# Convergence of an Iterative Algorithm for Common Solutions for Zeros of Maximal Accretive Operator with Applications

Authors: Uamporn Witthayarat; Yeol Je Cho; Poom Kumam
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The aim of this paper is to introduce an iterative algorithm for finding a common solution of the sets(A+M2)−1(0) and (B+M1)−1(0), where M is a maximal accretive operator in a Banach space and, by using the proposed algorithm, to establish some strong convergence theorems for common solutions of the two sets above in a uniformly convex and 2-uniformly smooth Banach space. The results obtained in this paper extend and improve the corresponding results of Qin et al. 2011 from Hilbert spaces to Banach spaces and Petrot et al. 2011. Moreover, we also apply our results to some applications for solving convex feasibility problems.


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## Body

## 1. Introduction

LetE be a real Banach space with norm ∥·∥ with the dual space E* and let 〈·,·〉 denote the pairing between E and E*. Let C be a nonempty closed convex subset of E. We define the generalized duality mappingJq:E→2E* by(1.1)Jq(x)={f∈E*:〈x,f〉=‖x‖q,‖f‖=‖x‖q-1},∀x∈E,
for all q>1. In the special case, for q=2, we called the mapping J2 as the normalized duality mapping and as usual we write J2=J. The following is the well-known properties of the generalized duality mapping Jq:(1)
Jq(x)=∥x∥q-2J2(x) for all x∈E with x≠0;(2)
Jq(tx)=tq-1Jq(x) for all x∈E and t∈[0,∞);(3)
Jq(-x)=-Jq(x) for all x∈E.It is well known that ifX is smooth, then J is single valued, which is denoted by j. Recall that the duality mapping j is said to be weakly sequentially continuous if, for each sequence {xn} with xn→x weakly, we have j(xn)→j(x) weakly*. We know that, if X admits a weakly sequentially continuous duality mapping, then X is smooth. For the details, see [1–3].LetU={x∈E:∥x∥=1}. A Banach space E is said to be(1)
uniformly convex if there exists δ>0 such that, for any x,y∈U and, for any ϵ∈(0,2], ∥x-y∥≥ϵ implies ∥(x+y)/2∥≤1-δ.We can see that every uniformly convex Banach space is also reflexive and strictly convex.(2)
Smooth if limt→0(∥x+ty∥-∥x∥)/t exists for all x,y∈U.(3)
Uniformly smooth if the limit is attained uniformly for x,y∈U. The modulus of smoothness of E is defined by
(1.2)ρ(τ)=sup⁡{12(‖x+y‖+‖x-y‖)-1:x,y∈E,‖x‖=1,‖y‖=τ},
where ρ:[0,∞)→[0,∞) is a function. In the other way, E is uniformly smooth if and only if limτ→0ρ(τ)/τ=0.(4)
q-uniformly smooth if there exists a constant c>0 such that ρ(τ)≤cτq for all τ>0 where q is a fixed real number with 1<q≤2. (see, for instance, [1, 4]).We note thatE is a uniformly smooth Banach space if and only if Jq is single valued and uniformly continuous on any bounded subset of E. Examples of both uniformly convex and uniformly smooth Banach spaces are Lp, where p>1. More precisely, Lp is min{p,2}-uniformly smooth for any p>1. Note also that no Banach space is q-uniformly smooth for q>2 (see [1, 5] for more details).LetA:C→E be a nonlinear mapping. The mapping A is said to be(1)
accretive if
(1.3)〈Ax-Ay,J(x-y)〉≥0,∀x,y∈C,(2)
λ-strongly accretive if there exists a constant α>0 such that
(1.4)〈Ax-Ay,J(x-y)〉≥λ‖x-y‖2,∀x,y∈C,(3)
λ-inverse-strongly accretive if there exists a constant α>0 such that(1.5)〈Ax-Ay,J(x-y)〉≥λ‖Ax-Ay‖2,∀x,y∈C,Definition 1.1.
LetM:E→2E be a multivalued maximal accretive mapping. The single-valued mapping JM,ρ:E→E defined by
(1.6)JM,ρ(u)=(I+ρM)-1(u),∀u∈E,
is called the resolvent operator associated with M, where ρ is any positive number and I is the identity mapping.LetT be a mapping from E into itself. We use F(T) to denote the set of fixed points of the mapping T. Recall that the mapping T is said to be nonexpansive if(1.7)‖Tx-Ty‖≤‖x-y‖,∀x,y∈E.A mappingf:C→C is said to be contractive if there exists a constant α∈(0,1) such that(1.8)‖f(x)-f(y)‖≤α‖x-y‖,∀x,y∈C.Recently, Aoyama et al. [4] considered the following generalized variational inequality problem in a smooth Banach space: Find a point x∈C such that(1.9)〈Ax,j(y-x)〉≥0,∀y∈C,
where A is an accretive operator of C into E. This problem is related to the fixed point problem for nonlinear mappings, the problem of finding a zero point of an accretive operator, and so on. For the problem of finding a zero point of an accretive operator by the proximal point algorithm, see Agarwal et al. [6], Cho et al. [7, 8], Kamimura and Takahashi [9, 10], Qin et al. [11], Song et al. [12], and Wei and Cho [13]. In order to find a solution of the variational inequality (1.9), Aoyama et al. [4] studied the weak convergence theorem for accretive operators in Banach spaces, which is a generalization of the result by Iiduka et al. [14] from the class of Hilbert spaces.Theorem AIT (see [4], Aoyama et al. Theorem  3.1).
LetE be a uniformly convex and 2-uniformly smooth Banach space and let  C be a nonempty closed convex subset of  E. Let  QC be a sunny nonexpansive retraction from  E onto  C,  α>0 and  A be an  α-inverse strongly accretive operator of  C into  E with  S(C,A)≠∅, where
(1.10)S(C,A)={x*∈C:〈Ax*,j(x-x*)〉≥0,∀x∈C}.
If {λn} and {αn} are chosen such that λn∈[a,α/K2] for some a>0 and αn∈[b,c] for some b,c with 0<b<c<1, then the sequence {xn} defined by the following manners: x1=x∈C and
(1.11)xn+1=αnxn+(1-αn)QC(xn-λnAxn),∀n≥1,
converges weakly to an element z of S(C,A), where K is the 2-uniformly smoothness constant of E and QC is a sunny nonexpansive retraction.In 2011, Katchang and Kumam [15] presented an iterative algorithm for finding a common solution of fixed point problems and a general system of variational inequality problems for two accretive operators as shown in the following: for all n≥0,(1.12)x0=u∈C,yn=QC(xn-μBxn),xn+1=αnf(xn)+βnxn+γnSQC(yn-λAyn).
They proved that the sequence {xn} generated by the above algorithm converges strongly to a point x¯=Qℱf(x¯). Moreover, they apply their theorem to find zeros of accretive operators and the class of k-strictly pseudocontractive mappings.Recently, Petrot et al. [16] considered the problem so-called quasivariational inclusion problem, that is, determine an element u∈H such that(1.13)0∈A(u)+M(u),
where A:H→H is a single-valued nonlinear mapping and M:H→2H is a multivalued mapping. The set of solutions of the above problem is denoted by VI(H,A,M). Therefore, they presented a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inclusion problem with a multivalued maximal monotone mapping and an α-inverse-strongly monotone mapping by using the iterative sequence {xn} defined as follows:(1.14)x0∈H,chosenarbitrarily,xn+1=αnf(xn)+βnxn+γnSzn,zn=JMλ(yn-λAyn),yn=JMλ(xn-λAxn),∀n≥0,
and, under appropriated conditions, they proved the the sequence {xn} generated by (1.14) converges strongly to a point z0∈H, which is the unique solution in F(S)∩VI(H,A,M) to the following variational inequality:(1.15)〈(f-I)z0,z0-z〉≤0,∀z∈F(S)∩VI⁡(H,A,M).Very recently, Qin et al. [17] introduced an iterative scheme for a general variational inequality (VI) and proved the strong convergence theorems of common solutions of two variational inequalities in a uniformly convex and 2-uniformly smooth Banach space by using the following iterative sequence {xn}:(1.16)x0=u∈C,yn=δnQC(xn-ρBxn)+(1-δn)QC(xn-λAxn),xn+1=αnu+βnxn+γnyn,∀n≥0.
They proved that the sequence {xn} generated by the above algorithm converges strongly to a point q=QVIu, where QVI is the unique sunny nonexpansive retraction from C onto VI.Motivated and inspired by the above recent works, in this paper, we introduce an iterative scheme for finding zeros of maximal accretive operators. Furthermore, we prove some strong convergence theorems and also propose applications for solving the convex feasibility problems. Our results improve and extend the corresponding results of Qin et al. [17] and Katchang and Kumam [15], Petrot et al. [16], and many others.


## 2. Preliminaries

Note that, ifC and D are nonempty subsets of a Banach space E such that D is a subset of a closed convex subset C and Q:C→D. Then Q is said to be sunny if(2.1)Q(Qx+t(x-Qx))=Qx,
whenever Qx+t(x-Qx)∈C for any x∈C and t≥0. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction Q of C onto D. A mapping Q:C→C is called a retraction if Q2=Q. If a mapping Q:C→C is a retraction, then Qz=z for all z is in the range of Q (see [4, 18] for more details).The following result describes a characterization of sunny nonexpansive retractions on a smooth Banach space.Proposition 2.1 (see [19]).
LetE be a smooth Banach space and let C be a nonempty subset of E. Let Q:E→C be a retraction and let J be the normalized duality mapping on E. Then the following are equivalent:(1)
Q is sunny and nonexpansive;(2)
∥Qx-Qy∥2≤〈x-y,J(Qx-Qy)〉 for all x,y∈E;(3)
〈x-Qx,J(y-Qx)〉≤0 for all x∈E and y∈C.Proposition 2.2 (see [20]).
LetC be a nonempty closed convex subset of a uniformly convex and let uniformly smooth Banach space E and T be a nonexpansive mapping of C into itself with F(T)≠∅. Then the set F(T) is a sunny nonexpansive retract of C.We need the following lemmas in order to prove our main results.Lemma 2.3 (see [5]).
LetE be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:
(2.2)‖x+y‖2≤‖x‖2+2〈y,Jx〉+2‖Ky‖2,∀x,y∈E.Lemma 2.4 (see [21]).
Let{xn} and {yn} be bounded sequences in a Banach space E and {βn} be a sequence in [0,1] with
(2.3)0<liminf⁡n→∞βn≤limsup⁡n→∞βn<1.
Suppose that xn+1=(1-βn)yn+βnxn for all n≥0 and
(2.4)limsup⁡n→∞(‖yn+1-yn‖-‖xn+1-xn‖)≤0.
Then limn→∞∥yn-xn∥=0.Lemma 2.5 (see [22]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.5)an+1≤(1-αn)an+δn,∀n≥0,
where {αn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that(1)
∑n=1∞αn=∞;(2)
limsupn→∞δn/αn≤0 or ∑n=1∞|δn|<∞. Then limn→∞an=0.Lemma 2.6 (see [23]).
LetC be a closed convex subset of a strictly convex Banach space E. Let Tm:C→C be a nonexpansive mappings for each 1≤m≤r, where r is some integer. Suppose that ∩m=1rF(Tm) is nonempty. Let {λn} be a sequence of positive numbers with ∑m=1rλn=1. Then the mapping S:C→C defined by
(2.6)Sx=∑m=1rλmTmx,∀x∈C,
is well defined, nonexpansive, and F(S)=∩m=1rF(Tm) holds.Lemma 2.7 (see [24]).
LetC be a nonempty bounded closed convex subset of a uniformly convex Banach space E and T be nonexpansive mapping of C into itself. If {xn} is a sequence in C such that xn→x weakly and xn-Txn→0 strongly, then x is a fixed point of T.Lemma 2.8 (see [3, 4]).
LetC be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E. Let a mapping A:C→E be λ-inverse-strongly accretive. Then one has
(2.7)‖(I-ρ2A)x-(I-ρ2A)y‖2≤‖x-y‖2+2ρ2(ρ2K2-λ)‖Ax-Ay‖2.
Ifλ≥ρ2K2, then I-ρ2A is nonexpansive.Proof.
For anyx,y∈C, it follows from Lemma 2.3 that
(2.8)‖(I-ρ2A)x-(I-ρ2A)y‖2=‖(x-y)-ρ2(Ax-Ay)‖2≤‖x-y‖2-2ρ2〈Ax-Ay,j(x-y)〉+2ρ22K2‖Ax-Ay‖2≤‖x-y‖2-2ρ2λ‖Ax-Ay‖2+2ρ22K2‖Ax-Ay‖2=‖x-y‖2+2ρ2(ρ2K2-λ)‖Ax-Ay‖2.
If λ≥ρ2K2, then I-ρ2A is nonexpansive. This completes the proof.Lemma 2.9.
LetC be a nonempty subset of a Banach space E. Let A be a mapping of C into E, M be a maximal accretive operator on E and JM,ρ=(I+ρM)-1 be the resolvent of M for any ρ>0. Then F(JM,ρ(I-ρA))=(A+M)-1(0) for all ρ>0.Proof.
Letρ>0 be fixed. Then we have
(2.9)u∈F(JM,ρ(I-ρA))u⟺u=JM,ρ(I-ρA)u=(I+ρM)-1(I-ρA)u⟺(I+ρM)u∋(I-ρA)u⟺ρMu∋-ρAu⟺Mu∋-Au⟺(A+M)u∋0⟺u∈(A+M)-1(0).
This completes the proof.Lemma 2.10.
LetE be a Banach space. Then for all x,y∈E,
(2.10)‖x+y‖2≤‖x‖2+2〈y,J(x+y)〉.


## 3. Main Results

In this section, we prove strong convergence theorems for aλ-inverse-strongly accretive mapping A:C→E and a β-inverse-strongly accretive B:C→E in a real 2-uniformly smooth Banach space E.In order to prove our main results, we need the following lemma.Lemma 3.1.
LetC be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E with the best smooth constant K. Let JM1,ρ1,JM2,ρ2 be a resolvent operator associated with M1,M2, where M1,M2:E→2E is a multivalued maximal accretive mapping. Let the mappings A,B:C→E be λ-inverse-strongly accretive and β-inverse-strongly accretive, respectively. Let G:C→C be a mapping defined by
(3.1)Gx=δJM1,ρ1(x-ρ1Bx)+(1-δ)JM2,ρ2(x-ρ2Ax),∀x∈C.
If λ≥ρ2K2 and β≥ρ1K2, then G is nonexpansive.Proof.
SinceJM1,ρ1 and JM2,ρ2 are nonexpansive, for any x,y∈C, it follows from Lemma 2.8 that
(3.2)‖G(x)-G(y)‖=‖δJM1,ρ1(x-ρ1Bx)+(1-δ)JM2,ρ2(x-ρ2Ax)-δJM1,ρ1(y-ρ1By)-(1-δ)JM2,ρ2(y-ρ2Ay)‖≤δ‖JM1,ρ1(x-ρ1Bx)-JM1,ρ1(y-ρ1By)‖+(1-δ)‖JM2,ρ2(x-ρ2Ax)-JM2,ρ2(y-ρ2Ay)‖≤δ‖JM1,ρ1(I-ρ1B)x-JM1,ρ1(I-ρ1B)y‖+(1-δ)‖JM2,ρ2(I-ρ2A)x-JM2,ρ2(I-ρ2A)y‖≤‖x-y‖.
Therefore, G is nonexpansive. This completes the proof.Next, we state the main result of this work.Theorem 3.2.
LetE be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and C be a nonempty closed convex subset of E. Let A,B:C→E be λ-inverse-strongly accretive and β-inverse-strongly accretive, respectively, and K be the best smooth constant. Let f be a contraction of E into itself with coefficient α∈[0,1). Suppose that Ω:=(A+M2)-1(0)∩(B+M1)-1(0)≠∅ and G is a mapping defined by Lemma 3.1. Let ρ1, ρ2 be any positive real numbers such that ρ1≤β/K2 and ρ2≤λ/K2. For arbitrary x0=x∈C, define the iterative sequence {xn} as follows:
(3.3)x0=u∈C,yn=δnJM1,ρ1(xn-ρ1Bxn)+(1-δn)JM2,ρ2(xn-ρ2Axn),xn+1=αnf(xn)+βnxn+γnyn,
where the sequences {αn}, {βn}, and {γn} in (0,1) satisfy the following conditions:(C1)
αn+βn+γn=1;(C2)
limn→∞αn=0 and ∑n=0∞αn=∞;(C3)
0<liminfn→∞βn≤limsupn→∞βn<1;(C4)
limn→∞δn=δ∈(0,1).Then the sequence {xn} generated by (3.3) converges strongly to a point q=QΩf(q), where QΩ is a sunny nonexpansive retraction on Ω.Proof.
First, we prove thatJM1,ρ1(I-ρ1B) and JM2,ρ2(I-ρ2A) are nonexpansive mappings. Consider the following:
(3.4)‖JM1,ρ1(I-ρ1B)x-JM1,ρ1(I-ρ1B)y‖2≤‖(x-y)-ρ1(Bx-By)‖2≤‖x-y‖2-2ρ1〈Bx-By,J(x-y)〉+2K2ρ12‖Bx-By‖2≤‖x-y‖2-2ρ1β‖Bx-By‖2+2K2ρ12‖Bx-By‖2=‖x-y‖2+2ρ1(ρ1K2-β)‖Bx-By‖2≤‖x-y‖2.
Thus, it follows that JM1,ρ1(I-ρ1B) is nonexpansive and so is JM2,ρ2(I-ρ2A).Step 1.
We show that{xn} is bounded. For any p∈Ω, we have
(3.5)‖yn-p‖=‖δn[JM1,ρ1(xn-ρ1Bxn)-p]+(1-δn)[JM2,ρ2(xn-ρ2Axn)-p]‖≤δn‖JM1,ρ1(xn-ρ1Bxn)-p‖+(1-δn)‖JM2,ρ2(xn-ρ2Axn)-p‖≤δn‖xn-p‖+(1-δn)‖xn-p‖≤‖xn-p‖.
It follows by induction that
(3.6)‖xn+1-p‖=‖αnf(xn)+βnxn+γnyn-p‖≤αn‖f(xn)-p‖+βn‖xn-p‖+γn‖yn-p‖≤ααn‖xn-p‖+αn‖f(p)-p‖+βn‖xn-p‖+γn‖xn-p‖≤ααn‖xn-p‖+αn‖f(p)-p‖+βn‖xn-p‖+γn‖xn-p‖=(1-αn+ααn)‖xn-p‖+αn‖f(p)-p‖=(1-αn(1-α))‖xn-p‖+αn(1-α)‖f(p)-p‖1-α≤max⁡{‖xn-p‖,‖f(p)-p‖1-α}⋯≤max⁡{‖x1-p‖,‖f(p)-p‖1-α}.
Thus the sequence {xn} is bounded and so is {yn}.Step 2.
We show thatlimn→∞∥xn+1-xn∥=0. Let un=JM1,ρ1(I-ρ1B)xn and vn=JM2,ρ2(I-ρ2A)xn for each n≥0. Then we have
(3.7)yn+1-yn=(δn+1un+1+(1-δn+1)vn+1)-(δnun+(1-δn)vn)=δn+1(un+1-un)+(δn+1-δn)(un-vn)+(1-δn+1)(vn+1-vn),
and so
(3.8)‖yn+1-yn‖≤δn+1‖un+1-un‖+|δn+1-δn|‖un-vn‖+(1-δn+1)‖vn+1-vn‖≤δn+1‖xn+1-xn‖+|δn+1-δn|M1+(1-δn+1)‖xn+1-xn‖=‖xn+1-xn‖+|δn+1-δn|M1,
where M1 ia an appropriate constant such that M1≥supn≥0{∥un-vn∥}.
Next, letzn=(xn+1-βnxn)/(1-βn) for all n≥0. Then we have xn+1=(1-βn)zn+βnxn for all n≥0. Now, we compute
(3.9)zn+1-zn=xn+2-βn+1xn+11-βn+1-xn+1-βnxn1-βn=αn+1f(xn+1)+γn+1yn+11-βn+1-αnf(xn)+γnyn1-βn=αn+1f(xn+1)+(1-αn+1-βn+1)yn+11-βn+1-αnf(xn)+(1-αn-βn)yn1-βn=αn+1f(xn+1)-αn+1yn+11-βn+1-αnf(xn)-αnyn1-βn+yn+1-yn,
and so
(3.10)‖zn+1-zn‖≤αn+11-βn+1‖f(xn+1)-yn+1‖+αn1-βn‖f(xn)-yn‖+‖yn+1-yn‖.
Substituting (3.8) into (3.10), we get
(3.11)‖zn+1-zn‖≤αn+11-βn+1‖f(xn+1)-yn+1‖+αn1-βn‖f(xn)-yn‖+‖xn+1-xn‖+|δn+1-δn|M1,
that is,
(3.12)‖zn+1-zn‖-‖xn+1-xn‖≤αn+11-βn+1‖f(xn+1)-yn+1‖+αn1-βn‖f(xn)-yn‖+|δn+1-δn|M1.
From the conditions (C2) and (C3), it follows that
(3.13)limsup⁡n→∞(‖zn+1-zn‖-‖xn+1-xn‖)≤0.
Thus, from Lemma 2.4, it follows that
(3.14)lim⁡n→∞‖zn-xn‖=0.
From the definition of xn+1 in this step, we observe that xn+1-xn=(1-βn)(zn-xn). Then we have
(3.15)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞‖(1-βn)(zn-xn)‖=0.Step 3.
We show thatlimsupn→∞〈(f-I)q,J(xn-q)〉≤0, where q=QΩf(q). Define a mapping G:C→C by Lemma 3.1 Then, it follows that G is a nonexpansive mapping such that
(3.16)F(G)=F(JM1,ρ1(I-ρ1B))∩F(JM2,ρ2(I-ρ2A))=(B+M)-1(0)∩(A+M)-1(0)=Ω.
Consider the following:
(3.17)yn-Gxn=δnvn+(1-δn)un-(δvn+(1-δ)un)=(δn-δ)(vn-un).
From the condition (C4), we have
(3.18)lim⁡n→∞‖yn-Gxn‖=0.
Next, we consider
(3.19)‖xn-Gxn‖=‖xn-xn+1+xn+1-yn+yn-Gxn‖≤‖xn-xn+1‖+‖xn+1-yn‖+‖yn-Gxn‖≤‖xn-xn+1‖+αn‖f(xn)-yn‖+βn‖xn-yn‖+‖yn-Gxn‖≤‖xn-xn+1‖+αn‖f(xn)-yn‖+βn‖xn-Gxn‖+βn‖Gxn-yn‖+‖yn-Gxn‖.
Therefore, we have
(3.20)(1-βn)‖xn-Gxn‖≤‖xn-xn+1‖+αn‖f(xn)-yn‖+(βn+1)‖Gxn-yn‖.
From the conditions (C2), (C3), (3.15), (3.18), and the inequality above, we obtain
(3.21)lim⁡n→∞‖xn-Gxn‖=0.
Thus, since QΩf(q) is a contraction, there exists a unique fixed point. We denote that q is the unique fixed point to the mapping QΩf(q) which means that q=QΩf(q).
Since{xn} is bounded, there exists a subsequence {xni} of {xn} such that xni⇀p, it follows from (3.21) that
(3.22)lim⁡n→∞‖xni-Gxni‖=0.
Since G is nonexpansive, it follows from Lemma  2.7 that p=Gp we obtain that p∈F(G). By (3.22), we have p∈Ω.
Furthermore, with the reason that{xn} is bounded, we can choose the sequence {xni} of {xn} which {xni}⇀p such that
(3.23)limsup⁡n→∞〈(f-I)q,J(xn-q)〉=lim⁡i→∞〈(f-I)q,J(xni-q)〉.
Now, from (3.23) and Proposition 2.1(3) and the weakly sequential continuity of the duality mapping J, we have
(3.24)limsup⁡n→∞〈(f-I)q,J(xn-q)〉=lim⁡i→∞〈(f-I)q,J(xni-q)〉=〈(f-I)q,J(p-q)〉≤0.
From (3.15), it follows that
(3.25)limsup⁡n→∞〈(f-I)q,J(xn+1-q)〉≤0.Step 4.
We show that{xn} converges strongly to a point q=QΩf(q). In fact, observe that
(3.26)‖xn+1-q‖2=〈xn+1-q,J(xn+1-q)〉=〈αn(f(xn)-q)+βn(xn-q)+γn(yn-q),J(xn+1-q)〉≤ααn〈xn-q,J(xn+1-q)〉+αn〈f(q)-q,J(xn+1-q)〉+βn〈xn-q,J(xn+1-q)〉+γn〈yn-q,J(xn+1-q)〉≤ααn‖xn-q‖‖xn+1-q‖+αn〈f(q)-q,J(xn+1-q)〉+βn‖xn-q‖‖xn+1-q‖+γn‖yn-q‖‖xn+1-q‖≤(1-αn(1-α))‖xn-q‖‖xn+1-q‖+αn〈f(q)-q,J(xn+1-q)〉≤1-αn(1-α)2(‖xn-q‖2+‖xn+1-q‖2)+αn〈f(q)-q,J(xn+1-q)〉≤1-αn(1-α)2‖xn-q‖2+12‖xn+1-q‖2+αn〈f(q)-q,J(xn+1-q)〉.
Thus it follows that
(3.27)‖xn+1-q‖2≤(1-αn(1-α))‖xn-q‖2+2αn〈f(q)-q,J(xn+1-q)〉.
Therefore, from Condition (C2), (3.25), and Lemma 2.5, we get ∥xn-q∥→0 as n→∞. This completes the proof.Corollary 3.3.
LetE be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and let C be a nonempty closed convex subset of E. Let A,B:C→E be λ-inverse-strongly accretive and β-inverse-strongly accretive, respectively, and K be the best smooth constant. Suppose that Ω:=(A+M2)-1(0)∩(B+M1)-1(0)≠∅, where G is a mapping defined by Lemma 3.1. Let ρ1, ρ2 be any positive real numbers such that ρ1≤β/K2 and ρ2≤λ/K2. For arbitrary x0=x∈C, define the iterative sequence {xn} by
(3.28)x0=u∈C,yn=δnJM1,ρ1(xn-ρ1Bxn)+(1-δn)JM2,ρ2(xn-ρ2Axn),xn+1=αnu+βnxn+γnyn,∀n≥0,
where the sequences {αn}, {βn}, and {γn} in (0,1) satisfy the following conditions:(C1)
αn+βn+γn=1;(C2)
limn→∞αn=0 and ∑n=0∞αn=∞;(C3)
0<liminfn→∞βn≤limsupn→∞βn<1;(C4)
limn→∞δn=δ∈(0,1).Then the sequence {xn} generated by (3.28) converges strongly to a point q=QΩf(q), where QΩ is a sunny nonexpansive retraction on Ω.Proof.
Takef(xn)=u for all n≥1 for any fixed u∈C in (3.3). Then, by Theorem 3.2, we can conclude the desired conclusion easily.


## 4. Applications

### 4.1. Application to Convex Feasibility Problems

In this part, we consider the following convex feasibility problem (CFP): findx∈∩j=1NCj, where j∈{1,2,…,N} and Cj denotes the set of zeros of a maximal accretive operator.The following result can be obtained from Theorem3.2.Theorem 4.1.
LetE be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and let C be a nonempty closed convex subset of E. Let {Ai}i=1N:C→E be an λi-inverse-strongly accretive and K be the best smooth constant. Let f be a contraction of E into itself with coefficient α∈[0,1). Suppose that Ω:=⋂i=1N(Ai+Mi)-1(0)≠∅, where G is a mapping defined by Lemma 3.1. Let ρi be any positive real numbers such that ρi≤λi/K2, i=1,2,3,…,N. For arbitrary x0=x∈C, define the iterative sequence {xn} by
(4.1)x0=u∈C,yn=∑i=1Nδi,nJMi,ρi(xn-ρiAixn),xn+1=αnf(xn)+βnxn+γnyn,∀n≥0,
where the sequences {αn}, {βn}, and {γn} in (0,1) satisfy the following conditions:(C1)
αn+βn+γn=1 and ∑i=1Nδi,n=1;(C2)
limn→∞αn=0 and ∑n=0∞αn=∞;(C3)
0<liminfn→∞βn≤limsupn→∞βn<1;(C4)
limn→∞δi,n=δ∈(0,1).Then the sequence {xn} generated by (4.1) converges strongly to a point q=QΩ, where QΩ is a sunny nonexpansive retraction on Ω.


### 4.2. Application to Hilbert Spaces

Assume thatH is a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let A:H→H be a single-valued nonlinear mapping and M:H→2H be a multivalued mapping. The problem of finding u∈H such that(4.2)θ∈A(u)+M(u)
is called the quasivariational inclusion problem, and we denote the set of solutions of the above variational inclusion by VI(H,A,M).IfM=∂δC, where C is a nonempty closed convex subset of H and δC:H→[0,∞] is the indicator function of C, that is,(4.3)δC(x)={0,x∈C,+∞,x∉C.
Then the variational inclusion problem (4.2) is equivalent to the problem of finding u∈C such that(4.4)〈A(u),v-u〉≥0,∀v∈C,
which is the well-known Hartman-Stampacchia variational inequality problem [25].Theorem 4.2.
LetC be a closed convex subset of a real Hilbert space H. Let A,B:C→H be λ-inverse-strongly monotone and β-inverse-strongly monotone, respectively. Let f be a contraction of E into itself with coefficient α∈[0,1). Suppose that Ω:=VI(C,A)∩VI(C,B)≠∅, where VI(C,A) and VI(C,B) are the sets of solutions of variational inequality (4.4). For arbitrary x0=x∈C, define the iterative sequence {xn} by
(4.5)x0=u∈C,yn=δnPC(xn-ρ1Bxn)+(1-δn)PC(xn-ρ2Axn),xn+1=αnf(xn)+βnxn+γnyn,∀n≥0,
where the sequences {αn}, {βn}, and {γn} in (0,1) satisfy the following conditions:(C1)
αn+βn+γn=1;(C2)
limn→∞αn=0 and ∑n=0∞αn=∞;(C3)
0<liminfn→∞βn≤limsupn→∞βn<1;(C4)
limn→∞δn=δ∈(0,1).Then the sequence {xn} generated by (4.5) converges strongly to a point PΩx0.Proof.
TakeM=∂δC:H→2H, where δC:H→[0,∞] is the indicator function of C. Let J(M,ρ)=I. Then we get
(4.6)PC(xn-ρ1Axn)=JM,ρ1PC(xn-ρ1Axn),PC(xn-ρ2Bxn)=JM,ρ2PC(xn-ρ2Bxn).
Thus the conclusion can be obtained from Theorem 3.2 immediately.


## 4.1. Application to Convex Feasibility Problems

In this part, we consider the following convex feasibility problem (CFP): findx∈∩j=1NCj, where j∈{1,2,…,N} and Cj denotes the set of zeros of a maximal accretive operator.The following result can be obtained from Theorem3.2.Theorem 4.1.
LetE be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and let C be a nonempty closed convex subset of E. Let {Ai}i=1N:C→E be an λi-inverse-strongly accretive and K be the best smooth constant. Let f be a contraction of E into itself with coefficient α∈[0,1). Suppose that Ω:=⋂i=1N(Ai+Mi)-1(0)≠∅, where G is a mapping defined by Lemma 3.1. Let ρi be any positive real numbers such that ρi≤λi/K2, i=1,2,3,…,N. For arbitrary x0=x∈C, define the iterative sequence {xn} by
(4.1)x0=u∈C,yn=∑i=1Nδi,nJMi,ρi(xn-ρiAixn),xn+1=αnf(xn)+βnxn+γnyn,∀n≥0,
where the sequences {αn}, {βn}, and {γn} in (0,1) satisfy the following conditions:(C1)
αn+βn+γn=1 and ∑i=1Nδi,n=1;(C2)
limn→∞αn=0 and ∑n=0∞αn=∞;(C3)
0<liminfn→∞βn≤limsupn→∞βn<1;(C4)
limn→∞δi,n=δ∈(0,1).Then the sequence {xn} generated by (4.1) converges strongly to a point q=QΩ, where QΩ is a sunny nonexpansive retraction on Ω.


## 4.2. Application to Hilbert Spaces

Assume thatH is a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let A:H→H be a single-valued nonlinear mapping and M:H→2H be a multivalued mapping. The problem of finding u∈H such that(4.2)θ∈A(u)+M(u)
is called the quasivariational inclusion problem, and we denote the set of solutions of the above variational inclusion by VI(H,A,M).IfM=∂δC, where C is a nonempty closed convex subset of H and δC:H→[0,∞] is the indicator function of C, that is,(4.3)δC(x)={0,x∈C,+∞,x∉C.
Then the variational inclusion problem (4.2) is equivalent to the problem of finding u∈C such that(4.4)〈A(u),v-u〉≥0,∀v∈C,
which is the well-known Hartman-Stampacchia variational inequality problem [25].Theorem 4.2.
LetC be a closed convex subset of a real Hilbert space H. Let A,B:C→H be λ-inverse-strongly monotone and β-inverse-strongly monotone, respectively. Let f be a contraction of E into itself with coefficient α∈[0,1). Suppose that Ω:=VI(C,A)∩VI(C,B)≠∅, where VI(C,A) and VI(C,B) are the sets of solutions of variational inequality (4.4). For arbitrary x0=x∈C, define the iterative sequence {xn} by
(4.5)x0=u∈C,yn=δnPC(xn-ρ1Bxn)+(1-δn)PC(xn-ρ2Axn),xn+1=αnf(xn)+βnxn+γnyn,∀n≥0,
where the sequences {αn}, {βn}, and {γn} in (0,1) satisfy the following conditions:(C1)
αn+βn+γn=1;(C2)
limn→∞αn=0 and ∑n=0∞αn=∞;(C3)
0<liminfn→∞βn≤limsupn→∞βn<1;(C4)
limn→∞δn=δ∈(0,1).Then the sequence {xn} generated by (4.5) converges strongly to a point PΩx0.Proof.
TakeM=∂δC:H→2H, where δC:H→[0,∞] is the indicator function of C. Let J(M,ρ)=I. Then we get
(4.6)PC(xn-ρ1Axn)=JM,ρ1PC(xn-ρ1Axn),PC(xn-ρ2Bxn)=JM,ρ2PC(xn-ρ2Bxn).
Thus the conclusion can be obtained from Theorem 3.2 immediately.



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*Source: 185104-2012-04-10.xml*