# Strong Convergence of Iterative Algorithm for a New System of GeneralizedH·,·-η-Cocoercive Operator Inclusions in Banach Spaces

Authors: Saud M. Alsulami; Eskandar Naraghirad; Nawab Hussain
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduce and study a new system of generalizedH·,·-η-cocoercive operator inclusions in Banach spaces. Using the resolvent operator technique associated with H·,·-η-cocoercive operators, we suggest and analyze a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method. We highlight the applicability of our results by examples in function spaces.


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## Body

## 1. Introduction

The resolvent operator technique is a powerful tool to study the approximation solvability of nonlinear variational inequalities and variational inclusions, which have been applied widely to optimization and control, mechanics and physics, economics and transportation equilibrium, and engineering sciences, see, for example, [1–4] and the references therein.In a series of papers [5–8], the authors investigated (A,η)-accretive and H(·,·)-accretive operators for solving variational inclusions in Banach spaces. Convergence and stability of iterative algorithms for the systems of (A,η)-accretive operators have been studied in [9, 10]. The notion of (H,ϕ)-η-monotone operators has been introduced and investigated by the authors in [11]. Generalized mixed variational inclusions involving (H(·,·),η)-monotone operators have been discussed in [12]. Some results on H((·,·),η)-accretive operators and application for solving set-valued variational inclusions in Banach spaces have been proved in [7]. Some other related articles on the variational inclusion problems can be found in [13–22].Very recently, Ahmad et al. [23] introduced a new H(·,·)-η-cocoercive operator and its resolvent operator in the setting of Banach spaces. The authors proposed concrete examples in support of H(·,·)-η-cocoercive operators and they also proved the Lipschitz continuity of resolvent operator associated with H(·,·)-η-cocoercive operator. Motivated and inspired by the research works mentioned above, in this paper, we introduce and study a new system of H(·,·)-η-cocoercive mapping inclusions in Banach spaces. Using the resolvent operator associated with H(·,·)-η-cocoercive mapping, we suggest and analyze a new general algorithm and establish the existence and uniqueness of solutions for this system of H(·,·)-η-cocoercive mappings.


## 2. Preliminaries

Throughout this paper, we denote the set of positive integers byℕ. Let X be a Banach space with the norm ∥·∥ and the dual space X*. For any x∈X, we denote the value of x*∈X* at x by 〈x,x*〉. When {xn} is a sequence in X, we denote the strong convergence of {xn} to x∈X by xn→x as n→∞. We denote by 2X the family of all nonempty subsets of X. Let CB(X) be the family of all nonempty, closed, and bounded subsets of X. The Hausdörff metric on CB(X) [24] is defined by
(1)D(A,B)=max{supx∈Ad(x,B),supy∈Bd(A,y)},A,B∈CB(X),
where d(x,B)=infb∈B∥x-b∥ and d(A,y)=infa∈A∥a-y∥.Definition 1 (see [25]).
A continuous and strictly increasing functionϕ:[0,+∞)→[0,∞) such that ϕ(0)=0 and limt→∞ϕ(t)=∞ is called a gauge function.Definition 2 (see [25]).
LetX be a Banach space. Given a gauge function ϕ, the mapping Jϕ:X→2X* corresponding to ϕ defined by
(2)Jϕ(x)={x*∈X*:〈x,x*〉=∥x∥∥x*∥,∥x*∥=ϕ(∥x∥)},∀x∈X,
is called the duality mapping with gauge function ϕ.
In particular, ifϕ(t)=t, the duality map  J=Jϕ is called the normalized duality mapping.Lemma 3 (see [26]).
LetX be a real Banach space and J:X→2X* be the normalized duality mapping. Then, for any x,y∈X,
(3)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,
for all j(x+y)∈J(x+y).Definition 4.
LetX be a Banach space. Let A:X→X and η:X×X→X be two mappings and  J:X→2X* be the normalized duality mapping. Then, A is called(i)
η-cocoercive, if there exists a constant μ1>0 such that
(4)〈Ax-Ay,j(η(x,y))〉≥μ1∥Ax-Ay∥2,∀x,y∈X,j(η(x,y))∈J(η(x,y)),(ii)
η-accretive, if
(5)〈Ax-Ay,j(η(x,y))〉≥0,∀x,y∈X,j(η(x,y))∈J(η(x,y)),(iii)
η-strongly accretive, if there exists a constant β1>0 such that
(6)〈Ax-Ay,j(η(x,y))〉≥β1∥x-y∥2,∀x,y∈X,j(η(x,y))∈J(η(x,y)),(iv)
η-relaxed cocoercive, if there exists a constant γ1>0 such that
(7)〈Ax-Ay,j(η(x,y))〉≥(-γ1)∥Ax-Ay∥2,∀x,y∈X,j(η(x,y))∈J(η(x,y)),(v)
Lipschitz continuous, if there exists a constantλA>0 such that
(8)∥Ax-Ay∥≤λA∥x-y∥,∀x,y∈X,(vi)
α-expansive, if there exists a constant α>0 such that
(9)∥Ax-Ay∥≥α∥x-y∥,∀x,y∈X,(vii)
η is said to be Lipschitz continuous, if there exists a constant τ>0 such that
(10)∥η(x,y)∥≤τ∥x-y∥,∀x,y∈X.Definition 5.
LetX be a Banach space. Let A,B:X→X, H:X×X→X, η:X×X→X be four single-valued mappings and J:X→2X* be the normalized duality mapping. Then,(i)
H(A,·) is said to be η-cocoercive with respect to A, if there exists a constant μ>0 such that
(11)〈H(Ax,u)-H(Ay,u),j(η(x,y))〉≥μ∥Ax-Ay∥2,∀x,y,u∈X,j(η(x,y))∈J(η(x,y)),(ii)
H(·,B) is said to be η-relaxed cocoercive with respect to B, if there exists a constant γ>0 such that
(12)〈H(u,Bx)-H(u,By),j(η(x,y))〉≥(-γ)∥Bx-By∥2,∀x,y,u∈X,j(η(x,y))∈J(η(x,y)),(iii)
H(A,·) is said to be r1-Lipschitz continuous with respect to A, if there exists a constant r1>0 such that
(13)∥H(Ax,u)-H(Ay,u)∥≤r1∥x-y∥,∀x,y,u∈X,(iv)
H(·,B) is said to be r2-Lipschitz continuous with respect to B, if there exists a constant r2>0 such that
(14)∥H(u,Bx)-H(u,By)∥≤r2∥x-y∥,∀x,y,u∈X.Definition 6.
LetX be a Banach space. A set-valued mapping M:X→2X is said to be η-cocoercive, if there exists a constant μ2>0 such that
(15)〈u-v,j(η(x,y))〉≥μ2∥u-v∥2,∀x,y∈X,u∈M(x),v∈M(y),j(η(x,y))∈J(η(x,y)).Definition 7.
LetX be a Banach space. A mapping T:X→CB(X) is said to be 𝒟-Lipschitz continuous, if there exists a constant λT>0 such that
(16)𝒟(Tx,Ty)≤λT∥x-y∥,x,y∈X.Definition 8.
LetX be a Banach space. Let T,Q:X→CB(X) be the mappings. A mapping N:X×X→X is said to be(i)
Lipschitz continuous in the first argument with respect toT, if there exists a constant t1>0 such that
(17)∥N(w1,·)-N(w2,·)∥≤t1∥w1-w2∥,∀u1,u2∈X,w1∈T(u1),w2∈T(u2),(ii)
Lipschitz continuous in the second argument with respect toQ, if there exists a constant t2>0 such that
(18)∥N(·,v1)-N(·,v2)∥≤t2∥v1-v2∥,∀u1,u2∈X,v1∈Q(u1),v2∈Q(u2),(iii)
η-relaxed Lipschitz in the first argument with respect to T, if there exists a constant τ1>0 such that
(19)〈N(w1,·)-N(w2,·),j(η(u1,u2))〉≤(-τ1)∥u1-u2∥2,∀u1,u2∈X,w1∈T(u1),w2∈T(u2),j(η(u1,u2))∈J(η(u1,u2)),(iv)
η-relaxed Lipschitz in the second argument with respect to Q, if there exists a constant τ2>0 such that
(20)〈N(·,v1)-N(·,v2),j(η(u1,u2))〉≤(-τ2)∥u1-u2∥2,∀u1,u2∈X,v1∈Q(u1),v2∈Q(u2),j(η(u1,u2))∈J(η(u1,u2)).Definition 9.
LetX be a Banach space. Let A,B:X→X, H:X×X→X, η:X×X→X be four single-valued mappings. Let M:X→2X be a set-valued mapping. M is said to be H(·,·)-η-cocoercive operator with respect to A and B, if M is η-cocoercive and (H(A,B)+λM)(X)=X, for every λ>0.Example 10.
LetX=ℝ×ℝ and A,B:X→X be defined by
(21)A(x1,x2)=(2x1-x2,x1-x2),B(y1,y2)=(-2y2,y1-y2),∀(x1,x2),(y1,y2)∈X.
Assume now that H(A,B), η:X×X→X are defined by
(22)H(Ax,By)=Ax+By,η(x,y)=x-y,∀x,y∈X.
LetM=I, where I is the identity mapping. Then, M is H(·,·)-η-cocoercive with respect to A and B.Example 11.
LetX=C[0,1], the space of all real valued continuous functions defined on closed interval [0,1] with the norm
(23)∥f∥=maxt∈[0,1]|f(t)|.
Let A,B:X→X be defined by
(24)A(f)=sin2(f),B(g)=cos2(g),∀f,g∈X.
Let H(A,B):X×X→X be defined by
(25)H(A(f),B(g))=A(f)+B(g),∀f,g∈X.
Suppose that M(f)=f2, where f2(t)=f(t)f(t) for all t∈[0,1]. Then, for λ=1, we conclude that
(26)∥(H(A,B)+M)(f)∥=∥A(f)+B(f)+f2∥=maxt∈[0,1]|sin2(f(t))+cos2(f(t))+f2(t)|=1+f2(t)>0.
This proves that 0∉(H(A,B)+M)(X) and M is not H(A,B)-η-cocoercive with respect to A and B.Proposition 12 (see [23]).
LetH(A,B) be η-cocoercive with respect to A with constant μ>0 and η-relaxed cocoercive with respect to B with constant γ>0, A be α-expansive and B be β-Lipschitz continuous μ>γ and α>β. Let M:X→2X be H(A,B)-η-cocoercive operator. Suppose that
(27)〈x-y,j(η(u,v))〉≥0,∀(v,y)∈Graph(M),j(η(u,v))∈J(η(u,v)).
Then,  x∈Mu, where Graph(M)={(u,x)∈X×X:x∈Mu}.Theorem 13 (see [23]).
LetH(A,B) be η-cocoercive with respect to A with constant μ>0 and η-relaxed cocoercive with respect to B with constant γ>0, A be α-expansive and B be β-Lipschitz continuous, μ>γ and α>β. Let M be an H(·,·)-η-cocoercive operator with respect to A and B. Then, for each λ>0, the operator (H(A,B)+λM)-1 is single-valued.Definition 14.
LetX be a Banach space. Let H(A,B) be η-cocoercive with respect to A with constant μ>0 and η-relaxed cocoercive with respect to B with constant γ>0, A be α-expansive B be β-Lipschitz continuous and η be β-Lipschitz continuous, μ>γ, and α>β. Let M be a H(·,·)-η-cocoercive operator with respect to A and B. Then, the resolvent Rλ,MH(·,·)-η:X→X is defined by
(28)Rλ,MH(·,·)-η(u)=(H(A,B)+λM)-1(u),∀u∈X.Theorem 15 (see [23]).
LetX be a Banach space. Let H(A,B) be η-cocoercive with respect to A with constant μ>0 and η-relaxed cocoercive with respect to B with constant γ>0, A be α-expansive B be β-Lipschitz continuous, and η be ρ-Lipschitz continuous; μ>γ and α>β. Let M be H(·,·)-η-cocoercive operator with respect to A and B. Then, the resolvent operator Rλ,MH(·,·)-η:X→X is ρ/(μα2-γβ2)-Lipschitz continuous, that is,
(29)∥Rλ,MH(·,·)-η(u)-Rλ,MH(·,·)-η(v)∥≤ρμα2-γβ2∥u-v∥,∀u,v∈X.


## 3. Strong Convergence Theorem

In this section, using the resolvent operator technique associated withH(·,·)-η-cocoercive operators, we propose a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method.Fori=1,2, let Xi be real Banach spaces with the norm ∥·∥i. Let Ai,Bi:Xi→Xi, Hi:Xi×Xi→Xi, ηi:Xi×Xi→Xi, F:X1×X2→X1, and G:X1×X2→X2 be single-valued mappings, and T:X1→CB(X1), Q:X2→CB(X2) be set-valued mappings. Let M:X1×X1→2X1, N:X2×X2→2X2 be H1(·,·)-η1-cocoercive and H2(·,·)-η2-cocoercive operators with respect to (A1,B1) and (A2,B2), respectively. We consider the following problem.Find(x,y)∈X1×X2, w∈T(x), and v∈Q(y) such that
(30)0∈M(x,x)+F(w,v),0∈N(y,y)+G(w,v).
We call problem (30) a system of generalized H(·,·)-η-cocoercive operator inclusions.Under the assumptions mentioned above, we have the following key and simple lemma.Lemma 16.
(
x
,
y
)
∈
X
1
×
X
2, w∈T(x), v∈Q(y) is a solution of problem (30) if and only if
(31)x=Rλ1,M(·,x)H1(·,·)-η[H1(A1x,B1x)-λ1F(w,v)],y=Rλ2,N(·,y)H2(·,·)-η[H2(A2y,B2y)-λ2G(w,v)],
where Rλ1,M(·,x)H1(·,·)-η=(H1(A1x,B1x)+λ1M(·,x))-1, Rλ1,N(·,y)H2(·,·)-η=(H2(A2y,B2y)+λ2N(·,y))-1, and λ1,λ2>0 are constants.Proof .
This is an easy and direct consequence of Definition14.Algorithm 17.
Fori=1,2, let Xi be real Banach spaces with the norm ∥·∥i. Let Ai,Bi:Xi→Xi, Hi:Xi×Xi→Xi, ηi:Xi×Xi→Xi, F:X1×X2→X1, and G:X1×X2→X2 be single-valued mappings, and T:X1→CB(X1), Q:X2→CB(X2) be set-valued mappings. Let M:X1×X1→→2X1, N:X2×X2→2X2 be such that, for each fixed x∈X1, y∈X2, M(·,x) and N(·,y) are H1(·,·)-η1-cocoercive and H2(·,·)-η2-cocoercive operators with respect to (A1,B1) and (A2,B2), respectively. For any given constants λi>0(i=1,2), define the mappings S1:X1×X2→X1 and S2:X1×X2→X2 by
(32)S1(x,y)=⋃w∈T(x)⋃v∈Q(y)Rλ1,M(·,x)H1(·,·)-η1[H1(A1x,B1x)-λ1F(w,v)],S2(x,y)=⋃w∈T(x)⋃v∈Q(y)Rλ2,N(·,y)H2(·,·)-η2[H2(A2y,B2y)-λ2G(w,v)].
For any given (x0,y0)∈X1×X2, w0∈T(x0), v0∈Q(y0), let
(33)z0=Rλ1,M(·,x0)H1(·,·)-η1×[H1(A1x0,B1x0)-λ1F(w0,v0)]∈S1(x0,y0),u0=Rλ2,N(·,y0)H2(·,·)-η2×[H2(A2y0,B2y0)-λ2G(w0,v0)]∈S2(x0,y0).
Since w0∈T(x0)⊂CB(X1) and v0∈Q(y0)⊂CB(X2), in view of Nadler's theorem [24], there exist w1∈T(x1) and v1∈Q(y1) such that
(34)∥w1-w0∥1≤(1+1)D(T(x1),T(x0)),∥v1-v0∥2≤(1+1)D(Q(y1),Q(y0)).
By induction, we define iterative sequences {xn}, {yn}, {wn}, and {vn} as follows:
(35)xn+1=Rλ1,M(·,xn)H1(·,·)-η1×[H1(A1xn,B1xn)-λ1F(wn,vn)]∈S1(xn,yn),yn+1=Rλ2,N(·,yn)H2(·,·)-η2×[H2(A2yn,B2yn)-λ2G(wn,vn)]∈S2(xn,yn),wn∈T(xn),∥wn+1-wn∥1≤(1+1n+1)D(T(xn+1),T(xn)),vn∈Q(yn),∥vn+1-vn∥2≤(1+1n+1)D(Q(yn+1),Q(yn)),
where n=0,1,2,…, and λ1,λ2>0 are constants.Theorem 18.
Fori=1,2, let Xi be real Banach spaces with the norm ∥·∥i. Let Ai,Bi:Xi→Xi, Hi:Xi×Xi→Xi, ηi:Xi×Xi→Xi, F:X1×X2→X1, and G:X1×X2→X2 be single-valued mappings, and T:X1→CB(X1), Q:X2→CB(X2) be set-valued mappings. Let M:X1×X1→2X1 and N:X2×X2→2X2 be such that, for each fixed x∈X1,y∈X2, M(·,x) and N(·,y) are H1(·,·)-η1-cocoercive and H2(·,·)-η2-cocoercive operators with respect to (A1,B1) and (A2,B2), respectively. Suppose that the following conditions are satisfied.(i)
Hi(Ai,Bi) is ηi-cocoercive with respect to Ai with constant μi and ηi-relaxed cocoercive with respect to Bi with constant γi, i=1,2.(ii)
Ai is αi-expansive and Bi is βi-Lipschitz continuous, i=1,2.(iii)
Hi(Ai,Bi) is ri-Lipschitz continuous with respect to Ai and si-Lipschitz continuous with respect to Bi, i=1,2.(iv)
T is D-Lipschitz continuous with constant λT and Q is D-Lipschitz continuous with constant λQ.(v)
F is t1-Lipschitz continuous with respect to T in the first argument and t2-Lipschitz continuous with respect to Q in the second argument.(vi)
G is l1-Lipschitz continuous with respect to T in the first argument and l2-Lipschitz continuous with respect to Q in the second argument.(vii)
ηi is ρi-Lipschitz continuous, i=1,2.(viii)
F is η1-relaxed Lipschitz continuous with respect to T in the first argument and η1-relaxed Lipschitz continuous with respect to Q in the second argument with constants τ1 and τ2, respectively.(ix)
G is η2-relaxed Lipschitz continuous with respect to T in the first argument and η2-relaxed Lipschitz continuous with respect to Q in the second argument with constants ϵ1 and ϵ2, respectively. Furthermore, assume that there exist constants σ1,σ2>0 such that
(36)∥Rλ1,M(·,x1)H1(·,·)-η1(x)-Rλ1,M(·,x2)H1(·,·)-η1(x)∥1≤σ1∥x1-x2∥1,∀x,x1,x2∈X1,(37)∥Rλ2,N(·,y1)H2(·,·)-η2(y)-Rλ2,N(·,y2)H2(·,·)-η2(y)∥2≤σ2∥y1-y2∥2,∀y,y1,y2∈X2,and λ1,λ2>0 are constants satisfying the following conditions:
(38)p0=r12+2λ1t1λT[r1+λ1t1λT+ρ1]-2λ1τ1,q0=r22+2λ2l2λQ[r2+λ2l2λQ+ρ2]-2λ2ϵ2,r12+2λ1t1λT[r1+λ1t1λT+ρ1]>2λ1τ1,r22+2λ2l2λQ[r2+λ2l2λQ+ρ2]>2λ2ϵ2,μi>γi,αi>βi,i=1,2,θ1=ρ1μ1α12-γ1β12,θ2=ρ2μ2α22-γ2β22,σ1+θ1p0+θ1s1+λ2θ2l1λT<1,σ2+θ2q0+θ2s2+λ1θ1t2λQ<1.
Then, the iterative sequences {xn}, {yn}, {wn}, and {vn} generated by Algorithm 17 converge strongly to x, y, w, and v, respectively, and (x,y,w,v) is a solution of problem (30).Proof.
In view of Theorem13, the resolvent operator Rλ1,MH1(·,·)-η1 is θ1-Lipschitz continuous. This, together with Algorithm 17 and (36), implies that(39)∥xn+1-xn∥1=∥Rλ1,M(·,xn)H1(·,·)-η1[H1(A1xn,B1xn)-λ1F(wn,vn)]-Rλ1,M(·,xn-1)H1(·,·)-η1×[H1(A1xn-1,B1xn-1)-λ1F(wn-1,vn-1)]Rλ1,M(.,xn-1)H1(.,.)-η1∥1≤∥Rλ1,M(·,xn)H1(·,·)-η1[H1(A1xn,B1xn)-λ1F(wn,vn)]-Rλ1,M(·,xn-1)H1(·,·)-η1[H1(A1xn,B1xn)-λ1F(wn,vn)]∥1+∥Rλ1,M(·,xn-1)H1(·,·)-η1[H1(A1xn,B1xn)-λ1F(wn,vn)]-Rλ1,M(·,xn-1)H1(·,·)-η1×[H1(A1xn-1,B1xn-1)-λ1F(wn-1,vn-1)]Rλ1,M(.,xn-1)H1(.,.)-η1∥1≤σ1∥xn-xn-1∥1+θ1∥H1(A1xn,B1xn)-H1(A1xn-1,B1xn-1)-λ1(F(wn,vn)-F(wn-1,vn-1))∥1≤σ1∥xn-xn-1∥1+θ1∥H1(A1xn,B1xn)-H1(A1xn-1,B1xn-1)-λ1(F(wn,vn)-F(wn-1,vn))∥1+λ1θ1∥F(wn-1,vn)-F(wn-1,vn-1)∥1≤σ1∥xn-xn-1∥1+θ1∥H1(A1xn,B1xn)-H1(A1xn-1,B1xn)-λ1(F(wn,vn)-F(wn-1,vn))∥1+θ1∥H1(A1xn-1,B1xn)-H1(A1xn-1,B1xn-1)∥1+λ1θ1∥F(wn-1,vn)-F(wn-1,vn-1)∥1.
Since F is t1-Lipschitz continuous with respect to T in the first argument and t2-Lipschitz continuous in the second argument, T is λT-Lipschitz continuous, and Q is λQ-Lipschitz continuous, by Algorithm 17, we get
(40)∥F(wn,vn)-F(wn-1,vn)∥1≤t1∥wn-wn-1∥1≤t1(1+1n)D(T(xn),T(xn-1))≤t1λT(1+1n)∥xn-xn-1∥1,(41)∥F(wn-1,vn)-F(wn-1,vn-1)∥1≤t2∥vn-vn-1∥2≤t2(1+1n)D(Q(yn),Q(yn-1))≤t2λQ(1+1n)∥yn-yn-1∥2.
As H1(·,·) is r1-Lipschitz continuous with respect to A1, we obtain
(42)∥H1(A1xn,B1xn)-H1(A1xn-1,B1xn)∥1≤r1∥xn-xn-1∥1.
Since η1 is ρ1-Lipschitz continuous, we conclude that
(43)∥η1(xn,xn-1)∥1≤ρ1∥xn-xn-1∥1.
SinceH1(·,·) is η1-relaxed Lipschitz continuous with respect to T and η2-relaxed Lipschitz continuous with respect to Q in the first and second arguments with constants τ1 and τ2, respectively, we have(44)〈F(wn,vn)-F(wn-1,vn),j(η1(xn,xn-1))〉≤-τ1∥xn-xn-1∥12.
Employing Lemma 3 and taking into account (39)–(44), we obtain
(45)∥(F(wn,vn)-F(wn-1,vn))H1(A1xn,B1xn)-H1(A1xn-1,B1xn)-λ1(F(wn,vn)-F(wn-1,vn))∥12≤∥H1(A1xn,B1xn)-H1(A1xn-1,B1xn)∥12-2λ1〈-λ1(F(wn,vn)-F(wn-1,vn))]F(wn,vn)-F(wn-1,vn),j[(F(wn,vn)-F(wn-1,vn))H1(A1xn,B1xn)-H1(A1xn-1,B1xn)-λ1(F(wn,vn)-F(wn-1,vn))]〈-λ1(F(wn,vn)-F(wn-1,vn))]F(wn,vn)-F(wn-1,vn),〉=∥H1(A1xn,B1xn)-H1(A1xn-1,B1xn)∥12-2λ1〈(η1(xn,xn-1))F(wn,vn)-F(wn-1,vn),j[(F(wn,vn)-F(wn-1,vn))H1(A1xn,B1xn)-H1(A1xn-1,B1xn)-λ1(F(wn,vn)-F(wn-1,vn))]+j(η1(xn,xn-1))〈(η1(xn,xn-1))F(wn,vn)-F(wn-1,vn),〉+2λ1〈F(wn,vn)-F(wn-1,vn),j(η1(xn,xn-1))〉≤∥H1(A1xn,B1xn)-H1(A1xn-1,B1xn)∥12+2λ1∥F(wn,vn)-F(wn-1,vn)∥1×[∥η1(xn,xn-1)∥1∥H1(A1xn,B1xn)-H1(A1xn-1,B1xn)∥1+λ1∥F(wn,vn)-F(wn-1,vn)∥1+∥η1(xn,xn-1)∥1∥η1(xn,xn-1)∥1∥H1(A1xn,B1xn)-H1(A1xn-1,B1xn)∥1]+2λ1〈F(wn,vn)-F(wn-1,vn),j(η1(xn,xn-1))〉≤r12∥xn-xn-1∥12+2λ1t1λT(1+1n)∥xn-xn-1∥1×[r1∥xn-xn-1∥1+λ1t1λT(1+1n)×∥xn-xn-1∥1+ρ1∥xn-xn-1∥1(1+1n)]-2λ1τ1∥xn-xn-1∥12=[r12+2λ1t1λT(1+1n)×[r1+λ1t1λT(1+1n)+ρ1]-2λ1τ1]∥xn-xn-1∥12.
This implies that
(46)∥(F(wn,vn)-F(wn-1,vn))H1(A1xn,B1xn)-H1(A1xn-1,B1xn)-λ1(F(wn,vn)-F(wn-1,vn))∥1≤r12+2λ1t1λT(1+1n)[r1+λ1t1λT(1+1n)+ρ1]-2λ1τ1×∥xn-xn-1∥1=pn∥xn-xn-1∥1,
where
(47)pn=r12+2λ1t1λT(1+1n)[[r1+λ1t1λT(1+1n)+ρ1]-2λ1τ1].
Usings1-Lipschitz continuity of H1(·,·) with respect to B1, we deduce that
(48)∥H1(A1xn-1,B1xn)-H1(A1xn-1,B1xn-1)∥1≤s1∥xn-xn-1∥1.
In view of (41), (46), (48), (39) becomes
(49)∥xn+1-xn∥1≤σ1∥xn-xn-1∥1+θ1pn∥xn-xn-1∥1+θ1s1∥xn-xn-1∥1+λ1θ1t2λQ(1+1n)∥yn-yn-1∥2=(σ1+θ1pn+θ1s1)∥xn-xn-1∥1+λ1θ1t2λQ(1+1n)∥yn-yn-1∥2.
Similarly, we have
(50)∥yn+1-yn∥2=∥Rλ2,N(·,yn)H2(·,·)-η2[H2(A2yn,B2yn)-λ2G(wn,vn)]-Rλ2,N(·,yn-1)H2(·,·)-η2[H2(A2yn-1,B2yn-1)-λ2G(wn-1,vn-1)H2(A2yn,B2yn)]Rλ2,N(.,yn)H2(.,.)-η2∥2≤∥Rλ2,N(·,yn-1)H2(·,·)-η2[H2(A2yn,B2yn)-λ2G(wn,vn)]Rλ2,N(·,yn)H2(·,·)-η2[H2(A2yn,B2yn)-λ2G(wn,vn)]-Rλ2,N(·,yn-1)H2(·,·)-η2[H2(A2yn,B2yn)-λ2G(wn,vn)]∥2+∥Rλ2,N(·,yn-1)H2(·,·)-η2[H2(A2yn,B2yn)-λ2G(wn,vn)]-Rλ2,N(·,yn-1)H2(·,·)-η2[H2(A2yn-1,B2yn-1)-λ2G(wn-1,vn-1)[H2(A2yn-1,B2yn-1)]Rλ2,N(.,yn-1)H2(.,.)-η2∥2≤σ2∥yn-yn-1∥2+θ2∥H2(A2yn,B2yn)-H2(A2yn-1,B2yn-1)-λ2(G(wn,vn)-G(wn-1,vn-1))∥2≤σ2∥yn-yn-1∥2+θ2∥H2(A2yn,B2yn)-H2(A2yn-1,B2yn-1)-λ2(G(wn,vn)-G(wn,vn-1))∥2+λ2θ2∥G(wn,vn-1)-G(wn-1,vn-1)∥2≤σ2∥yn-yn-1∥2+θ2∥H2(A2yn,B2yn)-H2(A2yn-1,B2yn)-λ2(G(wn,vn)-G(wn,vn-1))∥2+θ2∥H2(A2yn-1,B2yn)-H2(A2yn-1,B2yn-1)∥2+λ2θ2∥G(wn,vn-1)-G(wn-1,vn-1)∥2≤σ2∥yn-yn-1∥2+θ2qn∥yn-yn-1∥2+θ2s2∥yn-yn-1∥2+λ2θ2t1λT(1+1n)∥xn-xn-1∥1=(σ2+θ2qn+θ2s2)∥yn-yn-1∥2+λ2θ2l1λT(1+1n)∥xn-xn-1∥1,
where
(51)qn=r22+2λ2l2λQ(1+1n)[r2+λ2l2λQ(1+1n)+ρ2]-2λ2ϵ2.
In view of (49) and (50), we obtain
(52)∥xn+1-xn∥1+∥yn+1-yn∥2≤cn∥xn-xn-1∥1+dn∥yn-yn-1∥2≤kn(∥xn-xn-1∥1+∥yn-yn-1∥2),
where cn=σ1+θ1pn+θ1s1+λ2θ2l1λT(1+1/n), dn=σ2+θ2qn+θ2s2+λ1θ1t2λQ(1+1/n),  and kn=max{cn,dn}.
Lettingn→∞, we obtain kn→k, where
(53)k=max{θ2q0+θ2s2+λ1θ1t2λQσ1+θ1p0+θ1s1+λ2θ2l1λT,σ2+θ2q0+θ2s2+λ1θ1t2λQ}.
Next, we define the norm∥·∥ on X1×X2 by
(54)∥(x,y)∥=∥x∥1+∥y∥2,(x,y)∈X1×X2.
One can easily check that (X1×X2,∥·∥) is a Banach space.
Definean+1=(xn+1,yn+1). Then, we have
(55)∥an+1-an∥=∥xn+1-xn∥1+∥yn+1-yn∥2.
In view of (38), we conclude that 0<k<1. This implies that there exist n0∈ℕ and k0∈(0,1) such that kn≤k0 for all n≥n0. It follows from (52) and (54) that
(56)∥an+1-an∥≤k0∥an-an-1∥,∀n≥n0.
In view of (56), we obtain
(57)∥an+1-an∥≤k0n-n0∥an0+1-an0∥,∀n≥n0.
This implies that for any m≥n≥n0,
(58)∥xm-xn∥1≤∥am-an∥≤∑i=nm-1∥ai+1-ai∥≤∑i=nm-1k0i-n0∥an0+1-an0∥.
Since 0<k0<1, it follows from (58) that ∥xm-xn∥1→0 and n→∞. This proves that {xn} is a Cauchy sequence in X1. Similarly, we conclude that {yn} is a Cauchy sequence in X2. Thus, there exist x∈X1 and y∈X2 such that xn→x and yn→y as n→∞.
Next, we prove thatwn→w∈T(x) and vn→v∈Q(y). In view of Lipschitz continuity of T and Q and Algorithm 17, we obtain
(59)∥wn-wn-1∥1≤(1+1n)λT∥xn-xn-1∥1,∥vn-vn-1∥2≤(1+1n)λQ∥yn-yn-1∥2.
From (59), we deduce that {wn},{vn} are Cauchy sequences in X1 and X2, respectively. Thus, there exist w∈T(x) and v∈Q(y) such that wn→w and vn→v as n→∞. Since T is D-Lipschitz continuous with constant λT, it is obvious that
(60)d(w,T(x))≤∥w-wn∥1+d(wn,T(x))d(w,T(x))≤∥w-wn-1∥1+D(T(xn),T(x))d(w,T(x))≤∥w-wn∥1+λT∥xn-x∥1⟶0(n⟶∞).
By the closedness of T(x), we conclude that w∈T(x). Similarly, we have v∈Q(y).
Assume now that(61)x0=Rλ1,M(·,x)H1(·,·)-η[H1(A1x,B1x)-λ1F(w,v)],y0=Rλ2,N(·,y)H2(·,·)-η[H2(A2y,B2y)-λ2G(w,v)].
Then, we have
(62)∥xn+1-x0∥1≤∥Rλ1,M(·,xn)H1(·,·)-η1[H1(A1xn,B1xn)-λ1F(wn,vn)]-Rλ1,M(·,x)H1(·,·)-η1[H1(A1x,B1x)-λ1F(w,v)]∥1≤∥Rλ1,M(·.,xn)H1(·,·)-η1[H1(A1xn,B1xn)-λ1F(wn,vn)]-Rλ1,M(·,x)H1(·,·)-η1[H1(A1xn,B1xn)-λ1F(wn,vn)]∥1+∥Rλ1,M(·,x)H1(·,·)-η1[H1(A1xn,B1xn)-λ1F(wn,vn)]-Rλ1,M(·,x)H1(·,·)-η1[H1(A1x,B1x)-λ1F(w,v)]∥1≤σ1∥xn-x∥1+θ1∥H1(A1xn,B1xn)-λ1F(wn,vn)-[H1(A1x,B1x)-λ1F(w,v)]∥1≤σ1∥xn-x∥1+θ1[∥H1(A1xn,B1xn)-H1(A1x,B1x)∥1+λ1∥F(wn,vn)-F(w,v)∥1]≤σ1∥xn-x∥1+θ1[∥H1(A1xn,B1xn)-H1(A1x,B1xn)∥1+∥H1(A1x,B1xn)-H1(A1x,B1x)∥1]+λ1θ1[∥F(wn,vn)-F(w,vn)∥1+∥F(w,vn)-F(w,v)∥1]≤σ1∥xn-x∥1+θ1[r1+s1]∥xn-x∥1+λ1θ1[t1∥wn-w∥1+t2∥vn-v∥2]=(σ1+θ1r1+θ1s1)∥xn-x∥1+λ1θ1[t1∥wn-w∥1+t2∥vn-v∥2].
Since xn→x, wn→w, and vn→v as n→∞, it follows from (62) that
(63)limn→∞∥xn-x0∥1=0,
and hence x0=x.
A similar argument shows thaty0=y. Therefore,
(64)x=x0=Rλ1,M(·,x)H1(·,·)-η[H1(A1x,B1x)-λ1F(w,v)],y=y0=Rλ2,N(·,y)H2(·,·)-η[H2(A2y,B2y)-λ2G(w,v)].
In view of Lemma 16, we conclude that (x,y,w,v) is a solution of problem (30), which completes the proof.At the end of this paper, we include the following simple example in support of Theorem18.Example 19.
LetX=ℝ2 with the usual inner product. We define two mappings A,B:ℝ2→ℝ2 by
(65)A(x):=(14x1-x2,x1+14x2),B(x):=(-14x1+14x2,-14x1-14x2),∀x=(x1,x2)∈ℝ2.
Let a mapping H:ℝ2×ℝ2→ℝ2 be defined by
(66)H(Ax,By):=Ax+By,∀x,y∈ℝ2.
By similar arguments, as in Example 4.1 of [27], we can prove the following.(1)
H(A,B) is 4/17-cocoercive with respect to A and 1-relaxed cocoercive with respect to B.(2)
A is 17/n-expansive, for n=4,5.(3)
B is 1/n-expansive, for n=1,2.(4)
H(A,B) is 17/n-Lipschitz continuous with constant 17/n with respect to A and B, for n=1,2,…,15,16.(5)
Letf,g:ℝ2→ℝ2 be defined by
(67)f(x):=(8x1-85x2,85x1+8x2),g(x):=(-178x1+58x2,58x1+178x2),∀x=(x1,x2)∈ℝ2.(6)
Now, we define a mappingM:ℝ2×ℝ2→ℝ2 by
(68)M(fx,gy):=fx-gy,∀x,y∈ℝ2.Let R,S,T:ℝ2→ℝ2 be the identity mappings. It is obvious that these mappings are D-Lipschitz continuous.(7)
Assume thatF,G:ℝ2→ℝ2 are defined by
(69)F(x):=(-14x1-18x2,18x1-14x2),G(x):=(-18x1+15x2,-15x1-18x2),∀x=(x1,x2)∈ℝ2. It could easily be seen that all the aspects of the hypotheses of Theorem 18 are satisfied, so we have the desired conclusion.



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*Source: 540108-2013-12-26.xml*