# Mixed Variational Inequality Problem Involving Generalized Yosida Approximation Operator in q-Uniformly Smooth Banach Space

Authors: Arvind Kumar Rajpoot; Rais Ahmad; Mohd Ishtyak; Ching-Feng Wen
Journal: Journal of Mathematics
 (2022)

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## Abstract

A mixed variational inequality problem involving generalized Yosida approximation operator is considered and studied inq-uniformly smooth Banach space. We have shown that mixed variational inequality problem involving generalized Yosida approximation operator is equivalent to a fixed-point equation. The fixed-point formulation is applied to establish an algorithm to obtain the solution of mixed variational inequality problem involving generalized Yosida approximation operator. Convergence criteria are also discussed. In support of our main result, we provide an example using Matlab program together with a computation table and convergence graphs. To check the validity of mixed variational inequality problem involving generalized Yosida approximation operator and its fixed-point formulation, we construct one more example.


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## Body

## 1. Introduction

In 1959, Antonio Signorini posed the Signorini problem in the form of a variational inequality problem and solved by Gaetano Fichera in 1963, which was the first paper related to variational inequalities. After that, in order to study the regularity problem for partial differential equations, Guido Stampacchia proved a theorem called Lax–Milgram theorem and invented the name “variational inequality” for all the problems of the same nature involving the inequalities.In fact, variational inequality is an inequality which embroils a functional, which is to be solved for all feasible values of a given variable, generally associated with a convex set. At first, this theory was developed to solve some equilibrium problems. In this problem, the functional involved was obtained as the first variation of the involved potential energy. That is why it has a variational origin.Mixed variational inequality problem was originally studied by Lescarret [1] and Browder [2] due to their various applications. Konnov and Volotskaya [3] considered general equilibrium problems and oligopolistic equilibrium problems which can be formulated as mixed variational inequality problems. Mixed variational inequalities involve a nonlinear term, and due to this nonlinear term, all projection methods fail to solve the mixed variational inequalities. In this case, the technique of resolvent of a maximal monotone operator is applicable.Many authors have shown that the applications of variational inequalities and its generalized forms can be found in fluid flow through porous media, elasticity, economics, transportation, regional and engineering sciences, odd-order obstacle problems, problems of lubrication, computational procedures, spatial price equilibrium problems, oligopolistic market equilibrium problems, environment network problems, etc., see [4–13] and references therein.Moreau’s theorem demonstrates that convex functions are well behaved on Hilbert spaces and are differentiable. In this case, derivatives are approximated by Yosida approximation, which is explained with reference to the resolvent operator. Yosida approximation operator are applicable to solve first-order differential evolution inclusions, nonlinear nonhomogeneous evolution inclusions, heat and wave equations, etc.If we choose Hilbert space structure to solve problems related to variational inequalities, then it is easy for us to use tools of functional analysis. That is, it is easy to use polarization identity, parallelogram law, and many other concepts. In view of applications of several problems, the Hilbert space structure is not much suitable. The Banach space structure is appropriate from the application point of view because of availability of the same geometric structure such as polarization identity and parallelogram law. Moreover, in Banach spaces, we can use the property of uniform convexity or uniform smoothness, duality mapping etc. For more details and applications, see [13–18].Motivated by the abovementioned facts discussed above, in this paper, we introduce and study a mixed variational inequality problem involving generalized Yosida approximation operator inq-uniformly smooth Banach space. An algorithm is suggested which is based on a fixed-point formulation to obtain the solution of our problem. An existence and convergence is proved. Two examples are constructed.


## 2. Basic Tools and Required Results

LetE be a real Banach space with its norm ⋅, E∗ be its topological dual, ⋅,⋅ be the pairing between E and E∗, d be the metric induced by the norm, CBE be the family of all nonempty closed and bounded subsets of E, and D⋅,⋅ be the Hausdorff metric on CBE defined by(1)DA,B=maxsupx∈Adx,B,supy∈BdA,y,where dx,B=infy∈Bdx,y and dA,y=infx∈Adx,y.The generalized duality mappingJq:E⟶2E∗ is defined by(2)Jqx=f∈E∗:x,f=xqandf=xq−1,forallx∈E,where q>1 is a constant. For q=2, the generalized duality mapping coincides with normalized duality mapping. It is well known that Jqx=xq−2J2x, for all x≠0, and Jq is single-valued if E∗ is strictly convex.The modulus of smoothness ofE is the function ρE:0,∞⟶0,∞ defined by(3)ρEt=x+y+x−y2−1:x≤1,y≤t.A Banach spaceE is called uniformly smooth if(4)limt⟶0ρEtt=0.E is called q-uniformly smooth if there exists a constant C>0 such that(5)ρEt≤Ctq,q>1.The following important result of Xu [19] is instrumental to prove our main result.Theorem 1.
LetE be a real uniformly smooth Banach space. Then, E is q-uniformly smooth if and only if there exists a constant Cq>0 such that, for all x,y∈E,(6)x+yq≤xq+qy,Jqx+Cqyq.Definition 1.
Letϕ:E⟶ℝ∪+∞ be a functional. A vector w∗∈E∗ is called subgradient of ϕ at x∈domϕ if(7)w∗,y−x≤ϕy−ϕx,for ally∈E.
The subdifferential ofϕ denoted by ∂ϕ is defined as(8)∂ϕx=w∗∈E∗:w∗,y−x≤ϕy−ϕx,∀y∈E,x∈domϕ,∅,x∉domϕ.Definition 2.
LetH:E⟶E be a single-valued mapping and ϕ:E⟶ℝ∪+∞ be a proper and subdifferential functional. The generalized proximal operator associated with H and ϕ, for some ρ>0, is defined as(9)JH,ρ∂ϕx=H+ρ∂ϕ−1x,for allx∈E.Definition 3.
The generalized Yosida approximation operator associated with proximal operatorJH,ρ∂ϕ is defined as(10)YH,ρ∂ϕx=1ρH−JH,ρ∂ϕx,for allx∈E,where ρ>0 is a constant.Definition 4.
LetA:E⟶E be a mapping. Then,(i)
A is said to be Lipschitz continuous if there exists a constant λA>0 such that(11)Ax−Ay≤λAx−y,for allx,y∈E.(ii)
A is said to be accretive if(12)Ax−Ay,Jqx−y≥0,for allx,y∈E.(iii)
A is said to be strongly accretive if there exists a constant δA>0 such that(13)Ax−Ay,Jqx−y≥≥δAx−yq,for allx,y∈E.Definition 5.
A multivalued mappingF:E⟶CBE is said to be D-Lipschitz continuous if, for all x,y∈E, there exists a constant λDF>0, such that(14)DFx,Fy≤λDFx−y.
Now, we prove some results which are required in the sequel.Proposition 1.
LetH be strongly accretive with constant α, and the subdifferential operator ∂ϕ of ϕ is accretive. Then, the generalized proximal operator JH,ρ∂ϕ:E⟶E is 1/α-Lipschitz continuous, that is,(15)JH,ρ∂ϕx−JH,ρ∂ϕy≤1/αx−y,for allx,y∈E.Proof.
Letx and y be any given points in E. It follows from Definition 2 that(16)JH,ρ∂ϕx=H+ρ∂ϕ−1x,JH,ρ∂ϕy=H+ρ∂ϕ−1y.
From the above, we have(17)1ρx−HJH,ρ∂ϕx∈∂ϕJH,ρ∂ϕx1ρy−HJH,ρ∂ϕy∈∂ϕJH,ρ∂ϕy.
Since∂ϕ is accretive, we obtain(18)1ρx−HJH,ρ∂ϕx−y−HJH,ρ∂ϕy,JqJH,ρ∂ϕx−JH,ρ∂ϕy≥0,1ρx−y−HJH,ρ∂ϕx−HJH,ρ∂ϕy,JqJH,ρ∂ϕx−JH,ρ∂ϕy≥0.
Therefore, we have(19)x−y,JqJH,ρ∂ϕx−JH,ρ∂ϕy≥HJH,ρ∂ϕx−HJH,ρ∂ϕy,JqJH,ρ∂ϕx−JH,ρ∂ϕy.
SinceH is strongly accretive with constant α, we have(20)x−yJH,ρ∂ϕx−JH,ρ∂ϕyq−1≥αJH,ρ∂ϕx−JH,ρ∂ϕyq.
Thus,(21)JH,ρ∂ϕx−JH,ρ∂ϕy≤1αx−y.Proposition 2.
LetH be Lipschitz continuous with constant λH and the generalized proximal operator JH,ρ∂ϕ be Lipschitz continuous with constant 1/α. Then, the generalized Yosida approximation operator YH,ρ∂ϕ is λY-Lipschitz continuous, where λY=αλH+1/ρα.Proof.
Using Definition3, we have(22)YH,ρ∂ϕx−YH,ρ∂ϕy=1ρHx−JH,ρ∂ϕx−Hy−JH,ρ∂ϕy,=1ρHx−Hy−JH,ρ∂ϕx−JH,ρ∂ϕy,=1ρHx−Hy+1ρJH,ρ∂ϕx−JH,ρ∂ϕy,=1ρλHx−y+1ραx−y,=λYx−y,forallx,y∈E,where λY=αλH+1/ρα.Proposition 3.
LetH be strongly accretive with constant α>0 and the generalized proximal operator JH,ρ∂ϕ is Lipschitz continuous with constant 1/α. Then, the generalized Yosida approximation operator YH,ρ∂ϕ is strongly accretive with constant δY, where δY=α2−1/ρα.Proof.
Using Definition3, we have(23)YH,ρ∂ϕx−YH,ρ∂ϕy,Jqx−y=1ρHx−Hy−JH,ρ∂ϕx−JH,ρ∂ϕy,Jqx−y,=1ρHx−Hy,Jqx−y−1ρJH,ρ∂ϕx−JH,ρ∂ϕy,Jqx−y,≥1ραx−yq−JH,ρ∂ϕx−JH,ρ∂ϕyx−yq−1,≥αρx−yq−1αρx−yq,=α2−1ραx−yq,=δYx−yq,forallx,y∈E,where δY=α2−1/ρα and α2>1.


## 3. Construction of the Problem and Fixed-Point Formulation

LetA,B,C:E⟶E∗ and g,H:E⟶E be the single-valued mappings and F,T,S:E⟶CBE be the multivalued mappings. Let ϕ:E⟶ℝ∪+∞ be a functional on E such that g∩dom∂ϕ≠∅, where ∂ϕ is the subdifferential of ϕ. Let YH,ρ∂ϕ:E⟶E be the generalized Yosida approximation operator. We consider the following problem.Findx∈E,u∈Fx,v∈Tx, and w∈Sx such that(24)Au−Bv−Cw,gYH,ρ∂ϕy−gYH,ρ∂ϕx≥ϕgYH,ρ∂ϕx−ϕgYH,ρ∂ϕy.Problem (24) is called mixed variational inequality problem involving generalized Yosida approximation operator.It is to be noted that, for appropriate choices of operators, one can obtain many variant forms of problem (24) available in the literature. That is, problems studied by Verma [20], Lee et al.[21], Ding [22], Ahmad et al. [23], Ahmad and Siddiqi [24], etc. can be obtained from problem (24).The fixed-point formulation of mixed variational inequality problem involving generalized Yosida approximation operator is given below.Lemma 1.
The set of elementsx,u,v,w, where x∈E,u∈Fx,v∈Tx, and w∈Sx, is a solution of mixed variational inequality problem involving generalized Yosida approximation operator (24) if and only if the following equation is satisfied:(25)gYH,ρ∂ϕx=JH,ρ∂ϕHgYH,ρ∂ϕx−ρAu−Bv−Cw.Proof.
Letx∈E,u∈Ax,v∈Bx,andw∈Cx satisfy equation (25); then, we have(26)gYH,ρ∂ϕx=JH,ρ∂ϕHgYH,ρ∂ϕx−ρAu−Bv−Cw.
Applying the definition of generalized proximal operatorJH,ρ∂ϕ, we obtain(27)gYH,ρ∂ϕx=H+ρ∂ϕ−1HgYH,ρ∂ϕx−ρAu−Bv−Cw⇔HgYH,ρ∂ϕx+ρ∂ϕgYH,ρ∂ϕx=HgYH,ρ∂ϕx−ρAu−Bv−Cw⇔−ρAu−Bv−Cw∈ρ∂ϕgYH,ρ∂ϕx.
Using the definition of subdifferential ofϕ, we obtain the required mixed variational inequality problem involving generalized Yosida approximation operator (24). That is,(28)Au−Bv−Cw,gYH,ρ∂ϕy−gYH,ρ∂ϕx≥ϕgYH,ρ∂ϕx−ϕgYH,ρ∂ϕy,for all y∈E.
Based on Lemma1, we construct the following algorithm for solving mixed variational inequality problem involving generalized Yosida approximation operator (24).Algorithm 1.
For givenx0∈E, u0∈Fx0,v0∈Tx0,andw0∈Sx0, let(29)x1=x0−gYH,ρ∂ϕx0+JH,ρ∂ϕHgYH,ρ∂ϕx0−ρAu0−Bv0−Cw0.
Sinceu0∈Fx0∈CBE,v0∈Tx0∈CBE,andw0∈Sx0∈CBE, by Nadler [25], there exist u1∈Fx1,v1∈Tx1,andw1∈Sx1 such that(30)u0−u1≤1+1DFx0,Fx1,v0−v1≤1+1DTx0,Tx1,w0−w1≤1+1DSx0,Sx1.
Continuing the above process inductively, we can obtain the sequencesxn,un,vn, and wn as(31)xn+1=xn−gYH,ρ∂ϕxn+JH,ρ∂ϕHgYH,ρ∂ϕxn−ρAun−Bvn−Cwn.(32)un−un+1≤1+n+1−1DFxn,Fxn+1,(33)vn−vn+1≤1+n+1−1DTxn,Txn+1,(34)wn−wn+1≤1+n+1−1DSxn,Sxn+1,where n=0,1,2,3,⋯.Theorem 2.
LetE be a real q-uniformly smooth Banach space and g,H:E⟶E, A,B,C:E⟶E∗ be the single-valued mappings such that g is Lipschitz continuous with constant λg, H be Lipschitz continuous with constant λH, A be Lipschitz with constant λA, B be Lipschitz continuous with constant λB, and C be Lipschitz continuous with constant λC. Suppose ϕ:E⟶ℝ∪+∞ is a subdifferentiable, proper functional satisfying gx∈dom∂ϕ, where ∂ϕ is the subdifferential of ϕ.LetYH,ρ∂ϕ be the generalized Yosida approximation operator such that YH,ρ∂ϕ is Lipschitz continuous with constant λY and g°YH,ρ∂ϕ is strongly accretive with constant δgY. Let F,T, S:E⟶CBE be the multivalued mappings such that F,T, and S are D-Lipschitz continuous mappings with constants λDF,λDT, and λDS, respectively. If the following condition is satisfied,(35)0<1−qδgY+CqgqλYqq+1αλHλgλY+ρλAλDF1+n−1+λBλDT1+n−1+λCλDS1+n−1<1,where λY=αλH+1/ρα and Cq is the same as in Theorem 1.Then, the mixed variational inequality problem involving generalized Yosida approximation operator (24) has a solution.Proof.
Applying (31) of Algorithm 1, we have(36)xn+1−xn=xn+gYH,ρ∂ϕxn+JH,ρ∂ϕHgYH,ρ∂ϕxn−ρAun−Bvn−Cwn−xn−1+gYH,ρ∂ϕxn−1+JH,ρ∂ϕHgYH,ρ∂ϕxn−1−−ρAun−1−Bvn−1−Cwn−1=xn−xn−1+gYH,ρ∂ϕxn−gYH,ρ∂ϕxn−1,+JH,ρ∂ϕHgYH,ρ∂ϕxn−ρAun−1−Bvn−1−Cwn−1−JH,ρ∂ϕHgYH,ρ∂ϕxn−1−ρAun−1−Bvn−1−Cwn−1.
Using Theorem1, we have(37)xn−xn−1+gYH,ρ∂ϕxn−gYH,ρ∂ϕxn−1q≤xn−xn−1q+qgYH,ρ∂ϕxn−gYH,ρ∂ϕxn−1,Jqxn−xn−1+CqgYH,ρ∂ϕxn−gYH,ρ∂ϕxn−1q.
Sinceg°YH,ρ∂ϕ is strongly accretive with constant δgY, we have(38)gYH,ρ∂ϕxn−gYH,ρ∂ϕxn−1,Jqxn−xn−1≥δgYxn−xn−1q.
Asg and YH,ρ∂ϕ are Lipschitz continuous mappings with constants λg and λY, respectively, we have(39)gYH,ρ∂ϕxn−gYH,ρ∂ϕxn−1≤λgYH,ρ∂ϕxn−YH,ρ∂ϕxn−1,≤λgλYxn−xn−1.
Thus,(40)gYH,ρ∂ϕxn−gYH,ρ∂ϕxn−1≤λgqλYqxn−xn−1q.
Combining (38) and (40) with (37), we obtain(41)xn−xn−1+gYH,ρ∂ϕxn−gYH,ρ∂ϕxn−1q≤xn−xn−1q+qδgYxn−xn−1q+CqλgqλYqxn−xn−1q≤1+qδgY+CqλgqλYqxn−xn−1q,which gives us(42)xn−xn−1+gYH,ρ∂ϕxn−gYH,ρ∂ϕxn−1q≤1+qδgY+CqλgqλYqqxn−xn−1q.
Using the Lipschitz continuity ofJH,ρ∂ϕ,H,g, and YH,ρ∂ϕ, we obtain(43)JH,ρ∂ϕHgYH,ρ∂ϕxn−ρAun−Bvn−Cwn−JH,ρ∂ϕHgYH,ρ∂ϕxn−ρAun−1−Bvn−1−Cwn−1,≤1αHgYH,ρ∂ϕxn−HgYH,ρ∂ϕxn−ρAun−Bvn−Cwn−Aun−1−Bvn−1−Cwn−1≤1αλHλgλYxn−xn−1+1αρAun−Bvn−Cwn−Aun−1−Bvn−1−Cwn−1.
Applying Lipschitz continuity ofA,B and C, D-Lipschitz continuity of F,T,S and using (32), (37), and (38) of Algorithm 1, we obtain(44)Aun−Bvn−Cwn−Aun−1−Bvn−1−Cwn−1,≤Aun−Aun−1−Bvn−Bvn−1−Cwn−Cwn−1,≤λAun−un−1+λBvn−vn−1+λCwn−wn−1,≤λADFxn,Fxn+1+λBDTxn,Txn+1+λCDSxn,Sxn+1,≤λA1+n−1λDFxn−xn−1+λB1+n−1λDTxn−xn−1+λC1+n−1λDSxn−xn−1,≤λA1+n−1λDF+λB1+n−1λDT+λC1+n−1λDSxn−xn−1.
Combining (43) and (44), we obtain(45)JH,ρ∂ϕHgYH,ρ∂ϕxn−ρAun−Bvn−Cwn−JH,ρ∂ϕHgYH,ρ∂ϕxn−ρAun−1−Bvn−1−Cwn−1,≤1αλHλgλYxn−xn−1+1αρλA1+n−1λDF+λB1+n−1λDT+λC1+n−1λDSxn−xn−1≤1αλHλgλY+ρλA1+n−1λDF+λB1+n−1λDT+λC1+n−1λDSxn−xn−1.
Combining (42) and (43) with (36), we obtain(46)xn−xn−1≤1+qδgY+CqλgqλYqq+1αλHλgλY+ρλA1+n−1λDF+λB1+n−1λDT+λC1+n−1λDSxn−xn−1,Thusxn−xn−1≤θnxn−xn−1,where(47)θn=1+qδgY+CqλgqλYqq+1αλHλgλY+ρλA1+n−1λDF+λB1+n−1λDT+λC1+n−1λDS,Letθ=1+qδgY+CqλgqλYqq+1αλHλgλY+ρλAλDF+λBλDT+λCλDS.
Clearly,θn⟶θ, as n⟶∞. It follows from (35) that θ<1, and hence, θn<1, for n sufficiently large. From (46), it is clear that xn is a Cauchy sequence and xn⟶x∈E.
Furthermore, we will show thatun⟶u∈Fx,vn⟶v∈Tx, and wn⟶w∈Sx. It follows from (32), (33), and (34) of Algorithm 1 and D-Lipschitz continuity of F,T, and S that(48)un−un−1≤1+n−1DFxn,Fxn+1≤1+n−1λDFxn−xn−1,(49)vn−vn−1≤1+n−1DTxn,Txn+1≤1+n−1λDTxn−xn−1,(50)wn−wn−1≤1+n−1DSxn,Sxn+1≤1+n−1λDSxn−xn−1.
From (48), (49), and (50), we obtain that un,vn, and wn are all Cauchy sequences in E. Therefore, un⟶u,vn⟶v, and wn⟶w, as n⟶∞. We have(51)du,Fx=infu−y:y∈Fx,=u−un+dun,Fx,=u−un+DFxn,Fx,=u−un+λDFxn−x⟶0,asn⟶∞.
Hence,u∈Fx. Similarly, we can show that v∈Tx and w∈Sx. This completes the proof.


## 4. Numerical Example

In support of concepts used in Theorem2, we provide numerical example 1 using MATLAB R2018a with a computation table and convergence graphs. Thenceforward, in example 2, it is shown that mixed variational inequality problem involving generalized Yosida approximation operator (24) is satisfied together with its fixed-point formulation (25).Example 1.
LetE=ℝ=E∗ and g,H:ℝ⟶ℝ be the single-valued mappings such that, for all x∈ℝ,(52)gx=23x,Hx=53x+1.
Then,(i)
g is Lipschitz continuous mapping with constant λg=4/3. That is,(53)gx−gy=23x−23y,=23x−y,≤43x−y.(ii)
H is Lipschitz continuous mapping with constant λH=7/3. That is,(54)Hx−Hy=53x+1−53y+1,=53x−y,≤73x−y. Also, H is strongly accretive mapping with constant α=4/3. That is,(55)Hx−Hy,x−y=53x+1−53y+1,x−y,=53x−y,x−y,=53x−y2,≥43x−y2.(iii)
Suppose thatϕ:ℝ⟶ℝ∪+∞ be the operator such that, for all x∈ℝ,(56)ϕx=x2. Then, the subdifferential of ϕ, that is, ∂ϕx=2x.For ρ=1, we calculate the generalized proximal operator and generalized Yosida operator associated with H and ϕ:(57)JH,ρ∂ϕx=H+ρ∂ϕ−1x,=53x+1+1×2x−1x,=113x+1−1x,=311x−311,and(58)YH,ρ∂ϕx=1ρH−JH,ρ∂ϕx,=1153x+1−311x−311,=4633x+1411. It is easy to check that the generalized proximal operator JH,ρ∂ϕ and generalized Yosida operator YH,ρ∂ϕ are 1/α and λY-Lipschitz continuous, respectively, where α=4/3 and λY=37/12.(iv)
The generalized Yosida approximation operatorYH,ρ∂ϕ is strongly accretive with constant δY=7/12, for all x,y∈ℝ,(59)YH,ρ∂ϕx−YH,ρ∂ϕy,x−y=4633x+1411−4633y+1411,x−y,=4633x−y,x−y,=4633x−y2,≥712x−y2.(v)
g°YH,ρ∂ϕ is strongly accretive with constant δgY=4/5. As,(60)g°YH,ρ∂ϕx=gYH,ρ∂ϕx,=g4633x+1411,=234633x+1411,=9299x+2833,therefore,(61)g°YH,ρ∂ϕx−YH,ρ∂ϕy,x−y=9299x+2833−9299y+2833,x−y,=9299x−y,x−y,=9299x−y2,≥45x−y2.(vi)
LetA,B,C:ℝ⟶ℝ be the single-valued mappings and F,S,T:ℝ⟶CBℝ be the multivalued mapping such that(62)Ax=34x,Bx=25x,Cx=45x,Fx=12x−1,Tx=32x+1,Sx=14x−1.
Accordingly,(63)un=12xn−1∈Fxn,vn=32xn+1∈Txn,andwn=14xn−1∈Sxn,foralln=0,1,2,….
Using (31) of Algorithm 1, we have(64)xn+1=xn−gYH,ρ∂ϕxn+JH,ρ∂ϕHgYH,ρ∂ϕxn−ρAun−Bvn−Cwn,=xn−9299xn+2833+JH,ρ∂ϕ460297xn+23999−−140xn−3920,=799xn−2833+JH,ρ∂ϕ1869711880xn+86411980,which gives us(65)xn+1=725914520xn+1672420.
It is shown in Figures1, 2, and 3 that, for initial values x0=1,−2and3, the sequence xn converges to 0.137 997 5, respectively. In Figure 4, we combine all the graphs plotted in Figures 1–3 (Table 1).Figure 1
Case 1: for the initial valuex0 = 1.Figure 2
Case 2: for the initial valuex0 = −2.Figure 3
Case 3: for the initial valuex0 = 3.Figure 4
By combining all the above graphs, we get the following conjoining graph of convergence.Table 1
Computational results for different initial valuesx0 = 1, x0 = −2, and x0 = 3.
No. of iterationForx0=1Forx0=−2Forx0=3xnxnxnn = 11.000 0−2.000 03.000 0n = 20.568 939 393 939 394−0.930 853 994 490 3581.568 801 652 892 56n = 30.353 438 778 278 654−0.396 354 624 380 5450.853 301 046 718 120n = 40.245 703 312 088 482−0.129 141 750 577 0230.495 600 020 532 151n = 50.191 842 998 791 3420.004 446 283 234 2550.316 774 142 496 067n = 60.164 916 551 530 7410.071 231 099 862 084 10.227 373 519 309 845n = 70.151 455 182 338 9560.104 618 908 670 7210.182 679 364 784 447n = 80.144 725 424 834 6060.121 310 513 639 1710.160 335 365 631 563n = 90.141 361 009 564 3530.129 655 166 563 8250.149 164 904 898 038n = 100.139 679 033 638 2670.133 826 918 325 5380.143 580 443 846 753n = 120.138 417 783 362 8500.136 955 157 545 1550.139 392 867 241 313n = 140.138 102 557 650 9800.137 737 001 921 4240.138 346 261 470 683n = 160.138 023 772 931 2770.137 932 409 173 1680.138 084 682 103 349n = 180.138 004 082 176 9220.137 981 247 529 2320.138 019 305 275 383n = 200.137 999 160 844 3530.137 993 453 754 9560.138 002 965 570 617n = 250.137 997 572 212 3880.137 997 393 988 6390.137 997 691 028 221n = 300.137 997 522 601 8200.137 997 517 036 1630.137 997 526 312 259Example 2.
Let all the mappings be the same as considered in Example1; then, we check the validity of mixed variational inequality problem involving generalized Yosida approximation operator (24) and its fixed-point formulation (25):(66)Au−Bv−Cw,gYH,ρ∂ϕy−gYH,ρ∂ϕx==−140−3920,9299y+2833−9299x+2833=−140−3920,9299y−9299x=140x−39209299y−x=9299140xy−3920y+140x2+3920x=23990x2−xy+78x−78y,ϕgYH,ρ∂ϕx−ϕgYH,ρ∂ϕy==ϕ9299x+2833−ϕ9299y+2833=9299x+28332−9299y+28332=278980123x2−23y2+42x−42y.
We obtained in Example1 that x=0.1379975 is one of the solutions of mixed variational inequality problem involving generalized Yosida approximation operator (24). For x=0.1379975 and y=1, we obtain(67)Au−Bv−Cw,gYH,ρ∂ϕy−gYH,ρ∂ϕx=23990x2−xy+78x−78y=239900.13799752−0.1379975+780.1379975−78=−1.565and(68)ϕgYH,ρ∂ϕx−ϕgYH,ρ∂ϕy=278980123x2−23y2+42x−42y=2789801230.13799752−23+420.1379975−42=−1.666.
From the above computation, we have(69)Au−Bv−Cw,gYH,ρ∂ϕy−gYH,ρ∂ϕx>ϕgYH,ρ∂ϕx−ϕgYH,ρ∂ϕy.
Letx=1.0704 be another solution of mixed variational inequality problem involving generalized Yosida approximation operator (24); then,(70)Au−Bv−Cw,gYH,ρ∂ϕy−gYH,ρ∂ϕx=23990x2−xy+78x−78y=239901.07042−1.0704+781.0704−78=0.1792and(71)ϕgYH,ρ∂ϕx−ϕgYH,ρ∂ϕy=278980123x2−23y2+42x−42y=2789801231.07042−23+421.0704−42=0.1792.
That is,(72)Au−Bv−Cw,gYH,ρ∂ϕy−gYH,ρ∂ϕx=ϕgYH,ρ∂ϕx−ϕgYH,ρ∂ϕy.
Combining (69) and (72), we have(73)Au−Bv−Cw,gYH,ρ∂ϕy−gYH,ρ∂ϕx≥ϕgYH,ρ∂ϕx−ϕgYH,ρ∂ϕy.
We now verify the fixed-point formulation (25):(74)gYH,ρ∂ϕx=9299x+2833,JH,ρ∂ϕHgYH,ρ∂ϕx−ρAu−Bv−Cw=JH,ρ∂ϕ1869711880x+86411980=56091130680x+219813239580.
Forx=0.1379975, we observe that the fixed-point formulation (25) is fulfilled:(75)gYH,ρ∂ϕx=92990.1379975+2833,=0.97672,JH,ρ∂ϕHgYH,ρ∂ϕx−ρAu−Bv−Cw=560911306800.1379975+219813239580,=0.97672.
That is,(76)gYH,ρ∂ϕx=JH,ρ∂ϕHgYH,ρ∂ϕx−ρAu−Bv−Cw.


## 5. Conclusion

In this work, we study a new problem, that is, a mixed variational inequality problem involving a generalized Yosida apoproximation operator. A fixed-point formulation is provided in order to define an iterative algorithm.Finally, an existence and convergence result is obtained for mixed variational inequality problem involving generalized Yosida approximation operator inq-uniformly smooth Banach space. A numerical example is provided in support of concepts used in our main result, and another numerical example is furnished to check the validity of mixed variational inequality problem involving generalized Yosida approximation operator (24) and its fixed-point formulation (25).Furthermore, we remark that our result can be extended in higher dimensional spaces.



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