# Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions with Finitely Many Variational Inequality Constraints in Banach Spaces

Authors: Lu-Chuan Ceng; Abdul Latif; Abdullah E. Al-Mazrooei
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative 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 Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. 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 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,
(2)∥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
(3)limt→0∥x+ty∥-∥x∥t
exists for all x,y∈U; in this case, X is also said to have a Gáteaux differentiable norm. X is said to have a uniformly Gáteaux 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 Fréchet differential if, for each x∈U, this limit is attained uniformly for y∈U. In addition, we define a function ρ:[0,∞)→[0,∞) called the modulus of smoothness of X as follows:
(4)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. It is well-known that 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 Gáteaux differentiable norm then the duality mapping J is norm-to-weak* uniformly continuous on bounded subsets of X.LetC be a nonempty closed convex subset of a real Banach space X. A mapping T:C→C is called nonexpansive if
(5)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.The set of fixed points ofT is denoted by Fix(T). We use the notation ⇀ to indicate the weak convergence and the one → to indicate the strong convergence.Definition 1.
LetA:C→X be a mapping of C into X. Then A is said to be(i)
accretive if for eachx,y∈C there exists j(x-y)∈J(x-y) such that
(6)〈Ax-Ay,j(x-y)〉≥0,
where J is the normalized duality mapping;(ii)
α-strongly accretive if for each x,y∈C there exists j(x-y)∈J(x-y) such that
(7)〈Ax-Ay,j(x-y)〉≥α∥x-y∥2,
for some α∈(0,1);(iii)
β-inverse strongly accretive if for each x,y∈C there exists j(x-y)∈J(x-y) such that
(8)〈Ax-Ay,j(x-y)〉≥β∥Ax-Ay∥2,
for some β>0;(iv)
λ-strictly pseudocontractive if for each x,y∈C there exists j(x-y)∈J(x-y) such that
(9)〈Ax-Ay,j(x-y)〉≤∥x-y∥2-λ∥x-y-(Ax-Ay)∥2
for some λ∈(0,1).LetX be a real smooth Banach space. Let C be a nonempty closed convex subset of X and let A:C→X be a nonlinear mapping. The so-called variational inequality problem (VIP) is the problem of finding x*∈C such that
(10)〈Ax*,J(x-x*)〉≥0,∀x∈C,
which was considered by Aoyama et al. [1]. Note that VIP (10) is connected with the fixed point problem for nonlinear mapping (see e.g., [2]), the problem of finding a zero point of a nonlinear operator (see e.g., [3]), and so on. In particular, whenever X=H a Hilbert space, the VIP (10) reduces to the classical VIP of finding x*∈C such that
(11)〈Ax*,x-x*〉≥0,∀x∈C,whose solution set is denoted byVI(C,A). Recently, in order to find a solution of VIP (10), Aoyama et al. [1] introduced Mann-type iterative scheme for an accretive operator A as follows:
(12)xn+1=αnxn+(1-αn)ΠC(xn-λnAxn),∀n≥1,
where ΠC is a sunny nonexpansive retraction from X onto C. Then they proved a weak convergence theorem.Definition 2.
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:
(13)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 mapping K is called the K-mapping generated by T1,…,TN and λ1,…,λN.Lemma 3 (see [4]).
LetC be a nonempty closed convex subset of a strictly convex Banach space. Let {Ti}i=1N be a finite family of nonexpansive mappings of C into itself with ∩i=1NFix(Ti)≠∅ and let λ1,…,λN be real numbers such that 0<λi<1 for every i=1,…,N-1 and 0<λN≤1. Let K be the K-mapping generated by T1,…,TN and λ1,…,λN. Then Fix(K)=∩i=1NFix(Ti).From Lemma3, it is easy to see that the K-mapping is a nonexpansive mapping.On the other hand, letCB(X) be the family of all nonempty, closed, and bounded subsets of a real smooth Banach space X. Also, we denote by H(·,·) the Hausdorff metric on CB(X) defined by
(14)H(A,B):=max{supx∈Binfy∈Ad(x,y),supx∈Ainfy∈Bd(x,y)},∀A,B∈CB(X).LetT,F:X→CB(X) be two multivalued mappings, let A:D(A)⊂X→2X be an m-accretive mapping, let g:X→D(A) be a single-valued mapping, and let N(·,·):X×X→X be a nonlinear mapping. Then for any given v∈X, λ>0, Chidume et al. [5] introduced and studied the multivalued variational inclusion (MVVI) of finding x∈D(A) such that (x,w,k) is a solution of the following:
(15)v∈N(w,k)+λA(g(x)),∀w∈Tx,k∈Fx.Ifv=0 and λ=1, then the MVVI (15) reduces to the problem of finding x∈D(A) such that (x,w,k) is a solution of the following:
(16)0∈N(w,k)+A(g(x)),∀w∈Tx,k∈Fx.We denote byΓ the set of such solutions x for MVVI (16).The authors [5] established an existence theorem for MVVI (15) in a smooth Banach space X and then proved that the sequence generated by their iterative algorithm converges strongly to a solution of MVVI (16).Theorem 4 (see [5, Theorem 3.2]).
LetX be a real smooth Banach space. Let T,F:X→CB(X), and A:D(A)⊂X→2X be three multivalued mappings, let g:X→D(A) be a single-valued mapping, and let N(·,·):X×X→X be a single-valued continuous mapping satisfying the following conditions:(C1)
A∘g:X→2X is m-accretive and H-uniformly continuous;(C2)
T:X→CB(X) is H-uniformly continuous;(C3)
F:X→CB(X) is H-uniformly continuous;(C4)
the mappingx↦N(x,y) is ϕ-strongly accretive and μ-H-Lipschitz with respect to the mapping T, where ϕ:[0,∞)→[0,∞) is a strictly increasing function with ϕ(0)=0;(C5)
the mappingy↦N(x,y) is accretive and ξ-H-Lipschitz with respect to the mapping F.
For arbitraryx0∈D(A), define the sequence {xn} iteratively by
(17)xn+1=xn-σn(N(wn,kn)+un),un∈A(g(xn)),
where {un} is defined by
(18)∥un-un+1∥≤(1+ɛ)H(A(g(xn+1)),A(g(xn))),∀n≥0,
for any wn∈Txn, kn∈Fxn, and some ɛ>0, where {σn} is a positive real sequence such that limn→∞σn=0,∑n=0∞‍σn=∞. Then, there exists d¯>0 such that, for 0<σn≤d¯ and for all n≥0, {xn} converges strongly to x¯∈Γ, and, for any w∈Tx¯ and k∈Fx¯, (x¯,w,k) is a solution of the MVVI (16).LetC be a nonempty closed convex subset of a real smooth Banach space X and let ΠC be a sunny nonexpansive retraction from X onto C. Let f:C→C be a contraction with coefficient ρ∈(0,1). Motivated and inspired by the research going on this area, we introduce Mann-type viscosity approximation methods for finding solutions of the MVVI (16) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings. Here, the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative 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 Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of ηi-strictly pseudocontractive mappings (i=1,…,N) and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. 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, [6–11].


## 2. Preliminaries

LetX be a real Banach space with dual X*. We denote by J the normalized duality mapping from X to 2X* defined by
(19)J(x)={x*∈X*:〈x,x*〉=∥x∥2=∥x*∥2},
where 〈·,·〉 denotes the generalized duality pairing. Throughout this paper, the single-valued normalized duality map is still denoted by J. Unless otherwise stated, we assume that X is a smooth Banach space with dual X*.A multivalued mappingA:D(A)⊆X→2X is said to be(i)
accretive, if(20)〈u-v,J(x-y)〉≥0,∀u∈Ax,v∈Ay;(ii)
m-accretive, if A is accretive and (I+rA)(D(A))=X, for all r>0, where I is the identity mapping;(iii)
ζ-inverse strongly accretive, if there exists a constant ζ>0 such that
(21)〈u-v,J(x-y)〉≥ζ∥u-v∥2,∀u∈Ax,v∈Ay;(iv)
ϕ-strongly accretive, if there exists a strictly increasing continuous function ϕ:[0,∞)→[0,∞) with ϕ(0)=0 such that
(22)〈u-v,J(x-y)〉,≥ϕ(∥x-y∥)∥x-y∥,∀u∈Ax,v∈Ay;(v)
ϕ-expansive, if
(23)∥u-v∥≥ϕ(∥x-y∥),∀u∈Ax,v∈Ay.It is easy to see that ifA is ϕ-strongly accretive, then A is ϕ-expansive.A mappingT:X→CB(X) is said to be H-uniformly continuous, if for any given ɛ>0, there exists a δ>0 such that whenever ∥x-y∥<δ then H(Tx,Ty)<ɛ.A mappingN:X×X→X is ϕ-strongly accretive, with respect to T:X→CB(X), in the first argument if
(24)〈N(u,z)-N(v,z),J(x-y)〉≥ϕ(∥x-y∥)∥x-y∥,∀u∈Tx,v∈Ty.A mappingS:X→2X is called lower semicontinuous, if S-1(O):={x∈X:Sx∩O≠∅} is open in X whenever O⊂Y is open.We list some propositions and lemmas that will be used in the sequel.Proposition 5 (see [12]).
Let{λn} and {bn} be sequences of nonnegative numbers and {αn}⊂(0,1) a sequence satisfying the conditions that {λn} is bounded, ∑n=0∞‍αn=∞, and bn→0, as n→∞. Let the recursive inequality
(25)λn+12≤λn2-2αnψ(λn+1)+2αnbnλn+1,∀n≥0,
be given where ψ:[0,∞)→[0,∞) is a strictly increasing function such that it is positive on (0,∞) and ψ(0)=0. Then λn→0, as n→∞.Proposition 6 (see [13]).
LetX be a real smooth Banach space. Let T, and F:X→2X be two multivalued mappings, and let N(·,·):X×X→X be a nonlinear mapping satisfying the following conditions:(i)
the mapping  x↦N(x,y)is  ϕ-strongly accretive with respect to the mapping  T;(ii)
the mapping  y↦N(x,y)2009  is accretive with respect to the mapping  F.Then the mappingS:X→2X defined by Sx=N(Tx,Fx) is ϕ-strongly accretive.Proposition 7 (see [14]).
LetX be a real Banach space and let S:X→2X∖{∅} be a lower semicontinuous and ϕ-strongly accretive mapping; then, for any x∈X,Sx is a one-point set; that is, S is a single-valued mapping.Lemma8 can be found in [15]. Lemma 9 is an immediate consequence of the subdifferential inequality of the function (1/2)∥·∥2.Lemma 8.
Let{sn} be a sequence of nonnegative real numbers satisfying
(26)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.Lemma 9.
In a smooth Banach spaceX, there holds the inequality
(27)∥x+y∥2≤∥x∥2+2〈y,J(x+y)〉,∀x,y∈X.Lemma 10 (see [1]).
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 A be an accretive operator of C into X. Then, for all λ>0,
(28)VI(C,A)=Fix(ΠC(I-λA)).LetD be a subset of C and let Π be a mapping of C into D. Then Π is said to be sunny if
(29)Π[Π(x)+t(x-Π(x))]=Π(x),
whenever Π(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 11 (see [16]).
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 12 (see [17]).
LetX be a uniformly convex Banach space and B¯r(0):={x∈X:∥x∥≤r}, r>0. Then there exists a continuous, strictly increasing, and convex function φ:[0,∞]→[0,∞], φ(0)=0 such that
(30)∥αx+βy+γz∥2≤α∥x∥2+β∥y∥2+γ∥z∥2-αβφ(∥x-y∥),
for all x,y,z∈B¯r(0) and all α,β,γ∈[0,1] with α+β+γ=1.Lemma 13 (see [18]).
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 let T:C→C be 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,
(31)xt=tf(xt)+(1-t)Txt.Lemma 14 (see [19]).
LetX be a uniformly smooth Banach space or a reflexive and strictly convex Banach space with a uniformly Gáteaux differentiable norm. Let C be a nonempty closed convex subset of X, let T:C→C be a nonexpansive mapping with Fix(T)≠∅, and let f∈ΞC. Then the net {xt} defined by xt=tf(xt)+(1-t)Txt converges strongly to a point in Fix(T). If one defines a mapping Q:ΞC→Fix(T) by Q(f):=s-limt→0xt, for all f∈ΞC, then Q(f) solves the VIP as follows:
(32)〈(I-f)Q(f),J(Q(f)-p)〉≤0,∀f∈ΞC,p∈Fix(T).Lemma 15 (see [20]).
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 ⋂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 and nonexpansive, and Fix(S)=⋂n=0∞‍Fix(Tn) holds.Lemma 16 (see [21]).
Given a numberr>0. A real Banach space X is uniformly convex if and only if there exists a continuous strictly increasing function φ:[0,∞)→[0,∞),φ(0)=0, such that
(33)∥λx+(1-λ)y∥2≤λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)φ(∥x-y∥),
for all λ∈[0,1] and x,y∈X such that ∥x∥≤r and ∥y∥≤r.


## 3. Mann-Type Viscosity Algorithms in Uniformly Convex and 2-Uniformly Smooth Banach Spaces

In this section, we introduce Mann-type viscosity iterative algorithms in uniformly convex and2-uniformly smooth Banach spaces and show strong convergence theorems. We will use the following useful lemma.Lemma 17.
LetC be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X. Let A:C→X be an α-inverse strongly accretive mapping. Then, one has
(34)∥(I-λA)x-(I-λA)y∥2≤∥x-y∥2+2λ(λκ2-α)∥Ax-Ay∥2,∀x,y∈C,
where λ>0. In particular, if 0<λ≤α/κ2, then I-λA is nonexpansive.Theorem 18.
LetX be a uniformly convex and 2-uniformly smooth Banach space and let C be a nonempty closed convex subset of X such that C±C⊂C. Let ΠC be a sunny nonexpansive retraction from X onto C. Let T, F:X→CB(X), and A:C→2C be three multivalued mappings, let g:X→C be a single-valued mapping, and let N(·,·):X×X→C be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4. Consider that
(C6)N(Tx,Fx)+A(g(x)):X→2C∖{∅} is ζ-inverse strongly accretive with ζ≥κ2.LetAi: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 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 Δ:=⋂i=0∞‍Fix(Si)∩Γ∩(∩i=1NVI(C,Ai))≠∅. Suppose that {αn},{βn},{γn},{δn},{σn}, and {ϵn} are the sequences in [0,1],αn+βn+γn+δn=1 and satisfy the following conditions:(i)
∑n=1∞‍(|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|+|σn-σn-1|+|ϵn-ϵn-1|)<∞;(ii)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(iii)
{γn},{δn}⊂[c,d] for some c,d∈(0,1);(iv)
0<liminfn→∞βn≤limsupn→∞βn<1;(v)
0<liminfn→∞σn≤limsupn→∞σn<1;(vi)
0<liminfn→∞ϵn≤limsupn→∞ϵn<1.For arbitraryx0∈C, define the sequence {xn} iteratively by
(35)yn=αnf(xn)+βnxn+γnBxn+δnSnxn,xn+1=ϵn[xn-σn(N(wn,kn)+un)]+(1-ϵn)yn,un∈A(g(xn)),∀n≥0,
where {un} is defined by
(36)∥un-un+1∥≤(1+ɛ)H(A(g(xn+1)),A(g(xn))),∀n≥0,
for any wn∈Txn, kn∈Fxn, and some ɛ>0. 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 {xn} converges strongly to q∈Δ, which solves the following VIP:
(37)〈q-f(q),J(q-p)〉≤0,∀p∈Δ,
and, for any w∈Tq and k∈Fq, (q,w,k) is a solution of the MVVI (16).Proof.
First of all, by Lemma17 we know that I-λiAi is a nonexpansive mapping, where λi∈(0,αi/κ2) for each i=1,…,N. Hence, from the nonexpansivity of ΠC, it follows that Gi is a nonexpansive mapping for each i=1,…,N. Since B:C→C is the K-mapping generated by G1,…,GN and ρ1,…,ρN, by Lemma 3, we deduce that Fix(B)=∩i=1NFix(Gi). Utilizing Lemma 10, and the definition of Gi, we get Fix(Gi)=VI(C,Ai) for each i=1,…,N. Thus, we have
(38)Fix(B)=⋂i=1N‍Fix(Gi)=⋂i=1N‍VI(C,Ai).
Now, let us show that for anyv∈C, λ>0, there exists a point x~∈C such that (x~,w,k) is a solution of the MVVI (15), for any w∈Tx~ and k∈Fx~. Indeed, following the argument idea in the proof of Chidume et al. [5, Theorem 3.1], we put Vx:=N(Tx,Fx) for all x∈X. Then by Proposition 6, V is ϕ-strongly accretive. Since T and F are H-uniformly continuous and N(·,·) is continuous, Vx is continuous and hence lower semicontinuous. Thus, by Proposition 7,  Vx is single-valued. Moreover, since V is ϕ-strongly accretive and by assumption A∘g:X→2C is m-accretive, we have that V+λA∘g is an m-accretive and ϕ-strongly accretive mapping, and hence by Cioranescu [22, page 184], for any x∈X, we have that (V+λA∘g)(x) is closed and bounded. Therefore, by Morales [23], V+λA∘g is surjective. Hence, for any v∈X and λ>0, there exists x~∈D(A)=C such that v∈Vx~+λA(g(x~))=N(w,k)+λA(g(x~)), where w∈Tx~ and k∈Fx~. In addition, in terms of Proposition 7, we know that V+λA∘g is a single-valued mapping. Assume that N(Tx,Fx)+λA(g(x)):X→C is ζ-inverse strongly accretive with ζ≥κ2. Then by Lemma 17, we conclude that the mapping x↦x-(N(Tx,Fx)+λA(g(x))) is nonexpansive.
Without loss of generality, we may assume thatv=0 and λ=1. Let p∈Δ and let r(≥∥f(p)-p∥/(1-ρ)) be sufficiently large such that x0∈B¯r(p)=:B. Then p∈D(A)=C such that 0∈N(w,k)+A∘g(p) for any w∈Tp and k∈Fp. Let M:=sup{∥u∥:u∈N(w,k)+A(g(x)), x∈B, w∈Tx, k∈Fx}. Then as A∘g,T, and F are H-uniformly continuous on X, for ɛ1:=ϕ(r)/8(1+ɛ), ɛ2:=ϕ(r)/8μ(1+ɛ), and ɛ3:=ϕ(r)/8ξ(1+ɛ), there exist δ1, δ2, δ3>0 such that for any x,y∈X,∥x-y∥<δ1,∥x-y∥<δ2 and ∥x-y∥<δ3 imply H(A∘g(x),A∘g(y))<ɛ1,H(Tx,Ty)<ɛ2 and H(Fx,Fy)<ɛ3, respectively.
Let us show thatxn∈B for all n≥0. We show this by induction. First, x0∈B by construction. Assume that xn∈B. We show that xn+1∈B. If possible we assume that xn+1∉B, then ∥xn+1-p∥>r. Further from (35) it follows that
(39)∥yn-p∥=∥αn(f(xn)-p)+βn(xn-p)+γn(Bxn-p)+δn(Snxn-p)∥≤αn∥f(xn)-p∥+βn∥xn-p∥+γn∥Bxn-p∥+δn∥Snxn-p∥≤αn(∥f(xn)-f(p)∥+∥f(p)-p∥)+βn∥xn-p∥+γn∥Bxn-p∥+δn∥Snxn-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∥=(1-αn(1-ρ))∥xn-p∥+αn(1-ρ)∥f(p)-p∥1-ρ≤max{∥xn-p∥,∥f(p)-p∥1-ρ},
and hence
(40)∥xn+1-p∥2=〈ϵn[xn-p-σn(N(wn,kn)+un)]+(1-ϵn)(yn-p),J(xn+1-p)〉=〈ϵn(xn-p)+(1-ϵn)(yn-p),J(xn+1-p)〉-ϵnσn〈N(wn,kn)+un,J(xn+1-p)〉≤∥αn(xn-p)+(1-αn)(yn-p)∥∥xn+1-p∥-ϵnσn〈N(wn,kn)+un,J(xn+1-p)〉≤(ϵn∥xn-p∥+(1-ϵn)∥yn-p∥)∥xn+1-p∥-αnσn〈N(wn,kn)+un,J(xn+1-p)〉≤(∥f(p)-p∥1-ρϵn∥xn-p∥+(1-ϵn)max{∥xn-p∥,∥f(p)-p∥1-ρ})∥xn+1-p∥-αnσn〈N(wn,kn)+un,J(xn+1-p)〉≤max{∥xn-p∥,∥f(p)-p∥1-ρ}∥xn+1-p∥-αnσn〈N(wn,kn)+un,J(xn+1-p)〉≤12(max{∥xn-p∥2,∥f(p)-p1-ρ∥2}+∥xn+1-p∥2)-αnσn〈N(wn,kn)+un,J(xn+1-p)〉,
which immediately yields(41)∥xn+1-p∥2≤max{∥xn-p∥2,∥f(p)-p1-ρ∥2}-2αnσn〈N(wn,kn)+un,J(xn+1-p)〉=max{∥xn-p∥2,∥f(p)-p1-ρ∥2}-2αnσn〈N(wn+1,kn+1)+un+1,J(xn+1-p)〉-2αnσn〈N(wn,kn)+un-(N(wn+1,kn+1)+un+1),J(xn+1-p)〉.
SinceN(·,·) is ϕ-strongly accretive with respect to T and A(g(·)) is accretive, we deduce from (41) that
(42)∥xn+1-p∥2≤max{∥xn-p∥2,∥f(p)-p1-ρ∥2}-2αnσnϕ(∥xn+1-p∥)∥xn+1-p∥+2αnσn[∥N(wn+1,kn+1)-N(wn,kn)∥+∥un+1-un∥]∥xn+1-p∥≤max{∥xn-p∥2,∥f(p)-p1-ρ∥2}-2αnσnϕ(∥xn+1-p∥)∥xn+1-p∥+2αnσn[∥N(wn+1,kn+1)-N(wn+1,kn)∥+∥N(wn+1,kn)-N(wn,kn)∥+∥un+1-un∥]∥xn+1-p∥.
Again from (35), we have that
(43)∥xn+1-p∥≤ϵn∥xn-p-σn(N(wn,kn)+un)∥+(1-ϵn)∥yn-p∥≤ϵn[∥xn-p∥+σn∥N(wn,kn)+un∥]+(1-ϵn)max{∥xn-p∥,∥f(p)-p∥1-ρ}≤ϵn[r+σnM]+(1-ϵn)r≤2r.
Also, from Proposition7, Vx=N(Tx,Fx) is a single-valued mapping; that is, for any k,k′∈Fx and w,w′∈Tx, we have N(w,k)=N(w,k′) and N(w,k)=N(w′,k). On the other hand, it follows from Nadler [24] that, for kn+1∈Fxn+1 and wn+1∈Txn+1, there exist kn′∈Fxn and wn′∈Txn such that
(44)∥kn+1-kn′∥≤(1+ɛ)H(Fxn+1,Fxn),(45)∥wn+1-wn′∥≤(1+ɛ)H(Txn+1,Txn),
respectively. Therefore, from (42) and (36), we have
(46)∥xn+1-p∥2≤max{∥xn-p∥2,∥f(p)-p1-ρ∥2}-2αnσnϕ(r)r+2αnσn[∥N(wn+1,kn+1)-N(wn+1,kn′)∥+∥N(wn+1,kn)-N(wn′,kn)∥+∥un+1-un∥kn′]2r≤max{∥xn-p∥2,∥f(p)-p1-ρ∥2}-2αnσnϕ(r)r+2αnσn[ξ(1+ɛ)H(Fxn+1,Fxn)+μ(1+ɛ)H(Txn+1,Txn)+(1+ɛ)H(A(g(xn+1)),A(g(xn)))]2r≤max{∥xn-p∥2,∥f(p)-p1-ρ∥2}-2αnσnϕ(r)r+2αnσn[ϕ(r)8+ϕ(r)8+ϕ(r)8]2r=max{∥xn-p∥2,∥f(p)-p1-ρ∥2}-2αnσnϕ(r)r+αnσn32ϕ(r)r≤max{∥xn-p∥2,∥f(p)-p1-ρ∥2}.
So, we get∥xn+1-p∥≤r, a contradiction. Therefore, {xn} is bounded.
Let us show thatlimn→∞∥xn-xn+1∥=0 and limn→∞∥xn-yn∥=0.
Indeed, we defineG:C→C by Gx:=x-(N(Tx,Fx)+A(g(x))) for all x∈C. Then, G is a nonexpansive mapping and the iterative scheme (35) can be rewritten as follows:
(47)yn=αnf(xn)+βnxn+γnBxn+δnSnxn,xn+1=ϵn[(1-σn)xn+σnGxn]+(1-ϵn)yn,∀n≥0.
Taking into account condition (iv), we may assume that{βn}⊂[a,b] for some a,b∈(0,1). From (47), we can rewrite yn by
(48)yn=βnxn+(1-βn)zn,
where zn=(αnf(xn)+γnBxn+δnSnxn)/(1-βn). Now, we have
(49)∥zn+1-zn∥=∥αn+1f(xn+1)+γn+1Bxn+1+δn+1Sn+1xn+11-βn+1-αnf(xn)+γnBxn+δnSnxn1-βn∥=∥yn+1-βn+1xn+11-βn+1-yn-βnxn1-βn∥=∥yn+1-βn+1xn+11-βn+1-yn-βnxn1-βn+1+yn-βnxn1-βn+1-yn-βnxn1-βn∥≤∥yn+1-βn+1xn+11-βn+1-yn-βnxn1-βn+1∥+∥yn-βnxn1-βn+1-yn-βnxn1-βn∥=11-βn+1∥yn+1-βn+1xn+1-(yn-βnxn)∥+|11-βn+1-11-βn|∥yn-βnxn∥=11-βn+1∥yn+1-βn+1xn+1-(yn-βnxn)∥+|βn+1-βn|(1-βn)(1-βn+1)∥yn-βnxn∥=11-βn+1∥αn+1f(xn+1)+γn+1Bxn+1+δn+1Sn+1xn+1-(αnf(xn)+γnBxn+δnSnxn)∥+|βn+1-βn|(1-βn)(1-βn+1)∥yn-βnxn∥≤11-βn+1(αn+1∥f(xn+1)-f(xn)∥+γn+1∥Bxn+1-Bxn∥+δn+1∥Sn+1xn+1-Snxn∥+|αn+1-αn|∥f(xn)∥+|γn+1-γn|∥Bxn∥+|δn+1-δn|∥Snxn∥)+|βn+1-βn|(1-βn)(1-βn+1)∥yn-βnxn∥≤11-βn+1[αn+1∥f(xn+1)-f(xn)∥+γn+1∥xn+1-xn∥+δn+1(∥Sn+1xn+1-Sn+1xn∥+∥Sn+1xn-Snxn∥)+|αn+1-αn|∥f(xn)∥+|γn+1-γn|×∥Bxn∥+|δn+1-δn|∥Snxn∥]+|βn+1-βn|(1-βn)(1-βn+1)∥yn-βnxn∥≤11-βn+1[αn+1ρ∥xn+1-xn∥+γn+1∥xn+1-xn∥+δn+1(∥xn+1-xn∥+∥Sn+1xn-Snxn∥)+|αn+1-αn|∥f(xn)∥+|γn+1-γn|∥Bxn∥+|δn+1-δn|∥Snxn∥]+|βn+1-βn|(1-βn)(1-βn+1)∥αnf(xn)+γnBxn+δnSnxn∥=1-βn+1-αn+1(1-ρ)1-βn+1∥xn+1-xn∥+δn+11-βn+1∥Sn+1xn-Snxn∥+11-βn+1[|αn+1-αn|∥f(xn)∥+|γn+1-γn|×∥Bxn∥+|δn+1-δn|∥Snxn∥]+|βn+1-βn|(1-βn)(1-βn+1)∥αnf(xn)+γnBxn+δnSnxn∥≤(1-αn+1(1-ρ)1-βn+1)∥xn+1-xn∥+∥Sn+1xn-Snxn∥+M0[|αn+1-αn|+|βn+1-βn|+|γn+1-γn|+|δn+1-δn|],
where 1/(1-b)2supn≥0{∥f(xn)∥+∥Bxn∥+∥Snxn∥}≤M0 for some M0>0. By simple calculation, we have
(50)yn-yn-1=βn(xn-xn-1)+(βn-βn-1)×(xn-1-zn-1)+(1-βn)(zn-zn-1).
So, from (49), we get
(51)∥yn-yn-1∥≤βn∥xn-xn-1∥+|βn-βn-1|∥xn-1-zn-1∥+(1-βn)∥zn-zn-1∥≤βn∥xn-xn-1∥+|βn-βn-1|∥xn-1-zn-1∥+(1-βn){(1-αn(1-ρ)1-βn)∥xn-xn-1∥+∥Snxn-1-Sn-1xn-1∥+M0[|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|]αn(1-ρ)1-βn}≤(1-αn(1-ρ))∥xn-xn-1∥+|βn-βn-1|∥xn-1-zn-1∥+∥Snxn-1-Sn-1xn-1∥+M0[|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|].
Also, for convenience, we write(52)xn+1=ϵnz^n+(1-ϵn)yn,z^n=σnGxn+(1-σn)xn.
By simple calculation, we get(53)xn+1-xn=ϵn(z^n-z^n-1)+(ϵn-ϵn-1)×(z^n-1-yn-1)+(1-ϵn)(yn-yn-1),z^n-z^n-1=σn(Gxn-Gxn-1)+(σn-σn-1)×(Gxn-1-xn-1)+(1-σn)(xn-xn-1).
From (51) and (53), we deduce that
(54)∥z^n-z^n-1∥≤σn∥Gxn-Gxn-1∥+|σn-σn-1|×∥Gxn-1-xn-1∥+(1-σn)∥xn-xn-1∥≤σn∥xn-xn-1∥+|σn-σn-1|×∥Gxn-1-xn-1∥+(1-σn)∥xn-xn-1∥≤∥xn-xn-1∥+|σn-σn-1|∥Gxn-1-xn-1∥,
and hence
(55)∥xn+1-xn∥≤ϵn∥z^n-z^n-1∥+|ϵn-ϵn-1|×∥z^n-1-yn-1∥+(1-ϵn)∥yn-yn-1∥≤ϵn[∥xn-xn-1∥+|σn-σn-1|∥Gxn-1-xn-1∥]+|ϵn-ϵn-1|∥z^n-1-yn-1∥+(1-ϵn){(1-αn(1-ρ))∥xn-xn-1∥+|βn-βn-1|∥xn-1-zn-1∥+∥Snxn-1-Sn-1xn-1∥+M0[|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|]}≤[1-(1-ϵn)αn(1-ρ)]∥xn-xn-1∥+|σn-σn-1|∥Gxn-1-xn-1∥+|ϵn-ϵn-1|∥z^n-1-yn-1∥+|βn-βn-1|×∥xn-1-zn-1∥+∥Snxn-1-Sn-1xn-1∥+M0[|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|]≤[1-(1-ϵn)αn(1-ρ)]∥xn-xn-1∥+∥Snxn-1-Sn-1xn-1∥+M1[|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|+|σn-σn-1|+|ϵn-ϵn-1|],
where supn≥1{∥Gxn-1-xn-1∥+∥z^n-1-yn-1∥+∥xn-1-zn-1∥+M0}≤M1 for some M1>0. Utilizing Lemma 17, we conclude from (55), conditions (i), (ii), and (vi), and the assumption on {Sn} that
(56)limn→∞∥xn-xn+1∥=0.
Furthermore, utilizing Lemma16, we obtain from (39) and (47) that
(57)∥xn+1-p∥2=∥ϵn[(1-σn)(xn-p)+σn(Gxn-p)]+(1-ϵn)(yn-p)∥2≤ϵn∥(1-σn)(xn-p)+σn(Gxn-p)∥2+(1-ϵn)∥yn-p∥2-ϵn(1-ϵn)×φ(∥(1-σn)(xn-yn)+σn(Gxn-yn)∥)≤ϵn[(1-σn)∥xn-p∥2+σn∥Gxn-p∥2-σn(1-σn)φ1(∥xn-Gxn∥)∥xn-p∥2]+(1-ϵn)∥yn-p∥2-ϵn(1-ϵn)×φ(∥(1-σn)(xn-yn)+σn(Gxn-yn)∥)≤ϵn[(1-σn)∥xn-p∥2+σn∥xn-p∥2-σn(1-σn)φ1(∥xn-Gxn∥)∥xn-p∥2]+(1-ϵn)[∥xn-p∥+αn∥f(p)-p∥]2-ϵn(1-ϵn)φ(∥(1-σn)(xn-yn)+σn(Gxn-yn)∥)=ϵn[∥xn-p∥2-σn(1-σn)φ1(∥xn-Gxn∥)]+(1-ϵn)[∥xn-p∥2+αn∥f(p)-p∥×(2∥xn-p∥+αn∥f(p)-p∥)∥xn-p∥2]-ϵn(1-ϵn)φ(∥(1-σn)(xn-yn)+σn(Gxn-yn)∥)≤∥xn-p∥2-ϵnσn(1-σn)φ1(∥xn-Gxn∥)+αn∥f(p)-p∥(2∥xn-p∥+αn∥f(p)-p∥)-ϵn(1-ϵn)φ(∥(1-σn)(xn-yn)+σn(Gxn-yn)∥),
which immediately yields
(58)ϵnσn(1-σn)φ1(∥xn-Gxn∥)+ϵn(1-ϵn)×φ(∥(1-σn)(xn-yn)+σn(Gxn-yn)∥)≤∥xn-p∥2-∥xn+1-p∥2+αn∥f(p)-p∥×(2∥xn-p∥+αn∥f(p)-p∥)≤(∥xn-p∥+∥xn+1-p∥)∥xn-xn+1∥+αn∥f(p)-p∥(2∥xn-p∥+αn∥f(p)-p∥).
So, from (56) and conditions (ii), (v), and (vi), we get
(59)limn→∞φ1(∥xn-Gxn∥)=0,limn→∞φ(∥(1-σn)(xn-yn)+σn(Gxn-yn)∥)=0,
which together with the properties of φ and φ1 implies that
(60)limn→∞∥xn-Gxn∥=0,limn→∞∥(1-σn)(xn-yn)+σn(Gxn-yn)∥=0.
Note that(61)∥xn-yn∥=∥(1-σn)(xn-yn)+σn(Gxn-yn)+σn(xn-Gxn)∥≤∥(1-σn)(xn-yn)+σn(Gxn-yn)∥+σn∥xn-Gxn∥≤∥(1-σn)(xn-yn)+σn(Gxn-yn)∥+∥xn-Gxn∥.
Hence, from (60), it follows that
(62)limn→∞∥xn-yn∥=0.
Let us show thatlimn→∞∥xn-Bxn∥=0 and limn→∞∥xn-Sxn∥=0.
Indeed, from the definition ofyn, we can rewrite yn by
(63)yn=αnf(xn)+βnxn+γnBxn+δnSnxn=αnf(xn)+βnxn+(γn+δn)γnBxn+δnSnxnγn+δn=αnf(xn)+βnxn+enzn′,
where en=γn+δn and zn′=(γnBxn+δnSnxn)/(γn+δn).
Utilizing Lemma12, from (63) we have
(64)∥yn-p∥2=∥αn(f(xn)-p)+βn(xn-p)+en(zn′-p)∥2≤αn∥f(xn)-p∥2+βn∥xn-p∥2+en∥zn′-p∥2-βnenφ2(∥zn′-xn∥)=αn∥f(xn)-p∥2+βn∥xn-p∥2-βnenφ2(∥zn′-xn∥)+en∥γnBxn+δnSnxnγn+δn-p∥2=αn∥f(xn)-p∥2+βn∥xn-p∥2-βnenφ2(∥zn′-xn∥)+en∥(1-δnγn+δn)(Bxn-p)+δnγn+δn(Snxn-p)∥2≤αn∥f(xn)-p∥2+βn∥xn-p∥2-βnenφ2(∥zn′-xn∥)+en((1-δnγn+δn)∥Bxn-p∥+δnγn+δn∥Snxn-p∥)2≤αn∥f(xn)-p∥2+βn∥xn-p∥2-βnenφ2(∥zn′-xn∥)+en∥xn-p∥2≤αn∥f(xn)-p∥2+∥xn-p∥2-βnenφ2(∥zn′-xn∥),
which implies that
(65)βnenφ2(∥zn′-xn∥)≤αn∥f(xn)-p∥2+∥xn-p∥2-∥yn-p∥2≤αn∥f(xn)-p∥2+(∥xn-p∥+∥yn-p∥)∥xn-yn∥.
From (62) and conditions (ii), (iii), and (iv), we have
(66)limn→∞φ2(∥zn′-xn∥)=0.
From the properties of φ2, we have
(67)limn→∞∥zn′-xn∥=0.
By Lemma 16, we deduce from the definition of zn′ the following
(68)∥zn′-p∥2=∥γnBxn+δnSnxnγn+δn-p∥2=∥(1-δnδn+γn)(Bxn-p)+δnδn+γn(Snxn-p)∥2≤(1-δnδn+γn)∥Bxn-p∥2+δnδn+γn∥Snxn-p∥2-δnδn+γn(1-δnδn+γn)φ3(∥Bxn-Snxn∥)≤∥xn-p∥2-δnδn+γn(1-δnδn+γn)φ3(∥Bxn-Snxn∥),
which implies that
(69)δnδn+γn(1-δnδn+γn)φ3(∥Bxn-Snxn∥)≤∥xn-p∥2-∥zn′-p∥2≤(∥xn-p∥+∥zn′-p∥)∥xn-zn′∥.
From (67) and condition (iii), we have
(70)limn→∞φ3(∥Bxn-Snxn∥)=0.
From the properties of φ3, we have
(71)limn→∞∥Bxn-Snxn∥=0.
From the definition of yn, we can rewrite yn by
(72)yn=αnf(xn)+βnxn+γnBxn+δnSnxn=βnxn+γnBxn+(αn+δn)αnf(xn)+δnSnxnαn+δn=βnxn+γnBxn+dnzn′′,
where dn=αn+δn and zn′′=(αnf(xn)+δnSnxn)/(αn+δn).
Utilizing Lemma12, from (72) and the convexity of ∥·∥2, we have
(73)∥yn-p∥2=∥βn(xn-p)+γn(Bxn-p)+dn(zn′′-p)∥2≤βn∥xn-p∥2+γn∥Bxn-p∥2+dn∥zn′′-p∥2-βnγnφ4(∥xn-Bxn∥)=βn∥xn-p∥2+γn∥Bxn-p∥2+dn∥αnf(xn)+δnSnxnαn+δn-p∥2-βnγnφ4(∥xn-Bxn∥)=βn∥xn-p∥2+γn∥Bxn-p∥2+dn∥αnαn+δn(f(xn)-p)+(1-αnαn+δn)(Snxn-p)∥2-βnγnφ4(∥xn-Bxn∥)≤βn∥xn-p∥2+γn∥Bxn-p∥2+dn[αnαn+δn∥f(xn)-p∥2+(1-αnαn+δn)∥Snxn-p∥2]-βnγnφ4(∥xn-Bxn∥)=βn∥xn-p∥2+γn∥Bxn-p∥2+αn∥f(xn)-p∥2+δn∥Snxn-p∥2-βnγnφ4(∥xn-Bxn∥)≤βn∥xn-p∥2+γn∥xn-p∥2+αn∥f(xn)-p∥2+δn∥xn-p∥2-βnγnφ4(∥xn-Bxn∥)≤∥xn-p∥2+αn∥f(xn)-p∥2-βnγnφ4(∥xn-Bxn∥),
which implies that
(74)βnγnφ4(∥xn-Bxn∥)≤∥xn-p∥2-∥yn-p∥2+αn∥f(xn)-p∥2≤(∥xn-p∥+∥yn-p∥)∥xn-yn∥+αn∥f(xn)-p∥2.
From (62), (74), and conditions (ii), (iii), and (iv), we have
(75)limn→∞φ4(∥xn-Bxn∥)=0.
By the properties ofφ4, we have
(76)limn→∞∥xn-Bxn∥=0.
From (71), (76), and
(77)∥xn-Snxn∥≤∥xn-Bxn∥+∥Bxn-Snxn∥,
we have
(78)limn→∞∥xn-Snxn∥=0.
Observe that(79)∥xn-Sxn∥≤∥xn-Snxn∥+∥Snxn-Sxn∥.
Utilizing Lemma 13, we conclude from (78) that
(80)limn→∞∥xn-Sxn∥=0.
Define a mappingWx=(1-θ1-θ2)Bx+θ1Sx+θ2Gx, where θ1,θ2∈(0,1) are two constants with θ1+θ2<1. Then by Lemma 15, we have Fix(W)=Fix(B)∩Fix(S)∩Fix(G)=Δ. We observe that
(81)∥xn-Wxn∥=∥(1-θ1-θ2)(xn-Bxn)+θ1(xn-Sxn)+θ2(xn-Gxn)∥≤(1-θ1-θ2)∥xn-Bxn∥+θ1∥xn-Sxn∥+θ2∥xn-Gxn∥.
From (60), (76), and (80), we obtain
(82)limn→∞∥xn-Wxn∥=0.
Now, we claim that(83)limsupn→∞〈f(q)-q,J(xn-q)〉≤0,
where q=s-limt→0xt with xt being the fixed point of the contraction
(84)x⟼tf(x)+(1-t)Wx.
Then xt solves the fixed point equation xt=tf(xt)+(1-t)Wxt. Thus we have
(85)xt-xn=(1-t)(Wxt-xn)+t(f(xt)-xn).
By Lemma 9, we conclude that
(86)∥xt-xn∥2=∥(1-t)(Wxt-xn)+t(f(xt)-xn)∥2≤(1-t)2∥Wxt-xn∥2+2t〈f(xt)-xn,J(xt-xn)〉≤(1-t)2(∥Wxt-Wxn∥+∥Wxn-xn∥)2+2t〈f(xt)-xn,J(xt-xn)〉≤(1-t)2(∥xt-xn∥+∥Wxn-xn∥)2+2t〈f(xt)-xn,J(xt-xn)〉=(1-t)2[∥xt-xn∥2+2∥xt-xn∥×∥Wxn-xn∥+∥Wxn-xn∥2]+2t〈f(xt)-xt,J(xt-xn)〉+2t〈xt-xn,J(xt-xn)〉=(1-2t+t2)∥xt-xn∥2+fn(t)+2t〈f(xt)-xt,J(xt-xn)〉+2t∥xt-xn∥2,
where
(87)fn(t)=(1-t)2(2∥xt-xn∥+∥xn-Wxn∥)×∥xn-Wxn∥⟶0,asn⟶∞.
It follows from (86) that
(88)〈xt-f(xt),J(xt-xn)〉≤t2∥xt-xn∥2+12tfn(t).
Lettingn→∞ in (88) and noticing (87), we derive
(89)limsupn→∞〈xt-f(xt),J(xt-xn)〉≤t2M2,
where M2>0 is a constant such that ∥xt-xn∥2≤M2 for all t∈(0,1) and n≥0. Taking t→0 in (89), we have
(90)limsupt→0limsupn→∞〈xt-f(xt),J(xt-xn)〉≤0.
On the other hand, we have(91)〈f(q)-q,J(xn-q)〉=〈f(q)-q,J(xn-q)〉-〈f(q)-q,J(xn-xt)〉+〈f(q)-q,J(xn-xt)〉-〈f(q)-xt,J(xn-xt)〉+〈f(q)-xt,J(xn-xt)〉-〈f(xt)-xt,J(xn-xt)〉+〈f(xt)-xt,J(xn-xt)〉=〈f(q)-q,J(xn-q)-J(xn-xt)〉+〈xt-q,J(xn-xt)〉+〈f(q)-f(xt),J(xn-xt)〉+〈f(xt)-xt,J(xn-xt)〉.
It follows that(92)limsupn→∞〈f(q)-q,J(xn-q)〉≤limsupn→∞〈f(q)-q,J(xn-q)-J(xn-xt)〉+∥xt-q∥limsupn→∞∥xn-xt∥+ρ∥q-xt∥limsupn→∞∥xn-xt∥+limsupn→∞〈f(xt)-xt,J(xn-xt)〉.
Taking into account thatxt→q as t→0, we have
(93)limsupn→∞〈f(q)-q,J(xn-q)〉=limsupt→0limsupn→∞〈f(q)-q,J(xn-q)〉≤limsupt→0limsupn→∞〈f(q)-q,J(xn-q)-J(xn-xt)〉.
SinceX has a uniformly Fréchet differentiable norm, the duality mapping J is norm-to-norm uniformly continuous on bounded subsets of X. Consequently, the two limits are interchangeable and hence (83) holds. Noticing that J is norm-to-norm uniformly continuous on bounded subsets of X, we deduce from (62) that
(94)limsupn→∞〈f(q)-q,J(yn-q)〉=limsupn→∞(〈f(q)-q,J(xn-q)+〈f(q)-q,J(yn-q)-J(xn-q)〉)=limsupn→∞〈f(q)-q,J(xn-q)〉≤0.
Finally, let us show thatxn→q as n→∞. Indeed, utilizing Lemma 9, we obtain from (47) that
(95)∥yn-q∥2=∥αn(f(xn)-f(q))+βn(xn-q)+γn(Bxn-q)+δn(Snxn-q)+αn(f(q)-q)∥2≤∥αn(f(xn)-f(q))+βn(xn-q)+γn(Bxn-q)+δn(Snxn-q)∥2+2αn〈f(q)-q,J(yn-q)〉≤αn∥f(xn)-f(q)∥2+βn∥xn-q∥2+γn∥Bxn-q∥2+δn∥Snxn-q∥2+2αn〈f(q)-q,J(yn-q)〉≤αnρ∥xn-q∥2+βn∥xn-q∥2+γn∥xn-q∥+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
(96)∥xn+1q∥2=∥ϵn[(1-σn)(xn-q)+σn(Gxn-q)]+(1-ϵn)(yn-q)∥2≤ϵn∥(1-σn)(xn-q)+σn(Gxn-q)∥2+(1-ϵn)∥yn-q∥2≤ϵn[(1-σn)∥xn-q∥2+σn∥Gxn-q∥2]+(1-ϵn)∥yn-q∥2≤ϵn[(1-σn)∥xn-q∥2+σn∥xn-q∥2]+(1-ϵn)∥yn-q∥2=ϵn∥xn-q∥2+(1-ϵn)∥yn-q∥2≤ϵn∥xn-q∥2+(1-ϵn)×[(1-αn(1-ρ))∥xn-q∥2+2αn〈f(q)-q,J(yn-q)〉]=[1-(1-ϵn)αn(1-ρ)]∥xn-q∥2+2(1-ϵn)αn〈f(q)-q,J(yn-q)〉=[1-(1-ϵn)αn(1-ρ)]∥xn-q∥2+(1-ϵn)αn(1-ρ)2〈f(q)-q,J(yn-q)〉1-ρ.
Applying Lemma8 to (96), we conclude from conditions (ii) and (vi) and (94) that xn→q as n→∞. This completes the proof.Corollary 19.
LetX be a uniformly convex and 2-uniformly smooth Banach space and let C be a nonempty closed convex subset of X such that C±C⊂C. Let ΠC be a sunny nonexpansive retraction from X onto C. Let T,F:X→CB(X), and A:C→2C be three multivalued mappings, let g:X→C be a single-valued mapping, and let N(·,·):X×X→C be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4 and (C6) N(Tx,Fx)+A(g(x)):X→2C∖{∅} is ζ-inverse strongly accretive with ζ≥κ2.LetTi:C→X be a ηi-strictly pseudocontractive mapping for each i=1,…,N. Define the mapping Gi:C→C by Gi=ΠC(I-λi(I-Ti)) for i=1,…,N, where λi∈(0,ηi/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 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 Δ:=⋂i=0∞‍Fix(Si)∩Γ∩(∩i=1NFix(Ti))≠∅. Suppose that {αn},{βn},{γn},{δn},{σn}, and {ϵn} are the sequences in [0,1],αn+βn+γn+δn=1 and satisfy the following conditions:(i)
∑n=1∞‍(|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|+|σn-σn-1|+|ϵn-ϵn-1|)<∞;(ii)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(iii)
{γn},{δn}⊂[c,d] for some c,d∈(0,1);(iv)
0<liminfn→∞βn≤limsupn→∞βn<1;(v)
0<liminfn→∞σn≤limsupn→∞σn<1;(vi)
0<liminfn→∞ϵn≤limsupn→∞ϵn<1.For arbitraryx0∈C, define the sequence {xn} iteratively by
(97)yn=αnf(xn)+βnxn+γnBxn+δnSnxn,xn+1=ϵn[xn-σn(N(wn,kn)+un)]+(1-ϵn)yn,un∈A(g(xn)),∀n≥0,
where {un} is defined by
(98)∥un-un+1∥≤(1+ɛ)H(A(g(xn+1)),A(g(xn))),∀n≥0,
for any wn∈Txn, kn∈Fxn and some ɛ>0. 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 {xn} converges strongly to q∈Δ, which solves the following VIP:
(99)〈q-f(q),J(q-p)〉≤0,∀p∈Δ,
and, for any w∈Tq and k∈Fq,(q,w,k) is a solution of the MVVI (16).Proof.
SinceTi is a ηi-strictly pseudocontractive mapping for each i=1,…,N, it is known that Ai:=I-Ti is ηi-inverse strongly accretive for each i=1,…,N. In Theorem 18, we put Gi=ΠC(I-λiAi) for i=1,…,N, where λi∈(0,ηi/κ2). It is not hard to see that Fix(Ti)=VI(C,Ai). As a matter of fact, we have, for λi>0,
(100)u∈VI(C,Ai)⟺〈Aiu,J(y-u)〉≥0∀y∈C⟺〈u-λiAiu-u,J(u-y)〉≥0∀y∈C⟺u=ΠC(u-λiAiu)⟺u=ΠC(u-λiu+λiTiu)⟺〈u-λiu+λiTiu-u,J(u-y)〉≥0∀y∈C⟺〈u-Tiu,J(u-y)〉≤0∀y∈C⟺u=Tiu⟺u∈Fix(Ti).
Accordingly, we conclude that Δ:=⋂i=0∞‍Fix(Si)∩Γ∩(∩i=1NVI(C,Ai))=  ⋂i=0∞‍Fix(Si)∩Γ∩(∩i=1NFix(Ti)). Therefore, the desired result follows from Theorem 18.Remark 20.
Theorem18 improves, extends, supplements, and develops [5, Theorem 3.2] and [25, Theorem 3.1] in the following aspects.(i) Kangtunyakarn's problem of finding a point ofFix(S)∩Fix(V)∩(∩i=1NVI(C,Ai)) (see [25, Theorem 1.1]) is extended to develop our problem of finding a point of ⋂i=0∞‍Fix(Si)∩Γ∩(∩i=1NVI(C,Ai)) in Theorem 18 because BA:=S((1-α)I+αV) is nonexpansive with α∈(0,η/κ2) and Fix(BA)=Fix(S)∩Fix(V) (see [25, Lemma2.12]). It is clear that the problem of finding a point of ⋂i=0∞‍Fix(Si)∩Γ∩(∩i=1NVI(C,Ai)) in Theorem 18 is more general and more subtle than the problem of finding a point of Γ in [5, Theorem 3.2].(ii) The iterative scheme in [25, Theorem 3.1] is extended to develop the iterative scheme (35) of Theorem 18 by virtue of the iterative schemes of [5, Theorem 3.2]. The iterative scheme (35) of Theorem 18 is more advantageous and more flexible than the iterative scheme of [10, Theorem 3.2] because it can be applied to solving three problems (i.e., MVVI (16), a finite family of VIPs, and the fixed point problem of {Sn}) and involves several parameter sequences {αn}, {βn}, {γn}, {δn}, {σn}, and {ϵn}.(iii) Theorem18 extends and generalizes [5, Theorems 3.2] to the setting of a countable family of nonexpansive mappings and a finite family of VIPs. In the meantime, Theorem 18 extends and generalizes Kangtunyakarn [25, Theorem 3.1] to the setting of the MVVI (16).(iv) The iterative scheme (35) in Theorem 18 is very different from every one in [5, Theorem 3.2] and [25, Theorem 3.1] because every iterative scheme in [25, Theorem 3.1] and [5, Theorem 3.2] is one-step iterative scheme and the iterative scheme (35) in Theorem 18 is the combination of two iterative schemes in [25, Theorem 3.1] and [5, Theorem 3.2].(v) No boundedness condition on the rangesR(I-N(T(·),F(·))) and R(A(g(·))) is imposed in Theorems 18.


## 4. Mann-Type Viscosity Algorithms in a Uniformly Convex Banach Space Having a Uniformly Gáteaux Differentiable Norm

In this section, we introduce Mann-type viscosity iterative algorithms in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm and show strong convergence theorems. First, we give the following useful lemma.Lemma 21.
LetC be a nonempty closed convex subset of a smooth Banach space X and let A:C→X be a ξ-strictly pseudocontractive and ν-strongly accretive mapping with ξ+ν≥1. Then, for λ∈(0,1], one has
(101)∥(I-λA)x-(I-λA)y∥≤{1-νξ+(1-λ)(1+1ξ)}∥x-y∥,∀x,y∈C.
In particular, if 1-(ξ/(1+ξ))(1-(1-ν)/ξ)≤λ≤1, then I-λA is nonexpansive.Theorem 22.
LetX be a nonempty closed convex subset of a uniformly convex Banach space which has a uniformly Gáteaux differentiable norm and let C be a nonempty closed convex subset of X such that C±C⊂C. Let ΠC be a sunny nonexpansive retraction from X onto C. Let T,F:X→CB(X), and A:C→2C be three multivalued mappings, let g:X→C be a single-valued mapping, and let N(·,·):X×X→C be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4. Consider that
(H6)N(Tx,Fx)+A(g(x)):X→C is ξ0-strictly pseudocontractive and ν0-strongly accretive with ξ0+ν0≥1.LetAi:C→X be ξi-strictly pseudocontractive and νi-strongly accretive with ξi+νi≥1 for each i=1,…,N. Define the mapping Gi:C→C by Gi=ΠC(I-λiAi) where 1-(ξi/(1+ξi))(1-(1-νi)/ξi)≤λi≤1 for each i=1,…,N. 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 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 Δ:=⋂i=0∞‍Fix(Si)∩Γ∩(∩i=1NVI(C,Ai))≠∅. Suppose that {αn},{βn},{γn},{δn},{σn}, and {ϵn} are the sequences in [0,1],αn+βn+γn+δn=1 and satisfy the following conditions:(i)
∑n=1∞‍(|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|+|σn-σn-1|+|ϵn-ϵn-1|)<∞;(ii)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(iii)
{γn},{δn}⊂[c,d] for some c,d∈(0,1);(iv)
0<liminfn→∞βn≤limsupn→∞βn<1;(v)
0<liminfn→∞σn≤limsupn→∞σn<1;(vi)
0<liminfn→∞ϵn≤limsupn→∞ϵn<1.For arbitraryx0∈C, define the sequence {xn} iteratively by
(102)yn=αnf(xn)+βnxn+γnBxn+δnSnxn,xn+1=ϵn[xn-σn(N(wn,kn)+un)]+(1-ϵn)yn,un∈A(g(xn)),∀n≥0,
where {un} is defined by
(103)∥un-un+1∥≤(1+ɛ)H(A(g(xn+1)),A(g(xn))),∀n≥0,
for any wn∈Txn, kn∈Fxn, and some ɛ>0. 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 {xn} converges strongly to q∈Δ, which solves the following VIP:
(104)〈q-f(q),J(q-p)〉≤0,∀p∈Δ,
and, for any w∈Tq and k∈Fq, (q,w,k) is a solution of the MVVI (16).Proof.
First of all, by Lemma21, we know that I-λiAi is a nonexpansive mapping, where 1-(ξi/(1+ξi))(1-(1-νi)/ξi)≤λi≤1 for each i=1,…,N. Hence, from the nonexpansivity of ΠC, it follows that Gi is a nonexpansive mapping for each i=1,…,N. Since B:C→C is the K-mapping generated by G1,…,GN and ρ1,…,ρN, by Lemma 3, we deduce that Fix(B)=∩i=1NFix(Gi). Utilizing Lemma 10 and the definition of Gi, we get Fix(Gi)=VI(C,Ai) for each i=1,…,N. Thus, we have
(105)Fix(B)=⋂i=1N‍Fix(Gi)=⋂i=1N‍VI(C,Ai).
Repeating the same arguments as those in the proof of Theorem18, we can prove that for any v∈C, λ>0, there exists a point x~∈C such that (x~,w,k) is a solution of the MVVI (15), for any w∈Tx~ and k∈Fx~. In addition, in terms of Proposition 7, we know that V+λA∘g is a single-valued mapping due to the fact that V+λA∘g is ϕ-strongly accretive. Assume that N(Tx,Fx)+A(g(x)):X→C is ξ0-strictly pseudocontractive and ν0-strongly accretive with ξ0+ν0≥1. Then by Lemma 21, we conclude that the mapping x↦x-(N(Tx,Fx)+λA(g(x))) is nonexpansive.
Without loss of generality, we may assume thatv=0 and λ=1. Let p∈Δ and let r(≥∥f(p)-p∥/(1-ρ)) be sufficiently large such that x0∈B¯r(p)=:B. Observe that
(106)∥yn-p∥≤αn∥f(xn)-p∥+βn∥xn-p∥+γn∥Bxn-p∥+δn∥Snxn-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∥≤max{∥xn-p∥,∥f(p)-p∥1-ρ}.
Utilizing (106) and repeating the same arguments as those in the proof of Theorem 18, we can derive xn∈B for all n≥0. Hence {xn} is bounded.
Let us show thatlimn→∞∥xn-xn+1∥=0 and limn→∞∥xn-yn∥=0.
Indeed, we defineG:C→C by Gx:=x-(N(Tx,Fx)+A(g(x))) for all x∈C. Then, G is a nonexpansive mapping and the iterative scheme (102) can be rewritten as follows:
(107)yn=αnf(xn)+βnxn+γnBxn+δnSnxn,xn+1=ϵn[(1-σn)xn+σnGxn]+(1-ϵn)yn,∀n≥0.
Repeating the same arguments as those of (56), (60), (62), (76), and (80) in the proof of Theorem 18, we can obtain that
(108)limn→∞∥xn-xn+1∥=0,limn→∞∥xn-yn∥=0,(109)limn→∞∥xn-Gxn∥=0,limn→∞∥xn-Bxn∥=0,limn→∞∥xn-Sxn∥=0.
Define a mappingWx=(1-θ1-θ2)Bx+θ1Sx+θ2Gx, where θ1,θ2∈(0,1) are two constants with θ1+θ2<1. Then by Lemma 15, we have that Fix(W)=Fix(B)∩Fix(S)∩Fix(G)=Δ. We observe that
(110)∥xn-Wxn∥=∥(1-θ1-θ2)(xn-Bxn)+θ1(xn-Sxn)+θ2(xn-Gxn)∥≤(1-θ1-θ2)∥xn-Bxn∥+θ1∥xn-Sxn∥+θ2∥xn-Gxn∥.
From (109), we obtain
(111)limn→∞∥xn-Wxn∥=0.
Now, we claim that(112)limsupn→∞〈f(q)-q,J(xn-q)〉≤0,
where q=s-limt→0xt with xt being the fixed point of the contraction
(113)x⟼tf(x)+(1-t)Wx.
Then xt solves the fixed point equation xt=tf(xt)+(1-t)Wxt. Repeating the same arguments as those of (93) in the proof of Theorem 18, we can deduce that
(114)limsupn→∞〈f(q)-q,J(xn-q)〉=limsupt→0limsupn→∞〈f(q)-q,J(xn-q)〉≤limsupt→0limsupn→∞〈f(q)-q,J(xn-q)-J(xn-xt)〉.
Since X has a uniformly Gáteaux differentiable norm, the duality mapping J is norm-to-weak* uniformly continuous on bounded subsets of X. Consequently, the two limits are interchangeable and hence (112) holds. Noticing that J is norm-to-weak* uniformly continuous on bounded subsets of X, we conclude from (108) that
(115)limsupn→∞〈f(q)-q,J(yn-q)〉=limsupn→∞(〈f(q)-q,J(xn-q)〉+〈f(q)-q,J(yn-q)-J(xn-q)〉)=limsupn→∞〈f(q)-q,J(xn-q)〉≤0.
Finally, let us show thatxn→q as n→∞. Indeed, repeating the same arguments as those (96) in the proof of Theorem 18, we can deduce from (107) that
(116)∥xn+1-q∥2≤[1-(1-ϵn)αn(1-ρ)]∥xn-q∥2+(1-ϵn)αn(1-ρ)2〈f(q)-q,J(yn-q)〉1-ρ.
Applying Lemma 8 to (116), we infer from conditions (ii) and (vi) and (115) that xn→q as n→∞. This completes the proof.Corollary 23.
LetX be a uniformly convex Banach space which has a uniformly Gáteaux differentiable norm and let C be a nonempty closed convex subset of X such that C±C⊂C. Let ΠC be a sunny nonexpansive retraction from X onto C. Let T,F:X→CB(X), and A:C→2C be three multivalued mappings, let g:X→C be a single-valued mapping, and let N(·,·):X×X→C be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4 and (H6) N(Tx,Fx)+A(g(x)):X→C is ξ0-strictly pseudocontractive and ν0-strongly accretive with ξ0+ν0≥1.For eachi=1,…,N, let Ti:C→C be a self-mapping such that I-Ti:C→X is ξi-strictly pseudocontractive and νi-strongly accretive with ξi+νi≥1. Define the mapping Gi:C→C by Gi=(1-λi)I+λiTi where 1-(ξi/(1+ξi))(1-(1-νi)/ξi)≤λi≤1 for each i=1,…,N. 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 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 Δ:=⋂i=0∞‍Fix(Si)∩Γ∩(∩i=1NFix(Ti))≠∅. Suppose that {αn},{βn},{γn},{δn},{σn}, and {ϵn} are the sequences in [0,1],αn+βn+γn+δn=1 and satisfy the following conditions:(i)
∑n=1∞‍(|αn-αn-1|+|βn-βn-1|+|γn-γn-1|+|δn-δn-1|+|σn-σn-1|+|ϵn-ϵn-1|)<∞;(ii)
limn→∞αn=0 and ∑n=0∞‍αn=∞;(iii)
{γn},{δn}⊂[c,d] for some c,d∈(0,1);(iv)
0<liminfn→∞βn≤limsupn→∞βn<1;(v)
0<liminfn→∞σn≤limsupn→∞σn<1;(vi)
0<liminfn→∞ϵn≤limsupn→∞ϵn<1.For arbitrary x0∈C, define the sequence {xn} iteratively by
(117)yn=αnf(xn)+βnxn+γnBxn+δnSnxn,xn+1=ϵn[xn-σn(N(wn,kn)+un)]+(1-ϵn)yn,un∈A(g(xn)),hhhhhhhhhhhhhhhhhhhhhh∀n≥0,
where {un} is defined by
(118)∥un-un+1∥≤(1+ɛ)H(A(g(xn+1)),A(g(xn))),∀n≥0,for anywn∈Txn, kn∈Fxn, and some ɛ>0. 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 {xn} converges strongly to q∈Δ, which solves the following VIP:
(119)〈q-f(q),J(q-p)〉≤0,∀p∈Δ
and, for any w∈Tq and k∈Fq, (q,w,k) is a solution of the MVVI (16).



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*Source: 328740-2013-12-05.xml*