# A Note on the Generalized Nonlinear Vector Variational-Like Inequality Problem

Authors: Ankit Gupta; Satish Kumar; Ratna Dev Sarma; Pankaj Kumar Garg; Reny George
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we discuss two variants of the generalized nonlinear vector variational-like inequality problem. We provide their solutions by adopting topological approach. Topological properties such as compactness, closedness, and net theory are used in the proof. The admissibility of the function space topology and KKM-Theorem have played important role in proving the results.


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## Body

## 1. Introduction

Variational inequalities have appeared as a working and important tool to investigate various fields of mathematics as well as of sciences including elasticity, vector equilibrium problems, and optimization problems [1–4]. In mid-sixties, Browder [5] formulated and proved the basic existence results for the solutions to a class of nonlinear variational inequality problems. He used a reflexive Banach space X and a monotone nonlinear map T from the space X to its dual space X∗, to set up the nonlinear variational inequality problem. Browder used the property of hemicontinuity and monotonicity of mapping T along with the lower semicontinuity of f, for providing the existence of the solution of nonlinear variational inequality problem. After that, this problem has been generalized and extended in various directions under different set-ups using different techniques. Liu et al. [6], Zhao et al. [7], and Ahmad and Irfan [8] are a few, who extended Browder’s results to more generalized nonlinear variational inequalities. In 2009, Farajzadeh et al. [9] considered new kinds of generalized variational-like inequality problems under the frame work of topological vector spaces.In the subsequent period, generalized quasi-variational inequalities were studied by Hung and others [10–12]. In 2017, Irfan et al. introduced a new generalized variational-like inclusion problem involving relaxed monotone operators [13]. A class of η-generalized operator variational-like inequalities were introduced by Kim et al. in 2018 [14]. In the same year, Tavakoli et al. studied the C-pseudomonotone property for the set-valued mappings in order to solve a generalized variational inequality problems [15]. On the other hand, vector equilibrium problems for the set-valued mappings were studied by Farajzadeh et al. and Chen et al. during this period [16, 17]. This wide range of literature is a clear indication of the importance that variational inequality problems have gained in the recent years. In this paper, we further add to this literature by providing solutions to a generalized nonlinear vector variational-like inequality problem, using topological methods.Variational-like inequalities have number of applications which make it an interesting discipline for research. Vector variational inequality on flow equilibrium problem on a network has been discussed in [18]. Application of variational-like inequality in fuzzy optimization problem is discussed in [19]. More such studies are available in the literature [20–22].Motivated by these studies, here, we investigate a generalized nonlinear variational like inequality problem, which was proposed by Farajzadeh et al. [9] as follows:Generalized nonlinear variational-like inequality problem: let X,X∗ be a dual system of Hausdorff topological vector spaces and K be a nonempty convex subset of X. Given the mappings f,g,p:X∗⟶X∗ and η:K×K⟶X, set-valued map M,S,T:K⟶X∗, and a map h:K×K⟶ℝ, consider the following generalized nonlinear variational-like inequality problem (GNVLIP)(1)Findx∈Ksuchthatforeachy∈K,∃u∈Mx,v∈Sx,w∈Txsatisfyingpu−fv−gw,ηy,x≥hx,yIn this paper, we consider two variants of nonlinear vector variational-like inequality problems in a more general set-up as follows:LetX and Y be two topological vector spaces, and let K be a nonempty, closed, and convex subset of X and CLX,Y be the space of all continuous linear mappings from the space X to the space Y. Clearly, CLX,Y is nonempty as the zero mapping, that is, k:X⟶Y defined as kx=0 for all x∈X is always linear and continuous, hence belongs to CLX,Y.Further, letM,S,T:K⟶CLX,Y be set-valued mappings and f,g,p:CLX,Y⟶CLX,Y be single-valued mappings. Suppose the maps η:K×K⟶X and h:K×K⟶Y are two bifunctions.Problem 1.
SupposeC⊆Y is a closed, convex, pointed cone with intC≠∅. Then, the generalized nonlinear vector variational-like inequality problem (I) (GNVVLIP (I)) is to find x0∈K, such that for each y∈K, there exist u∈Mx0, v∈Sx0, and w∈Tx0 such that(2)pu−fv−gwηy,x0−hx0,y∉−intC.Problem 2.
SupposeC:K⟶Y is a set-valued map such that for every x∈K, Cx is a proper, closed, convex, pointed cone with nonempty interior. Then, the generalized nonlinear vector variational-like inequality problem (II) (GNVVLI (II)) is to find x0∈K such that for each y∈K, ∃u∈Mx0, v∈Sx0, and w∈Tx0 such that(3)pu−fv−gwηy,x0−hx0,y∉−intCx0.Here, we are trying to provide solutions to the above stated generalized nonlinear vector variational-like inequality problems (I and II) from a topological point of view. We considerX and Y to be any topological vector spaces and use the concept of admissibility of the function space topology along with net theory to prove the existence of solutions of these generalized nonlinear vector variational-like inequality problems.


## 2. Preliminaries

Below, we provide some definitions and results related mainly to set-valued maps between topological spaces.Definition 3 [23].
SupposeX,τ1 and Y,τ2 are two topological spaces and F:X⟶Y is a set-valued map.(i)
F is called upper semicontinuous (in short, u.s.c.) at a point x∈X, if for every open set V in Y such that Fx⊆V, there exists an open set U in X with x∈U such that FU⊆V;(ii)
F is called lower semicontinuous (in short, l.s.c.) at a point x∈X, if for every open set V in Y such that Fx∩V≠∅, there exists an open set U in X with x∈U such that for each u∈U, Fu∩V≠∅;(iii)
F is said to be continuous at x∈X if it is both upper semicontinuous and lower semicontinuous at x;(iv)
F is said to be continuous (resp. u.s.c. and l.s.c.) if it is so at each point of X.Lemma 4 [24].
SupposeX,τ1 and Y,τ2 are topological spaces and F:X⟶Y is a set-valued map. Then,(i)
ifF is upper semicontinuous at x∈X and Fx is compact then for every net xαα∈D and yα∈Fxα with xα⟶x and yα⟶y, we have y∈Fx;(ii)
F is lower semicontinuous at x∈X if and only if for every y∈Fx and every net xαα∈D with xα⟶x, there exists a subnet xββ∈Ω, where Ω is a directed subset of D and a net yββ∈Ω such that yβ∈Fxβ with yβ⟶y.Theorem 5 [23].
LetX,τ and Y,μ be two topological spaces. Let F:X⟶Y be a set-valued map. Then, F is lower semicontinuous at x∈X if for any net xnn∈Δ in X converging to x∈X, the image net Fxnn∈Δ converges to Fx.Lemma 6 [25].
SupposeX and Y are topological spaces and F:X⟶Y is a set-valued upper semicontinuous function. If Fx is compact for each x∈X, then image of every compact subset of X under F is compact.Definition 7 [23, 26].
LetY,μ1 and Z,μ2 be two topological spaces. Let CY,Z be the space of all continuous mappings from Y to Z. A topology τ on CY,Z is called admissible, if the evaluation map e:CY,Z×Y⟶Z, defined by ef,y=fy, is continuous.Definition 8 [26].
Letfnn∈Δ be a net in CY,Z. Then, fnn∈Δ is said to continuously converge to f if for each net ymm∈σ in Y converging to y, fnymn,m∈Δ×σ converges to fy in Z.Theorem 9 [23].
LetY,τ and Z,μ be two topological spaces. A topology T on CY,Z, the family of continuous mappings from Y to Z, is admissible if and only if for any net fnn∈Δ in CY,Z, fnn∈Δ converges to f in T implies continuous convergence of fnn∈Δ to f.Definition 10 [27].
SupposeF:X⟶Y is a set-valued map from X to Y. The graph of F, denoted by GF, is(4)GF=x,y∈X×Yx∈X,y∈Fx.Definition 11 [28].
SupposeS is a nonempty subset of some topological vector space X. A set-valued map F:S⟶X is called a KKM-mapping if for every nonempty finite set x1,x2,⋯,xn of S, we have(5)convx1,x2,⋯,xn⊆⋃j=1nFxj.The following result is taken from [28].Lemma 12 (KKM-Theorem).
SupposeS is a nonempty subset of some topological vector space X and F:S⟶X is a KKM-mapping such that for every x∈S, Fx is a closed subset of X. If there exists a point x0∈S such that Fx0 is compact, then ⋂x∈SFx≠∅.


## 3. Main Results

Theorem 13.
SupposeX and Y are two topological vector spaces and CLX,Y is the space of all continuous linear mappings from the space X to the space Y equipped with an admissible topology. Let K be a nonempty, compact, closed, and convex subset of X. Suppose C⊆Y is a closed, convex, pointed cone with intC≠∅. Further, let M,S,T:K⟶CLX,Y be set-valued lower semicontinuous mappings and f,g,p:CLX,Y⟶CLX,Y be continuous mappings. Suppose the maps η:K×K⟶X and h:K×K⟶Y are affine mappings such that η is continuous in the second argument and h is continuous in the first argument, respectively, with ηx,x=hx,x=0 for all x∈K. Then, the generalized nonlinear vector variational-like inequality problem (I) has a solution. That is, there exists x0∈K such that for each y∈K, there exist u∈Mx0, v∈Sx0, and w∈Tx0 such that(6)pu−fv−gwηy,x0−hx0,y∉−intC.Proof.
We define a set-valued mapF:K⟶K by(7)Fy=x∈K∃u∈Mx,v∈Sx,w∈Txsuchthatpu−fv−gwηy,x−hx,y∉−intC.
Clearly,Fy is nonempty as y∈Fy. As y∈K, we have ηy,y=hy,y=0. Thus, for each u∈My, v∈Sy, and w∈Tx, we have pu−fv−gwηy,y−hy,y=pu−fv−gw0−0=0. Since C is a closed convex and pointed cone, thus 0∉−intC.
The proof of the theorem is divided into two parts:(i)
F is a KKM-mapping on K:
LetA=x1,x2,⋯,xn⊆K be any finite subset of K.
We show thatconvx1,x2,⋯,xn⊆⋃i=1nFxi. Let, if possible, x′∉⋃i=1nFxi for some x′∈convx1,x2,⋯,xn. Then, we have x′=∑i=1nμixi for some μi≥0 and ∑i=1nμi=1. Also, as x′∉Fxi, for all u∈Mx′, v∈Sx′, and w∈Tx′, we have pu−fv−gwηxi,x′−hx′,xi∈−intC, for each i=1,2,⋯,n. Since −intC is convex and μi≥0 with ∑i=1nμi=1, therefore ∑i=1nμipu−fv−gwηxi,x′−hx′,xi∈−intC. As pu, fv, and gw belong to CLX,Y, they are linear. Therefore, we have ∑i=1nμipu−fv−gwηxi,x′−hx′,xi=pu−fv−gw∑i=1nμiηxi,x′−∑i=1nμihx′,xi. Again, η and h are affine; hence, ∑i=1nμiηxi,x′−∑i=1nμihx′,xi=η∑i=1nμixi,∑i=1nμix′−h∑i=1nμix′,∑i=1nμixi=ηx′,x′−hx′,x′=0 as ηx′,x′=0=hx′,x′ by the given hypothesis. Therefore, ∑i=1nμipu−fv−gwηxi,x′−hx′,xi=0. Thus, we have 0∈−intC, where 0 is the zero vector in Y. Thus, 0=−0∈intC, which is a contradiction. Therefore, we have convx1,x2,⋯,xn⊆⋃i=1nFxi. Hence, F is a KKM-mapping on K.(ii)
Fy is closed for each y∈K:
Letxαα∈D be a net in Fy, converging to some z0 in X. As K is closed, z0∈K. We have to show that z0∈Fy, that is, there exist u0∈Mz0, v0∈Sz0, and w0∈Tz0 such that pu0−fv0−gw0ηy,z0−hz0,y∉−intC. Since xα∈Fy, therefore there exist some uα∈Mxα, vα∈Sxα, and wα∈Txα such that puα−fvα−gwαηy,xα−hxα,y∉−intC. Now, the maps M, S, and T are set-valued lower semicontinuous functions; therefore, for each u0∈Mz0 and xα converging to z0, there exists a subnet xαk of xα with uαk∈Mxαk such that uαk converges to u0, in view of Lemma 4. Now, xαk is a net in itself converging to z0 and w0∈Tz0; therefore, there exists a subnet xαkl of xαk with vαkl∈Sxαkl. Similarly, we have a subnet xαklm of xαkl with wαklm∈Txαklm such that the subnets uαk, vαkl, and wαklm converge to u0, v0, and w0, respectively, in view of Lemma 4. As these nets are subnets of uαk, vαk, and wαk, respectively, thus without loss of generality, we denote the subnets uαk, vαkl, and wαklm by uαk, vαk, and wαk, respectively, which converge to u0, v0, and w0, respectively.
Since the single-valued mapp, f, and g are continuous, therefore we have puαk, fvαk, and gwαk converge to pu0, fv0, and gw0, respectively.
By the given hypothesis, that is,η is continuous in the second argument, we have ηy,xαk converges to ηy,z0. Since the space CLX,Y is given to be admissible, thus we have puαk−fvαk−gwαkηy,xαk converges to pu0−fv0−gw0ηy,z0, by Theorem 9. As the map h is continuous in the first component, therefore hxαk,y converges to hz0,y. Hence, puαk−fvαk−gwαkηy,xαk−hxαk,y converges to pu0−fv0−gw0ηy,z0−hz0,y, in view of the fact that CLX,Y is admissible.
Now, we will show thatpu0−fv0−gw0ηy,z0−hz0,y∉−intC.
Let, if possible,pu0−fv0−gw0ηy,z0−hz0,y∈−intC. Then, by the convergence of net, we have puαk−fvαk−gwαkηy,xαk−hxαk,y∈−intC eventually, which leads to contradiction. Hence, pu0−fv0−gw0ηy,z0−hz0,y∉−intC. Thus, we have z0∈Fy.
NowFy is closed, and K is compact. This implies that Fy is a compact subset of K. Therefore, by KKM-Theorem, ⋂y∈KFy≠∅. Hence, there exists some x0∈K such that x0∈⋂y∈KFy. That is, for each y∈K, there exist u0∈Mx0, v0∈Sx0, and w0∈Tx0 such that pu0−fv0−gw0ηy,x0−hx0,y∉−intC, hence the result.☐In the above theorem, we have proved that, along with other conditions, lower semicontinuity ofM,S,T ensures existence of solutions for GNVVLIP (I).In the next theorem, we are providing another set of conditions for the existence of solutions for these class of problems.Theorem 14.
SupposeX and Y are two topological vector spaces and CLX,Y is the space of all continuous linear mappings from the space X to the space Y equipped with an admissible topology. Let K be a nonempty, closed, compact, and convex subset of X. Suppose C⊆Y is a closed, convex, pointed cone with intC≠∅. Further, let M,S,T:K⟶CLX,Y be set-valued upper semicontinuous functions with nonempty compact values, that is, Mx, Sx, and Tx are compact for every x∈K. Let f,g,p:CLX,Y⟶CLX,Y be continuous mappings. Suppose the maps η:K×K⟶X and h:K×K⟶Y are affine mappings such that η is continuous in the second argument and h is continuous in the first argument, respectively, with ηx,x=hx,x=0 for all x∈K. Then, there exists a solution to the generalized nonlinear vector variational inequality problem. That is, there exists x0∈K such that for each y∈K, there exist u∈Mx0, v∈Sx0, and w∈Tx0 such that(8)pu−fv−gwηy,x0−hx0,y∉−intC.Proof.
Consider a set-valued mapF:K⟶K defined as(9)Fy=x∈K∃u∈Mx,v∈Sx,w∈Txsuchthatpu−fv−gwηy,x−hx,y∉−intC.
The proof of the theorem is divided into two parts:(i)
F is a KKM-mapping on K;(ii)
Fy is closed for each y∈K.
Proof of part (i) is similar to that of Theorem13. Therefore, we are providing the proof of part (ii) only.
Letxαα∈D be a net in Fy, converging to some z0∈X. As K is closed, z0∈K. We have to show that z0∈Fy, that is, there exist u0∈Mz0, v0∈Sz0, and w0∈Tz0 such that pu0−fv0−gw0ηy,z0−hz0,y∉−intC. Since xα∈Fy, therefore there exist some uα∈Mxα, wα∈Txα, and vα∈Sxα such that puα−fvα−gwαηy,xα−hxα,y∉−intC. Now, K is a compact subset of X, and M, S, and T are set-valued upper semicontinuous functions such that Mx, Sx, and Tx are compact. Therefore, MK, SK, and TK are also compact by Lemma 6. As uα, vα, and wα are nets in MK, SK, and TK, respectively, there exist subnets uαkαk∈D1, vαlαl∈D2, and wαmαm∈D3 such that uαk, vαl, and wαm converge to some u0∈MK, v0∈SK, and w0∈TK, respectively.
Now, we construct a directed setD4⊂D, defined in the following way:
By the order property of the directed setD, for each triplet αk,αl,αm∈D, there exists some αδ∈D such that αδ≥αk, αδ≥αl, and αδ≥αm. We denote the collection of such αδ’s by D4. It can be easily verified that D4 is a directed set under the induced ordering of D.
Thus, we have subnetsuδδ∈D4, vδδ∈D4, and wδδ∈D4 of uαk, vαl, and wαm, respectively, such that uδ, vδ, and wδ converge to u0∈MK, v0∈SK, and w0∈TK, respectively.
Then, proceeding as in Theorem13, we have z0∈Fy. Thus, Fy is closed, and K is compact. This implies that Fy is a compact subset of K. Therefore, by KKM-Theorem, there exists some x0∈Fy for each y∈K. That is, for each y∈K, there exist u0∈Mx0, v0∈Sx0, and w0∈Tx0 such that pu0−fv0−gw0ηy,x0−hx0,y∉−intC, hence the result.☐In the next theorem, we investigate the properties of solution sets of GNVVLIP (I).Theorem 15.
LetS⊆K be the set of all solutions of a generalized nonlinear vector variational-like inequality problem as obtained in Theorem 13 (respectively, Theorem 14). Then, S is closed and compact in K.Proof.
SupposeS⊆K is the solution set of the generalized nonlinear vector variational-like inequality problem. Then, by Theorem 13, we have S=⋂y∈KFy, where(10)Fy=x∈K∃u∈Mx,v∈Sx,w∈Txsuchthatpu−fv−gwηy,x−hx,y∉−intC.
It has been proved in Theorem13 that each Fy is closed. Therefore, S is a closed set and hence a closed subset of K. Since K is compact, S is compact as well.☐In the next set of theorems, we consider the other variant of the generalized nonlinear vector variational-like inequality problem and provide the conditions for the solution.Theorem 16.
SupposeX and Y are two topological vector spaces and CLX,Y is the space of all continuous linear mappings from the space X to the space Y equipped with an admissible topology. Let K be a nonempty, closed, compact, and convex subset of X. Suppose C:K⟶Y is a set-valued map such that for every x∈K, Cx is a proper closed, convex, pointed cone with intCx≠∅. Further, suppose that G:K⟶Y is also a set-valued map defined by Gx=Y\−intCx such that the graph of G, GG, is a closed set in X×Y. Further, let M,S,T:K⟶CLX,Y be set-valued lower semicontinuous mappings and f,g,p:CLX,Y⟶CLX,Y be continuous mappings. Suppose the maps η:K×K⟶X and h:K×K⟶Y are affine mappings such that η is continuous in the second argument and h is continuous in the first argument, respectively, with ηx,x=hx,x=0 for all x∈K. Then, the generalized nonlinear vector variational-like inequality problem (II) has a solution. That is, there exists x0∈K such that for each y∈K, ∃u∈Mx0, v∈Sx0, and w∈Tx0 such that(11)pu−fv−gwηy,x0−hx0,y∉−intCx0.Proof.
Consider a set-valued mapF:K⟶K defined as(12)Fy=x∈K∣∃u∈Mx,v∈Sx,w∈Txsuchthatpu−fv−gwηy,x−hx,y∉−intCx0.
Likewise in Theorem13, the proof of this theorem is also divided into two parts:(i)
F is a KKM-mapping on K;(ii)
Fy is closed for each y∈K.
Proof of part (i) is similar to that of Theorem13. Therefore, we are providing the proof of part (ii) only.As in Theorem13, it follows that the net puαk−fvαk−gwαkηy,xαk−hxαk,y converges to pu0−fv0−gw0ηy,z0−hz0,y, in view of admissibility of CLX,Y, lower semicontinuity of T, and the other given conditions of the hypothesis.Now, we will show thatpu0−fv0−gw0ηy,z0−hz0,y∉−intCz0. As in Theorem 13, we have the net xα,puαk−fvαk−gwαkηy,xαk−hxαk,y is convergent and converges to z0,pu0−fv0−gw0ηy,z0−hz0,y. Also, the net is contained in GG, and the graph GG is closed; therefore, z0,pu0−fv0−gw0ηy,z0−hz0,y∈GG. Hence, we have pu0−fv0−gw0ηy,z0−hz0,y∉−intCz0. Thus, we have z0∈Fy. Therefore, Fy is closed.NowFy is closed, and K is compact. This implies that Fy is a compact subset of K. Therefore, by KKM-Theorem, ⋂y∈KFy≠∅. Hence, there exists some x0∈K such that x0∈⋂y∈KFy. That is, for each y∈K, there exist u0∈Mx0, v0∈Sx0, and w0∈Tx0 such that pu0−fv0−gw0ηy,x0−hx0,y∉−intCx0, hence the result.☐Another solution of the generalized nonlinear vector variational-like inequality problem (II) is provided below.Theorem 17.
SupposeX and Y are two topological vector spaces and CLX,Y is the space of all continuous linear mappings from the space X to the space Y equipped with an admissible topology. Let K be a nonempty, closed, compact, and convex subset of X. Suppose C:K⟶Y is a set-valued map such that for every x∈K, Cx is a proper, closed, convex, pointed cone with intCx≠∅. Further, suppose that G:K⟶Y is also a set-valued map defined by Gx=Y\−intCx such that graph of G, GG, is a closed set in X×Y. Further, let M,S,T:K⟶CLX,Y be set-valued upper semicontinuous functions with nonempty compact values, that is, Mx, Sx, and Tx are compact for every x∈K and let f,g,p:CLX,Y⟶CLX,Y be continuous mappings. Suppose the maps η:K×K⟶X and h:K×K⟶Y are affine mappings such that η is continuous in the second argument and h is continuous in the first argument with ηx,x=hx,x=0, respectively, for all x∈K. Then, the generalized nonlinear vector variational-like inequality problem (II) has a solution. That is, there exists x0∈K such that for each y∈K, ∃u∈Mx0, v∈Sx0, and w∈Tx0 such that(13)pu−fv−gwηy,x0−hx0,y∉−intCx0.Proof.
The result can be proved on the similar lines as that of Theorem14 and Theorem 16.☐We can draw the following conclusions from the results obtained so far:(i)
Supposeh:K×K⟶Y is defined as the constant map with hx,y=0 for all x,y∈K×K and η:K×K⟶Y is defined as ηx,y=x−y, and M and S are taken as zero functions. If f,p:CLX,Y⟶CLX,Y are also defined as zero functions, that if fx=px=0 for all x∈CLX,Y and g:CLX,Y⟶CLX,Y is taken as an identity map, that is, gx=x for all x∈CLX,Y. Then, Theorem 16 ensures solution to the generalized vector variational inequality problem discussed in [29]. Also, we have(a)
Theorem16 reduces to Theorem 3.6 of [29];(b)
Theorem17 reduces to Theorem 3.1 of [29].Further, ifT is single-valued, then Theorem 13 reduces to Theorem 3.1 and Theorem 17 reduces to Theorem 3.2 of [30], respectively.


## 4. Conclusion

In this paper, we have provided solutions to two variants of the generalized vector variational-like inequality problem. Our approach and the result obtained here differ significantly from those of the existing literature. To be precise,(i)
The spacesX and Y considered in this paper are topological vector spaces. In [15], the space Y is taken to be ℝ, whereas [13] deals with real Hilbert spaces;(ii)
We have used conditions of upper semicontinuity as well as lower semicontinuity on the set-valued mappings to obtain our results. In [15], the concepts of transfer closed and intersectionally closed are used along with an assumption milder than C-pseudomonotonicity on the set-valued mappings. On the other hand, relaxed monotonicity and relaxed Lipschitz’s continuity are used in [13] and generalized C-quasi-convexity is used in [17] for the set-valued mappings.The approach adopted in our paper is topological and varies significantly from the rest literature. Net theory is extensively used in all the main results. We have used admissibility of the function space topology to obtain our results. The authors are not aware of any such results in the literature of variational inequality which are proved adopting similar techniques.



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