# Algorithms for General System of Generalized Resolvent Equations with Corresponding System of Variational Inclusions

Authors: Lu-Chuan Ceng; Ching-Feng Wen
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Very recently, Ahmad and Yao (2009) introduced and considered a system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. In this paper we introduce and study a general system of generalized resolvent equations with corresponding general system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between general system of generalized resolvent equations and general system of variational inclusions. The iterative algorithms for finding the approximate solutions of general system of generalized resolvent equations are proposed. The convergence criteria of approximate solutions of general system of generalized resolvent equations obtained by the proposed iterative algorithm are also presented. Our results represent the generalization, improvement, supplement, and development of Ahmad and Yao corresponding ones.


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## Body

## 1. Introduction and Preliminaries

It is well known that the theory of variational inequalities has played an important role in the investigation of a wide class of problems arising in mechanics, physics, optimization and control, nonlinear programming, elasticity, and applied sciences and so on; see, for example, [1–7] and the references therein. In recent years variational inequalities have been extended and generalized in different directions. A useful and significant generalization of variational inequalities is called mixed variational inequalities involving the nonlinear term [8], which enables us to study free, moving, obstacle, equilibrium problems arising in pure and applied sciences in a unified and general framework. Due to the presence of the nonlinear term, the projection method and its variant forms including the technique of the Wiener-Hopf equations cannot be extended to suggest the iterative methods for solving mixed variational inequalities. To overcome these drawbacks, Hassouni and Moudafi [9] introduced variational inclusions which contain mixed variational inequalities as special cases. They studied the perturbed method for solving variational inclusions. Subsequently, M. A. Noor and K. I. Noor [10] introduced and considered the resolvent equations by virtue of the resolvent operator concept and established the equivalence between the mixed variational inequalities and the resolvent equations. The technique of resolvent equations is being used to develop powerful and efficient numerical techniques for solving mixed (quasi)variational inequalities and related optimization problems. At the same time, some iterative algorithms for approximating a solution of some system of variational inequalities are also introduced and studied in Verma [11]. Pang [12], Cohen and Chaplais [13], Binachi [14], Ansari and Yao [15] considered a system of scalar variational inequalities and Pang showed that the traffic equilibrium problem, the Nash equilibrium, and the general equilibrium programming problem can be modeled as a system of variational inequalities. As generalizations of system of variational inequalities, Agarwal et al. [16] introduced a system of generalized nonlinear mixed quasi-variational inclusions and investigated the sensitivity analysis of solutions for the system of generalized mixed quasi-variational inclusions in Hilbert spaces. In 2007, Peng and Zhu [17] considered and studied a new system of generalized mixed quasi-variational inclusions with (H,η)-monotone operators and Lan et al. [18] studied a new system of nonlinear A-monotone multivalued variational inclusions. Furthermore, for more details in the related research work of this field, we invoke the readers to see, for instance, [19–30]. Very recently, Ahmad and Yao [31] introduced and considered a new system of variational inclusions in uniformly smooth Banach spaces, which covers the system of variational inclusions in Hilbert spaces considered by [18]. They established an equivalence relation between this system of variational inclusions and a system of generalized resolvent equations, proposed a number of iterative algorithms for this system of variational inclusions, and also gave the convergence criteria.LetE be a real Banach space with its norm ∥·∥,E* the topological dual of E, and d the metric induced by the norm ∥·∥. Let CB(E) (resp., 2E) be the family of all nonempty closed and bounded subsets (resp., all nonempty subsets) of E and D(·,·) the Hausdorff metric on CB(E) defined by(1.1)D(A,B)=max⁡{sup⁡x∈Ad(x,B),sup⁡y∈Bd(A,y)},
where d(x,B)=infy∈Bd(x,y) and d(A,y)=infx∈Ad(x,y). We write by J:E→2E* the normalized duality mapping defined as(1.2)J(x)={f∈E*:〈x,f〉=‖x‖2=‖f‖2},∀x∈E,
where 〈·,·〉 denotes the duality pairing between E and E*.The uniform convexity of a Banach spaceE means that for any ϵ>0, there exists δ>0, such that for any x,y∈E,∥x∥≤1,∥y∥≤1,∥x-y∥=ϵ ensure the following inequality:(1.3)‖x+y‖≤2(1-δ).
The function(1.4)δE(ϵ)=inf⁡{1-‖x+y‖2:‖x‖=1,‖y‖=1,‖x-y‖=ϵ}
is called the modulus of convexity of E.The uniform smoothness of a Banach spaceE means that for any given ϵ>0, there exists δ>0 such that(1.5)‖x+y‖+‖x-y‖2-1≤ϵ‖y‖
holds. The function(1.6)τE(t)=sup⁡{‖x+y‖+‖x-y‖2-1:‖x‖=1,‖y‖=t}
is called the modulus of smoothness of E.It is well known that the Banach spaceE is uniformly convex if and only if δE(ϵ)>0 for all ϵ>0, and it is uniformly smooth if and only if limt→0τE(t)/t=0. All Hilbert spaces, Lp (or lp) spaces (p≥2), and the Sobolov spaces Wmp(p≥2) are 2-uniformly smooth, while, for 1<p≤2,Lp (or lp) and Wmp spaces are p-uniformly smooth.Proposition 1.1 (see [15]).
LetE be a uniformly smooth Banach space. Then the normalized duality mapping  J:E→2E* is single-valued, and for any x,y∈E there holds the following:(i)
∥x+y∥2≤∥x∥2+2〈y,J(x+y)〉,(ii)
〈x-y,J(x)-J(y)〉≤2C2τE(4∥x-y∥/C), where C=∥x∥2+∥y∥2/2.Definition 1.2 (see [32]).
A mappingg:E→E is said to be
(i)
k-strongly accretive, k∈(0,1), if for any x,y∈E, there exists j(x-y)∈J(x-y) such that(1.7)〈g(x)-g(y),j(x-y)〉≥k‖x-y‖2;(ii)
Lipschitz continuous if for anyx,y∈E, there exists a constant λg>0, such that(1.8)‖g(x)-g(y)‖≤λg‖x-y‖.Definition 1.3 (see [13]).
A set-valued mappingA:E→2E is said to be
(i)
accretive, if for anyx,y∈E, there exists j(x-y)∈J(x-y) such that for all u∈A(x) and v∈A(y),
(1.9)〈u-v,j(x-y)〉≥0;(ii)
k-strongly accretive, k∈(0,1), if for any x,y∈E, there exists j(x-y)∈J(x-y), such that for all u∈A(x) and v∈A(y),
(1.10)〈u-v,j(x-y)〉≥k‖x-y‖2;(iii)
m-accretive if A is accretive and (I+ρA)(E)=E, for every (equivalently, for some) ρ>0, where I is the identity mapping (equivalently, if A is accretive and (I+A)(E)=E). In particular, it is clear from [9] that if E=H is a Hilbert space, than A:E→2E is an m-accretive mapping if and only if it is a maximal monotone mapping.Definition 1.4 (see [31]).
LetM:E→2E be an m-accretive mapping. For any ρ>0, the mapping JMρ:E→E associated with M defined by
(1.11)JMρ(x)=(I+ρM)-1(x),∀x∈E
is called the resolvent operator.Definition 1.5 (see [33]).
The resolvent operatorJMρ:E→E is said to be a retraction if
(1.12)(I+ρM)-1∘(I+ρM)-1(x)=(I+ρM)-1(x),∀x∈E.
It is well known thatJMρ is a single-valued and nonexpansive mapping.Definition 1.6 (see [10]).
A set-valued mappingH:E→CB(E) is said to be D-Lipschitz continuous if for any x,y∈E, there exists a constant λDH>0 such that
(1.13)D(H(x),H(y))≤λDH‖x-y‖.LetE1 and E2 be two real Banach spaces, S:E1×E2→E1 and T:E1×E2→E2 single-valued mappings, and G:E1→2E1,F:E2→2E2,H:E1→2E1 and V:E2→2E2 any four multivalued mappings. Let M:E1→2E1 and N:E2→2E2 be any nonlinear mappings, m:E2→E1,n:E1→E2,f:E1→E1 and g:E2→E2 nonlinear mappings with f(E1)∩D(M)≠∅ and g(E2)∩D(N)≠∅, respectively. Then we consider the problem of finding (x,y)∈E1×E2,(s,v)∈G(x)×F(y),(u,t)∈H(x)×V(y) such that(1.14)m(y)∈S(s,v)+M(f(x)),n(x)∈T(u,t)+N(g(y)),
which is called a general system of variational inclusions. In particular, if m(y)=0∈E1,n(x)=0∈E2,G(x)=p(x) and V(y)=q(y), where p:E1→E1 and q:E2→E2 are single-valued mappings, then the general system of variational inclusions (1.14) reduces to the following system of variational inclusions(1.15)0∈S(p(x),v)+M(f(x)),0∈T(u,q(y))+N(g(y)),
which was considered by Lan et al. [18] in Hilbert spaces and studied by Ahmad and Yao [31] in Banach spaces, respectively.Proposition 1.7 (see [31, Lemma  2.1]).
(x,y)∈E1×E2,u∈H(x),v∈F(y) is a solution of the system of variational inclusions (1.15) if and only if (x,y,u,v) satisfies
(1.16)f(x)=JMρ(f(x)-ρS(p(x),v)),ρ>0,g(y)=JNγ(g(y)-γT(u,q(y))),γ>0.Proposition 1.8 (see [31, Proposition  3.1]).
The system of variational inclusions (1.15) has a solution (x,y,u,v) with (x,y)∈E1×E2,u∈H(x) and v∈F(y) if and only if the following system of generalized resolvent equations
(1.17)S(p(x),v)+ρ-1RMρ(z′)=0,RMρ=I-JMρ,ρ>0,T(u,q(y))+γ-1RNγ(z′′)=0,RNγ=I-JNγ,γ>0,
has a solution (z′,z′′,x,y,u,v) with (x,y)∈E1×E2,u∈H(x),v∈F(y),z′∈E1 and z′′∈E2, where f(x)=JMρ(z′),g(y)=JNγ(z′′) and z′=f(x)-ρS(p(x),v),z′′=g(y)-γT(u,q(y)).Based on the above Propositions1.7 and 1.8, Ahmad and Yao [31] presented the following algorithm and established the following strong convergence result for the sequences generated by the algorithm.Algorithm 1.9 (see [31, Algorithm  3.1]).
For given(x0,y0)∈E1×E2,u0∈H(x0),v0∈F(y0),z0′∈E1 and z0′′∈E2, compute {zk′},{zk′′},{xk},{yk},{uk}, and {vk} by the iterative scheme:
(1.18)f(xk)=JMρ(zk′),g(yk)=JNγ(zk′′),uk∈H(xk):‖uk+1-uk‖≤D(H(xk+1),H(xk)),vk∈F(yk):‖vk+1-vk‖≤D(F(yk+1),F(yk)),zk+1′=f(xk)-ρS(p(xk),vk),zk+1′′=g(yk)-γT(uk,q(yk)),k=0,1,2,….Theorem 1.10 (see [31, Theorem  3.1]).
LetE1 and E2 be two real uniformly smooth Banach spaces with modulus of smoothness τE1(t)≤C1t2 and τE2(t)≤C2t2 for C1,C2>0, respectively. Let H:E1→CB(E1),F:E2→CB(E2) be D-Lipschitz continuous mappings with constants λDH and λDF, respectively, and let M:E1→2E1,N:E2→2E2 be m-accretive mappings such that the resolvent operators associated with M and N are retractions. Let f:E1→E1,g:E2→E2 be both strong accretive with constants α and β, respectively, and Lipschitz continuous with constants δ1 and δ2, respectively. Let p:E1→E1,q:E2→E2 be Lipschitz continuous with constants λp and λq, respectively, and let S:E1×E2→E1,T:E1×E2→E2 be Lipschitz continuous in the first and second arguments with constants λS1,λS2 and λT1,λT2, respectively.
If there exist constantsρ>0 and γ>0, such that
(1.19)0<B′+θ1+θ41-B′<1,0<B′′+θ2+θ31-B′′<1,
where B′=1-2α+64C1δ12,B′′=1-2β+64C2δ22 and θ1=(1+ρλS1λp)/(1-ρ(λS1λp+λS2λDF)),θ2=(1+ρλS2λDF)/(1-ρ(λS1λp+λS2λDF)),θ3=(1+γλT2λq)/(1-γ(λT1λDH+λT2λq)),θ4=γλT1λDH/(1-γ(λT1λDH+λT2λq)), then there exist (x,y)∈E1×E2,u∈H(x),v∈F(y) and (z′,z′′)∈E1×E2 satisfying the system of generalized resolvent equations (1.17) (in this case, (x,y,u,v) is a solution of system of variational inclusions (1.15)), and the iterative sequences {zk′},{zk′′},{xk},{yk},{uk}, and {vk} generated by Algorithm 1.9 converge strongly to z′,z′′,x,y,u, and v, respectively.In this paper we introduce and study a general system of generalized resolvent equations with corresponding general system of variational inclusions in uniformly smooth Banach spaces. Motivated and inspired by the above Proposition1.8, we establish an equivalence relation between general system of generalized resolvent equations and general system of variational inclusions. By using Nadler [34] we propose some new iterative algorithms for finding the approximate solutions of general system of generalized resolvent equations, which include Ahmad and Yao’s corresponding algorithms as special cases to a great extent. Furthermore, the convergence criteria of approximate solutions of general system of generalized resolvent equations obtained by the proposed iterative algorithm are also presented. There is no doubt that our results represent the generalization, improvement, supplement, and development of Ahmad and Yao corresponding ones [31].


## 2. Main Results

LetE1 and E2 be two real Banach spaces, let S:E1×E2→E1 and T:E1×E2→E2 be single-valued mappings, and let G:E1→2E1,F:E2→2E2,H:E1→2E1 and V:E2→2E2 be any four multivalued mappings. Let M:E1→2E1 and N:E2→2E2 be any nonlinear mappings, m:E2→E1,n:E1→E2,f:E1→E1 and g:E2→E2 nonlinear mappings with f(E1)∩D(M)≠∅ and g(E2)∩D(N)≠∅, respectively. Then we consider the problem of finding (x,y)∈E1×E2,(s,v)∈G(x)×F(y),(u,t)∈H(x)×V(y),z′∈E1,z′′∈E2 such that(2.1)S(s,v)+ρ-1RMρ(z′)=m(y),ρ>0,T(u,t)+γ-1RNγ(z′′)=n(x),γ>0,
where RMρ=I-JMρ,RNγ=I-JNγ and JMρ,JNγ are the resolvent operators associated with M and N, respectively.The corresponding general system of variational inclusions of (2.1) is the problem (1.14), that is, find (x,y)∈E1×E2,(s,v)∈G(x)×F(y),(u,t)∈H(x)×V(y) such that(2.2)m(y)∈S(s,v)+M(f(x)),n(x)∈T(u,t)+N(g(y)).Proposition 2.1.
(x,y)∈E1×E2,(s,v)∈G(x)×F(y),(u,t)∈H(x)×V(y)  are solutions of general system of variational inclusions (1.14) if and only if (x,y,u,v,s,t) satisfies
(2.3)f(x)=JMρ[f(x)-ρ(S(s,v)-m(y))],ρ>0,g(y)=JNγ[g(y)-γ(T(u,t)-n(x))],γ>0.Proof.
The proof of Proposition2.1 is a direct consequence of the definition of resolvent operator, and hence, is omitted.Next we first establish an equivalence relation between general system of generalized resolvent equations (2.1) and general system of variational inclusions (1.14) and then prove the existence of a solution of (2.1) and convergence of sequences generated by the proposed algorithms.Proposition 2.2.
The general system of variational inclusions (1.14) has a solution (x,y,u,v,s,t) with (x,y)∈E1×E2,(s,v)∈G(x)×F(y) and (u,t)∈H(x)×V(y) if and only if general system of generalized resolvent equations (2.1) has a solution (z′,z′′,x,y,u,v,s,t) with (x,y)∈E1×E2,(s,v)∈G(x)×F(y),(u,t)∈H(x)×V(y),(z′,z′′)∈E1×E2, where
(2.4)f(x)=JMρ(z′),g(y)=JNγ(z′′),
and z′=f(x)-ρ(S(s,v)-m(y)) and z′′=g(y)-γ(T(u,t)-n(x)).Proof.
Let(x,y)∈E1×E2,(s,v)∈G(x)×F(y),(u,t)∈H(x)×V(y) be a solution of general system of variational inclusions (1.14). Then, by Proposition 2.1, it satisfies the following system of equations
(2.5)f(x)=JMρ[f(x)-ρ(S(s,v)-m(y))],g(y)=JNγ[g(y)-γ(T(u,t)-n(x))].
Letz′=f(x)-ρ(S(s,v)-m(y)) and z′′=g(y)-γ(T(u,t)-n(x)). Then we have
(2.6)f(x)=JMρ(z′),g(y)=JNγ(z′′),
and hence z′=JMρ(z′)-ρ(S(s,v)-m(y)) and z′′=JNγ(z′′)-γ(T(u,t)-n(x)). Thus it follows that
(2.7)(I-JMρ)(z′)=-ρ(S(s,v)-m(y)),(I-JNγ)(z′′)=-γ(T(u,t)-n(x)),
that is,
(2.8)S(s,v)+ρ-1RMρ(z′)=m(y),T(u,t)+γ-1RNγ(z′′)=n(x).
Therefore, (z′,z′′,x,y,u,v,s,t) is a solution of general system of generalized resolvent equations (2.1).
Conversely, let(z′,z′′,x,y,u,v,s,t) be a solution of general system of generalized resolvent equations (2.1). Then
(2.9)ρ(S(s,v)-m(y))=-RMρ(z′),γ(T(u,t)-n(x))=-RNγ(z′′).
Now observe that(2.10)ρ(S(s,v)-m(y))=-RMρ(z′)=-(I-JMρ)(z′)=JMρ(z′)-z′=JMρ[f(x)-ρ(S(s,v)-m(y))]-[f(x)-ρ(S(s,v)-m(y))],
which leads to
(2.11)f(x)=JMρ[f(x)-ρ(S(s,v)-m(y))],
and also that
(2.12)γ(T(u,t)-n(x))=-RNγ(z′′)=-(I-JNγ)(z′′)=JNγ(z′′)-z′′=JNγ[g(y)-γ(T(u,t)-n(x))]-[g(y)-γ(T(u,t)-n(x))],
which leads to
(2.13)g(y)=JNγ[g(y)-γ(T(u,t)-n(x))].
Consequently, we have
(2.14)f(x)=JMρ[f(x)-ρ(S(s,v)-m(y))],g(y)=JNγ[g(y)-γ(T(u,t)-n(x))].
Therefore, by Proposition 2.1, (x,y,u,v,s,t) is a solution of general system of variational inclusions (1.14).Proof (Alternative).
Let(2.15)z′=f(x)-ρ(S(s,v)-m(y)),z′′=g(y)-γ(T(u,t)-n(x)).
Then, utilizing (2.4), we can write
(2.16)z′=JMρ(z′)-ρ(S(s,v)-m(y)),z′′=JNγ(z′′)-γ(T(u,t)-n(x))
which yield that
(2.17)S(s,v)+ρ-1RMρ(z′)=m(y),T(u,t)+γ-1RNγ(z′′)=n(x),
the required general system of generalized resolvent equations.Algorithm 2.3.
For given(x0,y0)∈E1×E2,(s0,v0)∈G(x0)×F(y0),(u0,t0)∈H(x0)×V(y0),(z0′,z0′′)∈E1×E2, compute
(2.18)z1′=f(x0)-ρ(S(s0,v0)-m(y0)),z1′′=g(y0)-γ(T(u0,t0)-n(x0)).
For (z1′,z1′′)∈E1×E2, we take (x1,y1)∈E1×E2 such that f(x1)=JMρ(z1′) and g(y1)=JNγ(z1′′). Then, by Nadler [34], there exist (s1,v1)∈G(x1)×F(y1),(u1,t1)∈H(x1)×V(y1) such that
(2.19)‖u1-u0‖≤(1+1)D(H(x1),H(x0)),‖v1-v0‖≤(1+1)D(F(y1),F(y0)),‖s1-s0‖≤(1+1)D(G(x1),G(x0)),‖t1-t0‖≤(1+1)D(V(y1),V(y0)),
where D(·,·) is the Hausdorff metric on CB(E1) (for the sake of convenience, we also denote by D(·,·) the Hausdorff metric on CB(E2)). Compute
(2.20)z2′=f(x1)-ρ(S(s1,v1)-m(y1)),z2′′=g(y1)-γ(T(u1,t1)-n(x1)).
By induction, we can obtain sequences (xk,yk)∈E1×E2,(sk,vk)∈G(xk)×F(yk),(uk,tk)∈H(xk)×V(yk),(zk′,zk′′)∈E1×E2 by the iterative scheme:
(2.21)f(xk)=JMρ(zk′),g(yk)=JNγ(zk′′),(2.22)uk∈H(xk):‖uk+1-uk‖≤(1+1k+1)D(H(xk+1),H(xk)),vk∈F(yk):‖vk+1-vk‖≤(1+1k+1)D(F(yk+1),F(yk)),sk∈G(xk):‖sk+1-sk‖≤(1+1k+1)D(G(xk+1),G(xk)),tk∈V(yk):‖tk+1-tk‖≤(1+1k+1)D(V(yk+1),V(yk)),(2.23)zk+1′=f(xk)-ρ(S(sk,vk)-m(yk)),zk+1′′=g(yk)-γ(T(uk,tk)-n(xk)),
for k=0,1,2,….
The general system of generalized resolvent equations (2.1) can also be rewritten as
(2.24)z′=f(x)-S(s,v)+m(y)+(I-ρ-1RMρ)(z′),z′′=g(y)-T(u,t)+n(x)+(I-γ-1RNγ)(z′′).
Utilizing this fixed-point formulation, we suggest the following iterative algorithm.Algorithm 2.4.
For given(x0,y0)∈E1×E2,(s0,v0)∈G(x0)×F(y0),(u0,t0)∈H(x0)×V(y0),(z0′,z0′′)∈E1×E2, compute
(2.25)z1′=f(x0)-S(s0,v0)+m(y0)+(I-ρ-1RMρ)(z0′),z1′′=g(y0)-T(u0,t0)+n(x0)+(I-γ-1RNγ)(z0′′).
For (z1′,z1′′)∈E1×E2, we take (x1,y1)∈E1×E2 such that f(x1)=JMρ(z1′) and g(y1)=JNγ(z1′′). Then, by Nadler [34], there exist (s1,v1)∈G(x1)×F(y1),(u1,t1)∈H(x1)×V(y1) such that
(2.26)‖u1-u0‖≤(1+1)D(H(x1),H(x0)),‖v1-v0‖≤(1+1)D(F(y1),F(y0)),‖s1-s0‖≤(1+1)D(G(x1),G(x0)),‖t1-t0‖≤(1+1)D(V(y1),V(y0)),
where D(·,·) is the Hausdorff metric on CB(E1) (for the sake of convenience, we also denote by D(·,·) the Hausdorff metric on CB(E2)). Compute
(2.27)z2′=f(x1)-S(s1,v1)+m(y1)+(I-ρ-1RMρ)(z1′),z2′′=g(y1)-T(u1,t1)+n(x1)+(I-γ-1RNγ)(z1′′).
By induction, we can obtain sequences (xk,yk)∈E1×E2,(sk,vk)∈G(xk)×F(yk),(uk,tk)∈H(xk)×V(yk),(zk′,zk′′)∈E1×E2 by the iterative scheme:
(2.28)f(xk)=JMρ(zk′),g(yk)=JNγ(zk′′),uk∈H(xk):‖uk+1-uk‖≤(1+1k+1)D(H(xk+1),H(xk)),vk∈F(yk):‖vk+1-vk‖≤(1+1k+1)D(F(yk+1),F(yk)),sk∈G(xk):‖sk+1-sk‖≤(1+1k+1)D(G(xk+1),G(xk)),tk∈V(yk):‖tk+1-tk‖≤(1+1k+1)D(V(yk+1),V(yk)),zk+1′=f(xk)-S(sk,vk)+m(yk)+(I-ρ-1RMρ)(zk′),zk+1′′=g(yk)-T(uk,tk)+n(xk)+(I-γ-1RNγ)(zk′′),
for k=0,1,2,….
For positive stepsizeδ′,δ′′, the general system of generalized resolvent equations (2.1) can also be rewritten as
(2.29)f(x,z′)=f(x,z′)-δ′{z′-JMρ(z′)+ρ(S(s,v)-m(y))}=f(x,z′)-δ′{f(x)-JMρ(f(x))+ρ(S(s,v)-m(y))},g(y,z′′)=g(y,z′′)-δ′′{z′′-JNγ(z′′)+γ(T(u,t)-n(x))}=g(y,z′′)-δ′′{g(y)-JNγ(g(y))+γ(T(u,t)-n(x))}.
This fixed point formulation enables us to propose the following iterative algorithm.Algorithm 2.5.
For given(x0,y0)∈E1×E2,(s0,v0)∈G(x0)×F(y0),(u0,t0)∈H(x0)×V(y0),(z0′,z0′′)∈E1×E2, compute (x1,y1)∈E1×E2 and (z1′,z1′′)∈E1×E2 such that
(2.30)f(x1,z1′)=f(x0,z0′)-δ′{f(x0)-JMρ(f(x0))+ρ(S(s0,v0)-m(y0))},g(y1,z1′′)=g(y0,z0′′)-δ′′{g(y0)-JNγ(g(y0))+γ(T(u0,t0)-n(x0))}.
Then, by Nadler [34], there exist (s1,v1)∈G(x1)×F(y1),(u1,t1)∈H(x1)×V(y1) such that
(2.31)‖u1-u0‖≤(1+1)D(H(x1),H(x0)),‖v1-v0‖≤(1+1)D(F(y1),F(y0)),‖s1-s0‖≤(1+1)D(G(x1),G(x0)),‖t1-t0‖≤(1+1)D(V(y1),V(y0)),
where D(·,·) is the Hausdorff metric on CB(E1) (for the sake of convenience, we also denote by D(·,·) the Hausdorff metric on CB(E2)). Compute (x2,y2)∈E1×E2 and (z2′,z2′′)∈E1×E2 such that
(2.32)f(x2,z2′)=f(x1,z1′)-δ′{f(x1)-JMρ(f(x1))+ρ(S(s1,v1)-m(y1))},g(y2,z2′′)=g(y1,z1′′)-δ′′{g(y1)-JNγ(g(y1))+γ(T(u1,t1)-n(x1))}.
By induction, we can obtain sequences (xk,yk)∈E1×E2,(sk,vk)∈G(xk)×F(yk),(uk,tk)∈H(xk)×V(yk),(zk′,zk′′)∈E1×E2 by the iterative scheme:
(2.33)uk∈H(xk):∥uk+1-uk∥≤(1+1k+1)D(H(xk+1),H(xk)),vk∈F(yk):∥vk+1-vk∥≤(1+1k+1)D(F(yk+1),F(yk)),sk∈G(xk):∥sk+1-sk∥≤(1+1k+1)D(G(xk+1),G(xk)),tk∈V(yk):∥tk+1-tk∥≤(1+1k+1)D(V(yk+1),V(yk)),f(xk+1,zk+1′)=f(xk,zk′)-δ′{f(xk)-JMρ(f(xk))+ρ(S(sk,vk)-m(yk))},g(yk+1,zk+1′′)=g(yk,zk′′)-δ′′{g(yk)-JNγ(g(yk))+γ(T(uk,tk)-n(xk))},
for k=0,1,2,….
Note that forδ′=δ′′=1,f(xk,zk′)=f(xk),g(yk,zk′′)=g(yk), Algorithm 2.5 reduces to the following algorithm which solves the general system of variational inclusions (1.14).Algorithm 2.6.
For given(x0,y0)∈E1×E2,(s0,v0)∈G(x0)×F(y0),(u0,t0)∈H(x0)×V(y0), compute (x1,y1)∈E1×E2 such that
(2.34)f(x1)=JMρ[f(x0)-ρ(S(s0,v0)-m(y0))],g(y1)=JNγ[g(y0)-γ(T(u0,t0)-n(x0))].
Then, by Nadler [34], there exist (s1,v1)∈G(x1)×F(y1),(u1,t1)∈H(x1)×V(y1) such that
(2.35)‖u1-u0‖≤(1+1)D(H(x1),H(x0)),‖v1-v0‖≤(1+1)D(F(y1),F(y0)),‖s1-s0‖≤(1+1)D(G(x1),G(x0)),‖t1-t0‖≤(1+1)D(V(y1),V(y0)),
where D(·,·) is the Hausdorff metric on CB(E1) (for the sake of convenience, we also denote by D(·,·) the Hausdorff metric on CB(E2)). Compute (x2,y2)∈E1×E2 such that
(2.36)f(x2)=JMρ[f(x1)-ρ(S(s1,v1)-m(y1))],g(y2)=JNγ[g(y1)-γ(T(u1,t1)-n(x1))].
By induction, we can obtain sequences (xk,yk)∈E1×E2,(sk,vk)∈G(xk)×F(yk),(uk,tk)∈H(xk)×V(yk) by the iterative scheme:
(2.37)f(xk+1)=JMρ[f(xk)-ρ(S(sk,vk)-m(yk))],g(yk+1)=JNγ[g(yk)-γ(T(uk,tk)-n(xk))],uk∈H(xk):‖uk+1-uk‖≤(1+1k+1)D(H(xk+1),H(xk)),vk∈F(yk):‖vk+1-vk‖≤(1+1k+1)D(F(yk+1),F(yk)),sk∈G(xk):‖sk+1-sk‖≤(1+1k+1)D(G(xk+1),G(xk)),tk∈V(yk):‖tk+1-tk‖≤(1+1k+1)D(V(yk+1),V(yk)),
for k=0,1,2,….We now study the convergence analysis of Algorithm2.3. In a similar way, one can study the convergence of other algorithms.Theorem 2.7.
LetE1 and E2 be two real uniformly smooth Banach spaces with modulus of smoothness τE1(t)≤C1t2 and τE2(t)≤C2t2 for C1,C2>0, respectively. Let G:E1→CB(E1), F:E2→CB(E2),H:E1→CB(E1),V:E2→CB(E2) be D-Lipschitz continuous mappings with constants λDG,λDF,λDH, and λDV, respectively, and let M:E1→2E1,N:E2→2E2 be m-accretive mappings such that the resolvent operators associated with M and N are retractions. Let f:E1→E1,g:E2→E2 be both strong accretive with constants α and β, respectively, and Lipschitz continuous with constants δ1 and δ2, respectively. Let m:E2→E1,n:E1→E2 be Lipschitz continuous with constants λm and λn, respectively, and S:E1×E2→E1,T:E1×E2→E2 Lipschitz continuous in the first and second arguments with constants λS1,λS2 and λT1,λT2, respectively.
If there exist constantsρ>0 and γ>0, such that
(2.38)0<B′+θ1+θ41-B′<1,0<B′′+θ2+θ31-B′′<1,
where B′=1-2α+64C1δ12,B′′=1-2β+64C2δ22, and θ1=(1+ρλS1λDG)/(1-ρ(λS1λDG+λS2λDF+λm)),θ2=ρ(λS2λDF+λm)/(1-ρ(λS1λDG+λS2λDF+λm)),θ3=(1+γλT2λDV)/(1-γ(λT1λDH+λT2λDV+λn)),θ4=γ(λT1λDH+λn)/(1-γ(λT1λDH+λT2λDV+λn)), then there exist (x,y)∈E1×E2,(s,v)∈G(x)×F(y),(u,t)∈H(x)×V(y) and (z′,z′′)∈E1×E2 satisfying the general system of generalized resolvent equations (2.1) (in this case, (x,y,u,v,s,t) is a solution of general system of variational inclusions (1.14)), and the iterative sequences {zk′},{zk′′},{xk},{yk}, {uk},{vk},{sk}, and {tk} generated by Algorithm 2.3 converge strongly to z′,z′′,x,y,u,v,s, and t, respectively.Proof.
From Algorithm2.3 we have
(2.39)‖zk+1′-zk′‖=‖f(xk)-ρ(S(sk,vk)-m(yk))-[f(xk-1)-ρ(S(sk-1,vk-1)-m(yk-1))]‖≤‖xk-xk-1-(f(xk)-f(xk-1))‖+‖xk-xk-1-ρ[(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1))]‖.
By Proposition 1.1, we have (see, e.g., the proof of [32, Theorem  3])
(2.40)‖xk-xk-1-(f(xk)-f(xk-1))‖2≤(1-2α+64Cδ12)‖xk-xk-1‖2.
Since S is Lipschitz continuous in both arguments, G,F are D-Lipschitz continuous, and m is Lipschitz continuous, we have
(2.41)‖(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1))‖=‖S(sk,vk)-S(sk-1,vk)+S(sk-1,vk)-S(sk-1,vk-1)-(m(yk)-m(yk-1))‖≤‖S(sk,vk)-S(sk-1,vk)‖+‖S(sk-1,vk)-S(sk-1,vk-1)‖+‖m(yk)-m(yk-1)‖≤λS1‖sk-sk-1‖+λS2‖vk-vk-1‖+λm‖yk-yk-1‖≤λS1(1+1k)D(G(xk),G(xk-1))+λS2(1+1k)D(F(yk),F(yk-1))+λm‖yk-yk-1‖≤(1+1k)λS1λDG‖xk-xk-1‖+(1+1k)λS2λDF‖yk-yk-1‖+λm‖yk-yk-1‖=(1+1k)λS1λDG‖xk-xk-1‖+[(1+1k)λS2λDF+λm]‖yk-yk-1‖.
Utilizing (2.41) and Proposition 1.1, we have
(2.42)‖xk-xk-1-ρ[(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1))]‖2≤‖xk-xk-1‖2-2ρ〈(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1)),J(xk-xk-1-ρ[(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1))])〉≤‖xk-xk-1‖2+2ρ‖(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1))‖×‖xk-xk-1-ρ[(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1))]‖≤‖xk-xk-1‖2+2ρ{(1+1k)λS1λDG‖xk-xk-1‖+[(1+1k)λS2λDF+λm]‖yk-yk-1‖}×‖xk-xk-1-ρ[(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1))]‖≤‖xk-xk-1‖2+ρ(1+1k)λS1λDG×{‖xk-xk-1‖2+‖xk-xk-1-ρ[(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1))]‖2}+ρ[(1+1k)λS2λDF+λm]×{‖yk-yk-1‖2+‖xk-xk-1-ρ[(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1))]‖2}=[1+ρ(1+1k)λS1λDG]‖xk-xk-1‖2+ρ[(1+1k)(λS1λDG+λS2λDF)+λm]×‖xk-xk-1-ρ[(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1))]‖2+ρ[(1+1k)λS2λDF+λm]‖yk-yk-1‖2,
which implies that
(2.43)‖xk-xk-1-ρ[(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1))]‖2≤1+ρ(1+1/k)λS1λDG1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]‖xk-xk-1‖2+ρ[(1+1/k)λS2λDF+λm]1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]‖yk-yk-1‖2≤1+ρ(1+1/k)λS1λDG1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]‖xk-xk-1‖2+ρ[(1+1/k)λS2λDF+λm]1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]‖yk-yk-1‖2+21+ρ(1+1/k)λS1λDG1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]ρ[(1+1/k)λS2λDF+λm]1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]×‖xk-xk-1‖‖yk-yk-1‖={1+ρ(1+1/k)λS1λDG1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]‖xk-xk-1‖+ρ[(1+1/k)λS2λDF+λm]1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]‖yk-yk-1‖}2.
Thus, we have
(2.44)‖xk-xk-1-ρ[(S(sk,vk)-m(yk))-(S(sk-1,vk-1)-m(yk-1))]‖≤1+ρ(1+1/k)λS1λDG1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]‖xk-xk-1‖+ρ[(1+1/k)λS2λDF+λm]1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]‖yk-yk-1‖.
Note that
(2.45)lim⁡k→∞1+ρ(1+1/k)λS1λDG1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]=θ1,lim⁡k→∞ρ[(1+1/k)λS2λDF+λm]1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]=θ2,
where θ1=(1+ρλS1λDG)/(1-ρ(λS1λDG+λS2λDF+λm)) and θ2=ρ(λS2λDF+λm)/(1-ρ(λS1λDG+λS2λDF+λm)).
Utilizing (2.40) and (2.44), we deduce from (2.39) that
(2.46)‖zk+1′-zk′‖≤(1-2α+64Cδ12+1+ρ(1+1/k)λS1λDG1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm])‖xk-xk-1‖+ρ[(1+1/k)λS2λDF+λm]1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]‖yk-yk-1‖=(B′+1+ρ(1+1/k)λS1λDG1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm])‖xk-xk-1‖+ρ[(1+1/k)λS2λDF+λm]1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]‖yk-yk-1‖,
where B′:=1-2α+64Cδ12.
On the other hand, again from Algorithm2.3 we have
(2.47)‖zk+1′′-zk′′‖=‖g(yk)-γ(T(uk,tk)-n(xk))-[g(yk-1)-γ(T(uk-1,tk-1)-n(xk-1))]‖≤‖yk-yk-1-(g(yk)-g(yk-1))‖+‖yk-yk-1-γ[(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(xk-1))]‖.
Utilizing the same arguments as those for (2.40), we have
(2.48)‖yk-yk-1-(g(yk)-g(yk-1))‖2≤(1-2β+64C2δ22)‖yk-yk-1‖2.
Since T is Lipschitz continuous in both arguments, H,V are D-Lipschitz continuous, and n is Lipschitz continuous, we have
(2.49)‖(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(xk-1))‖=‖T(uk,tk)-T(uk-1,tk)+T(uk-1,tk)-T(uk-1,tk-1)-(n(xk)-n(xk-1))‖≤‖T(uk,tk)-T(uk-1,tk)‖+‖T(uk-1,tk)-T(uk-1,tk-1)‖+‖n(xk)-n(xk-1)‖≤λT1‖uk-uk-1‖+λT2‖tk-tk-1‖+λn‖xk-xk-1‖≤λT1(1+1k)D(H(xk),H(xk-1))+λT2(1+1k)D(V(yk),V(yk-1))+λn‖xk-xk-1‖≤(1+1k)λT1λDH‖xk-xk-1‖+(1+1k)λT2λDV‖yk-yk-1‖+λn‖xk-xk-1‖=[(1+1k)λT1λDH+λn]‖xk-xk-1‖+(1+1k)λT2λDV‖yk-yk-1‖.
Utilizing (2.49) and Proposition 1.1, we have
(2.50)‖yk-yk-1-γ[(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(xk-1))]‖2≤‖yk-yk-1‖2-2γ〈(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(yk-1)),J(yk-yk-1-γ[(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(xk-1))])〉≤‖yk-yk-1‖2+2γ‖(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(yk-1))‖×‖yk-yk-1-γ[(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(xk-1))]‖≤‖yk-yk-1‖2+2γ{[(1+1k)λT1λDH+λn]‖xk-xk-1‖+(1+1k)λT2λDV‖yk-yk-1‖}×‖yk-yk-1-γ[(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(xk-1))]‖≤‖yk-yk-1‖2+γ[(1+1k)λT1λDH+λn]×{‖xk-xk-1‖2+‖yk-yk-1-γ[(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(xk-1))]‖2}+γ(1+1k)λT2λDV×{‖yk-yk-1‖2+‖yk-yk-1-γ[(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(xk-1))]‖2}=[1+γ(1+1k)λT2λDV]‖yk-yk-1‖2+γ[(1+1k)(λT1λDH+λT2λDV)+λn]×‖yk-yk-1-γ[(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(xk-1))]‖2+γ[(1+1k)λT1λDH+λn]‖xk-xk-1‖2,
which implies that
(2.51)‖yk-yk-1-γ[(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(xk-1))]‖2≤1+γ(1+1/k)λT2λDV1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]‖yk-yk-1‖2+γ[(1+1/k)λT1λDH+λn]1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]‖xk-xk-1‖2≤1+γ(1+1/k)λT2λDV1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]‖yk-yk-1‖2+γ[(1+1/k)λT1λDH+λn]1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]‖xk-xk-1‖2+21+γ(1+1/k)λT2λDV1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]γ[(1+1/k)λT1λDH+λn]1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]×‖yk-yk-1‖‖xk-xk-1‖={1+γ(1+1/k)λT2λDV1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]‖yk-yk-1‖+γ[(1+1/k)λT1λDH+λn]1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]‖xk-xk-1‖}2.
Thus, we have
(2.52)‖yk-yk-1-γ[(T(uk,tk)-n(xk))-(T(uk-1,tk-1)-n(xk-1))]‖≤1+γ(1+1/k)λT2λDV1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]‖yk-yk-1‖+γ[(1+1/k)λT1λDH+λn]1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]‖xk-xk-1‖.
Note that
(2.53)lim⁡k→∞1+γ(1+1/k)λT2λDV1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]=θ3,lim⁡k→∞γ[(1+1/k)λT1λDH+λn]1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]=θ4,
where θ3=(1+γλT2λDV)/(1-γ(λT1λDH+λT2λDV+λn)) and θ4=γ(λT1λDH+λn)/(1-γ(λT1λDH+λT2λDV+λn)).
Utilizing (2.48) and (2.52), we deduce from (2.47) that
(2.54)‖zk+1′′-zk′′‖≤(1-2β+64Cδ22+1+γ(1+1/k)λT2λDV1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn])‖yk-yk-1‖+γ[(1+1/k)λT1λDH+λn]1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]‖xk-xk-1‖=(B′′+1+γ(1+1/k)λT2λDV1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn])‖yk-yk-1‖+γ[(1+1/k)λT1λDH+λn]1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]‖xk-xk-1‖,
where B′′:=1-2β+64Cδ22.
Adding (2.46) and (2.54), we have
(2.55)‖zk+1′-zk′‖+‖zk+1′′-zk′′‖≤(B′+1+ρ(1+1/k)λS1λDG1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]+γ[(1+1/k)λT1λDH+λn]1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn])‖xk-xk-1‖+(B′′+ρ[(1+1/k)λS2λDF+λm]1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]+1+γ(1+1/k)λT2λDV1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn])‖yk-yk-1‖.
Also from (2.21), we have
(2.56)‖xk-xk-1‖=‖xk-xk-1-(f(xk)-f(xk-1))+JMρ(zk′)-JMρ(zk-1′)‖≤‖xk-xk-1-(f(xk)-f(xk-1))‖+‖JMρ(zk′)-JMρ(zk-1′)‖≤B′∥xk-xk-1∥+∥zk′-zk-1′∥,
which implies that
(2.57)‖xk-xk-1‖≤11-B′‖zk′-zk-1′‖,(2.58)‖yk-yk-1‖=‖yk-yk-1-(g(yk)-g(yk-1))+JNγ(zk′′)-JNγ(zk-1′′)‖≤‖yk-yk-1-(g(yk)-g(yk-1))‖+‖JNγ(zk′′)-JNγ(zk-1′′)‖≤B′′‖yk-yk-1‖+‖zk′′-zk-1′′‖,
which implies that
(2.59)‖yk-yk-1‖≤11-B′′‖zk′′-zk-1′′‖.
Utilizing (2.57) and (2.59), we conclude from (2.55) that
(2.60)‖zk+1′-zk′‖+‖zk+1′′-zk′′‖≤{B′+1+ρ(1+1/k)λS1λDG1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]+γ[(1+1/k)λT1λDH+λn]1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]}11-B′‖zk′-zk-1′‖+{B′′+ρ[(1+1/k)λS2λDF+λm]1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]+1+γ(1+1/k)λT2λDV1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]}11-B′′‖zk′′-zk-1′′‖.
Observe that
(2.61)lim⁡k→∞{B′+1+ρ(1+1/k)λS1λDG1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]+γ[(1+1/k)λT1λDH+λn]1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]}11-B′=B′+θ1+θ41-B′≤θ,lim⁡k→∞{B′′+ρ[(1+1/k)λS2λDF+λm]1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]+1+γ(1+1/k)λT2λDV1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]}11-B′′=B′′+θ2+θ31-B′′≤θ,
where
(2.62)θ∶=max⁡{B′+θ1+θ41-B′,B′′+θ2+θ31-B′′}.
By (2.38), we know that 0<θ<1. Now take a fixed θ0∈(θ,1) arbitrarily. Then from (2.61) and (2.31) it follows that there exists an integer k̅≥1 such that for all k≥k̅,
(2.63){B′+1+ρ(1+1/k)λS1λDG1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]+γ[(1+1/k)λT1λDH+λn]1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]}11-B′<θ0,{B′′+ρ[(1+1/k)λS2λDF+λm]1-ρ[(1+1/k)(λS1λDG+λS2λDF)+λm]+1+γ(1+1/k)λT2λDV1-γ[(1+1/k)(λT1λDH+λT2λDV)+λn]}11-B′′<θ0.
So, we obtain from (2.60) that
(2.64)‖zk+1′-zk′‖+‖zk+1′′-zk′′‖≤θ0(‖zk′-zk-1′‖+‖zk′′-zk-1′′‖),∀k≥k̅,
which implies that {zk′} and {zk′′} are both Cauchy sequences. Thus, there exist z′∈E1 and z′′∈E2 such that zk′→z′ and zk′′→z′′ as k→∞. From (2.57) and (2.59) it follows that {xk} and {yk} are also Cauchy sequences in E1 and E2, respectively, that is, there exist x∈E1,y∈E2 such that xk→x and yk→y as k→∞.
Also from (2.22), we have
(2.65)‖uk+1-uk‖≤(1+1k+1)D(H(xk+1),H(xk))≤(1+1k+1)λDH‖xk+1-xk‖,‖vk+1-vk‖≤(1+1k+1)D(F(yk+1),F(yk))≤(1+1k+1)λDF‖yk+1-yk‖,‖sk+1-sk‖≤(1+1k+1)D(G(xk+1),G(xk))≤(1+1k+1)λDG‖xk+1-xk‖,‖tk+1-tk‖≤(1+1k+1)D(V(yk+1),V(yk))≤(1+1k+1)λDV‖yk+1-yk‖,
and hence, {uk},{vk},{sk}, and {tk} are also Cauchy sequences. Accordingly, there exist u,s∈E1 and v,t∈E2 such that uk→u,vk→v,sk→s, and tk→t, respectively.
Now, we will show thatu∈H(x),v∈F(y),s∈G(x), and t∈V(y). Indeed, since uk∈H(xk) and
(2.66)d(uk,H(x))≤max⁡{d(uk,H(x)),sup⁡w1∈H(x)d(H(xk),w1)}≤max⁡{sup⁡w2∈H(xk)d(w2,H(x)),sup⁡w1∈H(x)d(H(xk),w1)}=D(H(xk),H(x)),
we have
(2.67)d(u,H(x))≤‖u-uk‖+d(uk,H(x))≤‖u-uk‖+D(H(xk),H(x))≤‖u-uk‖+λDH‖xk-x‖→0ask→∞,
which implies that d(u,H(x))=0. Taking into account that H(x)∈CB(E1), we deduce that u∈H(x). Similarly, we can show that v∈F(y),s∈G(x) and t∈V(y). By the continuity of f,g,m,n,G,F,H,V,S,T,JMρ,JNγ and Algorithm 2.3, we have
(2.68)z′=f(x)-ρ(S(s,v)-m(y))=JMρ(z′)-ρ(S(s,v)-m(y))∈E1,z′′=g(y)-γ(T(u,t)-n(x))=JNγ(z′′)-γ(T(u,t)-n(x))∈E2.
By Propositions 2.1 and 2.2, the required result follows.



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*Source: 487394-2012-03-25.xml*