# Mann-Type Extragradient Methods for General Systems of Variational Inequalities with Multivalued Variational Inclusion Constraints in Banach Spaces

Authors: Lu-Chuan Ceng; Abdul Latif; Saleh A. Al-Mezel
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduce Mann-type extragradient methods for a general system of variational inequalities with solutions of a multivalued variational inclusion and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. 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 as follows:(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 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)∥2,for someλ∈(0,1).LetU={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 strictly convex. A Banach space X is said to be smooth if the limit
(7)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. 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 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∥)-1:0000x,y∈X,∥x∥=1,∥y∥=τ12}.
It is known that X 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 [1], 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 J is norm-to-norm uniformly continuous on bounded subsets of X.Very recently, Cai and Bu [2] 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
(9)〈μ1B1y*+x*-y*,J(x-x*)〉≥0,∀x∈C,〈μ2B2x*+y*-x*,J(x-y*)〉≥0,∀x∈C,
where C 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 (9) is denoted by GSVI(C,B1,B2). In particular, if X=H, a real Hilbert space, then GSVI (9) reduces to the following GSVI of finding (x*,y*)∈C×C such that
(10)〈μ1B1y*+x*-y*,x-x*〉≥0,∀x∈C,〈μ2B2x*+y*-x*,x-y*〉≥0,∀x∈C,
where μ1 and μ2 are two positive constants. The set of solutions of problem (10) is still denoted by GSVI(C,B1,B2).Recently, Ceng et al. [3] transformed problem (10) into a fixed point problem in the following way.Lemma 1 (see [3]).
For givenx-,y-∈C,(x-,y-) is a solution of problem (10) if and only if x- is a fixed point of the mapping G:C→C defined by
(11)G(x)=PC[PC(x-μ2B2x)-μ1B1PC(x-μ2B2x)],00000000000000000000000000000∀x∈C,
where y-=PC(x--μ2B2x-) and PC is the the projection of H onto C.In particular, if the mappingsBi: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.Define the mappingG:C→C as follows:
(12)G(x):=ΠC(I-μ1B1)ΠC(I-μ2B2)x,∀x∈C.
The fixed point set of G is denoted by Ω.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
(13)H(A,B)∶=max{supx∈Binfy∈Ad(x,y),supx∈Ainfy∈Bd(x,y)},00000000000000000000∀A,B∈CB(X).
Let T and 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. [4] introduced and studied the multivalued variational inclusion (MVVI) of finding x∈D(A) such that (x,w,k) is a solution of the following:
(14)v∈N(w,k)+λA(g(x)),∀w∈Tx,k∈Fx.Ifv=0 and λ=1, then the MVVI (14) reduces to the problem of finding x∈D(A) such that (x,w,k) is a solution of the following:
(15)0∈N(w,k)+A(g(x)),∀w∈Tx,k∈Fx.
We denote by Γ the set of such solutions x for MVVI (15).The authors [4] first established an existence theorem for MVVI (14) in smooth Banach space X and then proved that the sequence generated by their iterative algorithm converges strongly to a solution of MVVI (15).Theorem 2 (see [4, Theorem 3.2]).
LetX be a real smooth Banach space. Let T and F:X→CB(X), let 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 arbitrary x0∈D(A) define the sequence {xn} iteratively by
(16)xn+1=xn-σn(N(wn,kn)+un),un∈A(g(xn)),
where {un} is defined by
(17)∥un-un+1∥≤(1+ε)H(A(g(xn+1)),A(g(xn))),0000000000000000000000000∀n≥0,
for any wn∈Txn, kn∈Fxn and some ε>0, where {σn} is a positive real sequence such that limn→∞σn=0 and ∑n=0∞‍σn=∞. Then, there exists d->0 such that for 0<σn≤d-, for all n≥0, {xn} converges strongly to x-∈Γ; and for any w∈Tx-, k∈Fx-, (x-,w,k) is a solution of the MVVI (15).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. Motivated and inspired by the research going on this area, we introduce Mann-type extragradient methods for finding solutions of the GSVI (9) which are also ones of the MVVI (15) and common fixed points of a countable family of nonexpansive mappings. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space, and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. 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, [2–10].


## 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
(18)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 as follows:(i)
accretive, if(19)〈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
(20)〈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
(21)〈u-v,J(x-y)〉≥ϕ(∥x-y∥)∥x-y∥,∀u∈Ax,v∈Ay;(v)
ϕ-expansive, if
(22)∥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 ε>0 there exists a δ>0 such that whenever ∥x-y∥<δ then H(Tx,Ty)<ε.A mappingN:X×X→X is ϕ-strongly accretive, with respect to T:X→CB(X), in the first argument if
(23)〈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(V)∶={x∈X:Sx∩V≠∅} is open in X whenever V⊂Y is open.We list some propositions and lemmas that will be used in the sequel.Proposition 3 (see [11]).
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
(24)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 4 (see [12]).
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 mappingx↦N(x,y)isϕ-strongly accretive with respect to the mappingT;(ii)
the mappingy↦N(x,y)is accretive with respect to the mappingF. Then the mapping S:X→2X defined by Sx=N(Tx,Fx) is ϕ-strongly accretive.Proposition 5 (see [13]).
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.Recall that a Banach spaceX is said to satisfy Opial’s condition, if whenever {xn} is a sequence in X which converges weakly to x as n→∞, then
(25)limsupn→∞∥xn-x∥<limsupn→∞∥xn-y∥,0∀y∈Xwithx≠y.Lemma 6 (Demiclosedness principle; see [14, Lemma 2]).
LetC be a nonempty closed convex subset of a reflexive Banach space X that satisfies Opial’s condition and suppose that T:C→X is nonexpansive. Then the mapping I-T is demiclosed at zero, that is, xn⇀x and xn-Txn→0 imply x=Tx; that is, x∈Fix(T).The following lemma is an immediate consequence of the subdifferential inequality of the function(1/2)∥·∥2.Lemma 7.
In a real smooth Banach spaceX, there holds the inequality
(26)∥x+y∥2≤∥x∥2+2〈y,J(x+y)〉,∀x,y∈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
(27)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 8 (see [15]).
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. The following result is an easy consequence of Lemma 8.Lemma 9.
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 (9) if and only if x*=ΠC(y*-μ1B1y*), where y*=ΠC(x*-μ2B2x*).In terms of Lemma9, we observe that
(28)x*=ΠC[ΠC(x*-μ2B2x*)-μ1B1ΠC(x*-μ2B2x*)],
which implies that x* 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 10 (see [16]).
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
(29)∥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.Lemma 11 (see [17]).
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.


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

In this section, we introduce Mann-type extragradient algorithms in uniformly convex and 2-uniformly smooth Banach spaces and show weak and strong convergence theorems. We will use some useful lemmas in the sequel.Lemma 12 (see [2, Lemma 2.8]).
LetC be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X. Let the mapping Bi:C→X be αi-inverse-strongly accretive. Then, we have
(30)∥(I-μiBi)x-(I-μiBi)y∥2≤∥x-y∥2+2μi(μiκ2-αi)∥Bix-Biy∥2,0000000000000000000000000000∀x,y∈C,
for i=1,2, where μi>0. In particular, if 0<μi≤αi/κ2, then I-μiBi is nonexpansive for i=1,2.Lemma 13 (see [2, Lemma 2.9]).
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 αi-inverse-strongly accretive for i=1,2. Let G:C→C be the mapping defined by
(31)Gx=ΠC[ΠC(x-μ2B2x)-μ1B1ΠC(x-μ2B2x)],0000000000000000000000000000000∀x∈C.
If 0<μi≤αi/κ2 for i=1,2, then G:C→C is nonexpansive.Theorem 14.
LetX be a uniformly convex and 2-uniformly smooth Banach space satisfying Opial’s condition 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 and F:X→CB(X) and let 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 2 and(C6)
N(Tx,Fx)+A(g(x)):X→2C∖{∅} is ζ-inverse strongly accretive with ζ≥κ2. Let Bi:C→X be αi-inverse strongly accretive for i=1,2. Let {Si}i=0∞ be a countable family of nonexpansive mappings of C into itself such that Δ:=⋂i=0∞‍Fix(Si)∩Ω∩Γ≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2) with 0<μi<αi/κ2 for i=1,2. Assume that {αn}, {βn}, and {σn} are sequences in [0,1] such that(i)
0<liminfn→∞αn≤limsupn→∞αn<1;(ii)
0<liminfn→∞βn≤limsupn→∞βn<1;(iii)
0<liminfn→∞σn≤limsupn→∞σn<1. For arbitrary x0∈C define the sequence {xn} iteratively by
(32)yn=βnSnxn+(1-βn)ΠC(I-μ1B1)×ΠC(I-μ2B2)xn,xn+1=αn[xn-σn(N(wn,kn)+un)]+(1-αn)yn,00000000000000un∈A(g(xn)),∀n≥0,
where {un} is defined by
(33)∥un-un+1∥≤(1+ε)H(A(g(xn+1)),A(g(xn))),000000000000000000000000∀n≥0,
for any wn∈Txn, kn∈Fxn and some ε>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 weakly to some x-∈Δ, and for any w∈Tx-, k∈Fx-,(x-,w,k) is a solution of the MVVI (15).Proof.
First of all, 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 (14), for any w∈Tx~ and k∈Fx~. Indeed, following the argument idea in the proof of Chidume et al. [4, Theorem 3.1], we put Vx:=N(Tx,Fx) for all x∈X. Then by Proposition 4, 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 5  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 [18, page 184] for any x∈X we have (V+λA∘g)(x) is closed and bounded. Therefore, by Morales [19], 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 5 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 12, we conclude that the mapping x↦x-(N(Tx,Fx)+λA(g(x))) is nonexpansive. Meantime, by Lemma 13 we know that G:C→C is also nonexpansive.
Without loss of generality we may assume thatv=0 and λ=1. Let p∈Δ and let r>0 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 ε1∶=ϕ(r)/8(1+ε), and ε2∶=ϕ(r)/8μ(1+ε), ε3∶=ϕ(r)/8ξ(1+ε), 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))<ε1, H(Tx,Ty)<ε2 and H(Fx,Fy)<ε3, 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 (32) it follows that
(34)∥yn-p∥=∥βnSnxn+(1-βn)ΠC(I-μ1B1)ΠC0×(I-μ2B2)xn-p∥≤βn∥Snxn-p∥+(1-βn)∥Gxn-p∥≤βn∥xn-p∥+(1-βn)∥xn-p∥=∥xn-p∥,
and hence
(35)∥xn+1-p∥200=〈αn[xn-p-σn(N(wn,kn)+un)]0000+(1-αn)(yn-p),J(xn+1-p)〉00=〈αn(xn-p)+(1-αn)00000×(yn-p),J(xn+1-p)〉00000-αnσn〈N(wn,kn)+un,J(xn+1-p)〉00≤∥αn(xn-p)+(1-αn)(yn-p)∥0000×∥xn+1-p∥0000-αnσn〈N(wn,kn)+un,J(xn+1-p)〉00≤(αn∥xn-p∥+(1-αn)∥xn-p∥)0000×∥xn+1-p∥0000-αnσn〈N(wn,kn)+un,J(xn+1-p)〉00=∥xn-p∥∥xn+1-p∥0000-αnσn〈N(wn,kn)+un,J(xn+1-p)〉00≤12(∥xn-p∥2+∥xn+1-p∥2)0000-αnσn〈N(wn,kn)+un,J(xn+1-p)〉,
which immediately yields
(36)∥xn+1-p∥200≤∥xn-p∥2-2αnσn〈N(wn,kn)+un,J(xn+1-p)〉00=∥xn-p∥20000-2αnσn〈N(wn+1,kn+1)+un+1,J(xn+1-p)〉0000-2αnσn〈N(wn,kn)000000000000+un-(N(wn+1,kn+1)+un+1),000000000000J(xn+1-p)〉.
Since N(·,·) is ϕ-strongly accretive with respect to T and A(g(·)) is accretive, we deduce from (36) that
(37)∥xn+1-p∥200≤∥xn-p∥2-2αnσnϕ(∥xn+1-p∥)∥xn+1-p∥0000+2αnσn[∥N(wn+1,kn+1)-N(wn,kn)∥000000000000+∥un+1-un∥]∥xn+1-p∥00≤∥xn-p∥2-2αnσnϕ(∥xn+1-p∥)∥xn+1-p∥0000+2αnσn[∥N(wn+1,kn+1)-N(wn+1,kn)∥000000000000+∥N(wn+1,kn)-N(wn,kn)∥000000000000+∥un+1-un∥]∥xn+1-p∥.
Again from (32) we have that
(38)∥xn+1-p∥00≤αn∥xn-p-σn(N(wn,kn)+un)∥0000+(1-αn)∥yn-p∥00≤αn[∥xn-p∥+σn∥N(wn,kn)+un∥]0000+(1-αn)∥xn-p∥00≤αn[r+σnM]+(1-αn)r00≤2r.
Also, from Proposition 5, 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 [20] that, for kn+1∈Fxn+1 and wn+1∈Txn+1, there exist kn′∈Fxn and wn′∈Txn such that
(39)∥kn+1-kn′∥≤(1+ε)H(Fxn+1,Fxn),∥wn+1-wn′∥≤(1+ε)H(Txn+1,Txn),
respectively. Therefore, from (37) and (33), we have that
(40)∥xn+1-p∥200≤∥xn-p∥2-2αnσnϕ(r)r0000+2αnσn[∥N(wn+1,kn+1)-N(wn+1,kn′)∥00000000000+∥N(wn+1,kn)-N(wn′,kn)∥00000000000+∥un+1-un∥∥N(wn+1,kn+1)-N(wn+1,kn′)∥]2r00≤∥xn-p∥2-2αnσnϕ(r)r0000+2αnσn[(A(g(xn+1)),A(g(xn)))ξ(1+ε)H(Fxn+1,Fxn)00000000000+μ(1+ε)H(Txn+1,Txn)00000000000+(1+ε)H(A(g(xn+1)),A(g(xn)))]2r00≤∥xn-p∥2-2αnσnϕ(r)r0000+2αnσn[ϕ(r)8+ϕ(r)8+ϕ(r)8]2r00=∥xn-p∥2-2αnσnϕ(r)r0000+αnσn32ϕ(r)r≤∥xn-p∥2.
So, we get ∥xn+1-p∥≤r, a contradiction. Therefore, {xn} is bounded.
Let us show thatlimn→∞∥xn-yn∥=0 and limn→∞∥xn-xn+1∥=0.
Indeed, utilizing Lemma10 and the nonexpansivity of the mapping x↦x-(N(Tx,Fx)+A(g(x))), we obtain from (32) that for all n≥0(41)∥xn+1-p∥200=∥αn[xn-p-σn(N(wn,kn)+un)]00000+(1-αn)(yn-p)∥200≤αn∥xn-p-σn(N(wn,kn)+un)∥20000+(1-αn)∥yn-p∥2-αn(1-αn)φ0000×(∥xn-yn-σn(N(wn,kn)+un)∥)00=αn∥(1-σn)(xn-p)0000000+σn[xn-(N(wn,kn)+un)-p]∥20000+(1-αn)∥yn-p∥2-αn(1-αn)φ0000×(∥xn-yn-σn(N(wn,kn)+un)∥)00≤αn[(1-σn)∥xn-p∥20000000+σn∥xn-(N(wn,kn)+un)-p∥20000000-σn(1-σn)φ1(∥N(wn,kn)+un∥)∥xn-p∥2]0000+(1-αn)∥xn-p∥2-αn(1-αn)φ0000×(∥xn-yn-σn(N(wn,kn)+un)∥)00≤αn[(1-σn)∥xn-p∥2+σn∥xn-p∥20000000-σn(1-σn)φ1(∥N(wn,kn)+un∥)∥xn-p∥2]0000+(1-αn)∥xn-p∥2-αn(1-αn)φ0000×(∥xn-yn-σn(N(wn,kn)+un)∥)00=αn∥xn-p∥2-αnσn(1-σn)φ10000×(∥N(wn,kn)+un∥)0000+(1-αn)∥xn-p∥2-αn(1-αn)φ0000×(∥xn-yn-σn(N(wn,kn)+un)∥)00=∥xn-p∥20000-αnσn(1-σn)φ1(∥N(wn,kn)+un∥)0000-αn(1-αn)φ0000×(∥xn-yn-σn(N(wn,kn)+un)∥)00≤∥xn-p∥2.
It is easy to see that the limit limn→∞∥xn-p∥ exists. Meantime, it can be readily seen from (41) that
(42)αnσn(1-σn)φ1(∥N(wn,kn)+un∥)0000+αn(1-αn)φ0000×(∥xn-yn-σn(N(wn,kn)+un)∥)00≤∥xn-p∥2-∥xn+1-p∥2,
which together with conditions (i) and (iii) and the existence of limn→∞∥xn-p∥, implies that
(43)limn→∞φ1(∥N(wn,kn)+un∥)=0,limn→∞φ(∥xn-yn-σn(N(wn,kn)+un)∥)=0.
Utilizing the properties of φ and φ1, we get
(44)limn→∞∥N(wn,kn)+un∥=0,limn→∞∥xn-yn-σn(N(wn,kn)+un)∥=0.
Note that
(45)∥xn-yn∥≤∥xn-yn-σn(N(wn,kn)+un)∥+σn∥N(wn,kn)+un∥.
So, we have
(46)limn→∞∥xn-yn∥=0.
Also, observe that
(47)∥xn+1-xn∥≤αnσn∥N(wn,kn)+un∥+(1-αn)∥xn-yn∥≤∥N(wn,kn)+un∥+∥xn-yn∥.
Thus, from (44) and (46) it follows that
(48)limn→∞∥xn+1-xn∥=0.
Let us show thatlimn→∞∥xn-Gxn∥=limn→∞∥xn-Sxn∥=0.
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 12 we have
(49)∥un-q∥200=∥ΠC(xn-μ2B2xn)-ΠC(p-μ2B2p)∥200≤∥xn-p-μ2(B2xn-B2p)∥200≤∥xn-p∥2-2μ2(α2-κ2μ2)0000×∥B2xn-B2p∥2,(50)∥vn-p∥200=∥ΠC(un-μ1B1un)-ΠC(q-μ1B1q)∥200≤∥un-q-μ1(B1un-B1q)∥200≤∥un-q∥2-2μ1(α1-κ2μ1)0000×∥B1un-B1q∥2.
Substituting (49) for (50), we obtain
(51)∥vn-p∥200≤∥xn-p∥20000-2μ2(α2-κ2μ2)∥B2xn-B2p∥20000-2μ1(α1-κ2μ1)∥B1un-B1q∥2.
Utilizing [21, Proposition 1] and Lemma 10, from (32) and (51) we have
(52)∥yn-p∥200≤βn∥Snxn-p∥2+(1-βn)∥Gxn-p∥20000-βn(1-βn)φ2(∥Snxn-Gxn∥)00≤βn∥xn-p∥2+(1-βn)∥vn-p∥20000-βn(1-βn)φ2(∥Snxn-Gxn∥)00≤βn∥xn-p∥20000-βn(1-βn)φ2(∥Snxn-Gxn∥)0000+(1-βn)[2μ1(α1-κ2μ1)∥B1un-B1q∥2∥xn-p∥20000000000000-2μ2(α2-κ2μ2)∥B2xn-B2p∥20000000000000-2μ1(α1-κ2μ1)∥B1un-B1q∥2]00=∥xn-p∥20000-βn(1-βn)φ2(∥Snxn-Gxn∥)0000-2(1-βn)[μ2(α2-κ2μ2)∥B2xn-B2p∥200000000000000+2μ1(α1-κ2μ1)∥B1un-B1q∥2],
which hence implies that
(53)βn(1-βn)φ2(∥Snxn-Gxn∥)0000+2(1-βn)[μ2(α2-κ2μ2)000000000000000×∥B2xn-B2p∥2000000000000000+μ1(α1-κ2μ1)000000000000000×∥B1un-B1q∥2]00≤∥xn-p∥2-∥yn-p∥200≤(∥xn-p∥+∥yn-p∥)∥xn-yn∥.
Since 0<μi<αi/κ2 for i=1,2 and {xn}, {yn} are bounded, we obtain from (46), (53), condition (ii) and the properties of φ2 that
(54)limn→∞∥B2xn-B2p∥=limn→∞∥B1un-B1q∥=0,limn→∞∥Snxn-Gxn∥=0.
Utilizing Proposition 3 and Lemma 8, we have
(55)∥un-q∥200=∥ΠC(xn-μ2B2xn)-ΠC(p-μ2B2p)∥200≤〈xn-μ2B2xn-(p-μ2B2p),J(un-q)〉00=〈xn-p,J(un-q)〉0000+μ2〈B2p-B2xn,J(un-q)〉00≤12[∥xn-p∥2+∥un-q∥2000000-ψ1(∥xn-un-(p-q)∥)∥un-q∥2]0000+μ2∥B2p-B2xn∥∥un-q∥,
which implies that
(56)∥un-q∥2≤∥xn-p∥2-ψ1(∥xn-un-(p-q)∥)+2μ2∥B2p-B2xn∥∥un-q∥.
In the same way, we derive
(57)∥vn-p∥200=∥ΠC(un-μ1B1un)-ΠC(q-μ1B1q)∥200≤〈un-μ1B1un-(q-μ1B1q),J(vn-p)〉00=〈un-q,J(vn-p)〉0000+μ1〈B1q-B1un,J(vn-p)〉00≤12[∥un-q∥2+∥vn-p∥2000000-ψ2(∥un-vn+(p-q)∥)∥un-q∥2]0000+μ1∥B1q-B1un∥∥vn-p∥,
which implies that
(58)∥vn-p∥2≤∥un-q∥2-ψ2(∥un-vn+(p-q)∥)+2μ1∥B1q-B1un∥∥vn-p∥.
Substituting (56) for (58), we get
(59)∥vn-p∥2≤∥xn-p∥2-ψ1(∥xn-un-(p-q)∥)-ψ2(∥un-vn+(p-q)∥)+2μ2∥B2p-B2xn∥∥un-q∥+2μ1∥B1q-B1un∥∥vn-p∥.
By Lemma 7, we have from (52) and (59)
(60)∥yn-p∥200≤βn∥Snxn-p∥2+(1-βn)∥vn-p∥200≤βn∥xn-p∥2+(1-βn)0000×[∥xn-p∥2000000-ψ1(∥xn-un-(p-q)∥)000000-ψ2(∥un-vn+(p-q)∥)000000+2μ2∥B2p-B2xn∥∥un-q∥000000+2μ1∥B1q-B1un∥∥vn-p∥∥xn-p∥2+∥∥un-q∥2]00≤∥xn-p∥2-(1-βn)0000×[ψ1(∥xn-un-(p-q)∥)000000+ψ2(∥un-vn+(p-q)∥)]0000+2μ2∥B2p-B2xn∥∥un-q∥0000+2μ1∥B1q-B1un∥∥vn-p∥,
which hence leads to
(61)(1-βn)[ψ1(∥xn-un-(p-q)∥)0000000+ψ2(∥un-vn+(p-q)∥)]00≤∥xn-p∥2-∥yn-p∥20000+2μ2∥B2p-B2xn∥∥un-q∥0000+2μ1∥B1q-B1un∥∥vn-p∥00≤(∥xn-p∥+∥yn-p∥)∥xn-yn∥0000+2μ2∥B2p-B2xn∥∥un-q∥0000+2μ1∥B1q-B1un∥∥vn-p∥.
From (46), (54), (61), condition (ii), and the boundedness of {xn}, {yn}, {un}, and {vn}, we deduce that
(62)limn→∞ψ1(∥xn-un-(p-q)∥)=0,limn→∞ψ2(∥un-vn+(p-q)∥)=0.
Utilizing the properties of ψ1 and ψ2, we deduce that
(63)limn→∞∥xn-un-(p-q)∥=0,limn→∞∥un-vn+(p-q)∥=0.
From (63) we get
(64)∥xn-vn∥≤∥xn-un-(p-q)∥+∥un-vn+(p-q)∥⟶0asn⟶∞,
which together with (54), leads to
(65)limn→∞∥xn-Gxn∥=0,limn→∞∥xn-Snxn∥=0.
Since
(66)∥xn-Sxn∥≤∥xn-Snxn∥+∥Snxn-Sxn∥.
Utilizing the assumption on {Sn} and Lemma 11, from (65) we get
(67)limn→∞∥xn-Sxn∥=0.
Next, let us show that{xn} converges weakly to some x-∈Δ.
Indeed, sinceX is reflexive and {xn} is bounded, there exists a subsequence {xni} of {xn} such that xni⇀x-∈C. Then by Lemma 6, we obtain from (44), (65), and (67) that x-∈Γ, x-∈Fix(G)=Ω, and x-∈Fix(S)=⋂i=0∞‍Fix(Si). Thus, x-∈Δ. In addition, if {xmj} is another subsequence of {xn} such that xmj⇀x^, then by Lemma 6 we also deduce from (44), (65), and (67) that x^∈Δ. Thus, the limits limn→∞∥xn-x-∥ and limn→∞∥xn-x^∥ exist. Now we claim that x-=x^. Assume that x-≠x^. Then in terms of Opial’s condition, we get
(68)limn→∞∥xn-x-∥=limsupi→∞∥xni-x-∥<limsupi→∞∥xni-x^∥=limsupn→∞∥xn-x^∥=limsupj→∞∥xmj-x^∥<limsupj→∞∥xmj-x-∥=limn→∞∥xn-x-∥,
which leads to a contradiction. So, we must have x-=x^. Therefore, xn⇀x-∈Δ. This completes the proof.Theorem 15.
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 and F:X→CB(X) and let 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 2. Let Bi:C→X be αi-inverse strongly accretive for i=1,2. Let {Si}i=0∞ be a countable family of nonexpansive mappings of C into itself such that Δ:=⋂i=0∞‍Fix(Si)∩Ω∩Γ≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2) with 0<μi<αi/κ2 for i=1,2. Let {αn},{βn}⊂[0,1], and {σn}⊂(0,∞) such that limn→∞αnσn=0 and ∑n=0∞‍αnσn=∞. For arbitrary x0∈C define the sequence {xn} iteratively by (32), where {un} is defined by (33) for any wn∈Txn, kn∈Fxn and some ε>0. Then, there exists d->0 such that for 0<αnσn≤d-, for all n≥0, {xn} converges strongly to p∈Δ provided limn→∞∥xn+1-xn∥=0; in this case, for any w∈Tx-, k∈Fx-, (x-,w,k) is a solution of the MVVI (15).Proof.
First of all, repeating the same arguments as those in the proof of Theorem14, we can prove that for any v∈C, λ>0, there exists a point x~∈D(A)=C such that (x~,w,k) is a solution of the MVVI (14), for any w∈Tx~ and k∈Fx~. In addition, in terms of Proposition 5 we know that V+λA∘g is a single-valued mapping due to the fact that V+λA∘g is ϕ-strongly accretive. Meantime, by Lemma 13 we know that G:C→C is nonexpansive.
Without loss of generality we may assume thatv=0 and λ=1. Let p∈Δ and let r>0 be sufficiently large such that x0∈B¯r(p)=:B. 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 ε1∶=ϕ(r)/8(1+ε) and ε2∶=ϕ(r)/8μ(1+ε), ε3∶=ϕ(r)/8ξ(1+ε), 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))<ε1, H(Tx,Ty)<ε2 and H(Fx,Fy)<ε3, respectively. Let d-∶=(1/2)min{δ2/M,δ3/M,δ1/M,r/M}. Taking into account limn→∞αnσn=0 we may assume that 0<αnσn≤d-, for all n≥0.
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 (32) it follows that
(69)∥yn-p∥≤βn∥Snxn-p∥+(1-βn)∥Gxn-p∥≤βn∥xn-p∥+(1-βn)∥xn-p∥=∥xn-p∥.
Repeating the same arguments as those of (37) in the proof of Theorem 14, we can get
(70)∥xn+1-p∥2≤∥xn-p∥2-2αnσnϕ(∥xn+1-p∥)∥xn+1-p∥+2αnσn[∥N(wn+1,kn+1)-N(wn,kn)∥+∥un+1-un∥]∥xn+1-p∥≤∥xn-p∥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∥.
Utilizing (33) and (70) and repeating the same arguments as those of (40) in the proof of Theorem 14, we can derive ∥xn+1-p∥2≤∥xn-p∥2. So, we have ∥xn+1-p∥≤r, a contradiction. Therefore, {xn} is bounded.
Next let us show thatxn→x- as n→∞.
Indeed, since∥xn+1-xn∥→0 as n→∞, we have that H(A(g(xn+1)),A(g(xn)))→0 and ∥N(wn+1,kn+1)-N(wn,kn)∥→0 as n→∞. The conclusion now follows from inequality (70) with the use of Proposition 3 and hence (x-,w,k) is a solution of the MVVI (15) for any w∈Tx-, k∈Fx-. This completes the proof.Remark 16.
Theorems14 and 15 improve, extend, supplement, and develop [4, Theorem 3.2] and [2, Theorem 3.1] in the following aspects.(i)
The problem of finding a point of⋂i=0∞‍Fix(Si)∩Ω∩Γ in Theorems 14 and 15 is more general and more subtle than every one of the problem of finding a point of Γ in [4, Theorem 3.2] and the problem of finding a point of ⋂i=1∞‍Fix(Si)∩Ω in [2, Theorem 3.1].(ii)
The iterative scheme in [2, Theorem 3.1] is extended to develop the iterative scheme (32) of Theorems 14 and 15 by virtue of the iterative schemes of [4, Theorems 3.2]. The iterative scheme (32) of Theorems 14 and 15 is more advantageous and more flexible than the iterative scheme of [4, Theorem 3.2] because it can be applied to solving three problems (i.e., GSVI (9), MVVI (15), and the fixed point problem of {Sn}) and involves several parameter sequences {αn}, {βn}, and {σn}.(iii)
Theorems14 and 15 extend and generalize [4, Theorems 3.2] to the setting of a countable family of nonexpansive mappings and the GSVI (9) for two inverse-strongly accretive mappings. In the meantime, Theorems 14 and 15 extend and generalize [2, Theorem 3.1] to the setting of the MVVI (15).(iv)
The iterative scheme (32) in Theorems 14 and 15 is very different from everyone in [4, Theorem 3.2] and [2, Theorem 3.1] because the iterative scheme in [2, Theorem 3.1] involves the viscosity approximation method and the iterative scheme in [4, Theorem 3.2] is a one-step iterative scheme.(v)
No boundedness condition on the rangesR(I-N(T(·),F(·))) and R(A(g(·))) is imposed in Theorems 14 and 15.


## 4. Mann-Type Extragradient Algorithms in Smooth and Uniformly Convex Banach Spaces

In this section, we introduce Mann-type extragradient algorithms in smooth and uniformly convex Banach spaces and show weak and strong convergence theorems. First, we give some useful lemmas whose proofs will be omitted.Lemma 17.
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] we have
(71)∥(I-μiBi)x-(I-μiBi)y∥00≤{1-αiλi+(1-μi)(1+1λi)}0000×∥x-y∥,∀x,y∈C,
for i=1,2. In particular, if 1-(λi/(1+λi))(1-(1-αi)/λi)≤μi≤1, then I-μiBi is nonexpansive for i=1,2.Lemma 18.
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
(72)G(x)=ΠC[ΠC(x-μ2B2x)000-μ1B1ΠC(x-μ2B2x)],∀x∈C.
If 1-(λi/(1+λi))(1-(1-αi)/λi)≤μi≤1, then G:C→C is nonexpansive.Theorem 19.
LetX be a smooth and uniformly convex Banach space satisfying Opial’s condition and 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 and F:X→CB(X) and let 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 2 and(H6)
N(Tx,Fx)+A(g(x)):X→C is λ0-strictly pseudocontractive and α0-strongly accretive with λ0+α0≥1. Let Bi:C→X be λi-strictly pseudocontractive and αi-strongly accretive with λi+αi≥1 for i=1,2. Let {Si}i=0∞ be a countable family of nonexpansive mappings of C into itself such that Δ:=⋂i=0∞‍Fix(Si)∩Ω∩Γ≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2) with 1-(λi/(1+λi))(1-(1-αi)/λi)≤μi≤1 for i=1,2. Assume that {αn}, {βn}, {γn}, and {σn} are sequences in [0,1] such that(i)
0<liminfn→∞αn≤limsupn→∞αn<1;(ii)
{βn},{γn},{βn+γn}⊂[c,d] for some c,d∈(0,1);(iii)
0<liminfn→∞σn≤limsupn→∞σn<1. For arbitrary x0∈C define the sequence {xn} iteratively by
(73)yn=βnSnxn+γnxn+(1-βn-γn)ΠC(I-μ1B1)×ΠC(I-μ2B2)xn,xn+1=αn[xn-σn(N(wn,kn)+un)]+(1-αn)yn,un∈A(g(xn)),∀n≥0,
where {un} is defined by
(74)∥un-un+1∥≤(1+ε)H(A(g(xn+1)),A(g(xn))),000000000000000000000000∀n≥0,
for any wn∈Txn, kn∈Fxn and some ε>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 weakly to some x-∈Δ, and for any w∈Tx-, k∈Fx-, (x-,w,k) is a solution of the MVVI (15).Proof.
First of all, repeating the same arguments as those in the proof of Theorem14, 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 (14), for any w∈Tx~ and k∈Fx~. In addition, in terms of Proposition 5 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)+λA(g(x)):X→C is λ0 strictly pseudocontractive and α0 strongly accretive with λ0+α0≥1. Then by Lemma 17 we conclude that the mapping x↦x-(N(Tx,Fx)+λA(g(x))) is nonexpansive. Meantime, by Lemma 18 we know that G:C→C is also nonexpansive.
Without loss of generality we may assume thatv=0 and λ=1. Let p∈Δ and let r>0 be sufficiently large such that x0∈B¯r(p)=:B. Observe that
(75)∥yn-p∥≤βn∥Snxn-p∥+γn∥xn-p∥+(1-βn-γn)∥Gxn-p∥≤βn∥xn-p∥+γn∥xn-p∥+(1-βn-γn)∥xn-p∥=∥xn-p∥.
Utilizing (75) and repeating the same arguments as those in the proof of Theorem 14, we can derive xn∈B for all n≥0. Hence {xn} is bounded.
Let us show thatlimn→∞∥xn-yn∥=0 and limn→∞∥xn-xn+1∥=0.
Indeed, repeating the same arguments as those of (41) in the proof of Theorem 14, we obtain from (73) and (75) that for all n≥0(76)∥xn+1-p∥2≤∥xn-p∥2-αnσn(1-σn)φ1×(∥N(wn,kn)+un∥)-αn(1-αn)φ×(∥xn-yn-σn(N(wn,kn)+un)∥)≤∥xn-p∥2.
It is easy to see that the limit limn→∞∥xn-p∥ exists. Meantime, it can be readily seen from (76) that
(77)αnσn(1-σn)φ1(∥N(wn,kn)+un∥)0000+αn(1-αn)φ(∥xn-yn-σn(N(wn,kn)+un)∥)00≤∥xn-p∥2-∥xn+1-p∥2,
which together with conditions (i), (iii), and the existence of limn→∞∥xn-p∥, implies that
(78)limn→∞φ1(∥N(wn,kn)+un∥)=0,limn→∞φ(∥xn-yn-σn(N(wn,kn)+un)∥)=0.
Utilizing the properties of φ and φ1, we get
(79)Limn→∞∥N(wn,kn)+un∥=0,limn→∞∥xn-yn-σn(N(wn,kn)+un)∥=0.
Note that
(80)∥xn-yn∥≤∥xn-yn-σn(N(wn,kn)+un)∥+σn∥N(wn,kn)+un∥.
So, we have
(81)limn→∞∥xn-yn∥=0.
Also, observe that
(82)∥xn+1-xn∥≤αnσn∥N(wn,kn)+un∥+(1-αn)∥xn-yn∥≤∥N(wn,kn)+un∥+∥xn-yn∥.
Thus from (79) and (81) it follows that
(83)limn→∞∥xn+1-xn∥=0.
Let us show thatlimn→∞∥xn-Gxn∥=limn→∞∥xn-Sxn∥=0.
Indeed, from (73) we obtain that
(84)βnβn+γn(Snxn-Gxn)+γnβn+γn(xn-Gxn)=βnSnxn+γnxnβn+γn-Gxn=yn-(1-βn-γn)Gxnβn+γn-Gxn=yn-Gxnβn+γn.
Utilizing Lemma 10 we deduce from (73) and (84) that
(85)∥yn-p∥200=∥βn(Snxn-p)+γn(xn-p)00000+(1-βn-γn)(Gxn-p)∥200=∥(βn+γn)[βnβn+γn(Snxn-p)00000000000000+γnβn+γn(xn-p)]00000+(1-βn-γn)(Gxn-p)(βn+γn)[βn(βn+γn)(Snxn-p)∥200≤(βn+γn)∥βnβn+γn(Snxn-p)0000000000000+γnβn+γn(xn-p)∥20000+(1-βn-γn)∥Gxn-p∥20000-(βn+γn)(1-βn-γn)φ20000×(∥βnβn+γn(Snxn-Gxn)00000000+γnβn+γn(xn-Gxn)∥)00=(βn+γn)∥βnβn+γn(Snxn-p)0000000000000+γnβn+γn(xn-p)∥20000+(1-βn-γn)∥Gxn-p∥20000-(βn+γn)(1-βn-γn)φ20000×(1βn+γn∥yn-Gxn∥)00≤(βn+γn)[βnγn(βn+γn)2βnβn+γn∥Snxn-p∥2000000000000+γnβn+γn∥xn-p∥2000000000000-βnγn(βn+γn)2φ3(∥Snxn-xn∥)]0000+(1-βn-γn)∥Gxn-p∥20000-(βn+γn)(1-βn-γn)φ20000×(1βn+γn∥yn-Gxn∥)00≤βn∥Snxn-p∥2+γn∥xn-p∥20000-βnγnφ3(∥Snxn-xn∥)0000+(1-βn-γn)∥Gxn-p∥20000-(βn+γn)(1-βn-γn)φ20000×(1βn+γn∥yn-Gxn∥)00≤βn∥xn-p∥2+γn∥xn-p∥20000-βnγnφ3(∥Snxn-xn∥)0000+(1-βn-γn)∥xn-p∥20000-(βn+γn)(1-βn-γn)φ20000×(1βn+γn∥yn-Gxn∥)00=∥xn-p∥2-βnγnφ3(∥Snxn-xn∥)0000-(βn+γn)(1-βn-γn)φ20000×(1βn+γn∥yn-Gxn∥),
which hence implies that
(86)βnγnφ3(∥Snxn-xn∥)0000+(βn+γn)(1-βn-γn)φ20000×(1βn+γn∥yn-Gxn∥)00≤∥xn-p∥2-∥yn-p∥200≤(∥xn-p∥+∥yn-p∥)∥xn-yn∥.
Utilizing condition (ii) we conclude from (81) and (86) that
(87)limn→∞φ3(∥Snxn-xn∥)=0,limn→∞φ2(1βn+γn∥yn-Gxn∥)=0.
From the properties of φ2 and φ3 and condition (ii), it immediately follows that
(88)limn→∞∥Snxn-xn∥=0,limn→∞∥yn-Gxn∥=0.
Observe that
(89)∥xn-Sxn∥≤∥xn-Snxn∥+∥Snxn-Sxn∥,∥xn-Gxn∥≤∥xn-yn∥+∥yn-Gxn∥.
Thus, utilizing Lemma 11 and the assumption on {Sn}, from (81) and (88) we conclude that
(90)limn→∞∥xn-Sxn∥=0,limn→∞∥xn-Gxn∥=0.
Next, let us show that{xn} converges weakly to some x-∈Δ.
Indeed, repeating the same arguments as those in the proof of Theorem14, we can prove that {xn} converges weakly to some x-∈Δ. This completes the proof.Theorem 20.
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 and F:X→CB(X) and let 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 2. Let Bi:C→X be λi strictly pseudocontractive and αi strongly accretive with λi+αi≥1 for i=1,2. Let {Si}i=0∞ be a countable family of nonexpansive mappings of C into itself such that Δ:=⋂i=0∞‍Fix(Si)∩Ω∩Γ≠∅, where Ω is the fixed point set of the mapping G=ΠC(I-μ1B1)ΠC(I-μ2B2) with 1-(λi/(1+λi))(1-(1-αi)/λi)≤μi≤1 for i=1,2. Let {αn},{βn}⊂[0,1] and {σn}⊂(0,∞) such that limn→∞αnσn=0 and ∑n=0∞‍αnσn=∞. For arbitrary x0∈C define the sequence {xn} iteratively by (73), where {un} is defined by (74) for any wn∈Txn,kn∈Fxn and some ε>0. Then, there exists d->0 such that for 0<αnσn≤d-, for all n≥0, {xn} converges strongly to p∈Δ provided limn→∞∥xn+1-xn∥=0; in this case, for any w∈Tx-, k∈Fx-, (x-,w,k) is a solution of the MVVI (15).Proof.
First of all, repeating the same arguments as those in the proof of Theorem14, we can prove that for any v∈C,λ>0, there exists a point x~∈D(A)=C such that (x~,w,k) is a solution of the MVVI (14), for any w∈Tx~ and k∈Fx~. In addition, in terms of Proposition 5 we know that V+λA∘g is a single-valued mapping due to the fact that V+λA∘g is ϕ-strongly accretive. Meantime, by Lemma 18 we know that G:C→C is nonexpansive.
Without loss of generality we may assume thatv=0 and λ=1. Let p∈Δ. Then from (73) we deduce that for all n≥0(91)∥yn-p∥≤βn∥Snxn-p∥+γn∥xn-p∥+(1-βn-γn)∥Gxn-p∥≤βn∥xn-p∥+γn∥xn-p∥+(1-βn-γn)∥xn-p∥=∥xn-p∥.
Repeating the same arguments as those of the remainder in the proof of Theorem 15, we can derive the desired result.



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*Source: 616549-2013-12-02.xml*