# Hybrid and Relaxed Mann Iterations for General Systems of Variational Inequalities and Nonexpansive Mappings

Authors: L. C. Ceng; A. E. Al-Mazrooei; A. A. N. Abdou; A. Latif
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduce hybrid and relaxed Mann iteration methods for
a general system of variational inequalities with solutions being also common solutions of a
countable family of variational inequalities and common fixed points of a countable family
of nonexpansive mappings in real smooth and uniformly convex Banach spaces. Here, the
hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method,
viscosity approximation method, and Mann iteration method. Under suitable assumptions, we
derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms
not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also
in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. The
results presented in this paper improve, extend, supplement, and develop the corresponding
results announced in the earlier and very recent literature.


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## Body

## 1. Introduction

LetX be a real Banach space whose dual space is denoted by X*. The normalized duality mapping J:X→2X* is defined by
(1)J(x)={x*∈X*:〈x,x*〉=∥x∥2=∥x*∥2},∀x∈X,where〈·,·〉 denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that J(x) is nonempty for each x∈X. Let C be a nonempty closed convex subset of X. A mapping T:C→C is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for every x,y∈C. The set of fixed points of T is denoted by Fix(T). We use the notation ⇀ to indicate the weak convergence and the one → to indicate the strong convergence. A mapping A:C→X is said to be(i)
accretive if for eachx,y∈C there exists j(x-y)∈J(x-y) such that
(2)〈Ax-Ay,j(x-y)〉≥0;(ii)
α-strongly accretive if for each x,y∈C there exists j(x-y)∈J(x-y) such that
(3)〈Ax-Ay,j(x-y)〉≥α∥x-y∥2,for someα∈(0,1);(iii)
β-inverse strongly accretive if, for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(4)〈Ax-Ay,j(x-y)〉≥β∥Ax-Ay∥2,for someβ>0;(iv)
λ-strictly pseudocontractive [1] (see also [2]) if for each x,y∈C there exists j(x-y)∈J(x-y) such that
(5)〈Ax-Ay,j(x-y)〉≤∥x-y∥2-λ∥x-y-(Ax-Ay)∥2for someλ∈(0,1).It is worth emphasizing that the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, for example, [3–5]. Let U={x∈X:∥x∥=1} denote the unite sphere of X. A Banach space X is said to be uniformly convex if, for each ϵ∈(0,2], there exists δ>0 such that, for all x,y∈U,
(6)∥x-y∥≥ϵ⟹∥x+y∥2≤1-δ.It is known that a uniformly convex Banach space is reflexive and strict convex. A Banach spaceX is said to be smooth if the limit
(7)limt→0∥x+ty∥-∥x∥texists for allx,y∈U; in this case, X is also said to have a Gateaux differentiable norm. X is said to have a uniformly, Gateaux differentiable norm if, for each y∈U, the limit is attained uniformly for x∈U. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for x,y∈U. The norm of X is said to be the Frechet differential if for each x∈U, this limit is attained uniformly for y∈U. In the meantime, we define a function ρ:[0,∞)→[0,∞) called the modulus of smoothness of X as follows:
(8)ρ(τ)=sup{12(∥x+y∥+∥x-y∥)-1:x,y∈X,∥x∥=1,∥y∥=τ12}.It is known thatX is uniformly smooth if and only if limτ→0ρ(τ)/τ=0. Let q be a fixed real number with 1<q≤2. Then, a Banach space X is said to be q-uniformly smooth if there exists a constant c>0 such that ρ(τ)≤cτq for all τ>0. As pointed out in [6], no Banach space is q-uniformly smooth for q>2. In addition, it is also known that J is single valued if and only if X is smooth, whereas if X is uniformly smooth, then the mapping J is norm-to-norm uniformly continuous on bounded subsets of X. If X has a uniformly Gateaux differentiable norm, then the duality mapping J is norm-to-weak* uniformly continuous on bounded subsets of X.Recently, Yao et al. [7] combined the viscosity approximation method and Mann iteration method and gave the following hybrid viscosity approximation method.LetC be a nonempty closed convex subset of a real uniformly smooth Banach space X, T:C→C a nonexpansive mapping with Fix(T)≠∅, and f:C→C a contraction with coefficient ρ∈(0,1). For an arbitrary x0∈C, define {xn} in the following way:
(YCY)yn=αnxn+(1-αn)Txn,xn+1=βnf(xn)+(1-βn)yn,∀n≥0,where{αn} and {βn} are two sequences in (0,1).They proved under certain control conditions on the sequences{αn} and {βn} that {xn} converges strongly to a fixed point of T. Subsequently, under the following control conditions on {αn} and {βn}:(i)
1≤βn≤1-ρ, for all n≥n0 for some integer n0≥1,(ii)
∑n=0∞‍βn=∞,(iii)
0<liminfn→∞αn≤limsupn→∞αn<1,(iv)
limn→∞(βn+1/(1-(1-βn+1)αn+1)-βn/(1-(1-βn)αn))=0.Ceng and Yao [8] proved that
(9)xn⟶q⟺βn(f(xn)-xn)⟶0,whereq∈Fix(T) solves the variational inequality problem (VIP):
(10)〈q-f(q),J(q-p)〉≤0,∀p∈Fix(T).Such a result includes [7, Theorem  1] as a special case.LetC be a nonempty closed convex subset of a real Banach space X and f∈ΞC with a contractive coefficient ρ∈(0,1), where ΞC is the set of all contractive self-mappings on C. Let {Tn}n=0∞ be a sequence of nonexpansive self-mappings on C and {λn}n=0∞ a sequence of nonnegative numbers in [0,1]. For any n≥0, define a self-mapping Wn on C as follows:
(CY)Un,n+1=I,Un,n=λnTnUn,n+1+(1-λn)I,Un,n-1=λn-1Tn-1Un,n+(1-λn-1)I,⋮Un,k=λkTkUn,k+1+(1-λk)I,Un,k-1=λk-1Tk-1Un,k+(1-λk-1)I,⋮Un,1=λ1T1Un,2+(1-λ1)I,Wn=Un,0=λ0T0Un,1+(1-λ0)I.Such a mappingWn is called the W-mapping generated by Tn,Tn-1,…,T0, and λn,λn-1,…,λ0; see [9].In 2008, Ceng and Yao [10] introduced and analyzed the following relaxed viscosity approximation method for finding a common fixed point of an infinite family of nonexpansive mappings in a strictly convex and reflexive Banach space with a uniformly Gateaux differentiable norm.Theorem 1 (see [10]).
LetX be a strictly convex and reflexive Banach space with a uniformly Gateaux differentiable norm, C a nonempty closed convex subset of X, {Tn}n=0∞ a sequence of nonexpansive self-mappings on C such that the common fixed point set F:=⋂n=0∞‍Fix(Tn)≠∅, and f∈ΞC with a contractive coefficient ρ∈(1/2,1). For any given x0∈C, let {xn}n=0∞ be the iterative sequence defined by
(11)yn=(1-γn)xn+γnWnxn,xn+1=(1-αn-βn)xn+αnf(yn)+βnWnyn,∀n≥0,
where{αn}n=0∞ and {βn}n=0∞ are two sequences in (0,1) with αn+βn≤1(n≥0), {γn}n=0∞ is a sequence in [0,1], and Wn is the W-mapping generated by (CY). Assume that(i)
limn→∞αn=0, ∑n=0∞‍αn=∞ and 0<liminfn→∞βn≤limsupn→∞βn<1;(ii)
limn→∞|γn-γn-1|=0 and limsupn→∞γn<1.
Then, there hold the following:(i)
limn→∞∥xn+1-xn∥=0;(ii)
the sequence{xn}n=0∞ converges strongly to some p∈F which is the unique solution of the variational inequality problem (VIP)
(12)〈(I-f)q,J(q-p)〉≤0,∀f∈ΞC,p∈F,providedlimn→∞γn=0 and βn≡β for some fixed β∈(0,1).On the other hand, Cai and Bu [11] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space X, which involves finding (x*,y*)∈C×C such that
(13)〈μ1B1y*+x*-y*,J(x-x*)〉≥0,∀x∈C,〈μ2B2x*+y*-x*,J(x-y*)〉≥0,∀x∈C,whereC is a nonempty, closed, and convex subset of X, B1,B2:C→X are two nonlinear mappings, and μ1 and μ2 are two positive constants. Here, the set of solutions of GSVI (13) is denoted by GSVI(C,B1,B2). In particular, if X=H, a real Hilbert space, then GSVI (13) reduces to the following GSVI of finding (x*,y*)∈C×C such that
(14)〈μ1B1y*+x*-y*,x-x*〉≥0,∀x∈C,〈μ2B2x*+y*-x*,x-y*〉≥0,∀x∈C,in whichμ1 and μ2 are two positive constants. The set of solutions of problem (14) is still denoted by GSVI(C,B1,B2). In particular, if B1=B2=A, then problem (14) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [12]. Further, if x*=y* additionally, then the NSVI reduces to the classical variational inequality problem (VIP) of finding x*∈C such that
(15)〈Ax*,x-x*〉≥0,∀x∈C.The solution set of the VIP (15) is denoted by VI(C,A). Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [13]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous.Recently, Ceng et al. [14] transformed problem (14) into a fixed point problem in the following way.Lemma 2 (see [14]).
For givenx-,y-∈C, (x-,y-) is a solution of problem (14) if and only if x- is a fixed point of the mapping G:C→C defined by
(16)G(x)=PC[PC(x-μ2B2x)-μ1B1PC(x-μ2B2x)],∀x∈C,
wherey-=PC(x--μ2B2x-) and PC is the projection of H onto C.In particular, if the mappingBi:C→H is βi-inverse strongly monotone for i=1,2, then the mapping G is nonexpansive provided μi∈(0,2βi) for i=1,2.In 1976, Korpelevič [15] proposed an iterative algorithm for solving the VIP (15) in Euclidean space Rn:
(17)yn=PC(xn-τAxn),xn+1=PC(xn-τAyn),n≥0withτ>0 a given number, which is known as the extragradient method (see also [16]). The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [3, 11, 13, 17–33] and references therein, to name but a few.In particular, wheneverX is still a real smooth Banach space, B1=B2=A and x*=y*, then GSVI (13) reduces to the variational inequality problem (VIP) of finding x*∈C such that
(18)〈Ax*,J(x-x*)〉≥0,∀x∈C,which was considered by Aoyama et al. [34]. Note that VIP (18) is connected with the fixed point problem for nonlinear mapping (see, e.g., [35]), the problem of finding a zero point of a nonlinear operator (see, e.g., [36]), and so on. It is clear that VIP (18) extends VIP (15) from Hilbert spaces to Banach spaces.In order to find a solution of VIP (18), Aoyama et al. [34] introduced the following Mann-type iterative scheme for an accretive operator A:
(19)xn+1=αnxn+(1-αn)ΠC(xn-λnAxn),∀n≥1,whereΠC is a sunny nonexpansive retraction from X onto C. Then, they proved a weak convergence theorem. For the related work, see [37] and the references therein.LetC be a nonempty convex subset of a real Banach space X. Let {Ti}i=1N be a finite family of nonexpansive mappings of C into itself and let λ1,…,λN be real numbers such that 0≤λi≤1 for every i=1,…,N. Define a mapping K:C→C as follows:
(20)U1=λ1T1+(1-λ1)I,U2=λ2T2U1+(1-λ2)U1,U3=λ3T3U2+(1-λ3)U2,⋮UN-1=λN-1TN-1UN-2+(1-λN-1)UN-2,K=UN=λNTNUN-1+(1-λN)UN-1.Such a mappingK is called the K-mapping generated by T1,…,TN and λ1,…,λN.Very recently, Kangtunyakarn [38] introduced and analyzed an iterative algorithm by the modification of Mann’s iteration process for finding a common element of the set of solutions of a finite family of variational inequalities and the set of fixed points of an η-strictly pseudocontractive mapping and a nonexpansive mapping in uniformly convex and 2-uniformly smooth Banach spaces.Theorem 3 (see [38]).
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C. Let Ai:C→X be an αi-inverse-strongly accretive mapping for each i=1,…,N. Define the mapping Gi:C→C by Gi=ΠC(I-λiAi) for i=1,…,N, where λi∈(0,αi/κ2) and κ is the 2-uniformly smooth constant of X. Let B:C→C be the K-mapping generated by G1,…,GN and ρ1,…,ρN, where ρi∈(0,1), for all i=1,…,N-1, and ρN∈(0,1]. Let f:C→C a contraction with coefficient ρ∈(0,1). Let V:C→C be an η-strictly pseudocontractive mapping and S:C→C be a nonexpansive mapping such that F=Fix(S)∩Fix(V)∩(⋂i=1NVI(C,Ai))≠∅. For arbitrarily given x0∈C, let {xn} be the sequence generated by
(21)xn+1=αnf(xn)+βnxn+γnBxn+δnS((1-α)I+αV)xn,∀n≥0,
whereα∈(0,η/κ2). Suppose that {αn}, {βn}, {γn}, and {δn} are the sequences in [0,1], αn+βn+γn+δn=1 and satisfy the following conditions:(i)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(ii)
{γn},{δn}⊂[c,d] for some c,d∈(0,1);(iii)
∑n=1∞‍(|βn-βn-1|+|γn-γn-1|+|δn-δn-1|)<∞;(iv)
0<liminfn→∞βn≤limsupn→∞βn<1.
Then,{xn} converges strongly to q∈F, which solves the following VIP:
(22)〈q-f(q),J(q-p)〉≤0,∀p∈F.Beyond doubt, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (13) which contains VIP (18) as a special case. Very recently, Cai and Bu [11] constructed an iterative algorithm for solving GSVI (13) and a common fixed point problem of a countable family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a 2-uniformly smooth Banach space X.Lemma 4 (see [39]).
LetX be a 2-uniformly smooth Banach space. Then,
(23)∥x+y∥2≤∥x∥2+2〈y,J(x)〉+2∥κy∥2,∀x,y∈X,
whereκ is the 2-uniformly smooth constant of X and J is the normalized duality mapping from X into X*.Define the mappingG:C→C as follows:
(24)G(x):=ΠC(I-μ1B1)ΠC(I-μ2B2)x,∀x∈C.The fixed point set ofG is denoted by Ω. Then, their strong convergence theorem on the proposed method is stated as follows.Theorem 5 (see [11]).
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C. Let the mapping Bi:C→X be βi-inverse-strongly accretive with 0<μi<βi/κ2 for i=1,2. Let f be a contraction of C into itself with coefficient δ∈(0,1). Let {Tn}n=1∞ be a countable family of nonexpansive mappings of C into itself such that F=⋂i=1∞‍Fix(Ti)∩Ω≠∅, where Ω is the fixed point set of the mapping G defined by (24). For arbitrarily given x1∈C, let {xn} be the sequence generated by
(25)xn+1=βnxn+(1-βn)Snyn,yn=αnf(xn)+(1-αn)zn,zn=ΠC(un-μ1B1un),un=ΠC(xn-μ2B2xn),∀n≥1.Suppose that{αn} and {βn} are two sequences in (0,1) satisfying the following conditions:(i)
limn→∞αn=0 and ∑n=1∞‍αn=∞;(ii)
0<liminfn→∞βn≤limsupn→∞βn<1.Assume that∑n=1∞‍supx∈D∥Tn+1x-Tnx∥<∞ for any bounded subset D of C and let T be a mapping of C into X defined by Tx=limn→∞Tnx for all x∈C and suppose that Fix(T)=⋂n=1∞‍Fix(Tn). Then, {xn} converges strongly to q∈F, which solves the following VIP:
(26)〈q-f(q),J(q-p)〉≤0,∀p∈F.It is easy to see that the iterative scheme in Theorem5 is essentially equivalent to the following two-step iterative scheme:
(27)yn=αnf(xn)+(1-αn)Gxn,xn+1=βnxn+(1-βn)Tnyn,∀n≥1.For the convenience of implementing the argument techniques in [14], the authors of [11] have used the following inequality in a real smooth and uniform convex Banach space X.Proposition 6 (see [40]).
LetX be a real smooth and uniform convex Banach space and let r>0. Then, there exists a strictly increasing, continuous, and convex function g:[0,2r]→R, g(0)=0 such that
(28)g(∥x-y∥)≤∥x∥2-2〈x,J(y)〉+∥y∥2,∀x,y∈Br,
whereBr={x∈X:∥x∥≤r}.LetC be a nonempty closed convex subset of a real smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C and f:C→C a contraction with coefficient ρ∈(0,1). Motivated and inspired by the research going on this area, we consider and introduce hybrid and relaxed Mann iteration methods for finding solutions of the GSVI (13) which are also common solutions of a countable family of variational inequalities and common fixed points of a countable family of nonexpansive mappings in X. Here, the hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method, and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature; see, for example, [8, 10, 11, 14, 33, 38].


## 2. Preliminaries

We list some lemmas that will be used in the sequel.Lemma 7 (see [41]).
Let{sn} be a sequence of nonnegative real numbers satisfying
(29)sn+1≤(1-αn)sn+αnβn+γn,∀n≥0,
where{αn}, {βn}, and {γn} satisfy the following conditions:(i)
{αn}⊂[0,1] and ∑n=0∞‍αn=∞;(ii)
limsupn→∞βn≤0;(iii)
γn≥0, for all n≥0, and ∑n=0∞‍γn<∞.
Then,limsupn→∞sn=0.The following lemma is an immediate consequence of the subdifferential inequality of the function(1/2)∥·∥2.Lemma 8 (see [42]).
LetX be a real Banach space X. Then, for all x,y∈X(i)
∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉 for all j(x+y)∈J(x+y);(ii)
∥x+y∥2≥∥x∥2+2〈y,j(x)〉 for all j(x)∈J(x).LetD be a subset of C and let Π be a mapping of C into D. Then, Π is said to be sunny if
(30)Π[Π(x)+t(x-Π(x))]=Π(x),wheneverΠ(x)+t(x-Π(x))∈C for x∈C and t≥0. A mapping Π of C into itself is called a retraction if Π2=Π. If a mapping Π of C into itself is a retraction, then Π(z)=z for every z∈R(Π) where R(Π) is the range of Π. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D. The following lemma concerns the sunny nonexpansive retraction.Lemma 9 (see [43]).
LetC be a nonempty closed convex subset of a real smooth Banach space X. Let D be a nonempty subset of C. Let Π be a retraction of C onto D. Then, the following are equivalent:(i)
Π is sunny and nonexpansive;(ii)
∥Π(x)-Π(y)∥2≤〈x-y,J(Π(x)-Π(y))〉, for all x,y∈C;(iii)
〈x-Π(x),J(y-Π(x))〉≤0, for all x∈C, y∈D.It is well known that ifX=H a Hilbert space, then a sunny nonexpansive retraction ΠC is coincident with the metric projection from X onto C; that is, ΠC=PC. If C is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space X and if T:C→C is a nonexpansive mapping with the fixed point set Fix(T)≠∅, then the set Fix(T) is a sunny nonexpansive retract of C.Lemma 10.
LetC be a nonempty closed convex subset of a smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C and let B1,B2:C→X be nonlinear mappings. For given x*,y*∈C, (x*,y*) is a solution of GSVI (13) if and only if x*=ΠC(y*-μ1B1y*), where y*=ΠC(x*-μ2B2x*).Proof.
We can rewrite GSVI (13) as
(31)〈x*-(y*-μ1B1y*),J(x-x*)〉≥0,∀x∈C,〈y*-(x*-μ2B2x*),J(x-y*)〉≥0,∀x∈C,
which is obviously equivalent to(32)x*=ΠC(y*-μ1B1y*),y*=ΠC(x*-μ2B2x*),
because of Lemma9. This completes the proof.In terms of Lemma10, we observe that
(33)x*=ΠC[ΠC(x*-μ2B2x*)-μ1B1ΠC(x*-μ2B2x*)],which implies thatx* is a fixed point of the mapping G. Throughout this paper, the set of fixed points of the mapping G is denoted by Ω.Lemma 11 (see [44]).
LetX be a uniformly convex Banach space and Br={x∈X:∥x∥≤r}, r>0. Then, there exists a continuous, strictly increasing, and convex function g:[0,∞]→[0,∞], g(0)=0 such that
(34)∥αx+βy+γz∥2≤α∥x∥2+β∥y∥2+γ∥z∥2-αβg(∥x-y∥)
for allx,y,z∈Br, and all α,β,γ∈[0,1] with α+β+γ=1.Lemma 12 (see [45]).
LetC be a nonempty closed convex subset of a Banach space X. Let S0,S1,… be a sequence of mappings of C into itself. Suppose that ∑n=1∞‍sup{∥Snx-Sn-1x∥:x∈C}<∞. Then for each y∈C, {Sny} converges strongly to some point of C. Moreover, let S be a mapping of C into itself defined by Sy=limn→∞Sny for all y∈C. Then limn→∞sup{∥Sx-Snx∥:x∈C}=0.LetC be a nonempty closed convex subset of a Banach space X and T:C→C a nonexpansive mapping with Fix(T)≠∅. As previous, let ΞC be the set of all contractions on C. For t∈(0,1) and f∈ΞC, let xt∈C be the unique fixed point of the contraction x↦tf(x)+(1-t)Tx on C; that is,
(35)xt=tf(xt)+(1-t)Txt.Lemma 13 (see [35, 46]).
LetX be a uniformly smooth Banach space, or a reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm. Let C be a nonempty closed convex subset of X, T:C→C a nonexpansive mapping with Fix(T)≠∅, and f∈ΞC. Then, the net {xt} defined by xt=tf(xt)+(1-t)Txt converges strongly to a point in Fix(T). If we define a mapping Q:ΞC→Fix(T) by Q(f):=s-limt→0xt, for all f∈ΞC, then Q(f) solves the VIP:
(36)〈(I-f)Q(f),J(Q(f)-p)〉≤0,∀f∈ΞC,p∈Fix(T).Lemma 14 (see [47]).
LetC be a nonempty closed convex subset of a strictly convex Banach space X. Let {Tn}n=0∞ be a sequence of nonexpansive mappings on C. Suppose that  ⋂n=0∞‍Fix(Tn) is nonempty. Let {λn} be a sequence of positive numbers with ∑n=0∞‍λn=1. Then, a mapping S on C defined by Sx=∑n=0∞‍λnTnx for x∈C is defined well; nonexpansive and Fix(S)=⋂n=0∞‍Fix(Tn) holds.Lemma 15 (see [39]).
Given a numberr>0, A real Banach space X is uniformly convex if and only if there exists a continuous strictly increasing function g:[0,∞)→[0,∞), g(0)=0, such that
(37)∥λx+(1-λ)y∥2≤λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)g(∥x-y∥)
for allλ∈[0,1] and x,y∈X such that ∥x∥≤r and ∥y∥≤r.Lemma 16 (see [48, Lemma  3.2]).
LetC be a nonempty closed convex subset of a strictly convex Banach space X. Let {Tn}n=0∞ be a sequence of nonexpansive self-mappings on C such that ⋂n=0∞‍Fix(Tn)≠∅ and let {λn}n=0∞ be a sequence of positive numbers in (0,b] for some b∈(0,1). Then, for every x∈C and k≥0, the limit limn→∞Un,kx exists.Using Lemma16, one can define a mapping W:C→C as follows:
(38)Wx=limn→∞Wnx=limn→∞Un,0xfor everyx∈C. Such a W is called the W-mapping generated by the sequences {Tn}n=0∞ and {λn}n=0∞. Throughout this paper, we always assume that {λn}n=0∞ is a sequence of positive numbers in (0,b] for some b∈(0,1).Lemma 17 (see [48]).
LetC be a nonempty closed convex subset of a strictly convex Banach space X. Let {Tn}n=0∞ be a sequence of nonexpansive self-mappings on C such that ⋂n=0∞‍Fix(Tn)≠∅ and let {λn}n=0∞ be a sequence of positive numbers in (0,b] for some b∈(0,1). Then, Fix(W)=⋂n=0∞‍Fix(Tn).
Letμ be a continuous linear functional on l∞ and s=(a0,a1,…)∈l∞. One  writes μn(an) instead of μ(s). μ is called a Banach limit if μ satisfies ∥μ∥=μn(1)=1 and μn(an+1)=μn(an) for all (a0,a1,…)∈l∞. If μ is a Banach limit, then, there hold the following:(i)
for alln≥0, an≤cn implies μn(an)≤μn(cn);(ii)
μn(an+r)=μn(an) for any fixed positive integer r;(iii)
liminfn→∞an≤μn(an)≤limsupn→∞an for all (a0,a1,…)∈l∞.Lemma 18 (see [49]).
Leta∈R be a real number and a sequence {an}∈l∞ satisfy the condition μn(an)≤a for all Banach limit μ. If limsupn→∞(an+r-an)≤0, then limsupn→∞an≤a.In particular, ifr=1 in Lemma 18, then we immediately obtain the following corollary.Corollary 19 (see [50]).
Leta∈R be a real number and a sequence {an}∈l∞ satisfy the condition μn(an)≤a for all Banach limit μ. If limsupn→∞(an+1-an)≤0, then, limsupn→∞an≤a.Lemma 20 (see [51]).
Let{xn} and {zn} be bounded sequences in a Banach space X and let {βn} be a sequence of nonnegative numbers in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose that xn+1=βnxn+(1-βn)zn for all integers n≥0 and limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥xn-zn∥=0.Lemma 21 (see [34]).
LetC be a nonempty closed convex subset of a smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C and A:C→X an accretive mapping. Then for all λ>0,
(39)VI(C,A)=Fix(ΠC(I-λA)).Lemma 22 (see [11]).
LetC be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X. Let the mapping Bi:C→X be β^i-inverse-strongly accretive. Then, one has
(40)∥(I-μiBi)x-(I-μiBi)y∥2≤∥x-y∥2+2μi(μiκ2-β^i)∥Bix-Biy∥2,∀x,y∈C,
fori=1,2 where μi>0. In particular, if 0<μi≤β^i/κ2, then I-μiBi is nonexpansive for i=1,2.Lemma 23 (see [11]).
LetC be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C. Let the mapping Bi:C→X be β^i-inverse-strongly accretive for i=1,2. Let G:C→C be the mapping defined by
(41)Gx=ΠC[ΠC(x-μ2B2x)-μ1B1ΠC(x-μ2B2x)],∀x∈C.
If0<μi≤β^i/κ2 for i=1,2, then G:C→C is nonexpansive.


## 3. Hybrid Mann Iterations and Their Convergence Criteria

In this section, we introduce our hybrid Mann iteration algorithms in real smooth and uniformly convex Banach spaces and present their convergence criteria.Theorem 24.
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C. Let {ρn}n=0∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and Ai:C→E an α^i-inverse strongly accretive mapping for each i=0,1,…. Define a mapping Gi:C→C by ΠC(I-λiAi)x=Gix for all x∈C and i=0,1,…, where λi∈(0,α^i/κ2], κ is the 2-uniformly smooth constant of X. Let Bn:C→C be the W-mapping generated by Gn,Gn-1,…,G0 and ρn,ρn-1,…,ρ0. Let the mapping Bi:C→X be β^i-inverse strongly accretive for i=1,2. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let {Si}i=0∞ be a countable family of nonexpansive mappings of C into itself such that F=(⋂i=0∞‍Fix(Si))∩Ω∩(⋂i=0∞‍VI(C,Ai))≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2) with 0<μi<β^i/κ2 for i=1,2. For arbitrarily given x0∈C, let {xn} be the sequence generated by
(42)yn=βnxn+γnBnxn+δnSnGxn,xn+1=αnf(xn)+(1-αn)yn,∀n≥0,
where{αn}, {βn}, {γn}, and {δn} are the sequences in [0,1] such that βn+γn+δn=1 for all n≥0. Suppose that the following conditions hold:(i)
∑n=0∞‍αn=∞ and 0≤αn≤1-ρ, for all n≥n0 for some integer n0≥0;(ii)
liminfn→∞γn>0 and liminfn→∞δn>0;(iii)
limn→∞(|αn+1/(1-(1-αn+1)βn+1)-αn/(1-(1-αn)βn)|+|δn+1/(1-βn+1)-δn/(1-βn)|)=0;(iv)
0<liminfn→∞βn≤limsupn→∞βn<1.
Assume that∑n=0∞‍supx∈D∥Sn+1x-Snx∥<∞ for any bounded subset D of C and let S be a mapping of C into itself defined by Sx=limn→∞Snx for all x∈C and suppose that Fix(S)=⋂i=0∞‍Fix(Si). Then, there hold the following:(I)
limn→∞∥xn+1-xn∥=0;(II)
xn→q⇔αn(f(xn)-xn)→0 provided βn≡β for some fixed β∈(0,1), where q∈F solves the following VIP:
(43)〈q-f(q),J(q-p)〉≤0,∀p∈F.Proof.
First of all, since0<λi<α^i/κ2 for i=0,1,…, it is easy to see that Gi is a nonexpansive mapping for each i=0,1,…. Since Bn:C→C is the W-mapping generated by Gn,Gn-1,…,G0 and ρn,ρn-1,…,ρ0, by Lemma 16 we know that, for each x∈C and k≥0, the limit limn→∞Un,kx exists. Moreover, one can define a mapping B:C→C as follows:
(44)Bx=limn→∞Bnx=limn→∞Un,0x
for everyx∈C. That is, such a B is the W-mapping generated by the sequences {Gn}n=0∞ and {ρn}n=0∞. According to Lemma 17, we know that Fix(B)=⋂i=0∞‍Fix(Gi). From Lemma 15 and the definition of Gi, we have Fix(Gi)=VI(C,Ai) for each i=0,1,…. Hence, we have
(45)Fix(B)=⋂i=0∞‍Fix(Gi)=⋂i=0∞‍VI(C,Ai).
Next, let us show that the sequence{xn} is bounded. Indeed, take a fixed p∈F arbitrarily. Then, we get p=Gp, p=Bnp, and p=Snp for all n≥0. By Lemma 23 we know that G is nonexpansive. Then, from (42), we have
(46)∥yn-p∥≤βn∥xn-p∥+γn∥Bnxn-p∥+δn∥SnGxn-p∥≤βn∥xn-p∥+γn∥xn-p∥+δn∥Gxn-p∥≤βn∥xn-p∥+γn∥xn-p∥+δn∥xn-p∥=∥xn-p∥,
and hence(47)∥xn+1-p∥≤αn∥f(xn)-p∥+(1-αn)∥yn-p∥≤αn(∥f(xn)-f(p)∥+∥f(p)-p∥)+(1-αn)∥yn-p∥≤αn(ρ∥xn-p∥+∥f(p)-p∥)+(1-αn)∥xn-p∥=(1-αn(1-ρ))∥xn-p∥+αn(1-ρ)∥f(p)-p∥1-ρ≤max{∥xn-p∥,∥f(p)-p∥1-ρ}.
By induction, we obtain(48)∥xn-p∥≤max{∥x0-p∥,∥f(p)-p∥1-ρ},∀n≥0.
Thus,{xn} is bounded, and so are the sequences {yn}, {Gxn} and {f(xn)}.
Let us show that(49)limn→∞∥xn+1-xn∥=0.
As a matter of fact, putσn=(1-αn)βn, for all n≥0. Then, it follows from (i) and (iv) that
(50)βn≥σn=(1-αn)βn≥(1-(1-ρ))βn=ρβn,∀n≥n0,
and hence(51)0<liminfn→∞σn≤limsupn→∞σn<1.
Define(52)xn+1=σnxn+(1-σn)zn.
Observe that(53)zn+1-zn=xn+2-σn+1xn+11-σn+1-xn+1-σnxn1-σn=αn+1f(xn+1)+(1-αn+1)yn+1-σn+1xn+11-σn+1-αnf(xn)+(1-αn)yn-σnxn1-σn=(αn+1f(xn+1)1-σn+1-αnf(xn)1-σn)-(1-αn)[βnxn+γnBnxn+δnSnGxn]-σnxn1-σn+(1-αn+1)[βn+1xn+1+γn+1Bn+1xn+1+δn+1Sn+12Gxn+1+δn+1Sn+1Gxn+1]-σn+1xn+1×(1-σn+1)-1=(αn+1f(xn+1)1-σn+1-αnf(xn)1-σn)+1-αn+11-σn+1×(γn+1Bn+1xn+1+δn+1Sn+1Gxn+1)-1-αn1-σn(γnBnxn+δnSnGxn)=(αn+1f(xn+1)1-σn+1-αnf(xn)1-σn)+(1-αn+1)(1-βn+1)1-σn+1×[γn+1Bn+1xn+1+δn+1Sn+1Gxn+11-βn+1-γnBnxn+δnSnGxn1-βn]+[(1-αn+1)(1-βn+1)1-σn+1-(1-αn)(1-βn)1-σn]×γnBnxn+δnSnGxn1-βn=αn+11-σn+1(f(xn+1)-f(xn))+(αn+11-σn+1-αn1-σn)f(xn)+(1-αn+1)(1-βn+1)1-σn+1×[γn+1γn+1+δn+1(Bn+1xn+1-Bnxn)+(γn+1γn+1+δn+1-γnγn+δn)Bnxn+δn+1γn+1+δn+1(Sn+1Gxn+1-SnGxn)+(δn+1γn+1+δn+1-δnγn+δn)SnGxn]-(αn+11-σn+1-αn1-σn)γnBnxn+δnSnGxnγn+δn=αn+11-σn+1(f(xn+1)-f(xn))+(αn+11-σn+1-αn1-σn)×(f(xn)-γnBnxn+δnSnGxnγn+δn)+1-σn+1-αn+11-σn+1×[γn+1γn+1+δn+1(Bn+1xn+1-Bnxn)+(γn+1γn+1+δn+1-γnγn+δn)Bnxn+δn+1γn+1+δn+1(Sn+1Gxn+1-SnGxn)+(δn+1γn+1+δn+1-δnγn+δn)SnGxn],
and hence(54)∥zn+1-zn∥≤αn+11-σn+1∥f(xn+1)-f(xn)∥+|αn+11-σn+1-αn1-σn|∥f(xn)-γnBnxn+δnSnGxnγn+δn∥+1-σn+1-αn+11-σn+1×∥γn+1γn+1+δn+1(Bn+1xn+1-Bnxn)+(γn+1γn+1+δn+1-γnγn+δn)Bnxn+δn+1γn+1+δn+1(Sn+1Gxn+1-SnGxn)+(δn+1γn+1+δn+1-δnγn+δn)SnGxn∥≤ραn+11-σn+1∥xn+1-xn∥+|αn+11-σn+1-αn1-σn|×(∥f(xn)∥+∥Bnxn∥+∥SnGxn∥)+1-σn+1-αn+11-σn+1×[γn+1γn+1+δn+1∥Bn+1xn+1-Bnxn∥+|γn+1γn+1+δn+1-γnγn+δn|∥Bnxn∥+δn+1γn+1+δn+1∥Sn+1Gxn+1-SnGxn∥+|δn+1γn+1+δn+1-δnγn+δn|∥SnGxn∥].
On the other hand, we note that, for alln≥0,
(55)∥Sn+1Gxn+1-SnGxn∥≤∥Sn+1Gxn+1-Sn+1Gxn∥+∥Sn+1Gxn-SnGxn∥≤∥Gxn+1-Gxn∥+∥Sn+1Gxn-SnGxn∥≤∥xn+1-xn∥+∥Sn+1Gxn-SnGxn∥.
Furthermore, by(CY), since Gi and Un,i are nonexpansive, we deduce that for each n≥0(56)∥Bn+1xn+1-Bnxn∥≤∥Bn+1xn+1-Bn+1xn∥+∥Bn+1xn-Bnxn∥≤∥xn+1-xn∥+∥Bn+1xn-Bnxn∥=∥xn+1-xn∥+∥λ0G0Un+1,1xn-λ0G0Un,1xn∥≤∥xn+1-xn∥+λ0∥Un+1,1xn-Un,1xn∥=∥xn+1-xn∥+λ0∥λ1G1Un+1,2xn-λ1G1Un,2xn∥≤∥xn+1-xn∥+λ0λ1∥Un+1,2xn-Un,2xn∥⋮≤∥xn+1-xn∥+(∏i=0n‍λi)∥Un+1,n+1xn-Un,n+1xn∥≤∥xn+1-xn∥+M0∏i=0n‍λi,
for some constantM0>0. Utilizing (54)–(56), we have
(57)∥zn+1-zn∥≤ραn+11-σn+1∥xn+1-xn∥+|αn+11-σn+1-αn1-σn|×(∥f(xn)∥+∥Bnxn∥+∥SnGxn∥)+1-σn+1-αn+11-σn+1×[γn+1γn+1+δn+1(∥xn+1-xn∥+M0∏i=0n‍λi)+|γn+1γn+1+δn+1-γnγn+δn|∥Bnxn∥+δn+1γn+1+δn+1(∥xn+1-xn∥+∥Sn+1Gxn-SnGxn∥)+|δn+1γn+1+δn+1-δnγn+δn|∥SnGxn∥γn+1γn+1+δn+1(∥xn+1-xn∥+M0∏i=0n‍λi)]=1-σn+1-αn+1(1-ρ)1-σn+1∥xn+1-xn∥+|αn+11-σn+1-αn1-σn|(∥f(xn)∥+∥Bnxn∥+∥SnGxn∥)+1-σn+1-αn+11-σn+1×[γn+1γn+1+δn+1M0∏i=0n‍λi+δn+1γn+1+δn+1∥Sn+1Gxn-SnGxn∥+|δn+1γn+1+δn+1-δnγn+δn|(∥Bnxn∥+∥SnGxn∥)γn+1γn+1+δn+1M0∏i=0n‍λi]≤∥xn+1-xn∥+|αn+11-σn+1-αn1-σn|M+M∏i=0n‍λi+∥Sn+1Gxn-SnGxn∥+|δn+1γn+1+δn+1-δnγn+δn|M=∥xn+1-xn∥+M(|δn+1γn+1+δn+1-δnγn+δn|+∏i=0n‍λi|αn+11-σn+1-αn1-σn|+|δn+1γn+1+δn+1-δnγn+δn|+∏i=0n‍λi)+∥Sn+1Gxn-SnGxn∥,
which hence yields(58)∥zn+1-zn∥-∥xn+1-xn∥≤M(∏i=0n‍λi|αn+11-σn+1-αn1-σn|+|δn+1γn+1+δn+1-δnγn+δn|≤M+∏i=0n‍λi)+∥Sn+1Gxn-SnGxn∥,
wheresupn≥0{∥f(xn)∥+∥Bnxn∥+∥SnGxn∥+M0}≤M for some M>0. So, from (58), condition (iii), and the assumption on {Sn}, it follows that (noting that 0<λi≤b<1, for all i≥0)
(59)limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0.
Consequently, by Lemma20, we have
(60)limn→∞∥zn-xn∥=0.
It follows from (51) and (52) that
(61)limn→∞∥xn+1-xn∥=limn→∞(1-σn)∥zn-xn∥=0.
From (42), we have
(62)xn+1-xn=αn(f(xn)-xn)+(1-αn)(yn-xn),
which hence implies that(63)ρ∥yn-xn∥=(1-(1-ρ))∥yn-xn∥≤(1-αn)∥yn-xn∥=∥xn+1-xn-αn(f(xn)-xn)∥≤∥xn+1-xn∥+∥αn(f(xn)-xn)∥.
Sincexn+1-xn→0 and αn(f(xn)-xn)→0, we get
(64)limn→∞∥yn-xn∥=0.
Next, we show that∥xn-Gxn∥→0 as n→∞.
Indeed, for simplicity, putq=ΠC(p-μ2B2p), un=ΠC(xn-μ2B2xn) and vn=ΠC(un-μ1B1un). Then, vn=Gxn for all n≥0. From Lemma 22, we have
(65)∥un-q∥2=∥ΠC(xn-μ2B2xn)-ΠC(p-μ2B2p)∥2≤∥xn-p-μ2(B2xn-B2p)∥2≤∥xn-p∥2-2μ2(β^2-κ2μ2)∥B2xn-B2p∥2,(66)∥vn-p∥2=∥ΠC(un-μ1B1un)-ΠC(q-μ1B1q)∥2≤∥un-q-μ1(B1un-B1q)∥2≤∥un-q∥2-2μ1(β^1-κ2μ1)∥B1un-B1q∥2.
Substituting (65) for (66), we obtain
(67)∥vn-p∥2≤∥xn-p∥2-2μ2(β^2-κ2μ2)∥B2xn-B2p∥2-2μ1(β^1-κ2μ1)∥B1un-B1q∥2.
From (42) and (67), we have
(68)∥yn-p∥2≤βn∥xn-p∥2+γn∥Bnxn-p∥2+δn∥SnGxn-p∥2≤βn∥xn-p∥2+γn∥xn-p∥2+δn∥vn-p∥2≤βn∥xn-p∥2+γn∥xn-p∥2+δn[∥xn-p∥2-2μ2(β^2-κ2μ2)×∥B2xn-B2p∥2-2μ1(β^1-κ2μ1)∥B1un-B1q∥2]=∥xn-p∥2-2δn[μ2(β^2-κ2μ2)∥B2xn-B2p∥2+2μ1(β^1-κ2μ1)∥B1un-B1q∥2],
which hence implies that(69)2δn[μ2(β^2-κ2μ2)∥B2xn-B2p∥2+μ1(β^1-κ2μ1)∥B1un-B1q∥2]≤∥xn-p∥2-∥yn-p∥2≤(∥xn-p∥+∥yn-p∥)∥xn-yn∥.
Since0<μi<β^i/κ2 for i=1,2, and {xn}, {yn} are bounded, we obtain from (64), (69), and condition (ii) that
(70)limn→∞∥B2xn-B2p∥=0,limn→∞∥B1un-B1q∥=0.
Utilizing Proposition6 and Lemma 9, we have
(71)∥un-q∥2=∥ΠC(xn-μ2B2xn)-ΠC(p-μ2B2p)∥2≤〈xn-μ2B2xn-(p-μ2B2p),J(un-q)〉=〈xn-p,J(un-q)〉+μ2〈B2p-B2xn,J(un-q)〉≤12[∥xn-p∥2+∥un-q∥2-g1(∥xn-un-(p-q)∥)]+μ2∥B2p-B2xn∥∥un-q∥,
which implies that(72)∥un-q∥2≤∥xn-p∥2-g1(∥xn-un-(p-q)∥)+2μ2∥B2p-B2xn∥∥un-q∥.
In the same way, we derive(73)∥vn-p∥2=∥ΠC(un-μ1B1un)-ΠC(q-μ1B1q)∥2≤〈un-μ1B1un-(q-μ1B1q),J(vn-p)〉=〈un-q,J(vn-p)〉+μ1〈B1q-B1un,J(vn-p)〉≤12[∥un-q∥2+∥vn-p∥2-g2(∥un-vn+(p-q)∥)∥un-q∥2+∥vn-p∥2]+μ1∥B1q-B1un∥∥vn-p∥,
which implies that(74)∥vn-p∥2≤∥un-q∥2-g2(∥un-vn+(p-q)∥)+2μ1∥B1q-B1un∥∥vn-p∥.
Substituting (72) for (74), we get
(75)∥vn-p∥2≤∥xn-p∥2-g1(∥xn-un-(p-q)∥)-g2(∥un-vn+(p-q)∥)+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥.
By Lemma8(i), we have from (68) and (75)
(76)∥yn-p∥2≤βn∥xn-p∥2+γn∥xn-p∥2+δn∥vn-p∥2≤βn∥xn-p∥2+γn∥xn-p∥2+δn[∥xn-p∥2-g1(∥xn-un-(p-q)∥)-g2(∥un-vn+(p-q)∥)+2μ2∥B2p-B2xn∥×∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥∥xn-p∥2-g1(∥xn-un-(p-q)∥)]≤∥xn-p∥2-δn[g1(∥xn-un-(p-q)∥)+g2(∥un-vn+(p-q)∥)]+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥,
which hence leads to(77)δn[g1(∥xn-un-(p-q)∥)+g2(∥un-vn+(p-q)∥)]≤∥xn-p∥2-∥yn-p∥2+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥≤(∥xn-p∥+∥yn-p∥)∥xn-yn∥+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥.
From (70), (77), condition (ii), and the boundedness of {xn}, {yn}, {un}, and {vn}, we deduce that
(78)limn→∞g1(∥xn-un-(p-q)∥)=0,limn→∞g2(∥un-vn+(p-q)∥)=0.
Utilizing the properties ofg1 and g2, we deduce that
(79)limn→∞∥xn-un-(p-q)∥=0,limn→∞∥un-vn+(p-q)∥=0.
From (79), we get
(80)∥xn-vn∥≤∥xn-un-(p-q)∥+∥un-vn+(p-q)∥⟶0asn⟶∞.
That is,(81)limn→∞∥xn-Gxn∥=0.
Next, let us show that(82)limn→∞∥Bnxn-xn∥=0,limn→∞∥Sxn-xn∥=0.
Indeed, utilizing Lemma15 and (42), we have
(83)∥yn-p∥2=∥δn(SnGxn-p)+(βn+γn)(βnxn+γnBnxnβn+γn-p)∥2≤δn∥SnGxn-p∥2+(βn+γn)×∥βnxn+γnBnxnβn+γn-p∥2=δn∥SnGxn-p∥2+(βn+γn)×∥βnβn+γn(xn-p)+γnβn+γn(Bnxn-p)∥2≤δn∥Gxn-p∥2+(βn+γn)×[βnβn+γn∥xn-p∥2+γnβn+γn∥Bnxn-p∥2βnγn(βn+γn)2-βnγn(βn+γn)2g3(∥xn-Bnxn∥)]≤δn∥xn-p∥2+βn∥xn-p∥2+γn∥xn-p∥2-βnγnβn+γng3(∥xn-Bnxn∥)=∥xn-p∥2-βnγnβn+γng3(∥xn-Bnxn∥),
which immediately implies that(84)βnγng3(∥xn-Bnxn∥)≤βnγnβn+γng3(∥xn-Bnxn∥)≤∥xn-p∥2-∥yn-p∥2≤(∥xn-p∥+∥yn-p∥)∥xn-yn∥.
So, from (64), the boundedness of {xn}, {yn}, and conditions (ii), (iv), it follows that
(85)limn→∞g3(∥xn-Bnxn∥)=0.
From the properties ofg3, we have
(86)limn→∞∥xn-Bnxn∥=0.
Taking into account that(87)yn-xn=γn(Bnxn-xn)+δn(SnGxn-xn),
we have(88)δn∥SnGxn-xn∥=∥yn-xn-γn(Bnxn-xn)∥≤∥yn-xn∥+γn∥Bnxn-xn∥≤∥yn-xn∥+∥Bnxn-xn∥.
From (64), (86), and condition (ii), it follows that
(89)limn→∞∥SnGxn-xn∥=0,limn→∞∥SnGxn-Bnxn∥=0.
Note that(90)∥xn-Sxn∥≤∥xn-SnGxn∥+∥SnGxn-Snxn∥+∥Snxn-Sxn∥≤∥xn-SnGxn∥+∥Gxn-xn∥+∥Snxn-Sxn∥.
So, in terms of (81), (89), and Lemma 12, we have
(91)limn→∞∥xn-Sxn∥=0.
Suppose thatβn≡β for some fixed β∈(0,1) such that β+γn+δn=1 for all n≥0. Define a mapping Vx=(1-θ1-θ2)Sx+θ1Bx+θ2Gx, where θ1,θ2∈(0,1) are two constants with θ1+θ2<1. Then, by Lemmas 14 and 17, we have that Fix(V)=Fix(S)∩Fix(B)∩Fix(G)=F. For each k≥1, let {pk} be a unique element of C such that
(92)pk=1kf(pk)+(1-1k)Vpk.
From Lemma13, we conclude that pk→q∈Fix(V)=F as k→∞. Observe that for every n,k(93)∥yn-Bpk∥=∥β(xn-Bpk)+γn(Bnxn-Bpk)+δn(SnGxn-Bpk)∥≤β∥xn-Bpk∥+γn∥Bnxn-Bpk∥+δn(∥SnGxn-Bnxn∥+∥Bnxn-Bpk∥)=β∥xn-Bpk∥+(1-β)∥Bnxn-Bpk∥+δn∥SnGxn-Bnxn∥,
and hence(94)∥xn+1-Bpk∥≤αn∥f(xn)-Bpk∥+(1-αn)∥yn-Bpk∥≤αn(∥f(xn)-xn∥+∥xn-Bpk∥)+(1-αn)∥yn-Bpk∥≤αn∥f(xn)-xn∥+αn∥xn-Bpk∥+(1-αn)×[β∥xn-Bpk∥+(1-β)∥Bnxn-Bpk∥+δn∥SnGxn-Bnxn∥]≤αn∥f(xn)-xn∥+αn∥xn-Bpk∥+(1-αn)×[β∥xn-Bpk∥+(1-β)(∥Bnxn-Bnpk∥+∥Bnpk-Bpk∥)+δn∥SnGxn-Bnxn∥]≤αn∥f(xn)-xn∥+αn∥xn-Bpk∥+(1-αn)[β∥xn-Bpk∥+(1-β)(∥xn-pk∥+∥Bnpk-Bpk∥)+δn∥SnGxn-Bnxn∥]=αn∥f(xn)-xn∥+(β+αn(1-β))×∥xn-Bpk∥+(1-αn)(1-β)∥xn-pk∥+(1-αn)[(1-β)∥Bnpk-Bpk∥+δn∥SnGxn-Bnxn∥]≤αn∥f(xn)-xn∥+(β+αn(1-β))×(∥xn-xn+1∥+∥xn+1-Bpk∥)+(1-αn)(1-β)(∥xn-xn+1∥+∥xn+1-pk∥)+(1-αn)[(1-β)∥Bnpk-Bpk∥+δn∥SnGxn-Bnxn∥Bnpk]=αn∥f(xn)-xn∥+(β+αn(1-β))×∥xn+1-Bpk∥+(1-αn)(1-β)∥xn+1-pk∥+(1-αn)[(1-β)∥Bnpk-Bpk∥+δn∥SnGxn-Bnxn∥]+∥xn-xn+1∥.
So, it immediately follows from0≤αn≤1-ρ, for all n≥n0, that
(95)∥xn+1-Bpk∥≤∥xn+1-pk∥+∥Bnpk-Bpk∥+1(1-αn)(1-β)×(∥αn(xn-f(xn))∥+∥xn-xn+1∥)+δn1-β∥SnGxn-Bnxn∥≤∥xn+1-pk∥+∥Bnpk-Bpk∥+∥SnGxn-Bnxn∥+1ρ(1-β)×(∥αn(xn-f(xn))∥+∥xn-xn+1∥)=∥xn+1-pk∥+θn,∀n≥n0,
whereθn=∥Bnpk-Bpk∥+∥SnGxn-Bnxn∥+(1/ρ(1-β))(∥αn(xn-f(xn))∥+∥xn-xn+1∥). Since limn→∞∥Bnpk-Bpk∥=limn→∞∥SnGxn-Bnxn∥=limn→∞∥αn(xn-f(xn))∥=limn→∞∥xn-xn+1∥=0, we know that θn→0 as n→∞.
From (95), we obtain
(96)∥xn+1-Bpk∥2≤∥xn+1-pk∥2+θn(2∥xn+1-pk∥+θn),∀n≥n0.
For any Banach limitμ, from (96), we derive
(97)μn∥xn-Bpk∥2=μn∥xn+1-Bpk∥2≤μn∥xn+1-pk∥2=μn∥xn-pk∥2.
In addition, note that(98)∥xn-Gpk∥2≤∥xn-Gxn+Gxn-Gpk∥2≤(∥xn-Gxn∥+∥xn-pk∥)2=∥xn-pk∥2+∥xn-Gxn∥×(2∥xn-pk∥+∥xn-Gxn∥),∥xn-Spk∥2≤∥xn-Sxn+Sxn-Spk∥2≤(∥xn-Sxn∥+∥xn-pk∥)2=∥xn-pk∥2+∥xn-Sxn∥×(2∥xn-pk∥+∥xn-Sxn∥).
It is easy to see from (81) and (91) that
(99)μn∥xn-Gpk∥2≤μn∥xn-pk∥2,μn∥xn-Spk∥2≤μn∥xn-pk∥2.
Utilizing (97) and (99), we deduce that
(100)μn∥xn-Vpk∥2=μn∥(1-θ1-θ2)(xn-Spk)+θ1(xn-Bpk)+θ2(xn-Gpk)∥2≤(1-θ1-θ2)μn∥xn-Spk∥2+θ1μn∥xn-Bpk∥2+θ2μn∥xn-Gpk∥2≤μn∥xn-pk∥2.
Also, observe that(101)xn-pk=1k(xn-f(pk))+(1-1k)(xn-Vpk);
that is,(102)(1-1k)(xn-Vpk)=xn-pk-1k(xn-f(pk)).
It follows from Lemma8 (ii) and (102) that
(103)(1-1k)2∥xn-Vpk∥2≥∥xn-pk∥2-2k〈xn-pk+pk-f(pk),J(xn-pk)〉=(1-2k)∥xn-pk∥2+2k〈f(pk)-pk,J(xn-pk)〉.
So by (100) and (103), we have
(104)(1-1k)2μn∥xn-pk∥2≥(1-2k)μn∥xn-pk∥2+2kμn〈f(pk)-pk,J(xn-pk)〉,
and hence(105)1k2μn∥xn-pk∥2≥2kμn〈f(pk)-pk,J(xn-pk)〉.
This implies that(106)12kμn∥xn-pk∥2≥μn〈f(pk)-pk,J(xn-pk)〉.
Sincepk→q∈Fix(V)=F as k→∞, by the uniform Frechet differentiability of the norm of X we have
(107)μn〈f(q)-q,J(xn-q)〉≤0.
On the other hand, from (49) and the norm-to-norm uniform continuity of J on bounded subsets of X, it follows that
(108)limn→∞|〈f(q)-q,J(xn+1-q)〉-〈f(q)-q,J(xn-q)〉|=0.
So, utilizing Lemma18 we deduce from (107) and (108) that
(109)limsupn→∞〈f(q)-q,J(xn-q)〉≤0,
which together with (49) and the norm-to-norm uniform continuity of J on bounded subsets of X, implies that
(110)limsupn→∞〈f(q)-q,J(xn+1-q)〉≤0.
Finally, let us show thatxn→q as n→∞. Utilizing Lemma 8 (i), from (42) and the convexity of ∥·∥2, we get
(111)∥yn-q∥2≤βn∥xn-q∥2+γn∥Bnxn-q∥2+δn∥SnGxn-q∥2≤βn∥xn-q∥2+γn∥xn-q∥2+δn∥xn-q∥2=∥xn-q∥2,(112)∥xn+1-q∥2=∥αn(f(xn)-f(q))+(1-αn)(yn-q)+αn(f(q)-q)∥2≤∥αn(f(xn)-f(q))+(1-αn)(yn-q)∥2+2αn〈f(q)-q,J(xn+1-q)〉≤αn∥f(xn)-f(q)∥2+(1-αn)∥yn-q∥2+2αn〈f(q)-q,J(xn+1-q)〉≤αnρ∥xn-q∥2+(1-αn)∥xn-q∥2+2αn〈f(q)-q,J(xn+1-q)〉=(1-αn(1-ρ))∥xn-q∥2+2αn〈f(q)-q,J(xn+1-q)〉=(1-αn(1-ρ))∥xn-q∥2+αn(1-ρ)2〈f(q)-q,J(xn+1-q)〉1-ρ.
Applying Lemma7 to (112), we obtain that xn→q as n→∞.
Conversely, ifxn→q∈F as n→∞, then from (42) it follows that
(113)∥yn-q∥≤βn∥xn-q∥+γn∥Bnxn-q∥+δn∥SnGxn-q∥≤βn∥xn-q∥+γn∥xn-q∥+δn∥xn-q∥=∥xn-q∥⟶0asn⟶∞,
that is,yn→q. Again from (42) we obtain that
(114)∥αn(f(xn)-xn)∥=∥xn+1-xn-(1-αn)(yn-xn)∥≤∥xn+1-xn∥+(1-αn)∥yn-xn∥≤∥xn+1-q∥+∥xn-q∥+(1-αn)(∥yn-q∥+∥xn-q∥)≤∥xn+1-q∥+2∥xn-q∥+∥yn-q∥.
Sincexn→q and yn→q, we get αn(f(xn)-xn)→0. This completes the proof.Corollary 25.
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C. Let {ρn}n=0∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and Ai:C→E an α^i-inverse strongly accretive mapping for each i=0,1,…. Define a mapping Gi:C→C by ΠC(I-λiAi)x=Gix for all x∈C and i=0,1,…, where λi∈(0,α^i/κ2] and κ is the 2-uniformly smooth constant of X. Let Bn:C→C be the W-mapping generated by Gn,Gn-1,…,G0 and ρn,ρn-1,…,ρ0. Let V:C→C be an α-strictly pseudocontractive mapping. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let {Si}i=0∞ be a countable family of nonexpansive mappings of C into itself such that F=(⋂i=0∞‍Fix(Si))∩Fix(V)∩(⋂i=0∞‍VI(C,Ai))≠∅. For arbitrarily given x0∈C, let {xn} be the sequence generated by
(115)yn=βnxn+γnBnxn+δnSn((1-l)I+lV)xn,xn+1=αnf(xn)+(1-αn)yn,∀n≥0,
where0<l<α/κ2, {αn}, {βn}, {γn}, and {δn} are the sequences in [0,1] such that βn+γn+δn=1 for all n≥0. Suppose that the following conditions hold:(i)
∑n=0∞‍αn=∞ and 0≤αn≤1-ρ, for all n≥n0 for some integer n0≥0;(ii)
liminfn→∞γn>0 and liminfn→∞δn>0;(iii)
limn→∞(|αn+1/(1-(1-αn+1)βn+1)-αn/(1-(1-αn)βn)|+|δn+1/(1-βn+1)-δn/(1-βn)|)=0;(iv)
0<liminfn→∞βn≤limsupn→∞βn<1.
Assume that∑n=0∞‍supx∈D∥Sn+1x-Snx∥<∞ for any bounded subset D of C and let S be a mapping of C into itself defined by Sx=limn→∞Snx for all x∈C and suppose that Fix(S)=⋂i=0∞‍Fix(Si). Then, there hold the following:(I)
limn→∞∥xn+1-xn∥=0;(II)
xn→q⇔αn(f(xn)-xn)→0 provided βn≡β for some fixed β∈(0,1), where q∈F solves the following VIP
(116)〈q-f(q),J(q-p)〉≤0,∀p∈F.Proof.
In Theorem24, we put B1=I-V, B2=0, and μ1=l, where 0<l<α/κ2. Then, GSVI (13) is equivalent to the VIP of finding x*∈C such that
(117)〈B1x*,J(x-x*)〉≥0,∀x∈C.
In this case,B1:C→X is α-inverse strongly accretive. It is not hard to see that Fix(V)=VI(C,B1). As a matter of fact, we have, for l>0,
(118)u∈VI(C,B1)⟺〈B1u,J(y-u)〉≥0∀y∈C⟺〈u-lB1u-u,J(u-y)〉≥0∀y∈C⟺u=ΠC(u-lB1u)⟺u=ΠC(u-lu+lVu)⟺〈u-lu+lVu-u,J(u-y)〉≥0∀y∈C⟺〈u-Vu,J(u-y)〉≤0∀y∈C⟺u=Vu⟺u∈Fix(V).
Accordingly, we know thatF=(⋂i=0∞‍Fix(Si))∩Ω∩(⋂i=0∞‍VI(C,Ai))=(⋂i=0∞‍Fix(Si))∩Fix(V)∩(⋂i=0∞‍VI(C,Ai)), and
(119)ΠC(I-μ1B1)ΠC(I-μ2B2)xn=ΠC(I-μ1B1)xn=ΠC((1-l)xn+lVxn)=((1-l)I+lV)xn.
So, the scheme (42) reduces to (115). Therefore, the desired result follows from Theorem 24.Here, we prove the following important lemmas which will be used in the sequel.Lemma 26.
LetC be a nonempty closed convex subset of a smooth Banach space X and let the mapping Bi:C→X be λi-strictly pseudocontractive and αi-strongly accretive with αi+λi≥1 for i=1,2. Then, for μi∈(0,1] one has
(120)∥(I-μiBi)x-(I-μiBi)y∥≤{1-αiλi+(1-μi)(1+1λi)}∥x-y∥,∀x,y∈C,
fori=1,2. In particular, if 1-(λi/(1+λi))(1-(1-αi)/λi)≤μi≤1, then I-μiBi is nonexpansive for i=1,2.Proof.
Taking into account theλi-strict pseudocontractivity of Bi, we derive for every x,y∈C(121)λi∥(I-Bi)x-(I-Bi)y∥2≤〈(I-Bi)x-(I-Bi)y,J(x-y)〉≤∥(I-Bi)x-(I-Bi)y∥∥x-y∥,
which implies that(122)∥(I-Bi)x-(I-Bi)y∥≤1λi∥x-y∥.
Hence,(123)∥Bix-Biy∥≤∥(I-Bi)x-(I-Bi)y∥+∥x-y∥≤(1+1λi)∥x-y∥.
Utilizing theαi-strong accretivity and λi-strict pseudocontractivity of Bi, we get
(124)λi∥(I-Bi)x-(I-Bi)y∥2≤∥x-y∥2-〈Bix-Biy,J(x-y)〉≤(1-αi)∥x-y∥2.
So, we have(125)∥(I-Bi)x-(I-Bi)y∥≤1-αiλi∥x-y∥.
Therefore, forμi∈(0,1] we have
(126)∥(I-μiBi)x-(I-μiBi)y∥≤∥(I-Bi)x-(I-Bi)y∥+(1-μi)∥Bix-Biy∥≤1-αiλi∥x-y∥+(1-μi)(1+1λi)∥x-y∥={1-αiλi+(1-μi)(1+1λi)}∥x-y∥.
Since1-(λi/(1+λi))(1-(1-αi)/λi)≤μi≤1, it follows immediately that
(127)1-αiλi+(1-μi)(1+1λi)≤1.
This implies thatI-μiBi is nonexpansive for i=1,2.Lemma 27.
LetC be a nonempty closed convex subset of a smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C and let the mapping Bi:C→X be λi-strictly pseudocontractive and αi-strongly accretive with αi+λi≥1 for i=1,2. Let G:C→C be the mapping defined by
(128)G(x)=ΠC[ΠC(x-μ2B2x)-μ1B1ΠC(x-μ2B2x)],∀x∈C.
If1-(λi/(1+λi))(1-(1-αi)/λi)≤μi≤1, then G:C→C is nonexpansive.Proof.
According to Lemma26, we know that I-μiBi is nonexpansive for i=1,2. Hence, for all x,y∈C, we have
(129)∥G(x)-G(y)∥=∥ΠC[ΠC(x-μ2B2x)-μ1B1ΠC(x-μ2B2x)]-ΠC[ΠC(y-μ2B2y)-μ1B1ΠC(y-μ2B2y)]∥=∥ΠC(I-μ1B1)ΠC(I-μ2B2)x-ΠC(I-μ1B1)ΠC(I-μ2B2)y∥≤∥(I-μ1B1)ΠC(I-μ2B2)x-(I-μ1B1)ΠC(I-μ2B2)y∥≤∥ΠC(I-μ2B2)x-ΠC(I-μ2B2)y∥≤∥(I-μ2B2)x-(I-μ2B2)y∥≤∥x-y∥.
This shows thatG:C→C is nonexpansive. This completes the proof.Theorem 28.
LetC be a nonempty closed convex subset of a uniformly convex Banach space X which has a uniformly Gateaux differentiable norm. Let ΠC be a sunny nonexpansive retraction from X onto C. Let {ρn}n=0∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and Ai:C→X be ξi-strictly pseudocontractive and α^i-strongly accretive with ξi+α^i≥1 for each i=0,1,…. Define a mapping Gi:C→C by ΠC(I-λiAi)x=Gix for all x∈C and i=0,1,…, where 1-(ξi/(1+ξi))(1-(1-α^i)/ξi)≤λi≤1 for all i=0,1,…. Let Bn:C→C be the W-mapping generated by Gn,Gn-1,…,G0 and ρn,ρn-1,…,ρ0. Let the mapping Bi:C→X  ζi-strictly pseudocontractive and β^i-strongly accretive with ζi+β^i≥1 for i=1,2. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let {Si}i=0∞ be a countable family of nonexpansive mappings of C into itself such that F=(⋂i=0∞‍Fix(Si))∩Ω∩(⋂i=0∞‍VI(C,Ai))≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2) with 1-(ζi/(1+ζi))(1-(1-β^i)/ζi)≤μi≤1 for i=1,2. For arbitrarily given x0∈C, let {xn} be the sequence generated by
(130)yn=βnxn+γnBnxn+δnSnGxn,xn+1=αnf(xn)+σnGxn+(1-αn-σn)yn,∀n≥0,
where{σn}, {αn}, {βn}, {γn}, and {δn} are the sequences in [0,1] such that βn+γn+δn=1 and αn+σn≤1 for all n≥0. Suppose that the following conditions hold:(i)
∑n=0∞‍αn=∞ and 0≤αn+σn≤1-ρ, for all n≥n0 for some integer n0≥0;(ii)
liminfn→∞σn>0, liminfn→∞γn>0 and liminfn→∞δn>0;(iii)
limn→∞(|αn+1/(1-(1-αn+1-σn+1)βn+1)-(αn/(1-(1-αn-σn)βn))|+  |σn+1/(1-(1-αn+1-σn+1)βn+1)-(σn/(1-(1-αn-σn)βn))|+  |δn+1/(1-βn+1)-δn/(1-βn)|)=0;(iv)
0<liminfn→∞βn≤limsupn→∞βn<1.
Assume that∑n=0∞‍supx∈D∥Sn+1x-Snx∥<∞ for any bounded subset D of C and let S be a mapping of C into itself defined by Sx=limn→∞Snx for all x∈C and suppose that Fix(S)=⋂i=0∞‍Fix(Si). Then there hold the following:(I)
limn→∞∥xn+1-xn∥=0;(II)
xn→q⇔αn(f(xn)-xn)→0 provided βn≡β for some fixed β∈(0,1), where q∈F solves the following VIP
(131)〈q-f(q),J(q-p)〉≤0,∀p∈F.Proof.
First of all, take a fixedp∈F arbitrarily. Then we obtain p=Gp, p=Bnp and Snp=p for all n≥0. By Lemma 27, we get from (130)
(132)∥yn-p∥≤βn∥xn-p∥+γn∥Bnxn-p∥+δn∥SnGxn-p∥≤βn∥xn-p∥+γn∥xn-p∥+δn∥xn-p∥=∥xn-p∥,
and hence(133)∥xn+1-p∥≤αn∥f(xn)-p∥+σn∥Gxn-p∥+(1-αn-σn)∥yn-p∥≤αn(∥f(xn)-f(p)∥+∥f(p)-p∥)+σn∥xn-p∥+(1-αn-σn)∥xn-p∥≤αn(ρ∥xn-p∥+∥f(p)-p∥)+σn∥xn-p∥+(1-αn-σn)∥xn-p∥=(1-αn(1-ρ))∥xn-p∥+αn∥f(p)-p∥=(1-αn(1-ρ))∥xn-p∥+αn(1-ρ)∥f(p)-p∥1-ρ≤max{∥xn-p∥,∥f(p)-p∥1-ρ}.
By induction, we have(134)∥xn-p∥≤max{∥x0-p∥,∥f(p)-p∥1-ρ},∀n≥0,
which implies that{xn} is bounded and so are the sequences {yn}, {Gxn}, and {f(xn)}.
Let us show that(135)limn→∞∥xn+1-xn∥=0.
As a matter of fact, putθn=(1-αn-σn)βn, for all n≥0. Then, it follows from (i) and (iv) that
(136)βn≥θn=(1-αn-σn)βn≥(1-(1-ρ))βn=ρβn,βn≥θn=(1-αn-σn)βn≥(1-(1-ρ))∀n≥n0,
and hence(137)0<liminfn→∞θn≤limsupn→∞θn<1.
Define(138)xn+1=θnxn+(1-θn)zn.
Observe that(139)zn+1-zn=xn+2-θn+1xn+11-θn+1-xn+1-θnxn1-θn=(αn+1f(xn+1)+σn+1Gxn+1+(1-αn+1-σn+1)yn+1-θn+1xn+1)×(1-θn+1)-1-αnf(xn)+σnGxn+(1-αn-σn)yn-θnxn1-θn=(αn+1f(xn+1+σn+1Gxn+1)1-θn+1-αnf(xn)+σnGxn1-θn)-(1-αn-σn)[βnxn+γnBnxn+δnSnGxn]-θnxn1-θn+(1-αn+1-σn+1)×[βn+1xn+1+γn+1Bn+1xn+1+δn+1Sn+1Gxn+1]-θn+1xn+1×(1-θn+1)-1=(αn+1f(xn+1)+σn+1Gxn+11-θn+1-αnf(xn)+σnGxn1-θn)+1-αn+1-σn+11-θn+1(γn+1Bn+1xn+1+δn+1Sn+1Gxn+1)-1-αn-σn1-θn(γnBnxn+δnSnGxn)=(αn+1f(xn+1)+σn+1Gxn+11-θn+1-αnf(xn)+σnGxn1-θn)+(1-αn+1-σn+1)(1-βn+1)1-θn+1×[γn+1Bn+1xn+1+δn+1Sn+1Gxn+11-βn+1-γnBnxn+δnSnGxn1-βn]+[(1-αn+1-σn+1)(1-βn+1)1-θn+1-(1-αn-σn)(1-βn)1-θn]×γnBnxn+δnSnGxn1-βn=αn+11-θn+1(f(xn+1)-f(xn))+σn+11-θn+1(Gxn+1-Gxn)+(αn+11-θn+1-αn1-θn)f(xn)+(σn+11-θn+1-σn1-θn)Gxn+(1-αn+1-σn+1)(1-βn+1)1-θn+1×[γn+1γn+1+δn+1(Bn+1xn+1-Bnxn)+(γn+1γn+1+δn+1-γnγn+δn)Bnxn+δn+1γn+1+δn+1(Sn+1Gxn+1-SnGxn)+(δn+1γn+1+δn+1-δnγn+δn)SnGxn]-(αn+1+σn+11-θn+1-αn+σn1-θn)γnBnxn+δnSnGxnγn+δn=αn+11-θn+1(f(xn+1)-f(xn))+σn+11-θn+1(Gxn+1-Gxn)+(αn+11-θn+1-αn1-θn)(f(xn)-γnBnxn+δnSnGxnγn+δn)+(σn+11-θn+1-σn1-θn)(Gxn-γnBnxn+δnSnGxnγn+δn)+1-αn+1-σn+1-θn+11-θn+1×[γn+1γn+1+δn+1(Bn+1xn+1-Bnxn)+(γn+1γn+1+δn+1-γnγn+δn)Bnxn+δn+1γn+1+δn+1(Sn+1Gxn+1-SnGxn)+(δn+1γn+1+δn+1-δnγn+δn)SnGxn],
and hence(140)∥zn+1-zn∥≤αn+11-θn+1∥f(xn+1)-f(xn)∥+σn+11-θn+1∥Gxn+1-Gxn∥+|αn+11-θn+1-αn1-θn|∥f(xn)-γnBnxn+δnSnGxnγn+δn∥+|σn+11-θn+1-σn1-θn|∥Gxn-γnBnxn+δnSnGxnγn+δn∥+1-αn+1-σn+1-θn+11-θn+1×[γn+1γn+1+δn+1∥Bn+1xn+1-Bnxn∥+|γn+1γn+1+δn+1-γnγn+δn|∥Bnxn∥+δn+1γn+1+δn+1∥Sn+1Gxn+1-SnGxn∥+|δn+1γn+1+δn+1-δnγn+δn|∥SnGxn∥].
On the other hand, repeating the same arguments as those of (55) and (56) in the proof of Theorem 24, we can get
(141)∥Sn+1Gxn+1-SnGxn∥≤∥xn+1-xn∥+∥Sn+1Gxn-SnGxn∥,∥Bn+1xn+1-Bnxn∥≤∥xn+1-xn∥+M0∏i=0n‍λi,
for some constantM0>0. Utilizing (140)-(141), we have
(142)∥zn+1-zn∥≤αn+11-θn+1ρ∥xn+1-xn∥+σn+11-θn+1∥xn+1-xn∥+|αn+11-θn+1-αn1-θn|×(∥f(xn)∥+∥Bnxn∥+∥SnGxn∥)+|σn+11-θn+1-σn1-θn|×(∥Gxn∥+∥Bnxn∥+∥SnGxn∥)+1-αn+1-σn+1-θn+11-θn+1×[γn+1γn+1+δn+1(∥xn+1-xn∥+M0∏i=0n‍λi)+|γn+1γn+1+δn+1-γnγn+δn|∥Bnxn∥+δn+1γn+1+δn+1(∥xn+1-xn∥+∥Sn+1Gxn-SnGxn∥)+|δn+1γn+1+δn+1-δnγn+δn|∥SnGxn∥γn+1γn+1+δn+1(∥xn+1-xn∥+M0∏i=0n‍λi)]=1-αn+1(1-ρ)-θn+11-θn+1∥xn+1-xn∥+1-αn+1-σn+1-θn+11-θn+1×[γn+1γn+1+δn+1M0∏i=0n‍λi+|δn+1γn+1+δn+1-δnγn+δn|×(∥Bnxn∥+∥SnGxn∥)+δn+1γn+1+δn+1∥Sn+1Gxn-SnGxn∥γn+1γn+1+δn+1M0∏i=0n‍λi+|δn+1γn+1+δn+1-δnγn+δn|]+|αn+11-θn+1-αn1-θn|×(∥f(xn)∥+∥Bnxn∥+∥SnGxn∥)+|σn+11-θn+1-σn1-θn|×(∥Gxn∥+∥Bnxn∥+∥SnGxn∥)≤∥xn+1-xn∥+M(∏i=0n‍λi+|δn+11-βn+1-δn1-βn|+|αn+11-θn+1-αn1-θn|+|σn+11-θn+1-σn1-θn|∏i=0n‍λi+|δn+11-βn+1-δn1-βn|)+∥Sn+1Gxn-SnGxn∥,
wheresupn≥0{∥f(xn)∥+∥Gxn∥+∥Bnxn∥+∥SnGxn∥+M0}≤M for some M>0. So, from (142), condition (iii), and the assumption on {Sn} it follows that (noting that 0<λi≤b<1, for all i≥0)
(143)limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0.
Consequently, by Lemma20, we have
(144)limn→∞∥zn-xn∥=0.
It follows from (137) and (138) that
(145)limn→∞∥xn+1-xn∥=limn→∞(1-θn)∥zn-xn∥=0.
Next, we show that∥xn-Gxn∥→0 as n→∞.
Indeed, in terms of Lemma11, from (130), we have
(146)∥yn-p∥2≤βn∥xn-p∥2+γn∥Bnxn-p∥2+δn∥SnGxn-p∥2≤βn∥xn-p∥2+γn∥xn-p∥2+δn∥xn-p∥2=∥xn-p∥2,∥xn+1-p∥2=∥αn(xn-p)+σn(Gxn-p)+(1-αn-σn)(yn-p)+αn(f(xn)-xn)∥2≤[∥αn(xn-p)+σn(Gxn-p)+(1-αn-σn)(yn-p)∥+∥αn(f(xn)-xn)∥]2=∥αn(xn-p)+σn(Gxn-p)+(1-αn-σn)(yn-p)∥2+∥αn(f(xn)-xn)∥×[2∥αn(xn-p)+σn(Gxn-p)+(1-αn-σn)(yn-p)∥+∥αn(f(xn)-xn)∥]=∥[σn1-αn(Gxn-p)+1-αn-σn1-αn(yn-p)]αn(xn-p)+(1-αn)×[σn1-αn(Gxn-p)+1-αn-σn1-αn(yn-p)]∥2+∥αn(f(xn)-xn)∥×[2∥αn(xn-p)+σn(Gxn-p)+(1-αn-σn)(yn-p)∥+∥αn(f(xn)-xn)∥]≤αn∥xn-p∥2+(1-αn)∥σn1-αn(Gxn-p)+1-αn-σn1-αn(yn-p)∥2+∥αn(f(xn)-xn)∥×[2(αn∥xn-p∥+σn∥Gxn-p∥+(1-αn-σn)∥yn-p∥)+∥αn(f(xn)-xn)∥]≤αn∥xn-p∥2+(1-αn)×[-σn(1-αn-σn)(1-αn)2g(∥Gxn-yn∥)σn1-αn∥Gxn-p∥2+1-αn-σn1-αn∥yn-p∥2-σn(1-αn-σn)(1-αn)2g(∥Gxn-yn∥)]+∥αn(f(xn)-xn)∥×[2(αn∥xn-p∥+σn∥xn-p∥+(1-αn-σn)∥xn-p∥)+∥αn(f(xn)-xn)∥]≤αn∥xn-p∥2+(1-αn)×[σn(1-αn-σn)(1-αn)2σn1-αn∥xn-p∥2+1-αn-σn1-αn∥xn-p∥2-σn(1-αn-σn)(1-αn)2g(∥Gxn-yn∥)]+∥αn(f(xn)-xn)∥(2∥xn-p∥+∥αn(f(xn)-xn)∥)=∥xn-p∥2-σn(1-αn-σn)1-αng(∥Gxn-yn∥)+∥αn(f(xn)-xn)∥(2∥xn-p∥+∥αn(f(xn)-xn)∥).
Then, it immediately follows from0≤αn+σn≤1-ρ, for all n≥n0 that
(147)ρσng(∥Gxn-yn∥)≤σn(1-αn-σn)1-αng(∥Gxn-yn∥)≤∥xn-p∥2-∥xn+1-p∥2+∥αn(f(xn)-xn)∥(2∥xn-p∥+∥αn(f(xn)-xn)∥)≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+∥αn(f(xn)-xn)∥×(2∥xn-p∥+∥αn(f(xn)-xn)∥),
for alln≥n0. Since ∥αn(f(xn)-xn)∥→0 and {xn} is bounded, we deduce from (145) and condition (ii) that
(148)limn→∞g(∥Gxn-yn∥)=0.
Utilizing the properties ofg, we have
(149)limn→∞∥Gxn-yn∥=0.
Also, from (130) we have
(150)xn+1-xn=αn(f(xn)-xn)+σn(Gxn-xn)+(1-αn-σn)(yn-xn)=αn(f(xn)-xn)+σn(Gxn-yn+yn-xn)+(1-αn-σn)(yn-xn)=αn(f(xn)-xn)+σn(Gxn-yn)+(1-αn)(yn-xn),
which hence leads to(151)ρ∥yn-xn∥≤(1-αn-σn)∥yn-xn∥≤(1-αn)∥yn-xn∥=∥xn+1-xn-αn(f(xn)-xn)-σn(Gxn-yn)∥≤∥xn+1-xn∥+∥αn(f(xn)-xn)∥+σn∥Gxn-yn∥≤∥xn+1-xn∥+∥αn(f(xn)-xn)∥+∥Gxn-yn∥.
So, it is easy to see from (145), (149), and ∥αn(f(xn)-xn)∥→0 that
(152)limn→∞∥yn-xn∥=0.
We note that(153)∥Gxn-xn∥≤∥Gxn-yn∥+∥yn-xn∥.
Therefore, from (149) and (152) it follows that
(154)limn→∞∥Gxn-xn∥=0.
Repeating the same arguments as those of (86), (89), and (91) in the proof of Theorem 24, we can obtain
(155)limn→∞∥xn-Bnxn∥=limn→∞∥SnGxn-Bnxn∥=limn→∞∥xn-Sxn∥=0.
Suppose thatβn≡β for some fixed β∈(0,1) such that β+γn+δn=1 for all n≥0. Define a mapping Vx=(1-θ1-θ2)Sx+θ1Bx+θ2Gx, where θ1,θ2∈(0,1) are two constants with θ1+θ2<1. Then, by Lemmas 14 and 17, we have that Fix(V)=Fix(S)∩Fix(B)∩Fix(G)=F. For each k≥1, let {pk} be a unique element of C such that
(156)pk=1kf(pk)+(1-1k)Vpk.
From Lemma13, we conclude that pk→q∈Fix(V)=F as k→∞. Observe that for every n,k(157)∥yn-Bpk∥≤β∥xn-Bpk∥+γn∥Bnxn-Bpk∥+δn(∥SnGxn-Bnxn∥+∥Bnxn-Bpk∥)=β∥xn-Bpk∥+(1-β)∥Bnxn-Bpk∥+δn∥SnGxn-Bnxn∥,
and hence(158)∥xn+1-Bpk∥≤αn∥f(xn)-Bpk∥+σn∥Gxn-Bpk∥+(1-αn-σn)∥yn-Bpk∥≤αn(∥f(xn)-xn∥+∥xn-Bpk∥)+σn(∥Gxn-xn∥+∥xn-Bpk∥)+(1-αn-σn)×[β∥xn-Bpk∥+(1-β)∥Bnxn-Bpk∥+δn∥SnGxn-Bnxn∥]≤αn(∥f(xn)-xn∥+∥xn-Bpk∥)+σn(∥Gxn-xn∥+∥xn-Bpk∥)+(1-αn-σn)×[β∥xn-Bpk∥+(1-β)(∥Bnxn-Bnpk∥+∥Bnpk-Bpk∥)+δn∥SnGxn-Bnxn∥]≤αn(∥f(xn)-xn∥+∥xn-Bpk∥)+σn(∥Gxn-xn∥+∥xn-Bpk∥)+(1-αn-σn)×[β∥xn-Bpk∥+(1-β)(∥xn-pk∥+∥Bnpk-Bpk∥)+δn∥SnGxn-Bnxn∥]=αn∥f(xn)-xn∥+σn∥Gxn-xn∥+[β+(αn+σn)(1-β)]∥xn-Bpk∥+(1-αn-σn)(1-β)∥xn-pk∥+(1-αn-σn)×[(1-β)∥Bnpk-Bpk∥+δn∥SnGxn-Bnxn∥]≤αn∥f(xn)-xn∥+σn∥Gxn-xn∥+[β+(αn+σn)(1-β)]×(∥xn-xn+1∥+∥xn+1-Bpk∥)+(1-αn-σn)(1-β)×(∥xn-xn+1∥+∥xn+1-pk∥)+(1-αn-σn)×[(1-β)∥Bnpk-Bpk∥+δn∥SnGxn-Bnxn∥]=αn∥f(xn)-xn∥+σn∥Gxn-xn∥+[β+(αn+σn)(1-β)]∥xn+1-Bpk∥+(1-αn-σn)(1-β)∥xn+1-pk∥+(1-αn-σn)×[(1-β)∥Bnpk-Bpk∥+δn∥SnGxn-Bnxn∥]+∥xn-xn+1∥.
So, it immediately follows from0≤αn≤1-ρ, for all n≥n0 that
(159)∥xn+1-Bpk∥≤∥xn+1-pk∥+∥Bnpk-Bpk∥+1(1-αn-σn)(1-β)×(∥αn(xn-f(xn))∥+σn∥Gxn-xn∥+∥xn-xn+1∥)+δn1-β∥SnGxn-Bnxn∥≤∥xn+1-pk∥+∥Bnpk-Bpk∥+∥SnGxn-Bnxn∥+1ρ(1-β)×(∥αn(xn-f(xn))∥+∥Gxn-xn∥+∥xn-xn+1∥∥αn(xn-f(xn))∥)=∥xn+1-pk∥+τn,∀n≥n0,
whereθn=∥Bnpk-Bpk∥+  ∥SnGxn-Bnxn∥+  1/(ρ(1-β))(∥αn(xn-f(xn))∥+  ∥Gxn-xn∥+  ∥xn-xn+1∥). Since limn→∞∥Bnpk-Bpk∥=  limn→∞∥SnGxn-Bnxn∥=  limn→∞∥αn(xn-f(xn))∥=  limn→∞∥Gxn-xn∥=  limn→∞∥xn-xn+1∥=  0, we know that τn→0 as n→∞.
From (159), we obtain
(160)∥xn+1-Bpk∥2≤∥xn+1-pk∥2+τn(2∥xn+1-pk∥+τn),∀n≥n0.
For any Banach limitμ, from (160) we derive
(161)μn∥xn-Bpk∥2=μn∥xn+1-Bpk∥2≤μn∥xn+1-pk∥2=μn∥xn-pk∥2.
Repeating the same arguments as those of (99), in the proof of Theorem 24, we can get
(162)μn∥xn-Gpk∥2≤μn∥xn-pk∥2,μn∥xn-Spk∥2≤μn∥xn-pk∥2.
Utilizing (161) and (162), we deduce that
(163)μn∥xn-Vpk∥2≤(1-θ1-θ2)μn∥xn-Spk∥2+θ1μn∥xn-Bpk∥2+θ2μn∥xn-Gpk∥2≤μn∥xn-pk∥2.
Also, observe that(164)(1-1k)(xn-Vpk)=xn-pk-1k(xn-f(pk)).
Repeating the same arguments as those of (106) in the proof of Theorem 24, we can get
(165)12kμn∥xn-pk∥2≥μn〈f(pk)-pk,J(xn-pk)〉.
Sincepk→q∈Fix(V)=F as k→∞, by the uniform Gateaux differentiability of the norm of X, we have
(166)μn〈f(q)-q,J(xn-q)〉≤0.
On the other hand, from (135) and the norm-to-weak* uniform continuity of J on bounded subsets of X, it follows that
(167)limn→∞|〈f(q)-q,J(xn+1-q)〉-〈f(q)-q,J(xn-q)〉|=0.
So, utilizing Lemma18, we deduce from (166) and (167) that
(168)limsupn→∞〈f(q)-q,J(xn-q)〉≤0,
which, together with (135) and the norm-to-norm uniform continuity of J on bounded subsets of X, implies that
(169)limsupn→∞〈f(q)-q,J(xn+1-q)〉≤0.
Finally, let us show thatxn→q as n→∞. Utilizing Lemma 8 (i), from (130) and the convexity of ∥·∥2, we get
(170)∥yn-q∥2≤βn∥xn-q∥2+γn∥Bnxn-q∥2+δn∥SnGxn-q∥2≤∥xn-q∥2,(171)∥xn+1-q∥2=∥αn(f(xn)-f(q))+σn(Gxn-q)+(1-αn-σn)(yn-q)+αn(f(q)-q)(f(xn)-f(q))∥2≤∥αn(f(xn)-f(q))+σn(Gxn-q)+(1-αn-σn)(yn-q)(f(xn)-f(q))∥2+2αn〈f(q)-q,J(xn+1-q)〉≤αn∥f(xn)-f(q)∥2+σn∥Gxn-q∥2+(1-αn-σn)∥yn-q∥2+2αn〈f(q)-q,J(xn+1-q)〉≤αnρ∥xn-q∥2+σn∥xn-q∥2+(1-αn-σn)∥xn-q∥2+2αn〈f(q)-q,J(xn+1-q)〉=(1-αn(1-ρ))∥xn-q∥2+2αn〈f(q)-q,J(xn+1-q)〉=(1-αn(1-ρ))∥xn-q∥2+αn(1-ρ)2〈f(q)-q,J(xn+1-q)〉1-ρ.
Applying Lemma7 to (171), we obtain that xn→q as n→∞.
Conversely, ifxn→q∈F as n→∞, then from (130) it follows that
(172)∥yn-q∥≤βn∥xn-q∥+γn∥Bnxn-q∥+δn∥SnGxn-q∥≤∥xn-q∥⟶0
asn→∞; that is, yn→q. Again from (130) we obtain that
(173)∥αn(f(xn)-xn)∥=∥xn+1-xn-σn(Gxn-xn)-(1-αn-σn)(yn-xn)∥≤∥xn+1-xn∥+σn∥Gxn-xn∥+(1-αn-σn)∥yn-xn∥≤∥xn+1-q∥+∥xn-q∥+σn(∥Gxn-q∥+∥xn-q∥)+(1-αn-σn)(∥yn-q∥+∥xn-q∥)≤∥xn+1-q∥+∥xn-q∥+σn(∥xn-q∥+∥xn-q∥)+(1-αn-σn)(∥yn-q∥+∥xn-q∥)≤∥xn+1-q∥+3∥xn-q∥+∥yn-q∥.
Sincexn→q and yn→q, we get αn(f(xn)-xn)→0. This completes the proof.Corollary 29.
LetC be a nonempty closed convex subset of a uniformly convex Banach space X which has a uniformly Gateaux differentiable norm. Let ΠC be a sunny nonexpansive retraction from X onto C. Let {ρn}n=0∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and Ai:C→Xξi-strictly pseudocontractive and α^i-strongly accretive with ξi+α^i≥1 for each i=0,1,…. Define a mapping Gi:C→C by ΠC(I-λiAi)x=Gix for all x∈C and i=0,1,…, where 1-(ξi/(1+ξi))(1-(1-α^i)/ξi)≤λi≤1 for all i=0,1,…. Let Bn:C→C be the W-mapping generated by Gn,Gn-1,…,G0 and ρn,ρn-1,…,ρ0. Let V:C→C be a self-mapping such that I-V:C→X is λ-strictly pseudocontractive and α-strongly accretive with α+λ≥1. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let {Si}i=0∞ be a countable family of nonexpansive mappings of C into itself such that F=(⋂i=0∞‍Fix(Si))∩Fix(V)∩(⋂i=0∞‍VI(C,Ai))≠∅. For arbitrarily given x0∈C, let {xn} be the sequence generated by
(174)yn=βnxn+γnBnxn+δnSn((1-l)I+lV)xn,xn+1=αnf(xn)+σn((1-l)I+lV)xn+(1-αn-σn)yn,∀n≥0,
where1-(λ/(1+λ))(1-(1-α)/λ)≤l≤1 and {σn}, {αn}, {βn}, {γn}, and {δn} are the sequences in [0,1] such that βn+γn+δn=1 and αn+σn≤1 for all n≥0. Suppose that the following conditions hold:(i)
∑n=0∞‍αn=∞ and 0≤αn+σn≤1-ρ, for all n≥n0 for some integer n0≥0;(ii)
liminfn→∞σn>0, liminfn→∞γn>0 and liminfn→∞δn>0;(iii)
limn→∞(|αn+1/(1-(1-αn+1-σn+1)βn+1)-αn/(1-(1-αn-σn)βn)|+  |σn+1/(1-(1-αn+1-σn+1)βn+1)-σn/(1-(1-αn-σn)βn)|+  |δn+1/(1-βn+1)-δn/(1-βn)|)=  0;(iv)
0<liminfn→∞βn≤limsupn→∞βn<1.
Assume that∑n=0∞‍supx∈D∥Sn+1x-Snx∥<∞ for any bounded subset D of C and let S be a mapping of C into itself defined by Sx=limn→∞Snx for all x∈C and suppose that Fix(S)=⋂i=0∞‍Fix(Si). Then there hold the following:(I)
limn→∞∥xn+1-xn∥=0;(II)
xn→q⇔αn(f(xn)-xn)→0 provided βn≡β for some fixed β∈(0,1), where q∈F solves the following VIP
(175)〈q-f(q),J(q-p)〉≤0,∀p∈F.Proof.
In Theorem28, we put B1=I-V, B2=0, and μ1=l, where 1-(λ/(1+λ))(1-(1-α)/λ)≤l≤1. Then, GSVI (13) is equivalent to the VIP of finding x*∈C such that
(176)〈B1x*,J(x-x*)〉≥0,∀x∈C.
In this case,B1:C→X is λ-strictly pseudocontractive and α-strongly accretive. Repeating the same arguments as those in the proof of Corollary 25, we can infer that Fix(V)=VI(C,B1). Accordingly, F=⋂i=0∞‍Fix(Si)∩Ω∩(⋂i=0∞‍VI(C,Ai))=⋂i=0∞‍Fix(Si)∩Fix(V)∩(⋂i=0∞‍VI(C,Ai)), and
(177)Gxn=((1-l)I+lV)xn,∀n≥0.
So, scheme (130) reduces to (174). Therefore, the desired result follows from Theorem 31.Remark 30.
Our Theorems24 and 28 improve, extend, supplement and develop Ceng and Yao’s [10, Theorem  3.2], Cai and Bu’s [11, Theorem  3.1], Kangtunyakarn’s [38, Theorem  3.1], and Ceng and Yao’s [8, Theorem  3.1], in the following aspects.(i)
The problem of finding a pointq∈(⋂i=0∞‍Fix(Si))∩Ω∩(⋂i=0∞‍VI(C,Ai)) in our Theorems 24 and 28 is more general and more subtle than every one of the problem of finding a point q∈⋂i=0∞‍Fix(Ti) in [10, Theorem  3.2], the problem of finding a point q∈⋂i=1∞‍Fix(Ti)∩Ω in [11, Theorem  3.1], the problem of finding a point q∈Fix(S)∩Fix(V)∩(⋂i=1N‍VI(C,Ai)) in [38, Theorem  3.1], and the problem of finding a point q∈Fix(T) in [8, Theorem  3.1].(ii)
The iterative scheme in [8, Theorem  3.1] is extended to develop the iterative schemes (42) and (130) in our Theorems 24 and 28 by virtue of the iterative schemes of [11, Theorem  3.1] and [10, Theorems  3.2]. The iterative schemes (42) and (130) in our Theorems 24 and 28 are more advantageous and more flexible than the iterative scheme of [8, Theorem  3.1] because they can be applied to solving three problems (i.e., GSVI (13), fixed point problem and infinitely many VIPs), and involve several parameter sequences {αn}, {βn}, {γn}, {δn}, (and {σn}).(iii)
Our Theorems24 and 28 extend and generalize Ceng and Yao [8, Theorem  3.1] from a nonexpansive mapping to a countable family of nonexpansive mappings, and Ceng and Yao’s [10, Theorems  3.2], to the setting of the GSVI (13) and infinitely many VIPs, Kangtunyakarn [38, Theorem  3.1], from finitely many VIPs to infinitely many VIPs, from a nonexpansive mapping to a countable family of nonexpansive mappings and from a strict pseudocontraction to the GSVI (13). In the meantime, our Theorems 24 and 28 extend and generalize Cai and Bu’s [11, Theorem  3.1], to the setting of infinitely many VIPs.(iv)
The iterative schemes (42) and (130) in our Theorems 24 and 28 are very different from every one in [10, Theorem  3.2], [11, Theorem  3.1], [38, Theorem  3.1], and [8, Theorem  3.1] because the mappings G and Tn in [11, Theorem  3.1] and the mapping T in [8, Theorem  3.1] are replaced with the same composite mapping SnG in the iterative schemes (42) and (130) and the mapping Wn in [10, Theorem  3.2] is replaced with Bn.(v)
Cai and Bu’s proof in [11, Theorem  3.1] depends on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), and the inequality in smooth and uniform convex Banach spaces (see Proposition 6). Because the composite mapping SnG appears in the iterative scheme (42) of our Theorem 24, the proof of our Theorem 24 depends on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), the inequality in smooth and uniform convex Banach spaces (see Proposition 6), the inequality in uniform convex Banach spaces (see Lemma 15 in Section 2 of this paper), and the properties of the W-mapping and the Banach limit (see Lemmas 16–18 in Section 2 of this paper). However, the proof of our Theorem 28 does not depend on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), and the inequality in smooth and uniform convex Banach spaces (see Proposition 6). It depends on only the inequality in uniform convex Banach spaces (see Lemma 15 in Section 2 of this paper) and the properties of the W-mapping and the Banach limit (see Lemmas 16–18 in Section 2 of this paper).(vi)
The assumption of the uniformly convex and2-uniformly smooth Banach space X in [11, Theorem  3.1] is weakened to the one of the uniformly convex Banach space X having a uniformly Gateaux differentiable norm in our Theorem 28. Moreover, the assumption of the uniformly smooth Banach space X in [8, Theorem  3.1] is replaced with the one of the uniformly convex Banach space X having a uniformly Gateaux differentiable norm in our Theorem 28. It is worth emphasizing that there is no assumption on the convergence of parameter sequences {αn}, {βn}, {γn}, and {δn} (and {σn}) to zero in our Theorems 24 and 28.


## 4. Relaxed Mann Iterations and Their Convergence Criteria

In this section, we introduce our relaxed Mann iteration algorithms in real smooth and uniformly convex Banach spaces and present their convergence criteria.Theorem 31.
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C. Let {ρn}n=0∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and Ai:C→X an α^i-inverse strongly accretive mapping for each i=0,1,…. Define a mapping Gi:C→C by ΠC(I-λiAi)x=Gix for all x∈C and i=0,1,…, where λi∈(0,α^i/κ2] and κ is the 2-uniformly smooth constant of X. Let Bn:C→C be the W-mapping generated by Gn,Gn-1,…,G0 and ρn,ρn-1,…,ρ0. Let the mapping Bi:C→X be β^i-inverse strongly accretive for i=1,2. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let {Si}i=0∞ be a countable family of nonexpansive mappings of C into itself such that F=(⋂i=0∞‍Fix(Si))∩Ω∩(⋂i=0∞‍VI(C,Ai))≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2) with 0<μi<β^i/κ2 for i=1,2. For arbitrarily given x0∈C, let {xn} be the sequence generated by
(178)xn+1=αnf(xn)+βnxn+γnBnxn+δnSnGxn,∀n≥0,
where{αn}, {βn}, {γn}, and {δn} are the sequences in (0,1) such that αn+βn+γn+δn=1 for all n≥0. Suppose that the following conditions hold:(i)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(ii)
{γn},{δn}⊂[c,d] for some c,d∈(0,1);(iii)
limn→∞(|βn-βn-1|+|γn-γn-1|+|δn-δn-1|)=0;(iv)
0<liminfn→∞βn≤limsupn→∞βn<1.
Assume that∑n=1∞‍supx∈D∥Snx-Sn-1x∥<∞ for any bounded subset D of C and let S be a mapping of C into itself defined by Sx=limn→∞Snx for all x∈C and suppose that Fix(S)=⋂i=0∞‍Fix(Si). Then, there hold the following:(I)
limn→∞∥xn+1-xn∥=0;(II)
the sequence{xn}n=0∞ converges strongly to some q∈F which is the unique solution of the variational inequality problem (VIP)
(179)〈(I-f)q,J(q-p)〉≤0,∀p∈F,Proof.
First of all, since0<λi<[α^i/κ2] for i=0,1,…, it is easy to see that Gi is a nonexpansive mapping for each i=0,1,…. Since Bn:C→C is the W-mapping generated by Gn,Gn-1,…,G0, and ρn,ρn-1,…,ρ0, by Lemma 16 we know that, for each x∈C and k≥0, the limit limn→∞Un,kx exists. Moreover, one can define a mapping B:C→C as follows:
(180)Bx=limn→∞Bnx=limn→∞Un,0x
for everyx∈C. That is, such a B is the W-mapping generated by the sequences {Gn}n=0∞ and {ρn}n=0∞. According to Lemma 17, we know that Fix(B)=⋂i=0∞‍Fix(Gi). From Lemma 21 and the definition of Gi, we have Fix(Gi)=VI(C,Ai) for each i=0,1,…. Hence, we have
(181)Fix(B)=⋂i=0∞‍Fix(Gi)=⋂i=0∞‍VI(C,Ai).
Next, let us show that the sequence{xn} is bounded. Indeed, take a fixed p∈F arbitrarily. Then, we get p=Gp, p=Bnp, and p=Snp for all n≥0. By Lemma 23, we know that G is nonexpansive. Then, from (178), we have
(182)∥xn+1-p∥≤αn∥f(xn)-p∥+βn∥xn-p∥+γn∥Bnxn-p∥+δn∥SnGxn-p∥≤αn(∥f(xn)-f(p)∥+∥f(p)-p∥)+βn∥xn-p∥+γn∥xn-p∥+δn∥Gxn-p∥≤αn(ρ∥xn-p∥+∥f(p)-p∥)+βn∥xn-p∥+γn∥xn-p∥+δn∥xn-p∥=(1-αn(1-ρ))∥xn-p∥+αn(1-ρ)∥f(p)-p∥1-ρ≤max{∥xn-p∥,∥f(p)-p∥1-ρ}.
By induction, we obtain(183)∥xn-p∥≤max{∥x0-p∥,∥f(p)-p∥1-ρ},∀n≥0.
Hence,{xn} is bounded, and so are the sequences {Gxn} and {f(xn)}.
Let us show that(184)limn→∞∥xn+1-xn∥=0.
As a matter of fact, observe thatxn+1 can be rewritten as follows:
(185)xn+1=βnxn+(1-βn)zn,
wherezn=(αnf(xn)+γnBnxn+δnSnGxn)/(1-βn). Observe that
(186)∥zn-zn-1∥=∥αnf(xn)+γnBnxn+δnSnGxn1-βn-αn-1f(xn-1)+γn-1Bn-1xn-1+δn-1Sn-1Gxn-11-βn-1∥=∥xn+1-βnxn1-βn-xn-βn-1xn-11-βn-1∥=∥xn+1-βnxn1-βn-xn-βn-1xn-11-βn+xn-βn-1xn-11-βn-xn-βn-1xn-11-βn-1∥≤∥xn+1-βnxn1-βn-xn-βn-1xn-11-βn∥+∥xn-βn-1xn-11-βn-xn-βn-1xn-11-βn-1∥=11-βn∥xn+1-βnxn-(xn-βn-1xn-1)∥+|11-βn-11-βn-1|∥xn-βn-1xn-1∥=11-βn∥xn+1-βnxn-(xn-βn-1xn-1)∥+|βn-βn-1|(1-βn-1)(1-βn)∥xn-βn-1xn-1∥=11-βn∥αnf(xn)+γnBnxn+δnSnGxn-αn-1f(xn-1)-γn-1Bn-1xn-1-δn-1Sn-1Gxn-1∥+|βn-βn-1|(1-βn-1)(1-βn)∥xn-βn-1xn-1∥≤11-βn[αn∥f(xn)-f(xn-1)∥+γn∥Bnxn-Bn-1xn-1∥+δn∥SnGxn-Sn-1Gxn-1∥+|αn-αn-1|∥f(xn-1)∥+|γn-γn-1|∥Bn-1xn-1∥+|δn-δn-1|∥Sn-1Gxn-1∥]+|βn-βn-1|(1-βn-1)(1-βn)∥xn-βn-1xn-1∥.
On the other hand, we note that, for alln≥1,
(187)∥SnGxn-Sn-1Gxn-1∥≤∥SnGxn-SnGxn-1∥+∥SnGxn-1-Sn-1Gxn-1∥≤∥Gxn-Gxn-1∥+∥SnGxn-1-Sn-1Gxn-1∥≤∥xn-xn-1∥+∥SnGxn-1-Sn-1Gxn-1∥.
Furthermore, by(CY), since Gi and Un,i are nonexpansive, we deduce that for each n≥1(188)∥Bnxn-Bn-1xn-1∥≤∥Bnxn-Bnxn-1∥+∥Bnxn-1-Bn-1xn-1∥≤∥xn-xn-1∥+∥Bnxn-1-Bn-1xn-1∥=∥xn-xn-1∥+∥λ0G0Un,1xn-1-λ0G0Un-1,1xn-1∥≤∥xn-xn-1∥+λ0∥Un,1xn-1-Un-1,1xn-1∥=∥xn-xn-1∥+λ0∥λ1G1Un,2xn-1-λ1G1Un-1,2xn-1∥≤∥xn-xn-1∥+λ0λ1∥Un,2xn-1-Un-1,2xn-1∥⋮≤∥xn-xn-1∥+(∏i=0n-1‍λi)∥Un,nxn-1-Un-1,nxn-1∥≤∥xn-xn-1∥+M∏i=0n-1‍λi,
for some constantM>0. Taking into account 0<liminfn→∞βn≤limsupn→∞βn<1, we may assume, without loss of generality, that {βn}⊂[c^,d^]. Utilizing (186)–(188), we have
(189)∥zn-zn-1∥≤11-βn[αn∥f(xn)-f(xn-1)∥+γn∥Bnxn-Bn-1xn-1∥+δn∥SnGxn-Sn-1Gxn-1∥+|αn-αn-1|∥f(xn-1)∥+|γn-γn-1|∥Bn-1xn-1∥+|δn-δn-1|∥Sn-1Gxn-1∥]+|βn-βn-1|(1-βn-1)(1-βn)∥xn-βn-1xn-1∥≤11-βn{αnρ∥xn-xn-1∥+γn[∥xn-xn-1∥+M∏i=0n-1‍λi]+δn[∥xn-xn-1∥+∥SnGxn-1-Sn-1Gxn-1∥]+|αn-αn-1|∥f(xn-1)∥+|γn-γn-1|∥Bn-1xn-1∥+|δn-δn-1|∥Sn-1Gxn-1∥[∥xn-xn-1∥+M∏i=0n-1‍λi]}+|βn-βn-1|(1-βn-1)(1-βn)∥xn-βn-1xn-1∥=11-βn{∏i=0n-1‍(1-βn-αn(1-ρ))∥xn-xn-1∥+γnM∏i=0n-1‍λi+δn∥SnGxn-1-Sn-1Gxn-1∥+|αn-αn-1|∥f(xn-1)∥+|γn-γn-1|∥Bn-1xn-1∥+|δn-δn-1|∥Sn-1Gxn-1∥∏i=0n-1‍}+|βn-βn-1|(1-βn-1)(1-βn)∥xn-βn-1xn-1∥=(1-αn(1-ρ)1-βn)∥xn-xn-1∥+γnM1-βn∏i=0n-1‍λi+δn1-βn∥SnGxn-1-Sn-1Gxn-1∥+11-βn×[|αn-αn-1|∥f(xn-1)∥+|γn-γn-1|∥Bn-1xn-1∥+|δn-δn-1|∥Sn-1Gxn-1∥]+|βn-βn-1|(1-βn-1)(1-βn)∥xn-βn-1xn-1∥≤∥xn-xn-1∥+M∏i=0n-1‍λi+∥SnGxn-1-Sn-1Gxn-1∥+11-βn[|αn-αn-1|∥f(xn-1)∥+|γn-γn-1|×∥Bn-1xn-1∥+|δn-δn-1|∥Sn-1Gxn-1∥]+|βn-βn-1|(1-βn-1)(1-βn)×∥αn-1f(xn-1)+γn-1Bn-1xn-1+δn-1Sn-1Gxn-1∥≤∥xn-xn-1∥+M1[∏i=0n-1‍λi+|αn-αn-1|+|γn-γn-1|+|δn-δn-1|+|βn-βn-1|∏i=0n-1‍λi+|αn-αn-1|]+∥SnGxn-1-Sn-1Gxn-1∥,
wheresupn≥0{(1/(1-d^)2)(∥f(xn)∥+∥Bnxn∥+∥SnGxn∥+M)}≤M1 for some M1>0. Thus, from (189), conditions (i), (iii) and the assumption on {Sn}, it follows that (noting that 0<λi≤b<1, for all i≥0)
(190)limn→∞(∥zn-zn-1∥-∥xn-xn-1∥)≤0.
Since0<liminfn→∞βn≤limsupn→∞βn<1, by Lemma 20 we get
(191)limn→∞∥xn-zn∥=0.
Consequently,(192)limn→∞∥xn+1-xn∥=limn→∞(1-βn)∥zn-xn∥=0.
Next we show that∥xn-Gxn∥→0 as n→∞.
Indeed, for simplicity, putq=ΠC(p-μ2B2p), un=ΠC(xn-μ2B2xn) and vn=ΠC(un-μ1B1un). Then, vn=Gxn for all n≥0. From Lemma 26 we have
(193)∥un-q∥2=∥ΠC(xn-μ2B2xn)-ΠC(p-μ2B2p)∥2≤∥xn-p-μ2(B2xn-B2p)∥2≤∥xn-p∥2-2μ2(β^2-κ2μ2)∥B2xn-B2p∥2,(194)∥vn-p∥2=∥ΠC(un-μ1B1un)-ΠC(q-μ1B1q)∥2≤∥un-q-μ1(B1un-B1q)∥2≤∥un-q∥2-2μ1(β^1-κ2μ1)∥B1un-B1q∥2.
Substituting (193) for (194), we obtain
(195)∥vn-p∥2≤∥xn-p∥2-2μ2(β^2-κ2μ2)∥B2xn-B2p∥2-2μ1(β^1-κ2μ1)∥B1un-B1q∥2.
By Lemma8, we have from (178) and (195)
(196)∥xn+1-p∥2=∥αn(f(xn)-f(p))+βn(xn-p)+γn(Bnxn-p)+δn(SnGxn-p)+αn(f(p)-p)∥2≤∥αn(f(xn)-f(p))+βn(xn-p)+γn(Bnxn-p)+δn(SnGxn-p)∥2+2αn〈f(p)-p,J(xn+1-p)〉≤αn∥f(xn)-f(p)∥2+βn∥xn-p∥2+γn∥Bnxn-p∥2+δn∥SnGxn-p∥2+2αn〈f(p)-p,J(xn+1-p)〉≤αnρ2∥xn-p∥2+βn∥xn-p∥2+γn∥xn-p∥2+δn∥vn-p∥2+2αn〈f(p)-p,J(xn+1-p)〉≤αnρ∥xn-p∥2+βn∥xn-p∥2+γn∥xn-p∥2+δn[∥xn-p∥2-2μ2(β^2-κ2μ2)∥B2xn-B2p∥2-2μ1(β^1-κ2μ1)∥B1un-B1q∥2]+2αn〈f(p)-p,J(xn+1-p)〉=(1-αn(1-ρ))∥xn-p∥2-2δn[μ2(β^2-κ2μ2)∥B2xn-B2p∥2+μ1(β^1-κ2μ1)∥B1un-B1q∥2]+2αn〈f(p)-p,J(xn+1-p)〉≤∥xn-p∥2-2δn[μ2(β^2-κ2μ2)∥B2xn-B2p∥2+μ1(β^1-κ2μ1)∥B1un-B1q∥2]+2αn∥f(p)-p∥∥xn+1-p∥,
which hence implies that(197)2δn[μ2(β^2-κ2μ2)∥B2xn-B2p∥2+μ1(β^1-κ2μ1)∥B1un-B1q∥2]≤∥xn-p∥2-∥xn+1-p∥2+2αn∥f(p)-p∥∥xn+1-p∥≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+2αn∥f(p)-p∥∥xn+1-p∥.
Since∥xn-xn+1∥→0, 0<μi<β^i/κ2 for i=1,2, and {xn} is bounded, we obtain from conditions (i), (ii) that
(198)limn→∞∥B2xn-B2p∥=0,limn→∞∥B1un-B1q∥=0.
Utilizing Proposition6 and Lemma 9, we have
(199)∥un-q∥2=∥ΠC(xn-μ2B2xn)-ΠC(p-μ2B2p)∥2≤〈xn-μ2B2xn-(p-μ2B2p),J(un-q)〉=〈xn-p,J(un-q)〉+μ2〈B2p-B2xn,J(un-q)〉≤12[∥xn-p∥2+∥un-q∥2-g1(∥xn-un-(p-q)∥)∥xn-p∥2+∥un-q∥2]+μ2∥B2p-B2xn∥∥un-q∥,
which implies that(200)∥un-q∥2≤∥xn-p∥2-g1(∥xn-un-(p-q)∥)+2μ2∥B2p-B2xn∥∥un-q∥.
In the same way, we derive(201)∥vn-p∥2=∥ΠC(un-μ1B1un)-ΠC(q-μ1B1q)∥2≤〈un-μ1B1un-(q-μ1B1q),J(vn-p)〉=〈un-q,J(vn-p)〉+μ1〈B1q-B1un,J(vn-p)〉≤12[∥un-q∥2+∥vn-p∥2-g2(∥un-vn+(p-q)∥)∥un-q∥2+∥vn-p∥2]+μ1∥B1q-B1un∥∥vn-p∥,
which implies that(202)∥vn-p∥2≤∥un-q∥2-g2(∥un-vn+(p-q)∥)+2μ1∥B1q-B1un∥∥vn-p∥.
Substituting (200) for (202), we get
(203)∥vn-p∥2≤∥xn-p∥2-g1(∥xn-un-(p-q)∥)-g2(∥un-vn+(p-q)∥)+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥.
By Lemma8, we have from (196) and (203)
(204)∥xn+1-p∥2≤αnρ2∥xn-p∥2+βn∥xn-p∥2+γn∥xn-p∥2+δn∥vn-p∥2+2αn〈f(p)-p,J(xn+1-p)〉≤αnρ∥xn-p∥2+βn∥xn-p∥2+γn∥xn-p∥2+δn[∥xn-p∥2-g1(∥xn-un-(p-q)∥)-g2(∥un-vn+(p-q)∥)+2μ2∥B2p-B2xn∥×∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥∥xn-p∥2-g1(∥xn-un-(p-q)∥)]+2αn〈f(p)-p,J(xn+1-p)〉≤(1-αn(1-ρ))∥xn-p∥2-δn[g1(∥xn-un-(p-q)∥)+g2(∥un-vn+(p-q)∥)]+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥×∥vn-p∥+2αn∥f(p)-p∥∥xn+1-p∥≤∥xn-p∥2-δn[g1(∥xn-un-(p-q)∥)+g2(∥un-vn+(p-q)∥)]+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥×∥vn-p∥+2αn∥f(p)-p∥∥xn+1-p∥,
which hence leads to(205)δn[g1(∥xn-un-(p-q)∥)+g2(∥un-vn+(p-q)∥)]≤∥xn-p∥2-∥xn+1-p∥2+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥+2αn∥f(p)-p∥∥xn+1-p∥≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥+2αn∥f(p)-p∥∥xn+1-p∥.
From (198), (205), conditions (i), (ii) and the boundedness of {xn}, {un}, and {vn}, we deduce that
(206)limn→∞g1(∥xn-un-(p-q)∥)=0,limn→∞g2(∥un-vn+(p-q)∥)=0.
Utilizing the properties ofg1 and g2, we deduce that
(207)limn→∞∥xn-un-(p-q)∥=0,limn→∞∥un-vn+(p-q)∥=0.
From (207), we get
(208)∥xn-vn∥≤∥xn-un-(p-q)∥+∥un-vn+(p-q)∥⟶0asn→∞.
That is,(209)limn→∞∥xn-Gxn∥=0.
Next, let us show that(210)limn→∞∥SnGxn-xn∥=0,limn→∞∥Bnxn-xn∥=0.
Indeed, observe thatxn+1 can be rewritten as follows:
(211)xn+1=αnf(xn)+βnxn+γnBnxn+δnSnGxn=αnf(xn)+βnxn+(γn+δn)×γnBnxn+δnSnGxnγn+δn=αnf(xn)+βnxn+enz^n,
whereen=γn+δn and z^n=(γnBnxn+δnSnGxn)/(γn+δn). Utilizing Lemma 11 and (211), we have
(212)∥xn+1-p∥2=∥αn(f(xn)-p)+βn(xn-p)+en(z^n-p)∥2≤αn∥f(xn)-p∥2+βn∥xn-p∥2+en∥z^n-p∥2-βneng3(∥z^n-xn∥)=αn∥f(xn)-p∥2+βn∥xn-p∥2-βneng3(∥z^n-xn∥)+en∥γnBnxn+δnSnGxnγn+δn-p∥2=αn∥f(xn)-p∥2+βn∥xn-p∥2-βneng3(∥z^n-xn∥)+en∥γnγn+δn(Bnxn-p)+δnγn+δn(SnGxn-p)∥2≤αn∥f(xn)-p∥2+βn∥xn-p∥2-βneng3(∥z^n-xn∥)+en[γnγn+δn∥Bnxn-p∥2+δnγn+δn∥SnGxn-p∥2]≤αn∥f(xn)-p∥2+βn∥xn-p∥2-βneng3(∥z^n-xn∥)+en[γnγn+δn∥xn-p∥+δnγn+δn∥Gxn-p∥2]≤αn∥f(xn)-p∥2+βn∥xn-p∥2-βneng3(∥z^n-xn∥)+en[γnγn+δn∥xn-p∥+δnγn+δn∥xn-p∥2]=αn∥f(xn)-p∥2+(1-αn)∥xn-p∥2-βneng3(∥z^n-xn∥)≤αn∥f(xn)-p∥2+∥xn-p∥2-βneng3(∥z^n-xn∥),
which hence implies that(213)βneng3(∥z^n-xn∥)≤αn∥f(xn)-p∥2+∥xn-p∥2-∥xn+1-p∥2≤αn∥f(xn)-p∥2+(∥xn-p∥+∥xn+1-p∥)×∥xn-xn+1∥.
Utilizing (184), conditions (i), (ii), (iv), and the boundedness of {xn} and {f(xn)}, we get
(214)limn→∞g3(∥z^n-xn∥)=0.
From the properties ofg3, we have
(215)limn→∞∥z^n-xn∥=0.
Utilizing Lemma15 and the definition of z^n, we have
(216)∥z^n-p∥2=∥γnBnxn+δnSnGxnγn+δn-p∥2=∥γnγn+δn(Bnxn-p)+δnγn+δn(SnGxn-p)∥2≤γnγn+δn∥Bnxn-p∥2+δnγn+δn∥SnGxn-p∥2-γnδn(γn+δn)2g4(∥SnGxn-Bnxn∥)≤γnγn+δn∥xn-p∥2+δnγn+δn∥Gxn-p∥2-γnδn(γn+δn)2g4(∥SnGxn-Bnxn∥)≤γnγn+δn∥xn-p∥2+δnγn+δn∥xn-p∥2-γnδn(γn+δn)2g4(∥SnGxn-Bnxn∥)=∥xn-p∥2-γnδn(γn+δn)2g4(∥SnGxn-Bnxn∥),
which hence yields(217)γnδn(γn+δn)2g4(∥SnGxn-Bnxn∥)≤∥xn-p∥2-∥z^n-p∥2≤(∥xn-p∥+∥z^n-p∥)∥xn-z^n∥.
Since{xn} and {z^n} are bounded and ∥z^n-xn∥→0 as n→∞, we deduce from condition (ii) that
(218)limn→∞g4(∥SnGxn-Bnxn∥)=0.
From the properties ofg4, we have
(219)limn→∞∥SnGxn-Bnxn∥=0.
On the other hand,xn+1 can also be rewritten as follows:
(220)xn+1=αnf(xn)+βnxn+γnBnxn+δnSnGxnxn+1=βnxn+γnBnxn+(αn+δn)αnf(xn)+δnSnGxnαn+δnxn+1=βnxn+γnBnxn+dnz~n,
wheredn=αn+δn and z~n=(αnf(xn)+δnSnGxn)/(αn+δn). Utilizing Lemma 11 and the convexity of ∥·∥2, we have
(221)∥xn+1-p∥2=∥βn(xn-p)+γn(Bnxn-p)+dn(z~n-p)∥2≤βn∥xn-p∥2+γn∥Bnxn-p∥2+dn∥z~n-p∥2-βnγng5(∥xn-Bnxn∥)=βn∥xn-p∥2+γn∥Bnxn-p∥2+dn∥αnf(xn)+δnSnGxnαn+δn-p∥2-βnγng5(∥xn-Bnxn∥)=βn∥xn-p∥2+γn∥Bnxn-p∥2+dn∥αnαn+δn(f(xn)-p)+δnαn+δn(SnGxn-p)∥2-βnγng5(∥xn-Bnxn∥)≤βn∥xn-p∥2+γn∥xn-p∥2+dn[αnαn+δn∥f(xn)-p∥2+δnαn+δn∥SnGxn-p∥2]-βnγng5(∥xn-Bnxn∥)≤βn∥xn-p∥2+γn∥xn-p∥2+dn[αnαn+δn∥f(xn)-p∥2+δnαn+δn∥Gxn-p∥2]-βnγng5(∥xn-Bnxn∥)≤βn∥xn-p∥2+γn∥xn-p∥2+dn[αnαn+δn∥f(xn)-p∥2+δnαn+δn∥xn-p∥2]-βnγng5(∥xn-Bnxn∥)=αn∥f(xn)-p∥2+(1-αn)∥xn-p∥2-βnγng5(∥xn-Bnxn∥)≤αn∥f(xn)-p∥2+∥xn-p∥2-βnγng5(∥xn-Bnxn∥),
which hence implies that(222)βnγng5(∥xn-Bnxn∥)≤αn∥f(xn)-p∥2+∥xn-p∥2-∥xn+1-p∥2≤αn∥f(xn)-p∥2+(∥xn-p∥+∥xn+1-p∥)×∥xn-xn+1∥.
From (184), conditions (i), (ii), (iv), and the boundedness of {xn} and {f(xn)}, we have
(223)limn→∞g5(∥xn-Bnxn∥)=0.
Utilizing the properties ofg5, we have
(224)limn→∞∥xn-Bnxn∥=0,
which, together with (219), implies that
(225)∥SnGxn-xn∥≤∥SnGxn-Bnxn∥+∥Bnxn-xn∥⟶0asn→∞.
That is,(226)limn→∞∥SnGxn-xn∥=0.
We note that(227)∥xn-Sxn∥≤∥xn-SnGxn∥+∥SnGxn-Snxn∥+∥Snxn-Sxn∥≤∥xn-SnGxn∥+∥Gxn-xn∥+∥Snxn-Sxn∥.
So, in terms of (209), (226), and Lemma 12, we have
(228)limn→∞∥xn-Sxn∥=0.
Suppose thatβn≡β for some fixed β∈(0,1) such that αn+β+γn+δn=1 for all n≥0. Define a mapping Vx=(1-θ1-θ2)Sx+θ1Bx+θ2Gx, where θ1,θ2∈(0,1) are two constants with θ1+θ2<1. Then by Lemmas 14 and 17, we have that Fix(V)=Fix(S)∩Fix(B)∩Fix(G)=F. For each k≥1, let {pk} be a unique element of C such that
(229)pk=1kf(pk)+(1-1k)Vpk.
From Lemma13, we conclude that pk→q∈Fix(V)=F as k→∞. Observe that for every n,k(230)∥xn+1-Bpk∥=∥αn(f(xn)-Bpk)+β(xn-Bpk)+γn(Bnxn-Bpk)+δn(SnGxn-Bpk)∥≤αn∥f(xn)-Bpk∥+β∥xn-Bpk∥+γn∥Bnxn-Bpk∥+δn(∥SnGxn-Bnxn∥+∥Bnxn-Bpk∥)=αn∥f(xn)-Bpk∥+β∥xn-Bpk∥+(γn+δn)∥Bnxn-Bpk∥+δn∥SnGxn-Bnxn∥=αn∥f(xn)-Bpk∥+β∥xn-Bpk∥+(1-αn-β)∥Bnxn-Bpk∥+δn∥SnGxn-Bnxn∥≤αn∥f(xn)-Bpk∥+β∥xn-Bpk∥+(1-αn-β)×[∥Bnxn-Bnpk∥+∥Bnpk-Bpk∥]+δn∥SnGxn-Bnxn∥≤αn∥f(xn)-Bpk∥+β∥xn-Bpk∥+(1-αn-β)[∥xn-pk∥+∥Bnpk-Bpk∥]+δn∥SnGxn-Bnxn∥≤αn∥f(xn)-Bpk∥+β∥xn-Bpk∥+(1-β)[∥xn-pk∥+∥Bnpk-Bpk∥]+δn∥SnGxn-Bnxn∥=θn+β∥xn-Bpk∥+(1-β)∥xn-pk∥,
whereθn=αn∥f(xn)-Bpk∥+(1-β)∥Bnpk-Bpk∥+δn∥SnGxn-Bnxn∥. Since limn→∞αn=limn→∞∥Bnpk-Bpk∥=limn→∞∥SnGxn-Bnxn∥=0, we know that θn→0 as n→∞.
From (230), we obtain
(231)∥xn+1-Bpk∥2≤(β∥xn-Bpk∥+(1-β)∥xn-pk∥)2+θn[2(β∥xn-Bpk∥+(1-β)∥xn-pk∥)+θn]=β2∥xn-Bpk∥2+(1-β)2∥xn-pk∥2+2β(1-β)∥xn-Bpk∥∥xn-pk∥+τn≤β2∥xn-Bpk∥2+(1-β)2∥xn-pk∥2+β(1-β)(∥xn-Bpk∥2+∥xn-pk∥2)+τn=β∥xn-Bpk∥2+(1-β)∥xn-pk∥2+τn,
whereτn=θn[2(β∥xn-Bpk∥+(1-β)∥xn-pk∥)+θn]→0 as n→∞.
For any Banach limitμ, from (231) we derive
(232)μn∥xn-Bpk∥2=μn∥xn+1-Bpk∥2≤μn∥xn-pk∥2.
In addition, note that(233)∥xn-Gpk∥2≤∥xn-Gxn+Gxn-Gpk∥2≤(∥xn-Gxn∥+∥xn-pk∥)2=∥xn-pk∥2+∥xn-Gxn∥×(2∥xn-pk∥+∥xn-Gxn∥),∥xn-Spk∥2≤∥xn-Sxn+Sxn-Spk∥2≤(∥xn-Sxn∥+∥xn-pk∥)2=∥xn-pk∥2+∥xn-Sxn∥×(2∥xn-pk∥+∥xn-Sxn∥).
It is easy to see from (209) and (228) that
(234)μn∥xn-Gpk∥2≤μn∥xn-pk∥2,μn∥xn-Spk∥2≤μn∥xn-pk∥2.
Utilizing (232) and (234), we deduce that
(235)μn∥xn-Vpk∥2=μn∥(1-θ1-θ2)(xn-Spk)+θ1(xn-Bpk)+θ2(xn-Gpk)∥2≤(1-θ1-θ2)μn∥xn-Spk∥2+θ1μn∥xn-Bpk∥2+θ2μn∥xn-Gpk∥2≤μn∥xn-pk∥2.
Also, observe that(236)xn-pk=1k(xn-f(pk))+(1-1k)(xn-Vpk);
that is,(237)(1-1k)(xn-Vpk)=xn-pk-1k(xn-f(pk)).
It follows from Lemma8(ii) and (237) that
(238)(1-1k)2∥xn-Vpk∥2≥∥xn-pk∥2-2k〈xn-pk+pk-f(pk),J(xn-pk)〉=(1-2k)∥xn-pk∥2+2k〈f(pk)-pk,J(xn-pk)〉.
So by (235) and (238), we have
(239)(1-1k)2μn∥xn-pk∥2≥(1-2k)μn∥xn-pk∥2+2kμn〈f(pk)-pk,J(xn-pk)〉,
and hence(240)1k2μn∥xn-pk∥2≥2kμn〈f(pk)-pk,J(xn-pk)〉.
This implies that(241)12kμn∥xn-pk∥2≥μn〈f(pk)-pk,J(xn-pk)〉.
Sincepk→q∈Fix(V)=F as k→∞, by the uniform Frechet differentiability of the norm of X, we have
(242)μn〈f(q)-q,J(xn-q)〉≤0.
On the other hand, from (184) and the norm-to-norm uniform continuity of J on bounded subsets of X, it follows that
(243)limn→∞|〈f(q)-q,J(xn+1-q)〉-〈f(q)-q,J(xn-q)〉|=0.
So, utilizing Lemma18 we deduce from (242) and (243) that
(244)limsupn→∞〈f(q)-q,J(xn-q)〉≤0,
which, together with (184) and the norm-to-norm uniform continuity of J on bounded subsets of X, implies that
(245)limsupn→∞〈f(q)-q,J(xn+1-q)〉≤0.
Finally, let us show thatxn→q as n→∞. Utilizing Lemma 8 (i), from (178) and the convexity of ∥·∥2, we get
(246)∥xn+1-q∥2=∥αn(f(xn)-f(q))+βn(xn-q)+γn(Bnxn-q)+δn(SnGxn-q)+αn(f(q)-q)∥2≤∥αn(f(xn)-f(q))+βn(xn-q)(f(xn)-f(q))+γn(Bnxn-q)+δn(SnGxn-q)∥2+2αn〈f(q)-q,J(xn+1-q)〉≤αn∥f(xn)-f(q)∥2+βn∥xn-q∥2+γn∥Bnxn-q∥2+δn∥SnGxn-q∥2+2αn〈f(q)-q,J(xn+1-q)〉≤αn∥f(xn)-f(q)∥2+βn∥xn-q∥2+γn∥xn-q∥2+δn∥Gxn-q∥2+2αn〈f(q)-q,J(xn+1-q)〉≤αnρ∥xn-q∥2+βn∥xn-q∥2+γn∥xn-q∥2+δn∥xn-q∥2+2αn〈f(q)-q,J(xn+1-q)〉=(1-αn(1-ρ))∥xn-q∥2+2αn〈f(q)-q,J(xn+1-q)〉=(1-αn(1-ρ))∥xn-q∥2+αn(1-ρ)2〈f(q)-q,J(xn+1-q)〉1-ρ.
Applying Lemma7 to (246), we obtain that xn→q as n→∞. This completes the proof.Corollary 32.
LetC be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let ΠC be a sunny nonexpansive retraction from X onto C. Let {ρn}n=0∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and Ai:C→X an α^i-inverse strongly accretive mapping for each i=0,1,…. Define a mapping Gi:C→C by ΠC(I-λiAi)x=Gix for all x∈C and i=0,1,…, where λi∈(0,α^i/κ2] and κ is the 2-uniformly smooth constant of X. Let Bn:C→C be the W-mapping generated by Gn,Gn-1,…,G0, and ρn,ρn-1,…,ρ0. Let V:C→C be an α-strictly pseudocontractive mapping. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let {Si}i=0∞ be a countable family of nonexpansive mappings of C into itself such that F=(⋂i=0∞‍Fix(Si))∩Fix(V)∩(⋂i=0∞‍VI(C,Ai))≠∅. For arbitrarily given x0∈C, let {xn} be the sequence generated by
(247)xn+1=αnf(xn)+βnxn+γnBnxn+δnSn((1-l)I+lV)xn,∀n≥0,
where0<l<α/κ2 and {αn}, {βn}, {γn}, and {δn} are the sequences in (0,1) such that αn+βn+γn+δn=1 for all n≥0. Suppose that the following conditions hold:(i)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(ii)
{γn},{δn}⊂[c,d] for some c,d∈(0,1);(iii)
limn→∞(|βn-βn-1|+|γn-γn-1|+|δn-δn-1|)=0;(iv)
0<liminfn→∞βn≤limsupn→∞βn<1.
Assume that∑n=1∞‍supx∈D∥Snx-Sn-1x∥<∞ for any bounded subset D of C and let S be a mapping of C into itself defined by Sx=limn→∞Snx for all x∈C and suppose that Fix(S)=⋂i=0∞‍Fix(Si). Then, there hold the following:(I)
limn→∞∥xn+1-xn∥=0;(II)
the sequence{xn}n=0∞ converges strongly to some q∈F which is the unique solution of the variational inequality problem (VIP)
(248)〈(I-f)q,J(q-p)〉≤0,∀p∈F,providedβn≡β for some fixed β∈(0,1).Proof.
In Theorem31, we put B1=I-V, B2=0 and μ1=l where 0<l<α/κ2. Then GSVI (13) is equivalent to the VIP of finding x*∈C such that
(249)〈B1x*,J(x-x*)〉≥0,∀x∈C.
In this case,B1:C→X is α-inverse strongly accretive. Repeating the same arguments as those in the proof of Corollary 25, we can infer that Fix(V)=VI(C,B1). Accordingly, we know that F=⋂i=0∞‍Fix(Ti)∩Ω∩A-10=⋂i=0∞‍Fix(Ti)∩Fix(V)∩A-10, and
(250)ΠC(I-μ1B1)ΠC(I-μ2B2)xn=ΠC(I-μ1B1)xn=ΠC((1-l)xn+lVxn)=((1-l)I+lV)xn.
So, scheme (178) reduces to (247). Therefore, the desired result follows from Theorem 31.Theorem 33.
LetC be a nonempty closed convex subset of a uniformly convex Banach space X which has a uniformly Gateaux differentiable norm. Let ΠC be a sunny nonexpansive retraction from X onto C. Let {ρn}n=0∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and Ai:C→Xξi-strictly pseudocontractive and α^i-strongly accretive with ξi+α^i≥1 for each i=0,1,…. Define a mapping Gi:C→C by ΠC(I-λiAi)x=Gix for all x∈C and i=0,1,…, where 1-(ξi/(1+ξi))(1-(1-α^i)/ξi)≤λi≤1 for all i=0,1,…. Let Bn:C→C be the W-mapping generated by Gn,Gn-1,…,G0 and ρn,ρn-1,…,ρ0. Let the mapping Bi:C→X be ζi-strictly pseudocontractive and β^i-strongly accretive with ζi+β^i≥1 for i=1,2. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let {Si}i=0∞ be a countable family of nonexpansive mappings of C into itself such that F=(⋂i=0∞‍Fix(Si))∩Ω∩(⋂i=0∞‍VI(C,Ai))≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2) with 1-(ζi/(1+ζi))(1-(1-β^i)/ζi)≤μi≤1 for i=1,2. For arbitrarily given x0∈C, let {xn} be the sequence generated by
(251)xn+1=σnGxn+(1-σn)×[αnf(xn)+βnxn+γnBnxn+δnSnGxn],∀n≥0,
where{αn}, {βn}, {γn}, {δn}, and {σn} are the sequences in (0,1) such that αn+βn+γn+δn=1 for all n≥0. Suppose that the following conditions hold:(i)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(ii)
{γn},{δn}⊂[c,d] for some c,d∈(0,1);(iii)
∑n=1∞‍(|σn-σn-1|+|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|)<∞;(iv)
0<liminfn→∞βn≤limsupn→∞βn<1 and 0<liminfn→∞σn≤limsupn→∞σn<1.
Assume that∑n=1∞‍supx∈D∥Snx-Sn-1x∥<∞ for any bounded subset D of C and let S be a mapping of C into itself defined by Sx=limn→∞Snx for all x∈C and suppose that Fix(S)=⋂i=0∞‍Fix(Si). Then, there hold the following:(I)
limn→∞∥xn+1-xn∥=0;(II)
the sequence{xn}n=0∞ converges strongly to some q∈F which is the unique solution of the variational inequality problem (VIP)
(252)〈(I-f)q,J(q-p)〉≤0,∀p∈F,Proof.
First of all, it is easy to see that (251) can be rewritten as follows:
(253)yn=αnf(xn)+βnxn+γnBnxn+δnSnGxn,xx+1=σnGxn+(1-σn)yn,∀n≥0.
Take a fixedp∈F arbitrarily. Then, we obtain p=Gp, p=Bnp and Snp=p for all n≥0. Thus, we get from (253)
(254)∥yn-p∥≤αn∥f(xn)-p∥+βn∥xn-p∥+γn∥Bnxn-p∥+δn∥SnGxn-p∥≤αn(ρ∥xn-p∥+∥f(p)-p∥)+βn∥xn-p∥+γn∥xn-p∥+δn∥xn-p∥=(1-αn(1-ρ))∥xn-p∥+αn∥f(p)-p∥,
and hence(255)∥xn+1-p∥≤σn∥Gxn-p∥+(1-σn)∥yn-p∥≤σn∥xn-p∥+(1-σn)×[(1-αn(1-ρ))∥xn-p∥+αn∥f(p)-p∥]=(1-(1-σn)αn(1-ρ))∥xn-p∥+(1-σn)αn∥f(p)-p∥=(1-(1-σn)αn(1-ρ))∥xn-p∥+(1-σn)αn(1-ρ)∥f(p)-p∥1-ρ≤max{∥xn-p∥,∥f(p)-p∥1-ρ}.
By induction, we have(256)∥xn-p∥≤max{∥x0-p∥,∥f(p)-p∥1-ρ},∀n≥0.
which implies that{xn} is bounded and so are the sequences {yn}, {Gxn} and {f(xn)}.
Let us show that(257)limn→∞∥xn+1-xn∥=0.
As a matter of fact, observe thatyn can be rewritten as follows:
(258)yn=βnxn+(1-βn)zn,
wherezn=(αnf(xn)+γnBnxn+δnSnGxn)/(1-βn). Observe that
(259)∥zn-zn-1∥=∥αnf(xn)+γnBnxn+δnSnGxn1-βn-αn-1f(xn-1)+γn-1Bn-1xn-1+δn-1Sn-1Gxn-11-βn-1∥=∥yn-βnxn1-βn-yn-1-βn-1xn-11-βn-1∥=∥yn-βnxn1-βn-yn-1-βn-1xn-11-βn+yn-1-βn-1xn-11-βn-yn-1-βn-1xn-11-βn-1∥≤∥yn-βnxn1-βn-yn-1-βn-1xn-11-βn∥+∥yn-1-βn-1xn-11-βn-yn-1-βn-1xn-11-βn-1∥=11-βn∥yn-βnxn-(yn-1-βn-1xn-1)∥+|11-βn-11-βn-1|∥yn-1-βn-1xn-1∥=11-βn∥yn-βnxn-(yn-1-βn-1xn-1)∥+|βn-βn-1|(1-βn-1)(1-βn)∥yn-1-βn-1xn-1∥=11-βn∥αnf(xn)+γnBnxn+δnSnGxn-αn-1f(xn-1)-γn-1Bn-1xn-1-δn-1Sn-1Gxn-1∥+|βn-βn-1|(1-βn-1)(1-βn)∥yn-1-βn-1xn-1∥≤11-βn[αn∥f(xn)-f(xn-1)∥+γn∥Bnxn-Bn-1xn-1∥+δn∥SnGxn-Sn-1Gxn-1∥+|αn-αn-1|∥f(xn-1)∥+|γn-γn-1|∥Bn-1xn-1∥+|δn-δn-1|∥Sn-1Gxn-1∥]+|βn-βn-1|(1-βn-1)(1-βn)∥yn-1-βn-1xn-1∥.
On the other hand, repeating the same arguments as those of (52) and (54) in the proof of Theorem 24, we can deduce that for all n≥1(260)∥SnGxn-Sn-1Gxn-1∥≤∥xn-xn-1∥+∥SnGxn-1-Sn-1Gxn-1∥,∥Bnxn-Bn-1xn-1∥≤∥xn-xn-1∥+M∏i=0n-1‍λi,
for some constantM>0. Taking into account 0<liminfn→∞βn≤limsupn→∞βn<1, we may assume, without loss of generality, that {βn}⊂[c^,d^]. Utilizing (259)-(260) we have
(261)∥zn-zn-1∥≤11-βn×{αnρ∥xn-xn-1∥+γn[∥xn-xn-1∥+M∏i=0n-1‍λi]+δn[∥xn-xn-1∥+∥SnGxn-1-Sn-1Gxn-1∥]+|αn-αn-1|∥f(xn-1)∥+|γn-γn-1|∥Bn-1xn-1∥+|δn-δn-1|∥Sn-1Gxn-1∥[∥xn-xn-1∥+M∏i=0n-1‍λi]}+|βn-βn-1|(1-βn-1)(1-βn)∥yn-1-βn-1xn-1∥=11-βn{∏i=0n-1‍(1-βn-αn(1-ρ))∥xn-xn-1∥+γnM∏i=0n-1‍λi+δn∥SnGxn-1-Sn-1Gxn-1∥+|αn-αn-1|∥f(xn-1)∥+|γn-γn-1|∥Bn-1xn-1∥+|δn-δn-1|∥Sn-1Gxn-1∥∏i=0n-1‍}+|βn-βn-1|(1-βn-1)(1-βn)∥yn-1-βn-1xn-1∥=(1-αn(1-ρ)1-βn)∥xn-xn-1∥+γnM1-βn∏i=0n-1‍λi+δn1-βn∥SnGxn-1-Sn-1Gxn-1∥+11-βn[|αn-αn-1|∥f(xn-1)∥+|γn-γn-1|×∥Bn-1xn-1∥+|δn-δn-1|∥Sn-1Gxn-1∥]+|βn-βn-1|(1-βn-1)(1-βn)∥yn-1-βn-1xn-1∥≤(1-αn(1-ρ)1-βn)∥xn-xn-1∥+M∏i=0n-1‍λi+∥SnGxn-1-Sn-1Gxn-1∥+11-βn×[|αn-αn-1|∥f(xn-1)∥+|γn-γn-1|∥Bn-1xn-1∥+|δn-δn-1|∥Sn-1Gxn-1∥]+|βn-βn-1|(1-βn-1)(1-βn)×∥αn-1f(xn-1)+γn-1Bn-1xn-1+δn-1Sn-1Gxn-1∥≤(1-αn(1-ρ)1-βn)∥xn-xn-1∥+M1[∏i=0n-1‍λi+|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|∏i=0n-1‍]+∥SnGxn-1-Sn-1Gxn-1∥,
wheresupn≥0{(1/(1-d^)2)(∥f(xn)∥+∥Bnxn∥+∥SnGxn∥+M)}≤M1 for some M1>0. In the meantime, observe that
(262)xn+1-xn=σn(Gxn-Gxn-1)+(σn-σn-1)(Gxn-1-zn-1)+(1-σn)(zn-zn-1).
This together with (261), implies that
(263)∥xn+1-xn∥≤σn∥Gxn-Gxn-1∥+|σn-σn-1|∥Gxn-1-zn-1∥+(1-σn)∥zn-zn-1∥≤σn∥xn-xn-1∥+|σn-σn-1|∥Gxn-1-zn-1∥+(1-σn){(1-αn(1-ρ)1-βn)∥xn-xn-1∥+M1[∏i=0n-1‍λi+|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|∏i=0n-1‍]+∥SnGxn-1-Sn-1Gxn-1∥(1-αn(1-ρ)1-βn)}≤(1-(1-σn)αn(1-ρ)1-βn)∥xn-xn-1∥+|σn-σn-1|∥Gxn-1-zn-1∥+M1[∏i=0n-1‍λi+|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|∏i=0n-1‍]+∥SnGxn-1-Sn-1Gxn-1∥≤(1-(1-σn)αn(1-ρ)1-βn)∥xn-xn-1∥+M2[∏i=0n-1‍λi+|σn-σn-1|+|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|∏i=0n-1‍]+∥SnGxn-1-Sn-1Gxn-1∥,
wheresupn≥0{M1+∥Gxn-zn∥}≤M2 for some M2>0. Since ∑n=0∞‍αn=∞ and (1-σn)αn(1-ρ)/(1-βn)≥(1-σn)αn(1-ρ), we obtain from conditions (i) and (iv) that ∑n=0∞‍((1-σn)αn(1-ρ)/(1-βn))=∞. Thus, applying Lemma 7 to (263), we deduce from condition (iii) and the assumption on {Sn} that (noting that 0<λi≤b<1, for all i≥0)
(264)limn→∞∥xn+1-xn∥=0.
Next, we show that∥xn-Gxn∥→0 as n→∞.
Indeed, according to Lemma8 we have from (253)
(265)∥yn-p∥2=∥αn(f(xn)-f(p))+βn(xn-p)+γn(Bnxn-p)+δn(SnGxn-p)+αn(f(p)-p)∥2≤∥αn(f(xn)-f(p))+βn(xn-p)+γn(Bnxn-p)+δn(SnGxn-p)∥2+2αn〈f(p)-p,J(xn+1-p)〉≤αn∥f(xn)-f(p)∥2+βn∥xn-p∥2+γn∥Bnxn-p∥2+δn∥SnGxn-p∥2+2αn〈f(p)-p,J(xn+1-p)〉≤αnρ∥xn-p∥2+βn∥xn-p∥2+γn∥xn-p∥2+δn∥xn-p∥2+2αn∥f(p)-p∥∥xn+1-p∥=(1-αn(1-ρ))∥xn-p∥2+2αn∥f(p)-p∥∥xn+1-p∥≤∥xn-p∥2+2αn∥f(p)-p∥∥xn+1-p∥.
Utilizing Lemma15 we get from (253) and (265)
(266)∥xn+1-p∥2=∥σn(Gxn-p)+(1-σn)(yn-p)∥2≤σn∥Gxn-p∥2+(1-σn)∥yn-p∥2-σn(1-σn)g(∥Gxn-yn∥)≤σn∥xn-p∥2+(1-σn)×[∥xn-p∥2+2αn∥f(p)-p∥∥xn+1-p∥]-σn(1-σn)g(∥Gxn-yn∥)≤∥xn-p∥2+2αn∥f(p)-p∥∥xn+1-p∥-σn(1-σn)g(∥Gxn-yn∥),
which hence yields(267)σn(1-σn)g(∥Gxn-yn∥)≤∥xn-p∥2-∥xn+1-p∥2+2αn∥f(p)-p∥∥xn+1-p∥≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+2αn∥f(p)-p∥∥xn+1-p∥.
Sinceαn→0 and ∥xn+1-xn∥→0, from condition (iv) and the boundedness of {xn}, it follows that
(268)limn→∞g(∥Gxn-yn∥)=0.
Utilizing the properties ofg, we have
(269)limn→∞∥Gxn-yn∥=0,
which, together with (253) and (257), implies that
(270)∥xn-yn∥≤∥xn-xn+1∥+∥xn+1-yn∥=∥xn-xn+1∥+σn∥Gxn-yn∥⟶0asn⟶∞.
That is,(271)limn→∞∥xn-yn∥=0.
Since(272)∥xn-Gxn∥≤∥xn-yn∥+∥yn-Gxn∥,
it immediately follows from (269) and (271) that
(273)limn→∞∥xn-Gxn∥=0.
On the other hand, observe thatyn can be rewritten as follows:
(274)yn=αnf(xn)+βnxn+γnBnxn+δnSnGxnyn=αnf(xn)+βnxn+(γn+δn)γnBnxn+δnSnGxnγn+δnyn=αnf(xn)+βnxn+enz^n,
whereen=γn+δn and z^n=(γnBnxn+δnSnGxn)/(γn+δn). Utilizing Lemma 11, we have
(275)∥yn-p∥2=∥αn(f(xn)-p)+βn(xn-p)+en(z^n-p)∥2≤αn∥f(xn)-p∥2+βn∥xn-p∥2+en∥z^n-p∥2-βneng1(∥z^n-xn∥)=αn∥f(xn)-p∥2+βn∥xn-p∥2-βneng1(∥z^n-xn∥)+en∥γnBnxn+δnSnGxnγn+δn-p∥2=αn∥f(xn)-p∥2+βn∥xn-p∥2-βneng1(∥z^n-xn∥)+en∥γnγn+δn(Bnxn-p)+δnγn+δn(SnGxn-p)∥2≤αn∥f(xn)-p∥2+βn∥xn-p∥2-βneng1(∥z^n-xn∥)+en[γnγn+δn∥Bnxn-p∥2+δnγn+δn∥SnGxn-p∥2]≤αn∥f(xn)-p∥2+βn∥xn-p∥2-βneng1(∥z^n-xn∥)+en[γnγn+δn∥xn-p∥2+δnγn+δn∥xn-p∥2]=αn∥f(xn)-p∥2+(1-αn)∥xn-p∥2-βneng1(∥z^n-xn∥)≤αn∥f(xn)-p∥2+∥xn-p∥2-βneng1(∥z^n-xn∥),
which hence implies that(276)βneng1(∥z^n-xn∥)≤αn∥f(xn)-p∥2+∥xn-p∥2-∥yn-p∥2≤αn∥f(xn)-p∥2+(∥xn-p∥+∥yn-p∥)∥xn-yn∥.
Utilizing (271), conditions (i), (ii), (iv), and the boundedness of {xn}, {yn} and {f(xn)}, we get
(277)limn→∞g1(∥z^n-xn∥)=0.
From the properties ofg1, we have
(278)limn→∞∥z^n-xn∥=0.
Utilizing Lemma15 and the definition of z^n, we have
(279)∥z^n-p∥2=∥γnBnxn+δnSnGxnγn+δn-p∥2=∥γnγn+δn(Bnxn-p)+δnγn+δn(SnGxn-p)∥2≤γnγn+δn∥Bnxn-p∥2+δnγn+δn∥SnGxn-p∥2-γnδn(γn+δn)2g2(∥SnGxn-Bnxn∥)≤∥xn-p∥2-γnδn(γn+δn)2g2(∥SnGxn-Bnxn∥),
which leads to(280)γnδn(γn+δn)2g2(∥SnGxn-Bnxn∥)≤∥xn-p∥2-∥z^n-p∥2≤(∥xn-p∥+∥z^n-p∥)∥xn-z^n∥.
Since{xn} and {z^n} are bounded, we deduce from (278) and condition (ii) that
(281)limn→∞g2(∥Bnxn-SnGxn∥)=0.
From the properties ofg2, we have
(282)limn→∞∥Bnxn-SnGxn∥=0.
Furthermore,yn can also be rewritten as follows:
(283)yn=αnf(xn)+βnxn+γnBnxn+δnSnGxn=βnxn+γnBnxn+(αn+δn)αnf(xn)+δnSnGxnαn+δn=βnxn+γnBnxn+dnz~n,
wheredn=αn+δn and z~n=(αnf(xn)+δnSnGxn)/(αn+δn). Utilizing Lemma 11 and the convexity of ∥·∥2, we have
(284)∥yn-p∥2=∥βn(xn-p)+γn(Bnxn-p)+dn(z~n-p)∥2≤βn∥xn-p∥2+γn∥Bnxn-p∥2+dn∥z~n-p∥2-βnγng3(∥xn-Bnxn∥)=βn∥xn-p∥2+γn∥Bnxn-p∥2+dn∥αnf(xn)+δnSnGxnαn+δn-p∥2-βnγng3(∥xn-Bnxn∥)=βn∥xn-p∥2+γn∥Bnxn-p∥2+dn∥αnαn+δn(f(xn)-p)+δnαn+δn(SnGxn-p)∥2-βnγng3(∥xn-Bnxn∥)≤βn∥xn-p∥2+γn∥xn-p∥2+dn[αnαn+δn∥f(xn)-p∥2+δnαn+δn∥SnGxn-p∥2]-βnγng3(∥xn-Bnxn∥)≤αn∥f(xn)-p∥2+(βn+γn)∥xn-p∥2+δn∥xn-p∥2-βnγng3(∥xn-Bnxn∥)=αn∥f(xn)-p∥2+(1-αn)∥xn-p∥2-βnγng3(∥xn-Bnxn∥)≤αn∥f(xn)-p∥2+∥xn-p∥2-βnγng3(∥xn-Bnxn∥),
which hence implies that(285)βnγng3(∥xn-Bnxn∥)≤αn∥f(xn)-p∥2+∥xn-p∥2-∥yn-p∥2≤αn∥f(xn)-p∥2+(∥xn-p∥+∥yn-p∥)×∥xn-yn∥.
Utilizing (271), conditions (i), (ii), (iv), and the boundedness of {xn}, {yn} and {f(xn)}, we get
(286)limn→∞g3(∥xn-Bnxn∥)=0.
From the properties ofg3, we have
(287)limn→∞∥xn-Bnxn∥=0.
Thus, from (282) and (287), we get
(288)∥xn-SnGxn∥≤∥xn-Bnxn∥+∥Bnxn-SnGxn∥⟶0asn⟶∞.
That is,(289)limn→∞∥xn-SnGxn∥=0.
Therefore, from Lemma12, (273), and (289), it follows that
(290)∥xn-Sxn∥≤∥xn-SnGxn∥+∥SnGxn-Snxn∥+∥Snxn-Sxn∥≤∥xn-SnGxn∥+∥Gxn-xn∥+∥Snxn-Sxn∥⟶0asn⟶∞.
That is,(291)limn→∞∥xn-Sxn∥=0.
Suppose thatβn≡β for some fixed β∈(0,1) such that αn+β+γn+δn=1 for all n≥0. Define a mapping Vx=(1-θ1-θ2)Sx+θ1Bx+θ2Gx, where θ1,θ2∈(0,1) are two constants with θ1+θ2<1. Then, by Lemmas 14 and 17, we have that Fix(V)=Fix(S)∩Fix(B)∩Fix(G)=F. For each k≥1, let {pk} be a unique element of C such that
(292)pk=1kf(pk)+(1-1k)Vpk.
From Lemma13, we conclude that pk→q∈Fix(V)=F as k→∞. Repeating the same arguments as those of (81) in the proof of Theorem 24, we can conclude that for every n,k(293)∥yn-Bpk∥≤αn∥f(xn)-Bpk∥+β∥xn-Bpk∥+γn∥Bnxn-Bpk∥+δn∥SnGxn-Bpk∥≤θn+β∥xn-Bpk∥+(1-β)∥xn-pk∥,
whereθn=αn∥f(xn)-Bpk∥+(1-β)∥Bnpk-Bpk∥+δn∥SnGxn-Bnxn∥. Since limn→∞αn=limn→∞∥Bnpk-Bpk∥=limn→∞∥SnGxn-Bnxn∥=0, we know that θn→0 as n→∞. So, it immediately follows that
(294)∥xn+1-Bpk∥2=∥yn-Bpk∥2+∥xn+1-yn∥×[2∥yn-Bpk∥+∥xn+1-yn∥]≤(β∥xn-Bpk∥+(1-β)∥xn-pk∥)2+θn[2(β∥xn-Bpk∥+(1-β)∥xn-pk∥)+θn]+∥xn+1-yn∥[2∥yn-Bpk∥+∥xn+1-yn∥]=β2∥xn-Bpk∥2+(1-β)2∥xn-pk∥2+2β(1-β)∥xn-Bpk∥∥xn-pk∥+τn≤β2∥xn-Bpk∥2+(1-β)2∥xn-pk∥2+β(1-β)(∥xn-Bpk∥2+∥xn-pk∥2)+τn=β∥xn-Bpk∥2+(1-β)∥xn-pk∥2+τn,
whereτn=θn[2(β∥xn-Bpk∥+(1-β)∥xn-pk∥)+θn]+∥xn+1-yn∥[2∥yn-Bpk∥+∥xn+1-yn∥]→0 as n→∞.
For any Banach limitμ, from (294), we derive
(295)μn∥xn-Bpk∥2=μn∥xn+1-Bpk∥2≤μn∥xn-pk∥2.
In addition, note that(296)∥xn-Gpk∥2≤∥xn-Gxn+Gxn-Gpk∥2≤(∥xn-Gxn∥+∥xn-pk∥)2=∥xn-pk∥2+∥xn-Gxn∥×(2∥xn-pk∥+∥xn-Gxn∥),∥xn-Spk∥2≤∥xn-Sxn+Sxn-Spk∥2≤(∥xn-Sxn∥+∥xn-pk∥)2=∥xn-pk∥2+∥xn-Sxn∥×(2∥xn-pk∥+∥xn-Sxn∥).
It is easy to see from (273) and (291) that
(297)μn∥xn-Gpk∥2≤μn∥xn-pk∥2,μn∥xn-Spk∥2≤μn∥xn-pk∥2.
Utilizing (295) and (297), we deduce that
(298)μn∥xn-Vpk∥2=μn∥(1-θ1-θ2)(xn-Spk)+θ1(xn-Bpk)+θ2(xn-Gpk)∥2≤(1-θ1-θ2)μn∥xn-Spk∥2+θ1μn∥xn-Bpk∥2+θ2μn∥xn-Gpk∥2≤μn∥xn-pk∥2.
Repeating the same arguments as those of (99) in the proof of Theorem 24, we can obtain that
(299)12kμn∥xn-pk∥2≥μn〈f(pk)-pk,J(xn-pk)〉.
Sincepk→q∈Fix(V)=F as k→∞, by the uniform Gateaux differentiability of the norm of X we have
(300)μn〈f(q)-q,J(xn-q)〉≤0.
On the other hand, from (257) and the norm-to-weak* uniform continuity of J on bounded subsets of X, it follows that
(301)limn→∞|〈f(q)-q,J(xn+1-q)〉-〈f(q)-q,J(xn-q)〉|=0.
So, utilizing Lemma18, we deduce from (300) and (301) that
(302)limsupn→∞〈f(q)-q,J(xn-q)〉≤0,
which together with (271) and the norm-to-weak* uniform continuity of J on bounded subsets of X, implies that
(303)limsupn→∞〈f(q)-q,J(yn-q)〉≤0.
Finally, let us show thatxn→q as n→∞. Utilizing Lemma 8 (i), from (253) and the convexity of ∥·∥, we get
(304)∥yn-q∥2≤∥αn(f(xn)-f(q))+βn(xn-q)+γn(Bnxn-q)+δn(SnGxn-q)∥2+2αn〈f(q)-q,J(yn-q)〉≤αn∥f(xn)-f(q)∥2+βn∥xn-q∥2+γn∥Bnxn-q∥2+δn∥SnGxn-q∥2+2αn〈f(q)-q,J(yn-q)〉≤αnρ∥xn-q∥2+βn∥xn-q∥2+γn∥xn-q∥2+δn∥xn-q∥2+2αn〈f(q)-q,J(yn-q)〉=(1-αn(1-ρ))∥xn-q∥2+2αn〈f(q)-q,J(yn-q)〉,
and hence(305)∥xn+1-q∥2≤σn∥Gxn-q∥2+(1-σn)∥yn-q∥2≤σn∥xn-q∥2+(1-σn)×[(1-αn(1-ρ))∥xn-q∥2+2αn〈f(q)-q,J(yn-q)〉(1-αn(1-ρ))∥xn-q∥2]=[1-(1-σn)αn(1-ρ)]∥xn-q∥2+2(1-σn)αn〈f(q)-q,J(yn-q)〉=[1-(1-σn)αn(1-ρ)]∥xn-q∥2+(1-σn)αn(1-ρ)2〈f(q)-q,J(yn-q)〉1-ρ.
From conditions (i) and (iv), it is easy to see that∑n=0∞‍(1-σn)αn(1-ρ)=∞. Applying Lemma 7 to (305), we infer that xn→q as n→∞. This completes the proof.Corollary 34.
LetC be a nonempty closed convex subset of a uniformly convex Banach space X which has a uniformly Gateaux differentiable norm. Let ΠC be a sunny nonexpansive retraction from X onto C. Let {ρn}n=0∞ be a sequence of positive numbers in (0,b] for some b∈(0,1) and Ai:C→Xξi-strictly pseudocontractive and α^i-strongly accretive with ξi+α^i≥1 for each i=0,1,…. Define a mapping Gi:C→C by ΠC(I-λiAi)x=Gix for all x∈C and i=0,1,…, where 1-(ξi/(1+ξi))(1-(1-α^i)/ξi)≤λi≤1 for all i=0,1,…. Let Bn:C→C be the W-mapping generated by Gn,Gn-1,…,G0 and ρn,ρn-1,…,ρ0. Let V:C→C be a self-mapping such that I-V:C→X is ζ-strictly pseudocontractive and θ-strongly accretive with θ+ζ≥1. Let f:C→C be a contraction with coefficient ρ∈(0,1). Let {Si}i=0∞ be a countable family of nonexpansive mappings of C into itself such that F=(⋂i=0∞‍Fix(Si))∩Fix(V)∩(⋂i=0∞‍VI(C,Ai))≠∅. For arbitrarily given x0∈C, let {xn} be the sequence generated by
(306)xn+1=σn((1-l)I+lV)xn+(1-σn)×[αnf(xn)+βnxn+γnBnxn+δnSn((1-l)I+lV)xn],∀n≥0,
where1-(ζ/(1+ζ))(1-(1-θ)/ζ)≤l≤1 and {αn}, {βn}, {γn}, {δn}, and {σn} are the sequences in (0,1) such that αn+βn+γn+δn=1 for all n≥0. Suppose that the following conditions hold:(i)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(ii)
{γn},{δn}⊂[c,d] for some c,d∈(0,1);(iii)
∑n=1∞‍(|σn-σn-1|+|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|)<∞;(iv)
0<liminfn→∞βn≤limsupn→∞βn<1 and 0<liminfn→∞σn≤limsupn→∞σn<1.
Assume that∑n=1∞‍supx∈D∥Snx-Sn-1x∥<∞ for any bounded subset D of C and let S be a mapping of C into itself defined by Sx=limn→∞Snx for all x∈C and suppose that Fix(S)=⋂i=0∞‍Fix(Si). Then, there hold the following:(I)
limn→∞∥xn+1-xn∥=0;(II)
the sequence{xn}n=0∞ converges strongly to some q∈F which is the unique solution of the variational inequality problem (VIP)
(307)〈(I-f)q,J(q-p)〉≤0,∀p∈F,providedβn≡β for some fixed β∈(0,1).Proof.
In Theorem33, we put B1=I-V, B2=0 and μ1=l where 1-(ζ/(1+ζ))(1-(1-θ)/ζ)≤l≤1. Then, GSVI (13) is equivalent to the VIP of finding x*∈C such that
(308)〈B1x*,J(x-x*)〉≥0,∀x∈C.
In this case,B1:C→X is ζ-strictly pseudocontractive and θ-strongly accretive. Repeating the same arguments as those in the proof of Corollary 25, we can infer that Fix(V)=VI(C,B1). Accordingly, F=(⋂i=0∞‍Fix(Si))∩Ω∩(⋂i=0∞‍VI(C,Ai))=(⋂i=0∞‍Fix(Si))∩Fix(V)∩(⋂i=0∞‍VI(C,Ai)),
(309)Gxn=((1-l)I+lV)xn,
So, the scheme (251) reduces to (306). Therefore, the desired result follows from Theorem 33.Remark 35.
Our Theorems31 and 33 improve, extend, supplement and develop Ceng and Yao’s [10, Theorem  3.2], Cai and Bu’s [11, Theorem  3.1], Kangtunyakarn’s [38, Theorem  3.1], and Ceng and Yao’s [8, Theorem  3.1], in the following aspects.(i)
The problem of finding a pointq∈(⋂i=0∞‍Fix(Si))∩Ω∩(⋂i=0∞‍VI(C,Ai)) in our Theorems 31 and 33 is more general and more subtle than every one of the problem of finding a point q∈⋂i=0∞‍Fix(Ti) in [10, Theorem  3.2], the problem of finding a point q∈⋂i=1∞‍Fix(Ti)∩Ω in [11, Theorem  3.1], the problem of finding a point q∈Fix(S)∩Fix(V)∩(⋂i=1N‍VI(C,Ai)) in [38, Theorem  3.1], and the problem of finding a point q∈Fix(T) in [8, Theorem  3.1].(ii)
The iterative scheme in [38, Theorem  3.1] is extended to develop the iterative scheme (178) of our Theorem 31, and the iterative scheme in [11, Theorem  3.1] is extended to develop the iterative scheme (251) of our Theorem 33. Iterative schemes (178) and (181) in our Theorems 31 and 33 are more advantageous and more flexible than the iterative scheme of [11, Theorem  3.1] because they both are one-step iteration schemes and involve several parameter sequences {αn}, {βn}, {γn}, {δn}, (and {σn}).(iii)
Our Theorems31 and 33 extend and generalize Ceng and Yao’s [8, Theorem  3.1] from a nonexpansive mapping to a countable family of nonexpansive mappings, and Ceng and Yao’s [10, Theorems  3.2] to the setting of the GSVI (13) and infinitely many VIPs, Kangtunyakarn’s [38, Theorem  3.1] from finitely many VIPs to infinitely many VIPs, from a nonexpansive mapping to a countable family of nonexpansive mappings and from a strict pseudocontraction to the GSVI (13). In the meantime, our Theorems 31 and 33 extend and generalize Cai and Bu’s [11, Theorem  3.1] to the setting of infinitely many VIPs.(iv)
The iterative schemes (178) and (251) in our Theorems 31 and 33 are very different from every one in [10, Theorem  3.2], [11, Theorem  3.1], [38, Theorem  3.1], and [8, Theorem  3.1] because the mappings G and Tn in [11, Theorem  3.1] and the mapping T in [8, Theorem  3.1] are replaced with the same composite mapping SnG in the iterative schemes (42) and (130) and the mapping Wn in [10, Theorem  3.2] is replaced by Bn.(v)
Cai and Bu’s proof in [11, Theorem  3.1] depends on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), and the inequality in smooth and uniform convex Banach spaces (see Proposition 6). Because the composite mapping SnG appears in the iterative scheme (178) of our Theorem 31, the proof of our Theorem 31 depends on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), the inequality in smooth and uniform convex Banach spaces (see Proposition 6), the inequalities in uniform convex Banach spaces (see Lemmas 11 and 15 in Section 2 of this paper), and the properties of the W-mapping and the Banach limit (see Lemmas 16, 17, and 18 in Section 2 of this paper). However, the proof of our Theorem 33 does not depend on the argument techniques in [14], the inequality in 2-uniformly smooth Banach spaces (see Lemma 4), and the inequality in smooth and uniform convex Banach spaces (see Proposition 6). It depends on only the inequalities in uniform convex Banach spaces (see Lemmas 11 and 15 in Section 2 of this paper) and the properties of the W-mapping and the Banach limit (see Lemmas 16–18 in Section 2 of this paper).(vi)
The assumption of the uniformly convex and2-uniformly smooth Banach space X in [11, Theorem  3.1] is weakened to the one of the uniformly convex Banach space X having a uniformly Gateaux differentiable norm in our Theorem 33. Moreover, the assumption of the uniformly smooth Banach space X in [8, Theorem  3.1] is replaced with the one of the uniformly convex Banach space X having a uniformly Gateaux differentiable norm in our Theorem 33.



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