# System of Nonlinear Set-Valued Variational Inclusions Involving a Finite Family ofH(·,·)-Accretive Operators

Authors: Prapairat Junlouchai; Somyot Plubtieng
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We study a new system of nonlinear set-valued variational inclusions involving a finite family
ofH(·,·)-accretive operators in Banach spaces. By using the resolvent operator technique associated with a finite family
of H(·,·)-accretive operators, we prove the existence of the solution for the system of nonlinear set-valued variational
inclusions. Moreover, we introduce a new iterative scheme and prove a strong convergence theorem for finding solutions for this system.


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## Body

## 1. Introduction

Variational inequality theory has become a very effective and powerful tool for studying a wide range of problems arising in pure and applied sciences which include work on differential equations, control problems, mechanics, general equilibrium problems in transportation and economics. In 1994, Hassouni and Moudafi [1] introduced and studied a class of variational inclusions and developed a perturbed algorithm for finding approximate solutions of the variational inclusions. In 1996, Adly [2] obtained some important extensions and generalizations of the results in [1] for nonlinear variational inclusions. Recently, Ding [3] introduced and studied a class of generalized quasivariational inclusions and Kazmi [4] introduced and studied another class of quasivariational inclusions in the same year. In [5, 6], Ansari et al. introduced the system of vector equilibrium problems and they proved the existence of solutions for such problems (see also in [7–9]). In 2004, Verma [10] studied nonlinear variational inclusion problems based on the generalized resolvent operator technique involving A-monotone mapping. For existence result and approximating solution of the system of set-valued variational inclusions and the class of nonlinear relaxed cocoercive variational inclusions, we refer the reader to Yan et al. [11], Plubtieng and Sriprad [12], Verma [13] and Cho et al. [14].Very recently, Verma [15] introduced and studied approximation solvability of a general class of nonlinear variational inclusion problems based on (A,η)-resolvent operator technique in a Hilbert space. On the other hand, Zou and Huang [16] studied the Lipschitz continuity of resolvent operator for the H(·,·)-accretive operator in Banach spaces. Moreover, they also applied these new concepts to solve a variational-like inclusion problem. One year later, Zou and Huang [17] introduced and studied a new class of system of variational inclusions involving H(·,·)-accretive operator in Banach spaces. By using the resolvent operator technique associated with H(·,·)-accretive operator, they proved the existence of the solution for the system of inclusions. Moreover, they also develop a step-controlled iterative algorithm to approach the unique solution.In this paper, we introduce a new system of nonlinear set-valued variational inclusions involving a finite family ofH(·,·)-accretive operators in Banach spaces. By using the resolvent operators technique associated with a finite family of H(·,·)-accretive operator, we prove the existence of the solution for the system of nonlinear set-valued variational inclusions. Moreover, we introduce a new iterative scheme and prove a strong convergence theorem for finding solutions of this system.


## 2. Preliminaries

LetX be a real Banach space with dual space X*,〈·,·〉 the dual pair between X and X* and 2X and C(X) denote the family of all the nonempty subsets of X and the family of all closed subsets of X, respectively. The generalized duality mapping Jq:X→2X* is defined by(2.1)Jq(X)={f*∈X*:〈x,f*〉=‖x‖q,‖f*‖=‖x‖q-1},∀x∈X,
where q>1 is a constant. It is known that, in general, Jq(x)=∥x∥q-1J2(x) for all x≠0 and Jq is single-valued if X* is strictly convex. In the sequel, we always assume that X is a real Banach space such that Jq is single-valued.The modulus of smoothness ofX is the function ρX:[0,∞)→[0,∞) defined by(2.2)ρX(t)=sup⁡{‖x+y‖+‖x-y‖2-1:‖x‖≤1,‖y‖≤t}.
A Banach space X is called uniformly smooth if(2.3)lim⁡t→0ρX(t)t=0.X is called q-uniformly smooth if there exists a constant c>0 such that(2.4)ρX(t)≤ctq,q>1.Note thatJq is single valued if X is uniformly smooth. In the study of characteristic inequalities in q-uniformly smooth Banach spaces, Xu [18] proved the following result.Definition 2.1.
LetH,η:X×X→X be two single-valued mappings and A,B:X→X two single-valued mappings.(i)
A is said to be accretive if(2.5)〈Ax-Ay,Jq(x-y)〉≥0,∀x,y∈X,(ii)
A is said to be strictly accretive if A is accretive and
(2.6)〈Ax-Ay,Jq(x-y)〉=0,∀x,y∈X,
if and only if x=y;(iii)
H(A,·) is said to be α-strongly accretive with respect to A if there exists a constant α>0 such that(2.7)〈H(Ax,u)-H(Ay,u),Jq(x-y)〉≥α‖x-y‖q,∀x,y,u∈X;(iv)
H(·,B) is said to be β-relaxed accretive with respect to B if there exists a constant β>0 such that(2.8)〈H(u,Bx)-H(u,By),Jq(x-y)〉≥-β‖x-y‖q,∀x,y,u∈X;(v)
H(·,·) is said to be γ-Lipschitz continuous with respect to A if there exists a constant γ>0 such that(2.9)‖H(Ax,u)-H(Ay,u)‖≤γ‖x-y‖q,∀x,y,u∈X;(vi)
A is said to be θ-Lipschitz continuous if there exists a constant θ>0 such that(2.10)‖Ax-Ay‖≤θ‖x-y‖q,∀x,y∈X;(vii)
η(·,·) is said to be strongly accretive with respect to H(A,B) if there exists a constant ρ>0 such that(2.11)〈η(x,u)-η(y,u),Jq(H(Ax,Bx)-H(Ay,By))〉≥ρ‖x-y‖q,∀x,y,u∈X.Definition 2.2.
Letη:X×X→X be single-valued mapping. Let M:X→2X be a set-valued mapping.(i)
η is said to be 𝒯-Lipschitz continuous if there exists a constant 𝒯>0 such that(2.12)‖η(x,y)‖≤T‖x-y‖,∀x,y∈X;(ii)
M is said to be accretive if(2.13)〈u-v,Jq(x-y)〉≥0,∀x,y∈X,u∈M(x),v∈M(y);(iii)
M is said to be η-accretive if(2.14)〈u-v,Jq(η(x,y))〉≥0,∀x,y∈X,u∈M(x),v∈M(y);(iv)
M is said to be strictly η-accretive if M is η-accretive and equality holds if and only if x=y;(v)
M is said to be γ-strongly η-accretive if there exists a positive constant γ>0 such that(2.15)〈u-v,Jq(η(x,y))〉≥γ‖x-y‖q,∀x,y∈X,u∈M(x),v∈M(y);(vi)
M is said to be α-relaxed η-accretive if there exists a positive constant α>0 such that(2.16)〈u-v,Jq(η(x,y))〉≥-α‖x-y‖q,∀x,y∈X,u∈M(x),v∈M(y).Definition 2.3.
LetA,B:X→X, H:X×X→X be three single-valued mappings. Let M:X→2X be a set-valued mapping. M is said to be H(·,·)-accretive with respect to A and B (or simply H(·,·)-accretive in the sequel), if M is accretive and (H(A,B)+λM)(X)=X for every λ>0.Lemma 2.4.
LetX be a real uniformly smooth Banach space. Then X is q-uniformly smooth if and only if there exists a constant cq>0 such that for all x,y∈X(2.17)‖x+y‖q≤‖x‖q+q〈y,Jq(x)〉+cq‖y‖q.Lemma 2.5 (see[16]).
LetH(A,B) be α-strongly accretive with respect to A, β-relaxed accretive with respect to B, and α>β. Let M be an H(·,·)-accretive operator with respect to A and B. Then, the operator H((A,B)+λM)-1 is single valued. Based on Lemma 2.4, one can define the resolvent operator RM,λH(·,·) as follows.Definition 2.6.
LetH,A,B,M be defined as in Definition 2.3. Let H(A,B) be α-strongly accretive with respect to A, β-relaxed accretive with respect to B, and α>β. Let M be an H(·,·)-accretive operator with respect to A and B. The resolvent operator RM,λH(·,·):X→X is defined by
(2.18)RM,λH(⋅,⋅)(z)=(H(A,B)+λM)-1(z),∀z∈X,
where λ>0 is a constant.Lemma 2.7 (see [16]).
LetH,A,B,M be defined as in Definition 2.3. Let H(A,B) be α-strongly accretive with respect to A, β-relaxed accretive with respect to B, and α>β. Suppose that M:X→2X is an H(·,·)-accretive operator. Then resolvent operator RM,λH(·,·) defined by (2.18) is 1/(α-β) Lipschitz continuous. That is,
(2.19)‖RM,λH(⋅,⋅)(x)-RM,λH(⋅,⋅)(y)‖≤1α-β‖x-y‖,∀x,y∈X.We define a Hausdorff pseudometricD:2X×2X→[0,+∞] by(2.20)D(U,V)=max⁡{sup⁡u∈Uinf⁡v∈V‖u-v‖,sup⁡u∈Vinf⁡v∈U‖u-v‖}
for any given U,V∈2X. Note that if the domain of D is restricted to closed bounded subsets, then D is the Hausdorff metric.Lemma 2.8 (see [19]).
Let{cn} and {kn} be two real sequences of nonnegative numbers that satisfy the following conditions:(i)
0<kn<1 for n=0,1,2,…, and limsupnkn<1;(ii)
cn+1≤kncn for n=0,1,2,….Then, cn converges to 0 as n→∞.


## 3. Main Result

LetX be q-uniformly smooth real Banach space and C(X) a nonempty closed convex set. Let Si,Hi:X×X→X, Ai,Bi:X→X be single-valued operators, for all i=1,2,…,N. For any fix i∈{1,2,…,N}, we let Mi:X→2X, Hi(Ai,Bi)-accretive set-valued operator and Ui:X→2X a set-valued mapping which nonempty values. The system of nonlinear set-valued variational inclusions is to find a1,…,aN∈X, u1∈U1(aN),…,uN∈UN(a1) such that(3.1)0∈Si(ai,ui)+Mi(ai),∀i=1,2,…,N.IfN=2, then system of nonlinear set-valued variational inclusions (3.1) becomes following system of variational inclusions: finding a1,a2∈X, u1∈U1(a2) and u2∈U2(a1) such that(3.2)0∈S1(a1,u1)+M1(a1),0∈S2(a2,u2)+M2(a2).IfN=1, then system of nonlinear set-valued variational inclusions (3.1) becomes the following class of nonlinear set-valued variational inclusions see [15]: finding a∈X, u∈U(a) such that(3.3)0∈S(a,u)+M(a).For solving the system of nonlinear set-valued variational inclusions involving a finite family ofH(·,·)-accretive operators in Banach spaces, let us give the following assumptions.For anyi∈{1,2,…,N}, we suppose that(A1)
H(Ai,Bi) is αi-strongly accretive with respect to Ai, βi-relaxed accretive with respect to Bi and αi>βi,(A2)
Mi:X→2X is an Hi(·,·)-accretive single-valued mapping,(A3)
Ui:X→C(X) is a contraction set-valued mapping with 0≤Li<1 and nonempty values,(A4)
Hi(Ai,Bi) is ri-Lipschitz continuous with respect to Ai and ti-Lipschitz continuous with respect to Bi,(A5)
Si:X×X→X is li-Lipschitz continuous with respect to its first argument and mi-Lipschitz continuous with respect to its second argument,(A6)
Si(·,u) is si-strongly accretive with respect to Hi(Ai,Bi).Theorem 3.1.
For givena1,…,aN∈X, u1∈U1(aN),…,uN∈UN(a1), it is a solution of problem (3.1) if and only if
(3.4)ai=RMi,λiHi(⋅,⋅)[Hi(Ai(ai),Bi(ai))-λiSi(ai,ui)],
where λi>0 are constants.Proof.
We note from the Definition2.6 that a1,…,aN∈X, u1∈U1(aN),…,uN∈UN(a1) is a solution of (3.1) if and only if, for each i∈{1,2,…,N}, we have
(3.5)ai=RMi,λiHi(⋅,⋅)[Hi(Ai(ai),Bi(ai))-λiSi(ai,ui)]⟺ai=[Hi(Ai,Bi)+λiMi]-1[Hi(Ai(ai),Bi(ai))-λiSi(ai,ui)]⟺[Hi(Ai(ai),Bi(ai))-λiSi(ai,ui)]∈[Hi(Ai,Bi)+λiMi](ai)⟺-λiSi(ai,ui)∈λiMi(ai)⟺0∈Si(ai,ui)+Mi(ai).Algorithm 3.2.
For givena01,…,a0N∈X, u01∈U1(a0N),…,u0N∈UN(a01), we let
(3.6)a1i=σ0a0i+(1-σ0)RMi,λiHi(⋅,⋅)[Hi(Ai(a0i),Bi(a0i))-λiSi(a0i,u0i)],
for all i=1,2,…,N, where 0<σ0≤1. By Nadler theorem [20], there exists u11∈U1(a1N),…,u1N∈UN(a11) such that
(3.7)‖u1i-u0i‖≤(1+1)D(Ui(a1N-(i-1)),Ui(a0N-(i-1))),∀i=1,2,…,N,
where D(·,·) is the Hausdorff pseudo metric on 2X. Continuing the above process inductively, we can obtain the sequences {ani} and {uni} such that
(3.8)an+1i=σnani+(1-σn)RMi,λiHi(⋅,⋅)[Hi(Ai(ani),Bi(ani))-λiSi(ani,uni)],
for all n=1,2,3,…,i=1,2,…,N, where 0<σn≤1 with limsupn→∞σn<1. Therefore, by Nadler theorem [20], there exists un+11∈U1(an+1N),…,un+1N∈UN(an+11) such that
(3.9)‖un+1i-uni‖≤(1+(1+n)-1)D(Ui(an+1N-(i-1)),Ui(anN-(i-1))),∀n=1,2,3,…,i=1,2,…,N.
The idea of the proof of the next theorem is contained in the paper of Verma [15] and Zou and Huang [17].Theorem 3.3.
LetX be q-uniformly smooth real Banach space. Let Ai,Bi:X→X be single-valued operators, Hi:X×X→X a single-valued operator satisfy (A1) and Mi, Ui, Hi(Ai,Bi), Si, Si(·,u) satisfy conditions (A2)–(A6), respectively. If there exists a constant cq,i such that
(3.10)(ri+ti)q-qλisi+cq,iλiqliqqαi-βi+λimiαi-βi<1
for all i=1,2,…,N, then problem (3.1) has a solution a1,…,aN, u1∈U1(aN),…,uN∈UN(a1).Proof.
For anyi∈{1,2,…,N} and λi>0, we define Fi:X×X→X by
(3.11)Fi(u,v)=RMi,λiHi(⋅,⋅)[Hi(Ai(u),Bi(u))-λiSi(u,v)],
for all u,v∈X. Let Ji(x,y)=Hi(Ai(x),Bi(y)). For any (u1,v1),(u2,v2)∈X×X, we note by (3.11) and Lemma 2.7 that
(3.12)‖Fi(u1,v1)-Fi(u2,v2)‖=‖RMi,λiHi(⋅,⋅)[Hi(Ai(u1),Bi(u1))-λiSi(u1,v1)]-RMi,λiHi(⋅,⋅)[Hi(Ai(u2),Bi(u2))-λiSi(u2,v2)]‖=‖RMi,λiHi(⋅,⋅)[Ji(u1,u1)-λiSi(u1,v1)]-RMi,λiHi(⋅,⋅)[Ji(u2,u2)-λiSi(u2,v2)]‖≤1αi-βi‖[Ji(u1,u1)-λiSi(u1,v1)]-[Ji(u2,u2)-λiSi(u2,v2)]‖=1αi-βi‖[Ji(u1,u1)-Ji(u2,u2)]-λi[Si(u1,v1)-Si(u2,v2)]‖≤1αi-βi‖[Ji(u1,u1)-Ji(u2,u2)]-λi[Si(u1,v1)-Si(u2,v1)]‖+λiαi-βi‖[Si(u2,v1)-Si(u2,v2)]‖.
By Lemma 2.4, we have
(3.13)‖Ji(u1,u1)-Ji(u2,u2)-λi[Si(u1,v1)-Si(u2,v1)]‖q≤‖Ji(u1,u1)-Ji(u2,u2)‖q-qλi〈Si(u1,v1)-Si(u2,v1),Jq(Ji(u1,u1)-Ji(u2,u2))〉+cq,iλiq‖Si(u1,v1)-Si(u2,v1)‖q.
Moreover, by (A4), we obtain
(3.14)‖Ji(u1,u1)-Ji(u2,u2)‖≤‖Ji(u1,u1)-Ji(u2,u1)‖+‖Ji(u2,u1)-Ji(u2,u2)‖≤ri‖u1-u2‖+ti‖u1-u2‖≤(ri+ti)‖u1-u2‖.
From (A6), we have
(3.15)-qλi〈Si(u1,v1)-Si(u2,v1),Jq(Ji(u1,u1)-Ji(u2,u2))〉≤-qλisi‖u1-u2‖q.
Moreover, from (A5), we obtain
(3.16)‖Si(u1,v1)-Si(u2,v1)‖≤li‖u1-u2‖,(3.17)‖Si(u2,v1)-Si(u2,v2)‖≤mi‖v1-v2‖.
From (3.13)–(3.16), we have
(3.18)‖Ji(u1,u1)-Ji(u2,u2)-λi[Si(u1,v1)-Si(u2,v1)]‖q≤(ri+ti)q-qλisi+cq,iλiqliqq‖u1-u2‖.
It follows from (3.12), (3.17), and (3.18) that
(3.19)‖Fi(u1,v1)-Fi(u2,v2)‖≤(ri+ti)q-qλisi+cq,iλiqliqqαi-βi‖u1-u2‖+λimiαi-βi‖v1-v2‖.
Put
(3.20)θ1i=(ri+ti)q-qλisi+cq,iλiqliqqαi-βi,θ2i=λimiαi-βi.
Define ∥·∥ on X×⋯×X︸N-times by ∥(x1,…,xN)∥=∥x1∥+⋯∥xN∥ for all (x1,…,xN)∈X×⋯×X︸N-times. It is easy to see that (X×⋯×X︸N-times,∥·∥) is a Banach space. For any given x1,…,xN∈X, we choose a finite sequence w1∈U1(xN),…,wN∈UN(x1). Define Q:X×⋯×X︸N-times→X×⋯×X︸N-times by Q(x1,…,xN)=(F1(x1,w1),…,FN(xN,wN)). Set k=max{(θ11+θ2NLN),…,(θ21L1+θ1N)}, where L1,…,LN are contraction constants of U1,…,UN, respectively. We note that θ1i+θ2iLi<θ1i+θ2i<1, for all i=1,2,…,N, and so k<1. Let x1,…,xN∈X, w1∈U1(xN),…,wN∈UN(x1) and y1,…,yN∈X, z1∈U1(yN),…,zN∈UN(y1). By (A3), we get
(3.21)‖Q(x1,…,xN)-Q(y1,…,yN)‖=‖(F1(x1,w1),…,FN(xN,wN))-(F1(y1,z1),…,FN(yN,zN))‖=‖F1(x1,w1)-F1(y1,z1)‖+⋯+‖FN(xN,wN)-FN(yN,zN)‖≤(θ11‖x1-y1‖+θ21‖w1-z1‖)+⋯+(θ1N‖xN-yN‖+θ2N‖wN-zN‖)≤(θ11‖x1-y1‖+θ21L1‖xN-yN‖)+⋯+(θ1N‖xN-yN‖+θ2NLN‖x1-y1‖)=(θ11+θ2NLN)‖x1-y1‖+⋯+(θ1N+θ21L1)‖xN-yN‖≤k‖x1-y1‖+⋯+k‖xN-yN‖=k(‖x1-y1‖+⋯+‖xN-yN‖)=k‖(x1,…,xN)-(y1,…,yN)‖,
and so Q is a contraction on X×⋯×X︸N-times. Hence there exists a1,…,aN∈X,u1∈U1(aN),…,uN∈UN(a1) such that a1=F1(a1,u1),…,aN=FN(aN,uN). From Theorem 3.1, a1,…,aN∈X, u1∈U1(aN),…,uN∈UN(a1) is the solution of the problem (3.1).Theorem 3.4.
LetX be q-uniformly smooth real Banach space. For i=1,2,…,N. Let Ai,Bi:X→X be single-valued operators, Hi:X×X→X single-valued operator satisfy (A1) and suppose that Mi, Ui, Hi(Ai,Bi), Si, Si(·,u) satisfy conditions (A2)–(A6), respectively. Then, for any i∈{1,2,…,N}, the sequences {ani}n=1∞ and {uni}n=1∞ generated by Algorithm 3.2 converge strongly to ai, ui∈Ui(aN-(i-1)), respectively.Proof.
By Theorem3.3, the problem (3.1) has a solution a1,…,aN∈X, u1∈U1(aN),…,uN∈UN(a1). From Theorem 3.1, we note that
(3.22)ai=σnai+(1-σn)RMi,λiHi(⋅,⋅)[Hi(Ai(ai),Bi(ai)-λiSi(ai,ui))],
for all i=1,2,…,N. Hence, by (3.8) and (3.22), we have
(3.23)‖an+1i-ani‖=‖σnani+(1-σn)RMi,λiHi(⋅,⋅)[Hi(Ai(ani),Bi(ani))-λiSi(ani,uni)]-[σnan-1i+(1-σn)RMi,λiHi(⋅,⋅)[Hi(Ai(an-1i),Bi(an-1i))-λiSi(an-1i,un-1i)]]‖≤σn‖ani-an-1i‖+(1-σn)‖RMi,λiHi(⋅,⋅)[Hi(Ai(ani),Bi(ani))-λiSi(ani,uni)]-RMi,λiHi(⋅,⋅)[Hi(Ai(an-1i),Bi(an-1i))-λiSi(an-1i,un-1i)]‖=σn‖ani-an-1i‖+(1-σn)‖RMi,λiHi(⋅,⋅)[Ji(ani,ani)-λiSi(ani,uni)]-RMi,λiHi(⋅,⋅)[Ji(an-1i,an-1i)-λiSi(an-1i,un-1i)]‖≤σn‖ani-an-1i‖+(1-σn)1αi-βi‖[Ji(ani,ani)-λiSi(ani,uni)]-[Ji(an-1i,an-1i)-λiSi(an-1i,un-1i)]‖=σn‖ani-an-1i‖+(1-σn)1αi-βi‖[Ji(ani,ani)-Ji(an-1i,an-1i)]-λi[Si(ani,uni)-Si(an-1i,un-1i)]‖≤σn‖ani-an-1i‖+(1-σn)1αi-βi‖[Ji(ani,ani)-Ji(an-1i,an-1i)]-λi[Si(ani,uni)-Si(an-1i,uni)]‖+(1-σn)1αi-βi‖Si(an-1i,uni)-Si(an-1i,un-1i)‖.
By Lemma 2.4, we obtain
(3.24)‖Ji(ani,ani)-Ji(an-1i,an-1i)-λi[Si(ani,uni)-Si(an-1i,uni)]‖q≤‖Ji(ani,ani)-Ji(an-1i,an-1i)‖q-qλi〈Si(ani,uni)-Si(an-1i,uni),Jq,i(Ji(ani,ani)-Ji(an-1i,an-1i))〉+cq,iλiq‖Si(ani,uni)-Si(an-1i,uni)‖q.
From (A4), we note that
(3.25)‖Ji(ani,ani)-Ji(an-1i,an-1i)‖=‖Hi(Ai(ani),Bi(ani))-Hi(Ai(an-1i),Bi(an-1i))‖≤‖Hi(Ai(ani),Bi(ani))-Hi(Ai(an-1i),Bi(ani))‖+‖Hi(Ai(an-1i),Bi(ani))-Hi(Ai(an-1i),Bi(an-1i))‖≤(ri+ti)‖ani-an-1i‖.
From (3.24) and (A6), it follows that
(3.26)-qλi〈Si(ani,uni)-Si(an-1i,uni),Jq,1(Ji(ani,ani)-Ji(an-1i,an-1i))〉≤-qλisi‖ani-an-1i‖q.
By (3.23), (3.24), and (A5), we have
(3.27)‖Si(an-1i,uni)-Si(an-1i,un-1i)‖≤mi‖uni-un-1i‖≤midi(1+n-1)‖ani-an-1i‖,(3.28)‖Si(ani,uni)-Si(an-1i,uni)‖≤li‖ani-an-1i‖.
From (3.23)–(3.28), we obtain
(3.29)‖Ji(ani,ani)-Ji(an-1i,an-1i)-λi[Si(ani,uni)-Si(an-1i,uni)]‖q≤(ri+ti)q-qλisi+cq,iλiqliqqαi-βi‖ani-an-1i‖+λimiαi-βidi(1+n-1)‖ani-an-1i‖.
Hence, by (3.23), (3.28) and (3.29), we have
(3.30)‖an+1i-ani‖≤σn‖ani-an-1i‖+(1-σn)(ri+ti)q-qλisi+cq,iλiqliqqαi-βi‖ani-an-1i‖+(1-σn)λimiαi-βidi(1+n-1)‖ani-an-1i‖.
Put k=max{π1…,πN}, where
(3.31)πi=(ri+ti)q-qλisi+cq,iλiqliqqαi-βi+λimidi(1+n-1)αi-βi.
It follows from (3.30) that
(3.32)‖an+11-an1‖+⋯+‖an+1N-anN‖≤σn‖an1-an-11‖+(1-σn)k‖an1-an-11‖+⋯+σn‖anN-an-1N‖+(1-σn)k‖anN-an-1N‖.
Set cn=∥an1-an-11∥+⋯+∥anN-an-1N∥ and kn=k+(1-k)σn. From (3.32), we obtain
(3.33)cn+1≤kncn,∀n=0,1,2,….
Since limsupn→∞σn<1, we have limsupn→∞kn<1. Thus, it follows from Lemma 2.8 that cn+1→0 and hence limn→∞∥an+1i-ani∥=0. Therefore, {ani} is a Cauchy sequence and hence there exists ai∈X such that ani→ai as n→∞, for all i=1,2,…,N. Next, we will show that un1→u1∈U1(aN) as n→∞. Hence, it follows from (3.9) that {un1} is also a Cauchy sequence. Thus there exists u1∈X such that un1→u1 as n→∞. Consider
(3.34)d(u1,U1(aN))=inf⁡{‖u1-q‖:q∈U1(aN)}≤‖u1-un1‖+d(un1,U1(aN))≤‖u1-un1‖+D(U1(anN),U1(aN))≤‖u1-un1‖+d1‖anN-aN‖⟶0
as n→∞. Since U1(aN) is a closed set and d(u1,U1(aN))=0, we have u1∈U1(aN). By continuing the above process, there exist u2∈U2(aN-1),…,uN∈UN(a1) such that un2→u2,…,unN→uN as n→∞. Hence, by (3.8), we obtain
(3.35)ai=RMi,λiHi(⋅,⋅)[Hi(Ai(ai),Bi(ai))-λiSi(ai,ui)].
Therefore, it follows from Theorem 3.1 that a1,…,aN is a solution of problem (3.1).
SettingN=2 in Theorem 3.3, we have the following result.Corollary 3.5.
LetX be q-uniformly smooth real Banach spaces. Let Ai,Bi:X→X be singled valued operators, Hi:X×X→X a single-valued operator such that H(Ai,Bi) is αi-strongly accretive with respect to Ai, βi-relaxed accretive with respect to Bi and αi>βi and suppose that Mi:X→2X is an Hi(·,·)-accretive set-valued mapping and Ui:X→C(X) contraction set-valued mapping with 0≤Li<1 and nonempty values, for all i=1,2. Assume that Hi(Ai,Bi) is ri-Lipschitz continuous with respect to Ai and ti-Lipschitz continuous with respect to Bi, Si:X×X→X is li-Lipschitz continuous with respect to its first argument and mi-Lipschitz continuous with respect to its second argument, S1(·,y) is s1-strongly accretive with respect to H1(A1,B1), and S2(x,·) is s2-strongly accretive with respect to H2(A2,B2), for all i=1,2. If
(3.36)(ri+ti)q-qλisi+cq,iλiqliqqαi-βi+λimiαi-βi<1,
for all i∈{1,2}, then problem (3.2) has a solution a1,a2∈X, u1∈U1(a2),u2∈U2(a1).SettingN=1 in Theorem 3.3, we have the following result.Corollary 3.6.
LetX be q-uniformly smooth real Banach spaces. Let A,B:X→X be two singled valued operators, H:X×X→X a single-valued operator such that H(A,B) is α-strongly accretive with respect to A, β-relaxed accretive with respect to B, and α>β and suppose that M:X→2X is an H(·,·)-accretive set-valued mapping, U:X→C(X) is contraction set-valued mapping with 0≤L<1 and nonempty values. Assume that H(A,B) is r-Lipschitz continuous with respect to A and t-Lipschitz continuous with respect to B, S:X×X→X is l-Lipschitz continuous with respect to its first argument and m-Lipschitz continuous with respect to its second argument, S(·,y) is s-strongly accretive with respect to H(A,B). If
(3.37)(r+t)q-qλs+cq,λqlqqα-β+λmα-β<1,
then problem (3.3) has a solution a∈X and u∈U(a).



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*Source: 560248-2012-06-28.xml*