# Convergence Theorems for Accretive Operators with Nonlinear Mappings in Banach Spaces

Authors: Yan-Lai Song; Lu-Chuan Ceng
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The purpose of this paper is to present two new forward-backward splitting schemes with relaxations and errors for finding a common element of the set of solutions to the variational inclusion problem with two accretive operators and the set of fixed points of strict pseudocontractions in infinite-dimensional Banach spaces. Under mild conditions, some weak and strong convergence theorems for approximating these common elements are proved. The methods in the paper are novel and different from those in the early and recent literature. Further, we consider the problem of finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a strict pseudocontractions.


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## Body

## 1. Introduction

The theory of nonexpansive mappings is very important because it is applied to convex optimization, the theory of nonlinear evolution equations, and others. Browder and Petryshyn [1] introduced a class of nonlinear mappings, called strict pseudocontractions, which includes the class of nonexpansive mappings. For strict pseudocontractions, we are interested in finding fixed points of the mappings. We also know the class of inverse-strongly accretive operators which is related to nonexpansive mappings. For inverse-strongly accretive operators, we are interested in finding zero points of the mappings.LetC be a nonempty closed convex subset of a real Hilbert space H. Let A:C→H be a single-valued nonlinear mapping and let B:H→2H be a multivalued mapping. The so called quasi-variational inclusion problem is to find a z∈H such that
(1)0∈(A+B)z.
The set of solutions of (1) is denoted by (A+B)-1(0). A number of problems arising in structural analysis, mechanics, and economics can be studied in the framework of this kind of variational inclusions; see, for instance, [2–5]. The problem (1) includes many problems as special cases. (1)
IfB=∂ϕ:H→2H, where ϕ:H→ℝ∪+∞ is a proper convex lower semicontinuous function and ∂ϕ is the subdifferential and if only erential of ∂ϕ, then the variational inclusion problem (1) is equivalent to finding u∈H such that
(2)〈Au,y-u〉+ϕ(y)-ϕ(u)≥0,∀y∈H,which is called the mixed quasi-variational inequality (see, Noor [6]).(2)
IfB=∂δC, where C is a nonempty closed convex subset of H and δC:H→[0,∞] is the indicator function of C, that is,
(3)δC={0,x∈C,+∞,x∉C,then the variational inclusion problem (1) is equivalent to finding u∈C such that
(4)〈Au,v-u〉≥0,∀v∈C.
This problem is called Hartman-Stampacchia variational inequality (see, e.g., [7]).In [8], Zhang et al. investigated the problem of finding a common element of the set of solutions to the inclusion problem and the set of fixed points of nonexpansive mappings by considering the following iterative algorithm:
(5)yn=JM,λ(xn-λAxn),xn+1=αnx+(1-αn)Syn,
where A:H→H is an α-cocoercive mapping, M:H→2H is a maximal monotone mapping, S:H→H is a nonexpansive mapping, and {αn} is a sequence in [0,1]. Under mild conditions, they obtained a strong convergence theorem.In [9], Manaka and Takahashi introduced the following iteration:
(6)x1∈C,xn+1=αnxn+(1-αn)SJλn(I-λnA)xn,n≥1,
where {αn} is a sequence in (0, 1), {λn} is a positive sequence, S:C→C is a nonexpansive mapping, A:C→H is an inverse-strongly monotone mapping, B:D(B)⊂C→2H is a maximal monotone operator, and Jλn=(I+λnB)-1 is the resolvent of B. They showed that the sequence {xn} generated in (6) converges weakly to some z∈(A+B)-1(0)∩F(S) provided that the control sequence satisfies some restrictions.It is well known that the quasi-variational inclusion problem in the setting of Hilbert spaces has been extensively studied in the literature; see, for instance, [4–10]. However, there is little work in the existing literature on this problem in the setting of Banach spaces (though there was some work on finding a common zero of a finite family of accretive operators [11–13]). The main difficulties are due to the fact that the inner product structure of a Hilbert space fails to be true in a Banach space. To overcome these difficulties, López et al. [14] use the new technique to carry out certain initiative investigations on splitting methods for accretive operators in Banach spaces. They considered the following algorithms with errors in Banach spaces:
(7)xn+1=(1-αn)xn+αn(Jrn(xn-rn(Axn+an))+bn),(8)xn+1=αnu+(1-αn)(Jrn(xn-rn(Axn+an))+bn),
where u∈E,{an},{bn}⊂E, and Jrn=(I+rnB)-1 is the resolvent of B. Then they studied the weak and strong convergence of algorithms (7) and (8), respectively.Motivated and inspired by Zhang et al. [8], Manaka and Takahashi [9], Takahashi et al. [10], Chen and Fan [13], López et al. [14], and Cho et al. [15], the purpose of this paper is to introduce two iterative forward-backward splitting methods for finding a common element of the set of solutions of the variational inclusion problem (1) with m-accretive operators and inverse-strongly accretive operators and the set of fixed points of strict pseudocontractions in the setting of Banach spaces. Under suitable conditions, some weak and strong convergence theorems for approximating to these common elements are proved. The results presented in the paper improve and extend the corresponding results in [8–10, 13–15].


## 2. Preliminaries

Throughout this paper, we denote byE and E* a real Banach space and the dual space of E, respectively. Let C be a subset of E and let T be a mapping on C. We use F(T) to denote the set of fixed points of T. The expressions xn→x and xn⇀x denote the strong and weak convergence of the sequence {xn}, respectively, and ωw(xn) stands for the set of weak limit points of the sequence {xn}. ℬr will denote the closed ball with center zero and radius r.Letq>1 be a real number. The (generalized) duality mapping Jq:E→2E* is defined by
(9)Jq(x)={x*∈E*:〈x,x*〉=∥x∥q,∥x*∥=∥x∥q-1}
for all x∈E, where 〈·,·〉 denotes the generalized duality pairing between E and E*. In particular, J=J2 is called the normalized duality mapping and Jq(x)=∥x∥q-2J2(x) for x≠0. If E is a Hilbert space, then J=I where I is the identity mapping. It is well known that if E is smooth, then Jq is single-valued, which is denoted by jq.A Banach spaceE is said to be uniformly convex if, for any ε∈(0,2], there exists δ>0 such that, for any x,y∈S(E),∥x-y∥≥ε implies ∥(x+y)/2∥≤1-δ. It is known that a uniformly convex Banach space is reflexive and strictly convex.The norm of a Banach spaceE is said to be Gâteaux differentiable if the limit
(10)lim⁡t→0∥x+ty∥-∥x∥t
exists for all x,y on the unit sphere S(E)={x∈E:∥x∥=1}. If, for each y∈S(E), limit (10) is uniformly attained for x∈S(E), then the norm of E is said to be uniformly Gâteaux differentiable. The norm of E is said to be Fréchet differentiable if, for each x∈S(E), limit (10) is attained uniformly for y∈S(E).LetρE:[0,1)→[0,1) be the modulus of smoothness of E defined by
(11)ρE(t)=sup⁡{12(∥x+y∥+∥x-y∥)-1:x∈S(E),∥y∥≤t}.A Banach spaceE is said to be uniformly smooth if ρE(t)/t→0 as t→0. Let q>1. A Banach space E is said to be q-uniformly smooth, if there exists a fixed constant c>0 such that ρE(t)≤ctq. It is well known that E is uniformly smooth if and only if the norm of E is uniformly Fréchet differentiable. If E is q-uniformly smooth, then q≤2 and E is uniformly smooth, and hence the norm of E is uniformly Fréchet differentiable; in particular, the norm of E is Fréchet differentiable. Typical 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 every p>1.A Banach spaceE is said to satisfy Opial’s condition if for any sequence {xn} in E the condition that {xn} converges weakly to x∈E implies that the inequality
(12)liminf⁡n→∞∥xn-x∥<liminf⁡n→∞∥xn-y∥
holds for every y∈E with y≠x.A Banach spaceE is said to have the Kadec-Klee property if, for every sequence {xn} in E, xn⇀x and ∥xn∥→∥x∥ together imply ∥xn-x∥→0. As we know the duals of reflexive Banach spaces with Fréchet differentiable norms have the Kadec-Klee property.Definition 1.
A mappingT:C→E is said to be (1)
nonexpansive if(13)∥Tx-Ty∥≤∥x-y∥∀x,y∈C;(2)
k-Lipschitz if there exists k>0 such that
(14)∥Tx-Ty∥≤k∥x-y∥∀x,y∈C;in particular, if 0<k<1, then T is called contractive and if k=1, then T reduces to a nonexpansive mapping; (3)
λ-strict pseudocontractive in the terminology of Browder and Petryshyn if for all x,y∈C, there exists λ>0 and jq(x-y)∈Jq(x-y) such that
(15)〈Tx-Ty,jq(x-y)〉≤∥x-y∥q-λ∥(I-T)x-(I-T)y∥q;(4)
accretive if for allx,y∈C, there exists jq(x-y)∈Jq(x-y) such that
(16)〈Tx-Ty,jq(x-y)〉≥0;(5)
η-strongly accretive if for all x,y∈C, there exists η>0 and jq(x-y)∈Jq(x-y) such that
(17)〈Tx-Ty,jq(x-y)〉≥η∥x-y∥q;(6)
μ-inverse-strongly accretive if for all x,y∈C, there exists μ>0 and jq(x-y)∈Jq(x-y) such that
(18)〈Tx-Ty,jq(x-y)〉≥μ∥Tx-Ty∥q.Remark 2.
The conception of strict pseudocontractions was firstly introduced by Browder and Petryshyn [1] in a real Hilbert space. Let C be a nonempty subset of a real Hilbert space H, and let T:C→C be a mapping. In light of [1], T is said to be a K-strict pseudocontraction, if there exists a K∈[0,1) such that
(19)∥Tx-Ty∥2≤∥x-y∥2+K∥(I-T)x-(I-T)y∥2∀x,y∈C.Remark 3.
The class of strictly pseudocontractive mappings has been studied by several authors (see, e.g., [1, 16, 17]). However, their iterative methods are far less developed though Browder and Petryshyn [1] initiated their work in 1967. As a matter of fact, strictly pseudocontractive mappings have more powerful applications in solving inverse problems (see, e.g., [18]). Therefore it is interesting to develop the theory of iterative methods for strictly pseudocontractive mappings.Remark 4.
IfE:=H is a real Hilbert space, then accretive and strongly accretive operators coincide with monotone and strongly monotone operators, respectively.Definition 5.
A set-valued mappingT:D(T)⊆E→2E is said to be(1)
accretive if for anyx,y∈D(T), there exists j(x-y)∈J(x-y), such that for all u∈T(x) and v∈T(y),
(20)〈u-v,j(x-y)〉≥0;(2)
m-accretive if T is accretive and (I+rT)(D(T))=E for every (equivalently, for some) r>0, where I is the identity mapping. In real Hilbert spaces, m-accretive operators coincide with maximal monotone operators.
LetM:D(M)→2E be m-accretive. The mapping JrM:E→D(M) defined by
(21)JrM(u)=(I+rM)-1(u),∀u∈E,
is called the resolvent operator associated with M, where r is any positive number and I is the identity mapping. It is well known that JrM is single-valued and nonexpansive.In order to prove our main results, we need the following lemmas.Lemma 6 (see [19]).
LetE be a Banach space and let Jq be a generalized duality mapping. Then for any given x,y∈E, the following inequality holds:
(22)∥x+y∥q≤∥x∥q+q〈y,jq(x+y)〉,jq(x+y)∈Jq(x+y).
In particular, we have, for any given x,y∈E,
(23)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,j(x+y)∈J(x+y).Lemma 7 (see [19]).
Let1<p<∞,q∈(1,2],r>0 be given.(i)
IfE is uniformly convex, then there exists a continuous, strictly increasing, and convex function φ:[0,∞)→[0,∞) with φ(0)=0 such that
(24)∥λx+(1-λ)y∥p≤λ∥x∥p+λ∥y∥p-Wp(λ)φ(∥x-y∥),x,y∈ℬr,0≤λ≤1,whereWp(λ)=λp(1-λ)+(1-λ)pλ,ℬr={z∈E:∥z∥≤r}.(ii)
IfE is a real q-uniformly smooth Banach space, then there exists a constant Cq>0 such that
(25)∥x+y∥q≤∥x∥q+q〈y,Jq(x)〉+Cq∥y∥q,∀x,y∈E.Lemma 8 (see [20]).
Let{an},{bn}, and {δn} be sequences of nonnegative real numbers satisfying the inequality
(26)an+1≤(1+δn)an+bn,∀n=1,2,….
If ∑n=0∞‍δn<∞ and ∑n=0∞‍bn<∞, then lim⁡n→∞an exists. In particular, lim⁡n→∞an=0 whenever there exists a subsequence {ank} in {an} which strongly converges to zero.Lemma 9 (see [21]).
Let{αn} be a sequence of nonnegative numbers satisfying the following property:
(27)αn+1≤(1-γn)αn+bn+γncn,n∈ℕ,
where {γn},{bn},{cn} satisfy the restrictions (i)
∑n=1∞‍γn=∞,(ii)
bn≥0,∑n=1∞‍bn<∞,(iii)
limsup⁡n→∞cn≤0.Then, lim⁡n→∞αn=0.Lemma 10 (see [16]).
LetC be a nonempty convex subset of a real q-uniformly smooth Banach space E and let T:C→C be a λ-strict pseudocontraction. For α∈(0,1), we define Tαx=(1-α)x+αTx. Then, as α∈(0,ρ], ρ=min⁡{1,{qλ/Cq}1/(q-1)}, Tα:C→C is nonexpansive such that F(Tα)=F(T).Lemma 11 (see [22]).
LetE be a uniformly convex Banach space, C a closed convex subset of E, and T:C→E a nonexpansive mapping with F(T)≠∅. Then, I-T is demiclosed at zero.Lemma 12.
LetC be a nonempty closed convex subset of a real q-uniformly smooth Banach space E. Let the mapping A:C→E be an α-inverse-strongly accretive operator. Then the following inequality holds:
(28)∥(I-λA)x-(I-λA)y∥q≤∥x-y∥q-λ(qα-Cqλq-1)∥Ax-Ay∥q.
In particular, if 0<λ≤(qα/Cq)1/(q-1), then ∥I-λA∥ is nonexpansive.Proof.
Indeed, for allx,y∈C, it follows from Lemma 7 that
(29)∥(I-λA)x-(I-λA)y∥q=∥(x-y)-λ(Ax-Ay)∥q≤∥x-y∥q-qλ〈Ax-Ay,jq(x-y)〉+Cqλq∥Ax-Ay∥q≤∥x-y∥q-qαλ∥Ax-Ay∥q+Cqλq∥Ax-Ay∥q≤∥x-y∥q-λ(qα-Cqλq-1)∥Ax-Ay∥q.
It is clear that if 0<λ≤(qα/Cq)1/(q-1), then I-λA is nonexpansive. This completes the proof.Lemma 13 (see [23]).
IfE is a uniformly convex Banach space and C is a closed convex bounded subset of E, there is a continuous strictly increasing function g:[0,∞)→[0,∞) with g(0)=0 such that
(30)g(∥S(tx+(1-t)y)-(αSx+(1-α)Sy)∥)≤∥x-y∥-∥Sx-Sy∥
for all x,y∈C,t∈[0,1] and nonexpansive mapping S:C→E.Lemma 14 (see [24]).
LetE be a real reflexive Banach space such that its dual E* has the Kadec-Klee property. Let {xn} be a bounded sequence in E and x*,y*∈ωw(xn); here ωw(xn) denotes the weak w-limit set of {xn}. Suppose lim⁡n→∞∥txn+(1-t)x*-y*∥ exists for all t∈[0,1]. Then x*=y*.Lemma 15.
Assume thatE is a real uniformly convex and q-uniformly smooth Banach space. Suppose that A:E→E is α-inverse-strongly accretive operator for some α>0 and B:E→2E is an m-accretive operator. Moreover, denote Jr by
(31)Jr≡JrB=(I+rB)-1
and Tr by
(32)Tr=Jr(I-rA)=(I+rB)-1(I-rA).
Then, it holds for all r>0 that F(Tr)=(A+B)-1(0).Proof.
From the definition ofTr, we have
(33)x=Trx⟺x=(I+rB)-1(I-rA)x⟺(I-rA)x∈(I+rB)x⟺0∈(A+B)x.
This completes the proof.Lemma15 alludes to the fact that, in order to solve the inclusion problem (1), it suffices to find a fixed point of Tr. Since Tr is already split, an iterative algorithm for Tr corresponds to a splitting algorithm for (1). However, to guarantee convergence (weak or strong) of an iterative algorithm for Tr, we need good metric properties of Tr such as nonexpansivity. To this end, some related geometric conditions on the underlying space E are very necessary (see Lemmas 16 and 17 below).Lemma 16 (see [14]).
Assume thatE is a real uniformly convex and q-uniformly smooth Banach space. Suppose that A:E→E is α-inverse-strongly accretive operator for some α>0 and B:E→2E is an m-accretive operator. Then, the following relations hold.(i)
Given0<s≤r and x∈E,
(34)∥Tsx-Trx∥≤|1-sr|∥x-Trx∥,∥x-Tsx∥≤2∥x-Trx∥.(ii)
Givens>0, there exists a continuous, strictly increasing, and convex function ϕq:[0,∞)→[0,∞) with ϕq(0)=0 such that, for all x,y∈ℬs,
(35)∥Trx-Try∥q≤∥x-y∥q-r(αq-rq-1Cq)∥Ax-Ay∥q-ϕq(∥(I-Jr)(I-rA)x-(I-Jr)(I-rA)y∥).Lemma 17.
LetE be a real uniformly convex and q-uniformly smooth Banach space. Suppose that S:E→E is a nonexpansive mapping, A:E→E is an α-inverse-strongly accretive operator for some α>0, and B:E→2E is an m-accretive operator. Assume that 0<r≤(qα/Cq)1/(q-1). Then F(STr)=F(S)∩F(Tr).Proof.
Suppose thatx1∈F(STr); it is sufficient to show that x1∈F(S)∩F(Tr). Indeed, for x2∈F(S)∩F(Tr), we have by Lemma 16 that
(36)∥x1-x2∥q=∥STrx1-STrx2∥q≤∥Trx1-Trx2∥q≤∥x1-x2∥q-r(αq-rq-1Cq)∥Ax1-Ax2∥q-ϕq∥(I-Jr)(I-rA)x1-(I-Jr)(I-rA)x2∥.
The property of ϕ and the condition 0<r≤(qα/Cq)1/(q-1) together imply that
(37)∥Ax1-Ax2∥=∥(I-Jr)(I-rA)x1-(I-Jr)(I-rA)x2∥=0.
It turns out that
(38)∥x1-Trx1-x2+Trx2∥=0,
which imply
(39)Trx1=x1.
Noticing the assumption of x1=STrx1, we can deduce x1=Sx1. This means that x1∈F(S)∩F(Tr).Lemma 18 (see [25]).
LetC be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space E. Let V:C→E be a k-Lipschitz and η-strongly accretive operator with constants k,η>0. Let 0<μ<(qη/Cqkq)1/(q-1) and τ=μ(η-(Cqμq-1kq/q)). Then for t∈(0,min⁡{1,1/τ}), the mapping S:C→E defined by S:=(I-tμV) is a contraction with a constant 1-tτ.Next we give a weak convergence theorem in a Banach spaceE.


## 3. Main Results

Theorem 19.
LetE be a uniformly convex and q-uniformly smooth Banach space. Let A:E→E be α-inverse-strongly accretive, B:E→2Em-accretive, and S:E→Eλ-strict pseudocontractive. Assume that F(S)∩(A+B)-1(0)≠∅. Define a mapping Tx:=(1-σ)x+σSx for all x∈E. For arbitrarily given x1∈E and σ∈(0,ρ], where ρ=min⁡{1,{qλ/Cq}1/(q-1)}, let {xn} be the sequence generated iteratively by
(40)xn+1=(1-αn)xn+αnT(Jrn(xn-rn(Axn+an))+bn),∀n≥1,
where Jrn=(I+rnB)-1,{an},{bn}⊂E,{αn}⊂(0,1], and {rn}⊂(0,+∞). Assume that (i)
∑n=1∞∥an∥<∞and∑n=1∞∥bn∥<∞,(ii)
0<liminf⁡n→∞⁡αn≤limsup⁡n→∞⁡αn<1,(iii)
0<liminf⁡n→∞⁡rn≤limsup⁡n→∞⁡rn<(qα/Cq)1/(q-1). Then {xn} converges weakly to some point x∈F(S)∩(A+B)-1(0).Proof.
We divide the proof into several steps.
Step  1. We prove lim⁡n→∞∥xn-z∥ exists for any point z∈F(S)⋂‍(A+B)-1(0).
PuttingTn=Jrn(I-rnA)=(I+rnB)-1(I-rnA), one has
(41)T(Jrn(xn-rn(Axn+an))+bn)=TTnxn+gn,
where
(42)gn=T(Jrn(xn-rn(Axn+an))+bn)-TTnxn.
Then the iterative formula (40) turns into the form
(43)xn+1=(1-αn)xn+αn(TTnxn+gn).
Thus, by virtue of Lemmas 10 and 12 and nonexpansivity of Jrn, we have
(44)∥gn∥=∥T(Jrn(xn-rn(Axn+an))+bn)-TTnxn∥≤∥Jrn(xn-rn(Axn+an))-Tnxn∥+∥bn∥≤rn∥an∥+∥bn∥.
By (44) and condition (i), we have that
(45)∑n=1∞‍∥gn∥<∞.
Since z∈F(S)∩(A+B)-1(0), according to Lemmas 10 and 15, we can deduce z∈F(T)∩F(Tn). Lemma 16 and condition (iii) together imply Tn is nonexpansive. Therefore, we get from (43) that
(46)∥xn+1-z∥=(1-αn)∥xn-z∥+αn∥TTnxn+gn-z∥≤(1-αn)∥xn-z∥+αn∥TTnxn-z∥+αn∥gn∥≤(1-αn)∥xn-z∥+αn∥xn-z∥+αn∥gn∥≤∥xn-z∥+αn∥gn∥.
In view of (45), (46), and Lemma 8, we get that lim⁡n→∞∥xn-z∥ exists. Therefore {xn} is bounded.
Step  2. We show lim⁡n→∞∥Tnxn-xn∥=0.
LetM1>0 be such that ∥xn∥<M1, for all n∈ℕ and let s=q(M1+∥z∥)q-1. By (43), Lemmas 6, 10, and 16, we have
(47)∥xn+1-z∥q=∥(1-αn)(xn-z)+αn(TTnxn+gn-z)∥q≤∥(1-αn)(xn-z)+αn(TTnxn-z)∥q+αnq〈gn,jq(xn+1-z)〉≤(1-αn)∥xn-z∥q+αn∥TTnxn-z∥q+αnq∥gn∥∥xn+1-z∥q-1≤(1-αn)∥xn-z∥q+αn∥Tnxn-z∥q+αnq∥gn∥∥xn+1-z∥q-1≤∥xn-z∥q-αnrn(αq-rnq-1Cq)∥Axn-Az∥q-αnϕq(∥(I-Jrn)(I-rnA)xn-(I-Jrn)(I-rnA)z∥)+αns∥gn∥≤∥xn-z∥q-αnrn(αq-rnq-1Cq)∥Axn-Az∥q-αnϕq(∥xn-rnAxn-Tnxn+rnAz∥)+αns∥gn∥.
Meanwhile, by the fact that ar-br≤rar-1(a-b),∀r≥1 and (47), we get that
(48)αnrn(αq-rnq-1Cq)∥Axn-Az∥q+αnϕq(∥xn-rnAxn-Tnxn+rnAz∥)≤∥xn-z∥q-∥xn+1-z∥q+αns∥gn∥≤q∥xn-z∥q-1(∥xn-z∥-∥xn+1-z∥)+αns∥gn∥.
Thanks to (45), existence of lim⁡n→∞∥xn-z∥, (ii) and (iii), one has
(49)lim⁡n→∞∥Axn-Az∥=lim⁡n→∞∥xn-rnAxn-Tnxn+rnAz∥=0.
It turns out that
(50)lim⁡n→∞∥Tnxn-xn∥=0.Step  3. We prove lim⁡n→∞∥TTnxn-xn∥=0.
Noticing (45) and Lemma 7, we have
(51)∥xn+1-z∥2=∥(1-αn)(xn-z)+αn(TTnxn+gn-z)∥2≤(1-αn)∥xn-z∥2+αn∥TTnxn+gn-z∥2-W2(αn)φ(∥TTnxn+gn-xn∥)≤(1-αn)∥xn-z∥2+αn(∥TTnxn-z∥2+2∥TTnxn-z∥∥gn∥+∥gn∥2)-W2(αn)φ(∥TTnxn+gn-xn∥)≤(1-αn)∥xn-z∥2+αn(∥xn-z∥2+2∥xn-z∥∥gn∥+∥gn∥2)-W2(αn)φ(∥TTnxn+gn-xn∥)≤∥xn-z∥2+2αn∥xn-z∥∥gn∥+αn∥gn∥2-W2(αn)φ(∥TTnxn+gn-xn∥)≤∥xn-z∥2+2∥xn-z∥∥gn∥+∥gn∥2-W2(αn)φ(∥TTnxn+gn-xn∥),
which implies
(52)W2(αn)φ(∥TTnxn+gn-xn∥)≤∥xn-z∥2-∥xn+1-z∥2+2∥xn-z∥∥gn∥+∥gn∥2,
where W2(αn)=αn(1-αn). From (45), (52), (ii), and existence of lim⁡n→∞∥xn-z∥, it turns out that
(53)lim⁡n→∞φ(∥TTnxn+gn-xn∥)=0.
It follows from the property of φ and (45) that
(54)lim⁡n→∞∥TTnxn-xn∥=0.
Step  4. We prove ωw(xn)⊂F(S)∩(A+B)-1(0).
Since0<liminf⁡n→∞rn≤limsup⁡n→∞rn<1, there exists ε>0 such that rn≥ε for all n≥1. Then, by Lemma 16, we have
(55)lim⁡n→∞∥Tεxn-xn∥≤2lim⁡n→∞∥Tnxn-xn∥=0.
It follows from (50), (54), and (55) that
(56)∥TTεxn-xn∥≤∥TTεxn-TTnxn∥+∥TTnxn-xn∥≤∥Tεxn-Tnxn∥+∥TTnxn-xn∥≤∥Tεxn-xn∥+∥xn-Tnxn∥+∥TTnxn-xn∥⟶0.
By Lemmas 10, 11, and 17 and (56), we get
(57)ωw(xn)⊂F(TTε)=F(T)∩F(Tε)=F(S)∩(A+B)-1(0).Step  5. We show {xn} converges weakly to a fixed point of x∈F(S)∩(A+B)-1(0).
Indeed, it suffices to showωw(xn) consists of exactly only one point. To this end, we suppose that two different points x and y are in ωw(xn). Then there exist two different subsequences {ni} and {nj} such that xni⇀x and xnj⇀y as i→∞ and j→∞. Define Sn,m:E→E by
(58)Sn,m=Vn+m-1Vn+m-2⋯Vn,Vn=(1-αn)I+αnTTn.
Then xn can be written
(59)xn+m=Sn,mxn+cn,m,
where
(60)cn,m=Vn+m-1×(Vn+m-2(⋯Vn+1(Vnxn+αngn)+αn+1gn+1⋯)+αn+m-2gn+m-2)+αn+m-1gn+m-1-Sn,mxn.
Thanks to the nonexpansivity of Vn, we have
(61)∥cn,m∥≤∑k=nn+m-1‍∥αkgk∥≤∑k=nn+m-1‍∥gk∥.
It follows from (45) that
(62)lim⁡m,n→∞∥cn,m∥⟶0.
Let
(63)fn(t)=∥txn+(1-t)x-y∥,dn,m=Sn,m(txn+(1-t)x)-(tSn,mxn+(1-t)x).
Applying Lemma 13 to the closed convex bounded subset D:=co⁡-({xn}∪{x,y}), we obtain
(64)g(∥dn,m∥)≤∥xn-x∥-∥Sn,mxn-Sn,mx∥≤∥xn-x∥-∥xn+m-x-cn,m∥≤∥xn-x∥-∥xn+m-x∥+∥cn,m∥.
Since lim⁡n→∞∥xn-x∥ exists, (62), (64), and the property of g together imply that
(65)lim⁡m,n→∞∥dn,m∥⟶0.
Furthermore, we have
(66)fn+m(t)=∥txn+m+(1-t)x-y∥≤∥dn,m∥+∥Sn,m(txn+(1-t)x)-y∥+t∥cm,n∥≤∥dn,m∥+∥txn+(1-t)x-y∥+t∥cm,n∥=∥dn,m∥+fn(t)+t∥cm,n∥.
After taking first limsup⁡m→∞ and then liminf⁡n→∞ in (66) and using (62) and (65), we get
(67)limsup⁡m→∞fm(t)≤liminf⁡n→∞fn(t)+lim⁡m,n→∞(∥dm,n∥+∥cn,m∥)=liminf⁡n→∞fn(t).
So lim⁡n→∞∥txn+(1-t)x-y∥ exists for all t∈[0,1]. It follows from Lemma 14 that x=y. This completes the proof.Remark 20.
Compared with the known results in the literature, our results are very different from those in the following aspects.(i)
Theorem19 improves and extends Theorem 3 of Kamimura and Takahashi [4] and Theorem 3.1 of Manaka and Takahashi [9] from Hilbert spaces to uniformly convex and q-uniformly smooth Banach spaces.(ii)
Theorem19 also improves and extends Theorem 3.6 of López et al. [14] from the problem of finding an element of (A+B)-1(0) to the problem of finding an element of (A+B)-1(0)∩F(S), where S is λ-strictly pseudocontractive on E.
In the following, we give a strong convergence theorem in a Banach spaceE.Theorem 21.
LetE be a uniformly convex and q-uniformly smooth Banach space which admits a weakly sequentially continuous generalized duality mapping jq:E→E*. Let A:E→E be α-inverse-strongly accretive, B:E→2Em-accretive, G:E→Ek-Lipschitz and η-strongly accretive, ψ:E→EL-Lipschitz, and S:E→Eλ-strictly pseudocontractive. Define a mapping Tx:=(1-σ)x+σSx for all x∈E. For arbitrarily given x1∈E and σ∈(0,ρ], where ρ=min⁡{1,{qλ/Cq}1/(q-1)}, let {xn} be the sequence generated iteratively by
(68)xn+1=αnγψ(xn)+(I-αnμG)T×(Jrn(xn-rn(Axn+an))+bn),∀n≥1.
Assume that {αn}⊂[0,1],{rn}⊂(0,+∞) and {an},{bn}⊂E satisfying the following conditions: (i)
∑n=1∞∥an∥<∞and∑n=1∞∥bn∥<∞,(ii)
∑n=1∞αn=∞,lim⁡n→∞⁡αn=0and∑n=1∞|αn+1-αn|<∞,(iii)
0<liminf⁡n→∞⁡rn≤limsup⁡n→∞⁡rn<(qα/Cq)1/(q-1)and∑n=1∞|rn+1-rn|<∞. Suppose in addition that (S)∩(A+B)-1(0)≠∅, 0<μ<(qη/Cqkq)1/(q-1) and 0≤γL<τ, where τ=μ(η-(Cqμq-1kq/q)). Then {xn} converges strongly to some point z∈F(S)∩(A+B)-1(0) which solves the variational inequality: 〈γψ(z)-μG(z),jq(x-z)〉≤0,for allx∈F(S)∩(A+B)-1(0).Proof.
Let{yn} be a sequence generated by
(69)yn+1=αnγψ(yn)+(I-αnμG)TTnyn,
where Tn:=Jrn(I-rnA). We show ∥yn-xn∥→0.
It follows from Lemmas10, 12, and 18 that
(70)∥yn+1-xn+1∥≤∥αnγψ(yn)+(I-αnμG)TJrn(yn-rnAyn)-αnγψ(xn)-(I-αnμG)T×(Jrn(xn-rn(Axn+an))+bn)∥≤(1-αnτ)∥TJrn(yn-rnAyn)-T(Jrn(xn-rn(Axn+an))+bn)∥+αnγL∥yn-xn∥≤(1-αnτ)∥Jrn(yn-rnAyn)-Jrn(xn-rn(Axn+an))∥+∥bn∥+αnγL∥yn-xn∥≤[1-αn(τ-γL)]∥yn-xn∥+rn∥an∥+∥bn∥.
By virtue of Lemma 8, (i), and (70), we have lim⁡n→∞∥yn-xn∥=0.
Hence, to show the desired result, it suffices to prove thatyn→z.
Step  1. We prove that the sequence {yn} is bounded. Taking x∈F(S)∩(A+B)-1(0), it follows from Lemmas 10, 12, 15, and 16 and condition (iii) that
(71)∥yn+1-x∥=∥αnγψ(yn)+(I-αnμG)TTnyn-x∥=∥αnγ(ψ(yn)-ψ(x))+αn(γψ(x)-μG(x))+(I-αnμG)TTnyn-(I-αnμG)x∥≤αnγL∥yn-x∥+αn∥γψ(x)-μG(x)∥+(1-αnτ)∥yn-x∥=[1-αn(τ-γL)]∥yn-x∥+αn∥γψ(x)-μG(x)∥≤max⁡{∥γψ(x)-μG(x)∥τ-γL,∥yn-x∥}.
By induction, we have
(72)∥yn-x∥≤max⁡{∥γψ(x)-μG(x)∥τ-γL,∥y1-x∥},∀n≥1.
Hence, {yn} is bounded, and so are {ψ(yn)} and {Tn(yn)}.
Step  2. We prove that
(73)lim⁡n→∞∥yn+1-yn∥⟶0.
Putting zn=Tnyn=Jrn(I-rnA)yn, it follows from Lemma 16 that
(74)∥zn+1-zn∥=∥Tn+1yn+1-Tnyn∥≤∥Tn+1yn+1-Tnyn+1∥+∥Tnyn+1-Tnyn∥≤|1-rαnrβn|∥yn+1-Jrβn(1-rβnA)yn+1∥+∥yn+1-yn∥≤|rβn-rαn|∥yn+1-Jrβn(1-rβnA)yn+1∥rβn+∥yn+1-yn∥≤|rn+1-rn|M2+∥yn+1-yn∥,
where M2>sup⁡n≥1{∥yn+1-Jrβn(1-rβnA)yn+1∥/rβn}, rαn=min⁡{rn+1,rn}, and rβn=max⁡{rn+1,rn}. Hence from (69) and (74) we have
(75)∥yn+1-yn∥=∥αnγψ(yn)+(I-αnμG)Tzn-αn-1γψ(yn-1)-(I-αn-1μG)Tzn-1∥=∥αnγ(ψ(yn)-ψ(yn-1))+(I-αnμG)Tzn-(I-αnμG)Tzn-1+(αn-αn-1)×(γψ(yn-1)-μGTzn-1)∥≤(1-αnτ)∥Tzn-Tzn-1∥+αnγL∥yn-yn-1∥+|αn-αn-1|M3≤(1-αnτ)∥zn-zn-1∥+αnγL∥yn-yn-1∥+|αn-αn-1|M3≤[1-αn(τ-γL)]∥yn-yn-1∥+|αn-αn-1|M3+|rn-rn-1|M2,
where M3>sup⁡n≥1{∥γψ(yn)-μGTzn∥}. It follows from Lemma 9, (ii), and (iii) that lim⁡n→∞∥yn+1-yn∥=0.
Again from Lemmas6 and 16, we obtain
(76)∥yn+1-x∥q=∥αnγψ(yn)+(I-αnμG)Tzn-x∥q=∥αn(γψ(yn)-μG(x))+(I-αnμG)Tzn-(I-αnμG)x∥q≤(1-αnτ)∥Tnyn-x∥q+qαn〈γψ(yn)-μG(x),jq(yn+1-x)〉≤∥Tnyn-x∥q+qαnM4≤∥yn-x∥q-rn(αq-rnq-1Cq)∥Ayn-Ax∥q-ϕq(∥yn-rnAyn-Tnyn+rnAx∥)+qαnM4,
where M4>sup⁡n≥1{〈γψ(yn)-μG(x),jq(yn+1-x)〉}. Meanwhile, by the fact that ar-br≤rar-1(a-b) for all r≥1, we get that
(77)rn(αq-rnq-1Cq)∥Ayn-Ax∥q+ϕq(∥yn-rnAyn-Tnyn+rnAx∥)≤∥yn-x∥q-∥yn+1-x∥q+qαnM4≤q∥yn-x∥q-1(∥yn-x∥-∥yn+1-x∥)+qαnM4.
It follows immediately from (ii), (iii), (77), existence of lim⁡n→∞∥yn-x∥, and the property of ϕq that
(78)lim⁡n→∞∥Ayn-Ax∥=lim⁡n→∞∥yn-rnAyn-Tnyn+rnAx∥=0.
Hence we obtain that
(79)lim⁡n→∞∥Tnyn-yn∥=0.
By condition (iii), there exists ε>0 such that rn≥ε for all n≥1. Then, by Lemma 16, we get
(80)lim⁡n→∞∥Tεyn-yn∥≤lim⁡n→∞2∥Tnyn-yn∥=0.Step  3. We show lim⁡n→∞∥TTεyn-yn∥=0.
From (73), (79), (80), and (ii), we have
(81)∥TTεyn-yn∥≤∥TTεyn-TTnyn∥+∥TTnyn-yn∥≤∥Tεyn-Tnyn∥+∥TTnyn-yn∥≤∥Tεyn-yn∥+∥yn-Tnyn∥+∥TTnyn-yn+1∥+∥yn+1-yn∥≤∥Tεyn-yn∥+∥yn-Tnyn∥+αn∥γψ(yn)-μGTTnyn∥+∥yn+1-yn∥⟶0.
Lemmas 10, 11, and 17 and (81) together imply that
(82)ωw(yn)⊂F(TTε)=F(T)∩F(Tε)=F(S)∩(A+B)-1(0).
By Song’s Lemma 2.11 [25], we deduce directly that {zt} defined by zt=tγψ(zt)-(I-tμG)TTεzt converges strongly to some point z∈F(TTε) which is the unique solution of the variational inequality:
(83)〈γψ(z)-μG(z),jq(x-z)〉≤0,∀x∈F(TTε).Step  4. We prove that
(84)limsup⁡n→∞〈γψ(z)-μG(z),jq(yn-z)〉≤0.
We take a subsequence {yni} of {yn} such that
(85)limsup⁡n→∞〈γψ(z)-μG(z),jq(yn-z)〉=lim⁡i→∞〈γψ(z)-μG(z),jq(yni-z)〉.
Without loss of generality, we may further assume that yni⇀x~ due to reflexivity of the Banach space E and boundness of {yn}. It follows from (82) that x~∈F(TTε). Since Banach space E has a weakly sequentially continuous generalized duality mapping jp:E→E*, we obtain that
(86)limsup⁡n→∞〈γψ(z)-μG(z),jq(yn-z)〉=lim⁡i→∞〈γψ(z)-μG(z),jq(yni-z)〉=〈γψ(z)-μG(z),jq(x~-z)〉≤0.Step  5. We show ∥yn-z∥→0.
By Lemmas9 and 16 and the fact that ab≤(1/q)aq+((q-1)/q)bq/(q-1), we get
(87)∥yn+1-z∥q=∥αnγψ(yn)+(I-αnμG)TTnyn-z∥q=〈αnγψ(yn)+(I-αnμG)TTnyn-z,jq(yn+1-z)〉=αnγ〈ψ(yn)-ψ(z),jq(yn+1-z)〉+αn〈γψ(z)-μG(z),jq(yn+1-z)〉+〈(I-αnμG)TTnyn-(I-αnμG)z,jq(yn+1-z)〉≤αnγ∥ψ(yn)-ψ(z)∥∥yn+1-z∥q-1+αn〈γψ(z)-μG(z),jq(yn+1-z)〉+(1-αnτ)∥yn-z∥∥yn+1-z∥q-1≤αnL∥yn-z∥∥yn+1-z∥q-1+αn〈γψ(z)-μG(z),jq(yn+1-z)〉+(1-αnτ)∥yn-z∥∥yn+1-z∥q-1≤[1-αn(τ-γL)]∥yn-z∥∥yn+1-z∥q-1+αn〈γψ(z)-μG(z),jq(yn+1-z)〉≤[1-αn(τ-γL)]1q∥yn-z∥q+q-1q∥yn+1-z∥q+αn〈γψ(z)-μG(z),jq(yn+1-z)〉,
which implies that
(88)∥yn+1-z∥q≤[1-αn(τ-γL)]∥yn-z∥q+qαn〈γψ(z)-μG(z),jq(yn+1-z)〉.
Apply Lemma 9 to (88) to conclude yn→z as n→∞. This completes the proof.Remark 22.
Theorem21 improves and extends Theorem 3.7 of López et al. [14] in the following aspects: (i)
from the problem of finding an element of(A+B)-1(0) to the problem of finding an element of (A+B)-1(0)∩F(S), where S is λ-strictly pseudocontractive on E;(ii)
from a fixed elementu to a Lipschitz mapping ψ.Remark 23.
Theorem21 improves and extends Theorem 2.1 of Zhang et al. [8] in the following aspects: (i)
from Hilbert spaces to uniformly convex andq-uniformly smooth Banach spaces;(ii)
from finding a common element of the set of solutions to the variational inclusion problem and the set of fixed points of nonexpansive mappings to finding a common element of the set of solutions to the variational inclusion problem and the set of fixed points ofλ-strict pseudocontractions;(iii)
from a fixed elementu to a Lipschitz mapping ψ;(iv)
from a fixed positive numberλ to a sequence positive number {rn}.
As a direct consequence of Theorem21, we obtain the following result.Corollary 24.
LetH be a real Hilbert space. Let A:H→H be α-inverse-strongly monotone, B:H→2H maximal monotone, G:H→Hk-Lipschitz and η-strongly monotone, ψ:H→HL-Lipschitz, and S:H→HK-strictly pseudocontractive. Define a mapping Tx:=(1-σ)x+σSx for all x∈H. For arbitrarily given x1∈H and σ∈[K,1), let {xn} be the sequence generated iteratively by
(89)xn+1=αnγψ(xn)+(I-αnμG)TJrn(xn-rn(Axn+an)),∀n≥1.
Assume that {αn}⊂[0,1], {rn}⊂(0,+∞), and {an}⊂H satisfying the following conditions: (i)
∑n=1∞∥an∥<∞,(ii)
∑n=1∞αn=∞, lim⁡n→∞⁡αn=0and∑n=1∞|αn+1-αn|<∞,(iii)
0<liminf⁡n→∞⁡rn≤limsup⁡n→∞⁡rn<2αand∑n=1∞|rn+1-rn|<∞. Suppose in addition that F(S)∩(A+B)-1(0)≠∅, 0<μ<2η/k2 and 0≤γL<τ, where τ=μ(η-μk2/2). Then {xn} converges strongly to some point z∈F(S)∩(A+B)-1(0) which solves the variational inequality: 〈γψ(z)-μG(z),x-z〉≤0, for allx∈F(S)∩(A+B)-1(0).


## 4. Applications

Using Corollary24, we consider the problem for finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a strict pseudocontraction in a Hilbert space. Let C be a nonempty, closed, and convex subset of a Hilbert space and let f:C×C→ℝ be a bifunction satisfying the following conditions:(A1)
f(x,x)=0 for all x∈C;(A2)
f is monotone, that is, f(x,y)+f(y,x)≤0 for all x,y∈C;(A3)
for allx,y,z∈C,
(90)limsup⁡t↓0f(tz+(1-t)x,y)≤f(x,y);(A4)
for allx∈C,f(x,·) is convex and lower semicontinuous.Then, the mathematical model related to equilibrium problems (with respect toC) is to find x^∈C such that
(91)f(x^,y)=0
for all y∈C. The set of such solutions x^ is denoted by EP⁡(f).The following lemma appears implicitly in Blum and Oettli [26].Lemma 25.
LetC be a nonempty, closed, and convex subset of H and let f:C×C→ℝ be a bifunction satisfying (A1)–(A4). Let r>0 and x∈H. Then, there exists z∈C such that
(92)f(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.The following lemma was also given in Combettes and Hirstoaga [27].Lemma 26.
Assume thatf:C×C→ℝ satisfies (A1)–(A4). For r>0 and x∈H, define a mapping Sr:H→C as follows:
(93)Srx={z∈C:f(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}
for all x∈H. Then, the following hold:(1)
Sr is single-valued;(2)
Sr is a firmly nonexpansive mapping; that is, for all x,y∈H,∥Srx-Sry∥2≤〈Srx-Sry,x-y〉;(3)
F(Sr)=EP(f);(4)
EP(f) is closed and convex.We call suchSr the resolvent of f for r>0. Using Lemmas 25 and 26, Takahashi et al. [10] proved the following theorem. See [10] for a more general result.Theorem 27.
LetH be a Hilbert space and let C be a nonempty, closed, and convex subset of H. Let f:C×C→ℝ satisfy (A1)–(A4). Let Af be a multivalued mapping of H into itself defined by
(94)Afx={{z∈H:f(x,y)≥〈y-x,z〉,∀y∈C},x∈C,∅,x∉C.
Then, EP(f)=Af-10 and Af is a maximal monotone operator withdom⁡(Af)⊂C. Further, for any x∈H and r>0, the resolvent Sr of f coincides with the resolvent of Af; that is, Srx=(I+rAf)-1x.Theorem 28.
LetH be a real Hilbert space. Suppose f:H×H→ℝ is a bifunction satisfying the following conditions:(B1)
f(x,x)=0 for all x∈H;(B2)
f is monotone, that is, f(x,y)+f(y,x)≤0 for all x,y∈H;(B3)
for allx,y,z∈H,
(95)limsup⁡t↓0f(tz+(1-t)x,y)≤f(x,y);(B4)
for allx∈H,f(x,·) is convex and lower semicontinuous. Assume Sδ is the resolvent of f for δ>0,G:H→H is k-Lipschitz and η-strongly monotone, ψ:H→H is L-Lipschitz, and S:H→H is K-strictly pseudocontractive. Define a mapping Tx:=(1-σ)x+σSx for all x∈H. For arbitrarily given x1∈H and σ∈[K,1), let {xn} be the sequence generated iteratively by
(96)xn+1=αnγψ(xn)+(I-αnμG)TSrnxn,∀n≥1.Assume that {αn}⊂[0,1] and {rn}⊂(0,+∞) satisfying the following conditions: (i)
∑n=1∞αn=∞,lim⁡n→∞⁡αn=0and∑n=1∞|αn+1-αn|<∞,(ii)
∑n=1∞|rn+1-rn|<∞. Suppose in addition that F(S)∩EP(f)≠∅, 0<μ<2η/k2 and 0≤γL<τ, where τ=μ(η-μk2/2). Then {xn} converges strongly to some point z∈F(S)∩EP(f) which solves the variational inequality: 〈γψ(z)-μG(z),x-z〉≤0, for allx∈F(S)∩EP(f).Proof.
PutA=0 and an=0 for all n∈ℕ in Corollary 24. From Theorem 27, we also know that JrnAf=Srn for all n∈ℕ. So, we obtain the desired result by Corollary 24.



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