# Stability and Convergence Analysis for Set-Valued Extended Generalized Nonlinear Mixed Variational Inequality Problems and Generalized Resolvent Dynamical Systems

Authors: Iqbal Ahmad; Zahoor Ahmad Rather; Rais Ahmad; Ching-Feng Wen
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we study a set-valued extended generalized nonlinear mixed variational inequality problem and its generalized resolvent dynamical system. A three-step iterative algorithm is constructed for solving set-valued extended generalized nonlinear variational inequality problem. Convergence and stability analysis are also discussed. We have shown the globally exponential convergence of generalized resolvent dynamical system to a unique solution of set-valued extended generalized nonlinear mixed variational inequality problem. In support of our main result, we provide a numerical example with convergence graphs and computation tables. For illustration, a comparison of our three-step iterative algorithm with Ishikawa-type algorithm and Mann-type algorithm is shown.


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## Body

## 1. Introduction

Variational inequality theory was introduced by Hartmann and Stampacchia [1] in 1966 as a tool for the study of partial differential equations with applications principally drawn from mechanics. Variational inclusions are the generalized forms of variational inequalities and they have wide range of applications in industry, mathematical finance, and economics and in several branches of applied sciences; see [2–6] and the references therein.We would like to emphasize that the projection method cannot be applied for solving the mixed variational inequalities involving the nonlinear term. To overcome this drawback, it is assumed that the nonlinear term involved in the general mixed variational inequalities is a proper, convex, and lower-semicontinuous functional. It is well known that the subdifferential of a proper, convex, and lower-semicontinuous functional is a maximal monotone operator. This characterization enables researchers to define the resolvent operator associated with the maximal monotone operator; see, for example, [7–9] and the references therein. The resolvent operator technique is used to establish the equivalence between the variational inequalities and fixed-point problems; see, for example, [10–12] and the references therein.In recent years, considerable interest has been shown in developing various extensions and generalizations of variational inequalities and variational inclusions, both for their own sake and for their applications; see, for example, [13–16] and the references therein. There are significant developments of these problems related to set-valued operators, nonconvex optimization, iterative methods, and structural analysis. Glowinski and Tallec [17] suggested and analyzed some three-step splitting methods for solving variational inequality problems by using the Lagrange multipliers technique. They showed that three-step splitting methods are numerically more efficient as compared with one-step and two-step splitting methods. They studied the convergence of these splitting methods under the assumption that the underlying operator is monotone and Lipschitz continuous. For the applications and convergence analysis of projection-based splitting methods, we refer to [18–20]. Moreover, one can find the important aspects of generalized variational inclusions in [21–26].In this paper, we introduce and study a set-valued extended generalized nonlinear mixed variational inequality problem including many existing problems studied by several authors. A three-step iterative algorithm is constructed for solving our problem, which is more general than Ishikawa-type algorithm and Mann-type algorithm. Existence of solution and convergence of the iterative sequences as well as stability analysis are shown. We also study generalized resolvent dynamical system associated with set-valued extended generalized nonlinear mixed variational inequality problem. Using Matlab R2019a program, we construct a numerical example for illustration.


## 2. Preliminaries

LetH be a real Hilbert space whose inner product and norm are denoted by ⋅,⋅ and ⋅, respectively. Let CBH be the family of all nonempty bounded closed subsets of H. Let D˜⋅,⋅ be the Hausdörff metric on CBH defined by(1)D˜A,B=maxsupx∈Adx,B,supy∈BdA,y,∀A,B∈CBH,where dx,B=infy∈Bdx,y and dA,y=infx∈Adx,y.The following definitions are required to achieve the goal.Definition 1.
Letg:H⟶H and T:H⟶H be the single-valued mappings. Then, we have the following.(i)
g is said to be Lipschitz continuous, if there exists a constant λg>0 such that(2)gx−gy≤λgx−y,∀x,y∈H.(ii)
g is said to be strongly monotone, if there exists a constant ξ>0 such that(3)gx−gy,x−y≥ξx−y2,∀x,y∈H.(iii)
T is said to be relaxed Lipschitz continuous, if there exists a constant α>0 such that(4)Tx−Ty,x−y≤−αx−y2,∀x,y∈H.(iv)
T is said to be strongly monotone with respect to g, if there exists a constant λT>0 such that(5)Tgx−Tgy,gx−gy≥λTx−y2,∀x,y∈H.Definition 2.
(see [27, 28]) Let A:H⟶CBH be a set-valued mapping and let T:H⟶H be a single-valued mapping. Then, A is said to be(i)
D˜-Lipschitz continuous if there exists a constant δA>0 such that(6)D˜Ax,Ay≤δAx−y,∀x,y∈H.(ii)
Strongly monotone with respect toT if there exists a constant τ>0 such that(7)Tu1−Tu2,x−y≥τx−y2,∀u1∈Ax,u2∈Ay.(iii)
Relaxed Lipschitz continuous with respect toT if there exists a constant β>0 such that(8)Tu1−Tu2,x−y≤−βx−y2,∀u1∈Ax,u2∈Ay.(iv)
Relaxed monotone with respect toT if there exists a constant κ>0 such that(9)Tu1−Tu2,x−y≥−κx−y2,∀u1∈Ax,u2∈Ay.Definition 3.
(see [27, 28]) Let A:H⟶CBH be a set-valued mapping. For all x,y∈H, the mapping η⋅,⋅,⋅:H×H×H⟶H is called(i)
Lipschitz continuous in the first argument with respect toA if there exists a constant λη1>0 such that(10)ηu1,⋅,⋅−ηu2,⋅,⋅≤λη1u1−u2,∀u1∈Ax,u2∈Ay.(ii)
Lipschitz continuous in the second argument with respect toA if there exists a constant λη2>0 such that(11)η⋅,u1,⋅−η⋅,u2,⋅≤λη2u1−u2,∀u1∈Ax,u2∈Ay.(iii)
Relaxed Lipschitz continuous in the first argument with respect toA if there exists a constant r1>0 such that(12)ηu1,⋅,⋅−ηu2,⋅,⋅,x−y≤−r1x−y2,∀u1∈Ax,u2∈Ay.(iv)
Relaxed Lipschitz continuous in the second argument with respect toA if there exists a constant r2>0 such that(13)η⋅,u1,⋅−η⋅,u2,⋅,x−y≤−r2x−y2,∀u1∈Ax,u2∈Ay.
Similarly, we can define Lipschitz continuity and relaxed Lipschitz continuity ofη in the third argument with respect to A.Definition 4.
(see [29]). Let S,T:H⟶H be a single-valued mapping, x0∈H, and(14)xn+1=ST,xndefines an iterative sequence which yields a sequence of points xn in H. Suppose that FT=p∈H:Tp=p≠∅ and xn converges to a fixed point x∗ of T. Let un⊂H and(15)ϑn=un+1−ST,un.
limn⟶∞ϑn=0 implies that un⟶x∗, and consequently the iterative sequence xn is said to be T-stable or stable with respect to T.Definition 5.
A set-valued mappingM:H⟶CBH is said to be monotone if, for any x,y∈H,(16)u1−u2,x−y≥0,∀u1∈Mx,u2∈My.
A monotone set-valued mappingM is called maximal if its graph, GrM:=x,y∈H×H:y∈Mx, is not properly contained in the graph of any other monotone mapping. It is well defined that M is a maximal monotone mapping if and only if I+ωMH=H, for all ω>0, where I denotes the identity mapping on H.Definition 6.
Letψ:H⟶ℝ∪∞ be a proper functional. A vector ξ∗∈H is called a subgradient of ψ at x∈domψ if(17)ξ∗,y−x≤ψy−ψx,∀y∈H.
Eachψ can be associated with the following map ∂ψ, called subdifferential of ψ at x, defined by(18)∂ψx=ξ∗∈H:ξ∗,y−x≤ψy−ψx,∀y∈H.Definition 7.
(see [30]). If M is a maximal monotone mapping, then the resolvent operator associated with M of parameter ω is defined as(19)RωMx=I+ωM−1x,∀x∈H.
It is well known that the resolvent operatorRωM is single-valued and nonexpansive.
Ifψ is a proper, convex, and lower-semicontinuous functional, then its subdifferential ∂ψ is a maximal monotone operator. Thus, we can define the resolvent operator associated with the subdifferential ∂ψ of parameter ω as(20)Rω∂ψx=I+ω∂ψ−1x,∀x∈H.
The resolvent operatorRω∂ψ has the following useful characterization.Lemma 1 (see [30]).
For anyz∈H, x∈H satisfies the inequality(21)x−z,y−x≥ωψx−ωψy,∀y∈H,if and only if x=Rω∂ψz, where Rω∂ψ is the resolvent operator associated with ∂ψ of parameter ω>0.Lemma 2 (see [31]).
Letχn be a nonnegative real sequence and let ζn be a real sequence in 0,1 such that ∑n=0∞ζn=∞. If there exists a positive integer m such that(22)χn≤1−ζnχn+ζnηn,∀n≥m,where ηn≥0, for all n≥0 and ηn⟶0n⟶0, then limn⟶∞χn=0.


## 3. Formulation of the Problem and Existence of Solution

LetS,T,g:H⟶H and η:H×H×H⟶H be the single-valued mappings, and let A,B,C,D,E:H⟶CBH be the set-valued mappings. Let ψ:H×H⟶ℝ∪+∞ be such that, for each fixed x∈H, ψ⋅,x is a proper convex lower-semicontinuous functional on H satisfying gH∩dom∂ψ⋅,x≠∅, where ∂ψ⋅,x is a subdifferential of ψ⋅,x. We consider the following set-valued extended generalized nonlinear mixed variational inequality problem (in short, SEGNMVI).Findx∈H, a∈Ax, b∈Bx, c∈Cx, d∈Dx, and e∈Ex such that gx∈dom∂ψ⋅,x and(23)Sa−Tb−ωηc,d,e,y−gx≥ωψgx,x−ωψy,x,∀y∈H.Problem (23) is equivalent to finding x∈H, a∈Ax, b∈Bx, c∈Cx, d∈Dx, and e∈Ex such that(24)0∈Sa−Tb−ωηc,d,e+ω∂ψgx,x.Problem (24) is called the set-valued extended generalized nonlinear mixed variational inclusion problem.


### 3.1. Special Cases

(i)
IfC,D,E,η≡0 and ψ.,x=ψ., then problem (23) reduces to the following problem which is to find x∈H, a∈Ax, and b∈Bx such that(25)Sa−Tb,y−gx≥ψgx−ψy,∀y∈H.Problem (25) was introduced and studied by Huang [27].(ii)
Ifηc,d,e≡ηc,d, A,B,E,S,T≡0, and ψ.,x=ψ., then problem (23) reduces to the following problem which is to find x∈H, c∈Cx, and d∈Dx such that(26)ηc,d,y−gx≥ψgx−ψy,∀y∈H.Problem (26) was introduced and studied by Noor [32].(iii)
IfA,B,E,S,T≡0,ηc,d,e≡ηc,d, ψ.,x=ψ., and g=I (the identity mapping), then problem (23) reduces to the following problem which is to find x∈H, c∈Cx, and d∈Dx such that(27)ηc,d,y−x≥ψx−ψy,∀y∈H.Problem (27) was introduced and studied by Noor [33].(iv)
Ifηc,d,e≡ηc,d, A,B,E,S,T≡0, and ψ.,x=ψ. is the indicator function of the closed convex set K in H, that is,(28)ψx=δKx=0,ifx∈K,+∞,otherwise.Then, problem (23) reduces to the following problem which is to find x∈H, c∈Cx, d∈Dx, and gx∈K such that(29)ηc,d,y−gx≥0,∀y∈K.Problem (29) was introduced and studied by Noor [34].The following lemma ensures the equivalence between the set-valued extended generalized nonlinear mixed variational inequality problem (23) and a fixed-point problem.Lemma 3.
Let the mappingsS,T,g,η,A,B,C,D,E be the same as in problem (23). Then x,a,b,c,d,e is a solution of SEGNMVI (23) if and only if x,a,b,c,d,e satisfies the following relation:(30)gx=Rω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e,where x∈H, a∈Ax, b∈Bx, c∈Cx, d∈Dx, and e∈Ex; ψ, ω>0 and Hω∂ψ⋅,x is the resolvent operator associated with ∂ψ⋅,x of parameter ω.Proof.
Assume thatx∈H, a∈Ax, b∈Bx, c∈Cx, d∈Dx, and e∈Ex satisfy relation (23); that is,(31)gx=Rω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e.
SinceRω∂ψ⋅,x=I+ω∂ψ⋅,x−1, the above inequality holds if and only if(32)gx−Sa−Tb−ωηc,d,e∈gx+ω∂ψgx,x.
By using the definition of subdifferential ofψ.,x, the above relations hold if and only if(33)Tb−ωηc,d,e−Sa,y−gx≤ωψy,x−ωψgx,x,∀y∈H.
Hence, we have(34)Sa−Tb−ωηc,d,e,y−gx≥ωψgx,x−ωψy,x,∀y∈H,that is, x,a,b,c,d,e is a solution of SEGNMVI (23).
The following theorem guarantees the existence of a unique solution of SEGNMVI (23).Theorem 1.
LetS,T,g:H⟶H and η:H×H×H⟶H be the single-valued mappings such that S is λS-Lipschitz continuous, T is λT-Lipschitz continuous and μT-strongly monotone, g is λg-Lipschitz continuous and μg-strongly monotone such that gH=H, and η is Lipschitz continuous in the first argument with respect to C with constant λη1, Lipschitz continuous in the second argument with respect to D with constant λη2, and Lipschitz continuous in the third argument with respect to E with constant λη3, respectively. Let A,B,C,D,E:H⟶CBH be the D˜-Lipschitz continuous mappings with constants δA, δB, δC, δD, and δE, respectively, A is μS-strongly monotone with respect to g, B is μT-strongly monotone with respect to T, η is relaxed Lipschitz continuous with respect to C in the first argument with constant r1, η is relaxed Lipschitz continuous with respect to D in the second argument with constant r2, and η is relaxed Lipschitz continuous with respect to E in the third argument with constant r3, respectively. Let ψ:H×H⟶ℝ∪+∞ be such that, for each fixed x∈H, ψ⋅,x is a proper convex lower-semicontinuous functional on H satisfying gH∩dom∂ψ⋅,x≠∅, where ∂ψ⋅,x is a subdifferential of ψ⋅,x. Suppose that there exist constants ω>0 and ξ>0 such that, for each z∈H,(35)Rω∂ψ⋅,xz−Rω∂ψ⋅,yz≤ξx−y,and the following conditions are satisfied:(36)ω−t2t1<t22−t1ϑ2−ϑt1,ϑ−1>t12−t1t22t1,ϑ=1−2μg+λg2+1−2μT+λT2δB2+λg2−2μA+λS2δA2+ξ2μg<1+λg2,2μT<1+λT2δB2,2μA<λg2+λS2δA2,t1=λη1δC+λη2δD+λη3δE2,t2=r1+r2+r3.
Then, the SEGNMVI (23) admits a unique solution x∗,u∗,v∗,w∗,e∗, where x∗∈H, a∗∈Ax∗, b∗∈Bx∗, c∗∈Cx∗, d∗∈Dx∗, and e∗∈Ex∗.Proof.
By Lemma3, it is sufficient to prove that there exist x∗∈H, a∗∈Ax∗, b∗∈Bx∗, c∗∈Cx∗, d∗∈Dx∗, and e∗∈Ex∗ satisfying (30). Define a set-valued mapping:(37)Gx=∪a∈Ax,b∈Bx,c∈Cx,d∈Dx,e∈Exx−gx+Rω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e,∀x∈.
We show that mappingG is a contraction mapping. Suppose that i∈1,2, and for any xi∈H and ℘i∈Gxi there exist ai∈Axi,bi∈Bxi,ci∈Cxi,di∈Dxi, and ei∈Exi such that(38)℘i=xi−gxi+Rω∂ψ⋅,xigxi−Sai−Tbi−ωηci,di,ei.
Since the resolvent operatorRω∂ψ⋅,x is nonexpansive and using (35), it follows that(39)℘1−℘2≤x1−x2−gx1−gx2+Rω∂ψ⋅,x1gx1−Sa1−Tb1−ωηc1,d1,e1−Rω∂ψ⋅,x2gx2−Sa2−Tb2−ωηc2,d2,e2≤x1−x2−gx1−gx2+Rω∂ψ⋅,x1gx1−Sa1−Tb1−ωηc1,d1,e1−Rω∂ψ⋅,x1gx2−Sa2−Tb2−ωηc2,d2,e2+Rω∂ψ⋅,x1gx2−Sa2−Tb2−ωηc2,d2,e2−Rω∂ψ⋅,x2gx2−Sa2−Tb2−ωηc2,d2,e2≤x1−x2−gx1−gx2+gx1−Sa1−Tb1−ωηc1,d1,e1−gx2−Sa2−Tb2−ωηc2,d2,e2+ξx1−x2≤x1−x2−gx1−gx2+gx1−gx2−Sa1−Sa2+x1−x2−Tb1−Tb2+x1−x2+ωηc1,d1,e1−ηc2,d2,e2+ξx1−x2.
It follows fromμg-strong monotonicity and λg-Lipschitz continuity of g that(40)x1−x2−gx1−gx22=x1−x22−2gx1−gx2,x1−x2+gx1−gx22≤x1−x22−2μgx1−x22+λg2x1−x22≤1−2μg+λg2x1−x22,that is,(41)x1−x2−gx1−gx2≤1−2μg+λg2x1−x2.
By using theμB-strong monotonicity of B with respect to T and D˜-Lipschitz continuity of B with constant δB, we have(42)x1−x2−Tb1−Tb22=x1−x22−2Tb1−Tb2,x1−x2+Tb1−Sb22≤x1−x22−2μBx1−x22+λS2δB2x1−x22≤1−2μB+λS2δB2x1−x22,that is,(43)x1−x2−Tb1−Tb2≤1−2μB+λT2δB2x1−x2.
By using theμA-strong monotonicity of A with respect to S, D˜-Lipschitz continuity of A with constant δA, and λg-Lipschitz continuity of g, respectively, we evaluate(44)gx1−gx2−Sa1−Sa2≤λg2−2μA+λS2δA2x1−x2.
By using the relaxed Lipschitz continuity ofη with respect to C,D,E in the first, second, and third arguments with constants r1,r2, and r3, respectively, we have(45)x1−x2+ωηc1,d1,e1−ηc2,d2,e22=x1−x22+2ωηc1,d1,e1−ηc2,d2,e2,x1−x2+ω2ηc1,d1,e1−ηc2,d2,e22=x1−x22+2ωηc1,d1,e1−ηc2,d1,e1,x1−x2+ηc2,d1,e1−ηc2,d2,e1,x1−x2+ηc2,d2,e1−ηc2,d2,e2,x1−x2+ω2ηc1,d1,e1−ηc2,d2,e22≤x1−x22−2r1+r2+r3x1−x22+ω2ηc1,d1,e1−ηc2,d2,e22.
Since mappingη is Lipschitz continuous in the first argument with respect to C with constant λη1, Lipschitz continuous in the second argument with respect to D with constant λη2, Lipschitz continuous in the third argument with respect to E with constant λη3, and D˜-Lipschitz continuous of C,D,E with constants δC, δD, and δE, respectively, we have(46)ηc1,d1,e1−ηc2,d2,e2≤ηc1,d1,e1−ηc2,d1,e1+ηc2,d1,e1−ηc2,d2,e1+ηc2,d2,e1−ηc2,d2,e2≤λη1c1−c2+λη2d1−d2+λη3e1−e2≤λη1D˜Cx1,Cx2+λη2D˜Dx1,Dx2+λη3D˜Ex1,Ex2≤λη1δCx1−x2+λη2δDx1−x2+λη3δEx1−x2=λη1δC+λη2δD+λη3δEx1−x2.
Combining (45) and (46), we obtain(47)x1−x2+ωηc1,d1,e1−ηc2,d2,e2≤1−2ωr1+r2+r3+ω2λη1δC+λη2δD+λη3δE2x1−x2.
From (39)–(47), we have(48)℘1−℘2≤1−2μT+λB2δB2+λg2+2μA+λS2δA2+1−2ωr1+r2+r3+ω2λN1δC+λN2δD+λN3δE2+1−2μg+λg2+ξx1−x2,which implies that(49)℘1−℘2≤Θ✠x1−x2,where(50)Θ✠=ϑ+1−2ωr1+r2+r3+ω2λη1δC+λη2δD+λη3δE2,ϑ=1−2μg+λg2+1−2μB+λT2δB2+λg2−2μA+λS2δA2+ξ.
By condition (36), it is clear that 0≤Θ✠<1. Since ℘1∈Gx1 and ℘2∈Gx2 are arbitrary, we obtain(51)D˜Gx1,Gx2≤Θ✠x1−x2,∀x1,x2∈H.
By Theorem1 of Siddique and Ansari [35], it follows that G has a fixed point x∗∈H such that a∗∈Ax∗,b∗∈Bx∗,c∗∈Cx∗,d∗∈Dx∗, and e∗∈Ex∗; that is, Gx∗=x∗. From (30), we conclude that(52)gx∗=ℛω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗.
Thus, by Lemma1, x∗∈H such that a∗∈Ax∗,b∗∈Bx∗,c∗∈Cx∗,d∗∈Dx∗, and e∗∈Ex∗ is a solution of SEGNMVI (23).


## 3.1. Special Cases

(i)
IfC,D,E,η≡0 and ψ.,x=ψ., then problem (23) reduces to the following problem which is to find x∈H, a∈Ax, and b∈Bx such that(25)Sa−Tb,y−gx≥ψgx−ψy,∀y∈H.Problem (25) was introduced and studied by Huang [27].(ii)
Ifηc,d,e≡ηc,d, A,B,E,S,T≡0, and ψ.,x=ψ., then problem (23) reduces to the following problem which is to find x∈H, c∈Cx, and d∈Dx such that(26)ηc,d,y−gx≥ψgx−ψy,∀y∈H.Problem (26) was introduced and studied by Noor [32].(iii)
IfA,B,E,S,T≡0,ηc,d,e≡ηc,d, ψ.,x=ψ., and g=I (the identity mapping), then problem (23) reduces to the following problem which is to find x∈H, c∈Cx, and d∈Dx such that(27)ηc,d,y−x≥ψx−ψy,∀y∈H.Problem (27) was introduced and studied by Noor [33].(iv)
Ifηc,d,e≡ηc,d, A,B,E,S,T≡0, and ψ.,x=ψ. is the indicator function of the closed convex set K in H, that is,(28)ψx=δKx=0,ifx∈K,+∞,otherwise.Then, problem (23) reduces to the following problem which is to find x∈H, c∈Cx, d∈Dx, and gx∈K such that(29)ηc,d,y−gx≥0,∀y∈K.Problem (29) was introduced and studied by Noor [34].The following lemma ensures the equivalence between the set-valued extended generalized nonlinear mixed variational inequality problem (23) and a fixed-point problem.Lemma 3.
Let the mappingsS,T,g,η,A,B,C,D,E be the same as in problem (23). Then x,a,b,c,d,e is a solution of SEGNMVI (23) if and only if x,a,b,c,d,e satisfies the following relation:(30)gx=Rω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e,where x∈H, a∈Ax, b∈Bx, c∈Cx, d∈Dx, and e∈Ex; ψ, ω>0 and Hω∂ψ⋅,x is the resolvent operator associated with ∂ψ⋅,x of parameter ω.Proof.
Assume thatx∈H, a∈Ax, b∈Bx, c∈Cx, d∈Dx, and e∈Ex satisfy relation (23); that is,(31)gx=Rω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e.
SinceRω∂ψ⋅,x=I+ω∂ψ⋅,x−1, the above inequality holds if and only if(32)gx−Sa−Tb−ωηc,d,e∈gx+ω∂ψgx,x.
By using the definition of subdifferential ofψ.,x, the above relations hold if and only if(33)Tb−ωηc,d,e−Sa,y−gx≤ωψy,x−ωψgx,x,∀y∈H.
Hence, we have(34)Sa−Tb−ωηc,d,e,y−gx≥ωψgx,x−ωψy,x,∀y∈H,that is, x,a,b,c,d,e is a solution of SEGNMVI (23).
The following theorem guarantees the existence of a unique solution of SEGNMVI (23).Theorem 1.
LetS,T,g:H⟶H and η:H×H×H⟶H be the single-valued mappings such that S is λS-Lipschitz continuous, T is λT-Lipschitz continuous and μT-strongly monotone, g is λg-Lipschitz continuous and μg-strongly monotone such that gH=H, and η is Lipschitz continuous in the first argument with respect to C with constant λη1, Lipschitz continuous in the second argument with respect to D with constant λη2, and Lipschitz continuous in the third argument with respect to E with constant λη3, respectively. Let A,B,C,D,E:H⟶CBH be the D˜-Lipschitz continuous mappings with constants δA, δB, δC, δD, and δE, respectively, A is μS-strongly monotone with respect to g, B is μT-strongly monotone with respect to T, η is relaxed Lipschitz continuous with respect to C in the first argument with constant r1, η is relaxed Lipschitz continuous with respect to D in the second argument with constant r2, and η is relaxed Lipschitz continuous with respect to E in the third argument with constant r3, respectively. Let ψ:H×H⟶ℝ∪+∞ be such that, for each fixed x∈H, ψ⋅,x is a proper convex lower-semicontinuous functional on H satisfying gH∩dom∂ψ⋅,x≠∅, where ∂ψ⋅,x is a subdifferential of ψ⋅,x. Suppose that there exist constants ω>0 and ξ>0 such that, for each z∈H,(35)Rω∂ψ⋅,xz−Rω∂ψ⋅,yz≤ξx−y,and the following conditions are satisfied:(36)ω−t2t1<t22−t1ϑ2−ϑt1,ϑ−1>t12−t1t22t1,ϑ=1−2μg+λg2+1−2μT+λT2δB2+λg2−2μA+λS2δA2+ξ2μg<1+λg2,2μT<1+λT2δB2,2μA<λg2+λS2δA2,t1=λη1δC+λη2δD+λη3δE2,t2=r1+r2+r3.
Then, the SEGNMVI (23) admits a unique solution x∗,u∗,v∗,w∗,e∗, where x∗∈H, a∗∈Ax∗, b∗∈Bx∗, c∗∈Cx∗, d∗∈Dx∗, and e∗∈Ex∗.Proof.
By Lemma3, it is sufficient to prove that there exist x∗∈H, a∗∈Ax∗, b∗∈Bx∗, c∗∈Cx∗, d∗∈Dx∗, and e∗∈Ex∗ satisfying (30). Define a set-valued mapping:(37)Gx=∪a∈Ax,b∈Bx,c∈Cx,d∈Dx,e∈Exx−gx+Rω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e,∀x∈.
We show that mappingG is a contraction mapping. Suppose that i∈1,2, and for any xi∈H and ℘i∈Gxi there exist ai∈Axi,bi∈Bxi,ci∈Cxi,di∈Dxi, and ei∈Exi such that(38)℘i=xi−gxi+Rω∂ψ⋅,xigxi−Sai−Tbi−ωηci,di,ei.
Since the resolvent operatorRω∂ψ⋅,x is nonexpansive and using (35), it follows that(39)℘1−℘2≤x1−x2−gx1−gx2+Rω∂ψ⋅,x1gx1−Sa1−Tb1−ωηc1,d1,e1−Rω∂ψ⋅,x2gx2−Sa2−Tb2−ωηc2,d2,e2≤x1−x2−gx1−gx2+Rω∂ψ⋅,x1gx1−Sa1−Tb1−ωηc1,d1,e1−Rω∂ψ⋅,x1gx2−Sa2−Tb2−ωηc2,d2,e2+Rω∂ψ⋅,x1gx2−Sa2−Tb2−ωηc2,d2,e2−Rω∂ψ⋅,x2gx2−Sa2−Tb2−ωηc2,d2,e2≤x1−x2−gx1−gx2+gx1−Sa1−Tb1−ωηc1,d1,e1−gx2−Sa2−Tb2−ωηc2,d2,e2+ξx1−x2≤x1−x2−gx1−gx2+gx1−gx2−Sa1−Sa2+x1−x2−Tb1−Tb2+x1−x2+ωηc1,d1,e1−ηc2,d2,e2+ξx1−x2.
It follows fromμg-strong monotonicity and λg-Lipschitz continuity of g that(40)x1−x2−gx1−gx22=x1−x22−2gx1−gx2,x1−x2+gx1−gx22≤x1−x22−2μgx1−x22+λg2x1−x22≤1−2μg+λg2x1−x22,that is,(41)x1−x2−gx1−gx2≤1−2μg+λg2x1−x2.
By using theμB-strong monotonicity of B with respect to T and D˜-Lipschitz continuity of B with constant δB, we have(42)x1−x2−Tb1−Tb22=x1−x22−2Tb1−Tb2,x1−x2+Tb1−Sb22≤x1−x22−2μBx1−x22+λS2δB2x1−x22≤1−2μB+λS2δB2x1−x22,that is,(43)x1−x2−Tb1−Tb2≤1−2μB+λT2δB2x1−x2.
By using theμA-strong monotonicity of A with respect to S, D˜-Lipschitz continuity of A with constant δA, and λg-Lipschitz continuity of g, respectively, we evaluate(44)gx1−gx2−Sa1−Sa2≤λg2−2μA+λS2δA2x1−x2.
By using the relaxed Lipschitz continuity ofη with respect to C,D,E in the first, second, and third arguments with constants r1,r2, and r3, respectively, we have(45)x1−x2+ωηc1,d1,e1−ηc2,d2,e22=x1−x22+2ωηc1,d1,e1−ηc2,d2,e2,x1−x2+ω2ηc1,d1,e1−ηc2,d2,e22=x1−x22+2ωηc1,d1,e1−ηc2,d1,e1,x1−x2+ηc2,d1,e1−ηc2,d2,e1,x1−x2+ηc2,d2,e1−ηc2,d2,e2,x1−x2+ω2ηc1,d1,e1−ηc2,d2,e22≤x1−x22−2r1+r2+r3x1−x22+ω2ηc1,d1,e1−ηc2,d2,e22.
Since mappingη is Lipschitz continuous in the first argument with respect to C with constant λη1, Lipschitz continuous in the second argument with respect to D with constant λη2, Lipschitz continuous in the third argument with respect to E with constant λη3, and D˜-Lipschitz continuous of C,D,E with constants δC, δD, and δE, respectively, we have(46)ηc1,d1,e1−ηc2,d2,e2≤ηc1,d1,e1−ηc2,d1,e1+ηc2,d1,e1−ηc2,d2,e1+ηc2,d2,e1−ηc2,d2,e2≤λη1c1−c2+λη2d1−d2+λη3e1−e2≤λη1D˜Cx1,Cx2+λη2D˜Dx1,Dx2+λη3D˜Ex1,Ex2≤λη1δCx1−x2+λη2δDx1−x2+λη3δEx1−x2=λη1δC+λη2δD+λη3δEx1−x2.
Combining (45) and (46), we obtain(47)x1−x2+ωηc1,d1,e1−ηc2,d2,e2≤1−2ωr1+r2+r3+ω2λη1δC+λη2δD+λη3δE2x1−x2.
From (39)–(47), we have(48)℘1−℘2≤1−2μT+λB2δB2+λg2+2μA+λS2δA2+1−2ωr1+r2+r3+ω2λN1δC+λN2δD+λN3δE2+1−2μg+λg2+ξx1−x2,which implies that(49)℘1−℘2≤Θ✠x1−x2,where(50)Θ✠=ϑ+1−2ωr1+r2+r3+ω2λη1δC+λη2δD+λη3δE2,ϑ=1−2μg+λg2+1−2μB+λT2δB2+λg2−2μA+λS2δA2+ξ.
By condition (36), it is clear that 0≤Θ✠<1. Since ℘1∈Gx1 and ℘2∈Gx2 are arbitrary, we obtain(51)D˜Gx1,Gx2≤Θ✠x1−x2,∀x1,x2∈H.
By Theorem1 of Siddique and Ansari [35], it follows that G has a fixed point x∗∈H such that a∗∈Ax∗,b∗∈Bx∗,c∗∈Cx∗,d∗∈Dx∗, and e∗∈Ex∗; that is, Gx∗=x∗. From (30), we conclude that(52)gx∗=ℛω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗.
Thus, by Lemma1, x∗∈H such that a∗∈Ax∗,b∗∈Bx∗,c∗∈Cx∗,d∗∈Dx∗, and e∗∈Ex∗ is a solution of SEGNMVI (23).


## 4. Convergence and Stability Analysis

We establish a three-step iterative algorithm for solving SEGNMVI (23) and discuss the convergence and stability analysis.Algorithm 1.
Let the mappingsS,T,g,η,A,B,C,D,E be the same as in problem (23). For any x0∈H, a0∈Ax0, b0∈Bx0, c0∈Cx0, d0∈Dx0, and e0∈Dx0, as gH=H, compute the iterative sequences xn, an, bn, cn, dn, and en by the following iterative process:(53)xn+1=1−γnxn+γnyn−gyn+Rω∂ψ⋅,yngyn−Sa¯n−Tb¯n−ωηc¯n,d¯n,e¯n+γnln,yn=1−σnxn+σnzn−gzn+Rω∂ψ⋅,zngzn−Sa^n−Tb^n−ωηc^n,d^n,e^n+σnpn,zn=1−ςnxn+σnxn−gxn+Rω∂ψ⋅,xngxn−San−Tbn−ωηcn,dn,en+ςnqn,where ψ, ω>0,an∈Axn,bn∈Bxn,cn∈Cxn,dn∈Dxn,en∈Exn,a¯n∈Ayn,b¯n∈Byn,c¯n∈Cyn,d¯n∈Dyn,e¯n∈Eyn,a^n∈Azn,b^n∈Bzn,c^n∈Czn,d^n∈Dzn and e^n∈Ezn can be chosen arbitrarily.
Letun be any sequence in H and define φn by(54)φn=un+1−1−γnun+γnsn−gsn+Rω∂ψ⋅,sngsn−San′−Tbn′−ωηcn′,dn′,en′+γnln,sn=1−σnun+σntn−gtn+Rω∂ψ⋅,tngtn−San″−Tbn″−ωηcn″,dn″,en″+σnpn,zn=1−ςnun+σnun−gun+Rω∂ψ⋅,ungun−San−Tbn−ωηcn,dn,en+ςnqn,where an∈Aun,bn∈Bun,cn∈Cun,dn∈Dun,en∈Eun,an′∈Asn,bn′∈Bsn,cn′∈Csn, dn′∈Dsn,en′∈Esn,an″∈Atn,bn″∈Btn,cn″∈Ctn,dn″∈Dtn, and en″∈Etn can be chosen arbitrarily. Let ln, pn, and qn be the three sequences in H to take into account the possible inexact computation and the sequences γn,σn, and ςn satisfy the following conditions: 0≤γn,σn,ςn≤1,∀n≥0, and ∑n=0∞γn diverges.Remark 1.
Ifcn=0,∀n≥0, then Algorithm 1 becomes Ishikawa-type iterative algorithm. If bn,cn=0,∀n≥0, then Algorithm 1 becomes Mann-type iterative algorithm. Also, we remark that, for suitable choices of operators involved in Algorithm 1, we can easily obtain many algorithms studied by several authors; see, for example, [27, 33, 34].Theorem 2.
Suppose that Theorem1 holds. Additionally, if limn⟶∞ln=0, limn⟶∞pn=0, and limn⟶∞qn=0, then(i)
The sequencesxn, an, bn, cn, dn, and en generated by Algorithm 1 converge strongly to x∗,a∗,b∗,c∗,d∗, and e∗, respectively, and x∗,a∗,b∗,c∗,d∗,e∗ is the solution of SEGNMVI (23)(ii)
Furthermore, if0<ε<γn, then limn⟶∞un=x∗ if and only if limn⟶∞φn=0, where φn is given in (54); that is, the sequences xn, an, bn, cn, dn, and en generated by (53) are Rω∂ψ⋅,x-stableProof.
(i) Suppose thatx∗,a∗,b∗,c∗,d∗,e∗ is a unique solution of SEGNMVI (23). Then, we have(55)x∗=1−γnx∗+γnx∗−gx∗+Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗,=1−σnx∗+σnx∗−gx∗+Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗,=1−ςnx∗+ςnx∗−gx∗+Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗.
Using Algorithm1 and (49), we have(56)xn+1−x∗=1−γnxn+γnyn−gyn+Rω∂ψ⋅,yngyn−Sa¯n−Tb¯n−ωηc¯n,d¯n,e¯n+γnln−1−γnx∗+γnx∗−gx∗+Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗≤1−γnxn−x∗+γnyn−gyn+Rω∂ψ⋅,xngyn−Sa¯n−Tb¯n−ωηc¯n,d¯n,e¯n−x∗−gx∗+Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗+γnln≤1−γnxn−x∗+γnΘ✠yn−x∗+γnln,where(57)Θ✠=ϑ+1−2ωr1+r2+r3+ω2λη1δC+λη2δD+λη3δE2,ϑ=1−2μg+λg2+1−2μT+λT2δB2+λg2−2μA+λS2δA2+ξ.
Using the same argument as for (56), we estimate(58)yn−x∗=1−σnxn+σnzn−gzn+Rω∂ψ⋅,zngzn−Sa^n−Tb^n−ωηc^n,d^n,e^n+σnpn−1−σnx∗+σnx∗−gx∗+Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗≤1−σnxn−x∗+σnzn−gzn+Rω∂ψ⋅,zngzn−Sa^n−Tb^n−ωηc^n,d^n,e^n−x∗−gx∗+Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗+σnpn≤1−σnxn−x∗+σnΘ✠zn−x∗+σnpn,(59)zn−x∗=1−ςnxn+σnxn−gxn+Rω∂ψ⋅,xngxn−San−Tbn−ωηcn,dn,en+ςnqn−1−ςnx∗+ςnx∗−gx∗+Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗≤1−ςnxn−x∗+ςnxn−gxn+Rω∂ψ⋅,xngxn−San−Tbn−ωηcn,dn,en−x∗−gx∗+Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗+ςnqn≤1−ςnxn−x∗+ςnΘ✠xn−x∗+ςnqn≤1−ςn1−Θ✠xn−x∗+ςnqn.
Since1−ςn1−Θ✠<1, we have(60)zn−x∗≤xn−x∗+ςnqn.
Using (58) and (60),(61)yn−x∗≤1−σnxn−x∗+σnΘ✠xn−x∗+ςnqn+σnpn=1−σn1−Θ✠xn−x∗+σnςnΘ✠qn+σnpn≤xn−x∗+σnςnΘ✠qn+σnpn.
Since1−σn1−Θ✠<1, we have(62)yn−x∗≤xn−x∗+σnςnΘ✠qn+σnpn.
Using (56) and (62),(63)xn+1−x∗≤1−γnxn−x∗+γnΘ✠xn−x∗+σnςnΘ✠qn+σnpn+γnln≤1−γn1−Θ✠xn−x∗+Θ✠2γnσnςnqn+Θ✠γnσnpn+γnln≤1−γn1−Θ✠xn−x∗+γn1−Θ✠Θ✠2σnςnqn+Θ✠σnpn+γnln1−Θ✠.
LetΓn=Θ✠2σnςnqn+Θ✠σnpn+γnln/1−Θ✠, Δn=xn−x∗, and Υn=γn1−Θ✠; then, (63) can be rewritten as(64)Δn+1≤1−ΥnΔn+ΥnΓn.
By Lemma2 and the assumptions limn⟶∞ln=limn⟶∞pn=limn⟶∞qn=0, we conclude that Δn⟶0, as n⟶∞, and so we obtain xn⟶x, as n⟶∞. Since A is D˜-Lipschitz continuous with constant δA, we have(65)an−a∗≤D˜Axn,Ax∗≤δAxn−x∗⟶0,and soan⟶a∗,asn⟶∞.
Similarly, by using theD˜-Lipschitz continuity of B,C,D, and E, one can show that bn⟶b∗,cn⟶c∗,dn⟶d∗, and en⟶e∗, as n⟶∞. Thus, we conclude that the sequences xn,an,bn,cn, dn, and en generated by Algorithm 1 converge strongly to x∗,a∗,b∗,c∗,d∗, and e∗, respectively. It remains to show that a∗∈Ax∗, b∗∈Bx∗,c∗∈Cx∗,d∗∈Dx∗, and e∈Ex∗. In fact,(66)da∗,Ax∗≤a∗−an+dan,Ax∗≤a∗−an+D˜Axn,Ax∗≤a∗−an+δAxn−x∗⟶0,asn⟶∞.
Hence,a∗∈Ax∗. Similarly, we can show that b∗∈Bx∗,c∗∈Cx∗,d∗∈Dx∗, and e∈Ex∗. By Lemma 3, we conclude that x∗,a∗,b∗,c∗,d∗,e∗ is the solution of SEGNMVI (23).Proof.
(ii). LetGx∗=x∗−gx∗+Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗. Using Algorithm 1, Lemma 2, and Lemma 3, we obtain(67)un+1−x∗≤un+1−1−γnun+γnGtn+γnln+1−γnun+γnGtn+γnln−1−γnx∗+γnGx∗≤un+1−1−γnun+γnGtn+γnln+1−γnun−x∗+γnGtn−Gx∗+γnln≤un+1−1−γnun+γnGtn+γnln+1−γnun−x∗+γnΘ✠tn−x∗+γnln.
Using (54), we have(68)tn−x∗≤1−σnun+σnGsn+σnpn−1−σnx∗+σnGx∗≤1−σnun−x∗+σnGsn−Gx∗+σnpn≤1−σnun−x∗+σnΘ✠sn−x∗+σnpn≤1−σnun−x∗+σnΘ✠sn−x∗+σnpn.
Again, using (54), we have(69)sn−x∗≤1−ςnun+ςnGun+ςnqn−1−ςnx∗+ςnGx∗≤1−ςnun−x∗+ςnGun−Gx∗+ςnqn≤1−ςnun−x∗+ςnΘ✠un−x∗+ςnqn≤1−ςn1−Θ✠un−x∗+ςnqn≤un−x∗+ςnqn.
Using (58) and (60),(70)sn−x∗≤1−σnun−x∗+σnΘ✠un−x∗+ςnqn+σnpn=1−σn1−Θ✠un−x∗+σnςnΘ✠qn+σnpn≤un−x∗+σnςnΘ✠qn+σnpn.
Using (68) and (70),(71)un+1−x∗≤un+1−1−γnun+γnGtn+γnln+1−γnun−x∗+γnΘ✠un−x∗+σnςnΘ✠qn+σnpn+γnln≤φn+1−γn1−Θ✠un−x∗+Θ✠2γnσnςnqn+Θ✠γnσnpn+γnln≤φn+1−γn1−Θ✠un−x∗+γn1−Θ✠Θ✠2σnςnqn+Θ✠σnpn+γnln1−Θ✠.
As0<ϵ≤γn, (70) becomes(72)un+1−x∗≤1−γn1−Θ✠un−x∗+γn1−Θ✠×φnε1−Θ✠+Θ✠2σnςnqn+Θ✠σnpn+γnln1−Θ✠.
Assume thatlimn⟶∞φn=0; hence, limn⟶∞un=x∗ such that a∗∈Ax∗, b∗∈Bx∗,c∗∈Cx∗,d∗∈Dx∗, and e∈Ex∗, where limn⟶∞ln=limn⟶∞pn=limn⟶∞qn=0.
Conversely, assume thatlimn⟶∞un=x∗ such that a∗∈Ax∗, b∗∈Bx∗,c∗∈Cx∗,d∗∈Dx∗, and e∈Ex∗. From (55) and as limn⟶∞ln=limn⟶∞pn=limn⟶∞qn=0, we have(73)φn=un+1−1−γnun+γnGtn+γnln≤un+1−x∗+1−γnun+γnGtn+γnln−x∗≤un+1−x∗+1−γnun+γnGtn+γnln−1−γnx∗+γnGx∗≤un+1−x∗+1−γnun−x∗+γnGtn−Gx∗+γnln≤un+1−x∗+1−γnun−x∗+γnΘ✠tn−x∗+γnln≤un+1−x∗+1−γn1−Θ✠un−x∗+Θ✠2γnσnςnqn+Θ✠γnσnpn+γnln,which implies that(74)limn⟶∞φn=0.
Hence, the sequencesun, an, bn, cn, dn, and en generated by Algorithm 1 are Rω∂ψ⋅,x∗-stable.
If we takeS,T,A,B,E≡0, ηc,d,e≡ηc,d, and ψ.,x=ψ. in SEGNMVI (23), we obtain the following corollary which ensures the solvability and stability of problem (26).Corollary 1.
Letg:H⟶H and η:H×H⟶H be the single-valued mappings such that g is λg-Lipschitz continuous and μg-strongly monotone and gH=H; η is Lipschitz continuous in the first argument with respect to C with constant λη1 and Lipschitz continuous in the second argument with respect to D with constant λη2, respectively. Let C,D:H⟶CBH be the D˜-Lipschitz continuous mappings with constants δC and δD, respectively, η is relaxed Lipschitz continuous with respect to C in the first argument with constant r1, and η is relaxed Lipschitz continuous with respect to D in the second argument with constant r2, respectively. Let ψ:H⟶ℝ∪+∞ be a proper convex lower-semicontinuous functional on satisfying gH∩dom∂ψ≠∅, where ∂ψ is a subdifferential of ψ. Suppose that there exists constant ω>0 and the following conditions are satisfied:(75)ω−t2t1<t22−t1ϑ2−ϑt1,ϑ−1>t12−t1t22t1,ϑ=21−2μg+λg2,2μg<1+λg2,t1=λη1δC+λη2δD2,t2=r1+r2.
Let, for anyx0∈, c0∈Cx0 and d0∈Dx0, and the following iterative sequences xn, cn, and dn are obtained by the following iterative scheme:(76)xn+1=1−γnxn+γnyn−gyn+Rω∂ψgyn−ωηc¯n,d¯n+γnln,yn=1−σnxn+σngzn−gzn+Rω∂ψzn−ωηc^n,d^n+σnpn,zn=1−ςnxn+σnxn−gxn+Rω∂ψgxn−ωηcn,dn+ςnqn,where cn∈Cxn,dn∈Dxn,c¯n∈Cyn,d¯n∈Dyn,c^n∈Czn, and d^n∈Dzn can be chosen arbitrarily.
Letun be any sequence in H and define φn by(77)φn=un+1−1−γnun+γnsn−gsn+Rω∂ψgsn−ωηcn′,dn′+γnln,sn=1−σnun+σntn−gtn+Rω∂ψgtn−ωηcn″,dn″+σnpn,zn=1−ςnun+σnun−gun+Rω∂ψgun−ωηcn,dn+ςnqn,where cn∈Cun,dn∈Dun,cn′∈Csn,dn′∈Dsn,cn″∈Ctn, and dn″∈Dtn can be chosen arbitrarily. Let ln, pn, and qn be the three sequences in H to take into account the possible inexact computation and the sequences γn,σn, and ςn satisfy the following conditions: 0≤γn,σn,ςn≤1,∀n≥0, and ∑n=0∞γn diverges. If limn⟶∞ln=0, limn⟶∞pn=0, and limn⟶∞qn=0, then(i)
The sequencesxn, cn, and dn converge strongly to x∗,c∗, and d∗, respectively, and x∗,c∗,d∗ is a solution of problem (26)(ii)
Furthermore, if0<ε<γn, then limn⟶∞un=x∗, if and only if limn⟶∞φn=0, where φn is given by (77); that is, the sequences xn, cn, and dn generated by (76) are Rω∂ψ-stable.
If we takeC,D,E,η≡0 and ψ.,x=ψ. in SEGNMVI (23), we obtain the following corollary to study stability and convergence analysis of problem (25).Corollary 2.
LetS,T,g:H⟶H be the single-valued mappings such that S is λS-Lipschitz continuous, T is λT-Lipschitz continuous and μT-strongly monotone, g is λg-Lipschitz continuous and μg-strongly monotone, and gH=H. Let A,B:H⟶CBH be the D˜-Lipschitz continuous mappings with constants δA and δB, respectively, A is μS-strongly monotone with respect to S, and B is μT-strongly monotone with respect to T, respectively. Let ψ:H⟶ℝ∪+∞ be a proper convex lower-semicontinuous functional on H satisfying gH∩dom∂ψ≠∅, where ∂ψ is a subdifferential of ψ. Suppose that there exists a constant ω>0 and the following conditions are satisfied:(78)ω−μA+λTδBt1−1λS2δA2−λT2δB2<μA−1−t1λTδB2−λS2δA2−λT2δB2t12−t1λS2δA2−λT2δB2,λTδB+λS2δA2−λT2δB2t12−t1<μA+t1λTδB,λSδA>λTδBωλTδB<1−t1,t1=21−2μg+λg2,t1<1.
For anyx0∈H, a0∈Ax0, and b0∈Bx0, as gH=H, we can obtain the iterative sequences xn, an, and bn by the following iterative scheme:(79)xn+1=1−γnxn+γnyn−gyn+Rω∂ψgyn−ωSa¯n−Tb¯n+γnln,yn=1−σnxn+σnzn−gzn+Rω∂ψgzn−ωSa^n−Tb^n+σnpn,zn=1−ςnxn+σnxn−gxn+Rω∂ψgxn−ωSan−Tbn+ςnqn,where an∈Axn,bn∈Bxn,a¯n∈Ayn,b¯n∈Byn,a^n∈Azn, and b^n∈Bzn can be chosen arbitrarily.
Letun be any sequence in H and define φn by(80)φn=un+1−1−γnun+γnsn−gsn+Rω∂ψgsn−ωSan′−Tbn′+γnln,sn=1−σnun+σntn−gtn+Rω∂ψgtn−ωSan″−Tbn″+σnpn,zn=1−ςnun+σnun−gun+Rω∂ψgun−ωSan−Tbn+ςnqn,where an∈Aun,bn∈Bun,an′∈Asn,bn′∈Bsn,an″∈Atn, and bn″∈Btn can be chosen arbitrarily. Let ln, pn, and qn be the three sequences in H to take into account the possible inexact computation and the sequences γn,σn, and ςn satisfy the following conditions: 0≤γn,σn,ςn≤1,∀n≥0, and ∑n=0∞γn diverges. If limn⟶∞ln=0, limn⟶∞pn=0, and limn⟶∞qn=0, then(i)
The sequencesxn, an, and bn generated by the suggested Algorithm 1 converge strongly to x∗,a∗, and b∗, respectively, and x∗,a∗,b∗ is the solution of SEGNMVI (23)(ii)
Furthermore, if0<ε<γn, then limn⟶∞un=x∗ if and only if limn⟶∞φn=0, where φn is given in (80); that is, the sequences xn, an, and bn generated by (68) are Rω∂ψ-stable


## 5. Generalized Resolvent Dynamical System

In this section, we consider the dynamical system technique to study the existence of unique solution of SEGNMVI (23). Using Lemma 3, we suggest and analyze the following generalized resolvent dynamical system associated with SEGNMVI (23):(81)dxdt=ωRω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e−gx,xt0=x0∈H,where x∈H, a∈Ax, b∈Bx, c∈Cx, d∈Dx, e∈Ex, and ω>0 is a parameter. System (81) is called generalized resolvent dynamical system associated with the set-valued extended generalized nonlinear mixed variational inequality problem (23).Definition 8.
(see [38]) The dynamical system is said to converge to the solution set Ω∗ of problem (23), if, irrespective of the initial point, the trajectory of the dynamical system satisfies(82)limt⟶∞distxt,Ω∗=0,where distxt,Ω∗=infy∈Ω∗x−y.
If the solution setΩ∗ has a unique solution x∗∈H with a∗∈Ax∗, b∗∈Bx∗, c∗∈Cx∗, d∗∈Dx∗, and e∗∈Ex∗, then (81) implies that limt⟶∞xt=x∗.Definition 9.
(see [38]). The dynamical system is said to be globally exponentially stable with degree θ at x∗, if, irrespective of the initial point, the trajectory of the dynamical system satisfies(83)xt−x∗≤c0xt−x∗exp−θt−t0,∀t≥t0,where c0 and θ are positive constants independent of the initial point.Lemma 4.
(see [38]). Let x^ and y^ be real-valued nonnegative continuous functions with domain t:t≥t0 and let αt=α0t−t0, where α0 is a monotone increasing function. If, for all t≥t0,(84)x^t≤αt+∫t0tx^sy^sds,then(85)x^t≤αtexp∫t0ty^sds.Combining Lemma4 and Theorem 1, we obtain the unique solution of generalized resolvent dynamical system (81).Theorem 3.
Suppose that Theorem1 holds. Then, for each x0∈H such that a0∈Ax0, b0∈Bx0, c0∈Cx0, d0∈Dx0, and e0∈Ex0, there exists a unique continuous solution xt such that at∈Axt, bt∈Bxt, ct∈Cxt, dt∈Dxt, and et∈Ext of generalized resolvent dynamical system (81) with xt0=x0 over t0,∞.Proof.
By Lemma3, we have(86)gx∗=Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗,such that x∗∈H, a∗∈Ax∗,b∗∈Bx∗,c∗∈Cx∗,d∗∈Dx∗, and e∗∈Ex∗ is a solution of SEGNMVI (23). Let(87)Fx=ωRω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e−gx,∀x∈H.
Then, for allx,x′∈H, such that a∈Ax,b∈Bx,c∈Cx,d∈Dx,e∈Ex, a′∈Ax′,b′∈Bx′,c′∈Cx′,d′∈Dx′, and e∈Ex, and using the arguments as for (39), we have(88)Fx−Fx′≤ωRω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e−Rω∂ψ⋅,x′gx′−Sa′−Tb′−ωηc′,d′,e′+gx−gx′≤ωgx−Sa−Tb−ωηc,d,e−gx′−Sa′−Tb′−ωηc′,d′,e′+1+ξx−x′+x−x′−gx−gx′.
Using the same arguments as for (41), (43), (44), and (47), we have(89)x−x′−gx−gx′≤1−2μg+λg2x−x′,(90)x−x′−Tb−Tb′≤1−2μB+λT2δB2x−x′,(91)gx−gx′−Sa−Sa′≤λg2−2μA+λS2δA2x−x′,(92)−x,x′+ωηc,d,e−ηc′,d′,e′≤1−2ωr1+r2+r3+ω2λη1δC+λη2δD+λη3δE2x−x′.
Combining (89), (90), (91), and (92) with (88), we obtain(93)Fx−Fx′≤ω1+Θ✠x−x′,where(94)Θ✠=ϑ+1−2ωr1+r2+r3+ω2λη1δC+λη2δD+λη3δE2ϑ=1−2μg+λg2+1−2μB+λT2δB2+λg2−2μA+λS2δA2+ξ.
Hence, mappingF is locally Lipschitz continuous in H. Therefore, for each x0∈H, there exists a unique and continuous solution xt such that at∈Axt, bt∈Bxt, ct∈Cxt, dt∈Dxt, and et∈Ext of generalized resolvent dynamical system (81) defined in the interval t0≤t<T with the initial condition xt0=x0. Let t0,T be its maximal interval of existence. Now, we show that T=∞. For any x such that a∈Ax, b∈Bx, c∈Cx, d∈Dx, and e∈Ex, we have(95)Fx≤ωRω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e−gx≤ωRω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e−gx∗+ωgx−gx∗≤ωRω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e−Rω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e+x−x∗+x−x∗−gx−gx∗≤ωgx−Sa−Tb−ωηc,d,e−gx′−Sa′−Tb′−ωηc′,d′,e′+1+ξx−x′+x−x′−gx−gx′≤ω1+Θ✠x−x∗≤ω1+Θ✠x+ω1+Θ✠x∗,and then(96)xt≤x0+∫t0tFxsds≤x0+k1t−t0+k2∫t0txsds,where k1=ω1+Θ✠x∗ and k2=ω1+Θ✠. Using Lemma 4, we have(97)xt≤x0+k1t−t0ek2t−t0,∀t∈t0,T.
Hence, the solution is bounded fort∈t0,T, if T is finite, and thus, T=∞.
By using the technique of Xia and Wang [36, 37], we show that the trajectory of solution of generalized resolvent dynamical system (81) converges to a unique solution of the set-valued extended generalized nonlinear mixed variational inequality problem (23).Theorem 4.
Suppose that Theorem1 holds. Additionally, if 1−t1<μg−ξ<1−t21+t2, where t1 and t2 are the same as in (36), then the generalized resolvent dynamical system (81) converges globally exponentially to the unique solution of the set-valued extended generalized nonlinear mixed variational inequality problem (23).Proof.
By Theorem1, there exists a unique solution x∗∈H such that a∗∈Ax∗, b∗∈Bx∗, c∗∈Cx∗, d∗∈Dx∗, and e∗∈Ex∗ for problem (23). By applying Lemma 3, we have gx∗=Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗. By Theorem 4, there exists a unique solution of system (81), that is, xt, such that at∈Axt, bt∈Bxt, ct∈Cxt, dt∈Dxt, and et∈Ext over t0,T for any fixed x0∈H. Let xt=xt,t0:x0 be the solution of (81) with xt0=x0. Now, we consider the Lyapunov function L defined on H by(98)Lx=x−x∗2,∀x∈H.
From (81), (90)–(92), and (98) and using μg-strong monotonicity of g, we have(99)dLdt=dLdxdxdt=2xt−x∗,dxdt=2ωxt−x∗,Rω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e−gx=−2ωxt−x∗,gx−gx∗+2ωxt−x∗,Rω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e−gx∗≤−2ωμgxt−x∗2+2ωxt−x∗Rω∂ψ⋅,xgx−Sa−Tb−ωηc,d,e−Rω∂ψ⋅,x∗gx∗−Sa∗−Tb∗−ωηc∗,d∗,e∗≤−2ωμg−ξ−1−2μB+λT2δB2+λg2−2μA+λS2δA2+1−2ωr1+r2+r3+ω2λη1δC+λη2δD+λη3δE2xt−x∗2,that is,(100)ddtx−x∗2≤−2ωμg−Φxt−x∗2,where(101)Φ=1−2μB+λT2δB2+λg2−2μA+λS2δA2+ξ+1−2ωr1+r2+r3+ω2λη1δC+λη2δD+λη3δE2.
Therefore, we have(102)x−x∗≤x0−x∗e−ωλg−Φt−t0.
The condition in (35) and the fact that 1−t1<μg−ξ<1−t21+t2 guarantee that μg−Φ>0. Hence, the trajectory of the solution of generalized resolvent dynamical system (81) converges globally exponentially to the unique solution of the set-valued extended generalized nonlinear mixed variational inequality problem (23).


## 6. Numerical Example

In support of Theorem2, we provide a numerical example with convergence graphs and computation tables. We compare our three-step iterative Algorithm 1 with Ishikawa-type and Mann-type algorithms. The convergence graphs and the computation tables are provided for the sequences generated by Algorithm 1.Example 1.
LetH=ℝ with the usual inner product and norm. Let S,T,g:ℝ⟶ℝ, η:ℝ×ℝ×ℝ⟶ℝ, and A,B,C,D,E:ℝ⟶CBℝ be the mappings defined by(103)Sx=x2+1,Tx=255x−254,gx=x5,ηx,y,z=x+y+z,Ax=−x2,Bx=1−x3,Cx=−x+12,Dx=−x7−1,Ex=−x2,∀x∈ℝ.
It is easy to see thatS is 3/4-Lipschitz continuous mapping, T is 256-Lipschitz continuous and 254-strongly monotone mapping, g is 1/2-Lipschitz continuous and 1/4-strongly monotone mapping, η is Lipschitz continuous in first, second, and third arguments with respect to C,D, and E with constants 3/4,1/5,1/10, respectively, η is relaxed Lipschitz continuous in first, second, and third arguments with respect to C,D, and E with constants 3/5,1/6,1/10, respectively, and A,B,C,D, and E are D˜-Lipschitz continuous mappings with constants 3/4,1/2,3/4,1/6, and 1/10, respectively.
Letψ:ℝ×ℝ⟶ℝ∪+∞ be such that, for each fixed x∈ℝ, ψ⋅,x is a functional satisfying gℝ∩dom∂ψ⋅,x≠∅, let ∂ψ:ℝ×ℝ⟶2ℝ be the subdifferential of ψ, and Jω∂ψ.,x is the resolvent operator associated with the subdifferential ∂ψ such that(104)ψgx,x=25gx2+x2=2x2,∀x∈ℝ.
Then,(105)∂ψgx,x=4x,Jω∂ψ.,xx=x1+4ω,∀x∈ℝ.
Forω=1, mapping G defined in (37) has a fixed point x∗=0∈H such that 0∈A0,1∈B0,−1/2∈C0,−1∈D0, and 0∈E0 which is a solution of SEGNMVI (23). We verify the conditions of Algorithm 1.
Letγn=1/n, σn=n+1/2n2+n, ςn=2n+2/5n2+2n, ln=n2+1/3n3+n2, pn=1/n2+1, and qn=1/n3. It is easy to check that the sequences γn, σn, ςn, ln, pn, and qn satisfy the conditions 0≤γn,σn,ςn≤1, and ∑n=0∞γn=∞ given in Algorithm 1. For ω=1, we obtain the sequences xn, γn, and zn generated by Algorithm 1 as(106)zn=5n2−25n2+2nxn+2n+1n5n+265xn−1685100xn+2n+1n45n+2yn=2n2−12n2+nxn+n+12n2+n65zn−1685100zn+n+12n2+n1n2+1xn+1=n−1nxn+1n65yn−1685100yn+n2+13n33n+1.
All codes are written in MATLAB versionR2019a, for different choices of initial values x0=3,5,−3, and −5, which ensures that the sequence xn converges to x∗=0.
It is shown in Figure1 that the sequence xn converges to 0. In Table 1, comparing different initial values of xn and for various number of iterations, it is obtained that the sequence xn converges to 0.Figure 1
The convergence ofxn with initial values x0=3,5,−3, and −5.Table 1
The values ofxn with initial values x0=3,5,−3, and −5.
No. of iterationsForx0=3xnForx0=5xnForx0=−3xnForx0=−5xnn = 11.4403142.885622−1.440044−2.400163n = 20.6915721.665419−0.691171−1.152086n = 30.3321310.961242−0.331668−1.152086n = 40.1595780.554863−0.159085−0.265306n = 50.0767420.320343−0.076235−0.127227n = 60.0369760.185001−0.036462−0.060941n = 70.0178850.106896−0.017368−0.029120n = 80.0087210.061822−0.008202−0.013844n = 90.0043220.035810−0.003802−0.006510n = 100.0022100.020798−0.00169−0.006510n = 110.0011960.012135−0.001690−0.001300n = 120.0007090.004250−0.000489−0.000489n = 130.0000260.001624−0.00019−0.000189n = 1400.001070−0.000044−0.000025n = 1500.00031300n = 200000Remark 2.
Consider the mappings defined in Example1; we compare Algorithm 1 with the Ishikawa-type algorithm and Mann-type algorithm. By taking ςn=0,∀n≥0, Algorithm 1 becomes Ishikawa-type Algorithm; that is, we can compute the sequences xn and yn by the following Ishikawa-type algorithm:(107)yn=2n2−12n2+nxn−1565100n+12n2+nxn+n+12n2+n1n2+1xn+1=n−1nxn+1n65yn−1685100yn+n2+13n33n+1.
By takingσn=ςn=0,∀n≥0, Algorithm 1 becomes Mann-type algorithm; we can compute the sequence xn by the following Mann-type algorithm:(108)xn+1=n−1nxn−1565100nxn+n2+13n33n+1.
The iterative schemes will suspend when the stopping criteriaxn+1−xn≤10−6 are fulfilled. The comparison of Algorithm 1 with Ishikawa-type algorithm and Mann-type algorithm is given in Figure 2 and Table 2, for initial value x0=5.
We conclude that our algorithm is fast, efficient, and stable, and it takes average of 16–20 iterations to converge. Thus, algorithm1 has a faster rate of convergence than Ishikawa-type algorithm and Mann-type algorithm.Figure 2
The convergence ofxn with initial value x0=5.Table 2
The values ofxn with initial value x0=5.
No. of iterationsProposed algorithm 1xnIshikawa-type algorithmxnMann-type algorithmxnn = 12.8856223.4751953.896799n = 21.6654192.4154383.037037n = 30.9612421.6788942.366997n = 40.5548631.1669871.844812n = 50.3203430.8112051.437856n = 60.1850010.5639331.120701n = 70.1068960.3920760.873532n = 110.0121350.1319230.413791n = 150.0003130.0002660.000344n = 2000.0217660.119386n = 2500.0010070.010417n = 3000.0001090.001936


## 7. Conclusion

The purpose of this paper is to introduce and study a set-valued extended generalized nonlinear mixed variational inequality problem and a generalized resolvent dynamical system. A three-step iterative algorithm is defined to obtain the solution of set-valued extended generalized nonlinear mixed variational inequality problem and it is shown through a numerical example that the rate of convergence of our three-step algorithm is faster than Ishikawa-type algorithm and Mann-type algorithm. It is shown that the trajectory of the solution of generalized resolvent dynamical system converges globally exponentially to a unique solution of set-valued extended generalized nonlinear mixed variational inequality problem. We remark that there is still a wide scope of studying globally asymptotic stability of generalized resolvent dynamical system and one can explore many applications of our results in pure and applied sciences.



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