# (H,ϕ)-η-Accretive Mappings and a New System of Generalized Variational Inclusions with (H,ϕ)-η-Accretive Mappings in Banach Spaces

Authors: Sayyedeh Zahra Nazemi
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

We introduce a new class of generalized accretive mappings, named(H,ϕ)-η-accretive mappings, in Banach spaces. We define a resolvent operator associated with (H,ϕ)-η-accretive mappings and show its Lipschitz continuity. We also introduce and study a new system
of generalized variational inclusions with (H,ϕ)-η-accretive mappings in Banach spaces. By using
the resolvent operator technique associated with (H,ϕ)-η-accretive mappings, we construct a new
iterative algorithm for solving this system of generalized variational inclusions in Banach spaces. We
also prove the existence of solutions for the generalized variational inclusions and the convergence
of iterative sequences generated by algorithm. Our results improve and generalize many known
corresponding results.


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## Body

## 1. Introduction

Variational inequalities and variational inclusions are among the most interesting and important mathematical problems and have been studied intensively in the past years since they have wide applications in mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium, and engineering sciences, and so forth (see, e.g., [1–4]).Recently Ding [5], Huang and Fang [6], Verma [7], Fang and Huang [8, 9], Huang and Fang [10], Fang et al. [11], Kazmi and Khan [12], and Lan et al. [13, 14] introduced the concepts of η-subdifferential operators, maximal η-monotone operators, A-monotone operators, and (H,η)-monotone operators in Hilbert spaces, H-accretive operators, generalized m-accretive mappings, (H,η)-accretive operators, P-η-accretive operators, and (A,η)-accretive mappings in Banach spaces, and their resolvent operators, respectively. In [15], Luo and Huang introduced a new concept of (H,ϕ)-η-monotone mappings in Banach spaces and defined the proximal mapping associated with (H,ϕ)-η-monotone mappings.Motivated and inspired by the research work going on this field, in this paper, we introduce a new concept of(H,ϕ)-η-accretive mappings and give the definition of its resolvent operator in Banach spaces. We also introduce and study a new system of generalized variational inclusions with (H,ϕ)-η-accretive mappings in Banach spaces, and we construct a new iterative algorithm for solving this system of generalized variational inclusions in Banach spaces. We also prove the existence of solutions for the generalized variational inclusions and the convergence of iterative sequences generated by algorithm. The results in this paper improve and extend some known results in the literature.


## 2. Preliminaries

LetE be a real Banach space equipped with norm ∥·∥; let E* be the topological dual space of E; let 〈·,·〉 be the pair between E and E*; let 2E be the power set of E; let H̃(·,·) be the Hausdorff metric on CB(E) defined by(2.1)H̃(A,B)=max⁡{sup⁡x∈A⁡d(x,B),sup⁡y∈B⁡d(A,y)},∀A,B∈CB(E).Definition 2.1 (see [16, 17]).
Forq>1, a mapping Jq:E→2E* is said to be generalized duality mapping if it is defined by
(2.2)Jq(x)={f∈E*:〈x,f〉=‖x‖q,‖x‖q-1=‖f‖},∀x∈E.
In particular,J2 is the usual normalized duality mapping on E.
It is well known that(2.3)Jq(x)=‖x‖q-1J2(x),∀x(≠0)∈E.
Note that ifE=ℋ is a real Hilbert space, then J2 becomes the identity mapping on ℋ.Definition 2.2 (see [18]).
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 E is the function ρE:[0,∞)→[0,∞), defined by
(2.4)ρE(τ)=sup⁡{‖x+y‖+‖x-y‖2-1:x,y∈E,‖x‖=1,‖y‖=τ}.Definition 2.3 (see [17]).
The Banach spaceE is said to be(i)
uniformly smooth, if(2.5)lim⁡τ→0ρE(τ)τ=0;(ii)
q-uniformly smooth, for q>1, if there exists a constant τ>0 such that
(2.6)ρE(τ)≤cτq,τ∈[0,∞).
It is well known (see [16]) that(2.7)Lq(orlq)is{q-uniformlysmooth,if1<q≤2,2-uniformly smooth,ifq≥2.
Note that ifE is uniformly smooth, Jq becomes single-valued, and Jq is single-valued if E* is strictly convex. In the sequel, unless otherwise specified, we always suppose that E is a real Banach space such that Jq is single-valued.Lemma 2.4 (see [19]).
Letq>1 be a real number and let E be a smooth Banach space. Then E is q-uniformly smooth if and only if there exists a constant cq>0 such that for every x,y∈E,
(2.8)‖x+y‖q≤‖x‖q+q〈y,Jq(x)〉+cq‖y‖q.Definition 2.5 (see [9, 20]).
LetP:E→E be a single-valued mapping. P is said to be(i)
accretive if(2.9)〈P(x)-P(y),Jq(x-y)〉≥0,iffx,y∈E;(ii)
strictly accretive if P is accretive and(2.10)〈P(x)-P(y),Jq(x-y)〉=0ifx=y;(iii)
r-strongly accretive if there exists a constant r>0 such that
(2.11)〈P(x)-P(y),Jq(x-y)〉≥r‖x-y‖q,∀x,y∈E;(iv)
m-relaxed accretive if there exists a constant m>0 such that
(2.12)〈P(x)-P(y),Jq(x-y)〉≥-m‖x-y‖q,∀x,y∈E;(v)
(α,ξ)-relaxed cocoercive if there exist constants α,ξ>0 such that
(2.13)〈P(x)-P(y),Jq(x-y)〉≥-α‖P(x)-P(y)‖q+ξ‖x-y‖q,∀x,y∈E;(vi)
s-Lipschitz continuous if there exists a constant s>0 such that
(2.14)‖P(x)-P(y)‖≤s‖x-y‖,∀x,y∈E.Definition 2.6 (see [12]).
A mappingη:E×E→E is said to be τ-Lipschitz continuous if there exists a constant τ>0 such that
(2.15)‖η(x,y)‖≤τ‖x-y‖,∀x,y∈E.Definition 2.7 (see [20]).
Letη:E×E→E and P:E→E be single-valued mappings. Then a multi-valued mapping T:E→2E is said to be(i)
η-accretive if
(2.16)〈u-v,Jq(η(x,y))〉≥0,∀x,y∈E,∀u∈T(x),v∈T(y);(ii)
strictlyη-accretive if
(2.17)〈u-v,Jq(η(x,y))〉≥0,∀x,y∈E,∀u∈T(x),v∈T(y),
and equality holds if and only if x=y;(iii)
γ-strongly η-accretive if there exists a constant γ>0 such that
(2.18)〈u-v,Jq(η(x,y))〉≥γ‖x-y‖q,∀x,y∈E,∀u∈T(x),v∈T(y);(iv)
m-relaxedη-accretive if there exists a constant m>0 such that
(2.19)〈u-v,Jq(η(x,y))〉≥-m‖x-y‖q,∀x,y∈E,∀u∈T(x),v∈T(y);(v)
η-m-accretive if T is η-accretive and (I+ρT)(E)=E holds for all ρ>0;(vi)
P-η-accretive if T is η-accretive and (P+ρT)(E)=E holds for all ρ>0.Definition 2.8.
LetE1,E2,…,En be real Banach spaces, and, for i=1,2,…,n, let Ni:∏j=1nEj→Ei be a single-valued mapping. Then Ni is said to be ξij-Lipschitz continuous in the jth argument, if there exists a constant ξij>0 such that
(2.20)‖Ni(x1,…,xj-1,yj1,xj+1,…,xn)-Ni(x1,…,xj-1,yj2,xj+1,…,xn)‖≤ξij‖yj1-yj2‖,∀yj1,yj2∈Ej,xi∈Ei(i∈{1,2,…,n},i≠j).Definition 2.9.
LetA:E→CB(E) be a set-valued mapping. A is said to be H̃-Lipschitz continuous, if there exists a constant t>0 such that
(2.21)H̃(A(x),B(y))≤t‖x-y‖,∀x,y∈E,
where H̃(·,·) denotes the Hausdorff metric on CB(E).Definition 2.10.
Letℋ be a Hilbert space, and let H:ℋ→ℋ be a single-valued mapping. H is said to be(i)
coercive if(2.22)lim⁡‖x‖→∞〈H(x),x〉‖x‖=+∞;(ii)
hemicontinuous if for any fixedx,y,z∈ℋ, the function t→〈H(x+ty),z〉 is continuous at 0+.


## 3.(H,ϕ)-η-Accretive Mappings

In this section, we will introduce a new class of generalized accretive mappings,(H,ϕ)-η-accretive mappings, and discuss some properties of (H,ϕ)-η-accretive mappings.Definition 3.1.
LetE be a Banach space and let H,ϕ:E→E, η:E×E→E be single-valued mappings and M:E→2E a multi-valued mapping. The mapping M is said to be a (H,ϕ)-η-accretive mapping, if ϕ∘M is η-accretive and (H+ϕ∘M)(E)=E.Remark 3.2.
(i) IfM is η-accretive and ϕ(x)=λx, for all x∈E, λ>0, then (H,ϕ)-η-accretive mapping reduces to the P-η-accretive mapping studied by Kazmi and Khan [12].
(ii) Ifη(x,y)=x-y, for all x,y∈E, M is η-accretive, and ϕ(x)=λx, for all x∈E, λ>0, then (H,ϕ)-η-accretive mapping reduces to the H-accretive mapping studied by Fang and Huang [9].
(iii) IfM is m-relaxed η-accretive and ϕ(x)=λx, for all x∈E, λ>0, then (H,ϕ)-η-accretive mapping reduces to the (A,η)-accretive mapping studied by Lan et al. [14].
Similarly, we give the following definition.Definition 3.3.
LetH,ϕ:E→E be single-valued mappings and M:E→2E a multi-valued mapping. The mapping M is said to be a (H,ϕ)-accretive mapping, if ϕ∘M is accretive and (H+ϕ∘M)(E)=E.Example 3.4.
Letℋ be a Hilbert space and for every x,y∈ℋ, η(x,y)=x-y, ϕ(x)=λx, where λ>0 is a constant. Let M:ℋ→2ℋ be a maximal monotone mapping and H:ℋ→ℋ a bounded, coercive, hemicontinuous, and α-strongly η-accretive mapping. Then it follows from Corollary  32.26 of [21] that M is (H,ϕ)-η-accretive mapping.Theorem 3.5.
Letϕ:E→E, η:E×E→E be single-valued mappings, H:E→E a strictly η-accretive mapping, and M:E→2E a (H,ϕ)-η-accretive mapping. Then (H+ϕ∘M)-1 is a single-valued mapping.Proof.
For any givenz∈E, let x,y∈(H+ϕ∘M)-1(z). It follows that
(3.1)z-H(x)∈ϕ∘M(x),z-H(y)∈ϕ∘M(y).
Then η-accretivity of ϕ∘M implies that
(3.2)〈z-H(x)-(z-H(y)),Jq(η(x,y))〉=〈-H(x)+H(y),Jq(η(x,y))〉≥0.
This implies that x=y and so (H+ϕ∘M)-1 is a single-valued mapping. This completes the proof.By Theorem3.5, we can define the resolvent operator JM,ϕH,η associated with an (H,ϕ)-η-accretive mapping M as follows.Definition 3.6.
Letϕ:E→E, η:E×E→E be single-valued mappings, let H:E→E be a strictly η-accretive mapping, and let M:E→2E be a (H,ϕ)-η-accretive mapping. A resolvent operator JM,ϕH,η:E→E is defined by
(3.3)JM,ϕH,η(x)=(H+ϕ∘M)-1(x),∀x∈E.Remark 3.7.
(i) IfM is η-accretive and ϕ(x)=λx, for all x∈E, λ>0, then the resolvent operator JM,ϕH,η reduces to the P-η-proximal point mapping Jη,ρM introduced by Kazmi and Khan [12].
(ii) Ifη(x,y)=x-y, for all x,y∈E, M is η-accretive, and ϕ(x)=λx, for all x∈E, λ>0, then the resolvent operator JM,ϕH,η reduces to the proximal-point mapping introduced by Fang and Huang [9].
(iii) IfM is m-relaxed η-accretive and ϕ(x)=λx, for all x∈E, λ>0, then the resolvent operator JM,ϕH,η reduces to the resolvent operator Rη,Mρ,A introduced by Lan at el. [14].Theorem 3.8.
Letϕ:E→E be a single-valued mapping, let η:E×E→E be a τ-Lipschitz continuous mapping, let H:E→E be a γ-strongly η-accretive mapping, and let M:E→2E be a (H,ϕ)-η-accretive mapping. Then the resolvent operator JM,ϕH,η is Lipschitz continuous with constant τq-1/γ, that is,
(3.4)‖JM,ϕH,η(x)-JM,ϕH,η(y)‖≤τq-1γ‖x-y‖,∀x,y∈E.Proof.
Letx,y be any given points in E, it follows from Definition 3.6 that
(3.5)JM,ϕH,η(x)=(H+ϕ∘M)-1(x),JM,ϕH,η(y)=(H+ϕ∘M)-1(y).
This implies that
(3.6)x-H(JM,ϕH,η(x))∈ϕ∘M(JM,ϕH,η(x)),y-H(JM,ϕH,η(y))∈ϕ∘M(JM,ϕH,η(y)).
Since M is (H,ϕ)-η-accretive, we have
(3.7)〈x-H(JM,ϕH,η(x))-(y-H(JM,ϕH,η(y))),Jq(η(JM,ϕH,η(x),JM,ϕH,η(y)))〉=〈x-y-(H(JM,ϕH,η(x))-H(JM,ϕH,η(y))),Jq(η(JM,ϕH,η(x),JM,ϕH,η(y)))〉≥0.
The inequality above implies that
(3.8)‖x-y‖‖η(JM,ϕH,η(x),JM,ϕH,η(y))‖q-1=‖x-y‖‖Jq(η(JM,ϕH,η(x),JM,ϕH,η(y)))‖≥〈x-y,Jq(η(JM,ϕH,η(x),JM,ϕH,η(y)))〉≥〈H(JM,ϕH,η(x))-H(JM,ϕH,η(y)),Jq(η(JM,ϕH,η(x),JM,ϕH,η(y)))〉≥γ‖JM,ϕH,η(x)-JM,ϕH,η(y)‖q.
Since η is Lipschitz continuous with a constant τ, we have
(3.9)‖η(JM,ϕH,η(x),JM,ϕH,η(y))‖≤τ‖JM,ϕH,η(x)-JM,ϕH,η(y)‖.
It follows from (3.8) and (3.9) that
(3.10)‖x-y‖τq-1‖JM,ϕH,η(x)-JM,ϕH,η(y)‖q-1≥γ‖JM,ϕH,η(x)-JM,ϕH,η(y)‖q.
Hence, we get
(3.11)‖JM,ϕH,η(x)-JM,ϕH,η(y)‖≤τq-1γ‖x-y‖.
This completes the proof.


## 4. A New System of Generalized Variational Inclusions

In this section, we will introduce a new system of generalized variational inclusions with(H,ϕ)-η-accretive mappings and construct a new iterative algorithm for solving this system of generalized variational inclusions. In what follows, for each i=1,2,3, suppose that Ei is a Banach space, Hi,fi,ϕi,pi:Ei→Ei, ηi:Ei×Ei→Ei, Fi,Ni:∏k=13Ek→Ei are single-valued mappings, Tki:Ek→CB(Ei) is a set-valued mapping, and Mi:Ei×Ei→2Ei is a (Hi,ϕi)-ηi-accretive mapping in the second argument. Assume that fi(Ei)∩domM(wi,·)≠∅, for each wi∈Ei. We consider the following system of generalized variational inclusions. Find (x1,x2,x3,u11,u12,u13,u21,u22,u23,u31,u32,u33) such that for each i=1,2,3, xi∈Ei, u1i∈T1i(x1), u2i∈T2i(x2), u3i∈T3i(x3), and(4.1)0∈Fi(x1-p1(x1),x2-p2(x2),x3-p3(x3))+Ni(ui1,ui2,ui3)+Mi(xi,fi(xi)).
The following are some special cases of problem (4.1).(i) IfE1=ℋ1, E2=ℋ2, and E3=ℋ3 are three Hilbert spaces, and, for each i=1,2,3, pi=0, Ni=0, M1(x,f1(x))=∂η1φ1(x), for all x∈E1, M2(y,f2(y))=∂η2φ2(y), for all y∈E2, and M3(z,f3(z))=∂η3φ3(z), for all z∈E3, where φi:Ei→R∪{∞} is a proper lower semicontinuous and ηi-subdifferential function, ∂η1φ1(x) is the η1-subdifferential of φ1 at x, ∂η2φ2(y) is the η2-subdifferential of φ2 at y, and ∂η3φ3(z) is the η3-subdifferential of φ3 at z, then SGVIP (4.1) reduces to the following system of variational inequalities, which is to find (x1,x2,x3)∈E1×E2×E3 such that(4.2)〈F1(x1,x2,x3),η1(a,x1)〉+φ1(a)-φ1(x1)≥0,∀a∈E1,〈F2(x1,x2,x3),η2(b,x2)〉+φ2(b)-φ2(x2)≥0,∀b∈E2,〈F3(x1,x2,x3),η3(c,x3)〉+φ3(c)-φ3(x3)≥0,∀c∈E3.Ifη1(a,x)=a-x, for all a,x∈E1, η2(b,y)=b-y, for all b,y∈E2, η3(c,z)=c-z, for all c,z∈E3, and M1(x,f1(x))=∂φ1(x) is the subdifferential of φ1 at x, M2(y,f2(y))=∂φ2(y) is the subdifferential of φ2 at y, and M3(z,f3(z))=∂φ3(z) is the subdifferential of φ3 at z, then problem (4.2) reduces to the following system of variational inequalities, which is to find (x1,x2,x3)∈E1×E2×E3 such that(4.3)〈F1(x1,x2,x3),a-x1〉+φ1(a)-φ1(x1)≥0,∀a∈E1,〈F2(x1,x2,x3),b-x2〉+φ2(b)-φ2(x2)≥0,∀b∈E2,〈F3(x1,x2,x3),c-x3〉+φ3(c)-φ3(x3)≥0,∀c∈E3.IfM1(x,f1(x))=∂δk1(x), M2(y,f2(y))=∂δk2(y), and M3(z,f3(z))=∂δk3(z), for all x∈K1, y∈K2, and z∈K3, where K1⊂E1, K2⊂E2, and K3⊂E3 are three nonempty, closed, and convex subsets, δk1, δk2, and δk3 denote the indicator functions of K1, K2, and K3, respectively, then problem (4.3) reduces to the following system of variational inequalities, which is to find (x1,x2,x3)∈E1×E2×E3 such that(4.4)〈F1(x1,x2,x3),a-x1〉≥0,∀a∈K1,〈F2(x1,x2,x3),b-x2〉≥0,∀b∈K2,〈F3(x1,x2,x3),c-x3〉≥0,∀c∈K3.IfE1=E2=E3=ℋ is a Hilbert space, K1=K2=K3=K is a nonempty, closed, and convex subset, F1(x,y,z)=sT1(y,z,x)+x-y, F2(x,y,z)=tT2(z,x,y)+y-z, and F3(x,y,z)=rT3(x,y,z)+z-x, for all x,y,z∈K, where T1,T2,T3:K×K×K→ℋ are mappings on K×K×K, r,s,t>0 are three numbers, then problem (4.4) reduces to the following system of variational inequalities, which is to find x1,x2,x3∈K such that(4.5)〈sT1(x2,x3,x1)+x1-x2,a-x1〉≥0,∀a∈K,〈tT2(x3,x1,x2)+x2-x3,a-x2〉≥0,∀a∈K,〈rT3(x1,x2,x3)+x3-x1,a-x3〉≥0,∀a∈K.
Problem (4.5) was introduced and studied by Cho and Qin [22].(ii) IfE2=E3, F1(·,·,·)=F1(·,·), F2(·,·,·)=F2(·,·), and F2=F3 and for each i=1,2,3, pi=0, Ni=0, M2(y,f2(y))=M3(z,f3(z)), for all (y,z)∈E2×E3, then SGVIP (4.1) reduces to the following system of variational inclusions, which is to find (x1,x2)∈E1×E2 such that(4.6)0∈F1(x1,x2)+M1(x1,f1(x1)),0∈F2(x1,x2)+M2(x2,f2(x2)).IfE1=ℋ1 and E2=ℋ2 are two Hilbert spaces and M1(x,f1(x))=∂η1φ1(x), for all x∈E1, M2(y,f2(y))=∂η2φ2(y), for all y∈E2, then the problem (4.6) reduces to the following system of variational inequalities, which is to find (x1,x2)∈E1×E2 such that(4.7)〈F1(x1,x2),η1(a,x1)〉+φ1(a)-φ1(x1)≥0,∀a∈E1,〈F2(x1,x2),η2(b,x2)〉+φ2(b)-φ2(x2)≥0,∀b∈E2.Ifη1(a,x)=a-x, for all a,x∈E1, η2(b,y)=b-y, for all b,y∈E2, M1(x,f1(x))=∂φ1(x) is the subdifferential of φ1 at x, and M2(y,f2(y))=∂φ2(y) is the subdifferential of φ2 at y, then the problem (4.7) reduces to the following system of variational inequalities, which is to find (x1,x2)∈E1×E2 such that(4.8)〈F1(x1,x2),a-x1〉+φ1(a)-φ1(x1)≥0,∀a∈E1,〈F2(x1,x2),b-x2〉+φ2(b)-φ2(x2)≥0,∀b∈E2.
Problem (4.8) was introduced and studied by Cho et al. [23].IfM1(x,f1(x))=∂δk1(x), M2(y,f2(y))=∂δk2(y), for all x∈K1, y∈K2, where K1⊂E1, K2⊂E2 are two nonempty, closed, and convex subsets and ∂δk1 and ∂δk2 denote the indicator functions of K1, K2, respectively, then problem (4.8) reduces to the following system of variational inequalities, which is to find (x1,x2)∈E1×E2 such that(4.9)〈F1(x1,x2),a-x1〉≥0,∀a∈K1,〈F2(x1,x2),b-x2〉≥0,∀b∈K2.
Problem (4.9) is just the problem [24] with F1 and F2 being single-valued.IfE1=E2=ℋ is a Hilbert space and K1=K2=K is a nonempty, closed, and convex subset, F1(x,y)=ρT(y,x)+x-y and F2(x,y)=λT(x,y)+y-x, for all x,y∈K, where T:K×K→ℋ is a mapping on K×K, ρ,λ>0 are two numbers, then problem (4.9) reduces to the following problem, which is to find x1,x2∈K such that(4.10)〈ρT(x2,x1)+x1-x2,a-x1〉≥0,∀a∈K,〈λT(x1,x2)+x2-x1,a-x2〉≥0,∀a∈K.
Problem (4.10) was introduced and studied by Verma [25].Lemma 4.1.
Let, fori=1,2,3, ϕi:Ei→Ei be a single-valued mapping satisfying ϕi(x+y)=ϕi(x)+ϕi(y) and kerϕi={0}, ηi:Ei×Ei→Ei a single-valued mapping, Hi:Ei→Ei a strictly ηi-accretive mapping and Mi:Ei×Ei→2Ei a (Hi,ϕi)-ηi-accretive mapping in the second argument. Then (x1,x2,x3,u11,u12,u13,u21,u22,u23,u31,u32,u33) in which xi∈Ei, u1i∈T1i(x1), u2i∈T2i(x2), u3i∈T3i(x3)(i=1,2,3) is a solution of the problem (4.1) if and only if
(4.11)fi(xi)=JMi(xi,⋅),ϕiHi,ηi[Hi∘fi(xi)-ϕi∘Fi(x1-p1(x1),x2-p2(x2),x3-p3(x3))-ϕi∘Ni(ui1,ui2,ui3)],
where JMi(xi,·),ϕiHi,ηi=(Hi+ϕi∘Mi(xi,·))-1.Proof.
The fact directly follows from Definition3.6.Algorithm 4.2.
For anyxi0∈Ei, take u1i0∈T1i(x10), u2i0∈T2i(x20), and u3i0∈T3i(x30)(i=1,2,3). For i=1,2,3, let
(4.12)xi1=xi0-fi(xi0)+JMi(xi0,⋅),ϕiHi,ηi[Hi∘fi(xi0)-ϕi∘Fi(x10-p1(x10),x20-p2(x20),x30-p3(x30))-ϕi∘Ni(ui10,ui20,ui30)].
Since u1i0∈T1i(x10), u2i0∈T2i(x20), u3i0∈T3i(x30)(i=1,2,3), by Nodler's theorem [26], there exist u1i1∈T1i(x11), u2i1∈T2i(x21), u3i1∈T3i(x31)(i=1,2,3), such that, for each i=1,2,3,
(4.13)‖u1i1-u1i0‖≤(1+1)H̃(T1i(x11),T1i(x10)),‖u2i1-u2i0‖≤(1+1)H̃(T2i(x21),T2i(x20)),‖u3i1-u3i0‖≤(1+1)H̃(T3i(x31),T3i(x30)).
For i=1,2,3, let
(4.14)xi2=xi1-fi(xi1)+JMi(xi1,⋅),ϕiHi,ηi[Hi∘fi(xi1)-ϕi∘Fi(x11-p1(x11),x21-p2(x21),x31-p3(x31))-ϕi∘Ni(ui11,ui21,ui31)].
Again by Nodler's theorem [26], there exist u1i2∈T1i(x12), u2i2∈T2i(x22), u3i2∈T3i(x32)(i=1,2,3), such that, for each i=1,2,3,
(4.15)‖u1i2-u1i1‖≤(1+12)H̃(T1i(x12),T1i(x11)),‖u2i2-u2i1‖≤(1+12)H̃(T2i(x22),T2i(x21)),‖u3i2-u3i1‖≤(1+12)H̃(T3i(x32),T3i(x31)).
By induction, we can compute the sequences xin, u1in, u2in, u3in(i=1,2,3) by the following iterative schemes such that, for each i=1,2,3,
(4.16)xin+1=xin-fi(xin)+JMi(xin,⋅),ϕiHi,ηi[Hi∘fi(xin)-ϕi∘Fi(x1n-p1(x1n),x2n-p2(x2n),x3n-p3(x3n))-ϕi∘Ni(ui1n,ui2n,ui3n)],u1in∈T1i(x1n),‖u1in-u1in-1‖≤(1+1n)H̃(T1i(x1n),T1i(x1n-1)),u2in∈T2i(x2n),‖u2in-u2in-1‖≤(1+1n)H̃(T2i(x2n),T2i(x2n-1)),u3in∈T3i(x3n),‖u3in-u3in-1‖≤(1+1n)H̃(T3i(x3n),T3i(x3n-1)),
for all n=0,1,2,….Now we prove the existence of solution of the SGVIP (4.1) and the convergence of Algorithm 4.2.Theorem 4.3.
Let, fori=1,2,3, Ei be a q-uniformly smooth Banach space and let ϕi:Ei→Ei be a θi-Lipschitz continuous mapping satisfying ϕi(x+y)=ϕi(x)+ϕi(y) and kerϕi={0}. Let ηi:Ei×Ei→Ei be τi-Lipschitz continuous, let Hi:Ei→Ei be a γi-strongly ηi-accretive and si-Lipschitz continuous mapping, fi:Ei→Ei be (αi,μi)-relaxed cocoercive and ξi-Lipschitz continuous, and let pi:Ei→Ei be a strongly accretive mapping with constant δpi and Lipschitz continuous with constant λpi. Suppose that Fi:∏k=13Ek→Ei is βik-Lipschitz continuous in the kth argument and Ni:∏k=13Ek→Ei be ζik-Lipschitz continuous in the kth argument for k=1,2,3, Mi:Ei×Ei→2Ei is a (Hi,ϕi)-ηi-accretive mapping in the second argument, and set-valued mappings T1i:E1→CB(Ei), T2i:E2→CB(Ei), T3i:E3→CB(Ei) are H̃-Lipschitz continuous with constants t1i>0, t2i>0, t3i>0, respectively. In addition if for all i=1,2,3, one has
(4.17)‖JMi(xi1,⋅),ϕiHi,ηi(zi)-JMi(xi2,⋅),ϕiHi,ηi(zi)‖≤δi‖xi1-xi2‖,∀xi1,xi2,zi∈Ei,(4.18)0<(1+qα1ξ1q-qμ1+cqξ1q)1/q+δ1+τ1q-1γ1×(∑j=13(s1qξ1q+qθ1β11s1q-1ξ1q-1(1-qδp1+cqλp1q)1/q+cqθ1qβ11q(1-qδp1+cqλP1q))1/q+∑j=13θ1ζ1jt1j)+(θ2β21τ2q-1γ2+θ3β31τ3q-1γ3)(1-qδp1+cqλP1q)1/q<1,0<(1+qα2ξ2q-qμ2+cqξ2q)1/q+δ2+τ2q-1γ2×(∑j=13(s2qξ2q+qθ2β22s2q-1ξ2q-1(1-qδp2+cqλp2q)1/q+cqθ2qβ22q(1-qδp2+cqλP2q))1/q+∑j=13θ2ζ2jt2j)+(θ1β12τ1q-1γ1+θ3β32τ3q-1γ3)(1-qδp2+cqλP2q)1/q<1,0<(1+qα3ξ3q-qμ3+cqξ3q)1/q+δ3+τ3q-1γ3×(∑j=13(s3qξ3q+qθ3β33s3q-1ξ3q-1(1-qδp3+cqλp3q)1/q+cqθ3qβ33q(1-qδp3+cqλP3q))1/q+∑j=13θ3ζ3jt3j)+(θ1β13τ1q-1γ1+θ2β23τ2q-1γ2)(1-qδp3+cqλP3q)1/q<1.
Then the problem (4.1) admits a solution (x1,x2,x3,u11,u12,u13,u21,u22,u23,u31,u32,u33) and sequences x1n,x2n,x3n,u11n,u12n,u13n,u21n,u22n,u23n,u31n,u32n,u33n converge to x1,x2,x3, u11, u12, u13, u21, u22, u23, u31, u32, u33, respectively, where xin,u1in,u2in,u3in(i=1,2,3) are sequences generated by Algorithm 4.2.Proof.
Fori=1,2,3, let
(4.19)Din=Hi∘fi(xin)-ϕi∘Fi(x1n-p1(x1n),x2n-p2(x2n),x3n-p3(x3n))-ϕi∘Ni(ui1n,ui2n,ui3n).
By Algorithm 4.2 and (4.17), we have
(4.20)‖xin+1-xin‖=‖xin-fi(xin)+JMi(xin,⋅),ϕiHi,ηi[Hi∘fi(xin)-ϕi∘Fi(x1n-p1(x1n),x2n-p2(x2n),x3n-p3(x3n))-ϕi∘Ni(ui1n,ui2n,ui3n)]-(xin-1-fi(xin-1)+JMi(xin-1,⋅),ϕiHi,ηi[Hi∘fi(xin-1)-ϕi∘Fi(x1n-1-p1(x1n-1),x2n-1-p2(x2n-1),x3n-1-p3(x3n-1))-ϕi∘Ni(ui1n-1,ui2n-1,ui3n-1)])JMi(xin,⋅),ϕiHi,ηi‖≤‖xin-xin-1-[fi(xin)-fi(xin-1)]‖+‖JMi(xin,⋅),ϕiHi,ηi(Din)-JMi(xin,⋅),ϕiHi,ηi(Din-1)‖+‖JMi(xin,⋅),ϕiHi,ηi(Din-1)-JMi(xin-1,⋅),ϕiHi,ηi(Din-1)‖≤‖xin-xin-1-[fi(xin)-fi(xin-1)]‖+τiq-1γi‖Din-Din-1‖+δi‖xin-xin-1‖.
Since fi:Ei→Ei is (αi,μi)-relaxed cocoercive and ξi-Lipschitz continuous and pi:Ei→Ei is δpi-strongly accretive and λpi-Lipschitz continuous, we have
(4.21)‖xin-xin-1-[fi(xin)-fi(xin-1)]‖q≤‖xin-xin-1‖q-q〈fi(xin)-fi(xin-1),Jq(xin-xin-1)〉+cq‖fi(xin)-fi(xin-1)‖q≤(1+qαiξiq-qμi+cqξiq)‖xin-xin-1‖q,‖xin-xin-1-[pi(xin)-pi(xin-1)]‖q≤‖xin-xin-1‖q-q〈pi(xin)-pi(xin-1),Jq(xin-xin-1)〉+cq‖pi(xin)-pi(xin-1)‖q≤(1-qδpi+cqλpiq)‖xin-xin-1‖q.
From (4.19), we have
(4.22)‖Din-Din-1‖=‖-ϕi∘Ni(ui1n-1,ui2n-1,ui3n-1))Hi∘fi(xin)-ϕi∘Fi(x1n-p1(x1n),x2n-p2(x2n),x3n-p3(x3n))-ϕi∘Ni(ui1n,ui2n,ui3n)-(Hi∘fi(xin-1)-ϕi∘Fi(x1n-1-p1(x1n-1),x2n-1-p2(x2n-1),x3n-1-p3(x3n-1))-ϕi∘Ni(ui1n-1,ui2n-1,ui3n-1))‖≤‖Hi∘fi(xin)-Hi∘fi(xin-1)-(ϕi∘Fi(x1n-p1(x1n),x2n-p2(x2n),x3n-p3(x3n))-ϕi∘Fi(x1n-1-p1(x1n-1),x2n-1-p2(x2n-1),x3n-1-p3(x3n-1)))‖+‖ϕi∘Ni(ui1n,ui2n,ui3n)-ϕi∘Ni(ui1n-1,ui2n-1,ui3n-1)‖.
Since Fi:∏k=13Ek→Ei is βik-Lipschitz continuous in the kth argument and by continuity of Hi, fi, ϕi(i=1,2,3), we have
(4.23)‖Hi∘fi(xin)-Hi∘fi(xin-1)-(ϕi∘Fi(x1n-p1(x1n),x2n-p2(x2n),x3n-p3(x3n))-ϕi∘Fi(x1n-1-p1(x1n-1),x2n-1-p2(x2n-1),x3n-1-p3(x3n-1)))‖≤(siqξiq+qθiβiisiq-1ξiq-1(1-qδpi+cqλpiq)1/q+cqθiqβiiq(1-qδpi+cqλPiq))1/q‖xin-xin-1‖+∑j=1i-1θiβij(1-qδpj+cqλPjq)1/q‖xjn-xjn-1‖+∑j=i+13θiβij(1-qδpj+cqλPjq)1/q‖xjn-xjn-1‖.
It follows from the Lipschitz continuity of Ni and the H̃-Lipschitz continuity of Tij that
(4.24)‖Ni(ui1n,ui2n,ui3n)-Ni(ui1n-1,ui2n-1,ui3n-1)‖≤‖Ni(ui1n,ui2n,ui3n)-Ni(ui1n-1,ui2n,ui3n)‖+‖Ni(ui1n-1,ui2n,ui3n)-Ni(ui1n-1,ui2n-1,ui3n)‖+‖Ni(ui1n-1,ui2n-1,ui3n)-Ni(ui1n-1,ui2n-1,ui3n-1)‖≤ζi1‖ui1n-ui1n-1‖+ζi2‖ui2n-ui2n-1‖+ζi3‖ui3n-ui3n-1‖=∑j=13ζij‖uijn-uijn-1‖≤∑j=13ζij(1+1n)H̃(Tij(xin),Tij(xin-1))≤∑j=13ζijtij(1+1n)‖xin-xin-1‖,i=1,2,3.
It follows from (4.20)–(4.24) that, for each i=1,2,3,
(4.25)‖xin+1-xin‖≤[∑j=13θiζijtij(1+qαiξiq-qμi+cqξiq)1/q+δi+τiq-1γi(∑j=i+13(cqθiqβiiq(1-qδpi+cqλPiq)(1-qδpi+cqλpiq)1/qsiqξiq+qθiβiisiq-1ξiq-1(1-qδpi+cqλpiq)1/q+cqθiqβiiq(1-qδpi+cqλPiq)(1-qδpi+cqλpiq)1/q)1/q+∑j=13θiζijtij(1+1n))]×‖xin-xin-1‖+∑j=1i-1θiβijτiq-1γi(1-qδpj+cqλPjq)1/q‖xjn-xjn-1‖+∑j=i+13θiβijτiq-1γi(1-qδpj+cqλPjq)1/q‖xjn-xjn-1‖.
Therefore,
(4.26)∑i=13‖xin+1-xin‖≤∑i=13{[∑j=13θiζijtij(1+1n)(1+qαiξiq-qμi+cqξiq)1/q+δi+τiq-1γi(∑j=13(siqξiq+qθiβiisiq-1ξiq-1(1-qδpi+cqλpiq)1/q+cqθiqβiiq×(1-qδpi+cqλPiq)(1-qδpi+cqλpiq)1/q)1/q+∑j=13θiζijtij(1+1n))]×‖xin-xin-1‖+∑j=1i-1θiβijτiq-1γi(1-qδpj+cqλPjq)1/q‖xjn-xjn-1‖+∑j=i+13θiβijτiq-1γi(1-qδpj+cqλPjq)1/q‖xjn-xjn-1‖}=[(1+qα1ξ1q-qμ1+cqξ1q)1/q+δ1+τ1q-1γ1×(∑j=13(s1qξ1q+qθ1β11s1q-1ξ1q-1(1-qδp1+cqλp1q)1/q+cqθ1qβ11q(1-qδp1+cqλP1q))1/q+∑j=13θ1ζ1jt1j(1+1n))+(θ2β21τ2q-1γ2+θ3β31τ3q-1γ3)(1-qδp1+cqλP1q)1/q]×‖x1n-x1n-1‖+[(1+qα2ξ2q-qμ2+cqξ2q)1/q+δ2+τ2q-1γ2×(∑j=13(s2qξ2q+qθ2β22s2q-1ξ2q-1(1-qδp2+cqλp2q)1/q+cqθ2qβ22q(1-qδp2+cqλP2q))1/q+∑j=13θ2ζ2jt2j(1+1n))+(θ1β12τ1q-1γ1+θ3β32τ3q-1γ3)(1-qδp2+cqλP2q)1/q]×‖x2n-x2n-1‖+[(1+qα3ξ3q-qμ3+cqξ3q)1/q+δ3+τ3q-1γ3×(∑j=13θ3ζ3jt3j(1+1n)∑j=13θ3ζ3jt3j(1+1n)(s3qξ3q+qθ3β33s3q-1ξ3q-1(1-qδp3+cqλp3q)1/q+cqθ3qβ33q(1-qδp3+cqλP3q))1/q+∑j=13θ3ζ3jt3j(1+1n))+(θ1β13τ1q-1γ1+θ2β23τ2q-1γ2)(1-qδp3+cqλP3q)1/q]×‖x3n-x3n-1‖≤θn(∑i=1n‖xin-xin-1‖),
where
(4.27)θn=max⁡{(1+qα1ξ1q-qμ1+cqξ1q)1/q+δ1+τ1q-1γ1((s1qξ1q+qθ1β11s1q-1ξ1q-1(1-qδp1+cqλp1q)1/q+cqθ1qβ11q(1-qδp1+cqλP1q))1/q+∑j=13θ1ζ1jt1j(1+1n))+(θ2β21τ2q-1γ2+θ3β31τ3q-1γ3)(1-qδp1+cqλP1q)1/q,(1+qα2ξ2q-qμ2+cqξ2q)1/q+δ2+τ2q-1γ2(∑j=13(s2qξ2q+qθ2β22s2q-1ξ2q-1(1-qδp2+cqλp2q)1/q+cqθ2qβ22q(1-qδp2+cqλP2q)(1-qδp2+cqλp2q)1/q)1/q+∑j=13θ2ζ2jt2j(1+1n))+(θ1β12τ1q-1γ1+θ3β32τ3q-1γ3)(1-qδp2+cqλP2q)1/q,(1+qα3ξ3q-qμ3+cqξ3q)1/q+δ3+τ3q-1γ3(∑j=13(s3qξ3q+qθ3β33s3q-1ξ3q-1(1-qδp3+cqλp3q)1/q+cqθ3qβ33q(1-qδp3+cqλP3q)(1-qδp3+cqλp3q)1/q)1/q+∑j=13θ3ζ3jt3j(1+1n))+(θ1β13τ1q-1γ1+θ2β23τ2q-1γ2)(1-qδp3+cqλP3q)1/q}.
Let
(4.28)θ=max⁡{(∑j=13(s1qξ1q+qθ1β11s1q-1ξ1q-1(1-qδp1+cqλp1q)1/q+cqθ1qβ11q(1-qδp1+cqλP1q))1/q(1+qα1ξ1q-qμ1+cqξ1q)1/q+δ1+τ1q-1γ1×(∑j=13(s1qξ1q+qθ1β11s1q-1ξ1q-1(1-qδp1+cqλp1q)1/q+cqθ1qβ11q(1-qδp1+cqλP1q))1/q+∑j=13θ1ζ1jt1j)+(θ2β21τ2q-1γ2+θ3β31τ3q-1γ3)(1-qδp1+cqλP1q)1/q,(1+qα2ξ2q-qμ2+cqξ2q)1/q+δ2+τ2q-1γ2×(∑j=13(s2qξ2q+qθ2β22s2q-1ξ2q-1(1-qδp2+cqλp2q)1/q+cqθ2qβ22q(1-qδp2+cqλP2q))1/q+∑j=13θ2ζ2jt2j)+(θ1β12τ1q-1γ1+θ3β32τ3q-1γ3)(1-qδp2+cqλP2q)1/q,(1+qα3ξ3q-qμ3+cqξ3q)1/q+δ3+τ3q-1γ3×(∑j=13(s3qξ3q+qθ3β33s3q-1ξ3q-1(1-qδp3+cqλp3q)1/q+cqθ3qβ33q(1-qδp3+cqλP3q))1/q+∑j=13θ3ζ3jt3j)+(θ1β13τ1q-1γ1+θ2β23τ2q-1γ2)(1-qδp3+cqλP3q)1/q}.
We know that θn→θ as n→∞.
Define∥·∥* on E=∏k=13Ek by
(4.29)‖(z1,z2,z3)‖*=‖z1‖+‖z2‖+‖z3‖,∀(z1,z2,z3)∈E.
It is easy to see that (E,∥·∥*) is a Banach space. Define zn+1=(x1n+1,x2n+1,x3n+1). Then we have
(4.30)‖zn+1-zn‖*=‖x1n+1-x1n‖+‖x2n+1-x2n‖+‖x3n+1-x3n‖.
It follows from (4.18) that 0<θ<1, and hence there exists an n0>0 and θ0∈(0,1) such that θn≤θ0, for all n≥n0. Therefore, by (4.26) and (4.30), we have
(4.31)‖zn+1-zn‖*≤θ0n-no‖zn0+1-zn0‖*.
Hence, for any m≥n≥n0, it follows that
(4.32)‖x1m-x1n‖≤‖zm-zn‖*≤∑i=nm-1‖zi+1-zi‖*≤∑i=nm-1θ0i-n0‖zn0+1-zn0‖*.
Since 0<θ0<1, it follows from (4.32) that ∥x1m-x1n∥→0 as n→∞, and hence {x1n} is a Cauchy sequence in E1. By the same argument, we also have {xin} is a Cauchy sequence in Ei(i=2,3). Thus, there exist x1∈E1, x2∈E2, x3∈E3 such that x1n→x1, x2n→x2, x3n→x3 as n→∞.
Now we prove thatu1in→u1i∈T1i(x1), u2in→u2i∈T2i(x2), u3in→u3i∈T3i(x3)(i=1,2,3). In fact, it follows from the Lipschitz continuity of T1i, T2i, T3i that for, i=1,2,3,
(4.33)‖u1in-u1in-1‖≤(1+1n)t1i‖x1n-x1n-1‖,‖u2in-u2in-1‖≤(1+1n)t2i‖x2n-x2n-1‖,‖u3in-u3in-1‖≤(1+1n)t3i‖x3n-x3n-1‖.
From (4.33), we have that u1in, u2in, u3in(i=1,2,3) are also Cauchy sequences. Therefore, there exist u1i∈Ei, u2i∈Ei, u3i∈Ei such that u1in→u1i, u2in→u2i, u3in→u3i as n→∞. Further, for i=1,2,3,
(4.34)d(u1i,T1i(x1))≤‖u1i-u1in‖+d(u1in,T1i(x1))≤‖u1i-u1in‖+H̃(T1i(x1n),T1i(x1))≤‖u1i-u1in‖+t1i‖x1n-x1‖⟶0.
Since T1i(x1) is closed, we have u1i∈T1i(x1)(i=1,2,3). Similarly u2i∈T2i(x2), u3i∈T3i(x3)(i=1,2,3). By continuity of fi, Hi, ϕi, Fi, Ni, T1i, T2i, T3i, JMi(xi,·),ϕiHi,ηi and Algorithm 4.2, we know that x1,x2,x3, u11, u12, u13, u21, u22, u23, u31, u32, u33 satisfy the following relation:
(4.35)fi(xi)=JMi(xi,⋅),ϕiHi,ηi[Hi∘fi(xi)-ϕi∘Fi(x1-p1(x1),x2-p2(x2),x3-p3(x3))-ϕi∘Ni(ui1,ui2,ui3)],i=1,2,3.
By Lemma 4.1, (x1,x2,x3,u11,u12,u13,u21,u22,u23,u31,u32,u33) is a solution of the SGVIP (4.1). This completes the proof.



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*Source: 709715-2011-10-02.xml*