# General Split Variational Inclusion Problem in Hilbert Spaces

Authors: Li Yang; Fu Hai Zhao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider a general split variational inclusion problem (GSFVIP) and propose an algorithm for finding the solutions of GSFVIP in Hilbert space. We establish the strong convergence of the proposed algorithm to a solution of GSFVIP. Our results extend and improve the related results in the literature.


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## Body

## 1. Introduction

LetH be a real Hilbert space, and let B be a set-valued mapping with domain D(B):={x∈H:B(x)≠∅}. Recall that B is calledmonotone if 〈u-v,x-y〉≥0 for any u∈Bx and v∈By; B ismaximal monotone if its graph {(x,y):x∈D(B),y∈Bx} is not properly contained in the graph of any other monotone mapping. An important problem for set-valued monotone mappings is to find x*∈H such that 0∈Bx*. Here, x* is called azero point of B. A well-known method for approximating a zero point of a maximal monotone mapping defined in a real Hilbert space isthe proximal point algorithm first introduced by Martinet [1] and generated by Rockafellar [2]. This is an iterative procedure, which generates {xn} by x1=x∈H and
(1)xn+1=JβnBxn,n∈N,
where {βn}⊂(0,∞), B is a maximal monotone mapping in a real Hilbert space, and JrB is theresolvent mapping of B defined by JrB=(I+rB)-1 for each r>0. In 1976, Rockafellar [2] proved the following in the Hilbert space setting. If the solution set B-1(0) is nonempty and liminf⁡n→∞βn>0, then the sequence {xn} in (1) converges weakly to an element of B-1(0). Later, many researchers have studied the convergence theorems of the proximal point algorithm in Hilbert spaces. For example, one can refer to [3–8] and references therein.LetH1 and H2 be two real Hilbert spaces, B1:H1→H1 and B2:H2→H2 two set-valued maximal monotone mappings, A:H1→H2 a linear and bounded operator, and A* the adjoint of A. Chuang [9] considers the followingsplit variational inclusion problem:
(2)(SFVIP)Findx*∈H1suchthat0∈B1(x*),hhhhhhhhhhhhhhhhhhhhhhhh0∈B2(Ax*),which was introduced by Moudafi [10]. In this paper, motivated by the works in Chuang [9] and related literature, we consider the following general split variational inclusion problem.Definition 1.
LetH1 and H2 be two real Hilbert spaces, {Bi:H1→H1}i∈N and {Ki:H2→H2}i∈N two families of set-valued maximal monotone mappings, A:H1→H2 a linear and bounded operator, and A* the adjoint of A. The general split variational inclusion problem (GSFVIP) is formulated as the following problem:
(3)(GSFVIP)Findx*∈H1suchthatx*∈⋂i=1∞Bi-1(0),hhhhhhhhhhhhhhhhhhihhhhhhhAx*∈⋂i=1∞Ki-1(0).In this paper, we propose an algorithm for finding the solutions of GSFVIP in a Hilbert space and prove that the sequence generated by the proposed method converges strongly to a solution of GSFVIP. Our results extend and improve the related results in the literature.


## 2. Preliminaries

Throughout this paper, letN be the set of positive integers. Let H be a real Hilbert space with the inner product 〈·,·〉 and the norm ∥·∥, respectively. We also use “→” to stand for strong convergence and “⇀” to stand for weak convergence.Lemma 2 (see [11]).
LetH be a real Hilbert space, and let x, y∈H. Then
(4)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.Lemma 3 (see [12]).
LetH be a Hilbert space, and let {xn} be a sequence in H. Then, for any given sequence {αn}n=1∞⊂(0,1) with ∑n=1∞αn=1 and for any positive integer i,j with i<j,
(5)∥∑n=1∞‍αnxn∥2≤∑n=1∞‍αn∥xn∥2-αiαj∥xi-xj∥2.LetD be a nonempty closed convex subset of a real Hilbert space H, and let T:D→H be a mapping. Then T is said to be anonexpansive mapping if ∥Tx-Ty∥≤∥x-y∥ for every x,y∈D. It is easy to see that Fix⁡(T):={x∈D:Tx=x} is a closed convex subset of D if T is a nonexpansive mapping. Besides, T is said to be afirmly nonexpansive mapping if ∥Tx-Ty∥2≤〈x-y,Tx-Ty〉 for every x, y∈D.Lemma 4 (see [13]).
LetD be a nonempty closed convex subset of a real Hilbert space H. Let T:D→H be a nonexpansive mapping, and let {xn} be a sequence in D. If xn⇀w and lim⁡n→∞∥xn-Txn∥=0, then Tw=w.LetD be a nonempty closed convex subset of H. For every point x∈H, there exists a unique nearest point in D, denoted by PDx, such that
(6)∥x-PDx∥≤∥x-y∥,∀y∈D.PD is called themetric projection of H onto D. It is known that PD is a nonexpansive mapping of H onto D.Lemma 5 (see [14]).
LetD be a nonempty closed convex subset of a Hilbert space H. Let PD be the metric projection from H onto D. Then, for each x∈H and z∈D, we know that z=PDx if and only if 〈x-z,z-y〉≥0 for all y∈D.The following result is an important tool in this paper. For similar results, one can see [15].Lemma 6.
LetH be a real Hilbert space. Let B:H→H be a set-valued maximal monotone mapping, β>0, and let JβB be a resolvent mapping of B.(i)
For eachβ>0, JβB is a single-valued and firmly nonexpansive mapping.(ii)
D(JβB)=H and Fix⁡(JβB)={x∈D(B):0∈Bx}.(iii)
∥x-JβBx∥≤∥x-JαBx∥ for all 0<β≤α and for all x∈H.(iv)
(I-JβB) is a firmly nonexpansive mapping for each β>0.(v)
Suppose thatB-1(0)≠∅. Then ∥x-JβBx∥2+∥JβBx-x*∥2≤∥x-x*∥2 for each x∈H, each x*∈B-1(0), and each β>0.(vi)
Suppose thatB-1(0)≠∅. Then 〈x-JβBx,JβBx-w〉≥0 for each x∈H, each w∈B-1(0), and each β>0.Lemma 7 (see [9]).
LetH1 and H2 be two real Hilbert spaces, let A:H1→H2 be a linear operator, let A* be the adjoint of A, let β>0 be fixed, and let γ∈(0,2/∥A∥2). Let K:H2→H2 be a set-valued maximal monotone mapping and let JβK be a resolvent mapping of K. Then
(7)∥[x-γA*(I-JβK)Ax]-[y-γA*(I-JβK)Ay]∥2≤∥x-y∥2-(2γ-γ2∥A∥2)×∥(I-JβK)Ax-(I-JβK)Ay∥2,
for all x, y∈H. Furthermore, I-γA*(I-JβK)A is a nonexpansive mapping.The following is a very important result for various strong convergence theorems. Recently, many researchers have studied Halpern’s type strong convergence theorems by using the following lemma and get many generalized results. For example, one can see [9, 16, 17]. In this paper, we also use this result to get our strong convergence theorems.Lemma 8 (see [18]).
Let{an} be a sequence of real numbers such that there exists a subsequence {ni} of {n} such that ani<ani+1 for all i∈N. Then there exists a nondecreasing sequence {τ(n)}⊂N such that τ(n)→∞, aτ(n)≤aτ(n)+1, and an≤aτ(n)+1 are satisfied by all (sufficiently large) numbers n∈N. In fact, τ(n)=max⁡{k≤n:ak<ak+1}.Lemma 9 (see [19]).
Let{an}n∈N be a sequence of nonnegative real numbers, {αn} a sequence of real numbers in (0,1) with ∑n=1∞αn=∞, {ηn} a sequence of nonnegative real numbers with ∑n=1∞ηn<∞, and {δn} a sequence of real numbers with limsup⁡n→∞δn≤0. Suppose that an+1≤(1-αn)an+αnδn+ηn for each n∈N. Then lim⁡n→∞an=0.


## 3. Main Results

In this section, we first give the following result.Lemma 10.
LetH1 and H2 be two real Hilbert spaces, let A:H1→H2 be a linear and bounded operator, and let A* denote the adjoint of A. Let {Bi:H1→H1}i∈N and {Ki:H2→H2}i∈N be two families of set-valued maximal monotone mappings, and let βi>0 and γi>0 for all i∈N. Given any x*∈H1, we have the following.(i)
Ifx* is a solution of GSFVIP, then JβiBi(x*-γiA*(I-JβiKi)Ax*)=x*, for all i∈N.(ii)
Suppose thatJβiBi(x*-γiA*(I-JβiKi)Ax*)=x*, for all i∈N, and that the solution set of GSFVIP is nonempty; then x* is a solution of GSFVIP.Proof.
(i) Suppose thatx*∈H1 is a solution of GSFVIP. Then x*∈⋂i=1∞Bi-1(0) and Ax*∈⋂i=1∞Ki-1(0). By Lemma 6(ii), it is easy to see that
(8)JβiBi(x*-γiA*(I-JβiKi)Ax*)=JβiBi(x*-γiA*(Ax*-JβiKiAx*))=JβiBix*=x*,∀i∈N.
(ii) Suppose thatw* is a solution of GSFVIP and x*=JβiBi(x*-γiA*(I-JβiKi)Ax*), for all i∈N. By Lemma 6(vi),
(9)〈(x*-γiA*(I-JβiKi)Ax*)-x*,x*-w〉≥0,hhhhhhhhihhhhhihhhh∀w∈Bi-1(0),∀i∈N.
That is,
(10)〈A*(I-JβiKi)Ax*,x*-w〉≤0,∀w∈Bi-1(0),∀i∈N.
By (10) and the fact that A* is the adjoint of A,
(11)〈Ax*-JβiKiAx*,Ax*-Aw〉≤0,∀w∈Bi-1(0),∀i∈N.
On the other hand, by Lemma 6(vi) again,
(12)〈Ax*-JβiKiAx*,v-JβiKiAx*〉≤0,∀v∈Ki-1(0),∀i∈N.
By (11) and (12),
(13)〈Ax*-JβiKiAx*,v-JβiKiAx*+Ax*-Aw〉≤0,∀i∈N
for each w∈Bi-1(0) and each v∈Ki-1(0), for all i∈N. That is,
(14)∥Ax*-JβiKiAx*∥2≤〈Ax*-JβiKiAx*,Aw-v〉,∀i∈N
for each w∈Bi-1(0) and each v∈Ki-1(0), for all i∈N. Since w* is a solution of GSFVIP, w*∈⋂i=1∞Bi-1(0) and Aw*∈⋂i=1∞Ki-1(0). So, it follows from (14) that Ax*=JβiKiAx*, for all i∈N. Then, Ax*∈Fix⁡(JβiKi)=Ki-1(0), for all i∈N. Therefore Ax*∈⋂i=1∞Ki-1(0). Further,
(15)x*=JβiBi(x*-γiA*(I-JβiKi)Ax*)=JβiBix*,∀i∈N.
Then x*∈Fix⁡(JβiBi)=Bi-1(0), for all i∈N. So, x*∈⋂i=1∞Bi-1(0). Therefore, x* is a solution of GSFVIP.Theorem 11.
LetH1 and H2 be two real Hilbert spaces, let A:H1→H2 be a linear and bounded operator, and let A* denote the adjoint of A. Let {Bi:H1→H1}i∈N and {Ki:H2→H2}i∈N be two families of set-valued maximal monotone mappings. Let {an},{bn},{cn,i} be sequences of real numbers in (0,1) with an+bn+∑i=1∞cn,i=1. Let {βn,i} be a sequence in (0,∞) and {γn,i}⊂(0,2/(∥A∥2+1)) for each i∈N. Let Ω be the solution set of GSFVIP and suppose that Ω≠∅. Let f be a self k-contraction mapping of H1, k∈(0,1). Let {xn} be defined by
(16)xn+1=anxn+bnf(xn)+∑i=1∞‍cn,iJβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn,n≥0.
If the sequences{an}, {bn}, {cn,i}, {βn,i}, and {γn,i} satisfy the following conditions: (i)
lim⁡n→∞bn=0, ∑n=0∞bn=∞,(ii)
for eachi∈N, liminf⁡n→∞ancn,i>0, liminf⁡n→∞cn,iγn,i>0, liminf⁡n→∞βn,i>0, and limsup⁡n→∞γn,i<2/(∥A∥2+1),then the sequence {xn} converges strongly to PΩf(x*).Proof.
First, we show that{xn} is bounded. In fact, let z∈Ω; it follows from Lemmas 6(i) and 7 that Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A), for all i∈N, are nonexpansive, and by Lemma 10 we have
(17)∥xn+1-z∥=∥anxn+bnf(xn)∑i=1∞‍cn,iJβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-zi+∑i=1∞‍cn,iJβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-z∥≤an∥xn-z∥+bn∥f(xn)-z∥+∑i=1∞‍cn,i∥Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-z∥≤an∥xn-z∥+bn∥f(xn)-z∥+∑i=1∞‍cn,i∥xn-z∥≤(1-bn)∥xn-z∥+bn∥f(xn)-z∥≤(1-bn)∥xn-z∥+bn∥f(xn)-f(z)∥+bn∥f(z)-z∥≤(1-bn)∥xn-z∥+bnk∥xn-z∥+bn∥f(z)-z∥≤(1-(1-k)bn)∥xn-z∥+(1-k)bn11-k∥f(z)-z∥≤max⁡{∥xn-z∥,11-k∥f(z)-z∥}≤⋯≤max⁡{∥x0-z∥,11-k∥f(z)-z∥},
which implies that {xn} is bounded, and we also obtain that {f(xn)} is bounded.
Next, we show that there exists a uniquex*∈Ω such that x*=PΩf(x*).
Since, for alli∈N, liminf⁡n→∞βn,i>0, we may assume that βn,i>βi>0 for each n∈N. Since, for all i∈N, {γn,i} is bounded, there exists a converge subsequence. Without loss of generality, we can assume that γn,i→γi∈(0,2/(∥A∥2+1)) for each i∈N.
It follows from Lemma10 that x*∈Ω solves the GSFVIP if and only if x* solves the fixed point equation
(18)x*=JβiBi(I-γiA*(I-JβiKi)A)x*,i∈N
that is, the solution sets of fixed point equation (18) and GSFVIP are the same. By Lemmas 6(i) and 7, the operators JβiBi(I-γiA*(I-JβiKi)A), for all i∈N, are nonexpansive. Since the fixed point set of nonexpansive operators is closed and convex, the projection onto the solution set Ω is well defined whenever Ω≠∅. We observe that PΩf is a contraction of H1 into itself. Indeed, since PΩ is nonexpansive and f is a self k-contraction mapping H1,
(19)∥PΩ(f)(x)-PΩ(f)(y)∥≤∥f(x)-f(y)∥≤k∥x-y∥.
Hence, there exists a unique element x*∈Ω such that x*=PΩf(x*).
In order to prove thatxn→x* as n→∞, we consider two possible cases.Case 1.
There exists a natural numbern0 such that ∥xn+1-x*∥≤∥xn-x*∥ for each n≥n0. Since {∥xn-x*∥} is bounded, we have {∥xn-x*∥} is convergent.
Next, we show that, for eachi∈N,
(20)lim⁡n→∞∥JβiBi(I-γiA*(I-JβiKi)A)xn-xn∥=0.
By Lemmas 6(i) and 7, for every z∈Ω and i∈N, we have
(21)∥Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-z∥≤∥(xn-γn,iA*(I-Jβn,iKi)Axn)h-(z-γn,iA*(I-Jβn,iKi)Az)∥≤∥xn-z∥2-(2γn,i-γn,i2∥A∥2)×∥(I-Jβn,iKi)Axn-(I-Jβn,iKi)Az∥2=∥xn-z∥2-(2γn,i-γn,i2∥A∥2)×∥Axn-Jβn,iKiAxn∥2.
By using Lemma 3 and (21), for every z∈Ω and i∈N, we have
(22)∥xn+1-z∥2=∥anxn+bnf(xn)∑j=1∞‍i+∑j=1∞‍cn,jJβn,jBj(I-γn,jA*(I-Jβn,jKj)A)xn-z∥2≤an∥xn-z∥2+bn∥f(xn)-z∥2+∑j=1∞‍cn,j∥Jβn,jBj(I-γn,jA*(I-Jβn,jKj)A)xn-z∥2-ancn,i∥Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-xn∥2≤an∥xn-z∥2+bn∥f(xn)-z∥2+∑j=1∞‍cn,j(∥xn-z∥2∥Axn-Jβn,jKjAxn∥2hhhhhhh-(2γn,j-γn,j2∥A∥2)∥Axn-Jβn,jKjAxn∥2)-ancn,i∥Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-xn∥2≤(1-bn)∥xn-z∥2+bn∥f(xn)-z∥2-cn,i(2γn,i-γn,i2∥A∥2)∥Axn-Jβn,iKiAxn∥2-ancn,i∥Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-xn∥2.
Hence, for each i∈N, we have
(23)ancn,i∥Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-xn∥2≤∥xn-z∥2-∥xn+1-z∥2+bn∥f(xn)-z∥2,cn,i(2γn,i-γn,i2∥A∥2)∥Axn-Jβn,iKiAxn∥2≤∥xn-z∥2-∥xn+1-z∥2+bn∥f(xn)-z∥2.
Sincelim⁡n→∞bn=0 and f(xn) is bounded, from (23) we get that
(24)lim⁡n→∞ancn,i∥Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-xn∥2=0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀i∈N,(25)lim⁡n→∞cn,i(2γn,i-γn,i2∥A∥2)∥Axn-Jβn,iKiAxn∥2=0,∀i∈N.
By assuming that, for all i∈N, liminf⁡n→∞ancn,i>0, it follows from (24) that
(26)lim⁡n→∞∥Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-xn∥=0,∀i∈N.
Further, for all i∈N, we have
(27)∥Jβn,iBi(I-γiA*(I-Jβn,iKi)A)xn-xn∥≤∥Jβn,iBi(I-γiA*(I-Jβn,iKi)A)xnh-Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn∥+∥Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-xn∥≤∥(I-γiA*(I-Jβn,iKi)A)xnh-(I-γn,iA*(I-Jβn,iKi)A)xn∥+∥Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-xn∥≤|γi-γn,i|∥A*(I-Jβn,iKi)Axn∥+∥Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-xn∥⟶0hhhhhhhhhhhhhhhhhhhhhhhhhhhhh(n⟶∞).
Clearly, for all i∈N, cn,i(2γn,i-γn,i2∥A∥2)≥cn,iγn,i/(∥A∥2+1). Since, for all i∈N, liminf⁡n→∞cn,iγn,i>0, it follows from (25) that
(28)lim⁡n→∞∥Axn-Jβn,iKiAxn∥=0,∀i∈N,
and it follows from Lemma 6(iii) that
(29)lim⁡n→∞∥Axn-JβiKiAxn∥=0,∀i∈N.
Besides, by Lemma 6(i) and (28), for all i∈N, we have
(30)∥Jβn,iBi(I-γiA*(I-Jβn,iKi)A)xn-Jβn,iBixn∥≤γi∥A∥∥Axn-Jβn,iKiAxn∥⟶0(n⟶∞).
By (27) and (30), for all i∈N, we obtain
(31)∥xn-Jβn,iBixn∥≤∥xn-Jβn,iBi(I-γiA*(I-Jβn,iKi)A)xn∥+∥Jβn,iBi(I-γiA*(I-Jβn,iKi)A)xn-Jβn,iBixn∥⟶0hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh(n⟶∞).
It follows from Lemma 6(iii) that
(32)lim⁡n→∞∥xn-JβiBixn∥=0,∀i∈N.
By Lemma 6(i) and (29), for all i∈N, we have
(33)∥JβiBi(I-γiA*(I-JβiKi)A)xn-JβiBixn∥≤γi∥A∥∥Axn-JβiKiAxn∥⟶0(n⟶∞).
By (32) and (33), for all i∈N, we obtain
(34)lim⁡n→∞∥JβiBi(I-γiA*(I-JβiKi)A)xn-xn∥=0.
Now, we show that(35)limsup⁡n→∞〈f(x*)-x*,xn-x*〉≤0.
To show this inequality, we choose a subsequence {xnk} of {xn} such that
(36)lim⁡k→∞〈f(x*)-x*,xnk-x*〉=limsup⁡n→∞〈f(x*)-x*,xn-x*〉.
Since {xnk} is bounded, there exists a subsequence {xnkj} of {xnk} which converges weakly to w. Without loss of generality, we can assume that xnk⇀w. Notice that, for each i∈N, JβiBi(I-γiA*(I-JβiKi)A) is nonexpansive. Thus, from Lemma 4 and (34), we have w∈Ω. Therefore, it follows from Lemma 5 that
(37)limsup⁡n→∞〈f(x*)-x*,xn-x*〉=lim⁡k→∞〈f(x*)-x*,xnk-x*〉=〈f(x*)-x*,w-x*〉≤0.
Finally, we show thatxn→x*. Applying Lemma 2, we have that
(38)∥xn+1-x*∥2=∥anxn+bnf(xn)∑i=1∞‍cn,iJβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-x*i+∑i=1∞‍cn,iJβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-x*∥2≤∥an(xn-x*)i+∑i=1∞‍cn,i(Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-x*)i+bn(f(xn)-x*)∥2≤∥an(xn-x*)∑i=1∞‍cn,ii+∑i=1∞‍cn,i(Jβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn-x*)∥2+2bn〈f(xn)-x*,xn+1-x*〉≤(an∥xn-x*∥+∑i=1∞‍cn,i∥xn-x*∥)2+2bn〈f(xn)-x*,xn+1-x*〉≤(1-bn)2∥xn-x*∥2+2bn〈f(xn)-f(x*),xn+1-x*〉+2bn〈f(x*)-x*,xn+1-x*〉≤(1-bn)2∥xn-x*∥2+2bnk∥xn-x*∥∥xn+1-x*∥+2bn〈f(x*)-x*,xn+1-x*〉≤(1-bn)2∥xn-x*∥2+bnk(∥xn-x*∥2+∥xn+1-x*∥2)+2bn〈f(x*)-x*,xn+1-x*〉.
This implies that
(39)∥xn+1-x*∥2≤(1-bn)2+bnk1-bnk∥xn-x*∥2+2bn1-bnk〈f(x*)-x*,xn+1-x*〉=1-2bn+bnk1-bnk∥xn-x*∥2+bn21-bnk∥xn-x*∥2+2bn1-bnk〈f(x*)-x*,xn+1-x*〉=(1-2(1-k)bn1-bnk)∥xn-x*∥2+2(1-k)bn1-bnk{bnM2(1-k)hhhhhhhhhhhh+11-k〈f(x*)-x*,xn+1-x*〉bnM2(1-k)}=(1-αn)∥xn-x*∥2+αnδn,
where δn=bnM/2(1-k)+(1/(1-k))〈f(x*)-x*,xn+1-x*〉, M=sup⁡{∥xn-x*∥2:n≥0}, and αn=2(1-k)bn/(1-bnk). It is easy to see that lim⁡n→∞αn=0, ∑n=1∞αn=∞, and limsup⁡n→∞δn≤0. Hence, by Lemma 9, the sequence {xn} converges strongly to x*=PΩf(x*).Case 2.
Assume that{∥xn-x*∥} is not a monotone sequence. Then, we can define an integer sequence {τ(n)} for all n≥n0 (for some n0 large enough) by
(40)τ(n)=max⁡{k∈N;k≤n:∥xk-x*∥<∥xk+1-x*∥}.
Clearly, {τ(n)} is a nondecreasing sequence such that τ(n)→∞ as n→∞ for all n≥n0:
(41)∥xτ(n)-x*∥<∥xτ(n)+1-x*∥.
Following a similar argument as the proof of Case1, we have
(42)limsup⁡n→∞〈f(x*)-x*,xτ(n)-x*〉≤0.
And by similar argument, we have
(43)∥xτ(n)+1-x*∥2=(1-ατ(n))∥xτ(n)-x*∥2+ατ(n)δτ(n),
where lim⁡n→∞ατ(n)=0, ∑n=1∞ατ(n)=∞, and limsup⁡n→∞δτ(n)≤0. Hence, by Lemma 9, we obtain lim⁡n→∞∥xτ(n)-x*∥=0 and lim⁡n→∞∥xτ(n)+1-x*∥=0. Now, from Lemma 8, we have
(44)0≤∥xn-x*∥≤∥xτ(n)+1-x*∥.
Therefore, the sequence {xn} converges strongly to x*=PΩf(x*). This completes the proof.Corollary 12.
LetH1 and H2 be two real Hilbert spaces, let A:H1→H2 be a linear and bounded operator, and let A* denote the adjoint of A. Let {Bi:H1→H1}i∈N and {Ki:H2→H2}i∈N be two families of set-valued maximal monotone mappings for all i∈N. Let {an},{cn,i} be sequences of real numbers in (0,1) with an+∑i=1∞cn,i⊂(0,1). Let {βn,i} be a sequence in (0,∞) and {γn,i}⊂(0,2/(∥A∥2+1)) for each i∈N. Let Ω be the solution set of GSFVIP and suppose that Ω≠∅. Let {xn} be defined by
(45)xn+1=anxn+∑i=1∞‍cn,iJβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhn≥0.
If the sequences{an}, {cn,i}, {βn,i}, and {γn,i} satisfy the following conditions: (i)
lim⁡n→∞(an+∑i=1∞cn,i)=1, ∑n=0∞(1-an-∑i=1∞cn,i)=∞,(ii)
for eachi∈N, liminf⁡n→∞ancn,i>0, liminf⁡n→∞cn,iγn,i>0, liminf⁡n→∞βn,i>0, and limsup⁡n→∞γn,i<2/(∥A∥2+1),then the sequence {xn} converges strongly to a point x* which is the minimum norm solution of GSFVIP.Proof.
Letf=0 in Theorem 11. We have proved that the sequence {xn} converges strongly to x*=PΩf(x*). Then, for all z∈Ω, we have
(46)〈f(x*)-x*,z-x*〉=〈f(x*)-PΩf(x*),z-PΩf(x*)〉≤0.
Since f=0, then 〈-x*,z-x*〉≤0, for all z∈Ω. Hence, we obtain
(47)∥x*∥2≤|〈x*,z〉|≤∥x*∥∥z∥
that is, ∥x*∥≤∥z∥. Thus, x* is the minimum norm solution of GSFVIP. This completes the proof.In Theorem11, if we set f(x)=u, then we get the following result.Corollary 13.
LetH1 and H2 be two real Hilbert spaces, let A:H1→H2 be a linear and bounded operator, and let A* denote the adjoint of A. Let {Bi:H1→H1}i∈N and {Ki:H2→H2}i∈N be two families of set-valued maximal monotone mappings for all i∈N. Let {an},{bn},{cn,i} be sequences of real numbers in (0,1) with an+bn+∑i=1∞cn,i=1. Let {βn,i} be a sequence in (0,∞) and {γn,i}⊂(0,2/(∥A∥2+1)) for each i∈N. Let u∈H1 be fixed. Let Ω be the solution set of GSFVIP and suppose that Ω≠∅. Let {xn} be defined by
(48)xn+1=anxn+bnu+∑i=1∞‍cn,iJβn,iBi(I-γn,iA*(I-Jβn,iKi)A)xn,n≥0.
If the sequences{an}, {bn}, {cn,i}, {βn,i}, and {γn,i} satisfy the following conditions: (i)
lim⁡n→∞bn=0, ∑n=0∞bn=∞,(ii)
for eachi∈N, liminf⁡n→∞ancn,i>0, liminf⁡n→∞cn,iγn,i>0, liminf⁡n→∞βn,i>0, and limsup⁡n→∞γn,i<2/(∥A∥2+1),then the sequence {xn} converges strongly to PΩu.



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*Source: 816035-2014-10-14.xml*