# GeneralizedH(·,·,·)-η-Cocoercive Operators and Generalized Set-Valued Variational-Like Inclusions

Authors: Shamshad Husain; Sanjeev Gupta; Vishnu Narayan Mishra
Journal: Journal of Mathematics
 (2013)

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## Abstract

We investigate a new class of cocoercive operators named generalizedH(·,·,·)-η-cocoercive operators in Hilbert spaces. We prove that generalized H(·,·,·)-η-cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with H(·,·)-cocoercive operators to the generalized H(·,·,·)-η-cocoercive operators. Some examples are given to justify the definition of generalized H(·,·,·)-η-cocoercive operators. Further, we consider a generalized set-valued variational-like inclusion problem involving generalized H(·,·,·)-η-cocoercive operator. In terms of the new resolvent operator technique, we give the approximate solution and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.


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## Body

## 1. Introduction

Variational inclusions, as the generalization of variational inequalities, have been widely studied in recent years. One of the most interesting and important problems in the theory of variational inclusions include variational, quasi-variational, and variational-like inequalities as special cases. For applications of variational inclusions, see, for example, [1]. Various kinds of iterative methods have been studied to solve the variational inclusions. Among these methods, the resolvent operator technique for the study of variational inclusions has been widely used by many authors. For details, we refer to [2–16].Recently Fang and Huang, Kazmi and Khan, and  Lan et al.  investigated several resolvent operators for generalized operators such asH-monotone [5], H-accretive [6], (P,η)-proximal point [11], (P,η)-accretive [12], (H,η)-monotone [7], (A,η)-accretive [13], and mappings. Very recently, Zou and Huang [16] introduced and studied H(·,·)-accretive operators, Kazmi et al. [8–10] introduced and studied generalized H(·,·)-accretive operators, H(·,·)-η-proximal point mapping, Xu and Wang [15] introduced and studied (H(·,·),η)-monotone operators, and Ahmad et al. [2] introduced and studied H(·,·)-cocoercive operators.Motivated by the recent work going in this direction, we consider a class of cocoercive operators called generalizedH(·,·,·)-η-cocoercive, a natural generalization of monotone (accretive) operators in Hilbert (Banach) spaces. For details, we refer to [2, 5–7, 13–16]. We prove that generalized H(·,·,·)-η-cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with H(·,·)-cocoercive operators to the generalized H(·,·,·)-η-cocoercive operators. Further, we consider the generalized set-valued variational-like inclusion problem involving generalized H(·,·,·)-η-cocoercive operator in Hilbert spaces. Using new a resolvent operator technique, we prove the existence of solutions and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as an extension and generalization of some known results [2, 15, 16]. For illustration of Definitions 2 and 7 and examples 2.3 and 3.2 are given, respectively.


## 2. Preliminaries

Throughout this paper, we suppose thatX is a real Hilbert space endowed with a norm ∥·∥ and an inner product 〈·,·〉, 2X (resp., CB(X)) is the family of all the nonempty (resp., closed and bounded) subsets of X, and 𝒟(·,·) is the Hausdorff metric on CB(X) defined by
(1)𝒟(P,Q)=max{supx∈Pd(x,Q),supy∈Qd(P,y)},∀P,Q∈CB(X),
where d(x,Q)=infy∈Q∥x-y∥ and d(P,y)=infx∈P∥x-y∥.In the sequel, let us recall some concepts.Definition 1 (see [4, 17]).
LetP:X→X and η:X×X→X be two mappings. Then, P is said to be(i)
η-monotone if
(2)〈P(x)-P(y),η(x,y)〉≥0,∀x,y∈X,(ii)
δ1-η-strongly monotone if there exists a constant δ1>0 such that
(3)〈P(x)-P(y),η(x,y)〉≥δ1∥x-y∥2,∀x,y∈X,(iii)
μ1-η-cocoercive if there exists a constant μ1>0 such that
(4)〈P(x)-P(y),η(x,y)〉≥μ1∥P(x)-P(y)∥2,∀x,y∈X,(iv)
γ1-η-relaxed cocoercive if there exists a constant γ1>0 such that
(5)〈P(x)-P(y),η(x,y)〉≥(-γ1)∥P(x)-P(y)∥2,∀x,y∈X,(v)
λP-Lipschitz continuous, if there exists a constant λP>0 such that
(6)∥P(x)-P(y)∥≤λP∥x-y∥,∀x,y∈X,(vi)
α-expansive, if there exists a constant α>0 such that
(7)∥P(x)-P(y)∥≥α∥x-y∥,∀x,y∈X,ifα=1, then it is expansive;(vii)
η is said to be τ-Lipschitz continuous, if there exists a constant τ>0 such that
(8)∥η(x,y)∥≤τ∥x-y∥,∀x,y∈X.Ifη(x,y)=x-y, for all x,y∈X, then definitions (i) to (iv) reduce to the Definitions of monotonicity, strong monotonicity [18], cocoercivity [19], and relaxed cocoercive, respectively.Definition 2.
LetP,Q,R:X→X, η:X×X→X, and H:X×X×X→X be the single-valued mappings. Then,(i)
H(P,·,·) is said to be μ-η-cocoercive with respect to P if there exists a constant μ>0 such that
(9)〈H(Px,u,u)-H(Py,u,u),η(x,y)〉≥μ∥Px-Py∥2,∀x,y,u∈X,(ii)
H(·,Q,·) is said to be γ-η-relaxed cocoercive with respect to Q if there exists a constant γ>0 such that
(10)〈H(u,Qx,u)-H(u,Qy,u),η(x,y)〉≥(-γ)∥Qx-Qy∥2,∀x,y,u∈X,(iii)
H(·,·,R) is said to be δ-η-strongly monotone with respect to R if there exists a constant δ>0 such that
(11)〈H(u,u,Rx)-H(u,u,Ry),η(x,y)〉≥δ∥x-y∥2,∀x,y,u∈X,(iv)
H(P,·,·) is said to be r1-Lipschitz continuous with respect to P if there exists a constant r1>0 such that
(12)∥H(Px,·,·)-H(Py,·,·)∥≤r1∥x-y∥,∀x,y∈X,(v)
H(·,Q,·) is said to be r2-Lipschitz continuous with respect to Q if there exists a constant r2>0 such that
(13)∥H(·,Qx,·)-H(·,Qy,·)∥≤r2∥x-y∥,∀x,y∈X,(vi)
H(·,·,R) is said to be r3-Lipschitz continuous with respect to R if there exists a constant r3>0 such that
(14)∥H(·,·,Rx)-H(·,·,Ry)∥≤r3∥x-y∥,∀x,y∈X.Ifη(x,y)=x-y, for all x,y∈H and H(·,·,·)=H(·,·), then Definitions (i)-(ii) are reduced to the definition of cocoercivity and relaxed cocoercive [2], respectively, and (iii) reduces to strong accretivity [16].Example 3.
LetX=ℝ2 with usual inner product. Let η:ℝ2×ℝ2→ℝ2,P,Q,R:ℝ2→ℝ2 be defined by
(15)η(x,y)=(mx1+nx2-my1-ny2),Px=(mx1-mx2-mx1+nx2),Qy=(-my1+my2-ny2),Rz=(nz1mz2),
for all scalers n,m∈ℝ with n>m and for all x=(x1,x2),y=(y1,y2),z=(z1,z2)∈ℝ2.
Suppose thatH(Px,Qy,Rz)=Px+Qy+Rz. Then, H(P,Q,R) is (1/2)-η-cocoercive with respect to P, (1/2)-η-relaxed cocoercive with respect to Q, m2-η-strongly monotone with respect to R, and η(x,y) is n-Lipschitz continuous.Indeed, let for anyu∈X,
(16)〈H(Px,u,u)-H(Py,u,u),η(x,y)〉=〈Px-Py,η(x,y)〉=〈(mx1-mx2,-mx1+nx2)-(my1-my2,-my1+ny2),(m(x1-y1),n(x2-y2))〉=〈(m(x1-y1)-m(x2-y2),-m(x1-y1)+n(x2-y2)),(m(x1-y1),n(x2-y2))〉=m2(x1-y1)2-m(m+n)(x1-y1)(x2-y2)+n2(x2-y2)2,∥Px-Py∥2=〈Px-Py,Px-Py〉=〈(mx1-mx2,-mx1+nx2)-(my1-y2,-my1+ny2),(mx1-mx2,-mx1+nx2)-(my1-y2,-my1+ny2)〉=2m2(x1-y1)2-2m(m+n)(x1-y1)(x2-y2)+(m2+n2)(x2-y2)2≤2m2(x1-y1)2-2m(m+n)(x1-y1)(x2-y2)+2n2(x2-y2)2=2〈H(Px,u,u)-H(Py,u,u),η(x,y)〉,〈H(Px,u,u)-H(Py,u,u),η(x,y)〉≥12∥Px-Py∥2,
that is, H(P,Q,R) is (1/2)-η-cocoercive with respect to P. Consider
(17)〈H(u,Qx,u)-H(u,Qy,u),η(x,y)〉=〈Qx-Qy,η(x,y)〉=〈(-mx1+mx2,-nx2)-(-my1+my2,-ny2),(m(x1-y1),n(x2-y2))〉=〈(-m(x1-y1)+m(x2-y2),-n(x2-y2)),(m(x1-y1),n(x2-y2))〉=-m2(x1-y1)2+m2(x1-y1)(x2-y2)-n2(x2-y2)2,∥Qx-Qy∥2=〈Qx-Qy,Qx-Qy〉=〈(-mx1+mx2,-nx2)-(-my1+my2,-ny2),(-mx1+mx2,-nx2)-(-my1-my2,-ny2)〉=m2(x1-y1)2-2m2(x1-y1)(x2-y2)+(m2+n2)(x2-y2)2≤2m2(x1-y1)2-2m2(x1-y1)(x2-y2)+2n2(x2-y2)2=(-2)〈H(u,Qx,u)-H(u,Qy,u)η(x,y)〉,〈H(u,Qx,u)-H(u,Qy,u),η(x,y)〉≥-12∥Qx-Qy∥2,
that is, H(P,Q,R) is (1/2)-η-relaxed cocoercive with respect to Q. In addition,
(18)〈H(u,u,Rx)-H(u,u,Ry),η(x,y)〉=〈Rx-Ry,η(x,y)〉=〈(nx1,mx2)-(ny2,my2),(m(x1-y1),n(x2-y2))〉=mn(x1-y1)2+mn(x2-y2)2≥m2∥x-y∥2,〈H(u,u,Rx)-H(u,u,Ry),η(x,y)〉≥m2∥x-y∥2,
that is, H(P,Q,R) is m2-η-strongly monotone with respect to R. Moreover,
(19)∥η(x,y)∥=∥m(x1-y1),n(x2-y2)∥=m2(x1-y1)2+n2(x2-y2)2≤n2(x1-y1)2+n2(x2-y2)2≤n∥x-y∥,
that is, η(x,y) is n-Lipschitz continuous.Definition 4.
A set-valued mappingM:X→2X is said to be m-η-relaxed monotone if there exists a constant m>0 such that
(20)〈u-v,η(x,y)〉≥(-m)∥x-y∥2,∀x,y∈X,u∈M(x),v∈M(y).Definition 5.
A set-valued mappingT:X→CB(X) is said to be 𝒟-Lipschitz continuous, if there exists a constant l>0 such that
(21)𝒟(Tx,Ty)≤l∥x-y∥,∀x,y∈X.Definition 6.
LetS,T:X→CB(X) be the set-valued mappings. A mapping N:X×X→X is said to be(i)
ϵ1-Lipschitz continuous in the first argument with respect to S, if there exists a constant ϵ1>0 such that
(22)∥N(w1,·)-N(w2,·)∥≤ϵ1∥w1-w2∥,∀u1,u2∈X,w1∈S(u1),w2∈S(u2),(ii)
ϵ2-Lipschitz continuous in the second argument with respect to T, if there exists a constant ϵ2>0 such that
(23)∥N(·,v1)-N(·,v2)∥≤ϵ2∥v1-v2∥,∀u1,u2∈X,v1∈T(u1),v2∈T(u2).


## 3. GeneralizedH(·,·,·)-η-Cocoercive Operators

Definition 7.
LetH:X×X×X→X, P,Q,R:X→X, and η:X×X→X be the single-valued mappings. Let H(P,Q,R) be μ-η-cocoercive with respect to P, γ-η-relaxed cocoercive with respect to Q, and δ-η-strongly monotone with respect to R. Then, the set-valued mapping M:X→2X is said to be a generalized H(·,·,·)-η-cocoercive with respect to the mappings P,Q, and R, if(i)
M is m-η-relaxed monotone;(ii)
(H(P,Q,R)+λM)(X)=X, for all λ>0.Example 8.
LetX,H,P,Q,R, and η be the same as in  Example 3,  and let M:ℝ2→ℝ2 be defined by M(x)=(-nx1,-mx2), for all x=(x1,x2)∈ℝ2.We claim thatM is n2-η-relaxed monotone mapping. Indeed, for any x=(x1,x2),y=(y1,y2)∈ℝ2, we have
(24)〈Mx-My,η(x,y)〉=〈((-nx1-mx2)-(-ny1-my2)),(m(x1-y1),n(x2-y2))〉=〈(-n(x1-y1),-m(x2-y2)),(m(x1-y1),n(x2-y2))〉=-mn{(x1-y1)2+(x2-y2)2}≥-n2{(x1-y1)2+(x2-y2)2}=-n2∥x-y∥2,〈Mx-My,η(x,y)〉≥(-n2)∥x-y∥2.
Furthermore, M is also a generalized H(·,·,·)-η-cocoercive operator since (H(P,Q,R)+λM)(ℝ2)=ℝ2 for any λ>0.Remark 9.
IfH(P,Q,R)=H(P,Q), P is α-strongly monotone, and Q is β-relaxed monotone, then generalized H(·,·,·)-η-cocoercive operator reduces to H(·,·)-η-monotone operator introduced and studied by Xu and Wang [15].Proposition 10.
Let set-valued mappingM:X→2X be a generalized H(·,·,·)-η-cocoercive operator with respect to the mappings P,Q, and R. If P is α-expansive, Q is β-Lipschitz continuous, and μ>γ, α>β with r=μα2-γβ2+δ>m, then the following inequality
(25)〈u-v,η(x,y)〉≥0
holds for all (y,v)∈Graph(M) and implies u∈Mx, where
(26)Graph(M)={(a,b)∈X×X:b∈M(a)}.Proof.
Suppose on the contrary that there exists(x0,u0)∉GraphM such that
(27)〈u0-v,η(x0,y)〉≥0,∀(y,v)∈Graph(M).
Since M is a generalized H(·,·,·)-η-cocoercive, we know that (H(P,Q,R)+λM)(X)=X holds for all λ>0, and so there exists (x1,u1)∈Graph(M) such that
(28)H(Px0,Qx0,Rx0)+λu0=H(Px1,Qx1,Rx1)+λu1∈X.
Now, λu0-λu1=H(Px1,Qx1,Rx1)-H(Px0,Qx0,Rx0)∈X(29)〈λu0-λu1,η(x0,x1)〉=-〈H(Px0,Qx0,Rx0)-H(Px1,Qx1,Rx1),η(x0,x1)〉.
Setting (y,v)=(x1,u1) in (27), and then from the resultant, (28) and m-η-relaxed monotonicity of M, we obtain
(30)-m∥x0-x1∥2≤λ〈u0-u1,η(x0,x1)〉=-〈H(Px0,Qx0,Rx0)-H(Px1,Qx1,Rx1),η(x0,x1)〉=-〈H(Px0,Qx0,Rx0)-H(Px1,Qx0,Rx0),η(x0,x1)〉-〈H(Px1,Qx0,Rx0)-H(Px1,Qx1,Rx0),η(x0,x1)〉-〈H(Px1,Qx1,Rx0)-H(Px1,Qx1,Rx1),η(x0,x1)〉.
Since H(P,Q,R) is μ-η-cocoercive with respect to P, γ-η-relaxed cocoercive with respect to Q, δ-η-strongly monotone with respect to R, and P is α-expansive, Q is β-Lipschitz continuous, thus (30) becomes
(31)-m∥x0-x1∥2≤-μ∥Px0-Px1∥2+γ∥Qx0-Qx1∥2-δ∥x0-x1∥2≤-(μα2-γβ2+δ)∥x0-x1∥2=-r∥x0-x1∥2≤0,≤-(r-m)∥x0-x1∥2≤0,wherer=μα2-γβ2+δ,
which gives x0=x1 since r>m. By (27), we have u0=u1, a contradiction. This completes the proof.Theorem 11.
Let set-valued mappingM:X→2X be a generalized H(·,·,·)-η-cocoercive operator with respect to the mappings P,Q, and R. If P is α-expansive, Q is β-Lipschitz continuous, and μ>γ, α>β with r=μα2-γβ2+δ>λm, then (H(P,Q,R)+λM)-1 is single-valued.Proof.
For any givenx∈X, let u,v∈(H(P,Q,R)+λM)-1(x). It follows that
(32)-H(Pu,Qu,Ru)+x∈λMu,-H(Pv,Qv,Rv)+x∈λMv.
Since M is m-η-relaxed monotone, we have
(33)-m∥u-v∥2≤1λ〈-H(Pu,Qu,Ru)+x-(-H(Pv,Qv,Rv)+x),η(u,v)〉,-λm∥u-v∥2≤〈-H(Pu,Qu,Ru)+x-(-H(Pv,Qv,Rv)+x),η(u,v)〉=-〈H(Pu,Qu,Ru)-H(Pv,Qv,Rv),η(u,v)〉=-〈H(Pu,Qu,Ru)-H(Pv,Qu,Ru),η(u,v)〉-〈H(Pv,Qu,Ru)-H(Pv,Qv,Ru),η(u,v)〉-〈H(Pv,Qv,Ru)-H(Pv,Qv,Rv),η(u,v)〉.
Since H(P,Q,R) is μ-η-cocoercive with respect to P, γ-η-relaxed cocoercive with respect to Q, δ-η-strongly monotone with respect to R, and P is α-expansive, Q is β-Lipschitz continuous, thus (33) becomes
(34)-λm∥u-v∥2≤-μ∥Pu-Pv∥2+γ∥Qu-Qv∥2-δ∥u-v∥2≤-(μα2-γβ2+δ)∥u-v∥2=-r∥u-v∥2≤0,≤-(r-λm)∥u-v∥2≤0,wherer=μα2-γβ2+δ,
since r>λm. Hence, it follows that ∥u-v∥≤0. This implies that u=v, and so (H(P,Q,R)+λM)-1 is single-valued.Definition 12.
Let set-valued mappingM:X→2X be a generalized H(·,·,·)-η-cocoercive operator with respect to the mappings P,Q, and R. If P is α-expansive, Q is β-Lipschitz continuous, and μ>γ, α>β and r=μα2-γβ2+δ>λm, then the resolvent operator Rλ,MH(·,·,·)-η:X→X is defined by
(35)Rλ,MH(·,·,·)-η(u)=(H(P,Q,R)+λM)-1(u),∀u∈X.Now, we prove that the resolvent operator defined by (35) is Lipschitz continuous.Theorem 13.
Let set-valued mappingM:X→2X be a generalized H(·,·,·)-η-cocoercive operator with respect to the mappings P,Q, and R. If P is α-expansive, Q is β-Lipschitz continuous, η is τ-Lipschitz continuous, and μ>γ, α>β with r=μα2-γβ2+δ>λm, then the resolvent operator Rλ,MH(·,·,·)-η:X→X is (τ/(r-λm))-Lipschitz continuous, that is,
(36)∥Rλ,MH(·,·,·)-η(u)-Rλ,MH(·,·,·)-η(v)∥≤τr-λm∥u-v∥,∀u,v∈X.Proof.
Letu,v∈X be any given points. It follows from (35) that
(37)Rλ,MH(·,·,·)-η(u)=(H(P,Q,R)+λM)-1(u),Rλ,MH(·,·,·)-η(v)=(H(P,Q,R)+λM)-1(v).1λ(u-H(P(Rλ,MH(·,·,·)-η(u)),Q(Rλ,MH(·,·,·)-η(u)),R(Rλ,MH(·,·,·)-η(u))))∈M(Rλ,MH(·,·,·)-η(u)),1λ(v-H(P(Rλ,MH(·,·,·)-η(v)),Q(Rλ,MH(·,·,·)-η(v)),R(Rλ,MH(·,·,·)-η(v))))∈M(Rλ,MH(·,·,·)-η(v)).
Let z1=Rλ,MH(·,·,·)-η(u) and z2=Rλ,MH(·,·,·)-η(v).
SinceM is m-η-relaxed monotone, we have
(38)〈1λ(u-H(P(z1),Q(z1),R(z1))-(v-H(P(z2),Q(z2),R(z2))),η(z1,z2)1λ〉≥-m∥z1-z2∥2,〈1λ(u-v-H(P(z1),Q(z1),R(z1))+H(P(z2),Q(z2),R(z2)),η(z1,z2)1λ〉≥-m∥z1-z2∥2,
which implies
(39)〈u-v,η(z1,z2)〉≥〈H(P(z1),Q(z1),R(z1))-H(P(z2),Q(z2),R(z2)),η(z1,z2)〉-λm∥z1-z2∥2.
Further, we have
(40)τ∥u-v∥∥z1-z2∥≥∥u-v∥∥η(z1,z2)∥≥〈u-v,η(z1,z2)〉≥〈H(P(z1),Q(z1),R(z1))-H(P(z2),Q(z2),R(z2)),η(z1,z2)〉-λm∥z1-z2∥2=〈H(P(z1),Q(z1),R(z1))-H(P(z2),Q(z1),R(z1)),η(z1,z2)〉+〈H(P(z2),Q(z1),R(z1))-H(P(z2),Q(z2),R(z1)),η(z1,z2)〉+〈H(P(z2),Q(z2),R(z1))-H(P(z2),Q(z2),R(z2)),η(z1,z2)〉-λm∥z1-z2∥2≥μ∥P(z1)-P(z2)∥2-γ∥Q(z1)-Q(z2)∥2+δ∥R(z1)-R(z2)∥2-λm∥z1-z2∥2=(μα2-γβ2+δ)∥z1-z2∥2-λm∥z1-z2∥2=(r-λm)∥z1-z2∥2,wherer=μα2-γβ2+δ,
and hence,
(41)τ∥u-v∥∥z1-z2∥≥(r-λm)∥z1-z2∥2,
that is,
(42)∥Rλ,MH(·,·,·)-η(u)-Rλ,MH(·,·,·)-η(v)∥≤τr-λm∥u-v∥,∀u,v∈X.
This completes the proof.


## 4. An Application ofH(·,·,·)-η-Cocoercive Operators for Solving Generalized Variational Inclusions

In this section, we will show that under suitable assumptions, the generalizedH(·,·,·)-η-cocoercive operator can also play important roles for solving the variational inclusion problem in Hilbert spaces.LetN:X×X→X, η:X×X→X, H:X×X×X→X, P,Q,R:X→X, and p:X→X be the single-valued mappings, and S,T:X→CB(X),M:X→2X be the set-valued mappings such that M is generalized H(·,·,·)-η-cocoercive with respect to P,Q, and R and range (p)∩ dom M≠∅. Then we consider the problem to find u∈X,w∈S(u),v∈T(u) such that
(43)0∈N(w,v)+M(p(u)).The problem (43) is called generalized set-valued variational-like inclusion problem. The problem of type (43) was introduced and studied by Chidume et al. [3] by applying η-proximal mapping. If T=0 and η(u,v)=u-v, for all u,v∈X, and N(·,·)=S(·), where S:X→CB(X) is a set-valued mapping. Then, problem (43) reduces to the problem of finding u∈X,w∈S(u) such that
(44)0∈w+M(p(u)).
The problem of type (44) was studied by Ahmad et al. [2] by applying H(·,·)-cocoercive operators.IfS,T=0, N(·,·)=0, and η(u,v)=u-v, for all u,v∈X, then problem (43) reduces to the problem of finding u∈X such that
(45)0∈M(p(u)).
The problem of type (45) was studied by Verma [14] in the setting of Banach spaces when M is A-maximal-relaxed accretive.Lemma 14.
The(u,w,v), where u∈X,w∈S(u),v∈T(u) is a solution of the problem (43), if and only if (u,w,v) satisfies the following relation:
(46)p(u)=Rλ,MH(·,·,·)-η[H(P(p(u)),Q(p(u)),R(p(u)))-λN(w,v)(P(p(u)),Q(p(u)),R(p(u)))],
where Rλ,MH(·,·,·)-η(u)=(H(P,Q,R)+λM)-1(u) and λ>0 is a constant.Proof.
By using the definitions of resolvent operatorsRλ,MH(·,·,·)-η, the conclusion follows directly.Using Lemma14 and using the technique of Chidume et al. [3] and Nadler [20], we develop an iterative algorithm for finding the approximate solution of problem (43) as follows.Algorithm 15.
LetN:X×X→X, η:X×X→X, H:X×X×X→X, P,Q,R:X→X, p:X→X, and S,T:X→CB(X) be such that, for each u∈X, F(u)⊆p(X), where F:X→2X is the set-valued mappings defined by
(47)F(u)=⋃w∈S(u)‍⋃v∈T(u)‍(Rλ,MH(·,·,·)-η(H(P(p(u)),Q(p(u)),R(p(u)))-λN(w,v)(p(u)))Rλ,MH(·,·,·)-η(H(P(p(u)),Q(p(u)),R(p(u)))),
where M:X→2X is a set-valued mapping such that, M is generalized H(·,·,·)-η-cocoercive with respect to the mappings P,Q, and R.
For givenu0∈X, take w0∈S(u0) and v0∈T(u0). Let
(48)x0=Rλ,MH(·,·,·)-η(H(P(p(u0)),Q(p(u0)),R(p(u0)))-λN(w0,v0))∈F(u0)⊆p(X).
Hence, there exists u1∈X such that x0=p(u1). Since w0∈S(u0)∈CB(X) and v0∈T(u0)∈CB(X), then, by Nadler’s result [20], there exist w1∈S(u1) and v1∈T(u1) such that
(49)∥w0-w1∥≤(1+1-1)𝒟(S(u0),S(u1)),∥v0-v1∥≤(1+1-1)𝒟(T(u0),T(u1)).
Let
(50)x1=Rλ,MH(·,·,·)-η(H(P(p(u1)),Q(p(u1)),R(p(u1)))-λN(w1,v1))∈F(u1)⊆p(X).
Hence, there exists u2∈X such that x1=p(u2). By induction, we can define iterative sequences {un}, {p(un)}, {wn}, and {vn} as follows:
(51)p(un+1)=Rλ,MH(·,·,·)-η[H(P(p(un)),Q(p(un)),R(p(un)))-λN(wn,vn)],(52)wn∈S(un),∥wn-wn+1∥≤(1+1n+1)𝒟(S(un),S(un+1)),(53)vn∈T(un),∥vn-vn+1∥≤(1+1n+1)𝒟(T(un),T(un+1)),
for all n=0,1,2,…, and λ>0 is a constant.IfN(w,v)=w, for all w,v∈X, then Algorithm 15 reduces to the following algorithm for solving the problem (44).Algorithm 16.
For anyu0∈X, and w0∈S(u0), compute the sequence {un} and {wn} by the following:
(54)p(un)=Rλ,MH(·,·,·)-η[H(P(p(un)),Q(p(un)),R(p(un)))-λwn],wn∈S(un),∥wn-wn+1∥≤(1+1n+1)𝒟(S(un),S(un+1)),
for all n=0,1,2,…, and λ>0 is a constant.IfN(w,v)=0, for all w,v∈X, then Algorithm 15 reduces to the following algorithm for solving the problem (45).Algorithm 17.
For anyu0∈X, compute the sequence {un} by the following:
(55)p(un)=Rλ,MH(·,·,·)-η[H(P(p(un)),Q(p(un)),R(p(un)))],
for all n=0,1,2,…, and λ>0 is a constant.Now, we prove the existence of a solution of problem (43) and the convergence of Algorithm 15.Theorem 18.
LetX be a real Hilbert space. Let η,N:X×X→X, H:X×X×X→X, P,Q,R:X→X, and p:X→X be single-valued mappings, and S,T:X→CB(X) and M:X→2X be the set-valued mappings such that M is generalized H(P,Q,R)-η-cocoercive operator with respect to the mappings P,Q, and R and range (p)∩  dom M≠∅, and, for each u∈X, let F(u)⊆p(X), where F is defined by (47). Assume that(i)
S,T are 𝒟-Lipschitz continuous with constants l1 and l2, respectively;(ii)
η is τ-Lipschitz continuous, P is α-expansive, and Q is β-Lipschitz continuous;(iii)
p is λp-Lipschitz continuous and ξ-strongly monotone;(iv)
H(P,Q,R) is r1-Lipschitz continuous with respect to P, r2-Lipschitz continuous with respect to Q, and r3-Lipschitz continuous with respect to R;(v)
N is ϵ1-Lipschitz continuous in the first argument with respect to S and ϵ2-Lipschitz continuous in the second argument with respect to T.
In addition,(56)(r1+r2+r3)λpτ+λ(ϵ1l1+ϵ2l2)<ξ(r-λm),
where r=μα2-γβ2+δ>λm and μ>γ, α>β, δ>0.
Then, generalized set-valued variational inclusion problem (43) has a solution (u,w,v)∈X, and the iterative sequences {un}, {p(un)}, {wn}, and {vn} generated by Algorithm 15 converge strongly to u, p(u), w, and v, respectively.Proof.
SinceS,T are 𝒟-Lipschitz continuous with constants l1 and l2, respectively, it follows from (52) and (53) that
(57)∥wn+1-wn∥≤(1+1n+1)𝒟(S(un+1),S(un))≤(1+1n+1)l1∥un+1-un∥,∥vn+1-vn∥≤(1+1n+1)𝒟(T(un+1),T(un))≤(1+1n+1)l2∥un+1-un∥,
for n=0,1,2,….
It follows from (51) and  Theorem 13  that
(58)∥p(un+1)-p(un)∥=∥Rλ,MH(·,·,·)-η{H(P(p(un)),Q(p(un)),R(p(un)))-λN(wn,vn)}-[Rλ,MH(·,·,·)-η{H(P(p(un-1)),Q(p(un-1)),R(p(un-1)))-λN(wn-1,vn-1)(P(p(un-1)),Q(p(un-1)),R(p(un-1)))}Rλ,MH(·,·,·)-η{H(P(p(un-1)),Q(p(un-1)),R(p(un-1)))]Rλ,MH(·,·,·)-η{H(P(p(un)),Q(p(un)),R(p(un)))∥+τr-λm∥H(P(p(un)),Q(p(un)),R(p(un)))-H(P(p(un-1)),Q(p(un-1)),R(p(un-1)))∥+τλr-λm∥N(wn,vn)-N(wn-1,vn-1)∥.
Since H(P,Q,R) is r1-Lipschitz continuous with respect to P, r2-Lipschitz continuous with respect to Q, and r3-Lipschitz continuous with respect to R, and p is λp-Lipschitz continuous, we have
(59)∥H(P(p(un)),Q(p(un)),R(p(un)))-H(P(p(un-1)),Q(p(un-1)),R(p(un-1)))∥=∥H(P(p(un)),Q(p(un)),R(p(un)))-H(P(p(un-1)),Q(p(un)),R(p(un)))∥+∥H(P(p(un-1)),Q(p(un)),R(p(un)))-H(P(p(un-1)),Q(p(un-1)),R(p(un)))∥+∥H(P(p(un-1)),Q(p(un-1)),R(p(un)))-H(P(p(un-1)),Q(p(un-1)),R(p(un-1)))∥≤(r1+r2+r3)λp∥un-un-1∥.
SinceN is ϵ1-Lipschitz continuous in the first argument with respect to S and ϵ2- Lipschitz continuous in the second argument with respect to T, and S,T are 𝒟-Lipschitz continuous with constants l1 and l2, respectively, we have
(60)∥N(wn,vn)-N(wn-1,vn-1)∥≤∥N(wn,vn)-N(wn-1,vn)∥+∥N(wn-1,vn)-N(wn-1,vn-1)∥≤ϵ1∥wn-wn-1∥+ϵ2∥vn-vn-1∥≤ϵ1(1+1n)𝒟(S(un),S(un-1))+ϵ2(1+1n)𝒟(T(un),T(un-1))≤ϵ1(1+1n)l1∥un-un-1∥+ϵ2(1+1n)l2∥un-un-1∥=(ϵ1l1+ϵ2l2)(1+1n)∥un-un-1∥.
Using (59), (60) in (58), we have
(61)∥p(un+1)-p(un)∥≤{(r1+r2+r3)λpτ+λ(ϵ1l1+ϵ2l2)(1+(1/n))r-λm}×∥un-un-1∥.
Using theξ-strong monotonicity of p, we have
(62)∥p(un+1)-p(un)∥∥un+1-un∥≥〈p(un+1)-p(un),un+1-un〉≥ξ∥un+1-un∥,
which implies that
(63)∥un+1-un∥≤1ξ∥p(un+1)-p(un)∥.
Combining (61) and (63), we have
(64)∥un+1-un∥≤θn∥un-un-1∥,
where
(65)θn=(r1+r2+r3)λpτ+λ(ϵ1l1+ϵ2l2)(1+(1/n))ξ(r-λm).
Let
(66)θ=(r1+r2+r3)λpτ+λ(ϵ1l1+ϵ2l2)ξ(r-λm).
From (56), it is easy to see that θ<1. Therefore, (64) implies that {un} is a Cauchy sequence in X. Since X is a Hilbert space, there exists u∈X such that un→u as n→∞. From (57), {wn} and {vn} are also Cauchy sequences in X; thus, there exist w,v∈X such that wn→w and vn→v as n→∞. By the continuity of p,Rλ,MH(·,·,·)-η,H,η,N,P,Q,R, and (51) of Algorithm 15, we have
(67)p(u)=Rλ,MH(·,·,·)-η[H(P(p(u)),Q(p(u)),R(p(u)))-λN(w,v)].
Now, we prove thatw∈S(u). In fact, since wn∈S(un), we have
(68)d(w,S(u))≤∥w-wn∥+d(wn,S(u))≤∥w-wn∥+𝒟(S(un),S(u))≤∥w-wn∥+l1∥un-u∥⟶0asn⟶∞,
which implies that d(w,S(u))=0. Since S(u)∈CB(X), it follows that w∈S(u). Similarly, it is easy to see that v∈T(u). By Lemma 14, (u,w,v) is the solution of problem (43). This completes the proof.Based on Lemma14 and Algorithm 16, Theorem 18 reduced to the following result for solving problem (44).Theorem 19.
LetX be a real Hilbert space. Let η:X×X→X, H:X×X×X→X, P,Q,R:X→X, and p:X→X be single-valued mappings, and let S:X→CB(X) and M:X→2X be the set-valued mappings such that M is generalized H(P,Q,R)-η-cocoercive operator with respect to the mappings P,Q, and R and range (p)∩  dom M≠∅, and, for each u∈X, let F(u)⊆p(X), where F is defined by (47). Assume that(i)
S is 𝒟-Lipschitz continuous with constants l;(ii)
η is τ-Lipschitz continuous, P is α-expansive, and Q is β-Lipschitz continuous;(iii)
p is λp-Lipschitz continuous and ξ-strongly monotone;(iv)
H(P,Q,R) is r1-Lipschitz continuous with respect to P, r2-Lipschitz continuous with respect to Q, and r3-Lipschitz continuous with respect to R.
In addition,(69)(r1+r2+r3)λpτ+λl<ξ(r-λm),
where r=μα2-γβ2+δ>λm and μ>γ, α>β, δ>0.
Then, generalized set-valued variational inclusion problem (44) has a solution (u,w)∈X, and the iterative sequences {un}, {p(un)}, and {wn} generated by Algorithm 16 converge strongly to u, p(u) and w, respectively.Based on Lemma14 and Algorithm 17, Theorem 18 reduced to the following result for solving problem (45).Theorem 20.
LetX be a real Hilbert space. Let η:X×X→X, H:X×X×X→X, P,Q,R:X→X, and p:X→X be single-valued mappings, and let M:X→2X be the set-valued mappings such that M is generalized H(P,Q,R)-η-cocoercive operator with respect to the mappings P,Q, and R and range (p)∩  dom M≠∅, and, for each u∈X, let F(u)⊆p(X), where F is defined by (47). Assume that(i)
η is τ-Lipschitz continuous, P is α-expansive, and Q is β-Lipschitz continuous;(ii)
p is λp-Lipschitz continuous and ξ-strongly monotone;(iii)
H(P,Q,R) is r1-Lipschitz continuous with respect to P, r2-Lipschitz continuous with respect to Q, and r3-Lipschitz continuous with respect to R.
In addition,(70)(r1+r2+r3)λpτ<ξ(r-λm),
where r=μα2-γβ2+δ>λm and μ>γ, α>β, δ>0.
Then, generalized set-valued variational inclusion problem (45) has a solution u∈X, and the iterative sequence {un} and {p(un)} generated by Algorithm 17 converge strongly to u and p(u), respectively.



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