# A System of Generalized Variational Inclusions Involving a New Monotone Mapping in Banach Spaces

Authors: Jinlin Guan; Changsong Hu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduce a new monotone mapping in Banach spaces, which is an extension of theCn-monotone mapping studied by Nazemi (2012), and we generalize the variational inclusion involving the Cn-monotone mapping. Based on the new monotone mapping, we propose a new proximal mapping which combines the proximal mapping studied by Nazemi (2012) with the η mapping studied by Lan et al. (2011) and show its Lipschitz continuity. Based on the new proximal mapping, we give an iterative algorithm. Furthermore, we prove the convergence of iterative sequences generated by the algorithm under some appropriate conditions. Our results improve and extend corresponding ones announced by many others.


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## Body

## 1. Introduction

Variational inequality theory has emerged as a powerful tool for a wide class of unrelated problems arising in various branches of physical, engineering, pure, and applied sciences in a unified and general framework. As the generalization of variational inequalities, variational inclusions have been widely studied in recent years. One of the most important problems in the theory of variational inclusions is the development of an efficient and implementable iterative algorithm. Therefore, many iterative algorithms and existence results for various variational inclusions have been studied see, for example, [1–3].Several years ago, Xia and Huang [4] proposed the concept of general H-monotone operators in Banach spaces and studied a class of variational inclusions involving the general H-monotone operator in Banach spaces. In 2010, Luo and Huang [5] introduced a new notion of B-monotone operators in Banach spaces and gave a new proximal mapping related to these operators. Then, they used it to study a new class of variational inclusions in Banach spaces. Very recently, Nazemi [6] introduced the notion of a new class of Cn-monotone mappings which is an extension of B-monotone operators introduced in [5].Motivated and inspired by the work going on in this direction, in this paper, we propose a new monotone mapping in Banach spaces namedCn-η-monotone mapping which generalizes the Cn-monotone mapping introduced in [6] from the same n-dimensional product space to different n-dimensional product space and reduces the Cn mapping from strictly monotone mapping to monotone mapping. Further, we consider a new proximal mapping which associates a η mapping introduced in [7] and generalizes the proximal mapping introduced in [6]. Furthermore, in the process of proving the convergence of iterative sequences generated by the algorithm, we change the condition of a uniformly smooth Banach space with ρE(t)≤Ct2 to a q-uniformly smooth Banach space, which extends the proof of the convergence of iterative sequences in [6]. The results presented in this paper generalize many known and important results in the recent literature and the references therein.


## 2. Preliminaries

LetE be a real Banach space, let E* be the topological dual space of E, and let 〈u,v〉 be the dual pair between u∈E* and v∈E. Let CB(E) denote the family of all nonempty, closed, and bounded subsets of E. Set ∏i=1nEi=E1×E2×⋯×En. Let H(·,·) be the Hausdorff metric on CB(E) defined by
(1)H(A,D)∶=max{supx∈Ainfy∈D∥x-y∥,supy∈Dinfx∈A∥x-y∥},A,D∈CB(E).
We recall the following definitions and results which are needed in the sequel.Definition 1 (see [7]).
A single-valued mappingη:E×E→E is said to be k-Lipschitz continuous if there exists a constant k>0 such that
(2)∥η(x,y)∥≤k∥x-y∥,∀x,y∈E.Definition 2 (see [8]).
A Banach spaceE is called smooth if, for every x∈E with ∥x∥=1, there exists a unique f∈E* such that ∥f∥=f(x)=1. The modulus of smoothness of E is the function ρE:[0,∞)→[0,∞), defined by
(3)ρE(t)=sup{12(∥x+y∥+∥x-y∥)-1:x,y∈E,∥x∥=1,∥y∥=t12}.Definition 3 (see [8]).
The Banach spaceE is said to be (i)
uniformly smooth if(4)limt→0ρE(t)t=0;(ii)
q-uniformly smooth, for q>1, if there exists a constant c>0 such that
(5)ρE(t)≤ctq,t∈[0,∞).It is well known (see, e.g., [9]) that
(6)Lq(orlq)is{q-uniformly  smooth,if1<q≤2,2-uniformly  smooth,ifq≥2.Note that ifE is uniformly smooth, jq becomes single-valued. In the study of characteristic inequalities in q-uniformly smooth Banach space, Xu [8] established the following lemma.Lemma 4 (see [8]).
Letq>1 be a real number and let E be a smooth Banach space and Jq:E→2E* the normalized duality mapping. Then, E is q-uniformly smooth if and only if there exists a constant cq>0 such that for every x,y∈E,
(7)∥x+y∥q≤∥x∥q+q〈y,jq(x)〉+cq∥y∥q,∀jq(x)∈Jq(x).Definition 5.
A single-valued mappingg:E→E is said to be (γ,μ)-relaxed cocoercive if there exist jq(x-y)∈Jq(x-y) and γ,μ>0 such that
(8)〈jq(x-y),g(x)-g(y)〉≥-γ∥g(x)-g(y)∥q+μ∥x-y∥q,∀x,y∈E.Definition 6.
Letn≥3 and M:∏i=1nEi⇉E* be a multivalued mapping, fi:E→Ei, i=1,2,…,n, and η:E×E→E single-valued mappings.(i)
For each1≤i≤n, M(…,fi,…) is said to be αi-strongly η-monotone with respect to fi (in the ith argument) if there exists a constant αi>0 such that
(9)〈ωi-ωi′,η(x,y)〉≥αi∥x-y∥2,ωi∈M(u1,…,ui-1,fi(x),ui+1,…,un),ωi′∈M(u1,…,ui-1,fi(y),ui+1,…,un),∀x,y∈E,u1∈E1,u2∈E2,…,ui-1∈Ei-1,ui+1∈Ei+1,…,un∈En.(ii)
For each1≤i≤n,M(…,fi,…) is said to be βi-relaxed η-monotone with respect to fi (in the ith argument) if there exists a constant βi>0 such that
(10)〈ωi-ωi′,η(x,y)〉≥-βi∥x-y∥2,ωi∈M(u1,…,ui-1,fi(x),ui+1,…,un),ωi′∈M(u1,…,ui-1,fi(y),ui+1,…,un),∀x,y∈E,u1∈E1,u2∈E2,…,ui-1∈Ei-1,ui+1∈Ei+1,…,un∈En.(iii)
By assumption thatn is an even number, M is said to be α1β2α3β4⋯αn-1βn-symmetric η-monotone with respect to f1,f2,…,fn if, for each i∈{1,3,…,n-1},M(…,fi,…) is αi-strongly η-monotone with respect to fi (in the ith argument) and for each j∈{2,4,…,n},M(…,fj,…) is βj-relaxed η-monotone with respect to fj (in the jth argument) with
(11)α1+α3+⋯+αn-1>β2+β4+⋯+βn.(iv)
By assumption thatn is an odd number, M is said to be α1β2α3β4⋯αn-1βn-symmetric η-monotone with respect to f1,f2,…,fn if, for each i∈{1,3,…,n}, M(…,fi,…) is αi-strongly η-monotone with respect to fi (in the ith argument) and for each j∈{2,4,…,n-1}, M(…,fj,…) is βj-relaxed η-monotone with respect to fj (in the jth argument) with
(12)α1+α3+⋯+αn>β2+β4+⋯+βn-1.Definition 7 (see [10]).
LetE be a Banach space. A multivalued mapping A:E⇉CB(Ei) is said to be H-Lipschitz continuous if there exists a constant t>0 such that
(13)H(A(x),A(y))≤t∥x-y∥,∀x,y∈E,
where H(·,·) is the Hausdorff metric on CB(Ei).Definition 8.
Let, for eachi=1,2,…,n,Ti:E⇉CB(Ei) be a multivalued mapping. A single-valued mapping F:∏i=1nEi→E* is said to be λFi-Lipschitz continuous in the ith argument with respect to Ti(i=1,2,…,n) if there exists a constant λFi>0 such that
(14)∥F(x1,…,xi-1,ui1,xi+1,…,xn)-F(x1,…,xi-1,ui2,xi+1,…,xn)∥≤λFi∥ui1-ui2∥,∀y1,y2∈E,x1∈E1,…,xi-1∈Ei-1,xi+1∈Ei+1,…,xn∈En,ui1∈Ti(y1),ui2∈Ti(y2).Definition 9.
LetE be a Banach space with the dual space E*andη:E×E→E single-valued mappings; Cn:E→E* is said to be η-monotone mapping if
(15)〈Cn(x)-Cn(y),η(x,y)〉≥0,∀x,y∈E.


## 3.Cn-η-Monotone Mapping

First, we define the notion ofCn-η-monotone mapping.Definition 10.
LetE be a Banach space with the dual space E*. Let n≥3 and fi:E→Ei,i=1,2,…,n,Cn:E→E* be single-valued mappings and M:∏i=1nEi⇉E* a multivalued mapping.(i)
In case thatn is an even number, M is said to be a Cn-η-monotone mapping if M is α1β2α3β4⋯αn-1βn-symmetric η-monotone with respect to f1,f2,…,fn and (Cn+λM(f1,f2,…,fn))(E)=E*, for every λ>0.(ii)
In case thatn is an odd number, M is said to be a Cn-η-monotone mapping if M is α1β2α3β4⋯βn-1αn-symmetric η-monotone with respect to f1,f2,…,fn and (Cn+λM(f1,f2,…,fn))(E)=E*, for every λ>0.Remark 11.
(i) IfM(f1,f2,…,fn)=M, η(y,x)=y-x, and M is monotone, then the Cn-η-monotone mapping reduces to the general H-monotone mapping considered in [4].
(ii) IfM(f1,f2,…,fn)=M(f1,f2), η(y,x)=y-x, then the Cn-η-monotone mapping reduces to the B-monotone mapping considered in [5].
(iii) IfM(f1,f2,…,fn)=M, η(y,x)=y-x, and M are m-relaxed monotone, then the Cn-η-monotone mapping reduces to the A-monotone mapping considered in [11].
(iv) IfM reduces to E×E×⋯×E︸n→E*,  η(y,x)=y-x, and f1,f2,…,fn reduce to E→E, then the Cn-η-monotone mapping reduces to the Cn-monotone mapping considered in [6].Example 12.
LetE=l2 and then E*=l2; and assume n is an even number; let Ei=(E,∥·∥i),i=1,2,…,n, where ∥·∥i is the equivalent norm on l2 space, ei=(0,…,0︸i-1,1,0,…)∈Ei, for ∀x∈E; let f1(x)=α1x+e1∈E1, f2(x)=-β2x+e2∈E2, f3(x)=α3x+e3∈E3, f4(x)=-β4x+e4∈E4,…, fn-1(x)=αn-1x+en-1∈En-1, fn(x)=-βnx+en∈En, where α1,β2,α3,β4,…,αn-1, βn>0 are constants such that
(16)α1-β2+α3-β4+⋯+αn-1-βn=γ>0.
Let M(u1,u2,…,un)=(u1-e1)+(u2-e2)+⋯+(un-en), where ui∈Ei, i=1,2,…,n; let Cn(x)=x+en+1,x∈E, η(x,y)=x-y, ∀x,y∈E. Then M is a Cn-η-monotone mapping.With no loss of generality, we may assume thatn is an even number in the next text.Lemma 13.
Letη:E×E→E,fi:E→Ei, i=1,2,…,n, be single-valued mappings; M:∏i=1nEi⇉E* a α1β2α3β4⋯αn-1βn-symmetric η-monotone with respect to f1,f2,…,fn. Then for ∀x,y∈E one has
(17)〈u-v,η(y,x)〉≥Kn∥x-y∥2,u∈M(f1y,f2y,…,fny),v∈M(f1x,f2x,…,fnx),
where Kn=(α1+α3+⋯+αn-1)-(β2+β4+⋯+βn).Proof.
Settingω1∈M(f1x,f2y,…,fny), ω2∈M(f1x,f2x,…,fny),…, ωn-1∈M(f1x,…,fn-1x,fny). From Definition 10, we have
(18)〈u-v,η(y,x)〉≥〈u-ω1,η(y,x)〉+〈ω1-ω2,η(y,x)〉+⋯+〈ωn-1-v,η(y,x)〉≥α1∥x-y∥2-β2∥x-y∥2+⋯+αn-1∥x-y∥2-βn∥x-y∥2=Kn∥x-y∥2,
where Kn=(α1+α3+⋯+αn-1)-(β2+β4+⋯+βn).
This completes the proof.Theorem 14.
LetE be a Banach space with the dual space E*. Let n≥3 and fi:E→Ei, i=1,2,…,n, η:E×E→E single-valued mappings, Cn:E→E* a η-monotone mapping, and M:∏i=1nEi⇉E* a Cn-η-monotone mapping. Then, (Cn+λM(f1,f2,…,fn))-1  is a single-valued mapping.Proof.
Suppose, on the contrary, that there existsx1,x2∈E,y*∈E*, such that
(19)x1=(Cn+λM(f1,f2,…,fn))-1(y*),x2=(Cn+λM(f1,f2,…,fn))-1(y*);
then
(20)λ-1(y*-Cn(x1))∈M(f1,f2,…,fn)(x1),λ-1(y*-Cn(x2))∈M(f1,f2,…,fn)(x2).
Now, by using Lemma 13 and since Cn is a η-monotone mapping, we have
(21)0=λ〈λ-1(y*-Cn(x1))-λ-1(y*-Cn(x2)),η(x1,x2)〉+〈Cn(x1)-Cn(x2),η(x1,x2)〉≥λKn∥x1-x2∥2.
Thus, we have x1=x2, which implies that (Cn+λM(f1,f2,…,fn))-1 is single valued. This completes the proof.By Theorem14, we can define the proximal mapping RM(f1,f2,…,fn)Cn,λ,η as follows.Definition 15.
LetE be a Banach space with the dual space E*. Let n≥3 and fi:E→Ei, i=1,2,…,n, be single-valued mappings, Cn:E→E* a η-monotone mapping, and M:∏i=1nEi⇉E* a Cn-η-monotone mapping. A proximal mapping RM(f1,f2,…,fn)Cn,λ,η:E*→E is defined by
(22)RM(f1,f2,…,fn)Cn,λ,η(x*)=(Cn+λM(f1,f2,…,fn))-1(x*),∀x*∈E*.Theorem 16.
LetE be a Banach space with the dual space E*. Let η:E×E→E be a k-Lipschitz continuous mapping. Let n≥3 and fi:E→Ei, i=1,2,…,n, be single-valued mappings, Cn:E→E* a η-monotone mapping, and M:∏i=1nEi⇉E* a Cn-η-monotone mapping. Then, the proximal mapping RM(f1,f2,…,fn)Cn,λ,η:E*→E is k/λKn-Lipschitz continuous, where Kn=(α1+α3+⋯+αn-1)-(β2+β4+⋯+βn).Proof.
Letx*,y*∈E* be any given points. It follows from Definition 15 that
(23)RM(f1,f2,…,fn)Cn,λ,η(x*)=(Cn+λM(f1,f2,…,fn))-1(x*),RM(f1,f2,…,fn)Cn,λ,η(y*)=(Cn+λM(f1,f2,…,fn))-1(y*).
Setting
(24)x=RM(f1,f2,…,fn)Cn,λ,η(x*),y=RM(f1,f2,…,fn)Cn,λ,η(y*)x,y∈E.
This implies that
(25)1λ(x*-Cn(x))∈M(f1,f2,…,fn)(x);1λ(y*-Cn(y))∈M(f1,f2,…,fn)(y).
By using Lemma 13, we have
(26)〈1λ(x*-Cn(x))-1λ(y*-Cn(y)),η(x,y)〉≥Kn∥x-y∥2;
since η is K-Lipschitz continuous, we have
(27)λKn∥x-y∥2≤〈x*-y*,η(x,y)〉≤k∥x*-y*∥∥x-y∥;
thus
(28)∥x-y∥≤kλKn∥x*-y*∥;
that is,
(29)∥RM(f1,f2,…,fn)Cn,λ,η(x*)-RM(f1,f2,…,fn)Cn,λ,η(x*)∥≤kλKn∥x*-y*∥,∀x*,y*∈E*,
where Kn=(α1+α3+⋯+αn-1)-(β2+β4+⋯+βn).
This completes the proof.


## 4. System of Variational Inclusions: Iterative Algorithm

Letn≥3 and A:E→E*, p:E→E, fi:E→Ei, i=1,2,…,n, F:∏i=1nEi→E* be single-valued mappings and Ti:E⇉CB(Ei), i=1,2,…,n, M:∏i=1nEi⇉E* be multivalued mappings. We will study the following variational inclusion problem: for any given a∈E*, find x∈E, t1∈T1(x), t2∈T2(x),…,tn∈Tn(x), such that
(30)a∈A(x-p(x))+M(f1(x),f2(x),…,fn(x))-F(t1,t2,…,tn).We remark that problem (30) includes as special cases many kinds of variational inclusion and variational inequality of [4, 5, 10, 12, 13].Theorem 17.
Letn≥3 and A:E→E*, p:E→E, fn:E→Ei, i=1,2,…,n, F:∏i=1nEi→E* be single-valued mappings and let Ti:E⇉CB(Ei), i=1,2,…,n, be multivalued mappings. Let Cn:E→E* be a η-monotone mapping and M:∏i=1nEi⇉E* a Cn-η-monotone mapping with respect to f1,f2,…,fn. Then, (x,t1,t2,…,tn) is a solution of problem (30) if and only if
(31)x=RM(f1,f2,…,fn)Cn,λ,η[Cn(x)-λA(x-p(x))+λa+λF(t1,t2,…,tn)],
where t1∈T1(x), t2∈T2(x),…,tn∈Tn(x), and λ>0 is a constant.Proof.
Let(x,t1,t2,…,tn) be a solution of problem (30); then we have
(32)a∈A(x-p(x))+M(f1(x),f2(x),…,fn(x))-F(t1,t2,…,tn);
then
(33)λa+λF(t1,t2,…,tn)-λA(x-p(x))∈λM(f1,f2,…,fn)(x),λ>0;
thus
(34)Cn(x)-λA(x-p(x))+λa+λF(t1,t2,…,tn)∈Cn(x)+λM(f1,f2,…,fn)(x).
Setting x*=Cn(x)-λA(x-p(x))+λa+λF(t1,t2,…,tn), from the definition of RM(f1,f2,…,fn)Cn,λ,η, we have
(35)x=RM(f1,f2,…,fn)Cn,λ,η(x*)=RM(f1,f2,…,fn)Cn,λ,η×[Cn(x)-λA(x-p(x))+λa+λF(t1,t2,…,tn)].
Conversely, let x=RM(f1,f2,…,fn)Cn,λ,η[Cn(x)-λA(x-p(x))+λa+λF(t1,t2,…,tn)]; then
(36)Cn(x)-λA(x-p(x))+λa+λF(t1,t2,…,tn)∈(Cn+λM(f1,f2,…,fn))(x);
thus we have
(37)a∈A(x-p(x))+M(f1(x),f2(x),…,fn(x))-F(t1,t2,…,tn).
This completes the proof.Based on Theorem17, we construct the following iterative algorithm for solving problem (30).Iterative Algorithm  1
For any givenx0∈E, we choose t1,0∈T1(x0),t2,0∈T2(x0),…,tn,0∈Tn(x0) and compute {xm},{t1,m},{t2,m},…,{tn,m} by iterative schemes
(38)xm+1=RM(f1,f2,…,fn)Cn,λ,η[Cn(xm)-λA(xm-p(xm))+λa+λF(t1,m,t2,m,…,tn,m)];t1,m∈T1(xm),∥t1,m+1-t1,m∥≤(1+1m+1)×H(T1(xm+1),T1(xm));t2,m∈T2(xm),∥t2,m+1-t2,m∥≤(1+1m+1)×H(T2(xm+1),T2(xm));⋮tn,m∈Tn(xm),∥tn,m+1-tn,m∥≤(1+1m+1)×H(Tn(xm+1),Tn(xm)),
for all m=0,1,2,….Now, we give some sufficient conditions which guarantee the convergence of iterative sequences generated by Algorithm4.1.Theorem 18.
LetE be a q-uniformly smooth Banach space with q>1 and E* the dual space of E. Let η:E×E→E  k-Lipschitz continuous. Let n≥3 and fi:E→Ei, i=1,2,…,n, be single-valued mappings, Cn:E→E* a η-monotone and δ-Lipschitz continuous mapping, p:E→E a (γ,μ)-relaxed cocoercive and λp-Lipschitz continuous mapping, and M:∏i=1nEi⇉E* a Cn-η-monotone mapping. Let A:E→E* be a τ-Lipschitz continuous mapping and, for each i=1,2,…,n, let Ti:E⇉CB(Ei) be H-Lipschitz continuous with constant λti. Suppose that F:∏i=1nEi→E* is λFi-Lipschitz continuous in the ith argument with respect to Ti(i=1,2,…,n) and the following condition is satisfied:
(39)0<kλKn×[δ+λτ(1+qγλpq-qμ+cqλpq)1/q+λ∑i=1n‍λFiλti]<1,
where
(40)Kn=(α1+α3+⋯+αn-1)-(β2+β4+⋯+βn).Then, the iterative sequences{xm},{t1,m},{t2,m},…,{tn,m} generated by Algorithm 4.1 converge strongly to x,t1,t2,…,tn, respectively, and (x,t1,t2,…,tn) is a solution of problem (30).Proof.
By using Algorithm4.1 and Theorem 16, we have
(41)∥xm+1-xm∥=∥RM(f1,f2,…,fn)Cn,λ,η[Cn(xm)-λA(xm-p(xm))+λa+λF(t1,m,t2,m,…,tn,m)]-RM(f1,f2,…,fn)Cn,λ,η[Cn(xm-1)-λA(xm-1-p(xm-1))+λa+λF(t1,m-1,t2,m-1,…,tn,m-1)]RM(f1,f2,…,fn)Cn,λ,η∥≤kλKn×∥Cn(xm)-λA(xm-p(xm))+λa+λF(t1,m,t2,m,…,tn,m)-Cn(xm-1)+λA(xm-1-p(xm-1))-λa-λF(t1,m-1,t2,m-1,…,tn,m-1)∥≤kλKn×(-F(t1,m-1,t2,m-1,…,tn,m-1)∥∥Cn(xm)-Cn(xm-1)∥+λ∥A(xm-p(xm))-A(xm-1-p(xm-1))∥+λ∥F(t1,m,t2,m,…,tn,m)-F(t1,m-1,t2,m-1,…,tn,m-1)∥).
From the Lipschitz continuity of Cn,p,A, and (γ,μ)-relaxed cocoercivity of p and Lemma 4, we have
(42)∥Cn(xm)-Cn(xm-1)∥≤δ∥xm-xm-1∥,(43)∥A(xm-p(xm))-A(xm-1-p(xm-1))∥≤τ∥xm-xm-1-(p(xm)-p(xm-1))∥,(44)∥xm-xm-1-(p(xm)-p(xm-1))∥q≤∥xm-xm-1∥q-q〈p(xm)-p(xm-1),jq(xm-xm-1)〉+cq∥p(xm)-p(xm-1)∥q≤∥xm-xm-1∥q+qγ∥p(xm)-p(xm-1)∥q-qμ∥xm-xm-1∥q+cq∥p(xm)-p(xm-1)∥q≤∥xm-xm-1∥q+qγλpq∥xm-xm-1∥q-qμ∥xm-xm-1∥q+cqλpq∥xm-xm-1∥q=(1+qγλpq-qμ+cqλpq)∥xm-xm-1∥q,
where jq:E→2E* is the normalized duality mapping.
SinceF,T1,T2,…,Tn are Lipschitz continuous, we have
(45)∥F(t1,m,t2,m,…,tn,m)-F(t1,m-1,t2,m-1,…,tn,m-1)∥≤∥F(t1,m,t2,m,…,tn,m)-F(t1,m-1,t2,m,…,tn,m)∥+∥F(t1,m-1,t2,m,t3,m,…,tn,m)-F(t1,m-1,t2,m-1,t3,m,…,tn,m)∥+⋯+∥F(t1,m-1,t2,m-1,…,tn-1,m-1,tn,m)-F(t1,m-1,t2,m-1,…,tn-1,m-1,tn,m-1)∥≤λF1∥t1,m-t1,m-1∥+λF2∥t2,m-t2,m-1∥+⋯+λFn∥tn,m-tn,m-1∥≤λF1(1+1m)H(T1(xm),T1(xm-1))+λF2(1+1m)H(T2(xm),T2(xm-1))+⋯+λFn(1+1m)H(Tn(xm),Tn(xm-1))≤λF1λt1(1+1m)∥xm-xm-1∥+λF2λt2(1+1m)∥xm-xm-1∥+⋯+λFnλtn(1+1m)∥xm-xm-1∥=∑i=1n‍λFiλti(1+1m)∥xm-xm-1∥.
It follows from (41)–(45) that
(46)∥xm+1-xm∥≤θm∥xm-xm-1∥,
where
(47)θm=kλKn×[δ+λτ(1+qγλpq-qμ+cqλpq)1/q+λ(1+1m)∑i=1n‍λFiλti].
Letting m→∞, we obtain θm→θ, where
(48)θ=kλKn×[δ+λτ(1+qγλpq-qμ+cqλpq)1/q+λ∑i=1n‍λFiλti].
From condition (39), we know that 0<θ<1, and hence {xm} is a Cauchy sequence in E. Thus, there exists x∈E such that xm→x, as m→∞. Now, we prove that t1,m→t1∈T1(x). In fact, it follows from the Lipschitz continuity of T1 and Algorithm 4.1 that
(49)∥t1,m-t1,m-1∥≤(1+1m)H(T1(xm),T1(xm-1))≤(1+1m)λt1∥xm-xm-1∥.
From (49), we know that {t1,m} is also a Cauchy sequence. In a similar way, {t2,m},{t3,m},…,{tn,m} are Cauchy sequences. Thus, there exist t1∈E1, t2∈E2,…,tn∈En such that t1,m→t1, t2,m→t2,…,tn,m→tn, as m→∞. Furthermore,
(50)d(t1,T1(x))≤∥t1-t1,m∥+d(t1,m,T1(x))≤∥t1-t1,m∥+H(T1(xm),T1(x))≤∥t1-t1,m∥+λt1∥xm-x∥⟶0,asm⟶∞.
Since T1(x) is closed, we have t1∈T1(x). In a similar way, we can show that t2∈T2(x), t3∈T3(x),…,tn∈Tn(x). By continuity of A,p,Cn,F,T1,T2,…,Tn,RM(f1,f2,…,fn)Cn,λ,η and Algorithm 4.1, we have
(51)x=RM(f1,f2,…,fn)Cn,λ,η[Cn(x)-λA(x-p(x))+λa+λF(t1,t2,…,tn)].
By Theorem 17, (x,t1,t2,…,tn) is a solution of problem (30). This completes the proof.


## 5. Conclusions

The purpose of this paper is to study a new monotone mapping in Banach spaces, which generalizes theCn-monotone mapping in [6], and generalizes the concepts of many monotone mappings. Moreover, the result of Theorem 18 improves and generalizes the corresponding results of [4–6, 10, 12, 13].



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*Source: 654537-2013-08-07.xml*