# Strong and Weak Convergence of Modified Mann Iteration for New Resolvents of Maximal Monotone Operators in Banach Spaces

Authors: Somyot Plubtieng; Wanna Sriprad
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

We prove strong and weak convergence theorems for a new resolvent of maximal monotone operators in a Banach space and give an estimate of the convergence rate of the algorithm. Finally, we apply our convergence theorem to the convex minimization problem. The result present in this paper extend
and improve the corresponding result of Ibaraki and Takahashi (2007), and Kim and Xu (2005).


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## Body

## 1. Introduction

LetE be a Banach space with norm ∥·∥, let E* denote the dual of E, and let 〈x,f〉 denote the value of f∈E* at x∈E. Let  T:E→E* be an operator. The problem of finding v∈E satisfying 0∈Tv is connected with the convex minimization problems. When T is maximal monotone, a well-known method for solving the equation 0∈Tv in Hilbert space H is the proximal point algorithm (see [1]): x1=x∈H and
(1.1)xn+1=Jrnxn,n=1,2,…,
where rn⊂(0,∞) and Jr=(I+rT)-1 for all r>0 is the resolvent operator for T. Rockafellar [1] proved the weak convergence of the algorithm (1.1).The modifications of the proximal point algorithm for different operators have been investigated by many authors. Recently, Kohsaka and Takahashi [2] considered the algorithm (1.2) in a smooth and uniformly convex Banach space and Kamimura et al. [3] considered the algorithm (1.3) in a uniformly smooth and uniformly convex Banach space E; x1=x∈E,u∈E and
(1.2)xn+1=J-1(αnJu+(1-αn)JJrnxn),n=1,2,…,(1.3)xn+1=J-1(αnJxn+(1-αn)JJrnxn),n=1,2,…,
where Jr=(J+rT)-1J,J is the duality mapping of E. They showed that the algorithm (1.2) converges strongly to some element of T-10 and the algorithm (1.3) converges weakly to some element of T-10 provided that the sequences {αn} and {rn} of real numbers are chosen appropriately. These results extend the Kamimura and Takahashi [4] results in Hilbert spaces to those in Banach spaces.In 2008, motivated by Kim and Xu [5], Li and Song [6] studied a combination of the schemes of (1.2) and (1.3); x1=x∈E and
(1.4)yn=J-1(αnJxn+(1-αn)JJrnxn),xn+1=J-1(βnJx+(1-βn)Jyn),
for every n=1,2,…, where Jr=(J+rT)-1J,J is the duality mapping of E. They also proved strong and weak convergence theorems and give an estimate for the rate of convergence of the algorithm (1.4).Very recently, Ibaraki and Takahashi [7] introduced the Mann iteration and Harpern iteration for new resovents of maximal monotone operator in a uniformly smooth and uniformly convex Banach space E; x1=x∈Eu∈E and
(1.5)xn+1=αnu+(1-αn)Jrnxn,n=1,2,…,(1.6)xn+1=αnxn+(1-αn)Jrnxn,n=1,2,…,
where Jr=(I+rBJ)-1,J is the duality mapping of E, and B⊂E*×E is maximal monotone. They proved that Algorithm (1.5) converges strongly to some element of (BJ)-10 and Algorithm (1.6) converges weakly to some element of (BJ)-10 provided that the sequences {αn} and {rn} of real numbers are chosen appropriately.Inspired and motivated by Li and Song [6] and Ibaraki and Takahashi [7], we study a combination of the schemes of (1.5) and (1.6); x1=x∈E and
(1.7)yn=αnxn+(1-αn)Jrnxn,xn+1=βnx+(1-βn)yn,
for every n=1,2,…, where Jr=(I+rBJ)-1,J is the duality mapping of E, and B⊂E*×E is maximal monotone. When αn≡0, Algorithm (1.7) reduces to (1.5) and, when βn≡0, Algorithm (1.7) reduces to (1.6). Then, we prove strong and weak convergence theorems of the sequence and we also estimate the rate of the convergence of algorithm (1.7). Finally, by using our main result, we consider the problem of finding minimizes of convex functions defined on Banach spaces.


## 2. Preliminaries

LetE be a real Banach space with dual space E*. When {xn} is a sequence in E, we denote strong convergence of {xn} to x∈E by xn→x and weak convergence by xn⇀x, respectively. As usual, we denote the duality pairing of E* by 〈x,x*〉, when x*∈E* and x∈E, and the closed unit ball by UE, and denote by ℝ and ℕ the set of all real numbers and the set of all positive integers, respectively. The set ℝ+ stands for [0,+∞) and ℝ̅+=ℝ+∪{+∞}. An operator T⊂E×E* is said to be monotone if 〈x-y,x*-y*〉≥0 whenever (x,x*),(y,y*)∈T. We denote the set {x∈E:0∈Tx} by T-10. A monotone T is said to be maximal if its graph G(T)={(x,y):y∈Tx} is not properly contained in the graph of any other monotone operator. If T is maximal monotone, then the solution set T-10 is closed and convex. If E is reflexive and strictly convex, then a monotone operator T is maximal if and only if R(J+λT)=E* for each λ>0 (see [8, 9] for more details).The normalized duality mappingJ from E into E* is defined by
(2.1)J(x)={x*∈E*:〈x,x*〉=∥x∥2=∥x*∥2},∀x∈E.
We recall [10] that E is reflexive if and only if J is surjective; E is smooth if and only if J is single-valued.LetE be a smooth Banach space. Consider the following function: (see [11])
(2.2)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,∀x,y∈E.
It is obvious from the definition of ϕ that (∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2, for all x,y∈E. We also know that
(2.3)ϕ(x,y)+ϕ(y,x)=2〈x-y,Jx-Jy〉,foreachx,y∈E.
We recall [12] that the functional ∥·∥2 is called totally convex at x if the function ν(x,t):[0,∞)→[0,∞] defined by
(2.4)ν(x,t)=inf{ϕ(y,x):y∈E,∥y-x∥=t},
is positive whenever t>0. The functional ∥·∥2 is called totally convex on bounded sets if for each bounded nonempty subset A of E, the function ν(A,t):[0,∞)→[0,∞] defined by ν(A,t)=inf{ν(x,t):x∈A} is positive on (0,∞).It is well known that if a Banach spaceE is uniformly convex, then ∥·∥2 is totally convex on any bounded nonempty set. It is known that (see [12]) if ∥·∥2 is totally convex on a bounded set A, then ν(A,ct)≥cν(A,t) for c≥1 and t≥0, and ν(A,·) is strictly increasing on [0,∞).Lemma 2.1 (see [13]).
LetE be a uniformly convex, smooth Banach space, and let {xn} and {yn} be sequences in E. If {xn} or {yn} is bounded and limn→∞ϕ(xn,yn)=0, then limn→∞∥xn-yn∥=0.LetE be a reflexive, strictly convex, smooth Banach space, and J the duality mapping from E into E*. Then J-1 is also single-valued, one-to-one, surjective, and it is the duality mapping from E* into E. We make use of the following mapping V studied in Alber [11]:
(2.5)V(x,x*)=∥x∥2-2〈x,x*〉+∥x*∥2
for all x∈E and x*∈E*. In other words, V(x,x*)=ϕ(x,J-1(x)) for all x∈E and x*∈E*.Lemma 2.2 (see [7]).
LetE be a reflexive, strictly convex, smooth Banach space, and let V be as in (2.5). Then
(2.6)V(x,x*)+2〈y,Jx-x*〉≤V(x+y,x*)
for all x,y∈E and x*∈E*.LetE be a smooth Banach space and let D be a nonempty closed convex subset of E. A mapping R:D→D is called generalized nonexpansive if F(R)≠∅ and ϕ(Rx,y)≤ϕ(x,y) for each x∈D and y∈F(R), where F(R) is the set of fixed points of R. Let C be a nonempty closed subset of E. A mapping R:E→C is said to be sunny if
(2.7)R(Rx+t(x-Rx))=Rx,∀x∈E,∀t≥0.
A mapping R:E→C is said to be a retraction if Rx=x, for all x∈C. If E is smooth and strictly convex, then a sunny generalized nonexpansive retraction of E onto C is uniquely decided if it exists (see [14]). We also know that if E is reflexive, smooth, and strictly convex and C is a nonempty closed subset of E, then there exists a sunny generalized nonexpansive retraction RC of E onto C if and only if J(C) is closed and convex. In this case, RC is given by RC=J-1ΠJ(C)J see [15]. Let C be a nonempty closed subset of a Banach space E. Then C is said to be a sunny generalized nonexpansive retract (resp., a generalized nonexpansive retract) of E if there exists a sunny generalized nonexpansive retraction (resp, a generalized nonexpansive retraction) of E onto C (see [14] for more detials). The set of fixed points of such a generalized nonexpansive retraction is C. The following lemma was obtained in [14].Lemma 2.3 (see [14]).
LetC be a nonempty closed subset of a smooth and strictly convex Banach space E. Let RC be a retraction of E onto C. Then RC is sunny and generalized nonexpansive if and only if
(2.8)〈x-RCx,JRCx-Jy〉≥0,
for each x∈E and y∈C, where J is the duality mapping of E.      Let E be a reflexive, strictly convex, and smooth Banach space with its dual E*. If a monotone operator B⊂E*×E is maximal, then (BJ)-10 is closed and E=R(I+rBJ) for all r>0 (see [14]). So, for each r>0 and x∈E, we can consider the set Jr(x)={z∈E:x∈z+rBJz}. From [14], Jrx consists of one point. We denote such a Jr by (I+rBJ)-1. However Jr is called a generalized resolvent of B. We also know that (BJ)-10=F(Jr) for each r>0, where F(Jr) is the set of fixed points of Jr and Jr is generalized nonexpansive for each r>0 (see [14]). The Yosida approximtion of B is defined by Ar=(I-Jr)/r. We know that (JJrx,Arx)∈B; (see [14] for more detials). The following result was obtained in [14].Theorem 2.4 (see [14]).
LetE be a uniformly convex Banach space with a Fréchet differentiable norm and let B⊂E*×E be a maximal monotone operator with B-10≠∅. Then the following hold: (1)
for eachx∈E, limr→∞Jrx exists and belongs to (BJ)-10,(2)
ifRx:=limr→∞Jrx for each x∈E, then R is a sunny generalized nonexpansive retraction of E onto (BJ)-10.Lemma 2.5 (see [7]).
LetE be a reflexive, strictly convex, and smooth Banach space, let B⊂E*×E be a maximal monotone operator with B-10≠∅, and Jr=(I+rBJ)-1 for all r>0. Then
(2.9)ϕ(x,Jrx)+ϕ(Jrx,u)≤ϕ(x,u),
for all r>0, u∈(BJ)-10, and x∈E.Lemma 2.6 (see [16]).
Let{sn} be a sequence of nonnegative real numbers satisfying
(2.10)sn+1≤(1-αn)sn+αntn+rn,n≥1,
where {αn}, {tn}, and {rn} satisfy the conditions: {αn}⊂[0,1],  ∑n=1∞αn=∞, limsupn→∞tn≤0, and rn≥0, ∑n=1∞rn<∞. Then, limn→∞sn=0.Lemma 2.7 (see [17]).
Let{αn} and {βn} be sequence of nonnegative real numbers satisfying
(2.11)αn+1≤αn+βn,
for all n∈ℕ. If ∑n=1∞βn<+∞. Then {αn} has a limit in [0,+∞).


## 3. Convergence Theorems

In this section, we first prove a strong convergence theorem for the algorithm (1.7) which extends the previous result of Ibaraki and Takahashi [7] and we next prove a weak convergence theorem for algorithm (1.7) under different conditions on data, respectively.Theorem 3.1.
LetE be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable. Let B⊂E*×E be a maximal monotone operator with B-10≠∅ and let Jr=(I+rBJ)-1 for all r>0. Let {xn} be a sequence generated by x1=x∈E and
(3.1)yn=αnxn+(1-αn)Jrnxn,xn+1=βnx+(1-βn)yn,
for every n=1,2,…, where {αn}, {βn}⊂[0,1], {rn}⊂(0,∞) satisfy limn→∞αn=0, limn→∞βn=0, ∑n=1∞βn=∞ and limn→∞rn=∞. Then the sequence {xn} converges strongly to R(BJ)-10(x), where R(BJ)-10 is a sunny generalized nonexpansive retraction of E onto (BJ)-10.Proof.
Note thatB-10≠∅ implies (BJ)-10≠∅. In fact, if u*∈B-10, we obtain 0∈Bu* and hence 0∈BJJ-1u*. So, we have J-1u*∈(BJ)-10. We denote a sunny generalized nonexpansive retraction R(BJ)-10 of E onto (BJ)-10 by R. Let z∈(BJ)-10. We first prove that {xn} is bounded. From Lemma 2.5 and the convexity of ∥·∥2, we have
(3.2)ϕ(yn,z)=ϕ(αnxn+(1-αn)Jrnxn,z)≤αnϕ(xn,z)+(1-αn)ϕ(Jrnxn,z)≤αnϕ(xn,z)+(1-αn){ϕ(xn,z)-ϕ(xn,Jrnxn)}≤αnϕ(xn,z)+(1-αn)ϕ(xn,z)=ϕ(xn,z),
for all n∈ℕ. By (3.2), we have
(3.3)ϕ(xn+1,z)=ϕ(βnx+(1-βn)yn,z)≤βnϕ(x,z)+(1-βn)ϕ(yn,z)≤βnϕ(x,z)+(1-βn)ϕ(xn,z),
for all n∈ℕ. Hence, by induction, we have ϕ(xn,z)≤ϕ(x,z) for all n∈ℕ and, therefore, {ϕ(xn,z)} is bounded. This implies that {xn} is bounded. Since ϕ(yn,z)≤ϕ(xn,z) and ϕ(Jrnxn,z)≤ϕ(xn,z) for all n∈ℕ, it follows that {yn} and {Jrnxn} are also bounded. We next prove that
(3.4)limn→∞sup〈x-Rx,Jxn-JRx〉≤0.
Put un=xn+1 for all n∈ℕ. Since {Jun} is bounded, without loss of generality, we have a subsequence {Juni} of {Jun} such that
(3.5)limi→∞〈x-Rx,Juni-JRx〉=limn→∞sup〈x-Rx,Jun-JRx〉,
and {Juni} converges weakly to some v*∈E*. From the definition of {xn}, we have
(3.6)un-yn=βn(x-yn),yn-Jrnxn=αn(xn-Jrnxn)
for all n∈ℕ. Since {yn} is bounded and βn→0 as n→∞, it follows that
(3.7)limn→∞∥un-yn∥=limn→∞βn∥x-yn∥=0.
Moreover, we note that
(3.8)limn→∞∥yn-Jrnxn∥=limn→∞αn∥xn-Jrnxn∥=0.
By (3.7) and (3.8), we have
(3.9)limn→∞∥un-Jrnxn∥=0.
Since E has a uniformly Gâteaux differentiable norm, the duality mapping J is norm to weak* uniformly continuous on each bounded subset of E. Therefore, we obtain from (3.9) that
(3.10)Juni-JJrnixni⇀0,asi→∞.
This implies that JJrnixni⇀v* as i→∞. On the other hand, from rn→∞ as n→∞, we have
(3.11)limn→∞∥Arnxn∥=limn→∞1rn∥xn-Jrnxn∥=0.
If (y*,y)∈B, then it holds from the monotonicity of B that
(3.12)〈y-Arnixni,y*-JJrnixni〉≥0,
for all i∈ℕ. Letting i→∞, we get 〈y,y*-v*〉≥0. Then, the maximal of B implies v*∈B-10. Put v=J-1v*. Applying Lemma 2.3, we obtain
(3.13)limn→∞sup〈x-Rx,Jun-JRx〉=limi→∞〈x-Rx,Juni-JRx〉=〈x-Rx,v*-JRx〉=〈x-Rx,Jv-JRx〉≤0.
Finally, we prove that xn→Rx as n→∞. From Lemma 2.2, the convexity of ∥·∥2 and (3.2), we have
(3.14)ϕ(xn+1,Rx)=V(βnx+(1-βn)yn,JRx)≤V(βnx+(1-βn)yn-βn(x-Rx),JRx)-2〈-βn(x-Rx),Jxn+1-JRx〉=V(βnRx+(1-βn)yn,JRx)+2βn〈x-Rx,Jxn+1-JRx〉=ϕ(βnRx+(1-βn)yn,Rx)+2βn〈x-Rx,Jxn+1-JRx〉≤βnϕ(Rx,Rx)+(1-βn)ϕ(yn,Rx)+2βn〈x-Rx,Jxn+1-JRx〉≤(1-βn)ϕ(xn,Rx)+βnσn,
for all n∈ℕ, where σn=2〈x-Rx,Jxn+1-JRx〉. It easily verified from the assumption and (3.4) that ∑n=1∞βn=∞ and limsupn→∞σn≤0. Hence, by Lemma 2.6, limn→∞ϕ(xn,Rx)=0. Applying Lemma 2.1, we obtain limn→∞∥xn-Rx∥=0. Therefore, {xn} converges strongly to R(BJ)-10(x).   Put αn≡0 in Theorem 3.1, then we obtain the following result.Corollary 3.2 (see Ibaraki and Takahashi [7]).
LetE be a uniformly convex and uniformly smooth Banach space and let B⊂E*×E be a maximal monotone operator with B-10≠∅, let Jr=(I+rBJ)-1 for all r>0, and let {xn} be a sequence generated by x1=x∈E and
(3.15)xn+1=βnx+(1-βn)Jrnxn,
for every n=1,2,…, where {βn}⊂[0,1], {rn}⊂(0,∞) satisfy limn→∞βn=0, ∑n=1∞βn=∞ and limn→∞rn=∞.Then the sequence {xn} converges strongly to R(BJ)-10(x), where R(BJ)-10 is the generalized projection of E onto (BJ)-10.Theorem 3.3.
LetE be a uniformly convex and smooth Banach space whose duality mapping J is weakly sequentially continuous. Let B⊂E*×E be a maximal monotone operator with B-10≠∅ and let Jr=(I+rBJ)-1 for all r>0. Let {xn} be a sequence generated by x1=x∈E and
(3.16)yn=αnxn+(1-αn)Jrnxn,xn+1=βnx+(1-βn)yn,
for every n=1,2,…, where {αn}, {βn}⊂[0,1], {rn}⊂(0,∞) satisfy ∑n=1∞βn<∞, limsupn→∞αn<1 and liminfn→∞rn>0. Then the sequence {xn} converges weakly to an element of (BJ)-10.Proof.
Letv∈(BJ)-10. Then, from (3.3), we have
(3.17)ϕ(xn+1,v)≤(1-βn)ϕ(xn,v)+βnϕ(x,v)≤ϕ(xn,v)+βnϕ(x,v),
for all n∈ℕ. By Lemma 2.7, limn→∞ϕ(xn,v) exists. From (∥xn∥-∥v∥)2≤ϕ(xn,v) and ϕ(Jrnxn,v)≤ϕ(xn,v), we note that {xn} and {Jrnxn} are bounded. From (3.3) and (3.2), we have
(3.18)ϕ(xn+1,v)≤βnϕ(x,v)+(1-βn)ϕ(yn,v)≤βnϕ(x,v)+(1-βn){αnϕ(xn,v)+(1-αn)(ϕ(xn,v)-ϕ(xn,Jrnxn))}=βnϕ(x,v)+(1-βn){ϕ(xn,v)-(1-αn)ϕ(xn,Jrnxn)}=βnϕ(x,v)+(1-βn)ϕ(xn,v)-(1-βn)(1-αn)ϕ(xn,Jrnxn),
for all n∈ℕ and hence,
(3.19)(1-βn)(1-αn)ϕ(xn,Jrnxn)≤βnϕ(x,v)+(1-βn)ϕ(xn,v)-ϕ(xn+1,v)=βn(ϕ(x,v)-ϕ(xn,v))+ϕ(xn,v)-ϕ(xn+1,v),
for all n∈ℕ. Since limn→∞βn=0 and limsupn→∞αn<1, limn→∞ϕ(xn,Jrnxn)=0. Applying Lemma 2.1, we obtain
(3.20)limn→∞∥xn-Jrnxn∥=0.
Since {xn} is bounded, we have a subsequence {xni} of {xn} such that xni⇀w∈E as i→∞. Then it follows from (3.20) that Jrnixni⇀w as i→∞. On the other hand, from (3.20) and liminfn→∞rn>0, we have
(3.21)limn→∞∥Arnxn∥=limn→∞1rn∥xn-Jrnxn∥=0.
Let (z*,z)∈B. Then, it holds from monotonicity of B that
(3.22)〈z-Arnixni,z*-JJrnixni〉≥0,
for all i∈ℕ. Since J is weakly sequentially continuous, letting i→∞, we get 〈z,z*-Jw〉≥0. Then, the maximality of B implies Jw∈B-10. Thus, w∈(BJ)-10.
Let{xni} and {xnj} be two subsequences of {xn} such that xni⇀w1 and xnj⇀w2. By similar argument as above, we obtain w1,w2∈(BJ)-10. Put a:=limn→∞(ϕ(xn,w1)-ϕ(xn,w2)).
Note thatϕ(xn,w1)-ϕ(xn,w2)=2〈xn,Jw2-Jw1〉+∥w1∥2-∥w2∥2,n=1,2,…. From xni⇀w1 and xnj⇀w2, we have
(3.23)a=2〈w1,Jw2-Jw1〉+∥w1∥2-∥w2∥2,(3.24)a=2〈w2,Jw2-Jw1〉+∥w1∥2-∥w2∥2,
respectively. Combining (3.23) and (3.24), we have
(3.25)〈w1-w2,Jw1-Jw2〉=0.
Since J is strictly monotone, it follows that w1=w2. Therefore, {xn} converges weakly to an element of (BJ)-10.Putβn≡0 in Theorem 3.3, then we obtain the following result.Corollary 3.4 (see Ibaraki and Takahashi [7]).
LetE be a uniformly convex and smooth Banach space whose duality mapping J is weakly sequentially continuous. Let B⊂E*×E be a maximal monotone operator with B-10≠∅, let Jr=(I+rBJ)-1 for all r>0 and let {xn} be a sequence generated by x1=x∈E and
(3.26)xn+1=βnxn+(1-βn)Jrnxn,
for every n=1,2,…, where {αn},{βn}⊂[0,1], {rn}⊂(0,∞) satisfy ∑n=1∞βn<∞, limsupn→∞αn<1 and liminfn→∞rn>0. Then the sequence {xn} converges weakly to an element of (BJ)-10.


## 4. Rate of Convergence for the Algorithm

In this section, we study the rate of the convergence of the algorithm (1.7). We use the following notations in [6, 18]:(4.1)N0:={φ:ℝ+→ℝ̅+∣t↦φ(t)isnondecreasingfort≥0,φ(0)=0},Ω0:={φ:ℝ+→ℝ̅+∣φ(0)=0,limt→0+φ(t)=0},Γ0≔{φ:ℝ+→ℝ̅+∣φislscandconvexandφ(t)=0⟺t=0},Σ1:={φ:ℝ+→ℝ̅+∣φislscandconvex,φ(0)=0,limt→0+t-1φ(t)=0}.
We recall [18] that, for a function φ:ℝ+→ℝ̅+ satisfying φ(0)=0, its pseudoconjugate φ#:ℝ+→ℝ̅+, defined by
(4.2)φ#(s):=sup{st-φ(t)∣t≥0}∈ℝ̅,
is lower semicontinuous, convex and satisfies φ#(0)=0,φ#(s)≥0 for all s≥0.For a functionφ∈N0, its greatest quasi-inverse φh:ℝ+→ℝ̅+, defined by
(4.3)φh(s):=sup{t≥0∣φ(t)≤s},
is nondecreasing. It is known [18] that φh∈N0∩Ω0 if φ(t)=0⇔t=0.For a functionφ:ℝ→ℝ̅, its lower semicontinuous convex hull, denoted by co̅φ, is defined by epi(co̅φ)= cl(co(epiφ)). It is obvious that co̅φ is lower semicontinuous convex and co̅φ≤φ.Proposition 4.1.
LetE be uniformly convex and uniformly smooth. Then, for every r>0, there exists σr∈Σ1 such that, for all x,y∈rUE,
(4.4)〈y-x,Jy-Jx〉≤σr(∥Jy-Jx∥).Proof.
SinceE is uniformly convex, f(x)=(1/2)∥x∥2 is uniform convex on rUE for all r>0. Since the norm of E is Fréchet differentiable, its Fréchet derivative ∇f(x)=Jx. In [18, Proposition  3.6.5] for f and B=rUE, where r is an arbitrary positive real number, we get the function ϑr(·):[0,+∞)→[0,+∞), defined by
(4.5)ϑr(t):=inf{12∥y∥2-12∥x∥2-〈y-x,Jx〉:x∈rUE,y∈E,∥y-x∥=t},
satisfies that ϑr(t)=0 if and only if t=0, and t-1ϑr(t) is nondecreasing. Thus,
(4.6)12∥y∥2-12∥x∥2-〈y-x,Jx〉≥ϑr(∥y-x∥)
for all x∈rUE, y∈E and hence
(4.7)12∥y+x∥2≥12∥x∥2+〈y,Jx〉+co̅ϑr(∥y∥)
for all x∈rUE, y∈E. It follows that
(4.8)〈x,Jx〉-12∥x∥2+〈x,Jy-Jx〉+〈y,Jy-Jx〉-co̅φϑr(∥y∥)≥〈y+x,Jy〉-12∥y+x∥2
for all x∈rUE, y∈E. Since 〈x,Jx〉=(1/2)∥x∥2+(1/2)∥x∥2, we have
(4.9)12∥x∥2+〈y+x,Jy-Jx〉-co̅ϑr(∥y∥)≥〈y+x,Jy〉-12∥y+x∥2
for all x∈rUE and y∈E.
Taking the supremum on both sides of (4.9) over y∈E, by [18, Lemma  3.3.1(v)] (if f(x):=φ(∥x∥), where φ∈N0, then f*(x*)=φ#(∥x*∥)), we get that
(4.10)12∥x∥2+〈x,Jy-Jx〉+(co̅ϑr)#(∥Jy-Jx∥)≥12∥y∥2
for all x∈rUE and y∈E. Since ϑr(t) is nondecreasing and limt→∞t-1ϑr(t)≥ϑr(1)>0, we have co̅ϑr∈Γ0. It follows from [20, Lemma  3.3.1(iii)] that (co̅ϑr)#∈Σ1.
Interchangingx and y in (4.10) for x,y∈rUE, it also holds that
(4.11)12∥y∥2+〈y,Jx-Jy〉+(co̅ϑr)#(∥Jy-Jx∥)≥12∥x∥2.
Thus, by taking σr:=2(co̅ϑr)#∈Σ1, and adding side by side (4.10) and (4.11), we obtained
(4.12)〈y-x,Jy-Jx〉≤σr(∥Jy-Jx∥),∀x,y∈rUE.Theorem 4.2.
LetE be a uniformly convex and uniformly smooth Banach space. Suppose that B⊂E*×E is maximal monotone with B-10={v*} and B-1 is Lipschitz continuous at 0 with modulus l≥0. Let {xn} be a sequence generated by x1=x∈E and
(4.13)yn=αnxn+(1-αn)Jrnxn,xn+1=βnx+(1-βn)yn,
for every n=1,2,…, where {αn}, {βn}⊂[0,1], {rn}⊂(0,∞) satisfying  limn→∞rn=∞. If either ∑n=1∞βn<∞ and limsupn→∞αn<1 or limn→∞βn=0, ∑n=1∞βn=∞ and limn→∞αn=0, then {xn} converges strongly to v:=J-1v*and ϕ(xn,v) converges to 0.
Moreover, there exists an integerN>0 such that
(4.14)ϕ(xn+1,v)≤τnϕ(x,v)+θn+δn,∀n≥N,
where τn=βn+∑i=Nn-1βi∏j=i+1n(1-βj)αj, θn=∏i=Nn(1-βi)αi, δn=l((1-αn)/rn+∑i=Nn-1((1-αi)/ri)∏j=i+1n(1-βj-1)αj) and limn→∞τn=limn→∞θn=limn→∞δn=0. Also, one obtains
(4.15)ϕ(xn+1,v)≤βnϕ(x,v)+(αn+∘(lrn))k(ϕ(xn,v)),
for all n≥N, where k(t)=max{t,νrh(t)}∈N0∩Ω0, and νr(t):=ν(rUE,t), νrh is the greatest quasi-inverse of νr(t), and r is a positive number such that {v}∪{xn}∪{Jrnxn}⊂rUE.Proof.
Putv=J-1v*. Since B-10={v*}, we have (BJ)-10={v}. We separate the proof into two cases.Case 1.
∑n=1∞βn<∞, and limsupn→∞αn<1.
According to Theorem3.3, we have limn→∞ϕ(xn,v) exists, {xn} and {Jrn} are bounded, and hence {v}∪{xn}∪{Jrnxn}⊂rUE for some r>0. Since B-1 is Lipschitz continuous at 0 with modulus l≥0, for some τ>0, we have ∥z*-v*∥≤l∥w∥ whenever z*∈B-1(w) and ∥w∥≤τ. Since rn→∞, we may assume l/rn<1 for all n≥1. From Theorem 3.3, we have ∥xn-Jrnxn∥→0 and ∥Arnxn∥→0 as n→∞. Hence, there exists an integer N>0 such that ∥Arnxn∥≤τ for all n≥N. Since JJrnxn∈B-1Arnxn, we have
(4.16)∥JJrnxn-v*∥≤l∥Arnxn∥,∀n≥N.
By ∥Jxn-v*∥≤∥Jxn-JJrnxn∥+∥JJrnxn-v*∥ for all n∈ℕ. Since J is uniformly continuous on each bounded set, (3.20), (3.21) and (4.16), we obtain limn→∞∥Jxn-v*∥=0. By the uniform smoothness of E*, we have limn→∞∥xn-v∥=limn→∞∥J-1Jxn-J-1v*∥=0. Since ϕ(xn,v)+ϕ(v,xn)=2〈xn-v,Jxn-Jv〉=2〈xn-v,Jxn-v*〉 for all n∈ℕ, we get
(4.17)ϕ(xn,v)≤2〈xn-v,Jxn-v*〉≤2∥xn-v∥∥Jxn-v*∥→0,asn→∞.
Hence, ϕ(xn,v)→0 as n→∞. It follows from Proposition 4.1 and (4.16) that there exists σ(t)∈Σ1, which implies σ(lt)≤lσ(t) for all t≥0 and l∈[0,1], such that
(4.18)ϕ(Jrnxn,v)≤ϕ(Jrnxn,v)+ϕ(v,Jrnxn)=2〈Jrnxn-v,JJrnxn-Jv〉≤σ(∥JJrnxn-Jv∥)=2σ(∥JJrnxn-v*∥)≤2σ(l∥Arnxn∥)=2σ(l∥xn-Jrnxnrn∥)≤2lrnσ(∥xn-Jrnxn∥)
for all n≥N. It follows from σ(t)∈Σ1 and (3.20) that
(4.19)limn→∞σ(∥xn-Jrnxn∥)=0.
From (3.3), (3.2), and (4.18), we have
(4.20)ϕ(xn+1,v)≤βnϕ(x,v)+(1-βn){αnϕ(xn,v)+(1-αn)ϕ(Jrnxn,v)}≤βnϕ(x,v)+(1-βn)αnϕ(xn,v)+(1-βn)(1-αn)2lrnσ(∥xn-Jrnxn∥)
for all n≥N. Since ϕ(xn,v)→0 and (4.19), we may assume ϕ(xn,v)≤1 and 2σ(∥xn-Jrnxn∥)≤1 for all n≥N. By (4.20) and induction, we obtain
(4.21)ϕ(xn+1,v)≤(βn+∑i=Nn-1βi∏j=i+1n(1-βj)αj)ϕ(x,v)+(∏i=Nn(1-βi)αi)ϕ(xN,v)+(1-βn)l((1-αn)rn+∑i=Nn-1(1-αi)ri2σ(∥xn-Jrnxn∥)∏j=i+1n(1-βj-1)αj)≤τnϕ(x,v)+θn+δn,
for all n≥N, where τn=βn+∑i=Nn-1βi∏j=i+1n(1-βj)αj, θn=∏i=Nn(1-βi)αi, δn=l((1-αn)/rn+∑i=Nn-1((1-αi)/ri)∏j=i+1n(1-βj-1)αj).
Next, we proveτn, θn and δn tend to 0. By αn,βn∈[0,1] and ∑i=1∞βi<∞, we get
(4.22)0≤τn=βn+∑i=Nn-1βi∏j=i+1n(1-βj)αj≤βn+∑i=Nn-1βi=∑i=Nnβi≤∑i=1∞βi.
Thus {τn} is bounded. Since αn∈[0,1] and limsupn→∞αn<1, there exists some α(0<α<1) such that 0≤αn≤α whenever n≥N. Then, we get
(4.23)0≤limsupn→∞τn≤limsupn→∞βn+limsupn→∞∑i=Nn-1βi∏j=i+1n(1-βj)αj=limn→∞βn+limsupn→∞(1-βn)αn(βn-1+∑i=Nn-2βi∏j=i+1n-1(1-βj)αj)=limsupn→∞(1-βn)αnτn-1≤limsupn→∞αnτn-1≤αlimsupn→∞τn-1=αlimsupn→∞τn,
which implies
(4.24)limn→∞τn=limsupn→∞τn=0.
Meanwhile, we also have
(4.25)0≤θn=∏i=Nn(1-βi)αi≤∏i=Nnαi≤αn-N+1→0.
On the other hand,
(4.26)δn=l((1-αn)rn+∑i=Nn-1(1-αi)ri∏j=i+1n(1-βj-1)αj)≤(1-αn)lrn+αnl((1-αn-1)rn-1+∑i=Nn-2(1-αi)ri∏j=i+1n-1(1-βj-1)αj)=(1-αn)lrn+αnδn-1.
Since limsupn→∞αn<1, ∑n=1∞(1-αn)=+∞, it follows from Lemma 2.6 that δn→0.
Fromσ(t)∈Σ1, which implies that limt→0+(σ(t)/t)=0, and (3.20). It follows that
(4.27)limn→∞σ(∥xn-Jrnxn∥)∥xn-Jrnxn∥=0.
By (4.18), we have
(4.28)ϕ(Jrnxn,v)≤lrn2σ(∥xn-Jrnxn∥)=lrn2σ(∥xn-Jrnxn∥)∥xn-Jrnxn∥∥xn-Jrnxn∥=∘(lrn)∥xn-Jrnxn∥,
for all n≥N, where ∘(lrn):=lrn2σ(∥xn-Jrnxn∥)∥xn-Jrnxn∥.
SinceE is uniformly convex, ∥·∥2 is uniformly totally convex on each bounded set of E. Denote by νr(t):=ν(rUE,t) the modulus of uniformly total convexity on the bounded set rUE. Then νr(t)∈N0 and satisfies ν(∥xn-Jrnxn∥)≤ϕ(xn,Jrnxn). From the definition of the greatest quasi-inverse of ν , we deduce that
(4.29)∥xn-Jrnxn∥≤νrh(ϕ(xn,Jrnxn)).
From Lemma 2.5, we have
(4.30)ϕ(xn,Jrnxn)≤ϕ(xn,v)-ϕ(Jrnxn,v)≤ϕ(xn,v).
Since νrh∈N0, it holds by (4.28), (4.29), and (4.30) that
(4.31)ϕ(Jrnxn,v)≤∘(lrn)∥xn-Jrnxn∥≤∘(lrn)νrh(ϕ(xn,v)),
for all n∈ℕ. Let k(t)=max{t,νrh(t)}. Since νrh∈N0 and νr(t)≥0 for t>0, it follows in [18, Lemma  3.3.1(i)] that νrh(t)∈Ω0 and this implies that k(t)∈N0∩Ω0. By the first inequality of (4.20), (4.31) and the definition of k(t), we have
(4.32)ϕ(xn+1,v)≤βnϕ(x,v)+(αn+∘(lrn))k(ϕ(xn,v)),∀n≥N.Case 2.
limn→∞βn=0,∑n=1∞βn=∞, and limn→∞αn=0.
From the proof of Theorem3.1, we note that if (BJ)-10={v}, then {ϕ(xn,v)} converges to 0, {xn} converges strongly to v, and limn→∞∥xn-Jrnxn∥=0. By the same argument as in the proof of Case 1, we obtain ϕ(xn+1,v)≤τnϕ(x,v)+θn+δn for all n≥N, where τn, θn and δn are those of Case 1.
It remains to show that{τn}, {θn}, and {δn} converge to 0. Since αn,βn∈[0,1] and limn→∞αn=0, limn→∞βn=0, it follows that
(4.33)0≤τn=βn+∑i=Nn-1βi∏j=i+1n(1-βj)αj=βn(1-αn)+βnαn+∑i=Nn-1βi∏j=i+1n(1-βj)αj≤βn+αn(βn+∑i=Nn-1βi∏j=i+1n(1-βj))=βn+αn(1-∏i=Nn(1-βi))≤βn+αn→0,0≤θn=∏i=Nn(1-βi)αi≤αn∏i=Nn(1-βi)→0
whenever n≥N large enough.
On the other hand,(4.34)δn=l((1-αn)rn+∑i=Nn-1(1-αi)ri∏j=i+1n(1-βj-1)αj)≤(1-αn)lrn+αnδn-1.
Since limn→∞αn=0, ∑n=1∞(1-αn)=+∞. It follows from Lemma 2.6 that δn→0. Moreover, according to the proof of Case 1, we also have that, for all n≥N,
(4.35)ϕ(xn+1,v)≤βnϕ(x,v)+(αn+∘(lrn))k(ϕ(xn,v))
where k(t) is the same as that of Case 1. Hence, the conclusion follows.Ifαn≡0 for all n∈ℕ, then the algorithm (1.7) reduces to (1.5). Also, letting αn=0 in (4.15) we obtain
(4.36)ϕ(xn+1,v)≤βnϕ(x,v)+∘(lrn)k(ϕ(xn,v)),∀n≥N.Corollary 4.3 (see Li and Song [6]).
LetH be a Hilbert space and let B⊂H×H be a maximal monotone with B-10={v*} and B-1 is Lipschitz continuous at 0 with modulus l≥0. Let {xn} be a sequence generated by x1=x∈H and
(4.37)xn+1=αnxn+(1-αn)Jrnxn,
for every n=1,2,…, where {αn}⊂[0,1], {rn}⊂(0,∞) satisfy limsupn→∞αn<1, and limn→∞rn=∞. Then the sequence {xn} converges strongly to v. Moreover, there exists an integer N>0 such that
(4.38)∥xn+1-v∥≤θn+δn∀n≥N,
where θn=∏i=Nnαi, δn=l((1-αn)/rn+∑i=Nn-1((1-αi)/ri)∏j=i+1nαj), and limn→∞θn=limn→∞δn=0. Meanwhile, one obtains the estimate, for all n≥N,
(4.39)∥xn+1-v∥≤αn+∘(lrn)2∥xn-v∥.Proof.
Note that, forx,y∈H, we have J=I and ϕ(x,y)=∥x-y∥2. Under our assumptions, the iterative sequence (4.37) reduces to a special case of the algorithm (1.7) where βn=0. From the proof of Case 1 in Theorem 4.2, we know that {xn} converges strongly to v and there exists some N>0 such that (4.20) and (4.21) hold for n≥N.
Letβn=0 in the inequality (4.21). Then, τn=0. It follows from (4.21) that
(4.40)∥xn+1-v∥≤θn+δn,∀n≥N.
Since αn∈[0,1] and limsupn→∞αn<1, there exists some α(0<α<1) such that 0≤αn≤α whenever n≥N. Then, we get θn=∏i=Nnαi≤αn-N+1→0. Also,
(4.41)δn=l((1-αn)rn+∑i=Nn-1(1-αi)ri∏j=i+1nαj)=(1-αn)lrn+αnl((1-αn-1)rn-1+∑i=Nn-2(1-αi)ri∏j=i+1n-1αj)=(1-αn)lrn+αnδn-1.
Since limsupn→∞αn<1, ∑n=1∞(1-αn)=+∞. It follows from Lemma 2.6 that δn→0. Additionally, from (4.20), (4.16), and (4.30), we have
(4.42)∥xn+1-v∥2≤αn∥xn-v∥2+(1-αn)∥Jrnxn-v∥2≤αn∥xn-v∥2+(1-αn)(l∥Arnxn∥)2=αn∥xn-v∥2+(1-αn)l2∥xn-Jrnxnrn∥2≤αn∥xn-v∥2+(1-αn)(lrn)2∥xn-v∥2≤(αn+(lrn)2)∥xn-v∥2,
for all n≥N, which implies the equality (4.39).


## 5. Applications

In this section, we study the problem of finding a minimizer of a proper lower semicontinuous convex function in a Banach space.Theorem 5.1.
LetE be a uniformly convex and uniformly smooth Banach space and let f*:E*→(-∞,∞] be a proper lower semicontinuous convex function such that (∂f*)-10≠∅. Let {xn} be a sequence defined as follows: x1=x∈E and
(5.1)zn*=argminy*∈E*{f*(y*)+12rn∥y*∥2-1rn〈xn,y*〉};yn*=αnxn+(1-αn)J-1zn*;xn+1=βnx+(1-βn)yn*,
for every n=1,2,…, where {αn}, {βn}⊂[0,1], and {rn}⊂(0,∞) satisfy limn→∞αn=0, limn→∞βn=0, ∑n=1∞βn=∞, and limn→∞rn=∞. Then the sequence {xn} converges strongly to R(∂f*J)-10(x), where R(∂f*J)-10 is a sunny generalized nonexpansive retraction of E onto (∂f*J)-10.Proof.
By Rockafellar's theorem [19, 20], the subdifferential mapping ∂f*⊂E*×E is maximal monotone. Let Jr=(I+r∂fJ)-1 for all r>0. As in the proof of [7, Corollary  5.1], we have J-1zn*=Jrnxn for all n∈ℕ. Hence, by Theorem 3.1, {xn} converges strongly to R(∂f*J)-10(x).Whenαn=0 in Theorem 5.1 we obtain the following corollary.Corollary 5.2 (see Ibaraki and Takahashi [7]).
LetE be a uniformly convex and uniformly smooth Banach space and let f*:E*→(-∞,∞] be a proper lower semicontinuous convex function such that (∂f*)-10≠∅. Let {xn} be a sequence defined as follows: x1=x∈E and
(5.2)yn*=argminy*∈E*{f*(y*)+12rn∥y*∥2-1rn〈xn,y*〉};xn+1=βnx+(1-βn)J-1yn*,
for every n=1,2,…, where {βn}⊂[0,1] and {rn}⊂(0,∞) satisfy limn→∞βn=0,∑n=1∞βn=∞, and limn→∞rn=∞. Then the sequence {xn} converges strongly to R(∂f*J)-10(x), where R(∂f*J)-10 is a sunny generalized nonexpansive retraction of E onto (∂f*J)-10.



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*Source: 795432-2009-07-28.xml*