate conditions imposed on the parameters. The strong convergence theorem for the above two sets is obtained, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.

2. Preliminaries

We collect the following lemmas which will be used in the proof of the main results in the next section.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B14">11</xref>]).

Let H be a Hilbert space. For any x,y∈H and λ∈ℝ, one has
‖λx+(1-λ)y‖2=λ‖x‖2+(1-λ)‖y‖2-λ(1-λ)‖x-y‖2.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let C be a nonempty closed convex subset of a Hilbert space H. Then the following inequality holds:
〈x-PCx,PCx-y〉≥0,∀x∈H,y∈C.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B5">5</xref>]).

Let C be a nonempty closed convex subset of a Hilbert space H, define mapping Wn as (1.8), 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

Wn is nonexpansive and F(Wn)=⋂i=1nF(Si) for each n≥1,

for each x∈C and for each positive integer k, limn→∞Un,kx exists,

the mapping W:C→C defined by
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 2.4 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

The resolvent operator JM,λ associated with M is single valued and nonexpansive for all λ>0.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

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,
VI(H,B,M)=F(JM,λ(I-λB)),∀λ>0.

Lemma 2.6 (see [<xref ref-type="bibr" rid="B11">12</xref>]).

Let C 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
Sx=∑n=1∞αnTnx,
for x∈C, is well defined, nonexpansive and F(S)=⋂n=1∞F(Tn) holds.

Lemma 2.7 (see [<xref ref-type="bibr" rid="B12">13</xref>]).

Let C 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 2.8 (see [<xref ref-type="bibr" rid="B13">14</xref>]).

Let C 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 nonexpansive such that F(S)=F(T).

Lemma 2.9 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Every Hilbert space H has Radon-Riesz property or Kadec-Klee property, that is, for a sequence {xn}⊂H with xn⇀x and ∥xn∥→∥x∥ then xn→x.

3. Main ResultsTheorem 3.1.

Let H be a real Hilbert space, Mi:H→2H a maximal monotone mapping, and Bi:H→H a ξi-cocoercive mapping for each i=1,2,…,N. 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 Sn:H→H by
Snx=αx+(1-α)Tnx,∀x∈H,
for all n∈ℕ, where α∈[k,1). 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))∩(⋂i=1NVI(H,Bi,Mi))≠∅. For x1=u∈H chosen arbitrarily, suppose that {xn} is generated iteratively by
yn=αnWnxn+(1-αn)∑i=1NβiJMi,λi(xn-λiBixn),ϵn=αn(1-αn)‖Wnxn-∑i=1NβiJMi,λi(xn-λiBixn)‖2,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2-ϵn},Qn+1={z∈Cn∩Qn:〈xn-z,x1-xn〉≥0},C1=Q1=H,xn+1=PCn+1∩Qn+1(x1),∀n∈N,
where

(C1) {αn}⊂[a,b] such that 0<a<b<1,

(C2) βi∈(0,1) and λi∈(0,2ξi] for each i=1,2,…,N,

(C3) ∑i=1Nβi=1.

Then, the sequences {xn} and {yn} converge strongly to w=PΩ(x1).

Proof.

For any x,y∈H and for each i=1,2,…,N, by the ξi-cocoercivity of Bi, we have
‖(I-λiBi)x-(I-λiBi)y‖2=‖(x-y)-λi(Bix-Biy)‖2=‖x-y‖2-2λi〈x-y,Bix-Biy〉+λi2‖Bix-Biy‖2≤‖x-y‖2-(2ξi-λi)λi‖Bix-Biy‖2≤‖x-y‖2,
which implies that I-λiBi is nonexpansive. Pick p∈Ω. Therefore, by Lemma 2.5, we have
p=JMi,λi(I-λiBi)p,
for each i=1,2,…,N. Since Snx=αx+(1-α)Tnx, where α∈[k,1) and {Tn} is a family of k-strict pseudocontraction, therefore, by Lemma 2.8, we have that Sn is nonexpansive and F(Sn)=F(Tn). It follows from Lemma 2.3(1) that F(Wn)=⋂i=1nF(Si)=⋂i=1nF(Ti), which implies that Wnp=p. Therefore, by (C3), (3.4), Lemma 2.1, and the nonexpansiveness of Wn,JMi,λi, and I-λiBi, we have
‖yn-p‖2=‖αnWnxn+(1-αn)∑i=1NβiJMi,λi(xn-λiBixn)-p‖2=‖αn(Wnxn-p)+(1-αn)∑i=1Nβi(JMi,λi(xn-λiBixn)-p)‖2=αn‖Wnxn-Wnp‖2+(1-αn)‖∑i=1Nβi(JMi,λi(xn-λiBixn)-JMi,λi(p-λiBip))‖2-αn(1-αn)‖Wnxn-∑i=1NβiJMi,λi(xn-λiBixn)‖2≤αn‖xn-p‖2+(1-αn)(∑i=1Nβi‖(xn-λiBixn)-(p-λiBip)‖)2-ϵn≤αn‖xn-p‖2+(1-αn)‖xn-p‖2-ϵn=‖xn-p‖2-ϵn,
for all n∈ℕ. Firstly, we prove that Cn∩Qn is closed and convex for all n∈ℕ. It is obvious that C1∩Q1 is closed and, by mathematical induction, that Cn∩Qn is closed for all n≥2, that is Cn∩Qn is closed for all n∈ℕ. Since ∥yn-z∥2≤∥xn-z∥2-ϵn is equivalent to
‖yn-xn‖2+2〈yn-xn,xn-z〉+ϵn≤0,
for all n∈ℕ, therefore, for any z1,z2∈Cn+1∩Qn+1⊂Cn∩Qn and ϵ∈(0,1), we have
‖yn-xn‖2+2〈yn-xn,xn-(ϵz1+(1-ϵ)z2)〉+ϵn=ϵ(‖yn-xn‖2+2〈yn-xn,xn-z1〉+ϵn)+(1-ϵ)(‖yn-xn‖2+2〈yn-xn,xn-z2〉+ϵn)≤0,
for all n∈ℕ, and we have
〈xn-(ϵz1+(1-ϵ)z2),x1-xn〉=ϵ〈xn-z1,x1-xn〉+(1-ϵ)〈xn-z2,x1-xn〉≥0,
for all n∈ℕ. Since C1∩Q1 is convex, and by putting n=1 in (3.6), (3.7), and (3.8), we have that C2∩Q2 is convex. Suppose that xk is given and Ck∩Qk is convex for some k≥2. It follows by putting n=k in (3.6), (3.7), and (3.8) that Ck+1∩Qk+1 is convex. Therefore, by mathematical induction, we have that Cn∩Qn is convex for all n≥2, that is, Cn∩Qn is convex for all n∈ℕ. Hence, we obtain that Cn∩Qn is closed and convex for all n∈ℕ.

Next, we prove that Ω⊂Cn∩Qn for all n∈ℕ. It is obvious that p∈Ω⊂H=C1∩Q1. Therefore, by (3.2) and (3.5), we have p∈C2 and note that p∈H=Q2, and so p∈C2∩Q2. Hence, we have Ω⊂C2∩Q2. Since C2∩Q2 is a nonempty closed convex subset of H, there exists a unique element x2∈C2∩Q2 such that x2=PC2∩Q2(x1). Suppose that xk∈Ck∩Qk is given such that xk=PCk∩Qk(x1), and p∈Ω⊂Ck∩Qk for some k≥2. Therefore, by (3.2) and (3.5), we have p∈Ck+1. Since xk=PCk∩Qk(x1), therefore, by Lemma 2.2, we have
〈xk-z,x1-xk〉≥0
for all z∈Ck∩Qk. Thus, by (3.2), we have p∈Qk+1, and so p∈Ck+1∩Qk+1. Hence, we have Ω⊂Ck+1∩Qk+1. Since Ck+1∩Qk+1 is a nonempty closed convex subset of H, there exists a unique element xk+1∈Ck+1∩Qk+1 such that xk+1=PCk+1∩Qk+1(x1). Therefore, by mathematical induction, we obtain Ω⊂Cn∩Qn for all n≥2, and so Ω⊂Cn∩Qn for all n∈ℕ, and we can define xn+1=PCn+1∩Qn+1(x1) for all n∈ℕ. Hence, we obtain that the iteration (3.2) is well defined.

Next, we prove that {xn} is bounded. Since xn=PCn∩Qn(x1) for all n∈ℕ, we have
‖xn-x1‖≤‖z-x1‖,
for all z∈Cn∩Qn. It follows by p∈Ω⊂Cn∩Qn that ∥xn-x1∥≤∥p-x1∥ for all n∈ℕ. This implies that {xn} is bounded, and so is {yn}.

Next, we prove that ∥yn-xn∥→0 as n→∞. Since xn+1=PCn+1∩Qn+1(x1)∈Cn+1∩Qn+1⊂Cn∩Qn, therefore, by (3.10), we have ∥xn-x1∥≤∥xn+1-x1∥ for all n∈ℕ. This implies that {∥xn-x1∥} is a bounded nondecreasing sequence and there exists the limit of ∥xn-x1∥, that is,
limn→∞‖xn-x1‖=m,
for some m≥0. Since xn+1∈Qn+1, therefore, by (3.2), we have
〈xn-xn+1,x1-xn〉≥0.
It follows by (3.12) that
‖xn-xn+1‖2=‖(xn-x1)+(x1-xn+1)‖2=‖xn-x1‖2+2〈xn-x1,x1-xn〉+2〈xn-x1,xn-xn+1〉+‖xn+1-x1‖2≤‖xn+1-x1‖2-‖xn-x1‖2.
Therefore, by (3.11), we have
‖xn-xn+1‖⟶0asn⟶∞.
Since xn+1∈Cn+1, therefore, by (3.2), we have
‖yn-xn+1‖2≤‖xn-xn+1‖2-ϵn≤‖xn-xn+1‖2.
It follows by (3.15) that
‖yn-xn‖≤‖yn-xn+1‖+‖xn+1-xn‖≤‖xn-xn+1‖+‖xn+1-xn‖=2‖xn-xn+1‖.
Therefore, by (3.14), we obtain
‖yn-xn‖⟶0asn⟶∞.

Since {xn} is bounded, there exists a subsequence {xni} of {xn} which converges weakly to w¯. Next, we prove that w¯∈Ω. Define the sequence of mappings {Qn:H→H} and the mapping Q:H→H by
Qnx=αnWnx+(1-αn)∑i=1NβiJMi,λi(I-λiBi)x,∀x∈H,Qx=limn→∞Qnx,
for all n∈ℕ. Therefore, by (C1) and Lemma 2.3(3), we have
Qx=cWx+(1-c)∑i=1NβiJMi,λi(I-λiBi)x,∀x∈H,
where a≤c=limn→∞αn≤b. From (C3) and Lemma 2.3(3), we have that W and ∑i=1NβiJMi,λi(I-λiBi) are nonexpansive. Therefore, by (C2), (C3), Lemmas 2.3(3), 2.5, 2.6, and 2.8, we have
F(Q)=F(W)∩F(∑i=1NβiJMi,λi(I-λiBi))=(⋂i=1∞F(Si))∩(⋂i=1NF(JMi,λi(I-λiBi)))=(⋂i=1∞F(Ti))∩(⋂i=1NVI(H,Bi,Mi)),
that is, F(Q)=Ω. From (3.17), we have ∥yni-xni∥→0asi→∞. Thus, from (3.2) and (3.18), we get ∥Qxni-xni∥→0asi→∞. It follows from xni⇀w¯ and Lemma 2.7 that w¯∈F(Q), that is, w¯∈Ω.

Since Ω is a nonempty closed convex subset of H, there exists a unique w∈Ω such that w=PΩ(x1). Next, we prove that xn→w as n→∞. Since w=PΩ(x1), we have ∥x1-w∥≤∥x1-z∥ for all z∈Ω, and it follows that
‖x1-w‖≤‖x1-w¯‖.
Since w∈Ω⊂Cn∩Qn, therefore, by (3.10), we have
‖x1-xn‖≤‖x1-w‖.
Therefore, by (3.21), (3.22), and the weak lower semicontinuity of norm, we have
‖x1-w‖≤‖x1-w¯‖≤liminfi→∞‖x1-xni‖≤limsupi→∞‖x1-xni‖≤‖x1-w‖.
It follows that
‖x1-w‖=limi→∞‖x1-xni‖=‖x1-w¯‖.
Since xni⇀w¯ as i→∞, therefore, we have
(x1-xni)⇀(x1-w¯)asi⟶∞.
Hence, from (3.24), (3.25), the Kadec-Klee property, and the uniqueness of w=PΩ(x1), we obtain
xni⟶w¯=wasi⟶∞.
It follows that {xn} converges strongly to w, and so is {yn}. This completes the proof.

Remark 3.2.

The iteration (3.2) is the difference with some well known results as the following.

The sequence {xn} is the projection sequence of x1 onto Cn∩Qn for all n∈ℕ such that
C1∩Q1⊃C2∩Q2⊃⋯⊃Cn∩Qn⊃⋯⊃Ω.

The proof of w¯∈Ω is simple by the demiclosedness principle because the sequence {yn} is a linear nonexpansive mapping form of the mappings Wn and JMi,λi(I-λiBi).

Solving a common fixed point for an infinite family of strictly pseudocontractive mappings and a system of cocoercive quasivariational inclusions problems by iteration is obtained.

4. ApplicationsTheorem 4.1.

Let H be a real Hilbert space, Mi:H→2H a maximal monotone mapping, and Bi:H→H a ξi-cocoercive mapping for each i=1,2,…,N. Let {Tn:H→H} be an infinite family of nonexpansive mappings. Define a mapping Sn:H→H by
Snx=αx+(1-α)Tnx,∀x∈H,
for all n∈ℕ, where α∈[0,1). 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))∩(⋂i=1NVI(H,Bi,Mi))≠∅. For x1=u∈H chosen arbitrarily, suppose that {xn} is generated iteratively by
yn=αnWnxn+(1-αn)∑i=1NβiJMi,λi(xn-λiBixn),ϵn=αn(1-αn)‖Wnxn-∑i=1NβiJMi,λi(xn-λiBixn)‖2,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2-ϵn},Qn+1={z∈Cn∩Qn:〈xn-z,x1-xn〉≥0},C1=Q1=H,xn+1=PCn+1∩Qn+1(x1),∀n∈N,
where

(C1) {αn}⊂[a,b] such that 0<a<b<1,

(C2) βi∈(0,1) and λi∈(0,2ξi] for each i=1,2,…,N,

(C3) ∑i=1Nβi=1.

Then the sequences {xn} and {yn} converge strongly to w=PΩ(x1).

Proof.

It is concluded from Theorem 3.1 immediately, by putting k=0.

Theorem 4.2.

Let C be a nonempty closed convex subset of a real Hilbert space H and {Tn:C→C} an infinite family of k-strictly pseudocontractive mappings with a fixed point such that k∈[0,1). Define a mapping Sn:C→C by
Snx=αx+(1-α)Tnx,∀x∈C,
for all n∈ℕ, where α∈[k,1). Let Wn:C→C 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)≠∅. For x1=u∈C chosen arbitrarily, suppose that {xn} is generated iteratively by
yn=αnWnxn+(1-αn)xn,ϵn=αn(1-αn)‖Wnxn-xn‖2,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2-ϵn},Qn+1={z∈Cn∩Qn:〈xn-z,x1-xn〉≥0},C1=Q1=C,xn+1=PCn+1∩Qn+1(x1),∀n∈N,
where {αn}⊂[a,b] such that 0<a<b<1. Then the sequences {xn} and {yn} converge strongly to w=PΩ(x1).

Proof.

It is concluded from Theorem 3.1 immediately, by putting Mi≡Bi≡0 for all i=1,2,…,N.

Theorem 4.3.

Let C be a nonempty closed convex subset of a real Hilbert space H and T:C→C a nonexpansive mapping. Assume that F(T)≠∅. For x1=u∈C chosen arbitrarily, suppose that {xn} is generated iteratively by
yn=σnTxn+(1-σn)xnδn=σn(1-σn)‖Txn-xn‖2,Dn+1={z∈Dn∩Qn:‖yn-z‖2≤‖xn-z‖2-δn},Qn+1={z∈Dn∩Qn:〈xn-z,x1-xn〉≥0},D1=Q1=C,xn+1=PDn+1∩Qn+1(x1),∀n∈N,
where {σn}⊂[a,b] such that 0<a<b<1. Then the sequences {xn} and {yn} converge strongly to w=PF(T)(x1).

Proof.

It is concluded from Theorem 4.2, by putting α=0. Setting T1≡T, Tn≡I for all n=2,3,… and leting μn⊂(0,μ] for some μ∈(0,1), therefore, from the definition of Sn in Theorem 4.2, we have S1=T1=T and Sn=I for all n=2,3,…. Since Wn is a W-mapping generated by {Sn} and {μn}, therefore, by the definition of Un,i and Wn in (1.8), we have Un,i=I for all i=2,3,… and Wn=Un,1=μ1S1Un,2+(1-μ1)I=μ1T+(1-μ1)I. Hence, by Theorem 4.2, we obtain
yn=αnWnxn+(1-αn)xn=αn(μ1Txn+(1-μ1)xn)+(1-αn)xn=σnTxn+(1-σn)xn,
where σn:=αnμ1. Since, the same as in the proof of Theorem 3.1, we have that Dn∩Qn is a nonempty closed convex subset of C for all n∈ℕ and by Theorem 4.2, we have
ϵn=αn(1-αn)‖Wnxn-xn‖2=αn(1-αn)‖μ1Txn+(1-μ1)xn-xn‖2=(αnμ1)(μ1-μ1αn)‖Txn-xn‖2=σn(μ1-σn)‖Txn-xn‖2≤σn(1-σn)‖Txn-xn‖2=δn,
for all n∈ℕ. It follows that Dn⊂Cn for all n∈ℕ, where Cn is defined as in Theorem 4.2. Hence, by Theorem 4.2, we obtain the desired result. This completes the proof.

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